ELECTRONIC
TRANSFORMERS AND CIRCUITS
ELECTRONIC
TRANSFORMERS AND CIRCUITS
Reuben Lee
Advisory Engineer
Westinghouse Electric Corporation
SECOND EDITION
NEW YORK JOHN WILEY & SONS, INC.
LONDON CHAPMAN & HALL, LIMITED
CopTBiGHT © 1947, 1955
BY
John Wiley & Sons, Inc.
All Rights Reserved
This book or any part thereof must not
be reproduced in any form without
the written permission of the publisher.
Library of Congress Catalog Card Number: 55-10001
FEINTED IN THE UNITED STATES OP AMERICA
PREFACE TO SECOND EDITION
In the years since the first edition of this book was published, several
new developments have taken place. This second edition encompasses
such new material as will afford acquaintance with advances in the art.
Some old topics which were inadequately presented have received fuller
treatment. Several sections, especially those on electronic amplifiers and
wave filters, have been deleted because more thorough treatments of
these subjects are available in current literature. Thus the original ob-
jectives of a useful book on electronic transformers and related devices,
with a minimum of unnecessary material, have been pursued in the sec-
ond edition. Wherever the old material appeared adequate, it has been
left unchanged, and the general arrangement is still the same, except for
the addition of new Chapters 9 and 11. More information in chart
form, but few mathematical proofs, are included.
In a book of general coverage, there is room only for a brief treatment
of any phase of the subject. Thus the new chapter on magnetic ampli-
fiers is a condensed outline of the more common components and circuits
of this rapidly growing field. It is hoped that this chapter will be helpful
as a general introduction to circuit and transformer designers alike. Re-
cent circuit developments are reported in the AIEE Transactions.
In response to inquiry it should be stated that, where a mathematical
basis is given, graphical performance is always calculated. There has
been good general correspondence between the graphs and experimental
tests. This correspondence is quite close in all cases except pulse trans-
formers ; for these, the graphs presented in this book predict wave shape
with fair accuracy, but to predict exactly all the superposed ripples
would be impracticable. This is pointed out in Chapter 10.
Although technical words usually have the same meaning as in the
first edition, there are several new magnetic terms in the second edition.
These terms conform with ASTM Standard A127-48.
Pascal said that an author should always use the word "our" rather
than "my" in referring to his work, because there is in it usually more
of other people's than his own. Never was this more true than of the
present volume. Acknowledgment is due many Westinghouse engineers,
especially R. M. Baker, L. F. Deise, H. L. Jessup, J. W. Ogden, G. F.
Pittman, R. A. Ramey, T. F. Saffold, and D. S. Stephens, all of whom
vi PREFACE TO SECOND EDITION
assisted immeasurably by their constructive comments on the manu-
script. D. G. Little's continued interest was most encouraging.
Helpful comment has been received from men outside Westinghouse.
Mr. P. Fenoglio of the General Electric Co. kindly pointed out an omis-
sion in the first edition. Output wave shapes given for the front or
leading edge of a pulse transformer were accurate for a hard-tube modu-
lator, but not for a line-type modulator. The missing information is
included in the second edition.
Finally, to my wife Margaret, my heartfelt thanks not only for her
understanding of the long disruption of normal social Hfe but also for
her patience in checking proofs.
Reuben Lee
Baltimore, Maryland
August, 1965
ACKNOWLEDGMENTS
Figures 23 and 24 were furnished through the courtesy of the Armco
Steel Corp. Figures 50, 51, 52, 53, and 86 first appeared in a paper by
O. H. Schade, Proc. I.R.E., July, 1943, p. 341. Figure 150 is reprinted
from Proc. I.R.E., April, 1945. Figure 63 first appeared in the I.R.E.
Transactions on Component Parts, April, 1955.
Figure 71 is reprinted from Electronics, March, 1955. Figures 89, 90,
and 91 are reprinted from, and Section 52 (p. 123) is based on, "Solving
a Rectifier Problem," Electronics, April, 1938. Figures 100 and 101 are
reprinted from Electronics for September, 1949. Figure 180 and Section
97 (p. 232) are based on "A Study of R-F Chokes," which appeared in
Electronics in April, 1934. Sections 123, 124, 125, and 127 (p. 294 et
seq) are based on "Iron-Core Components in Pulse Amplifiers," Elec-
tronics, August, 1943. Figures 73, 258, and 259 are reprinted from this
article.
Figure 88 is reprinted from Tele-Tech and Electronic Industries, Octo-
ber, 1953 (copyright Caldwell-Clements, 480 Lexington Avenue, New
York).
Figures 107 and 110, and part of Section 67 (p. 153), first appeared in
Radio Engineering, June, 1937.
Figure 142 is reprinted from the General Radio Experimenter , Novem-
ber, 1936.
Figures 163, 164, and 165 are reprinted from "Magnetic Ferrites —
Core Materials for High Frequencies," by C. L. Snyder, E. Albers-
Schoenberg, and H. A. Goldsmith, Electrical Manufacturing, December,
1949. Figure 191 is reprinted from Electrical Manufacturing for Septem-
ber, 1954.
The magnetic amplifier analysis on p. 276 is based on an unpublished
paper by D. Lebell and B. Bussell, presented at the I.R.E. Convention,
New York, March, 1952.
Figures 235, 252, 254, and 255, and Table XVII, are reprinted from
Proc. I.R.E., August, 1954.
PREFACE TO FIRST EDITION
The purpose of this book is twofold : first, to provide a reference book
on the design of transformers for electronic apparatus and, second, to
furnish electronic equipment engineers with an understanding of the
effects of transformer characteristics on electronic circuits. Familiarity
with basic circuit theory and transformer principles is assumed. Con-
ventional transformer design is treated adequately in existing books, so
only such phases of it as are pertinent to electronic transformers are in-
cluded here. The same can be said of circuit theory ; only that which is
necessary to an understanding of transformer operation is given. It is
intended that in this way the book will be encumbered with a minimum
of unnecessary material. Mathematical proofs as such are kept to a
minimum, but the bases for quantitative results are indicated. The
A.I.E.E. "American Standard Definitions of Electrical Terms" gives the
meaning of technical words used. Circuit symbols conform to A.S.A.
Standards Z32.5— 1944 and Z32. 10—1944.
Chapter headings, except for the first two, are related to general types
of apparatus. This arrangement should make the book more useful.
Design data are included which would make tedious reading if grouped
together. For instance, the design of an inductor depends on whether
it is for power or wave filter work, and the factors peculiar to each are
best studied in connection with their respective apparatus.
Parts of the book are based on material already published in the Pro-
ceedings of the Institute of Radio Engineers, Electronics, and Communica-
tions. Much of it leans heavily upon work done by fellow engineers of
the Westinghouse Electric Corporation, the warmth of whose friend-
ship I am privileged to enjoy. To list all their names would be a difficult
and inadequate expression of gratitude, but I should be guilty of a gross
omission if I did not mention the encouragement given me by Mr.
D. G. Little, at whose suggestion this book was written.
R. L.
July 191,7
CONTENTS
List of Symbols xiii
1. Introduction 1
2. Transformer Construction, Materials, and Ratings ... 17
3. Rectifier Transformers and Reactors 61
4. Rectifier Performance Ill
5. Amplifier Transformers 140
6. Amplifier Circuits 178
7. Higher-Frequency Transformers 214
8. Electronic Control Transformers 237
9. Magnetic Amplifiers 259
10. Pulse and Video Transformers 292
11. Pulse Circuits 329
Bibliography 347
Index 351
xi
LIST OF SYMBOLS
Page numbers are those on which the corresponding symbol first
appears. A symbol formed from one of the tabulated letters, with a
subscript or prime added, is not listed unless it is frequently and prom-
inently used in the book. Sometimes the same symbol denotes more
than one property; the meaning is then determined by the context.
Units are given wherever symbols are used. Small letters indicate in-
stantaneous or varying electrical quantities, and capital letters indicate
steady, effective, or scalar values.
a
Coil radius, 228
a
Coil winding height, 75
a
N2/NU 147
A
Area, 172
A,
Core area, 10
An
Ripple amplitude, 114
b
Winding traverse, 76
B
Xc/Ri at frequency fr, 150
B
Core flux density, 10
"mi Bmux
Maximum operating flux density, 23, 97
Br
Residual flux density, 23
C
Insulation thickness, 75
c
Specific heat, 57
C
Capacitance, 64
Ci, Cp
Primary capacitance, 147
C2, Cs
Secondary capacitance, 147
Ce
Effective capacitance, 172
Cs
Capacitance of winding to ground, 245
^ w
Capacitance across winding, 245
d
Core tongue width, 38
d
Toroid diameter, 288
D
Winding height, 38
D
Xc/i?2 at frequency fr, 159
e
Voltage (instantaneous value), 5
eg
Alternating grid voltage, 141
ep
Alternating plate voltage, 141
LIST OF SYMBOLS
E
Emissivity, 57
E
Voltage (effective value), 6
Es
Plate voltage, 141
Eo
Output voltage, 178
El
Primary voltage, 7
E,
Secondary voltage, 7
Es
Secondary no-load voltage, 7
El
Secondary full-load 'voltage, 7
Epic
Peak value of alternating voltage, 111
Edc
D-c voltage, 111
Ea
Voltage at top of pulse, 295
f
Frequency, 6
U
Midband frequency, 190
fr
Resonance frequency, 150
fo
Cut-off frequency, 185
/( )
Function of, 114
F
Factor, 230
Qm
Mutual conductance, 144
G
Gap loss constant, 191
H
Magnetizing force, 10
He
Coercive force, 23
i
Current (instantaneous value), 10
^ J Ml?-' Tms
Current (effective value), 6, 15
Ijc
Direct component of current, 16
Ij Ipk
Peak value of current, 16, 66
-^ J -^av
Average value of current, 15, 66
Ip,Ib
Plate current (d-c), 142
II
Load current, 7
Ie
Loss component of exciting current, 10
Im
Magnetizing current, 9
In
Exciting current, 9
Ig
Grid current (d-c), 142
3
•\/— 1 (vector operator), 146
J
Low-frequency permeability/pulse permeability, 335
k
Thermal conductivity, 57
k
Coefficient of coupling, 225
k
}/2 ratio of impedance/circuit resistance = ■\/L/C/2R, 104
K
Constant, 82
la
Mean length of core (or magnetic path), 10
k
Air gap, 88
L
Inductance, 90
Le
Open-circuit inductance {OCL), 26
LIST OF SYMBOLS
La
Short-circuit inductance, 76
Lm
Mutual inductance, 224
m
Decrement, 104
m
Order of harmonic, 1 14
M
Modulation factor, 16
MT
Mean turn length, 38
n
Number (e.g., of anodes), 76
N
Turns, 5
N,
Primary turns, 5
N,
Secondary turns, 5
Nl
Number of layers (of wire in coil), 173
OCL
Open-circuit inductance, 106
V
Density, 26
V
Ratio of voltages (in autotransformer), 250
V
Rectifier ripple frequency /line frequency (number of
phases), 113
Pa
Volt-amperes per pound, 26
Pc
Core loss, 26
Pa
Ripple amplitude/i?<;c (in rectifier), 114
Pr
Ripple amplitude/i?jc (across load), 114
PEN
Pulse forming network, 332
PRF
Pulse repetition frequency, 338
Q
wL/R = coil reactance/coil a-c resistance, 106
r
Radius, 38
re.
Equivalent radius, 57
rp
Plate resistance, 144
R
Resistance, 6
Rx
Source resistance, 146
i?2
Load resistance, 146
Rl
Load resistance, 8
Re
Equivalent core-loss (shunt) resistance, 8
S
Secondary winding, 71
s
Core window width, 102
t
Time (independent variable), 5
t
Thickness of insulation, 172
T
Period of a wave, 15
T
2ir\/LsC2 (undamped period of oscillatory wave), 295
V
Commutation voltage, 120
V
Volume (of core), 91
w
Core-stacking dimension, 38
w.
Gap loss, 191
We
Core loss, 82
LIST OF SYMBOLS
Ws
Copper loss, 82
X
Reactance, 6
Xff
Open-circuit reactance = 2irfLe, 9
Xc
Capacitive reactance = l/(2ir/C), 112
Xl
Inductive reactance = 2ir/L, 112
z
Impedance, 8
Zg
Source impedance, 141
Zl
Load impedance, 141
z.
Characteristic impedance, 145
a
Amplifier gain, 174
a
\/Cg/C^, 245
a
Damping factor, 319
P
Feedback constant, 178
P
Natural angular frequency, 304
5
Small interval of time, 15
A
Increment (e.g., of flux), 25
A
Exciting current/load current, 299
€
Base of natural logarithms (= 2.718), 5
e
Dielectric constant of insulation, 172
V
Efficiency, 14
e
Temperature, 57
d
Phase angle, 120
ij'
Amplification factor, 141
ti
Permeability, 24
MA
Incremental permeability, 25
■K
3.1416, 6
Phase angle, 195
4>
Flux (varying), 6
*max
Peak value of flux, 6
s
Summation (of a series of elements), 38
T
Pulse duration, 298
W
2x/ (angular frequency), 6
1. INTRODUCTION
1. What Is a Transformer ? In its most elementary form, a trans-
former consists of two coils wound of wire and inductively coupled to
each other. When alternating current at a given frequency flows in
either coil, an alternating voltage of the same frequency is induced
in the other coil. The value of this voltage depends on the degree
of coupling and the flux linkages in the two coils. The coil connected
to a source of alternating voltage is usually called the primary coil,
and the voltage across this coil is the primary voltage. Voltage in-
duced in the secondary coil may be greater than or less than the pri-
mary voltage, depending on the ratio of primary to secondary turns.
A transformer is termed a step-up or a step-down transformer accord-
ingly.
Most transformers have stationary iron cores, around which the
primary and secondary coils are placed. Because of the high perme-
ability of iron, most of the flux is confined to the core, and a greater
degree of coupling between the coils is thereby obtained. So tight is
the coupling between the coils in some transformers that the primary
and secondary voltages bear almost exactly the same ratio to each
other as the turns in the respective coils or windings. Thus the turns
ratio of a transformer is a common index of its function in raising or
lowering voltage. This function makes the transformer an important
adjunct of modern electrical power systems. Raising the voltage
makes possible the economical transmission of power over long dis-
tances; lowering the voltage again makes this power available in use-
ful form. It is safe to say that, without transformers, modern industry
could not have reached its present state of development.
2. Electronic Transformers. Although no exact line of demarcation
can be drawn between power transformers and electronic transformers,
in general electronic transformers are smaller. The source of power
on a 60-cycle network is extremely large and may be the combined
generating capacity of half a continent. Power in electronic equipment
is limited to the capabilities of electron tubes, of which even the largest
is small compared to a power station generator.
1
2 ELECTRONIC TRANSFORMERS AND CIRCUITS
Transformers are needed in electronic apparatus to provide the
different values of plate, filament, and bias voltage required for proper
tube operation, to insulate circuits from each other, to furnish high
impedance to alternating but low impedance to direct current, and to
maintain or modify wave shape and frequency response at different
potentials. The very concept of impedance, so characteristic of elec-
tronics, almost necessarily presupposes a means of changing from one
impedance level to another, and that means is commonly a trans-
former.
Impedance levels are usually higher in electronic, as compared with
power, equipment. Consider the connected kva on an 11,000-volt power
line; it may easily total 1,000,000. Compare this with a large broad-
cast transmitter operating at the same voltage and drawing 70 kva.
The currents in the two cases are 90,000 amp and 6 amp, respectively.
For the power line, the load impedance is 11,000/90,000, or slightly
more than 0.1 ohm; for the transmitter it is 11,000/6, or nearly 2,000
ohms. Source impedances are approximately proportional to these
load impedances. In low-power electronic circuits the source imped-
ance often exceeds the load impedance and influences the transformer
performance even further.
Weight and space are usually at a premium in electronic equipment,
and reliability is of paramount importance. Transformers account
for a considerable portion of the weight and space, and form a prime
component of the reliability.
These and other differences of application render many power trans-
formers unsuitable for electronic circuit use. The design, construc-
tion, and testing of electronic transformers have become separate arts,
directed toward the most effective use of materials for electronic
applications.
3. New Materials. Like all electronic apparatus, transformers are
subject to continual change. This is especially so since the introduc-
tion of new materials such as
(a) Grain-oriented core steel.
(6) Solventless impregnating varnish.
(c) Inorganic insulating tape.
(d) Improved wire enamel.
(e) Low-loss, powdered iron cores.
(/) Ferrite cores.
Through the application of these materials, it has been possible to
INTRODUCTION 3
(a) Reduce the size of audio and power transformers and reactors.
(b) Increase the usefulness of saturable reactors as magnetic ampli-
fiers.
(c) Reduce the size of high-voltage units.
(d) Design filters and reactors having sharper cut-off and higher
Q than previously was thought possible.
(e) Make efficient transformers for the non-sinusoidal wave shapes
such as are encountered in pulse, video, and sweep amplifiers.
(/) Extend the upper operating frequency of transformers into the
high-frequency r-f range.
Occasionally someone asks why electronic transformers cannot be
designed according to curves or charts showing the relation between
volts, turns, wire size, and power rating. Such curves are very useful
in designing the simpler transformers. However, this idea has not been
found universally practicable for the following reasons:
(a) Regulation. This property is rarely negligible in electronic
circuits. It often requires care and thought to use the most advan-
tageous winding arrangement in order to obtain the proper IX and
IR voltage drops. Sometimes the size is dictated by such consider-
ations.
(6) Frequency Range. The low-frequency end of a wideband
transformer operating range in a given circuit is determined by the
transformer open-circuit inductance. The high-frequency end is gov-
erned by the leakage inductance and distributed capacitance. Jug-
gling the various factors, such as core size, number of turns, interleav-
ing, and insulation, in order to obtain the optimum design constitutes
a technical problem too complex to solve on charts.
(c) Voltage. It would be exceedingly difficult, if not impossible, to
reduce to chart form the use of high voltages in the restricted space of
a transformer. Circuit considerations are very important here, and
the transformer designer must be thoroughly familiar with the func-
tioning of the transformer to insure reliable operation, low cost, and
small dimensions.
(d) Size. Much electronic equipment is cramped for space, and,
since transformers often constitute the largest items in the equipment,
it is imperative that they, too, be of small size. An open-minded atti-
tude toward this condition and good judgment may make it possible
to meet the requirements which otherwise might not be fulfilled. New
materials, too, can be instrumental in reducing size, sometimes down
to a small fraction of former size.
In succeeding chapters the foregoing considerations will be applied
4 ELECTRONIC TRANSFORMERS AND CIRCUITS
to the performance and design of several general types of electronic
transformers. The remainder of this chapter is a brief review of
fundamental transformer principles. Only iron-core transformers with
closed magnetic paths are considered in this introduction. Air-core
transformers, with or without slugs of powdered iron, are discussed
in a later chapter on high-frequency transformers. Most transformers
operate at power frequencies ; it is therefore logical to begin with low-
frequency principles. These principles are modified for other condi-
tions in later chapters.
COIL FORM
PRIMARY WINDING
SECONDARY
WINDING
COIL
CORE
LAMINATIONS
CORE FLUX'
Fig. 1. Transformer coil and core.
PRIMARY
WINDING
SECONDARY
WINDING
A simple transformer coil and core arrangement is shown in Fig. 1.
The primary and secondary coils are wound one over the other on an
insulating coil tube or form. The core is laminated to reduce losses.
Flux flows in the core along the path indicated, so that all the core
flux threads through or links both windings. In a circuit diagram
the transformer is represented by the
circuit symbol of Fig. 2.
4. Transformer Fundamentals. The
simple transformer of Fig. 2 has two
windings. The left-hand winding is as-
sumed to be connected to a voltage
source and is called the primary winding.
The right-hand winding is connected to a load and is called the sec-
ondary. The transformer merely delivers to the load a voltage similar
to that impressed across its primary, except that it may be smaller or
greater in amplitude.
In order for a transformer to perform this function, the voltage across
it must vary with respect to time. A d-c voltage such as that of a
storage battery produces no voltage in the secondary winding or power
Fig. 2. Simple transformer.
INTRODUCTION 5
in the load. If both varying and d-c voltages are impressed across the
primary, only the varying part is delivered to the load. This comes
about because the voltage e in the secondary is induced in that winding
by the core flux <j> according to the law
Nd<l>
e = X 10-« (1)
at
This law may be stated in words as follows : The voltage induced in a
coil is proportional to the number of turns and to the time rate of
change of magnetic flux in the coil. This rate of change of flux may
be large or small. For a given voltage, if the rate of change of flux is
small, many turns must be used. Conversely, if a small number of
turns is used, a large rate of change of flux is necessary to produce a
given voltage. The rate of change of flux can be made large in two
ways, by increasing the maximum value of flux and by decreasing the
period of time over which the flux change takes place. At low fre-
quencies, the flux changes over a relatively large interval of time, and
therefore a large number of turns is required for a given voltage, even
though moderately large fluxes are used. As the frequency increases,
the time interval between voltage changes is decreased, and for a given
flux fewer turns are needed to produce a given voltage. And so it is
that low-frequency transformers are characterized by the use of a
large number of turns, whereas high-frequency transformers have but
few turns.
If the flux <j> did not vary with time, the induced voltage would be
zero. Equation 1 is thus the fundamental transformer equation. The
voltage variation with time may be of any kind: sinusoidal, exponen-
tial, sawtooth, or impulse. The essential condition for inducing a
voltage in the secondary is that there be a flux variation. Only that
part of the flux which links both coils induces a secondary voltage.
In equation 1, if (j> denotes maxwells of flux and t time in seconds, e
denotes volts induced.
If all the flux links both windings, equation 1 shows that equal volts
per turn are induced in the primary and secondary, or
ei Ni
62 N2
where ei = primary voltage
62 = secondary voltage
Ni = primary turns
N2 = secondary turns.
6 ELECTRONIC TRANSFORMERS AND CIRCUITS
5. Sinusoidal Voltage. If the flux variation is sinusoidal,
<^ = *max sin ut
where $max is the peak value of flux, co is angular frequency, and t is
time. Equation 1 becomes
e = -iV$maxW COS oit X 10~^ (3)
or the induced voltage also is sinusoidal. This voltage has an effective
value
E = 0.707 X 27r/Ar$„,ax X IQ-^
= 4.44/iV$n,ax X 10-« (4)
where / is the frequency of the sine wave. Equation 4 is the relation
between voltage and flux for sinusoidal voltage.
Sufficient current is drawn by the primary winding to produce the
flux required to maintain the winding voltage. The primary induced
voltage in an unloaded transformer is just enough lower than the
impressed voltage to allow this current to flow into the primary wind-
ing. If a load is connected across the secondary terminals, the pri-
mary induced voltage decreases further, to allow more current to flow
into the winding in order that there may be a load current. Thus the
primary of a loaded transformer carries both an exciting current and a
load current, but only the load part is transformed into secondary
load current.
Primary induced voltage would exactly equal primary impressed
voltage if there were no resistance and reactance in the winding. Pri-
mary current flowing through the winding causes a voltage drop IR,
the product of primary current / and winding resistance R. The wind-
ing also presents a reactance X which causes an IX drop. Reactance
X is caused by the leakage flux or flux which does not link both primary
and secondary windings. There is at least a small percentage of the
flux which is not common to both windings. Leakage flux flows in the
air spaces adjacent to the windings. Because the primary turns link
leakage flux an inductance is thereby introduced into the winding,
producing leakage reactance X at the line frequency. The larger the
primary current, the greater the leakage flux, and the greater the react-
ance drop IX. Thus the leakage reactance drop is a series effect, pro-
portional to primary current.
6. Equivalent Circuit and Vector Diagram. For purposes of analysis
the transformer may be represented by a 1:1 turns-ratio equivalent
circuit. This circuit is based on the following assumptions:
INTRODUCTION 7
(a) Primary and secondary turns are equal in number. One wind-
ing is chosen as the reference winding ; the other is the referred wind-
ing. The voltage in the referred winding is multiplied by the actual
turns ratio after it is computed from the equivalent circuit. The
choice between primary and secondary for the reference winding is a
matter of convenience.
(6) Core loss may be represented by a resistance across the termi-
nals of the reference winding.
(c) Core flux reactance may be represented by a reactance across
the terminals of the reference winding.
(d) Primary and secondary IR and IX voltage drops may be
lumped together; the voltage drops in the referred winding are multi-
plied by a factor derived at the end of this section, to give them the
correct equivalent value.
(e) Equivalent reactances and resistances are linear.
As will be shown later, some of these assumptions are approximate,
and the analysis based on them is only accurate so far as the assump-
tions are justified. With proper attention to this fact, practical use
can be made of the equivalent circuit.
With many sine-wave electronic transformers, the transformer load
is resistive. A tube filament heating load, for example, has 100 per
cent power factor. Under this condition the relations between voltages
and currents become appreciably simplified in comparison with the
same relations for reactive loads. In what follows, the secondary
winding will be chosen as the reference winding. At low frequencies
such a transformer may be represented by Fig. 3 (a) . The transformer
equivalent circuit is approximated by Fig. 3(6), and its vector dia-
gram for 100 per cent p-f load by Fig. 3(c). Secondary load voltage
Ei, and load current II are in phase. Secondary induced voltage Es
is greater than El because it must compensate for the winding resist-
ances and leakage reactances. The winding resistance and leakage
reactance voltage drops are shown in Fig. 3(c) as IR and IX, which are
respectively in phase and in quadrature with II and E^. These voltage
drops are the sum of secondary and primary winding voltage drops,
but the primary values are multiplied by a factor to be derived
later. If voltage drops and losses are temporarily forgotten, the same
power is delivered to the load as is taken from the line. Let subscripts
1 and 2 denote the respective primary and secondary quantities.
Eili = E2I2 (5)
ELECTRONIC TRANSFORMERS AND CIRCUITS
I,
Fig. 3. (a) Transformer with resistive load; (6) equivalent circuit; (c) vector
diagram.
or
E2
h
h
(6)
so that the voltages are inversely proportional to the currents. Also,
from equation 2, they are directly proportional to their respective turns.
El
E2
N2
(2a)
Now the transformer may be replaced by an impedance Zi drawing the
same current from the line, so that
Likewise
h = E^/Z,
I2 = E2/Z2
where Z2 is the secondary load impedance, in this case Rl- If these
expressions for current are substituted in equation 6,
Z2
\E2/ W2/
(7)
Equation 7 is strictly true only for negligible voltage drops and
losses. It is approximately true for voltage drops up to about 10 per
cent of the winding voltage or for losses less than 20 per cent of the
power delivered, but it is not true when the voltage drops approach
in value the winding voltage or when the losses constitute most of the
primary load.
Not only does the load impedance bear the relation of equation 7
INTRODUCTION
IX
to the equivalent primary load impedance; the winding reactance and
resistance may also be referred from one winding to the other by the
same ratio. This can be seen if the secondary winding resistance and
reactance are considered part of the load, across which the secondary
induced voltage Eg appears. Thus the factor by which the primary
reactance and resistance are multiplied, to refer them to the secondary
for addition to the secondary drops, is [Ni/Ni)^. If the primary had
been the reference winding, the secondary reactance and resistance
would have been multiplied by iNi/N2)^.
In Fig. 3(c) the IR voltage drop subtracts directly from the terminal
voltage across the resistive load, but the IX drop makes virtually no
difference. How much the IX drop may be before it becomes appreci-
able is shown in Fig. 4. If the IX drop is 30 per cent of the induced
voltage, 4 per cent reduction in load
voltage results; 15 per cent IX drop
causes but 1 per cent reduction.
7. Magnetizing Current. In addi-
tion to the current entering the pri-
mary because of the secondary load,
there is the core exciting current In
which flows in the primary whether
the secondary load is connected or
not. This current is drawn by the
primary core reactance X,v and
equivalent core-loss resistance Rb
and is multiplied by N1/N2 when it
is referred to the secondary side. It
has two components: Im, the mag-
netizing component which flows 90°
lagging behind induced voltage Eg; and Ie, the core-loss current which
is in phase with Es. Ordinarily this current is small and produces
negligible voltage drop in the winding.
Core-loss current is often divided into two components: eddy cur-
rent and hysteresis. Eddy-current loss is caused by current circulat-
ing in the core laminations. Hysteresis loss is the power required to
magnetize the core first in one direction and then in the other on alter-
nating half-cycles. Hysteresis loss and magnetization are intimately
connected, as can be seen from Fig. 5. Here induced voltage e is
plotted against time, and core flux 4> lags e by 90°, in accordance with
equation 3. This flux is also plotted against magnetizing current in
the loop at the right. This loop has the same shape as the B-H loop
1.0
.9
^^
"=
^
v...^
'v
.8
.7
S
\
s.
.6
.5
s
\
\
,4
.3
\
\
.2
\
°C
.5 1.
E
E
L
S
Fig. 4. Relation between reactive
voltage drop and load voltage.
10
ELECTRONIC TRANSFORMERS AND CIRCUITS
1.41 E-
FiQ. 5. Transformer voltage, flux, and exciting current.
for the grade of iron used in the core, but the scales are changed so
that
(8)
^ = BAo
i = HIc/OAtN
where B = core flux density in gauss
Ac = core cross-sectional area in cm^
H = core magnetizing force in oersteds
Ic = core flux path length in cm.
Current is projected from the <j>-i loop to obtain the alternating
current i at the bottom of Fig. 5. This current contains both the mag-
netizing and the hysteresis loss components of current. In core-mate-
rial research it is important to separate these components, for it is
mainly through reduction of the B-H loop area (and hence hysteresis
loss) that core materials have been improved. Techniques have been
developed to separate the exciting current components, but it is evident
that these components cannot be separated by current measurement
only. It is nevertheless convenient for analysis of measurements to
add the loss components and call their sum Is, and to regard the mag-
netizing component Im as a separate lagging current, as in Fig. 3. As
long as the core reactance is large, the vector sum I^ of Im and In is
INTRODUCTION
11
small, and the non-sinusoidal shape of In does not seriously affect the
accuracy of Fig. 3.
Core flux reactance may be found by measuring the magnetizing
current, i.e., the current component which lags the applied voltage
90° with the secondary circuit open. Because of the method of meas-
urement, this is often called the open-circuit reactance, and this re-
actance divided by the angular frequency is called the open-circuit
inductance. The secondary and primary winding leakage reactances
are found by short-circuiting the secondary winding and measuring
the primary voltage with rated current flowing. The component of
primary voltage which leads the current by 90° is divided by the
current; this is the sum of the leakage reactances, the secondary react-
ance being multiplied by the (turns ratio) ^, and is called the short-
circuit reactance.
Practical cases sometimes arise where the magnetizing component
becomes of the same order of magnitude as II- Because current In
flows only in the primary, a different equivalent circuit and vector dia-
gram are necessary, as shown in Fig. 6. Note that the leakage react-
FiG. 6. (a) Equivalent circuit and (b) vector diagram for transformer with high
magnetizing current.
ance voltage drop has a marked effect upon the load voltage, and this
effect is larger as Im increases relative to II- Therefore, the statement
that IX voltage drop causes negligible difference between secondary
induced and terminal voltages in transformers with resistive loads is
true only for small values of exciting current. Moreover, the total
primary current /i has a largely distorted shape, so that treating the
IR and IX voltage drops as vectors is a rough approximation. For
12 ELECTRONIC TRANSFORMERS AND CIRCUITS
accurate calculation of load voltage with large core exciting current, a
point-by-point analysis would be necessary.
8. Flux and Average Voltage. If the variables are separated in
equation 1, thus
edt= ~N X 10~^ d<i>
an expression for flux may be found :
fe d< = -A^ X 10-^ fd^
Now if we consider the time interval to tt/cc, we have
*^ni.ax
I edt= -N X 10-^ d4
= -2Ar$^ax X 10-« (9)
Equation 9 gives the relation between maximum flux and the time
integral of voltage. The left side of the equation is the area under
the voltage-time wave. For a given frequency, it is proportional to
the average voltage value. This is perfectly general and holds true
regardless of wave form. If the voltage wave form is alternating, the
average value of the time integral over a long period of time is zero. If
the voltage wave form is sinusoidal, the flux wave form is also sinus-
oidal but is displaced 90° as in Fig. 5, and the integral over a half-
cycle is
"cos cctV'" 2.82E
-lAlE
whence
1.41 X io^-e;
*max = (10)
Cx>N
Equation 10 is the relation between maximum flux, efi'ective voltage,
frequency, and turns. It is a transposed form of equation 4.
9. Ideal Transformer. The use of equivalent circuits enables an
engineer to calculate many transformer problems with comparative
ease. It is always necessary to multiply properties in the referred
winding by the proper ratio. This has led to the interposition of a
transformer of the right turns ratio somewhere in the equivalent cir-
cuit, usually across the load. The transformer thus used must intro-
duce no additional losses or voltage drops in the circuit. It is called an
INTRODUCTION 13
ideal transformer,'^ and it has negligibly small winding resistances,
leakage flux, core loss, magnetizing current, and winding capacitances.
Some power and audio transformers very nearly approach the ideal
transformer at some frequencies. For example, in a typical 50-kva
plate transformer, the winding resistance IR drops total 1 per cent
and the leakage reactance IX drops 3 per cent of rated voltage, the
core loss 0.6 per cent of output power and magnetizing current 2 per
cent of rated primary current. When the term ideal transformer is
used, it should be borne in mind that negligibly small is not zero. Par-
ticularly in electronic work, where frequency may vary, a limiting
frequency may be reached at which the transformer is no longer ideal.
Aloreover, even if the limiting frequency is very low, it is never zero.
There must be voltage variation if transformation is to take place.
The assumptions of equations 5 to 7 were the same as for an ideal
transformer.
10. Polarity. Let turns from equation 2a be substituted in equation
5. Then we have
iViZi = N2I2 (11)
or the primary and secondary ampere-turns are equal and opposite.
This equality holds for only the load component of h ; that is, exciting
current has been regarded as negligibly small. If there is a direct cur-
rent in the load, but not in the primary, or vice versa, equation 11 is
true for only the a-c components.
A 1 : 1 turns-ratio transformer is shown diagramatically in Fig. 7.
Impressed voltage is Ei, and primary current is Ii. Induced voltage
Ei is slightly less than Ei, and is the same in magnitude and direction
for both windings. Secondary current h flows in the opposite direction
to 7i. Instantaneous polarities are indicated by -|- and — signs. That
is, when Ei reaches positive maximum so do E, and E2. Dots are con-
ventionally used to indicate terminals of the same polarity; dots in
the circuit symbol at the right of Fig. 7 are used to indicate the same
winding directions as in the left-hand figure.
11. Regulation, Efficiency, and Power Factor. Transformer regula-
tion is the difference in the secondary terminal voltage at full load
and at no load, expressed as a percentage of the full-load voltage. For
the resistive load of Fig. 3(a), (b), and (c),
(Eg —El)
Per cent regulation = 100 (12)
El
1 See Magnetic Circuits and Transformers, M.I.T. Electrical Engineering Staff,
John Wiley & Sons, New York, 1943, p. 269.
14
ELECTRONIC TRANSFORMERS AND CIRCUITS
,CORE FLUX
+
o-
PRI. <
p-
+ ■
-Iz.
/
CORE
Fig. 7. Transformer polarity.
Since with low values of leakage reactance Eg ~ El = IR,
Per cent regulation = 100IR/El
(13)
provided that R includes the primary winding resistance multiplied by
the factor (N2/Ni)^ as well as the secondary winding resistance. If
leakage reactance is not negligibly small, approximately
1 /ix\^l
Per cent regulation = 100
Efficiency is the ratio
IR 1 /IXV
'K 2 W/
Output power
Output power plus losses
(14)
(15)
where losses include both core and winding losses.
A convenient way of expressing power factor is
Power factor =
Output power plus losses
(16)
Input volt-amperes
Equation 16 gives the power factor of a transformer plus its load.
One of the problems of transformer design is the proper choice of
induction to obtain low values of exciting current and high power
factor. Low power factor may cause excessive primary winding cop-
per loss, low efficiency, and overheating.
12. Wave Shapes. Transformers in electronic circuits may be sub-
jected to alternating and direct currents simultaneously, to modified
sine waves, or to other non-sinusoidal waves. Although there is a
relation between current and voltage wave shapes in a transformer,
the two are frequently not the same, as has already been seen in Fig.
INTRODUCTION 15
5. D-c components of primary voltage are not transformed; only
the varying a-c component is transformed. Secondary current may
be determined by the connection of the load. For example: if the
load is a rectifier, the current will be some form of rectified wave;
if the load is a modulator, the secondary current may be the super-
position of two waves. If the primary voltage is non-sinusoidal, then
the secondary current almost certainly will be non-sinusoidal.
If the primary voltage comes from an alternating source only, and
the load is a half-wave rectifier, the secondary current has a d-c com-
ponent, but the primary current has no d-c component except under
changing conditions. That is to say, in the steady state there is no
primary d-c component resulting from secondary d-c component alone.
This is true, because any direct current in the primary requires a d-c
source. But by the initial assumption there is no direct current present
in the primary. Under these conditions, the core flux may be very
much distorted because the flux excursions go into saturation in one
direction only.
In succeeding chapters, two values of current will be of interest
in circuits with non-sinusoidal waves, the average and the rms. Aver-
age current causes core saturation unless there is an air gap. Rms
current determines the heating of the windings and is limited by the
permissible temperature rise. Voltage wave form will be dealt with
in subsequent chapters. Common current wave forms are tabulated
here for convenience. (See Table I.)
Root-mean-square or rms current values are based upon the
equation
'k
T
,■2
dt (17)
where i = current at any instant
/ = frequency of repetition of current waves per second
T = duration of current waves in seconds
t = time in seconds.
Average current values are
•^0
i dt (18)
In the first wave shape, T = 1/f. In the fifth wave shape, T -1- 28 is
the current wave duration.
16 ELECTRONIC TRANSFORMERS AND CIRCUITS
Table I. Non-Sinusoidal Cuhrent Wavk Foems
Current Wave Shape
—.4 T 1- l/J ^
iErL__n_
-i/j
— M T h us — H
.^v
Description
Direct current with
superposed sine
wave
Half-sine loops of T
duration and /
repetition rate
Square waves of T
duration and /
repetition fre-
quency
Sawtooth wave of T
duration and /
repetition fre-
quency
Trapezoidal wave of
/ repetition fre-
quenc3'
i^dc
'pk
'pk
VfT
tpk .
t pk
V
/(2a + 37')
/rf.
'2rp,fT
IpkfT
IpkfT
Ipkf(i + T)
In both equations 17 and 18, T refers to a full period. This is in
contrast to steady-state sinusoidal alternating currents, the rms and
average values of which are developed over a half-period because of
the symmetry of such currents about the zero axis.
2. TRANSFORMER CONSTRUCTION, MATERIALS,
AND RATINGS
13. Construction. Most electronic transformers are small, and for
small transformers the shell-type core is usually most suitable because
only one coil is required. Figure 8 shows shell-type transformer as-
semblies.
I 'i t .>b
■•*:
..if^
Fig. 8. Transformers with shell-type core.
The magnetic path is divided, half the flux enclosing one side of the
coil and half the other. The coil opening is called the window. Be-
tween the windows is the core tongue, which is twice as wide as the
o 1
o
o
r 1 1 ^
III J
o
1 1
K— 1 WINDOW
/ ■< 1 — TONGUE
O
~i 1
_j
o 1
E-I
E-E F
Fig. 9. Shell-type laminations.
iron around the rest of the window. The core is built up of thin lamina-
tions to reduce eddy-current losses; typical shapes are shown in Fig. 9.
Alternate stacking of the lamination pairs may be used to reduce mag-
netic reluctance and keep magnetizing current small. To reduce as-
sembly cost, this alternate stacking is sometimes done in groups of
17
18
ELECTRONIC TRANSFORMERS AND CIRCUITS
I ,1
'^^^^^^^^^^m ;
Fig. 10. Core-type transformer.
two or more laminations, with some increase in magnetizing current.
A wide range of sizes of shell-type laminations is available. At 60
cycles, common thicknesses are 0.014 in., 0.019 in., and 0.025 in.
Shell-type laminations are made with proportions to suit the trans-
former. In the E-I shape a scrapless lamination is widely used. Two
E's facing each other are first punched, and the punched-out strips are
of the right dimensions to form two I's. Then the E's are cut apart.
This economy of material is not justified in transformers in which turns
per layer, and hence window width, must be reduced relative to window
height.
For some applications, the core-type transformer is preferable. In
these there is only one magnetic path, but there are two coils, one on
each leg of the core. A core-type transformer is shown in Fig. 10, and
some core-type laminations in Fig. 11.
Cores wound from continuous steel strip are widely used. One
common shape is illustrated in Fig. 12; it is known as the type C core.
Steel strip is first wound to the proper build-up on a mandrel. The
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 19
wound core is then annealed, impregnated with a bond, and cut in
two to permit assembly with the coil. After assembly with the coil,
the core is held together with a
steel band as in Fig. 10. Several
advantages accrue from this con-
struction, which will be discussed
in Section 15.
Typical assemblies using two
type C cores are shown in Figs. 13
and 14; they correspond to shell-
type laminations. Because it is
simpler to assemble a single-core
loop, a single core is often used,
especially in small sizes. See Fig.
15. In 60-cycle service the laminations are usually stacked alternately
to produce an overlapping joint. This is approximated in the type C
cores with ground gap surfaces which fit closely together. Either type
of core can be used with core gaps; laminations are stacked butting,
with no overlap. The desired amount of gap material, such as fish-
paper, is inserted between the gap surfaces.
1
__J
Core-type laminations.
4 \' "
■^
^^
.^
.^•>
V
Fig. 12. Type cores.
20
ELECTRONIC TRANSFORMERS AND CIRCUITS
Fig. 13. Partly assembled transformer.
f
Ibk
^
1 - ,-
i«^
'W'"-
Lg 1
^m^ *
^S
■'2.
^B ^
\-.
%
, /
^K'^
p?*
Fig. 14. Assembled type C cores and
coil.
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 21
Fig. 15. Single-coil, single-core assemblies.
« .,, -'**
Fig. 16. Transformers mounted on amplifier chassis.
22 ELECTRONIC TRANSFORMERS AND CIRCUITS
14. Mountings. Both types of cores may be built into neat assem-
blies with the laminations exposed, and the coils covered by end cases,
such as those in the amplifier of Fig. 16. When complete enclosure
is desired, assemblies like those in Fig. 17 are used.
Fig. 17. Fully enclosed transformers.
The degree of enclosure depends on many conditions, among them
the following:
(a) Climate. In a humid climate, especially in the tropics, copper
corrodes readily. Transformers containing fine wire may have open
circuits soon after exposure to tropical conditions, and it is preferable
to seal them against the entry of moisture.
(b) Temperature Rise. Transformers handling large amounts of
power may become hot because of the electrical losses. To seal them
in containers imposes additional obstacles to the dissipation of this
heat. Fortunately the wire size is large enough to withstand corrosion
without developing open circuits. Such units may be of the open type.
(c) Space. Sealing a transformer usually requires more space than
mounting the core and coil directly on the chassis or panel. End cases
like those in Fig. 16 do not require much space but do reduce cooling
by convection. When air is used to cool other apparatus, power tubes
for instance, it is very often circulated near or through the transformer
to prevent the coils from overheating.
(d) Voltage. In high-voltage dry-type transformers, enclosure in a
metal case may add to the difficulties of insulating the windings. In
oil-filled transformers, a tank is required for the oil and enclosure is
thereby provided.
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 23
(e) Appearance. Generally speaking, enclosed transformers are
neater than the open type. This fact is given consideration where
space is available, especially in broadcast apparatus.
15. Core Materials. Electronic transformers make use of a large
variety of core materials. In this chapter, the more useful magnetic
properties of several grades of core materials are presented for refer-
ence and comparison. To guard against possible ambiguity, definitions
of magnetic terms are first reviewed.
Referring to the typical hysteresis loop of Fig. 18, curve OB™ is
the manner in which completely unmagnetized steel becomes magne-
tized by a magnetizing force H gradually increasing up to value Hm-
Flux density or induction is not proportional to H but rises more gradu-
ally as it approaches //„„ B„j. Once the material reaches this state, it
does not retrace curve 05„ if H is reduced. Instead, it follows the
left side of the solid-line loop in the direction of the arrow until, with
negative //,„, it reaches the maximum negative induction —Bm. If H
is now reversed, the induction increases as indicated by the right side
of the loop, which is symmetrical in that the upper and lower halves
are equal in area and have the same shape.
In laboratory tests of magnetic material, the changes in H are made
slowly by means of a permeameter. The solid curve of Fig. 18 is then
called the d-c hysteresis loop. If the changes are made more rapidly,
for example at a 60-cycle rate, the loop is wider, as shown by the dotted
lines. If a higher frequency is used, the loop becomes still wider, as
shown by the dot-dash lines. At any frequency, energy is expended in
changing induction from B^ to —Bm and back to Bm', this energy is
called the hysteresis loss and is proportional to the area of the B-H
loop. Increase in loop width with frequency is usually attributed to
eddy currents which flow, even in laminated cores, to some degree.
If a closed magnetic core is magnetized to induction B„, and then
the magnetizing force completely removed, induction decreases to
residual induction Br and remains at this value in the absence of mag-
netizing force, or for H = 0. The value of H required to reduce B to
zero is called the coercive force (He). From Fig. 18 it is evident that
Br and He may change with frequency for the same S,„ and grade of
core material, and the design of transformers and reactors may be
affected by the influence of frequency on core steel properties.
According to equation 10, p. 12, the core flux is proportional to effec-
tive alternating voltage for a given frequency and number of turns,
and so is flux density in a given core. Therefore the largest loop of
24
ELECTRONIC TRANSFORMERS AND CIRCUITS
Fig. 18. A-o and d-c hysteresis
loops.
Fig. 19. Normal induction.
Fig. 19 corresponds to a definite effective voltage and frequency, ap-
plied across a coil linking a definite core, and magnetizing it to maxi-
mum flux density Bm- If effective voltage is reduced 20 per cent a
smaller B-H loop results, with lower maximum flux density B'm- If
effective voltage is reduced further, still lower maximum flux density
B"m is reached. The locus of points Bm, B'^, B"m, etc., is drawn in
Fig. 19, and is called the normal induction curve. It is similar in shape
to, but not identical with, the virgin curve OB^ of Fig. 18. Each time
the maximum flux density is lowered, a short time elapses before the
new loop is traced each cycle. Thus the loops of Fig. 19 represent
symmetrical steady-state or cyclic magnetization at different levels
of maximum induction.
A normal induction curve is drawn in Fig. 20. The ratio oi B to H
at any point on the curve is the normal permeability for that value of
B. For the maximum flux density B^, the normal permeability is
M = Bm/H„
(19)
It is the slope of a straight line drawn through the origin and Bm- A
similar line drawn tangent to the curve at its "knee" is called the maxi-
mum permeability and is the ratio /*„ = B'/H'. The slope Bq/Hq of
normal induction at the origin (enlarged in Fig. 20) is the permeability
for very low induction Bo', it is called initial permeability and is usually
much less than /j.^-
Maximum permeability as here defined is really the average slope
of the normal induction curve up to induction B'. Actual slope from
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 25
1
«-4H->
■Bm
ae
A
^
?^
T~
/'/
( '/
,
11
1
ij
1 1
1
ll
«-H
dc-^
Hm
Fig. 20. Normal permeabilities.
Fig. 21. Incremental
permeability.
to B' is greater at some points than maximum permeability, because
the curve is steepest below B'. The slope at any induction is called
differential permeability.
From inspection of Fig. 19 it will be noticed that, for H = 0, the
sides of the B-H loop are steeper than any part of the normal induc-
tion curve and hence the slopes exceed /xm- This fact has practical
significance in the design of magnetic amplifiers.
In the foregoing, symmetrical magnetization has been assumed. If
a core is magnetized with d-c magnetizing force Hao as in Fig. 21, and
a-c magnetization AH is superimposed, the cyclic magnetization follows
a minor loop AB„,. Decreasing induction follows the left side of a
major loop whose maximum induction is B,n, down to induction A =
Bm — aB. Increasing induction follows a line which joins the right
side of the major loop. The area of this loop is small, but so is the
average slope, or incremental permeability. This permeability is im-
portant in reactor design. It is defined by
fXA = AB/AH
(20)
and is generally smaller than /*„. The dotted line in Fig. 21 is the
normal induction curve, the locus of the tops of minor loops as H^c
is decreased.
Returning now to Fig. 19, if Hm is increased, an induction is finally
reached at which unit increase of H produces only unit increase in B„.
This is known as saturation induction Bg. The value of H at which
Bs is first reached is very large compared to He for most core materials.
26
ELECTRONIC TRANSFORMERS AND CIRCUITS
A striking development has been the production of core materials with
rectangular hysteresis loops. In such materials Bs is reached at small
values of H, as shown in Fig. 22. Core
material having a rectangular hysteresis
loop is especially useful in magnetic am-
plifiers, and is discussed in Chapter 9.
The volt-amperes per pound or appar-
ent core loss (Pa) of a magnetic material
is the product of rms induced voltage and
rms exciting current drawn from the
source when a pound of the material is
subjected to sinusoidally varying induc-
tion of a specified maximum value B^ and
of a specified frequency /. Exciting cur-
rent is non-sinusoidal, as can be seen from
Fig. 5, Chapter 1. The power component
of Pa is the core loss Pc. The reactive
component is usually the larger and is
It is related to permeability in the following
— ^
J
Fig. 22.
Rectangular hyster-
esis loop.
called VARS per pound
way:
Let it be assumed that for conditions B„, Hm in a core the magnetizing cur-
rent is approximately sinusoidal, of effective value Im, drawn from a supply of
frequency / and effective voltage E. If we combine
1. Open-circuit inductance Le = E/2TrflM
2. Magnetizing force H„
EImV
V'2 1m X 10*
QA-wNIuy/2
3. VARS/lb
■^ci'C
convert to inches, and put density p = 0.27 lb/in. ^, then
152/B„2
(21)
(22)
(23)
(24
(25
VARS/lb X 10*
Because of the non-linearity of Im, this
At 60 cycles, M=Yi;000^^;j^g^.
equation is approximate. Moreover, there is no allowance for core gap.
In usual electronic transformer practice, it is necessary to avoid
reaching saturation flux densities, because high exciting currents pro-
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 27
duce high winding IR drops, high losses, low efficiency, and large size.
Curves of induction and core loss are available from manufacturers of
laminations. Grades and thickness are designated by numbers such as
Armco Trancor M15 and Allegheny Transformer A. A wide choice
of silicon-steel laminations is available in 0.014-in., 0.019-in., and
0.025-in. thicknesses, with silicon content of approximately 3 to Afo,
TF
3TS MADE
MPLE CUT
MPLE CUT
ON EP
STFIh
<;aupi r<! A<:
SHEARED
DmECTION
rRANSVERS
— n —
il
il
1 1
1
1 1
SA
SA
PARALLEL TO
HALF PARALL
60 CYCLE
ROLLING
EL, HALF 1
S
E
/
CO
t
/
^
2
1
o
t
//
//
//
D
o
2
//
//
//
//
//
/ /
^^
^
^
CORE L(
SS- WATTS
PER f
OUN
)
OX)l 0.1 1.0 10
Fig. 23. Core loss at high induotion. Armco Trancor M15 grade, 29 gage.
and with core losses ranging from 0.6 to 1.2 watts per pound at 10,000
gauss, 60 cycles (64,500 lines per square inch). Figures 23 and 24 are
core-loss and exciting va/lb for a widely used grade of electronic trans-
former core steel at 60 cycles.
Much work has been done in developing grain-oriented core mate-
rials. These materials have a composition similar to that of older,
non-oriented core material, but grains in the material are oriented by
cold-rolling in the direction illustrated by Fig. 25. Magnified sections
of laminations are shown in this figure; (a) shows the random direc-
tions of "easy" magnetization in grains of non-oriented silicon steel.
When magnetic flux is established in the lamination, the grains must
be aligned in the same direction, as in Fig. 25(6). If the grains are
already oriented in this direction during the rolling process, much
28
ELECTRONIC TRANSFORMERS AND CIRCUITS
smaller magnetizing force is required to produce the desired flux.
Coercive force and hysteresis loss are smaller than in non-oriented
steel ; permeability is greater, and so is Br, so that the rectangular loop
of Fig. 22 is approached in grain-oriented steel.
TE
STS MADE
MPLE CUT
kMPLE CUT
NEG
ON EPSTEIN
SAMPLES. ;
\S SHEARE
DIRECTION
TRANSVER
•
•
D
''' —
r''"^
^
St
SI
PARALLEL TO ROLLING
HALF PARALLEL, HALF
LIGIBLE JOINT EFFECTS
60 CYCLES
P
>
y
•
/
/ y
•
• /
'■/
/
>
/
^
0.1
r.o 10
EXCITING R.M.S. VOLT-AMPERES PER, POUND
100
Fig. 24. Exciting rms volt-amperes per pound, Armco Trancor M15 grade, 29
gage.
Grain- oriented core materials are of two major types: silicon-steel
and nickel-iron alloy. Electronic power transformers (i.e., plate and
filament supply transformers) formerly comprised only hot-rolled sili-
con-steel cores. The development of grain-oriented silicon steel has
had a marked effect on size and performance of such transformers. To
STEEL ROLLED IN THIS DIRECTION
(a)
(b)
FiQ. 25
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 29
illustrate this effect, a comparison is made below between the older
non-oriented steel (termed, for simplicity, silicon steel) and Hipersil,
a cold-rolled steel in which grain orientation is carried out to a high
degree. If core flux flows in the grain-oriented direction, high core in-
ductions may be reahzed. Type C cores fulfill this requirement, be-
cause the strip is wound in the same direction as the flux path.
^
—
" ;::'; —
UJ
(fi
VIlPEKSiu
3
^'
tlCON
jrEei|-
/'
^
>•
t-
o
u
u.
\
3
O
(E
Q.
?}
1
tn
}-
1-
<
/
r
i
CO
tn
o
-I
^
/
A
UJ
rr
/.
y
o
o
^
^
MAGNETIZING FORCE-OERSTEDS
FLUX DENSITY-KIUOGAUSSES
Fig. 26. Induction and core-loss curves of silicon steel and Hipersil at 60 cycles.
The material is rolled in three major thicknesses:
No. 29 gage (about 12 to 14 mils thick) for frequencies up to 400
cycles.
5 mils thick for frequencies 400 cycles and higher.
2 mils thick for frequencies in the low and medium r-f bands.
Probably the most remarkable property of this material is its high
saturation point. In Fig. 26 the comparison is given in terms of a
hypothetical 60-cycle working induction using high-grade, conventional
silicon steel. If this value is assumed to be 100 per cent, the induc-
tion obtained with grain-oriented steel is 130 to 150 per cent, with no
increase in magnetizing force. Another way of expressing this im-
provement is shown in Fig. 27 as a comparison of the permeability of
the two steels. The permeability of grain-oriented steel is much higher
at the maximum point, and has the same percentage increase as in
Fig. 26 for normal working inductions. Iron loss in Hipersil is less
than in silicon steel, as Fig. 26 shows. The decrease in iron loss is
chiefly due to a reduction in hysteresis loss; the eddy-current loss is
less affected by grain orientation. Future comparisons may widen
these differences.
30
ELECTRONIC TRANSFORMERS AND CIRCUITS
The increase in induction is beneficial in several ways. First, it
permits a reduction of core area for the same magnetizing current.
Second, it results in a smaller mean length of turn and thus in a reduc-
tion in the amount of copper needed. In distribution and power trans-
formers, for maximum benefit the iron and copper losses are repropor-
tioned. In small electronic transformers, the iron loss is usually a
small part of the total loss, and the reduction in copper loss is of greater
significance. Within certain limits, the sum of the two losses deter-
mines the size of a transformer, and here the usefulness of grain-
oriented steel becomes most apparent.
"
'
^
^
\
GRAIN ORIENTED STEEL ^
^
\
\ '
^
\
^1
1
^
,
-
"
:
. -1
,
_
-
-
■k
-SILICON STEEL
"•■
■•4J
rr
zr
^
^
■:
"
1
1 1
.
.
|■-^
100 1,000
MAXIMUM ALTERNATING FLUX DENSITY B IN GAUSSES
10.000 20.000
Fig. 27. Permeability of silicon and grain-oriented silicon steel.
The foregoing was written with 60-cycle applications particularly in
mind. At higher power supply frequencies, such as the 400- and 800-
cycle supplies encountered in aircraft and portable equipment, the
results are somewhat different. The decrease in iron loss is not so
marked, because the eddy current loss forms a larger proportion of the
total iron loss. However, it is usual practice to use thin-gage lamina-
tions at these frequencies, and much better space factor can be ob-
tained in wound cores than in stacked cores. The increase in permea-
bility is just as effective in these higher frequency applications as at
60 cycles. The net result is a smaller transformer than was formerly
possible, though for different reasons and in different proportions.
Reactors which carry direct current are usually smaller when made
with grain-oriented than with ordinary silicon steel. At low voltages,
where low inductions are involved, grain-oriented steel has greater in-
cremental permeability, and maintains it at high flux densities also.
Consequently, a reduction of 50 per cent in weight is often feasible.
Grain-oriented silicon steel does not replace high nickel-iron alloys
for audio transformers, when they work at low inductions, and with
little or no direct current. Some nickel-iron alloys have higher permea-
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 31
bility at low flux densities, and their use for this purpose continues.
But at high inductions, or where considerable amounts of direct cur-
rent are involved, grain-oriented silicon steel is used. Lower distortion,
extended frequency range, or small size is the result, and sometimes
a combination of all three occurs.
i
^F
1
i
1
~j"
1 '
^
M
!l
~ ■ AU
D
9
1
-
f
i
i
>ov
IC
7
1
' POW
PF
-
"^ T
5-
CL
—
1
J 1
TT
JCY
i
FREQUET
IN
CY
ES'i
1
1__
1
10 10^ 10' 10* 10* 10*
Fig. 28. Use of Hipersil in various frequency zones.
Hipersil can be used for transformers in various applications in the
low and medium r-f bands, at power levels ranging up to hundreds of
kilowatts. The same is true of video and pulse transformers, which
may be regarded as covering an extended frequency range down into
the audio range and up into the medium r-f range. Such transformers
are grouped rather loosely together as r-f transformers in the diagram
shown in Fig. 28. In this figure the several classifications, r-f, audio,
and power transformers, are shown with respect to their frequency
ranges and the approximate gage of the material used for these ranges.
The gage is indicated by the symbol number in Table II.
Table II. Hipersil Core Data
Typical Space
Typical Hipersil
Factor for
[ipersil
Thickness
Space Factor *
Silicon Steel *
C-97
0.013 in.
95%
90%
C-95
0.005 in.
90%
80%
C-91
0.002 in.
85%
70%
* Refers to percentage of core volume occupied by metal. The Hipersil figure
is for type C cores, and the silicon steel figure is for punched laminations.
Core-loss and exciting va/lb for 29-gage Hipersil are plotted in Figs.
29 and 30. Joint reluctance is neglected in Fig. 30.
An example of specialized core materials is the development of a
new grain-oriented silicon steel especially for weight reduction in com-
32
ELECTRONIC TRANSFORMERS AND CIRCUITS
10000
1000
100
10
=
Ut 1
-^
^'
y
_.n^";
-
y
K
^
f'
■■^
/J
^
.■■■
J&=^ —
y\
^y
^^
^4#
;^^^
.-
•*
J?' .r,.^
,}(^\)i^ yyyo'
''
^ ^'
«>i^^
yy
fJifi?^
y
y
y^
/--
y)
y^^m^
/
_ ^
'" "5
1
w
>"■
/
z —
-/'—
— ^ - >-
:.; ;S^^^
'^^V:,
/ ,
Ztxi>-
.'V,
^///
0.01
ai
1.0 10
WATTS PER POUND
100
1000
Fig. 29. Core loss in C-97 Hipersil cores (29 gage).
10000
(eIOOO
100
10
±::
1-.^^
^
— '>'^~
i^
^
5^ =
^■-^
^
.-•
^
>-'
^
/
^ M 1"
_t .^
^/
,'■
y
.'
fc^
•-^
-^'
/
^
^-
^q5i
^
^^
/
/"
y-
-'^^
f^
^'
^.-
•f>
r 1
.0^
■•a.^"''^
/
y^
- P 'i^ — — ;—
Lj^Li
litr — '
M
fTrt=^
-^^^=m
oSJ^c-^^
z^'
= ^^<^^^
^?-^
.,
^
To"<fT?,K*
v'^?
7^ ,
/
-^
^
•^^S^A?^
f^Q^Oo'
/
^
y
y
^5
1
■fl'' G^^*"
^
/>J/y
c^
. /
,^''
y
y
Sp/^
/]
1-^
'^^
,^/
^
y
7
^
^
^
/
f
:zm
^ '^
-:^0^
^^;^^
VX-
::^fc
=:
^
::::
^o!i
^
/
r
tSt^
"f^^
■v^'i
/'
/
y
z'
<' /
^ /
/
y
^
/
7
/
0.01 0.1 1.0 10 100 1000
APPARENT WATTS PER POUND
Fig. 30. A-c excitation curve, typical data. C-97 Hipersil cores (29 gage).
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 33
ponents for 400-cycle applications. By means of large reduction in
core loss at 400 cycles and still larger increase in permeability at high
induction, a 0.004-in. -thick core material was developed which oper-
ates satisfactorily in many instances at 17,000 gauss, 400 cycles. As a
result, 40 per cent of the weight was eliminated in transformers de-
signed to take advantage of the 0.004-in-thick core material. At lower
inductions the core loss of this material tends to be larger than in the
older 0.005-in. -thick material. Hence it is only where 17,000 gauss is
a practicable working induction that the weight reduction is possible.
Grain-oriented steel alloys of approximately 50% nickel content
are extensively used in saturable reactors. Electrical properties of
cores wound from these materials are spoiled if the strip is bent
or constrained mechanically. Usually the nickel-alloy strip is wound
into cores in the form of a toroid, annealed, and enclosed in an insulat-
ing box to protect it from damage. Special machinery is then used
to wind turns of wire around the core. With the proper precautions,
it is possible to realize the advantages of a very rectangular, narrow
hysteresis loop in the finished reactor. These properties have been
found useful also in pulse transformers, and are discussed in Chapters
9 and 10 in detail.
In audio- or higher- frequency low-loss reactors or transformers, it
may be desirable to use powdered iron or nickel-alloy cores. These
cores are made of finely divided particles, coated with insulating
compound, which separates them and introduces many fine air gaps in
the magnetic path. The cores are molded into various shapes suitable
for the application. Effective permeability of such cores is reduced
to a figure much lower than that of laminations made from the same
material.
Magnetic ferrites likewise are used at higher frequencies. These
substances are characterized by high resistivity so that neither lami-
nations nor powder particles are necessary to reduce eddy-current loss.
Cores are molded and sintered at high temperature. After sintering
they have ceramic hardness but relatively low Curie temperature.^
Ferrites are useful at very high frequencies.
Some of the principal core materials are listed in Table III.
16. Windings. Current density in the winding copper is sometimes
estimated for design purposes by rules such as 1,000 cir mils per amp.
These rules are useful in picking out a first choice of wire size for a
given current requirement but should not be regarded as final. In-
1 The temperature at which a ferric substance loses its intrinsic permeability.
34
ELECTRONIC TRANSFORMERS AND CIRCUITS
Table III.
Typical
Maximum
Approximate
Permeability
Description
Trade Names
Mm
Silicon steel
Transformer
Trancor M15
Power 58
8,500
Grain-oriented
Hipersil
30,000
silicon steel
Trancor 3X
50% nickel steel
Hipernik
Allegheny Elec-
tric Metal
Nicaloi
50,000
50% nickel steel,
Conpernik
1,400
special heat
treatment
Grain-oriented
Hipernik V
50,000
50% nickel
Orthonol
steel
Orthonik
Deltamax
Permenorm
80%, nickel steel
Permalloy
Mumetal
Hymu
100,000
80%o nickel steel,
Supermalloy
200,000
special heat
treatment
Powdered iron
Crolite
Polyiron
125
Ferrite
Ceramag
Ferramic
Ferroxcube
1,000
* These materials
are used for low flux density, low
Core Materials
Maximum Coercive
Operating Force
Flux Density D-C Loop
Bm (gauss) (oersteds) Chief Uses
12,000 0.5 Small power and voice frequency au-
dio transformers
17,000
6,000
6,000
0, 4 Larger sizes of power and wide-range
audio transformers; low-frequency
r-f transformers; saturable reactors
0.06 Small, wide-range audio transform-
ers and reactors (may have small
d-c induction)
Extremely hnear and low-loss trans-
formers
0, 15 Saturable reactors
0.05 Small or wide-range audio transform-
ers (no d-c induction)
0.01 Very small or wide-range transform-
ers (no d-c induction)
Wave filter reactors; low and me-
dium r-f transformers
0. 2 Sweep circuit transformers; r-f trans-
formers and reactors
r-loss apphcations.
stead, the temperature rise, regulation, or other performance criterion
should govern the final choice of wire size. Regulation is calculated as
in Section 11, and temperature rise as in Sections 22 and 23. In Fig. 31
the circular mils per ampere are plotted for small enclosed dry-type
transformers with Hipersil cores and a winding temperature rise of
55 centigrade degrees; it can be seen to vary appreciably over this
range of sizes.
Space occupied by the wire depends on the wire insulation as well
as on the copper section. This is especially noticeable in small wire
sizes. Table IV gives the bare and insulation diameters for several
common kinds of wire and Table V the turns per square inch of wind-
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 35
ing space. Space usually can be saved by avoiding cotton or silk wire
covering, and instead using enameled wire with paper layer insulation
as in Fig. 32. Thickness of layer paper may be governed by layer
voltage; it is good practice to use 50 volts per mil of paper. In coils
where layer voltage is low, the paper thickness is determined by the
mechanical strength necessary to produce even layers and a tightly
DUU
y
700
y
y
y
y
y
^ 600
UJ
/
y
y
Z
< 500
q:
UJ
Q.
V) 400
_l
z
< 300
_l
3
^
^
y
^
^
•
/
y
^
/
/
O
5 200
^
100
n
10
100
VOLT AMPERES
Fig. 31. Wire size in windings of small enclosed 60-cycle transformers.
wound coil. Table VI gives the minimum paper thickness based on
this consideration.
Space factor may refer to linear spacing as across a layer, or to the
total coil section area. It is more convenient to use linear space factor
in designing layer-wound coils and area space factor in random-wound
coils. The values in each case depend largely on the method of wind-
ing. For example, it is possible to wind No. 30 enameled wire with
97 per cent linear space factor by hand, but with only 89 per cent on
an automatic multiple-coil winding machine. (See Fig. 33.) More-
over, values of space factor vary from plant to plant. An average for
multiple-coil machines is given in Table VI.
36
ELECTRONIC TRANSFORMERS AND CIRCUITS
Table IV. Insulated Wire Sizes
Bare
Diam-
eter
Diameter of Insulated Wire
Area
in
Circu-
lar
Ohms
per
1000
Feet
at
Feet
per
Ohm
at
Pounds
B&
S
Gage
Single
Double
Single
Cotton
Enamel
Single
Silk
Enamel
Single
Double
Single
Double
per
1000
Feet
Enamel
Enamel
Cotton
Cotton
Silk
Silk
Mils
26°C
25°C
44
.0020
.0023
4.00
2,700
.3850
.012
43
.0022
.0025
4.84
2,150
.4670
.015
42
.0025
.0029
6.25
1,700
.6050
.019
41
.0028
.0032
7.84
1,350
.7630
.024
40
.0031
.0036
.0039
9.61
1,103
.9560
.030
39
.0035
.0040
.0044
12.25
864
1.204
.038
38
.0040
.0046
.0050
16.00
659
1.519
.048
37
.0045
.0051
.0055
20.30
522
1.915
.060
36
.0050
.0057
.0061
.0095
.0076
.0090
.0130
.0070
.0090
26.00
424
2.414
.076
39
.0056
.0064
.0067
.0102
.0082
.0096
.0136
.0076
.0096
31.40
338
3.045
.096
34
.0063
.0072
.0077
.0109
.0089
.0103
.0143
.0083
.0103
39.70
266
3.839
.120
33
.0071
.0080
.0085
.0117
.0097
.0111
.0151
.0091
.0111
50.40
210
4.841
.162
32
.0080
.0090
.0095
.0127
.0107
.0120
.0160
.0100
.0120
64.00
165
6.105
.19
31
.0089
.0100
.0104
.0137
.0117
.0129
.0169
.0109
.0129
79.20
134
7.698
.24
30
.0100
.0111
.0117
.0148
.0128
.0140
.0180
.0120
.0140
100
106
9.707
.31
29
.0113
.0125
.0130
.0162
.0142
.0153
.0193
.0133
.0153
128
83.1
12.24
.38
28
.0126
.0139
.0145
.0175
.0165
.0166
.0206
.0146
.0166
159
66.4
15.43
.48
27
.0142
.0155
.0161
.0192
.0172
.0182
.0222
.0162
.0182
202
52.6
19.46
.61
26
.0159
.0172
.0178
.0210
.0190
.0199
.0239
.0179
.0199
263
41.7
24.64
.77
25
.0179
.0193
.0200
.0234
.0211
.0222
.0202
.0199
.0219
320
33.0
30.96
.97
24
.0201
.0216
.0222
.0256
.0233
.0244
.0284
.0221
.0241
404
26.2
39.02
1.23
23
.0226
.0242
.0247
.0282
.0259
.0269
.0309
.0246
.0266
511
20.7
49.21
1.54
22
.0253
.0271
.0278
.0310
.0287
.0296
.0336
.0273
.0293
645
16.4
62.05
1.95
21
.0286
.0302
.0310
.0344
.0319
.0330
.0370
.0305
.0325
812
13.0
78.25
2.45
20
.0320
.034
.0345
.0385
.0355
.0370
.0410
.0340
.0360
1,020
10.3
98.66
3.09
19
.0359
.038
.0387
.0425
.0395
.0409
.0449
.0379
.0399
1,300
8.14
124.4
3.89
18
.0403
.042
.0431
.0469
.0439
.0453
.0493
.0423
.0443
1,600
6.59
156.9
4.9
17
.0453
.047
.0481
.0521
.0491
.0503
.0543
.0473
.0493
2,030
5.22
197.8
6.2
16
.0508
.053
.0536
.0576
.0546
.0558
.0608
.0528
.0548
2,600
4.07
249.4
7.8
15
.0571
.059
.0605
.0640
.0610
.0621
.0671
.0691
.0611
3,250
3.26
314.5
9.9
14
.0641
.066
.0675
.0711
.0681
.0691
.0741
.0661
.0681
4,100
2.68
396.6
12.4
13
.0719
5,180
2.00
499.3
15.7
12
.0808
6,630
1.59
629.6
19.8
11
.0907
8,235
1.26
794.0
24.9
10
.1019
10,380
1.00
1,001
31.4
9
.1144
13,090
.792
1,262
40.0
8
.1285
16,610
.628
1,592
50.0
TRANSFORMER CONSTRUCTION. MATERIALS, RATINGS 37
Table V. Turns per Square Inch of Insulated Wire
O
Single
Enamel
T)onlilp
Single
Single
Single
Double
Single
Double
Enamel
Cotton
Silk
Cotton-
Cotton-
Silk-
Silk-
■^
Wire
Wire
Enamel
Enamel
Covered
Covered
Covered
Covered
m
Wire
Wire
Wire
Wire
Wire
Wire
42
119,000
41
96,000
40
77,000
66,200
39
62,400
51,800
38
47,300
40,000
37
38,400
33,100
36
30,900
26,900
11,100
17,900
12,350
5,920
20,400
12,350
35
24,500
22,300
9,600
14,900
10,900
5,430
17,200
10,900
34
19,300
16,900
8,430
12,700
9,430
4,900
14,500
9,430
33
15,600
13,900
7,280
10,650
8,130
4,380
12,100
8,130
32
12,350
11,100
6,210
8,740
6,940
3,900
10,000
6,940
31
10,000
9,260
5,330
7,300
5,900
3,510
7,780
5,900
30
8,180
7,300
4,580
6,100
5,100
3,090
6,940
5,100
29
6,430
5,920
3,810
4,950
4,270
2,760
6,670
4,270
28
5,200
4,770
3,280
4,170
3,640
2,360
4,690
3,640
27
4,170
3,880
2,720
3,390
3,030
2,080
3,810
3,030
26
3,380
3,160
2,270
2,780
2,520
1,940
3,120
2,620
25
2,690
2,500
1,820
2,240
2,080
1,460
2,530
2,080
24
2,150
2,030
1,530
1,850
1,690
1,230
2,050
1,720
23
1,710
1,650
1,260
1,490
1,380
1,050
1,650
1,420
22
1,370
1,300
1,045
1,220
1,140
883
1,345
1,160
21
1,100
1,045
846
925
915
729
1,075
943
20
860
850
675
793
730
595
862
836
19
693
668
555
640
597
495
700
628
18
568
540
455
518
490
412
563
610
17
455
432
368
417
395
340
450
412
16
357
350
303
338
320
270
360
336
15
288
273
244
270
260
222
287
268
14
230
220
198
216
210
182
229
222
13
179
176
12
143
141
11
114
113
10
90
90
9
72
72
8
57
57
38
ELECTRONIC TRANSFORMERS AND CIRCUITS
Mean length of turn must be calculated for a coil in order to find its
resistance in ohms. This may be found by referring to the side view
of Fig. 32. Note that there is a small clearance space between core
-CORE TONGUE
d = TONGUE WIDTH
W« STACK
r - COIL TUBE RADIUS
A - MARGINS
B - WINDING TRAVERSE
C -OVERALL LENGTH
D -BUILD UP
E -INSIDE DIMENSION OF TUBE
F -OUTSIDE DIMENSION OF COIL
G-TUBE THICKNESS
Fig. 32. Paper-insulated coil.
and coil form or tube. Let d be the core tongue and w the stack. Sup-
pose there are several concentric windings. The length of mean turn
of a winding V at distance r from the core and having height D, is
MT = 2w + 2d + 2x
(-f)
= 2(w + d) + 7r(2Si) + D) (26)
where ^D is the sum of all winding heights and insulation thicknesses
between winding V and the core.
The mean turn of the winding U just below V ordinarily is calculated
before that of winding V. This fact simplifies the calculation of wind-
ing V, the mean turn of which is
MTv = MTu + Tv{Du + Dv + 2c) (27)
where c is the thickness of insulation between U and V.
Allowance must be made, with many coil leads, for bulging of the
coil at the ends and consequent increase of mean turn length.
The placement, insulation, and soldering of leads constitute perhaps
the most important steps in the manufacture of a coil. When coils
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 39
Table VI. Papek-Insulated Coil Data
(Courtesy Phelps-Dodge Copper Products Corp.)
B&S
Layer
Turns
Space
Gage
Insulation
per Inch
Factor
44
.0005"
369
85%
43
.0006"
340
85%
42
.0006"
304
86%
41
.0007"
265
85%
40
.0007"
239
86%
39
.0007"
216
86%
38
.001"
193
87%
37
.001"
170
87%
36
.001"
165
87%
35
.001"
140
88%
34
.001"
124
88%
33
.0013"
110
88%
32
.0013"
98
88%
31
.0015"
88
88%
30
.0016"
80
89%
29
.0015"
71
89%
28
.0015"
64
89%
27
.0022"
67
89%
26
.0022"
62
89%
25
.0022"
47
90%
24
.0022'
42
90%
23
.005"
37
90%
22
.006"
33
90%
21
.005"
30
90%
20
.006"
26
90%
19
.007"
23
90%
18
.007"
21
90%
17
.007"
19
90%
16
.010"
17
90%
15
.010"
16
90%
14
.010"
13
90%
13
.010"
12
90%
12
.010"
10
90%
11
.010"
9
90%
10 .010" 8 90%
40 ELECTRONIC TRANSFORMERS AND CIRCUITS
are wound one at a time, the leads can be placed in the coil while it is
being wound. The start lead may be placed on the coil form, suitable
insulation may be placed over it, and coil turns may be wound over
the insulation. Tap leads can be arranged in the same way. Finish
leads must be anchored by means of tape, string, or yarn, because
• >. f.
rtB- ■
•-.■'iS*-.5J... .^
* -*'
^%M
^^
Fig. 33. Winding 20 coils in multiple machine: layer paper at right.
there are no turns of wire to wind over them. Typical lead anchoring
is shown in Fig. 34.
In multiple-wound coils, the leads must be attached after the coils
are wound. Extra wire on the start turn is pulled out of the coil and
run up the side as shown in Fig. 35, with separator insulation between
wire extension and coil. Outer insulation covers the wire extension
up to the lead joint. A pad of insulation is placed under the joint, and
one or more layers of insulation, which insulate and anchor the joint,
are wound over the entire coil and the lead insulation. Electrical-grade
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 41
scotch tape is widely used for anchoring leads. It is important to avoid
corrosive adhesives.
Leads should be large enough to introduce only a small amount of
voltage drop and should have insulation clearances adequate for the
test voltage. These clearances can be found as explained in Section 19.
h v^ X ^^ ^' ^"^
■treated cloth
'fishpaper
treated cloth
fishpaper
■soldered joint
WHEN FIRST PLACED ON TUBING
.FIRST LAYER OF WIRE
AFTER FIRST LAYER IS WOUND
Fig. 34. Stiart-lead insulation in hand-wound coils.
In high-voltage transformers it would often be possible to seal the
windings if there were no leads; hence lead placement calls for much
care and skill. Leads and joints should also be mechanically strong
enough to withstand winding, impregnating, and handling stresses
without breakage.
17. Insulation. Three classes of insulation are used in dry-type
transformers. Class A insulation is organic material such as paper,
cotton, silk, varnish, or wire enamel. Class B insulation is mica, as-
bestos, glass, porcelain, or other inorganic material with organic bind-
ers such as varnish for embedding the insulation. A small amount of
42
ELECTRONIC TRANSFORMERS AND CIRCUITS
other class A material is permissible in a class B coil "for structural
reasons," but it should be kept to a minimum.
In general, the vital difference between these classes of insulation
is one of operating temperature. Glass-covered wire is preferable to
asbestos for space reasons; it is available in approximately the same
dimensions as cotton-covered wire. Built-up mica is the usual insula-
tion wrapper material. With special bonds it is flexible enough to
TAPE ANCHOR
LEAD
^^^3SS^
J^p^ SEPARATOR INSULATION
OUTER INSULATION
WINDING EXTENSION,
Fig. 35. Start-lead insulation in multiple-wound coils.
wind over coils or layers of wire. Stiff mica plate for lead insulation
and mica tubing for coil forms are usually bonded with heat-resistant
varnish. Class B insulating material is more expensive than class A
and is used only when other advantages outweigh the cost.
The necessity for small size in aircraft or mobile apparatus is con-
tinually increasing the tendency to use materials at their fullest capa-
bilities. As size decreases, the ability of a transformer to radiate a
given number of watts loss also decreases. Hence, it operates at higher
temperature. Transformers for 400- and 800-cycle power supplies can
be made in smaller overall dimensions by using class B insulation (see
Section 20). As a result, from 30 to 50 per cent decrease in size
(as compared with class A insulation), in addition to increased ability
to withstand extremes of ambient temperature, humidity, and alti-
tude, is obtained. Class B insulation is thus of special importance in
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 43
aircraft apparatus. Usually at 60 cycles enough room is available to
use class A insulation, but mica may be used to reduce the size of high-
voltage units.
A third class of insulation is the silicones, organic silicates with re-
markable thermal and mechanical properties. These materials are
coming into use at operating temperatures approaching 200°C. Sili-
cone-treated cloth, silicone rubber, and silicone varnish are already in
use. Under development are silicone wire enamel and silicone-bonded
mica. They are generally designated as class H insulation.
For apparatus having long service life, AIEE Standard 1 limits the
"hottest spot" temperature of impregnated^ coils as follows:
Class A insulation 105 °C
Class B insulation 130 °C
Class H insulation 200 °C
Life test data are plotted in Fig. 36 for class A and class B insula-
tion. The temperature scale is special, based on T. W. Dakin's data,^
showing that insulation life is proportional to the reciprocal of abso-
lute temperature. The two lines indicate how operating temperature
may be increased for a given life when class B insulation is used.
Equal life is obtained when class A insulation is operated at 105°C
maximum (40°C ambient, 55°C rise, 10°C hottest spot gradient), and
when class B insulation is operated at 130°C maximum (40°C ambient,
80°C rise, 10°C hottest spot gradient). Intermittent load tempera-
tures may be high for short periods. These periods are additive. For
example, class A insulation has approximately the same life whether
it is operated at 115°C continuously or half the time at 123°C and
half the time at 25 °C. Figure 36 shows only the influence of tempera-
ture on insulation life. Life is further reduced by moisture, vibration,
and corona. It is therefore important that insulation be protected
against damage caused by all these factors. Such protection is dis-
cussed in Section 20.
18. Dielectric Strength. The usual figure given for dielectric strength
is the breakdown value in rms volts at 60 cycles in a 1-minute test.
It is not possible to operate class A insulation anywhere near this
value because of the cellular structure of all organic materials. Even
after these materials are treated with varnish, many small holes exist
throughout a coil structure which ionize and form corona at voltage
1 For the definition of impregnation, see Section 20.
2 See "Electrical Insulation Deterioration Treated as a Chemical Rate Phe-
nomenon," by T. W. Dakin, Trans. AIEE, 67, 113 (1948).
44
ELECTRONIC TRANSFORMERS AND CIRCUITS
far below breakdown. With class A insulation (organic materials),
the designer must be governed more by resistance of the insulation
to corona over a long period than by breakdown strength of the in-
sulation in a 1 -minute test. For example, a 20-mil thickness of
treated cloth will withstand 10,000 volts for 1 minute. However,
220
ui 200
< 160
140
s
\
N
\
N
S,
s
\
No
N
<
N
A
hu
\
s
^C/
^
h\
^
^
^
N
N.
'?'
"/n
s
s
1
4-
K
■v
\
S,
\
N
•v
N
^
s
s
^
\
100 1,000
LIFE IN DAYS
10,000
Fig. 36. Approximate life expectancy of electrical insulation.
corona starts at 1,250 volts, and operation at any higher voltage would
puncture the insulation in a few weeks. It is much wiser to keep a
reasonable margin, say 20 to 30 per cent, below the corona limit than
to use a fraction of the 1-minute breakdown test. Approximate volt-
ages at which corona is audible are plotted in Fig. 37 as a function of
insulation thickness.
Differences in hearing ability between persons make a corona meas-
urement desirable. This is done by means of the standard NEMA
circuit of Fig. 38.^ With the transformer connected as shown, receiver
1 See "Radio Influence Characteristics of Electrical Apparatus," by P. L. Bel-
laschi and C. V. Aggers, Tram. AIEE, 67, 626 (November, 1938).
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS
45
output meter is adjusted to half-scale by a volume control potentiom-
eter in the receiver. Next, the transformer is replaced by a modulated
1-mc signal generator, the output of which is varied until the noise
meter output is again half-scale. The signal generator output in micro-
volts is read on an attenuator; this is then a measurement of the corona
present.
20,000
10,000
1,000
100
y
y
^
/-'
o
^
y^
o
>
<
z
o
o:
y
/
/
O
o
UJ
y
^/^
to
s
IE
.01 .02 .04 .06 .08 .1 .2 .4 .6 .8
TOTAL INSULATION THICKNESS IN INCHES
Fig. 37. Corona limit for treated cloth and paper.
Class B insulation can be worked much closer to the ultimate di-
electric strength, but the latter is less a factor in determining size than
creepage distance to the core. For mica an approximate working
voltage rule is 100 volts rms per mil thickness.
Insulated coils in air are subject to a two-dielectric effect that is
peculiarly troublesome. If the path of electric stress is partly through
solid material and partly through air, the air may be overstressed be-
cause it has the lower dielectric constant (unity, compared with 3 to 5
for most coil materials) . If this condition exists, it is usually imprac-
46
ELECTRONIC TRANSFORMERS AND CIRCUITS
ticable to increase the air distance and so reduce the volts per inch
to a value below the corona limit. The addition of more solid insula-
tion over the whole coil may make it too large. Often the only feasible
remedy is to fill the air space with more solid material, either in the
form of filling compound or strips of insulation like micarta or press-
board.
It is important, when dealing with insulation voltage, to make a
RFC
:T
TESTING TRANSFORMER
TRANSFORMER UNDER TEST
COUPLING CAPACITOR
R
NM
RFC
INPUT RESISTOR (600 OHMS)
NOISE METER (RECEIVER WITH
METER OUTPUT)
RADIO FREQUENCY CHOKE
Fig. 38. Standard NEMA radio-influence measuring circuit.
distinction between test voltage and operating voltage. Of the two,
operating voltage is the better value to specify.
19. Creepage Distance. Although solid insulation dielectric strength
is important, the usual bottleneck for high voltage is creepage distance,
such as margins between wire and core along the layers of insulation,
or margins between lead joints and frame along the leads and coil
sides. A common way of increasing the direct creepage distance
across the margins is to use an insulating channel as in Fig. 39(a).
This is especially helpful when the part of the coil adjacent to the core
tongue is at low potential and the upper part is at high potential, as
in some plate transformers. When the whole coil is at high potential
it may be insulated by taping the coil, but taping is expensive and is
avoided wherever creepage safely provides the necessary insulation
strength.
Creepage distances over treated cloth or other organic material in
air are shown in Fig. 40 for breakdown voltages up to 100 kv. The
primary purpose of these curves is to find the proper margins for coils
adjacent to the core.
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 47
Insulation between the start (or finish) turn of the first layer and
the core consists of creepage along the margin plus the thickness of
the coil form. This is not a relevant distance, however, if the coil lead
MARGIN
o
o
o
o
■ INSULATING
CHANNEL
(a)
TAPE
Fig. 39. (a) Use of insulating channel; (5) taped coil.
is brought across the margin and up the side of the coil. In such a
case, the only creepage distance is the thickness of the coil form.
In low-voltage coils this may be enough; in higher-voltage coils, a
barrier of insulating material is needed between the coil form and the
core, under the spot where the lead is brought out of the coil. Such a
48
ELECTRONIC TRANSFORMERS AND CIRCUITS
barrier is provided by outer insulation in Fig. 35. Dimensions of the
insulating barrier should be such that a distance at least equal to
the coil margin should intervene between the start lead and the core
in all directions and the thickness may be the same as the coil form.
100
80
60
40
20
1
1 —
] —
1 — r
-r
r-
1 —
1 —
; —
1
^
-T™ ^ h--^'^"'^"^
— '
^
S4 ' 1 j / BARRIEK
^
, ^
^
"
i^GROUND PLATF 1
^
y
GROUND
^^
e"
■^
y
o
__^
a"
•-•
^
/
UJ
"
^■
?"
C-
.^
y
z
.. —
— ■
■—
^
y
y'
z
^
■-
—
-^'
v"
■-
-^
^
/
^
/
/
a:
^?
1
^
^^..*— •
-
^
X
^
^1
y
a:
, -
—
Va
y
>
"^
^
I
^
l^
-I
Ix.
,'
i^
h
>^
/
■^V
^
^
8 10
20
40
60 80 100
150
THICKNESS X 10 INCHES
Fig. 40. Creepage curves in air over smooth organic insulation.
In any coil where the finish lead is at the top of the coil, there is
less difficulty in insulating the finish lead. The finish lead has a longer
creepage distance to the core if the height of the coil is a greater
distance than the margin. It is necessary to avoid using materials on
the sides of the coil which would result in any decrease of dielectric
strength. In this respect, the creepage strength of some materials with
high puncture strength is not good. The last layer of wire may be
insulated from the core with a channel as in Fig. 39(a).
When practical coil margins, even with harriers, are insufficient to
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 49
support the induced or applied voltage, coils are taped as in Fig. 39(6).
Taping is the most time-consuming but the safest method of insulation.
Separate secondaries may be taped and then assembled over the pri-
mary. If the whole transformer winding is taped, the coil form must
be large enough to allow room for the taping between the core and
coil form. It is also important that the leads be taped, to prevent
breakdown from joints to ground.
Ordinarily, a winding is separated from the winding under it by
wraps of Kraft paper or other insulation. In the coil of Fig. 41 the
insulation thickness between winding 1 and winding 2 is shown divided
WINDING, NO. I
1<MARGIN-»J (^ 1
j INSULATION
- THICKNESS
LINES OF ^ ^
ELECTRICAL STRESS ^'^DING NO. 2 , INSULATION
INSULATION-' THICKNESS
Fig. 41. Adaptation of Fig. 40 for insulation between coils
by an imaginary center line. With equal margins in the two windings,
the voltage stress is symmetrical about this center line. Margins
should be such that there is sufficient creepage distance, in conjunction
with one-half the insulation thickness, to withstand one-half the test
voltage between these adjacent windings. That is, when full test volt-
age is applied between the windings, only half of it appears between
the first layer of winding 1 and the center line of the insulation be-
tween the windings. If the margins are unequal, the sum of the two
margins, in conjunction with the total insulation thickness, should be
large enough to withstand the full test voltage, in accordance with
Fig. 40.
Coils may be divided into "part coils" or sections, to reduce insula-
tion stresses, but such coils should be closely integrated with the circuit.
For this reason, part coils are discussed in later chapters.
20. Impregnation. After a coil is wound the best practice is to im-
pregnate it in some sort of insulating liquid which hardens after filling.
This is done for several reasons. First, it protects the wire from move-
ment and possible mechanical damage. Second, it prevents the en-
trance of moisture and foreign matter which might corrode the wire
or cause insulation deterioration. Third, it increases the dielectric
50 ELECTRONIC TRANSFORMERS AND CIRCUITS
strength of fibrous insulating materials. Fourth, it assists in heat dis-
sipation from the coil. Single-layer coils may be dipped in the liquid,
drained, and dried, but deeper, thicker coils require the use of vacuum
to remove air from the coil and admit the liquid to all parts of the
interior. The best mechanical result is obtained when coils are assem-
bled with cores before treatment.
Insulation is considered to be impregnated when a suitable sub-
stance replaces the air between its fibers, even if this substance does
not completely fill the spaces between the insulated conductors.
Coils having little or no temperature rise in normal use are impreg-
nated with chemically neutral mineral wax. The wax is melted in a
sealed tank and is drawn into another tank in which preheated coils
have been placed, and a vacuum is maintained. Coils are removed
from the tank, drained, and allowed to cool. Wax treatment provides
good dielectric qualities and moisture protection. It is a quick, simple
process.
Transformers having operating temperatures of 65°C or higher are
impregnated with varnish. Varnish of good grade and close control is
essential to achieve thorough filling and dry coils after impregnation.
Oleoresinous varnishes, which polymerize to a hard state by baking,
are notably useful for the purpose. A high degree of vacuum, fresh
varnish, and accurate baking temperature control are necessary for
good results. Plasticizers are sometimes added to the varnish to pre-
vent brittleness in finished coils. Varnish may attack wire enamel
(which itself is a kind of varnish), and so the soaking and baking time
periods must be regulated carefully.
Varnishes for impregnation of electrical coils have until lately been
diluted by solvents to lower the viscosity so as to permit full pene-
tration of the windings. When the coils are baked, the varnish dries
and the solvent is driven off. The drying leaves very small holes
through which moisture can penetrate and in which corona may form.
Eventually, the insulation deteriorates. It is, therefore, necessary to
allow large clearances for high voltages or to immerse the coils in oil.
Either of these alternatives increases the size of a high-voltage trans-
former in relation to that of a low-voltage transformer. For this rea-
son, solventless resins have come into use as filling compounds for dry-
type coils. They are known by trade names such as Fosterite, Para-
plex, and Stypol. These resins have the advantage of changing from
a liquid to a solid state by heat polymerization, so that small holes
formed by drying of the solvent are eliminated. Filling of the coil
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 51
may be accomplished by casting the transformer in a mold, or by
encapsulation. Encapsulation is readily adapted to irregular coil sur-
faces and is accomplished by a leak-proof coat before filling. In either
process, a good vacuum is necessary to insure complete filling.
Silicone materials are moisture-resistant. Basic insulation should
be inorganic, or silicone-treated cloth, tape, laminated sheets, and
tubes. Through the use of silicones, some transformers may be de-
signed to have very small dimensions for their ratings. This may be
achieved most successfully if the coil insulation comprises only sili-
cone or inorganic materials, including impregnation with silicone
varnish. Dielectric strength of silicones is about the same as class A
materials. Hence the thickness of silicone coil insulation is similar
to that for organic materials.
Continual development improves all classes of insulation ; present A,
B, and H insulation classes may be superseded eventually by new
classes based entirely on functional evaluation. Life tests have been
proposed ^ which classify a transformer according to its ability to
withstand the effects of voltage, moisture, and vibration, as well as
temperature.
In encased high-voltage units, air around the coils, bushings, and
leads is especially subject to the formation of corona. To reduce this
tendency, the containers are filled with asphaltic compound which re-
places the air with solid, non-ionizing material. A similar compound
is often used to fill containers of low-voltage transformers to avoid
the need for mechanically fastening the core to the case. This is a per-
missible practice if the melting point of the compound is higher than
the highest operating temperature and if its cracking point is below the
lowest operating temperature.
21. Oil Insulation. Although, in electronic apparatus, there is a
tendency toward the use of dry-type transformers, frequently voltages
are so high that air clearances are impracticable and oil-filled contain-
ers must be used. In Fig. 42 the curves show rms breakdown voltage
versus creepage distance under oil. An example will show the ad-
vantage of oil filling. From Figs. 40 and 42 it will be seen that 10-in.
creepage distance is required in air to withstand a 1-minute breakdown
test of 60 kv on insulation 0.5 in. thick, whereas in oil only 2-in.
creepage distance is required.
1 See "Functional Evaluation of Insulation for Small Dry-Type Transformers
Used in Electronic Equipment," by R. L. Hamilton and H. B. Harms, AIEE
Tech. Paper 54-121.
52
ELECTRONIC TRANSFORMERS AND CIRCUITS
Curves of Fig. 42 are for pressboard or Micarta under oil. Some
kinds of porcelain have less creepage strength than these materials.
On the other hand, some grades of glass and polystyrene are much
better and withstand 150 kv for 1 minute with 2 in. of creepage path.
INCHES THICKNESS
4 5 6 7 8 9 10
20
30 40 50
.4 .5 .6 .7 .8 .9 I
INCHES THICKNESS
Fig. 42. Creepage curves of solid insulation under oil.
In high-voltage low-current power supplies, these special materials
are used to save weight and space. At 50 kv or more, sharp edges
and points should be avoided by the use of round terminals, leads, and
coils.
Only high grades of insulating oil are used for this purpose. Tests
are run continually to check condition of the oil. Oil is stored in such
a manner as to keep out moisture and dirt and avoid extremes of
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 53
temperature. Where very high voltages are used, as in X-ray appara-
tus, oil filling is done under vacuum to remove air bubbles, and con-
tainers are sealed afterwards to prevent moisture from entering. Mica
insulation is not used in oil because oil dissolves flexible bonds.
Often a high-voltage transformer can be integrated with some other
component, such as a tube socket, capacitor, or another transformer.
This is desirable from the standpoint of space conservation, provided
that adequate clearances to the case are maintained. "Packaged"
power supplies are sometimes made in this fashion to facilitate assem-
bly and repair.
22. Size versus Rating. Core area depends upon voltage, induction,
frequency, and turns. For a given frequency and grade of core mate-
rial, core area depends upon the applied voltage. Window area de-
pends upon coil size, or for a given voltage upon the current drawn.
Since window area and core area determine size, there is a relation
between size and v-a rating.
With other factors, such as frequency and grade of iron, constant,
the larger transformers dissipate less heat per unit volume than the
smaller ones. This is true because dissipation area increases as the
square of the equivalent spherical radius, whereas volume increases as
its cube. Therefore larger units are more commonly of the open type,
whereas smaller units are totally enclosed. Where enclosure is feasi-
ble, it tends to cause size increase by limiting the heat dissipation.
Figure 43 shows the relation between size and rating for small, en-
closed, low-voltage, two-winding, 60-cycle transformers having Hi-
persil cores and class A insulation and operating continuously in a
40°C ambient. The size increases for the same volt-amperes over that
in Fig. 43 for any of the following reasons:
High voltage Silicon-steel cores
High ambient temperature Low regulation
Lower frequency More windings
The size decreases for
Higher frequencies Open-type units
Class B insulation Intermittent operation
If low-voltage insulation is assumed, two secondary windings reduce
the rating of a typical size by 10 per cent; six secondaries by 50 per
cent. The decreased rating is due partly to space occupied by insula-
tion and partly to poorer space factor. The effects of voltage, tem-
perature, and core steel on size have been discussed in preceding sec-
54
ELECTRONIC TRANSFORMERS AND CIRCUITS
tions. Frequency and regulation will be considered separately in suc-
ceeding chapters.
Open-type transformers like those in Fig. 8 have better heat dis-
sipation than enclosed units. The lamination-stacking dimension can
UJ
o
(S
z
140
120
100
80
o
z
s
3
o
>
60
40
20
VOLUME
/
f
>
/
12
/
/
WEIGHT
/
/
/
CD
-J
Z 8
1-
X
7
/.
A
•FICI
:ncy
^
^
^'
/
/
/
/
/
o
UJ
S 6
/
/
/
/
4
/
'/
//
2
/
/
/
/
/
VA
40 80 120 160 200
Fig. 43. Size of enclosed 60-cycle transformera.
be made to suit the rating, so that one size of lamination may cover a
range of v-a ratings. Heat dissipation from the end cases is independ-
ent of the stacking dimension, but that from the laminations is directly
proportional to it. This is shown in Fig. 44 for several lamination
sizes. For each size the horizontal line represents heat dissipation
from the end cases; the sloping line represents dissipation from end
cases, plus that from the lamination edges which is proportional to the
stacking dimension. At ordinary working temperature, heat is dis-
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 55
sipated at the rate of 0.008 watt per square inch per centigrade degree
rise. In Fig. 44 the watts per centigrade degree of temperature rise
are given as a function of lamination stack. This refers to temperature
rise at the core surface only. In addition, there is a temperature
o
<
I-
<
LAM
NATION DIMENSIONS -INCHES
W H
A 3.75 4.63
B 5.00 r.oo
C 6.88 7.50
D 7.50 11.50
1 1 1 1
^
D
X
< w »1
^
^
H
^
^
^
C
^
^
^
^
^
^
^
^
B
Y^
^
^
^
-^
^
^
^
^
■^
^^
—
A
D
^
^
^
.-^
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„-^
-^
C
^
"
^,_— -
-"^
8
_„,-
■■-^
A
01 234 5678 9 10 II 12
LAMINATION STACK - INCHES
Fig. 44. Heat dissipation from open-type transformers with end cases.
gradient between coil and core which is given in similar manner in
Fig. 45.
To find the average coil temperature rise, divide the copper loss by
the watts per centigrade degree from the sloping line of Fig. 45. To
this add the total of copper and iron losses divided by the appropriate
ordinate from Fig. 44. That is, the total coil temperature rise is equal
to the sum of the temperature drop across the insulation (marked Cu-
Fe gradient in Fig. 45) and the temperature drop from the core to the
ambient air. Data like those in Figs. 44 and 45 can be established for
any lamination by making a heat run on two transformers, one having
a core stack near the minimum and one near the maximum that is
likely to be used. Usually stacking dimensions lie between the ex-
56
ELECTRONIC TRANSFORMERS AND CIRCUITS
tremes of Yo to 3 times the lamination tongue width, and poor use of
space results from stacking outside these limits. If end cases are
omitted, coil dissipation is improved as much as 50 per cent.
The same method can be used for figuring type C Hipersil core de-
signs; here the strip width takes the place of the stacking dimension
of punched laminations, and the build-up corresponds to the tongue
\^
D
1 1 1 1 1 1
LAMINATION WINDOW ASSUMED
y
^
^
y
/
C
^
^
^
z'
y^
^
^
B
^
^
^
^
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^
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^
^
'^
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^
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^
2345 6 789
LAMINATION STACK- INCHES
10 II 12
Fig. 45. Winding-to-core gradient for open-type transformers with end oases.
For lamination sizes, see Fig. 44.
width. When two cores are used, as in Fig. 14, the heating can be
approximated by using data for the nearest punching.
For irregular or unknown heat dissipation surfaces, an approxima-
tion to the temperature rise can be found from the transformer weight,
as derived in the next section.
23. Intermittent Ratings. It often happens that electronic equip-
ment is operated for repeated short lengths of time, between which
the power is off. In such cases the average power determines the heat-
ing and size. Transformers operating intermittently can be built
smaller than if they were operated continuously at full rating.
Intermittent operation affects size only if the "on" periods are short
compared to the thermal time constant of the transformer; that is, small
transformers have less heat storage capacity and hence rise to final
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS 57
temperature more quickly than do large ones. It is important, there-
fore, to know the relation between size and thermal time constant, or
the time that would be required to bring a transformer to 63 per
cent of the temperature to which it would finally rise if the power were
applied continuously.
The exact determination of temperature rise time in objects such as
transformers, having irregular shapes and non-homogeneous mate-
rials, has not yet been attempted. Even in simple shapes of homo-
geneous material, and after further simplifying assumptions have been
made, the solution is too complicated ^ for rapid calculation. How-
ever, under certain conditions, a spherical object can be shown to cool
according to the simple law: ^
d = doe per (28)
where 6 = temperature above ambient at any instant t
do = initial temperature above ambient
E = emissivity in calories per second per centigrade degree per
square centimeter
p = density of material
c = specific heat of material
r = radius of sphere
e = 2.718.
The conditions involved in this formula are that the sphere is so
small or the cooling so slow that the temperature at any time is sensibly
uniform throughout the whole volume. Mathematically, this is ful-
filled when the expression Er/k (where fc is the thermal conductivity
of the material) is small compared to unity. Knowing the various
properties of the transformer material, we can tell (1) whether the re-
quired conditions are met, and (2) what the thermal time constant is.
The latter is arrived at by the relation
t, = pcr,/3E (29)
where r^ is the radius of the equivalent sphere.
In order to convert the non-homogeneous transformer into a homo-
geneous sphere the average product of density and specific heat pc is
1 See The Mathematical Tlienry of Heat Conduction, by L. R. Ingersoll and
O. J. Zobel, Ginn and Co., Boston, 1913, p. 142.
2 Ingersoll and Zobel, op. cit., p. 143.
58
ELECTRONIC TRANSFORMERS AND CIRCUITS
found. Figures on widely different transformers show a variation from
0.862 to 0.879 in this product; hence an average value of 0.87 can be
taken, with only 1 per cent deviation in any individual case.
Since the densities of iron and copper do not differ greatly, and in-
sulation brings the coil density closer to that of iron, it may be further
assumed that the transformer has material of uniform density 7.8
throughout. The equivalent spherical radius can then be found from
= (Weight/1.073)'^
(30)
where re is in inches and weight is in pounds. The time constant is
plotted from equations 29 and 30 in terms of weight in Fig. 46.
4.0
10
2.0
1.0
0.8
0.6
0.6
0.4
0.3
0.2
Ql
V
-
'
or
^^
^
O
X
,-'
^
^
1-
^
■^
iS
^
^
o
■^
s
1-
_J
1-
TF
ANS
-OF
ME
R
TC
)TAL WE
IGHT
-UBS.
8 10
60 80 100
Fig. 46. Transformer time constant, or time required to reach 63 per cent of final
temperature.
The condition that Er/k be small compared to unity is approxi-
mated by assuming that fc is the conductivity of iron — a safe assump-
tion, because the conductivity of copper is 7 to 10 times that of iron.
A transformer weighing as much as 60 lb has r^ = 5.45 in., E — 0.00028
cal per sec per sq cm/°C, and fc = 0.11. Changing r^ to metric units
TRANSFORMER CONSTRUCTION, MATERIALS, RATINGS
59
gives Er/k = (0.00028 X 5.45 X 2.54) /O.ll = 0.34, which is small
enough to meet the necessary condition of equation 28.
It will be noticed that equation 28 is a law for cooling, not tempera-
ture rise. But if the source of heat is steady (as it nearly is) the equa-
tion can be inverted to the form 6o — for temperature rise, and Oo
becomes the final temperature.
Temperature rise of a typical transformer is shown in Fig. 47, to-
1
===
TEMPERATURE filSE'--^,^^,^^^;^
^«^^
i5.»?ox
?=^
/
1
/r —
ACTUAL TEMPERATURE RISE
IN TRANSFORMER
//
/ /
/ /
1/
TIME IN MULTIPLES OF THERMAL TIME CONSTANT Tj
Fig. 47. Transformer temperature rise time.
gether with the exponential law which is ^o — ^, where B is the tem-
perature of equation 28. The actual rise is less at first than that of
the foregoing simplified theory, then more rapid, and with a more
pronounced "knee." The 63 per cent of final temperature is reached
in about 70 per cent of the theoretical time constant tc for transformers
weighing between 5 and 200 lb. This average correction factor is in-
cluded in Fig. 46 also.
If a transformer is operated for a short time and then allowed to
cool to room temperature before operating again, the temperature rise
can be found from Figs. 46 and 47. As an example, suppose that the
continuously operated final coil temperature rise is 100 centigrade
degrees, the total weight is 5 lb, and operating duty is infrequent
periods of 2 hr. From Fig. 46, the transformer has a thermal time
constant of 0.85 hr. This corresponds to i^ = 1 in Fig. 47. Two hours
are therefore 2 -^ 0.85 = 2.35 times ic, and the transformer rises to
90 per cent of final temperature, or a coil temperature rise of 90 cen-
tigrade degrees, in 2 hr.
If, on the other hand, the transformer has regular off and on intervals,
the average watts dissipated over a long period of time govern the
60 ELECTRONIC TRANSFORMERS AND CIRCUITS
temperature rise. A transformer is never so small that it heats up
more in the first operating interval than at the end of many intervals.
From equation 30 can be found a relation between weight, losses,
and final temperature rise. For, since heat is dissipated at 0.008 watt
per sq in./°C rise, and the area Ag of the equivalent sphere is 4nrr/,
Total watts loss Total watts loss
^0 =
0.0084,s /Total weight in poundsV
\ 1.073 /
(31)
where ^o is the final temperature rise in centigrade degrees. This equa-
tion is subject to the same approximations as equation 28; test results
show that it is most reliable for transformers weighing 20 lb or more,
with 55°C temperature rise at 40°C ambient.
3. RECTIFIER TRANSFORMERS AND REACTORS
^.
Fig. 48. High- vac-
uum rectifier volt-
age-current curve.
Rectifiers are used to convert alternating into direct current. The
tubes generally have two electrodes, the cathode and the anode. Both
high vacuum and gas-filled tubes are used. Sometimes for control
purposes the gas-filled tubes have grids, which are discussed in Chap-
ter 8.
A high-vacuum rectifier tube characteristic voltage-current curve is
shown in Fig. 48. Current flows only when the anode is positive with
respect to the cathode. The voltage on this curve
is the internal potential drop in the tube when cur-
rent is drawn through it. This voltage divided by
the current gives effective tube resistance at any
point. Tube resistance decreases as current in-
creases, up to the emission limit, where all the
electrons available from the cathode are used.
Filament voltage governs the emission limit and
must be closely controlled. If the filament voltage
is too high, the tube life is shortened; if too low,
the tube will not deliver rated current at the proper voltage.
Gas-filled rectifier tubes have internal voltage drop which is virtually
constant and independent of current. Usually this voltage drop is
much lower than that of high vacuum tubes. Consequently, gas-
filled tubes are used in high power rectifiers, where high efficiency and
low regulation are important. In some rectifiers, silicon or germanium
crystals or selenium disks are used as the rectifying elements.
In this chapter, the rectifier circuits are summarized and then
rectifier transformers and reactors are discussed.
24. Rectifiers with Reactor-Input Filters. Table VII gives com-
monly used rectifier circuits, together with current and voltage rela-
tions in the associated transformers. This table is based on the use of
a reactor-input filter to reduce ripple. The inductance of the choke is
assumed to be great enough to keep the output direct current con-
stant. With any finite inductance there is always some superposed
61
62
ELECTRONIC TRANSFORMERS AND CIRCUITS
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RECTIFIER TRANSFORMERS AND REACTORS 63
ripple current which is neglected in the table, and which is considered
further in Chapter 4.
The single-phase half -wave rectifier ordinarily has discontinuous
output current, and its output voltage is therefore highly dependent
upon the inductance of the input filter choke. For this reason, the
currents and voltages are given for this rectifier without a filter.
The difference between primary and secondary v-a ratings in several
of these rectifiers does not mean that instantaneous v-a values are
different; it means that because of differences in current wave form
the rms values of current may be different for primary and secondary.
Unbalanced direct current in the half-wave rectifiers requires larger
transformers than in the full-wave rectifiers. This is partly overcome
in three-phase transformers by the use of zigzag connections. The
three-phase full-wave rectifier can be delta-connected on both primary
and secondary if desired; the secondary current is multiplied by 0.577
and the secondary voltage by 1.732. Anode windings have more turns
of smaller wire in the delta connection. Single-phase bridge and three-
phase full-wave rectifiers require notably low a-c voltage for a given d-c
output, low inverse peak voltage on the tubes, and small transformers.
25. Rectifiers with Capacitor-Input Filters. When the filter has no
reactor intervening between rectifier and first capacitor, rectifier cur-
rent is not continuous throughout each cycle and the rectified wave
form changes. During the voltage peaks of each cycle, the capacitor
charges and draws current from the rectifier. During the rest of the
time, no current is drawn from the rectifier, and the capacitor dis-
charges into the load.
RECTIFIED VOLTAGE
cr
'^ — 11 — ' — I
INPUT CURRENT
(a) (6)
Fig. 49. Voltage and current comparisons in reactor-input and capacitor-input
circuits.
Comparison between the rectified voltage of reactor-input and
capacitor-input filters in a single-phase full-wave rectifier may be seen
in Figs. 49(a) and (6), respectively. The two tube currents /i and
I2 in (a) add to a constant d-c output, whereas in [h] the high-peaked
tube currents flow only while the rectified voltage is higher than the
64
ELECTRONIC TRANSFORMERS AND CIRCUITS
average d-c voltage. Average current per tube in both cases is half the
rectifier output. With large values of capacitance, the rectified voltage
o o o o o
0) h- CO m o y
^?
\ \
\\
\
1
\ \
1
\
\V
A\
\\\
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11
X\\
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X"
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(
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w
WW \\
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w
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<
in Fig. 49(6) increases to within a few per cent of the peak voltage.
Ripple, average rectified voltage output, and rectifier current are
dependent on the capacitance, the supply line frequency, and the load
resistance. They are dependent also on rectifier internal resistance
because it afl^ects the peak value of current which the filter capacitor
can draw during the charging interval A^.
RECTIFIER TRANSFORMERS AND REACTORS
65
Analysis of this charge-discharge action involves complicated
Fourier series which reciuire a long time to calculate.^ Satisfactory
M°=
r « - "
"J-
U)
oo o
o \n oioo o oP2oo
i
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voltage and current values have been obtained from experimental
measurements by Schade ^ and are shown in Figs. 50, 51, and 52 for
1 See "Diode Rectifying Circuits with Capacitance Filters," by D. L. Waidelich,
Trans. AIEE, 61, 1161 (December, 1941).
2 "Analysis of Rectifier Operation," by O. H. Schade, Proc. I.R.E., 31, 341
(July, 1943).
66
ELECTRONIC TRANSFORMERS AND CIRCUITS
single-phase half-wave and full-wave rectifiers. In these figures Ra is
the rectifier series resistance, including the transformer resistance.
o
a
s
|s«i
sa *
q
o
2 8
S°
O
1 o
2
o
s
CM O
in
1
1
o
Cvj
IT
1
o
o 1
o
o
\
\
v
\\
, p
\ \
T r
\\
fr
\\
V|
\
™
\
\\\
1
m
1
\
n = 1 FOR HALF WAVE 1 RECTIFIER
n = 2 FOR FULL WAVE 1 (jl RECTIFIER
n= 1/2 FOR VOLTAGE DOUBLING CIRCUIT
C= FARADS R=OHMS
UJ = 2TTf f = LINE FREO
'
~-
-
\U-
-
■
\
w
w
\
-
\
i
-
1
3
3
CO
o
T
1
O
o
q
o
1^
(3i»ld 3N0)
iNBaanp 3ivici o a
iN3aanD 3iv^6 swa 'tdii
_!- OlVld 3N0I
iN3aanD 3ivid do ^
iN3banQ 3IVld )IV3d "dj
Results accurate to within 5 per cent are obtained if the rectifier re-
sistance corresponding to peak current tp is used in finding Rs- The
process is cut-and-try, because Ip depends on Rg, and vice versa, but
two trials usually suffice. Resistance is in ohms, capacitance is in
RECTIFIER TRANSFORMERS AND REACTORS 67
farads, and <o is 2ir times the supply frequency. Three-phase rectifiers
are rarely capacitor-input because of their larger power.
In Fig. 52 the peak current indicates whether the peak current of a
given tube is exceeded, and the rms current determines the transformer
secondary heating. The v-a ratings are greater, but ratios of primary
to secondary v-a ratings given in Table VII hold for capacitor-input
transformers also.
26. Voltage Doublers. To obtain more d-c output voltage from a
rectifier tube, the circuit of Fig. 53 is often used. With proper values
of circuit elements the output is nearly double the a-c peak voltage.
Tube inverse peak voltage is little more than the d-c output voltage,
and no d-c unbalance exists in the anode transformer. Current output
available from this circuit is less than from the single-phase full-wave
circuit for a given rectifier tube. Current relations are given in Fig. 52.
Voltage tripling and quadrupling circuits also are used, either to
increase the d-c voltage or to avoid the use of a transformer.^
27. Filament Transformers. Low-voltage filament transformers are
used for heating tube filaments at or near ground potential. Often
the filament windings of several tubes are combined into one trans-
former. Sometimes this requires several secondary windings. In
terms of a single secondary transformer a 5 or 6 secondary unit requires
about 50 per cent greater size and weight. But these multiwinding
transformers are smaller than five or six separate units; this warrants
designing them specially in many instances.
Rectifier tube filaments often operate at high d-c voltages and re-
quire windings with high voltage insulation. It is usually not feasible
to combine high-voltage windings with low-voltage windings when the
high voltage is more than 3,000 volts direct current because of insula-
tion difficulties, particularly in the leads. Large rectifier filaments are
usually heated by separate transformers; in polyphase rectifiers, all
tube filaments are at high voltage, and some secondary windings may
be combined. See the three-phase full-wave rectifier in Table VII,
where the -\-HV lead connects to a winding which heats the filaments
of three tubes.
Low capacitance filament windings are sometimes required for high-
frequency circuits. The problem is not particularly difficult in small
v-a ratings and at moderate voltages. Here air occupies most of the
space between windings. In larger ratings the problem is more diffi-
cult, because the capacitance increases directly as the coil mean turn
1 See "Analyses of Voltage Tripling and Quadrupling Circuits," by D. L. Waide-
lich and H. A. Taskin, Proc. I.R.E., 33, 449 (July, 1945).
68
ELECTRONIC TRANSFORMERS AND CIRCUITS
length for a given spacing between windings. As voltage to ground
increases, there comes a point beyond which creepage effects necessitate
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oil-insulated windings, whereupon the capacitance jumps 2 to 1 for a
given size and spacing. There is a value of capacitance below which
it is impossible to go because of space limitations in the transformer.
What this value is in any given case may be estimated from the fact
RECTIFIER TRANSFORMERS AKD REACTORS 69
that the capacitance in n/xi of a body in free space is roughly equal
to one-half its largest dimension in centimeters.
Except for the differences just mentioned, the design of filament
transformers does not differ much from that of small 60-cycle power
Fig. 54. 15 kv filament transformer enclosed in insulating case.
transformers. The load is constant and of unity power factor. Leak-
age reactance plays practically no part, because of its quadrature rela-
tionship to the load. Output voltage may therefore be figured as in
Fig. 3(c) (p. 8). It should be accurately calculated, however, to
maintain the proper filament emission and life.
When a tube filament is cold, the filament resistance is a small
fraction of its operating value. In large tubes it is often necessary to
protect the tube filaments against the high initial current they would
draw at rated filament voltage. This is done by automatically reduc-
ing the starting voltage through the use of a current-limiting trans-
70 ELECTRONIC TRANSFORMERS AND CIRCUITS
former having magnetic shunts between primary and secondary wind-
ings. The design of these transformers is somewhat special, and is
included in Chapter 8.
High-voltage filament transformers are sometimes mounted in an
insulating case, as in Fig. 54, with the tube socket on top. This ar-
rangement eliminates the need for high-voltage wiring between the
transformer and the tube, and provides the insulation for the socket.
The problem of air pockets at the base of high-voltage bushings is also
eliminated. It is still necessary to insulate well between windings and
to fill the case fully with insulating compound in order to eliminate
corona.
28. Filament Transformer Design. It is important that design work
be done systematically to save the designer's time and to afford a
ready means of finding calculations at a later date. To attain these
ends a calculation form, such as that in Fig. 55, is used. The form is
usually made to cover several kinds of transformers, and only the
spaces applicable to a filament transformer are used.
Suppose that a transformer is required to supply filament power for
four single-phase full-wave rectifiers having output voltages of 2,000,
500, 250, and 250 volts, respectively, with choke-input filters, as follows:
Primary voltage 100
Frequency 60 cycles
Four secondaries for the following tube filaments:
2-
-872 tubes:
5 volts
13.
5 amp
Insulated for -1-2000 v d-c
2-
-866 tubes:
2.5 volts
10
amp
Insulated for -f- 500 v d-c
1-
-5U4G tube:
5 volts
3
amp
Insulated for + 250 v d-c
1-
-6Y3GT tube:
5 volts
2
amp
Insulated for -f- 250 v d-c
Ambient temperature: 40 °C
First comes the choice of a core. Data such as those in Fig. 43 are
helpful in this, and so is design experience in the modification of such
data by the specified requirements. The core used here is a 2-in. stack
of laminations A, Fig. 44, which is described more fully in Fig. 56, and
has enough heat dissipation surface for this rating. For silicon steel,
an induction of 70,000 lines per square inch is practical. The primary
turns can be figured from equation 4 by making the substitution
(j> = BAc and transposing to
EX 10^
A^i = (32)
4:A4:fA,B
RECTIFIER TRANSFORMERS AND REACTORS
71
where Ac is the core cross-sectional area, or product of the core tongue
width and stack dimension, and B is the core induction. In this trans-
former, with 90 per cent stacking factor, 4<; = 2 X 0.9 X 1-375 = 2.48
sq in., and the primary turns are found to be 216.
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Fig. 55. Filament transformer design calculations.
Below this calculation are set down the primary voltage and fre-
quency, and the voltage, current, volt-amperes, and insulation voltage
for all secondary windings. These are designated Si to 1S4 for identifi-
cation. From the sum of the individual v-a figures, the transformer
72
ELECTRONIC TRANSFORMERS AND CIRCUITS
rating is found. To it is added an estimate of losses to obtain the
input volt-amperes, and the primary current.
Next an estimate of the regulation is made (10 per cent) and added
to unity to obtain the multiplier 1.1 in the estimate of secondary turns
near the top of the calculation form. From the currents listed, the
wire size for each winding is chosen. Round enameled wire is used for
CORE STACK
COIL FORM
l^"x2iL"|.D.
THICK WALL
Fig. 56. Dimensions and coil section of filament transformer.
each winding except »Si, and for it No. 12 square wire is used to save
space. The largest wire is placed next to the coil form to prevent dam-
age in winding to the smaller wires.
The next task is to find out whether the wire chosen will fit in the
core window space. Winding height D is entered for each winding.
For each secondary this is the wire diameter, because the wire is
wound in a single layer. D for Si is slightly larger than the wire dimen-
sion to allow for the bulge that occurs when square wire is wound.
The twelve turns of >Si occupy about 1% in. of horizontal winding
space. The core window is 2% in. wide. From this is subtracted
4 in. total coil width. Margins on each
— 114) = %g in. According to Fig. 40
(p. 48) this provides over 8 kv breakdown strength, which is well above
the 5-kv test voltage for Si. Other secondary windings have lower
test voltages and wider margins, and hence have more than adequate
creepage distances.
% in. for clearance, leaving 2^
side of Si are therefore 72
(21/8
RECTIFIER TRANSFORMERS AND REACTORS 73
The %e-iii.-thick Micarta rectangular tube used for the coil form
has a corona voltage of 2,700 rms, which affords about 23 per cent safety
factor over the normal operating voltage at the tube filaments. Over
Si are wound six wraps of O.OlO-in.-thick treated cloth, which has
2,600-volt corona limit. Winding S2 supplies a filament at 500 volts of
the same polarity as Si. Hence only 1,500 volts direct current or 1,660
volts alternating current occur across this insulation. At the right of
the small sketch in Fig. 55 are listed the number of wraps of 0.010-in.-
thick treated cloth over each section of winding. These are added to-
gether to give the columnar figure of 0.150 for TC.
The primary winding is wound without layer insulation and with
an area space factor of 70 per cent. Cotton is wound in with the wire
to form walls %6 in. thick on either side of the primary; this accounts
for the low space factor and for the 1%-in. winding traverse. The coil
is finished with two layers of treated cloth, a layer of 0.010-in, fishpaper
for mechanical protection, and a 0.025-in. serving of untreated cotton
yarn or tape to hold it together. The total winding adds up to 0.751
in., leaving 0.124 in. clearance, about the right allowance for winding
slack for four secondaries.
Mean turns are figured from equations 26 and 27, with 5 per cent
incremental increase in S2, S3, and S^ for leads. With the mean turn
values the winding resistances, weights of copper, and IR and PR for
each winding can be found. To Si, S2, and S3 winding resistance is
added lead resistance, and the lower figure is the sum of the two in each
case. Total copper loss is multiplied by 1.3 to correct for 75°C operat-
ing temperature. The core weight is 6.8 lb, and the grade of steel used
has 1.17 watts per pound at 70,000 lines per square inch. This gives a
core loss of 8 watts, and a total of copper and core loss of 20 watts.
After these losses are divided by the appropriate ordinates from Figs. 44
and 45 (pp. 55 and 56) the coil temperature rise is figured at 48 centi-
grade degrees, which is safe for class A insulation.
We know by now that the design is safe, but secondary voltages still
must be checked. The method of equation 13 is used. Output voltages
on first trial range from to 4 per cent high. S2 voltage is correct but
out of line with the rest. Changing So leads to a larger size makes the
per cent voltage drops more nearly alike, and increasing the primary
turns to 223 brings all output voltages to correct value within 1.2 per
cent. Filament voltage should be kept within 2 per cent for these
tubes, to allow for meter error. Primary voltage per layer is checked
at the lower left; this is equivalent to 22.7 volts per mil of wire enamel,
which is safe practice.
74 ELECTRONIC TRANSFORMERS AND CIRCUITS
If the design were deficient in any respect, even down to the last
things figured, some change would have to be made which would re-
quire recalculation of all or part of the transformer; hence the impor-
tance of good estimating all the way along.
The filament transformer outlined above had a center tap (C.T.)
in each filament winding. Such taps are used with directly heated
cathodes, especially when plate current is large, to prevent uneven
distribution of filament emission. In windings for supplying filaments
of small tubes, center taps are sometimes omitted. Ripple in the
rectified output then increases, and transformer core flux density be-
comes asymmetrical. Whether these effects are permissible depends
on operating conditions. Usually plate current is much smaller than
filament current, so that center-tap leads may be smaller in copper
section than start and finish leads. A certain amount of space is
required for these leads; rectifier wiring is also more time-consuming
when there are center taps. Nevertheless, the extra work and size may
be justified by improved performance.
An even number of turns, such as were used in the transformer
windings described in this section, results in center-tap placement on
the same coil end as the start and finish leads; if there were an odd
number of turns, the tap lead would be at the opposite end. In a
single-core, single-coil design, an odd number of turns cannot be center-
tapped exactly. Usually the unbalance caused by the tap being a half-
turn off center is not serious, but it should not be disregarded without
calculation.
29. Anode Transformers. Anode transformers differ from filament
transformers in several respects.
(a) Currents are non-sinusoidal. In a single-phase full-wave recti-
fier, for instance, current flows through one half of the secondary during
each positive voltage excursion and through the other half during each
negative excursion. For half of the time each half-secondary winding
is idle.
(6) Leakage inductance not only determines output voltage but also
affects rectifier regulation in an entirely different manner than with a
straight a-c load. This is discussed in Chapter 4.
(c) Half -wave rectifiers carry unbalanced direct current; this may
necessitate less a-c flux density, hence larger transformers, than full-
wave rectifiers. Unbalance in the three-phase half-wave type can be
avoided by the use of zigzag connections, but an increase in size over
full-wave results because of the out-of-phase voltages. These connec-
RECTIFIER TRANSFORMERS AND REACTORS
75
tions are desirable in full-wave rectifiers when half voltage is obtained
from a center tap. See Table VII.
(d) Single-phase full-wave rectifiers with two anodes have higher
secondary volt-amperes for a given primary v-a rating than a filament
transformer. Bridge-type (four-anode) rectifiers have equal primary
and secondary volt-amperes, as well as balanced direct current, and
plate transformers for these rectifiers are smaller than for other types.
Three-phase rectifier transformers are smaller in total size but require
more coils. The three-phase full-wave type has equal primary and
secondary v-a ratings.
(e) Induced secondary voltage is much higher. Filament trans-
formers are insulated for this voltage but have a few secondary turns
Fio. 57. Dimensions and coil section of anode transformer. Construction shown
is for shell-type transformer with 2 Hipersil cores.
of large wire, whereas anode transformers have many turns of small
wire. For this reason the volts per layer are higher in anode trans-
formers, and core windows having proportionately greater height and
less width than those in Fig. 56 are often preferable. This trend runs
counter to the conditions for low leakage inductance and makes it
necessary to interleave the windings. Figure 57 shows the windings
of a single-phase full-wave rectifier transformer with the primary inter-
leaved between halves of the secondary. This arrangement is espe-
cially adaptable to transformers with grounded center tap. The
primary-secondary insulation can be reduced to the amount suitable
for primary to ground. This is called graded insulation.
In large power rectifiers of the gas-filled or pool types, anode current
under short-circuit conditions may be very great, and anode trans-
former windings must be braced to prevent damage. If the conductors
76 ELECTRONIC TRANSFORMERS AND CIRCUITS
are small, solventless varnish is useful for solidly embedding the con-
ductors.
30. Leakage Inductance. Flux set up by the primary winding which
does not link the secondary, or vice versa, gives rise to leakage or self-
inductance in each winding without contributing to the mutual flux.
The greater this leakage flux, the greater the leakage inductance, be-
cause the inductance of a winding equals the flux linkages with unit
current in the winding. In Fig. 57, all flux which follows the core path
Ic is mutual flux. Leakage flux is the relatively small flux which
threads the secondary winding sections, enters the core, and returns
to the other side of the secondaries, without linking the primary. The
same is true of flux linking only the primary winding. But it is al-
most impossible for flux to leave the primary winding, enter the core,
and re-enter the primary without linking part of the secondary also.
The more the primary and secondary windings are interleaved, the less
leakage flux there is, up to the limit imposed by flux in the spaces c
between sections. These spaces contain leakage flux also; indeed, if
there is much interleaving or if the spaces c are large, most of the leak-
age flux flows in them. Large coil mean turn length, short winding
traverse b, and tall window height a all increase leakage flux.
Several formulas have been derived for the calculation of leakage
inductance. That originated by Fortescue ^ is generally accurate, and
errs, if at all, on the conservative side:
10.6N^MT(2nc + a)
Ls = -~7 (33)
10V6
where Ls = leakage inductance of both windings in henrys, referred to
the winding having N turns
MT = mean length of turn for whole coil in inches
n = number of dielectrics between windings (n = 2 in Fig. 57)
c = thickness of dielectric between windings in inches
a = winding height in inches
h = winding traverse in inches.
The greatest gain from interleaving comes when the dielectric thick-
ness c is small compared to the window height; when nc is comparable
to the window height, the leakage inductance does not decrease much
as n is increased. It is often difficult to reduce the leakage inductance
which occurs in high-voltage transformers because of leakage flux in
1 See Standard Handbook for Electrical Engineers, McGraw-Hill Book Co.,
New York, 1922, 5th ed., p. 413.
RECTIFIER TRANSFORMERS AND REACTORS 77
spaces c. A small number of turns, short mean turn, and low, wide
core windows all contribute to a low value of leakage inductance.
31. Anode Transformer Design. Let the requirements of a rectifier
be
1,200 volts 115 ma rectifier d-c output
Single-phase full-wave circuit with 866 tubes
Primary 115 volts 60 cycles
Rectifier regulation 5 per cent maximum
Ambient 55°C
To fulfill these requirements, a reactor-input filter must be used. If
1 per cent is allowed for reactor IR drop, a maximum of 4 per cent
regulation is left in the anode transformer. The approximate secondary
output voltage is 1,200 X 2.22 = 2,660, say 2,700 volts. The center tap
may be grounded. Suppose that a transformer like the one in Fig. 57
is used. The calculations are given in Fig. 58. The various steps are
performed in the same order as in filament transformers. The grain-
oriented type C core is worked at 38 per cent higher induction, with
but 60 per cent of the core loss of Fig. 55; its strip width is 2% in.,
build-up % in., and window 1 in. by 3 in. for each core loop. Note
the difference in primary and secondary volt-amperes and winding
heights. Since the primary and secondary are symmetrical about the
primary horizontal center line, they have the same mean turn length.
Losses and temperature rise are low. Regulation governs size. Sec-
ondary layer voltage is high enough to require unusually thick layer
paper. This coil is wound on a multiple-coil machine. Winding height
is figured on the basis of layer paper adequate for the voltage instead
of from Table VI (p. 39), but turns per layer are taken from this
table. Since adjacent layers are wound with opposite directions of
traverse, the highest voltage across the layer insulation is twice the
volts per layer. Layer insulation is used at 46 volts per mil in the
secondary; this counts the 1.7 mils of double enamel, which must
withstand impregnation without damage. Anode leads and margins
withstand 5 kv rms test voltage. Since the secondary center tap is
grounded, two thicknesses of 0.010-in. insulation between windings
are sufficient. Clearance of 0.253 in. allows room for in-and-out coil
taping.
Secondary leakage inductance, from equation 33, is
10.6 X 4,200^ X 10.2(4 X 0.020 + 0.747)
= 0.166 henry
4 X 2.375 X 10"
78
ELECTRONIC TRANSFORMERS AND CIRCUITS
At 60 cycles this is 6.28 X 60 X 0.166 = 63 ohms, which would be
240 ohms if the secondary were a single section, and which would
increase regulation as set forth in Chapter 4. The regulation calcu-
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lated in Fig. 58 is that due to primary IR calculated in the normal
manner, plus I^o times one-half the secondary winding resistance.
When high voltage is induced in a winding, the layer insulation and
coil size may often be reduced by using the scheme shown in Fig. 59.
This is applicable to a plate transformer of the single-phase full-wave
RECTIFIER TRANSFORMERS AND REACTORS
79
PRESSBOARD SPACER
f(IF USED)
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v//j///frrr//////^z.
type with center tap grounded. It then becomes practical to make the
secondary in two separately wound vertical halves or part coils. One
of the part coils is assembled with the turns
in the same direction as those of the primary,
and the other part coil is reversed so that the
turns are in the opposite direction. The two
start leads are connected together and to
ground as in Fig. 59. It is necessary then
to provide only sufficient insulation between
windings to withstand the primary test volt-
age. Channels may be used to insulate the
secondaries from the core. With higher volt-
ages, it may be necessary to provide pressboard spacers between the
secondary part coils, or to tape the secondary coils separately, bvrt
margins must be provided sufficient to prevent creepagc across the
edges of the spacers.
32. Combined Anode and Filament Transformers. Anode and fila-
ment windings are combined into a single transformer mainly in low-
power ratings such as those in receivers and grid bias power supplies.
^COIL FORM
Fio. 59. Anode transtormfir
with C.T. grounded.
2X2
I 15V.
50/60 CY.
Fig. 60. Power supply transformer.
One widely used combination includes the anode and filament windings
for a rectifier and a filament winding for the amplifier tubes. Figures
60 and 61 show how winding insulation sometimes may be graded to
require a minimum of insulation and space. The high-voltage filament
winding Si is placed over the coil form to take advantage of its thick
insulation. Layer insulation is sufficient between Si and S2, and be-
tween S2 and S3. Over and under the primary winding is 115-volt
ELECTRONIC TRANSFORMERS AND CIRCUITS
P
S3 ~^
S2
SI
6
= SI
82
S3 —
P
Fig. 61.
Winding arrangement to
save insulation.
insulation. Thus Fig. 61 is a high-voltage transformer with no high-
voltage insulation in it except what is incidental to the coil form.
Combined anode and filament transformers are difficult to test for
regulation or output voltage aside from operation in the rectifier cir-
cuit itself, because a-c loads do not
duplicate rectifier action. Most trans-
formers of this kind are used in recti-
fiers with capacitor-input filters or
with fixed loads in which regulation is
not important.
Ratings are easier to predict.
Anode secondary v-a rating is the
product of rms voltage and current,
but the corresponding portion of pri-
mary v-a rating depends on the recti-
fier and is found as mentioned in
Sections 24 and 25. To this is added
the sum of filament winding v-a ratings, and the primary current can
then be calculated from the total volt-amperes.
33. Power Supply Frequency. Foregoing examples were based on a
60-cycle supply. Twenty-five-cycle transformer losses are lower for a
given induction. It follows that induction can be increased somewhat
over the 60-cycle value, but saturation currents prevent a decided
increase. Larger size results, nearly 2:1 in volume. Otherwise 25-
cycle transformers are not appreciably different from 60-cycle trans-
formers.
Power supply frequencies of 400 and 800 cycles are used mainly in
aircraft and portable equipment to save weight and space. Silicon-
steel core materials 0.005 in. thick are principally used at these fre-
quencies to reduce eddy currents. Losses at 400 and 800 cycles for
three core materials are shown in Fig. 62. These losses can be the con-
trolling factors in determining transformer size, because a given mate-
rial saturates at nearly the same induction whether the frequency
is 60 cycles or 800 cycles, but the core loss is so high at 800 cycles that
the core material cannot be used near the saturation density. The
higher the induction the higher the core heating. For this reason,
class B insulation can be used in many 400- and 800-cycle designs to
reduce size still further. If advantage is taken of both the core mate-
rial and insulation, 800-cycle transformers can be reduced to 10 per
cent of the size of 60-cycle transformers of the same rating. Typical
RECTIFIER TRANSFORMERS AND REACTORS
<n 8
o
1
.
<0/
°7
//
SOLID LINES -800CY.
DOTTED LINES-400CY.
A.
^—/c
/?
f
y —
3
J^
/
9//
o~
d>~'
>^/
/
/
/
:^
9^
^^^
/
^
>^^^i^
/
=-;oO
t';^°
^
y
^
.0»*>
fO^
^^
^^
*-^'
4
2
I 2 3 4 5 6 7 8 9 10 II 12 13 14
INDUCTION- KILOGAUSS
Fig. 62. Silicon-steel core loss at 400 and 800 cycles.
combinations of grain-oriented core material and insulation are as
follows :
Strip
Class of Operating
Frequency
Thickness
B-Gauss
Insulation Temperature
60
0.014
15,000
A 95°C
400
0.005
12,500
B 140°C
800
0.005
8,500
B 140°C
In very small units, these flux densities may be used at lower tem-
peratures and with class A insulation because of regulation. The
special 4-mil steel developed for 400 cycles makes possible size reduc-
tion comparable to that for 800 cycles. The necessity for small di-
mensions, especially in aircraft apparatus, continually increases the
tendency to use materials at their fullest capabilities.
Many small 60-cycle transformers have core loss which is small com-
pared to winding or copper loss. This condition occurs because in-
ductance is limited by exciting current rather than by core loss. As
size or frequency increases, this limitation disappears, and core loss is
limited only by design considerations. Under such circumstances, the
ratio of core to copper loss for maximum rating in a given size may be
found as follows. Let
82 ELECTRONIC TRANSFORMERS AND CIRCUITS
We = core loss
Ws = copper loss
Ki, K2, etc. = constants
E = secondary voltage
/ = secondary current
For a transformer with a given core, winding, volt-ampere rating, and
frequency, We « KiE^. For a given winding, Wg = K2P. Also, for a
given size. We + Ws = ^3, a quantity determined by the permissible
temperature rise. Hence the transformer volt-ampere rating is approx-
imately
EI= l—_
K,Ko
= KiVWeiKs - We)
For a maximum, the rating may be differentiated with respect to We,
and the derivative equated to zero:
= Ks- 2We
whence
We = 7^3/2
so that Ws — Ks/2, or copper and core losses are equal for maximum
rating.
Although this equality is not critical, and is subject to many limi-
tations such as core shape, voltage rating, and method of cooling, it
does serve as a guidepost to the designer. If a transformer design is
such that a large disparity exists between core and copper losses, size
or temperature rise often may be reduced by a redesign in the direc-
tion of equal losses.
34. An 800-Cycle Transformer Design.
Primary 120 volts 800 cycles
Rectifier to deliver 0.2 amp at +450 volts using 5U4G in single-
phase full-wave circuit with 0.5-/tfd capacitor input filter.
Figures 51 and 52 tell whether the product mCRl will produce the
necessary d-c output without exceeding the rectifier tube peak inverse
voltage rating and peak current rating.
oiCRl = 6.28 X 800 X 0.5 X lO^** X (450/0.2) = 5.65
For Rs assume a peak current of 0.5 amp. Average anode character-
RECTIFIER TRANSFORMERS AND REACTORS 83
istics show 97 volts tube drop, or 97 -i- 0.5 = 194 ohms at peak current.
Rs/Rl = 194/2,250 = 0.086. Add 5 per cent for transformer windings;
estimated Rs/Rl = 13.6 per cent.
Check on Peak Current from Fig. 52.
m>CRL = 11.3
Ip = 5Ip = 5X 0.1 = 0.5 amp
the peak value assumed. Rms current in tube plates and secondary
windings is 2 X 0.1 = 0.2 amp. Output voltage, from Fig. 51, is 0.69
peak a-c voltage per side. Hence secondary rms voltage per side is
450 X 0.707 ~ 0.69 = 460 volts, and secondary volt-amperes = 2 X
460 X 0.2 = 184. The anode transformer must deliver 2 X 460 = 920
volts at 0.2 amp rms. Primary volt-amperes = 0.707 X 184 = 130.
Inverse peak voltage is the peak value of this voltage plus the d-c
output, because the tube filament is at d-c value, plus a small amount
of ripple, while one anode has a maximum of peak negative voltage,
during the non-conducting interval. Thus peak inverse voltage is
460 X 1-41 + 450 = 1,100 volts, which is within the tube rating.
Choice of core for this transformer is governed by size and cost con-
siderations. Assume that the core works at 8,500 gauss. The loss per
pound for 0.005-in. silicon steel and grain-oriented steel is 12.2 and 6.6,
respectively. (See Fig. 62.) But punchings have 80 per cent stacking
factor, whereas the type C core has 90 per cent. In this thickness
0.005-in. grain-oriented steel compares still better with ordinary silicon
steel than Fig. 62 would indicate and so will be used for the core.
Let two type C cores be used with the following dimensions:
Strip width
M in.
Window height
Vsin.
Build
% in.
Window width
IK in.
Total net core area
0.506 sq in.
Core weight
0.75 lb
Turns could be figured from equation 32, except that the induction is
in gauss. Since many core data are given in gauss, equation 32 is
changed for convenience to
3.4QE X 10**
Ni = (34)
fA^B
where dimensions are in inches and B is in gauss. Primary turns are
then
84 ELECTRONIC TEANSFORMERS AND CIRCUITS
3.49 X 120 X W
122
800 X 0.506 X 8,500
Final design figures are:
Primary 122 turns No. 26 glass-covered wire d-c resistance 1.8 ohms
Secondary 900 turns No. 29 glass-covered wire d-c resistance 38 ohms
Primary copper loss at 100° C = 3.35 watts
Secondary copper loss at 100° C = 2.04 watts
Core loss 6.6 X 0.75 = 4.95 watts
Total losses 10.34 watts
With an open-type mounting and mica insulation this transformer
has a temperature rise of 75 centigrade degrees.
35. Polyphase Transformers. In large power rectifiers three-phase
supplies are generally used. Accurate phase voltages must be main-
tained to avoid supply frequency ripple in the output. Delta-con-
nected primaries are shown in Table VII for the various rectifiers;
these are preferable to open-delta because phase balance is better, and
to Y-connections because of possibly high third harmonics. Open-
delta connections require only two single-phase transformers instead of
three, but a similar saving may be had by using a single core-type
three-phase unit which retains the phase-balance advantage. The
main drawback to a three-phase core is its special dimensions. Often,
to use standard parts, three single-phase units are employed in the
smaller power ratings. But if the power is hundreds or thousands of
kilowatts, the cores are built to order, and the weight saving in a three-
phase core is significant.
Two- and three-phase filament transformers are used with output
tubes for large broadcast stations to heat filaments uniformly and re-
duce hum in the r-f output.
36. Design Chart. In preceding sections, it has been stated that
special conditions require tailored designs. Windings for simple low-
voltage 60-cycle transformers may be chosen from the chart of Fig. 63.
This chart is based upon the following conditions:
(a) Two untapped concentric windings; primary wound first.
(b) Operating voltage in both windings less than 1,000 volts.
(c) Power supply frequency 60 cycles.
(d) Maximum temperature rise 40°C in 65°C ambient.
(e) Resistive loads.
RECTIFIER TRANSFORMERS AKD REACTORS 85
(/) Equal PR losses in primary and secondary,
(g) Solventless resin impregnated coils.
{h) Open-type assemblies like those of Fig. 15.
{i) Grain-oriented silicon-steel cores.
It was found that 40 °C rise in the four smallest sizes resulted in
excessive voltage regulation. For example, a small filament trans-
former would deliver correct filament voltage at room ambient tem-
perature of 25 °C, but at 105 °C this voltage dropped to less than the
published tube limit. Hence the winding regulation in the two smallest
transformers was limited to 15 per cent, and in the next two larger sizes
to 10 per cent. In still larger sizes, the 40°C temperature limit held the
regulation to less than 10 per cent.
In using the chart, ratings rarely fall exactly on the v-a values
assigned to each core. Hence a core is generally chosen with some-
what greater than required rating. Lower regulation and temperature
rise than maximum then result. Wire size in quadrant I also increases
in discrete sizes, and if the chart indication falls between two sizes the
smaller size should be used.
Instructions for Using Fig. 63.
1. Choose a core from Table VIII which has a v-a rating equal to
or greater than that required.
2. From rated primary and secondary voltages, find number of
turns for both windings in quadrant IV.
3. From rated primary and secondary currents, find wire size for
both windings in quadrant I.
4. Project turns across to quadrant III to obtain winding resistances.
Table VIII. Transformer Size, Rating, and Regulation
Maximum
%
Total
V-A
Regu-
Weight
Overall Dimen-
Core
Rating
lation
(lb)
sions
1 (inches)
1
5
15
0.38
mx
WiX\K
2
10
15
0.68
IJ^X
■iVsXlVi
3
25
10
1.2
2MX
2J^X2M
4
50
10
2.2
2JiX
3}^X23^
5
100
8
3.8
33^ X
3MX3
6
200
6
6.4
^VsX
4J€X3%
7
350
4
11.0
WsX
5^X4
8
500
3
15
5ys X
61^X5
9
1,000
2.2
24
5VsX
6MX6M
10
1,600
1.8
36
7MX
8MX7>^
11
3,200
1.2
75
9H X \2% X8
ELECTRONIC TRANSFORMERS AND CIRCUITS
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1-^
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^
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1 1
Fw. 63. Low-voltage 60-cycle transformer design chart.
RECTIFIER TRANSFORMERS AND REACTORS
87
y
V
-<
V
y
(O
^
<;
::^
-^
y
X
X*
/
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x
/
/
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y
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^
200
350
500
1000
1600
3200
200 300 600 1000
32
3 4 6 8 10 20 30 40 60 80 100 200 400 600 1000
VOLTS \ "
Fig. 63. {Continued)
88 ELECTRONIC TRANSFORMERS AND CIRCUITS
Departures from the assumed conditions preclude direct application
of Fig. 63, but the chart is still useful as a starting point in design.
For some common modifications, the following notes apply:
1. For each additional secondary winding reduce core maximum
rated volt-amperes by 10 per cent. Choose wire size from quadrant II.
2. For 50-cycle transformers, reduce core maximum rated v-a 10
per cent.
3. When permissible temperature rise is higher than 40°C, core maxi-
mum volt-amperes equal (v-a in table) X Vtemperature rise/40°C.
Example. A transformer is required for 115/390 volts, 60 cycles, to de-
liver 77 volt-amperes. This rating falls between the maxima for cores 4 and
5. Using core 5 at 115 volts, we read, from Fig. 63, for the primary, 440
turns of No. 22 wire and 3 ohms d-c resistance; for the secondary, 1,700 turns
of No. 27 wire and 40 ohms d-c resistance.
37. Reactors. Reactors are used in electronic power equipment to
smooth out ripple voltage in d-c supplies, so they carry direct current
in the coils. It is common practice to build such reactors with air gaps
in the core to prevent d-c saturation. The air gap, size of the core,
and number of turns depend upon three interrelated factors: induct-
ance desired; direct current in the winding; and a-c volts across the
winding.
The number of turns, the direct current, and the air gap determine
the d-c flux density, whereas the number of turns, the volts, and the
core size determine the a-c flux density. If the sum of these two flux
densities exceeds saturation value, noise, low inductance, and non-
linearity result. Therefore a reactor must be designed with knowledge
of all three of the conditions above.
Magnetic flux through the coil has two component lengths of path:
the air gap Ig, and the length of the core Ic. The core length Ic is much
greater geometrically than the air gap Ig, as indicated in Fig. 57, but
the two components do not add directly because their permeabilities
are different. In the air gap, the permeability is unity, whereas in the
core its value depends on the degree of saturation of the iron. The
effective length of the magnetic path is Ig + (Ic/f^), where /x is the
permeability for the steady or d-c component of flux.
Reactor design is, to a large extent, the proportioning of values of air
gap and magnetic path length divided by permeability. If the air
gap is relatively large, the reactor inductance is not much affected by
RECTIFIER TRANSFORMERS AND REACTORS 89
changes in /i ; it is then called a linear reactor. If the air gap is small,
changes in /j. due to current or voltage variations cause inductance to
vary; then the reactor is non-linear.
When direct current flows in an iron-core reactor, a fixed magnetizing
force Hdc is maintained in the core. This is shown in Fig. 64 as the
vertical line Hdc to the right of zero H in a, typical a-c hysteresis loop,
the upper half DB^D' of which corresponds to that in Fig. 21. Incre-
ment AH of a-c magnetization, su-
perposed on Hdc, causes flux den-
sity increment AB, with permeabil-
ity HA equal to the slope of dotted
line ABm- AB is twice the peak
a-c induction Bac- It will be re-
called from Fig. 19 that the nor-
mal induction curve OB^ is the lo-
cus of the end points of a series of
successively smaller major hystere-
sis loops. Since the top of the mi-
nor loop always follows the left side
of a major loop, as Hdc is reduced in
successive steps the upper ends of
corresponding minor loops termi-
nate on the normal induction curve.
Dotted-line slopes of a series of
minor loops are shown in Fig. 64,
the midpoints of which are C, C, C" , and C" . Increment of induction
AB is the same for each minor loop. It will be seen that the width of
the loop AH is smaller, and hence ma is greater, as Hdc is made smaller.
Midpoints C, C, etc., form the locus of d-c induction. The slope of
straight line OC is the d-c permeability for core magnetization Hdc- It
is much greater than the slope of ABm- Hence incremental permeability
is much smaller than d-c permeability. This is true in varying degree
for all the minor loops. The smaller AB is, the less the slope of a minor
loop becomes, and consequently the smaller the value of incremental
permeability /ja- The curve in Fig. 65 marked m is the normal per-
meability of 4% silicon steel for steady values of flux, in other words,
for the d-c flux in the core. It is 4 to 20 times as great as the incre-
mental permeability ma for a small alternating flux superposed upon the
d-c flux. The ratio of /x to ha gradually increases as d-c flux density
increases.
B
< AH >
1
B„
^x:^^^^
AB
A
y^^^'^^ /
T"
m/\
/
\jj 1
b'i^i/l 1
11
1
' / /
J \
j
„iit '''
%/l \
/
^^■IT
J \
I
^ 1
^ / 1
D' D Hdc Hm "
Fig. 64. Incremental permeability with
different amounts of d-c magnetization.
90
ELECTRONIC TRANSFORMERS AND CIRCUITS
Because of the low value of /xa for minute alternating voltages, the
effective length of magnetic path Zg + (Zc/ma) is considerably greater for
alternating than for steady flux. But the inductance varies inversely as
the length of a-c flux path. If, therefore, the incremental permeability
is small enough to make Zc/aia large compared to Ig, it follows that small
12,000
10,000
8,000 -t
6,000
2,000
\maX
10
V
__\
V
3
m
<
UJ
S
-IT -
UJ
Q.
1
a.
/
N
\ u
/
^
\
^
8 10 12
B-KILOGAUSS
14
18
Fig. 65. Normal and incremental permeability of 4% silicon steel.
variations in Ig do not affect the inductance much. For this reason the
exact value of the air gap is not important with small alternating
voltages.
Reactor size, with a given voltage and ratio of inductance to resist-
ance, is proportional to the stored energy LP. For the design of reac-
tors carrying direct current, that is, the selection of the right number
of turns, air gap, and so on, a simple method was originated by C. R.
Hanna.' By this method, magnetic data are reduced to curves such as
Fig. 66, plotted between LP/V and NI/lc from which reactors can be
designed directly. The various symbols in the coordinates are:
^ "Design of Reactances and Transformers Which Carry Direct Current," by
C. R. Hanna, J. AIEE, 46, 128 (February, 1927).
RECTIFIER TRANSFORMERS AND REACTORS
91
L = a-c inductance in lienrys
I = direct current in amperes
V = volume of iron core in cubic inches
= Aclc (see Fig. 57 for core dimen-
sions)
Ac = cross section of core in square
inches
Ic = length of core in inches
A^ = number of turns in winding
Ig = air gap in inches
(XIO"
*)
k
/
^
1
o
/
/
400
i
1
A.'
/
J
/
/=
1
v
o
o
/
300
?/
/
CO
/
^ 1
s7
u^
-iV
/ c
?
s
200
>
I
/
/
i
is
' 9
/
1
/
100
/
>
/f
/
/
/
/
^
j
k}.
/^
A
f¥-
NI
To
20 40 60 80 100 120
Fig. 66. Reactor energy per unit volume versus ampere-turns per inch of core.
Each curve of Fig. 66 is the envelope of a family of fixed air-gap
curves such as those shown in Fig. 67. These curves are plots of data
based upon a constant small a-c flux (10 gauss) in the core but a large
02
ELECTRONIC TRANSFORMERS AND CIRCUITS
and variable d-c flux. Each curve has a region of optimum usefulness,
beyond which saturation sets in and its place is taken by a succeeding
curve having a larger air gap. A curve tangent to the series of fixed
air-gap curves is plotted as in Fig. 66, and the regions of optimum use-
16
12
V
/
/
^;
-V
/
/
/ /
Y'
t /i
^
* X>^
Y
/
.^
<i3 //
7
/
/ .
^^
-^
k
fe
A-ZERC
B-0.00<
G-O.OOf
) AIR G/
^"AIR G/
3" AIR G/
\p _
XP
^-»*j
^
^
/
■"B
16
NI
OcH
Fig. 67. Fixed air-gap curves. For B^^. :s> B„^., aii' gap is not critical.
fulness are indicated by the scale Ig/lc- Hence Fig. 66 is determined
mainly by the d-c flux conditions in the core and represents the most
LP for a given amount of material.
Figure 67 illustrates how the exact value of air gap is of little con-
sequence in the final result. The dotted curve connecting B and C is
for a 6-mil gap. Point Y' represents the maximum inductance that
could be obtained from a given core for 'Nl/lo = 19. Point Y is the
inductance obtained if a gap of either 4 or 8 mils is used. The differ-
RECTIFIER TRANSFORMERS AKD REACTORS 93
ence in inductance between Y and Y' is 4 per cent, for a difference in
air gap of 33 per cent.
An example will show how easy it is to make a reactor according to
this method.
Example. Assume a stack of silicon-steel laminations having a cross section
Ji in. by % in., and with iron filling 92 pei- cent of the space. The length of
the flux path Ic in this core is 73/2 in. It is desired to know how many turns of
wire and what air gap are necessary to produce 70 henrys when 20 ma direct
current are flowing in the winding.
This problem is solved as follows :
Ac = (0.875)2 X 092 = o.71 sq in.
V = 0.71 X 7.5 = 5..3
LP 70 X 4 X I0-*
V 5.3
.53 X 10-*
In Fig. 66 the abscissa corresponding to LP/Y = 53 X 10^* is Nl/l^ = 25
for silicon steel. The ratio of air gap to core length Ig/lc is between 0.0005 and
0.001.
A^//?, = 25
A' = (25 X 7.5)/0.020 = 9,350 turns
The total air gap is nearly 0.001 X IVi or 7.5 mils; the gap at each joint is half
of this value, or 3.75 mils.
The conditions underlying Hanna's method of design are met in
most applications. In receivers and amplifiers working at low audio
levels, the alternating voltage is small and hence the alternating flux
is small compared to the steady flux. Even if the alternating voltage is
of the same order as the direct voltage, the alternating flux may be
small, especially if a large number of turns is necessary to produce the
required inductance; for a given core the alternating flux is inversely
proportional to the number of turns. D-c resistance of the coil is
usually fixed by the regulation or size requirements. Heating seldom
affects size.
38. Reactors with Large A-C Flux. With the increasing use of
higher voltages, it often happens that the a-c flux is no longer small
compared to the d-c flux. This occurs in high-impedance circuits where
the direct current has a low value and the alternating voltage has a
high value. The inductance increases by an amount depending on the
values of a-c and d-c fluxes. Typical increase of inductance is shown
94
ELECTRONIC TRANSFORMERS AND CIRCUITS
in Fig. 68 for a reactor working near the saturation point. Increasing
a-c flux soon adds to the saturation, which prevents further inductance
40
VXI04
—
^^
'
/
^
/
/
/
DAT
kTAK
EN A-
■t
= 10,
I-
3005
REF,
FIG.:
/
E
n^-G
AUSS
100 200 300 400 500 600
Fig. 68. Increase of inductance with a-c induction.
increase and accounts for the flattening off in Fig. 68. Saturation of
this sort may be avoided by limiting the value of the d-c flux.
To illustrate the effect of these latter conditions, suppose that a
reactor has already been designed for negligibly
small alternating flux and operates as shown by
the minor loop with center at G, Fig. 69. With-
out changing anything else, suppose that the
alternating voltage across the reactor is greatly
increased, so that the total a-c flux change is
from zero to Bm- (Assume that the reactor still
operates about point G.) The hysteresis loop,
however, becomes the unsymmetrical figure
OBmD'O. The average permeability during the
positive flux swing is represented by the line
GBm, and during the negative flux swing by OG.
The slope of GB^ is greater than that of the
minor loop; hence, the first effect exhibited by the reactor is an increase
of inductance.
The increase of inductance is non-linear, and this has a decided
Fig. 69. Change of per-
meability with a-c in-
duction.
RECTIFIER TRANSFORMERS AND REACTORS 95
effect upon the performance of the apparatus. An inductance bridge
measuring such a reactor at the higher a-c voltage would show an in-
ductance corresponding to the average slope of lines OG and GBm-
That is, the average permeability during a whole cycle is the average
of the permeabilities which obtain during the positive and negative
increments of induction, and it is represented by the average of the
slopes of lines OG and GB^. But if the reactor were put in the filter
of a rectifier, the measured ripple would be higher than a calculated
value based upon the bridge value of inductance. This occurs be-
cause the positive peaks of ripple have less impedance presented to
them than do the negative peaks, and hence they create a greater
ripple at the load. Suppose, for example, that the ripple output of the
rectifier is 500 volts and that this would be attenuated to 10 volts
across the load by a linear reactor having a value of inductance corre-
sponding to the average slope of lines OG and GBm. With the reactor
working between zero and i?„, suppose that the slope of OG is 5 times
that of GB,n. The expected average ripple attenuation of 50:1 be-
comes 16.7:1 for positive flux swings, and 83.3:1 for negative, and the
load ripple is
1 /500 500 \
1 = 18 volts
2 \16.7 83.3/
or an increase of nearly 2 : 1 over what would be anticipated from the
measured value of inductance.
This non-linearity could be reduced by increasing the air gap some-
what, thereby reducing Hac- Moreover, the average permeability in-
creases, and so does the inductance. It will be apparent that decreas-
ing Hge further means approaching in value the normal permeability.
This can be done only if the maximum flux density is kept low enough
to avoid saturation. Conversely, it follows that, if saturation is present
in a reactor, it is manifested by a decrease in inductance as the direct
current through the winding is increased from zero to full-load value.
In a reactor having high a-c permeability the equivalent length of
core Ic/ii is likely to be small compared to the air gap Ig. Hence, it is
vitally important to keep the air gap close to its proper value. This is,
of course, in marked contrast to reactors not subject to high a-c induc-
tion.
If a choke is to be checked to sec that no saturation effects are
{jresent, access must be had to an inductance bridge. With the proper
values of alternating voltage across the reactor, measurements of in-
ductance can be made with various values of direct current through it.
96 ELECTRONIC TRANSFORMERS AND CIRCUITS
If the inductance remains nearly constant up to normal direct current,
no saturation is present, and the reactor is suitable for the purpose.
If, on the other hand, the inductance drops considerably from zero
direct current to normal direct current, the reactor very probably is
non-linear. Increasing the air gap may improve it; otherwise, it
should be discarded in favor of a reactor which has been correctly de-
signed for the purpose.
Filter reactors subject to the most alternating voltage for a given
direct voltage are those used in choke-input filters of single-phase recti-
fiers. The inductance of this type of reactor influences the following:
Value of ripple in rectified output.
No-load to full-load regulation.
Transient voltage dip when load is suddenly applied, as in keyed
loads.
Peak current through tubes during each cycle.
Transient current through rectifier tubes when voltage is first applied
to rectifier.
It is important that the inductance be the right value. Several of
these effects can be improved by the use of swinging or tuned reactors.
In a swinging reactor, saturation is present at full load; therefore the
inductance is lower at full load than at no load. The higher inductance
at no load is available for the purpose of decreasing voltage regulation.
The same result is obtained by shunt-tuning the reactor, but here the
inductance should be constant from no load to full load to preserve
the tuned condition.
In swinging reactors, all or part of the core is purposely allowed to
saturate at the higher values of direct current to obtain high inductance
at low values of direct current. They are characterized by smaller
gaps, more turns, and larger size than reactors with constant induct-
ance ratings. Sometimes two parallel gaps are used, the smaller of
which saturates at full direct current. When the function of the reac-
tor is to control current by means of large inductance changes, no
air gap is used. Design of such reactors is discussed in Chapter 9.
The insulation of a reactor depends on the type of rectifier and how
it is used in the circuit. Three-phase rectifiers, with their low ripple
voltage, do not require the turn and layer insulation that single-phase
rectifiers do. If the reactor is placed in the ground side of the circuit
one terminal requires little or no insulation to ground, but the other
terminal may operate at a high voltage to ground. In single-phase
RECTIFIER TRANSFORMERS AND REACTORS 97
rectifiers tlie pealc voltage across the reactor is Eac, so the equivalent
rms voltage on the insulation is 0.707Eac- But for figuring 5max the
rms voltage is 0.707 X 0.67Eic- Reactor voltages are discussed in
Chapter 4.
39. Linear Reactor Design. A method of design for linear reactors
is based on three assumptions which are justified in the foregoing:
(a) The air gap is large compared to Ic/n, n being the d-c per-
meability.
(6) A-c flux density depends on alternating voltage and frequency,
(c) A-c and d-c fluxes can be added or subtracted arithmetically.
From (a) the relation B = ixH becomes B = H. Because of fringing
of flux around the gap, an average of 0.855 crosses over the gap. Hence
Bic = 0.4:TNIdc/0-85lg. With Ig in inches this becomes
Bdc = 0.6NTdc/lg gauss (35)
Transposing equation 34
Bac = (3.49E X 10^)/fAcN gauss (36)
The sum of Bac and Bjc is Bmax, which should not exceed 11,000 gauss
for 4% silicon steel, 16,000 gauss for grain-oriented steel, or 10,000
gauss for a 50% nickel alloy. Curves are obtainable from steel manu-
facturers which give incremental permeability ma for various combina-
tions of these two fluxes. Figure 70 shows values for 4% silicon steel.
By definition, inductance is the flux linkages per ampere or, in cgs
units,
</,A' _ R^^A.N
But
OAtNIm
Bac =
If this is substituted in equation 37
3A9N^Ac X 10-^
L = henrys (38)
k + (Ic/i^a)
provided that dimensions are in inches. The term Ac in equation 38 is
greater than in equation 36 because of the space factor of the lamina-
tions; if the gap is large Ac is greater still because the flux across it
98
ELECTRONIC TRANSFORMERS AND CIRCUITS
fringes. With large gaps, inductance is nearly independent of jua, at
least with moderate values of Bmax- With small gaps, permeability
9000
a.
3000
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
V
/
^
/
/
/
/
/
/
ft'
if
K/
o/
/
/
/
/
A
/
/
/
r
.->/
/
/
/
/
v
/
r
/
/
\4
^^
/
/
[/'
^4
w
.
V
/
'/
tf
y
//
/
f-y
o
V.
V
/
/,
/,
^A
5^
/
c
f
/
/.
'/
y
J
//
/.
^
y
/'
:^
<^
y
/A
^
/
//,
^
/
w
Y
—
3 4
Boc-KILOGAUSS
Fig. 70. Incremental permeability for 4% silicon steel with high a-c induction.
largely controls. There is always a certain amount of gap even with
punchings stacked alternately in groups of 1. Table IX gives the ap-
proximate gap equivalent of various degrees of interleaving laminations
for magnetic path Ic of 5.5 in.
RECTIFIER TRANSFORMERS AND REACTORS 99
Table IX. Equivalent Gaps with Interleaved Laminations
0.014-in. Laminations Equivalent, Air Cap in Ineh(!S
Alternately Stacked (Total) with Carcf-al Stacking
In groups of 1 0.0005
In groups of 4 . 001
In groups of 8 0.002
In groups of 12 0.003
In groups of 16 0.004
Butt stacking with zero gap . 005
Example. An input reactor is required for the filter of a 1,300-volt, 34-amp,
single-phase, full-wave, 60-cycle rectifier. Let N = 2,800 turns, net Ac = 2.48
sq in., gross Ac = 2.76 sq in., Ic = 9 in., Ig = 0.050 in. The 120-cycIe voltage
for figuring Bac is 0.707 X 0.67 X 1,300 = 605 volts.
_ 0.6 X 2,800 X 0.25 _
^'' 0^50 - ^'*°°
_ 3.49X605X10"
"^"^ ~ 120 X 2.48 X 2,800 " ^^^
Bmax = 10,940 gauss
Figure 70 shows
^x^ = 2,650
^ 3.19 X (2,800)2 X 2.76 X 10^
0.050 +
= 13.0 henrys
40. Linear Reactor Chart. In the preceding section, it was assumed
that the core air gap is large compared to Ic/ix, where m is the d-c per-
meability. In grain-oriented steel cores the air gap may be large com-
pared to Zc/ma, because of the high incremental permeabihty of these
cores. When this is true, variations in ;u do not affect the total effective
magnetic path length or the inductance to substantial degree. Reactor
properties may then be taken from Fig. 71. In order to keep the reac-
tor linear, it is necessary to limit the flux density. For grain-oriented
silicon-steel cores, inductance is usually linear within 10 per cent if the
d-c component of flux Bdc is limited to 12,000 gauss and the a-c com-
ponent Bac to 3,000 gauss.
Dotted lines in quadrant I are plots of turns vs. core area for a
given wire size and for low-voltage coils, where insulation and margins
are governed largely by mechanical considerations. Core numbers in
Fig. 71 have the same dimensions and weight as in Table VIII.
100
ELECTRONIC TRANSFORMERS AND CIRCUITS
N »0 lO CJ O
O O* O Q o' O
s§
o o o o
o o o o
o, q. o. o. ,
sNani=N
RECTIFIER TRANSFORMERS AND REACTORS 101
If the cores increased in each dimension by exactly the same amount,
the lines in quadrant I would be straight. In an actual line of cores,
several factors cause the lines to be wavy :
(a) Ratios of core window height to window width and core area
deviate from constancy.
(b) Coil margins increase stepwise.
(c) Insulation thickness increases stepwise.
A-c flux density in the core may be calculated by equation 36, and
Bac by equation 35. If £,„ materially exceeds 15,000 gauss, saturation
is reached, and the reactor may become non-linear or noisy.
Instructions for Using Fig. 71.
1. Estimate core to be used.
2. Divide required inductance by area (Ac) of estimated core to
obtain a value of L/sq in.
3. In second quadrant, locate intersection of L/sq in. and rated lac-
4. On this intersection, read total gap length (Ig) and number of
turns (N) . Gap per leg = lg/2.
5. Project intersection horizontally into first quadrant to intersect
vertical line which corresponds to estimated core. This second inter-
section gives d-c resistance and wire size.
Example. Required : 15 henrys at I^c = 50 ma.
Estimate core No. 1.
L/sq in. = 84.3, Ig = 0.015 in,, A' = 6,000, DCR = 800 ohms.
Wire size = No. 36.
(Example shown starting with dotted circle.)
A similar chart may be drawn for silicon-steel laminations, but to
maintain linearity lower values of flux density should be used.
41. Air-Gap Flux Fringing. In Section 39, equation 38 was de-
veloped for inductance of a linear reactor with an air gap. It is as-
sumed that 85 per cent of the core flux is confined to the cross section
of core face adjoining the gap. The remaining 15 per cent of the core
flux "fringes" or leaves the sides of the core, thus shunting the gap.
Fringing flux decreases the total reluctance of the magnetic path and
increases the inductance to a value greater than that calculated from
equation 38. Fringing flux is a larger percentage of the total for
larger gaps. Very large gaps are sometimes broken up into several
smaller ones to reduce fringing.
If it is again assumed that the air gap is large compared to lc/i>., the
102
ELECTRONIC TRANSFORMERS AND CIRCUITS
reluctance of the iron can be neglected in comparison with that of the
air gap. For a square stack of punchings, the increase of inductance
due to fringing is
= {1 + ^4= log.— (39)
Vii.
IJ
Equation 39 is plotted in Fig. 72 with core shape V^c/S as abscissas
and gap ratio Ig/B as parameter.^
L
3.0
2.5
2.0
1.5
COIL
u
--
CORE
->
« S
— ^
V^c
■v
X
v
N
\
V 1-
■eg/i
\
^
\
\
s?
3
■^
^
^
'\,
-v^
^
■~~^
^
^
-
0.1
0.2 0.3 0.4
0.6 0.8 1.0
1.5
Fig. 72. Increase of reactor inductance with flux fringing at core gap.
If the air gap is enclosed by a coil, as at the top of Fig. 72, flux
fringing is reduced because of the magnetizing force set up near the
gap by the ampere-turns of the coil. A coil fitting tightly all around
the core would produce no fringing at all. As the distance from inside
of the coil to the core increases, so does the fringing. Fringing there-
fore depends upon the coil form thickness; if it materially exceeds the
air gap per leg, fringing is nearly the same as it would be in a core
gap which is not enclosed by a coil. Figure 72 is based on a thick
coil form.
42. Similitude in Design. Charts such as Fig. 63 show that ratings
are related to size in an orderly sequence, provided that certain pro-
portions between core dimensions are maintained. Figure 63 is for 60
iSee G. F. Partridge, Fhil. Mag., 22 (7th series), 675 (July-December, 1936).
RECTIFIER TRANSFORMERS AND REACTORS 103
cycles. If a transformer is desired for another frequency, its size may
be estimated from Table VIII, provided that the same core propor-
tions apply, and similar values of induction and temperature rise are
used. If the new conditions are widely different, due allowance must
be made for them or the estimate will not be accurate.
Table VIII and Figs. 63 and 71 are examples of similitude. If all
variations between ratings are taken into account, similitude provides
a very accurate basis for estimating new sizes; for the transformer de-
signer there is no better basis for starting a new design.
43. Reactor Current Interruption. Sudden interruption of current
through a reactor may cause high voltages to develop in the winding.
This may be seen by considering the voltage across a reactor with
linear inductance L and varying current i in the winding. Let current
i be substituted for Im in equation 37 ; it may be transposed to give
</. = l(fU/N (37a)
where L is in henrys and i in amperes. If this expression for 4> be sub-
stituted in equation 1, we obtain
di
e = -L— (40)
dt
Equation 40 states that the magnitude of voltage across a reactor is
equal to the inductance multiplied by the rate of current change with
time. The sense or direction of this voltage is always such as to
oppose the current change. Therefore, if current interruption takes
place instantaneously, inductive voltage is infinitely large. In an
actual reactor, losses and capacitance are always present; hence inter-
ruption of reactor current forces the reactor voltage to discharge into
its own capacitance and loss resistance. The curves of Fig. 73 show
how the reactor voltage e rises when steady current I flowing in the
reactor is suddenly interrupted. The maximum value to which voltage
e could rise under any condition is IR2, where R2 is the equivalent loss
resistance. R2 depends mostly on the reactor iron loss at the resonance
frequency determined by reactor inductance L and capacitance C.
This frequency is l/T, where T is 2Tr-\JLC Conditions for high
voltage across the reactor occur with high values of fc, the ratio of
'\/L/C to 27^2- If subject to sudden current interruptions, reactors
must be insulated to withstand this voltage, or must be protected by
spark gaps or other means. The curves of Fig. 73 are based on equa-
tion 41:
104
ELECTRONIC TRANSFORMERS AND CIRCUITS
R2
__i
e = VOLTAGE ACROSS REACTOR
1= INITIAL CURRENT THROUGH
REACTOR
R2=REACT0R LOSS EQUIVALENT
RESISTANCE
C= REACTOR DISTRIBUTED CAPA-
CITANCE
L= REACTOR INDUCTANCE
T=2TrVLC
2 Re
where
0.2 0.4 0.6 0.8 1.0
TIME EXPRESSED AS A FRACTION OF TIME CONSTANT T
Fig. 73. Reactor voltage rise.
6 k
YRz
Vfc2
-27r
1
(f™2' _
m^t -:"*i^'\
(41)
nil, ni2
(k ± Vfc2 - 1 )
If there is appreciable circuit or wiring capacitance shunting the reactor
after it is disconnected, this contributes to the total reactor capaci-
tance C.
RECTIFIER TRANSFORMERS AND REACTORS 105
44. Transformers with D-C Flux. When there is a net d-c flux in
the core, as in single-phase half-wave anode transformers, the choice
of core depends on the same principles as in reactors with large a-c
flux. The windings carry non-sinusoidal load current, the form of
which depends on the circuit. Winding currents may be calculated
with the aid of Table I (p. 16). Generally the heating effects of these
currents are large. Maximum flux density should be limited as de-
scribed in Section 39. This precaution is essential in limited power
supplies like aircraft or portable generators, lest the generator voltage
wave form be badly distorted. On large power systems the rectifier
is a minor part of the total load and has no influence on voltage wave
form. The chief limitation then is primary winding current, and maxi-
mum induction may exceed the usual limits.
In single-phase half-wave transformers, air gaps are sometimes pro-
vided in the cores to reduce the core flux asymmetry described in Sec-
tion 12. Transformers designed in this manner resemble reactors in
that core induction is calculated as in Sections 37 to 41, depending on
the operating conditions. Even in transformers with no air gap, there
is a certain amount of incidental reluctance at the joints in both
stacked laminations and type C cores. This small gap reduces the
degree of core saturation that would exist in half-wave transformers
with unbroken magnetic paths.
45. Power Transformer Tests. A power transformer is tested to
discover whether the transformer will perform as required, or whether
it will give reliable service life. Some tests perform both functions.
(a) D-C Resistance. This test is usually made on transformers at
the factory as a check on the correctness of wire size in each winding.
Variations are caused by wire tolei'ances, and by difference in winding
tension between two lots of coils or lietween two coil machine opera-
tors. About 10 per cent variation can be expected in the d-c resist-
ance of most coils, but this value increases to 20 per cent rather sud-
denly in sizes smaller than No. 40. The test is made by means of a
resistance bridge or specially calibrated meter.
(b) Turns Ratio. Once the correct number of turns in each winding
is established, correct output voltage can be assured for a coil of given
design by measuring the turns. A simple way of doing this is by use
of the turns-ratio bridge in Fig. 74. If the turns are correct, the null
indicated by the meter occurs at a ratio of resistances
R1/R2 = N,/N2 (42)
106
ELECTRONIC TRANSFORMERS AND CIRCUITS
If there is an error in the number of turns of one winding, the null
occurs at the wrong value R1/R2. A source of 1,000 cycles is preferable
to one of 60 cycles for this test. The smaller current drawn by the
transformer reduces IR and IX errors. Harmonics in the source ob-
scure the null, and so the source should be filtered. The null is often
made sharper by switching a small variable resistor in series with i?i
or R2 to offset any lack of proportion in resistances of windings iVi
or N2.
< 1,000 CYCLE SOURCE >\
10,000 n , , , , 10,000 a
1,000 A
^ ^
lOOA [
T__ __F
1,000 A
10
Rl
.5 ^
] loon
lOA
Fig. 74. Turns-ratio bridge.
An accuracy of 0.1 per cent can usually be attained with four-decade
resistances. Polarity of winding is also checked by this test, because
the bridge will not balance if one winding is reversed.
(c) Open-Circuit Inductance (OCL). There are several ways of
measuring inductance. If the Q (or ratio of coil reactance to a-c resist-
ance) is high, the check may be made by measuring the current drawn
by an appropriate winding connected across a source of known voltage
and frequency. This method is limited to those cases where the amount
of current drawn can be measured. A more general method makes use
of an inductance bridge, of which one form is shown in Fig. 75.
If direct current normally flows in the winding, it can be applied
through a large choke as shown. Inductance is then measured under
the conditions of use. Source voltage should be adjustable for the
same reason and should be filtered to produce a sharp null. Re is
provided to compensate for coil a-c resistance. Without it an accurate
measurement is rarely attained. Enough data are provided by the
test to calculate a-c resistance as well as inductance.
RECTIFIER TRANSFORMERS AND REACTORS
107
When Q is low, as it is in coils with high resistance, better accu-
racy is obtained with the Maxwell bridge, which is like the Hay bridge
except that Xc and Be are paralleled. Then the equations for bridge
balance become
Lx = R1R2C Rx = RxRi/Rc
(43)
The Maxwell bridge has the further advantage that the null is inde-
pendent of the source frequency.
Fig. 75. Modified Hay bridge for measuring inductance.
(d) Temperature Rise. Tests to determine whether a transformer
overheats are made by measuring the winding resistances before and
after a heat run, during which the transformer is loaded up to its rating.
Where several secondaries are involved, each should deliver rated
voltage and current. Power is applied long enough to allow the trans-
former temperature to become stable ; this is indicated by thermometer
readings of core or case temperature taken every half hour until suc-
cessive readings are the same. Ambient temperature at a nearby loca-
tion should also be measured throughout the test. The average in-
crease in winding resistance furnishes an indication of the average
winding temperature. Figure 76 furnishes a convenient means for
finding this temperature.
(e) Regulation. It is possible to measure voltage regulation by
connecting a voltmeter across the output winding and reading the
voltage with load off and on. This method is not accurate because the
regulation is usually the difference between two relatively large quan-
tities. Better accuracy can be obtained by multiplying the rated
108
ELECTRONIC TRANSFORMERS AND CIRCUITS
winding currents by the measured winding resistances and using equa-
tion 13. If the winding reactance drop is small this equation works well
for resistive loads. To measure winding reactance drop, a short-
circuit test is used. With the secondary short-circuited, sufficient
voltage is applied to the primary to cause rated primary current to
-60 -40 -20 20 40 60 80 100 120 140 160 180 200
TEMPERATURE "C
Fig. 76. Copper resistance versus temperature in terms of resistance at 25°C.
flow. The quotient E/I is the vector sum of winding resistances and
reactances. Reactance is found from
X = V^- i?2
(44)
where R includes the resistance of both windings and the meter.
Sometimes it is more convenient to measure the leakage inductance
with secondary short-circuited on a bridge and multiply by 27r/.
(/) Output Voltage. Although the method described under (e) above
is accurate for two-winding transformers, it is not applicable to multi-
secondary transformers unless they are tested first with newly cali-
brated meters to see that all windings deliver proper voltage at full
load. Once this is established, values of winding resistance and react-
ance thereafter can be checked to control the voltage. The interde-
RECTIFIER TRANSFORMERS AND REACTORS 109
pendence of secondary voltages when there is a common primary wind-
ing makes such an initial test desirable. This is particularly true in
combined filament and plate transformers, for which the best test is
the actual rectifier circuit.
(g) Losses. Often it is possible to reduce the number of time-
consuming heat runs by measuring losses. The copper loss is readily
calculated by multiplying the measured values of winding resistance
(corrected for operating temperature) by the squares of the respective
rated currents. Core loss is measured with open secondary by means
of a low-reading wattmeter at rated voltage in the primary circuit. If
these losses correspond to the allowable temperature rise, the trans-
former is safely rated.
(h) Insulation. There is no test to which a transformer is subjected
which has such a shaky theoretical basis as the insulation test. Yet it
is the one test it must pass to be any good. Large quantities of trans-
formers can be built with little or no insulation trouble, but the empiri-
cal nature of standard test voltages does not assure insulation ade-
quacy. It has been found over a period of years that, if insulation
withstands the standard rule of twice normal voltage plus 1,000 volts
rms at 60 cycles for 1 minute, reasonable insulation life is usually
obtained. It is possible for a transformer to be extremely under-
insulated and still pass this test (see p. 44) ; conversely, there are con-
ditions under which the rule would be a handicap. Therefore it can
only be considered as a rough guide.
The manner of making insulation tests depends upon the trans-
former. Low-voltage windings categorically can be tested by short-
circuiting the terminals and applying the test voltage from each wind-
ing to core or case with other windings grounded. Filament trans-
formers with secondaries insulated for high voltage may be tested in
similar manner. But a high-voltage plate transformer with grounded
center tap requires unnecessary insulation if it is tested by this method.
Instead, a nominal voltage of, say, 1,500 volts is applied between the
whole winding and ground; after that the center tap is grounded and
a voltage is applied across the primary of such value as to test the end
terminals at twice normal plus 1,000 volts. Similar test values can be
calculated for windings operating at d-c voltages other than zero.
Such a test is called an induced voltage test. It is performed at higher
than normal frequency to avoid saturation. An advantage of induced
voltage testing is that it tests the layer insulation.
If insulation tests are repeated one or more times they may destroy
the insulation, because insulation breakdown values decrease with
110 ELECTRONIC TRANSFORMERS AND CIRCUITS
time. Successive applications of test voltage are usually made at
either decreased voltage or decreased time. In view of their dubious
value, repeated insulation tests are best omitted.
Corona tests are not open to this objection. A voltage 5 per cent
higher than normal is applied to the winding, and the leads are run
through blocking capacitors to the input of a sensitive radio re-
ceiver as in Fig. 38.^ RETMA standard noise values for this test are
based primarily on radio reception, but they do indicate whether
standard insulation practice is followed. See Table X.
Table X.
CoEONA Voltage
RMS Working Voltage
Corona Level
(kilovolts)
(microvolts)
Up to 8.6
1,000
8.61 to 15
2,500
Transformers which are subjected to voltage surges may be given
impulse tests to determine whether the insulation will withstand the
surges. Power line surges are the most difficult to insulate for. The
electric power industry has standardized on certain impulse voltage
magnitudes and wave shapes for this testing.^ The ratio of impulse
voltage magnitude to 60-cycle, 1 -minute insulation test voltage is
called the impulse ratio. This ratio is much greater for oil-insulated
transformers than for dry-type transformers, and is discussed further
in Chapter 4.
iSee RETMA Standard TR-102-B, "Power Transformers for Radio Trans-
mitters."
2 See ASA Standard C57 .22-1948, paragraph 22.116.
4. RECTIFIER PERFORMANCE
46. Ripple. Filters used with rectifiers allow the rectified direct
current to pass through to the load without appreciable loss, but ripple
in the rectified output is attenuated to the point where it is not objec-
tionable. Filtering sometimes must be carried out to a high degree.
From the microphone to the antenna of a high-power broadcast sta-
tion, there may be a power amplification of 2 X lO^''. The introduc-
tion of a ripple as great as 0.005 per cent of output voltage at the micro-
phone would produce a noise in the received wave loud enough to spoil
the transmitted program. A rectifier used at the low-power levels must
be unusually well filtered to prevent noticeable hum from being trans-
mitted.
Different types of rectifiers have differing output voltage waves,
which affect the filter design to a large extent. Certain assumptions,
generally permissible from the standpoint of the filter, will be made in
order to simplify the discussion. These assumptions are:
1. The alternating voltage to be rectified is a sine wave.
2. The rectifying device passes current in one direction but pre-
vents any current flow in the other direction.
3. Transformer and rectifier voltage drops are negligibly small.
4. Filter condenser and reactor losses are negligible.
47. Single-Phase Rectifiers. Single-phase half-wave rectified volt-
age across a resistive load R is shown in Fig. 77. It may be resolved by
Fourier analysis into the direct component whose value is 0.318-Bj,fc or
0.45£'oe, and a series of alternating components. The fundamental
alternating component has the same frequency as that of the supply.
Single-phase half-wave rectifiers are used only when the low average
value of load voltage and the presence of large variations in this voltage
are permissible. The chief advantage of this type of rectifier is its
simplicity. A method of overcoming both its disadvantages is illus-
trated in Fig. 78 where a capacitor C shunts the load. By using the
proper capacitor, it is often possible to increase the value of E^ to
111
112
ELECTRONIC TRANSFORMERS AND CIRCUITS
within a few per cent of the peak voltage Ej,,c- The principal disadvan-
tage of this method of filtering is the large current drawn by the capaci-
tor during the charging interval as shown in Fig. 49(6) (p. 63). This
current is limited only by transformer and rectifier regulation; yet it
must not be so large as to cause damage to the rectifier. The higher
the value of E^c with respect to Eac, the larger is the charging current
taken by C. (See Figs. 50 and 52, pp. 64 and 66.) Therefore, if a
smooth current wave is desired, some other method of filtering must
be used.
EpK.
Edc ° .318 Epk
T
Fig. 77. Half-wave rectifier voltage.
Fig. 78. Capacitor filter.
To obtain less voltage variation or ripple amplitude, after the limit-
ing capacitor size has been reached, an inductive reactor may be em-
ployed. It may be placed on either the rectifier or the load side of the
capacitor, depending on whether the load resistance R is high or low
respectively. See Figs. 79(a) and (b). In the former, the voltage
Eao has less than the average value QA5E„c, because the inductor de-
lays the build-up of current during the positive half-cycle of voltage,
and yet the inductor in this case should have a high value of reactance
Xi„ compared to the capacitive reactance Xr, in order to filter effec-
tively. When R is low, reactance Xr, should be high compared to R.
(a) (b)
Fig. 79. (a) Inductor-input filter; (6) capaiutor-input filter.
In Fig. 79(a) the ripple amplitude across R is —Xo/{Xj, — Xo) times
the amplitude generated by the rectifier, if 7? is high compared to Xo-
Also, in Fig. 79(6), the ripple amplitude across R is R/Xi, times the
ripple obtained with capacitor only. R here is small compared to X^.
Large values of inductance are rec}uired to cause continuous current
flow when the inductor is on the rectifier side of the capacitor in a half-
rectifie:r performance
113
wave rectifier circuit. Since current tends to flow only half the time,
the rectified output is reduced accordingly. This difficulty is elimi-
nated by the use of the full-wave rectifier of Fig. 80. The alternating
(a) (b)
Fig. 80. (o) Single-phase full-wave rectifier; (6) rectified fvill-wave voltage.
components of the output voltage have a fundamental frequency
double that of the supply, and the amplitudes of these components
are much less than for the half-wave rectifier. The higher ripple fre-
quency causes L and C to be doubly effective; the smaller amplitude
results in smaller percentage of ripple input to the filter. Current
flow is continuous and E^,. has double the value that it had in Fig. 77.
For these reasons, this type of rectifier is widely used.
A full- wave rectifier uses only one-half of the transformer winding
at a time; that is, E^c is only half the transformer secondary voltage.
A circuit which utilizes the whole of this volt-
age in producing E^^. is the single-phase bridge
rectifier shown in Fig. 81. The output voltage
relations are the same as those of Fig. 80(5).
Although this circuit requires more rectifying
tubes, it eliminates the need for a transformer
midtap.
48. Polyphase Rectifiers. The effect of rec-
tifying more than one phase is to superpose more voltages of the same
peak value but in different time relation to each other. Figures 82(a)
and (6) give a comparative picture of the rectified output voltage for
three-phase half-wave and full-wave rectifiers. Increasing the number
of phases increases the value of E^p, increases the frequency of the
alternating components, and decreases the amplitude of these compo-
nents. Ripple frequency is p times that of the unrectified alternating
voltage, p being 1, 2, 3, and 6 for the respective waves. Roughly speak-
ing, p may be taken to represent the number of phases, provided that
due allowance is made for the type of circuit, as in Fig. 83. Rectifiers
with p = 3 or 6 are derived from three-phase supply lines, and, by
special connections, rectifiers with p = 9, 12, or more are obtained.
Fig. 81. Bridge rectifier.
114 ELECTRONIC TRANSFORMERS AND CIRCUITS
The frequency of any ripple harmonic is mp, where m is the order of
the harmonic.
THREE-PHASE HALF WAVE THREE PHASE FULL WAVE OR DOUBLE Y
SIX PHASE
(0) (b)
Fig. 82. Polyphase rectifier output waves.
Ripple voltage for any of these rectifiers can be found by the Fourier
relation:
2 r''/2
^n = ^ I f{t) cos noit dt (45)
where A^ = amplitude of nth ripple harmonic
T — ripple fundamental period
t — time (with peak of rectified wave as / = 0)
CO = 2t/ Tp = 2ir X supply line frequency
/(<) = ripple as a function of time
= Epk cos cct, T/2 > cot> - T/2.
The voltage peak is chosen as i = to obtain a symmetrical func-
tion fit) and eliminate a second set of harmonic terms like those in
equation 45, but with sin nwi under the integral.
Ripple amplitude is given in Fig. 83 for the ripple fundamental, and
second and third harmonics with reactor-input filters. In this curve,
the ratio Pa of ripple amplitude to direct output voltage is plotted
against the number of phases p. It should be noted that Pa diminishes
by a considerable amount for the second and third harmonics. In
general, if a filter effectively reduces the percentage of fundamental
ripple across the load, the harmonics may be considered negligibly
small.
49. Multistage Filters. In the inductor-input filter shown in Fig.
79(a), the rectifier is a source of non-sinusoidal alternating voltage
connected across the filter. It is possible to replace the usual circuit
representation by Fig. 84(a). For any harmonic, say the nth, the
voltage across the whole circuit is the harmonic amplitude A„, and
the voltage across the load is PbE,ic, Pr being ripple allowable across
the load, expressed as a fraction of the average voltage. Since the load
RECTIFIER PERFORMANCE
115
5.0
LO
.05
.001
\
1 PH HALF WAVE, p= i
1 PH FULL WAVE, p= 2
S PH HALF WAVE, p=3
2 PH FULL WAVE, p-- 4
5 PH FULL WAVE, P' 6
\
\
6 PH HALF WAVE, p= 6
12 PH HALF WAVE, p=l2
\
\
\
\
\
V
K
RIPPLE
FUNDAMENTAL
\
\
RIPPLE -
SECOND
HARMONIC
"<^
RIPPLE \
THIRD —^X
HARMONIC
\
^
p=NUMBE
R OF PHASES
^
12 3 4 6 9
Fig. 83. Rectifier ripple voltage.
An
4:Xc R
PR
2Xc
y^-'f
R p
riy_rj..
4:Xc-
3Xo
Xl
R Pr
(0) (b) (c)
Fig. 84. Inductor-input filter circuits.
116 ELECTRONIC TRANSFORMERS AND CIRCUITS
resistance R is high compared to Xc, the two voltages are nearly in
phase, and they bear the same ratio to each other as their respective
reactances, or
Pa Xl — Xc Xl , ^
— = = 1 (46)
Pr Xc Xc
From the type of rectifier to be used, and the permissible amount of
ripple in the load voltage, it is possible to determine the ratio of induc-
tive to capacitive reactance.
When the magnitude Pjt must be kept very small, the single-stage
filter of Fig. 84(a) may require the inductor and the capacitor to be
abnormally large. It is preferable under this condition to split both
the inductor and the capacitor into two separate equal units, and
connect them like the two-stage filter of Fig. 84(6). A much smaller
total amount of inductance and of capacitance will then be necessary.
For this filter
Pr \ X'c /
X'l and X'c being the reactances of each inductor and capacitor in the
circuit. Likewise, the three-stage filter of Fig. 84(c) may be more prac-
ticable for still smaller values of Pr. In the latter filter,
P„ V X'c I
and, in general, for an w-stage filter,
Pr \ X'c )
(49)
It is advantageous to use more than one stage only if the ratio
Pa/Pr is high. That the gain from multistage filters is realized only
for certain values of Pa/Pr is shown by Fig. 85. The lower curve
shows the relation between Pa/Pr and X^/Xc for a single-stage filter.
The second curve shows the increase in Pa/Pr gained by splitting up
the same amount of L and C into a two-stage filter; as indicated in
Fig. 84(6), the inductor and capacitor both have one-half their
"lumped" value. The upper curve indicates the same increase for a
three-stage filter, each inductor and capacitor of which have one-third
RECTIFIER PERFORMANCE
117
of their "lumped" or single-stage filter value. The attenuation in
multistaging is enormous for high XjJXc. For lower ratios there
100,000
50,000
10,000
1000 "J
500
100
NOTE ^ APPLIES DIRECTLY FOR SINGLE STAGE FILTER
DIVIDE BY 4 FOR -*= PER STAGE FOR TWO -STAGE
I X <= . , : ^
DIVIDE BY 9 FOR -ri PER STAGE FOR THREE-STAGE
50 100 150 200 250 300 350
Xc
INDUCTIVE REACTANCE
(LUMPED VALUES)
CAPACITIVE REACTANCE
Fig. 85. Attenuation in one-, two-, and three-stage filters.
may be a loss instead of a gain, as shown by the intersection of the
two upper curves. These curves intersect the lower curve if all are
prolonged further to the left. This may be a puzzling condition; but
consider that, for Xj^/Xg = 50 in the single-stage filter, the ratio is
118 ELECTRONIC TRANSFORMERS AND CIRCUITS
y^XjJSXc or 5% in the three-stage filter; the rather small advantage
in the latter is not difficult to account for.
Other factors may influence the number of filter stages. In some
applications, modulation or keying may require that a definite size of
filter capacitor be used across the load. Usually these conditions re-
sult in a single-stage filter, where otherwise more stages might be most
economical.
Table VII (p. 62) shows filter reactors in the negative lead, which
may be either at ground or high potential. If low ripple is required
in the filtered output, it is usually preferable to locate the filter reac-
tors in the high-voltage lead. Otherwise, there is a ripple current
path through the anode transformer winding capacitance to ground
which bypasses the filter reactor. Ripple then has a residual value
which cannot be reduced by additional filtering. In the three-phase,
zigzag, full-wave circuit, with center tap used for half-voltage output,
separate reactors should be used in the positive leads; placing a com-
mon reactor in the negative lead introduces high amplitude ripple in
the high-voltage output.
In rectifiers with low ripple requirements, both filament and anode
windings should be accurately center-tapped to avoid low-frequency
ripple, which is difficult to filter. Three-phase leg voltages should be
balanced for the same reason.
50. Capacitor-Input Filters. One of the assumptions implied at the
beginning of this chapter, namely, that transformer and rectifier volt-
age drops are negligibly small, cannot usually be made when capacitor-
input filters are used, because of the large peak currents drawn by the
capacitor during the charging interval. Such charging currents drawn
through finite resistances affect both the d-c output voltage and the
ripple in a complicated manner, and simple analysis such as that given
for inductor-input filters is no longer possible. Figure 86 is a plot of
the ripple in the load of capacitor-input filters with various ratios of
source to load resistance, and for three types of single-phase rectifiers.
These curves are useful also when resistance is used in place of an in-
ductor at the input of a filter. « is 27r times the a-c supply frequency,
C is the capacitance, Rl is the load resistance, and Rg the source re-
sistance.
When L-C filter stages follow a capacitor-input filter, the ripple of
the latter is reduced as in Fig. 85, except that the value of P^ must be
taken from Fig. 86. When an R-C filter stage follows any type of filter,
RECTIFIER PERFORMANCE
110
UJ ^ 11// till
/
CIRCUIT
HALF- WAVE
VOLTAGE
DOUBLER
FULL WAVE
y//
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y^ / // .' ,
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SnOA 31ddlll %
o
120
ELECTRONIC TRANSFORMERS AND CIRCUITS
the ripple is reduced in the ratio R/Xc represented by the R-C stage.
51. Rectifier Regulation. The regulation of a rectifier comprises
three distinct components:
1. The d-c resistance or IR drop.
2. The commutation reactance or IX drop.
3. The capacitor charging effect.
The first component can be reduced to a small value by the use of
tubes, transformers, and inductors having low resistance. Mercury-
„^3MVCM:\::e
AXIS OF Igc
SHORT CIRCUIT CURRENT Igc
UNRECTIFIED A-C VOLTAGE
RECTIFIED VOLTAGE
Fig. 87. Commutation current effect on rectifier voltage.
vapor tubes are of noteworthy usefulness in this respect, as the internal
voltage drop is low and almost independent of load current variations.
Commutation reactance can be kept to a low value by proper trans-
former design, particularly where the ratio of short-circuit current to
normal load is high.
During part of each cycle, both tubes of a single-phase full-wave
rectifier are conducting. During this interval one tube loses its cur-
rent and the other one builds up to normal current. Because of the
inevitable reactance in the transformer, this change does not take
place immediately but during an angle 6 as in Fig. 87. Short-circuit
current is initiated which would rise as shown by the dotted lines of
Fig. 87, if it could pass through the rectifier tubes; it prevents the
rectified voltage wave from retaining its normal shape, so that for a
portion of each cycle the rectified output is zero.
Let the transformer winding resistance be temporarily neglected; if
the current could rise to maximum, the short-circuit value would be
RECTIFIER PERFORMANCE 121
2Ep]c/X, where X is the leakage reactance of the whole secondary,
but it is limited by the rectifier to I,u-. The short-circuit current
rises to (1 — cos 6) times maximum in the commutation period, or
[2Ej,k{l - cos d)yX = /rf.
The average voltage from zero to the re-ignition point V is
(iJp,A)(l -cose)
Combining these relations gives, for the average voltage cut out of
the rectified voltage wave by commutation,
F,. = hcX/2ir (50)
By similar reasoning, the commutation reactance drop for polyphase
rectifiers is
VhcX'/2ir (51)
where X' = the transformer leakage reactance from line to neutral on
the secondary side, and p = the number of phases in Fig. 83.
In this formula, the leakage reactance per winding is associated with
the voltage across that winding. This is accurate when each phase is
supplied by a separate transformer. But it fails for p = 2 in the single-
phase full-wave rectifier, using one plate transformer, where half of the
secondary voltage is rectified each half-cycle. In such a rectifier, dur-
ing commutation the whole secondary voltage is effective, and so is the
leakage reactance of the whole secondary. This reactance has 4 times
the leakage reactance of each secondary half-winding, but only twice
the half-winding voltage acts across it. Hence ec^uation 50 must be
used for the single-phase rectifier ; here A' = the reactance of the entire
secondary.
When high winding resistance limits short-circuit current, commuta-
tion has less effect than equation 50 would indicate. This condition
prevails in small rectifiers; the IX drop is negligibly small because of
the small transformer dimensions. For example, in the plate trans-
former designed in Fig. 58 the leakage inductance is 0.166 henry. The
commutation reactance drop is, from equation 50,
0.115 X 0.166 X 2ir X 60/27r = 1.15 volts
or 0.1 per cent. This is negligible compared to the 3.7 per cent regula-
tion calculated in Fig. 58. In this case the short-circuit current would
be limited by winding resistance rather than by leakage inductance.
122
ELECTRONIC TRANSFORMERS AND CIRCUITS
In large rectifiers, all rectifier components have low losses to pre-
vent power wastage or overheating, and the IR drop is a very small
percentage of the total. At the same time, a large transformer re-
quires careful design in order to keep the IX drop reasonably small.
Therefore, in large rectifiers the IX drop is the dominant cause of regu-
lation. An example with 60 kva rating has 0.7 per cent IR drop and
6 per cent IX drop.
In medium-size rectifiers the IR and IX drops may be of equal, or at
least comparable, value. In such rectifiers these two components of
5 13 2 1 .5 ••- ^
I t^ 1 it s^
M ill I.:. :m ^
tl , / 1 1--- ^/ ^^
tl ! t. ^fZLl^^X-
\i" 1 Xi 4^^^^
u I'J'i r J 4m^
■^ ; -'-f- j -V^]^^
X'i T'tl \fu
y j"rv\--w^
3 2 1 0.1 0.2 0.3 0.4
TOTAL REGULATION IjcR
Fig. 88. Increase in rectifier regulation due to transformer reactance.
regulation do not add arithmetically. Commutation interval (9, Fig. 87,
depends on the short-circuited reactance when resistance is negligible,
but if resistance is appreciable G is related to the ratio X/R exponenti-
ally.^ The increase in regulation caused by commutation reactance
may be found from Fig. 88, in terms of d-c output voltage E^c- In
this figure the regulation of three widely used rectifiers (single-phase
full-wave, three-phase half-wave, and three-phase full-wave) is given
in a manner which enables one to proceed directly from the IR com-
ponent of regulation to total regulation.
X and R are ohms per phase except X/R ratio is for the whole sec-
ondary in single-phase full-wave rectifiers. R in X/R ratio includes
primary J? in all cases. R in IdoR/Edc is for two windings in three-
phase full-wave rectifiers. To obtain total regulation, project
lacR/Eae vertically to one-phase or three-phase line. Project this
point to the left to proper X/R line. Abscissa at left gives total regu-
1 See Mercury-Arc Rectifiers and Their Circuits, by D. C. Prince and P. B.
Vogdes, McGraw-Hill Book Co., New York, 1927, p. 216.
RECTIFIER PERFORMANCE 123
lation. An example is indicated by the dotted line. In this example,
the rectifier is three-phase full-wave.
Edc = 2,000 volts X
Idc = 1 amp R
R = m ohms IdcR 60
X = 120 ohms Edc 2,000
3 per cent
Total regulation = 1.68 X 3 = 5.04 per cent. If the IX regulation had
been added directly to IR it would be 6 per cent + 3 per cent = 9 per
cent, and the calculated regulation would be nearly 4 per cent higher
than actual.
52. Capacitor Effect. If the rectifier had no filter capacitor, the
rectifier would deliver the average value of the rectified voltage wave,
less regulation components 1 and 2 of Section 51. But with a filter
capacitor, there is a tendency at light loads for the capacitor to charge
up to the peak value of the rectified wave. At zero load, this amounts
to 1.57 times the average value, or a possible regulation of 57 per cent
in addition to the IR and IX components, for single-phase full-wave
rectifiers. This effect is smaller in magnitude for polyphase rectifiers,
although it is present in all rectifiers to some extent.
Suppose that the rectifier circuit shown in Fig. 80(a) delivers single-
phase full-wave rectifier output as shown in Fig. 80(6) to an inductor-
input filter and thence to a variable load. In such a circuit, the filter
inductor keeps the capacitor from charging to a value greater than the
average E^c of the rectified voltage wave at heavy loads. At light loads
the d-c output voltage rises above the average of the rectified wave,
as shown by the typical regulation curve of Fig. 89.
Starting at zero load, the d-c output voltage Eq is 1.57 times the
average of the rectified wave. As the load increases, the output
voltage falls rapidly to Ei as the current h is reached. For any load
greater than /i, the regulation is composed only of the two com-
ponents IR and IX. It is good practice to use a bleeder load /i so
that the rectifier operates between 7i and I2.
Filter elements Xl and Xo determine the load h below which
voltage rises rapidly. The filter, if it is effective, attenuates the a-c
ripple voltage so that across the load there exists a d-c voltage with a
small ripple voltage superposed. A choke-input filter attenuates the
harmonic voltages much more than the fundamental, and, since the
harmonics are smaller to begin with, the main function of the filter is
124
ELECTRONIC TRANSFORMERS AND CIRCUITS
to take out the fundamental ripple voltage. This has a peak value,
according to Fig. 83, of 66.7 per cent of the average rectified d-c voltage
for a single-phase full-wave rectifier. Since this ripple is purely a-c it
encounters a-c impedances in its circuit. If we designate the choke
impedance as Xl, and the capacitor impedance as Xc, both at the
fundamental ripple frequency, the impedance to the fundamental
component is Xl — Xc, the load resistance being negligibly high com-
pared to Xc in an effective filter. The d-c voltage, on the other hand,
produces a current limited mainly by the load resistance, provided
that the choke IR drop is small.
Fig. 89. Rectifier regulation curve.
A-C and d-c components are shown in Fig. 90, with the ripple cur-
rent lao superposed on the load direct current I^c- If the direct cur-
rent is made smaller by increased load resistance, the a-c component
is not affected because load resistance has practically no influence in
determining its value. Hence a point will be reached, as the d-c load
current is diminished, where the peak value of ripple current just
equals the load direct current. Such a condition is given by d-c load
7i which is equal to /„„• If the d-c load is reduced further, say to the
value Ix, no current flows from the rectifier in the interval A-B of each
ripple cycle. The ripple current is not a sine wave, but is cut off on
the lower halves, as in the heavy line of Fig. 91. Now the average
value of this current is not I^: but a somewhat higher current Ip. That
is, the load direct current is higher than the average value of the rec-
tified sine wave voltage divided by the load resistance. This increased
current is caused by the tendency of the capacitor to charge up to the
peak of the voltage wave between such intervals as A-B; hence the
RECTIFIER PERFORMANCE
125
term capacitor effect which is appUed to the voltage increase. The
limiting value of voltage is the peak value of the rectified voltage,
which is 1.57 times the sine-wave average, at zero load current.
Fig. 90. A-c and d-c com-
ponents of filter current.
Fig. 91. Capacitor effect at
light load.
To prevent capacitor effect the choke must be large enough so that
lac is equal to or less than the bleeder current /]. This consideration
leads directly to the value of choke inductance. The bleeder current
Ix is Ei/Rj, where i?i is the value of bleeder resistance. The ripple
current is the fundamental ripple voltage divided by the ripple circuit
impedance, or
0.667£'i
■* ac
Xr
Xc
Equating /i and lac we have, for a single-phase full-wave rectifier,
R, """-^^
0.667
(52)
Here we see that the value of capacitance also has an effect, but it is
minor relative to that of the choke. In a well-designed filter, the choke
reactance X^ is high compared to Xc- Therefore, the predominant
element in fixing the value of Ri (and of 7i) is the filter reactor.
Polyphase rectifiers have similar effects, but the rise in voltage is
not so great because of the smaller difference between peak and aver-
age d-c output. The bleeder resistance for eliminating capacitor effect
can be found in general from
Ki =
X/Jti— Xc
Pi
(53)
where Pi is the fundamental ripple peak amplitude from Fig. 83, and
Xl and Xc are the filter reactances at fundamental ripple frequency.
Between load Zi and zero load, the rate of voltage rise depends upon
the filter. Figure 92 shows the voltage rise as a function of the ratio
126
ELECTRONIC TRANSFORMERS AND CIRCUITS
(Xi — Xc) /Rl for a single-phase full-wave rectifier. A curve of ripple
in terms of ripple at full load is given. Figure 92 is a plot of experi-
mental data taken on a rectifier with IR -\- IX regulation of 5 per cent.
Reactances Xl and Xc are computed for the fundamental ripple fre-
quency.
Capacitor-input filters have the voltage regulation curves shown in
Figs. 50, 51, and 53) (pp. 64, 65, and 68) for their respective circuits.
At light loads these filters may give reasonably good regulation, but it
60
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75 3
_J
50 =>
u
Li.
25 o
*~ 0.2 0.4 0.6 0.8
Xl~Xc
Fig. 92. Voltage rise in single-phase full-wave rectifier at light loads.
is possible to get very poor regulation at heavier loads, as can be seen
from the curves. Rectifier series resistance plays an important part
in the voltage regulation of this type of filter. The effect of anode
transformer leakage inductance can be found from Fig. 92.
53. Tuned Rectifier Filters. Sometimes an inductor-input filter is
tuned as in Fig. 93. The addition of capacitor C\ increases the effec-
tive reactance of the inductor to the fundamental ripple frequency.
Both regulation and ripple of this type of filter are improved. The
filter is not tuned for the ripple harmonics, so the use of high-Q filter
inductors is unnecessary. An increase in effectiveness of the filter in-
ductor of about 3 to 1 can be realized in a single-phase full-wave
rectifier circuit. Tuned filters are less effective with three-phase recti-
fiers because slight phase unbalance introduces low-frequency ripple
which the filter does not attenuate.
Filters may be tuned as in Fig. 94, where the filter capacitor C\ is
connected to a tap near the right end of inductor L, and the other
filter capacitor Ci is chosen to give series resonance and hence zero
RECTIFIER PERFORMANCE
127
reactance across the load at the fundamental ripple frequency. Be-
cause of choke losses, the impedance across Rr, is not zero, but the
resulting ripple across load resistor Rl can be made lower than without
if
Ci ^c
Fig. 93. Shunt-tuned filter.
;ci ^c
Fig. 94. Series-tuned filter.
the use of capacitor Ci. Ripple is attenuated more than in the usual
inductor-input filter, but regulation is not substantially different.
54. Rectifier Currents. If the inductor in an inductor-input filter
were infinitely large, the current through it would remain constant.
If the commutation reactance effect is not considered, the current
through each tube of a single-phase rectifier would be a square wave,
as shown by 7i and I2 of Fig. 49(a) (p. 63). The peak value of this
current wave is the same as the d-c output of the rectifier, and the rms
value is 0.707/^0- With finite values of inductance, an appreciable
amount of ripple current flows through the inductor and effectively
modulates /i and h, thus producing a larger rms inductor current like
the first wave of Table I (p. 16) .
Capacitor-input filters draw current from the rectifier only during
certain portions of the cycle, as shown in Fig. 49(5). For a given aver-
age direct current, the peak and rms values of these current weaves are
much higher than for inductor-input filters. Values for the single-
phase rectifiers are given in Fig. 52 (p. 66). If an L-C filter stage fol-
lows the input capacitor, the inductor rms current is the output direct
current plus the ripple current in quadrature.
Polyphase rectifiers arc ordinarily of the choke-input type, because
they are used mostly for larger power, and therefore any appreciable
amount of series resistance cannot be tolerated. For this reason, the
low IR drop tubes, such as mercury-vapor rectifiers, are commonly
used. Such tubes do not possess sufficient internal drop to restrict
the peak currents drawn by capacitor-input filters to the proper values.
In a shunt-tuned power supply filter such as shown in Fig. 93, the
current drawn from the rectifier is likely to be peaked because two
capacitors Ci and C^ are in series, without intervening resistance or
inductance. This peak quickly subsides because of the influence of
inductor L, but an oscillation may take place on top of the tube cur-
128
ELECTRONIC TRANSFORMERS AND CIRCUITS
rent wave as shown in Fig. 95. The rectifier tube must be rated to
withstand this peak current. At the end of commutation the voltage
Fig. 95. Anode current with shunt-tuned filter.
jumps suddenly from zero to V (Fig. 87). Peak rectifier current may
be as much as
Ipk = V/coL, (54)
Lg is half the transformer leakage inductance, and w = 2ir X fre-
quency of oscillation determined by Ls in series with Ci and C2. This
peak current is superposed on I^c- It flows through the anode trans-
former and tube, but the current in choke L (Fig. 93) is determined
by ripple voltage amplitude and choke reactance. Series resistance
Rg reduces this peak current to the value
Ipk —
V
wR,
(55)
It is obtained by applying a step function voltage to the series RsLgC
circuit. The criterion for oscillations is
R, < 2
C
(56)
where C is the capacitance of Ci and C2 in series. Many rectifier tubes
have peak current ratings which must not be exceeded by such cur-
rents.
Currents shown in Table VII (p. 62) and Figs. 49 and 95 are
reflected back into the a-c power supply line, except that alternate
current waves are of reverse polarity. Small rectifiers have little
effect on the power system, but large rectifiers may produce excessive
interference in nearby telephone lines because of the large harmonic
currents inherent in rectifier loads. High values of commutation re-
actance reduce these line current harmonics, but, since good regula-
tion requires low commutation reactance, there is a limit to the con-
trol possible by this means. A-c line filters are used to attenuate the
RECTIFIER PERFORMANCE
129
line current harmonics. A large rectifier, with three-phase series
resonant circuits designed to eliminate the eleventh, thirteenth,
seventeenth, and nineteenth harmonics of a 60-cycle system, is shown
in Fig. 96. Smaller rectifiers sometimes have filter sections such as
A-C SUPPLY
A
A
L(-V>i^| j| K'YV^I il J^nO-JI il J^'V^I S
A
A
RECTIFIER
1140 CY
1020 CY
780 CY
660 CY
RESONANT FILTERS
Fig. 96. A-o lino filter for large power reetifier.
those in Fig. 97; these are rarely used in large installations because of
the excessive voltage regulation introduced by the line inductors.
Filters designed to keep r-f currents out of the a-c lines are often
SHUNT
CAPACITORS
Fig. 97. A-c line filter for medium-sized power rectifier.
used with high-voltage rectifiers. Even if the anode transformer has
low radio influence, commutation may cause r-f currents to flow in the
supply lines unless there is a filter.
55. Rectifier Transients. The shunt-tuned filter currents mentioned
in the preceding section are transient. Since the tube current is cut off
during each cycle, a transient current may occur in each cycle. When
power is first applied to the rectifier, another transient occurs, which
may be smaller or larger than the cyclic transient, depending on the
130 ELECTRONIC TRANSFORMERS AND CIRCUITS
filter elements. In reactor-input filters the transient current can be
approximated by the formula given in Section 54 for a step function
applied to the series circuit comprising filter L and C plus Rg- This
circuit is valid because the shunting effect of the load is slight in a
well-proportioned filter. In capacitor-input filters, the same method
can be used, but here the inductance is the leakage inductance of the
anode transformer. Therefore, equation 55 applies, except that the
maximum step function voltage is Ej,^:.
Transients which occur when power is first applied differ from cyclic
transients in that they are spasmodic. Power may be applied at any
instant of the alternating voltage cycle, and the suddenly impressed
rectifier voltage ranges from zero to Ej,^,. Starting transients are dif-
ficult to observe on an oscilloscope because of their random character.
It is necessary to start the rectifier several times for one observation
of maximum amplitude, and the trace is faint because it appears for a
very brief time.
Excessive current inrush, which occurs when a power transformer
is connected to a supply line, plagues rectifier design. The phenom-
enon is associated with core saturation. For example, suppose that
the core induction is at the top of the hysteresis loop in Fig. 18 (p. 24)
at the instant when power is removed from the rectifier, and that it
decreases to residual value B,- for H = 0. Suppose that the next
application of power is at such a point in the voltage cycle that the
normal induction would be Bm- This added to Br requires a total in-
duction far above saturation value; therefore heavy initial magnetiz-
ing current is drawn from the line, limited only by primary winding
resistance and leakage inductance. This heavy current has a peaked
wave form which may induce momentary high voltages by internal
resonance in the secondary coils and damage the rectifier tubes. Or it
may trip a-c overload relays. The problem is especially acute in large
transformers with low regulation. A common remedy is to start the
rectifier with external resistors in the primary circuit and short-circuit
them a few cycles later. Some rectifiers are equipped with voltage
regulators which reduce the primary voltage to a low value before
restarting.
A-c line transients may cause trouble in three-phase rectifiers, espe-
cially those having balance coils, by shifting the floating neutral volt-
age. Filters like that in Fig. 97 prevent such transients from appearing
in the rectified output.
In some applications the load is varied or removed periodically.
Examples of this are keyed or modulated amplifiers. Transients occur
RECTIFIER PERFORMANCE 131
when the load is applied (key down) or removed (key up), causing
respectively a momentary drop or rise in plate voltage. If the load is
a device which transmits intelligence, the variation in filter output
voltage produced by these transients results in the following undesir-
able effects:
1. Modulation of the transmitted signal.
2. Frequency variation in oscillators, if they are connected to the
same plate supply.
3. Greater tendency for key clicks, especially if the transient initial
dip is sharp.
4. Loss of signal power.
A filter which attenuates ripple effectively is normally oscillatory;
hence damping out the oscillations is not practicable. Nor would it
remedy the transient dip in voltage, which may increase with non-
oscillatory circuits. The filter capacitor next to the load should be
large enough to keep the voltage dip reasonably small. An approxima-
tion for transient dip in load voltage which neglects the damping effect
of load and series resistance is
A«. = S j-C <'^''
where AEj, is the transient dip expressed as a fraction of the steady-
state voltage across R, and L, C, and R are as shown in Fig. 79(a).
The accuracy of this approximation is
poor for dips in transient voltage greater ^■^'■<-^
than 20 per cent. c.
Although the tendency for key clicks in ■ 1^
the signal may be reduced by attention to .
the d-c supply filter elements, the clicks liL jlz
may not be entirely eliminated. Where
key-click elimination is necessary, some
sort of key-click filter is used, of which
Fig. 98 is an example. This filter has ' to key circuit
inductance and capacitance enough to j-j^ gg xey-dick filter
round off the top and back of a wave and
eliminate sharp, click-producing corners. Figure 99 is an oscillogram
showing a keyed wave shape with and without such a filter.
In a choke-input filter, voltage surges are developed across the choke
under the following conditions:
132
ELECTRONIC TRANSFORMERS AND CIRCUITS
1. Ripple Voltage. With large rectifier commutation angles, or
with grid-controlled rectifiers, a surge occurs once each ripple cycle.
In the limit, this surge equals the rectifier peak voltage.
2. Initial Starting Surge. This surge adds to output d-c voltage.
Under the worst conditions it raises the voltage at this point to twice
normal and occurs every time rectifier plate voltage is applied.
60 CYCLES
KEYED WAVE, NO FILTER
ZERO
KEYED WAVE WITH FILTER
Fig. 99. Keyed wave shape with and without key-click filter.
3. Keying or Modulation Transient. Surge value depends upon con-
stants L, C, and Rl, and is limited by considerations of wave shape.
This occurs each time the key is opened or closed, or load is varied.
4. Short-Circuit Surge. If load Rl is suddenly short-circuited, it
causes full d-c voltage to appear across the filter reactor until the cir-
cuit breaker opens. This occurs occasionally. Rectifiers are some-
times arranged so that, if the short circuit persists, the circuit breaker
recloses 3 times and then remains open.
5. Interruption of Reactor Current. This surge voltage is limited
only by losses and capacitance of the circuit, and it may be large,
as shown by Fig. 73. Unless the reactor is designed to produce this
voltage, it occurs only through accident.
Conceivably, surges 1, 2, and 3 may occur simultaneously and add
arithmetically. A reactor insulated to withstand surges 1 plus 2 plus
3 also would withstand surge 4. A reasonable value of peak surge volt-
age comprising these factors is 2% times the full d-c working voltage.
RECTIFIER PERFORMANCE 133
If surges 1 and 5 are too much for reasonable insulation, the reactor
is protected by a gap or other means.
If a rectifier is disconnected from the supply line while the load is
off, interruption of plate transformer peak magnetizing current may
cause high voltages to appear at random in the windings in much the
same way as reactor current interruption causes high voltages. This
is especially true if the transformer operates at high core induction.
The effect is partly mitigated by the arc energy incident to the opening
of the disconnecting switch. But unless the plate transformer is insu-
lated specifically to prevent dangerously high voltages, protective ele-
ments may have to be added in a rectifier subject to switching at light
loads. The necessity for such protection may be estimated from ex-
citing volt-ampere data plus the curves of Fig. 73.
Insufficient attention sometimes is given to the manner in which
power supply lines are brought into buildings. This is particularly
important where a rectifier is supplied by overhead high-voltage lines.
Because of their relatively high surge impedance, lightning and switch-
ing surges occurring on such lines may cause abnormally high voltages
to appear in a rectifier and break down the insulation of transformers
or other component parts. The likelihood of such surges occurring
should be taken into account before the transformers are designed.
Underground cable power lines impose much less severe hazards:
first because they are protected from lightning strokes, and second be-
cause they have much lower impedance (about one-tenth that of over-
head lines). Surges on these cables have much lower values compared
to those on overhead lines carrying the same rated voltage. Protection
against these surges varies with the type of installation.
The best protection of all is provided by an indoor power system
with an underground cable connecting it to the rectifier. Good pro-
tection is afforded by oil-insulated outdoor surge-proof distribution
transformers, stepping down to the rectifier a-c power supply voltage,
with an underground cable between the distribution transformer and
rectifier. No protection at all is provided when overhead lines come
directly into the rectifier building.
With the trend to dry-type insulation, it is desirable to use lightning
arresters on overhead lines where they enter the building. Because
of their low impulse ratio, dry-type transformers require additional
arresters inside the building. When a line surge is discharged by a
lightning arrester, there is no power interruption.
56. Rectifier Filter Charts. From the preceding sections, it can be
seen that various properties of rectifier filters, such as ripple, regula-
134 ELECTRONIC TRANSFORMERS AND CIRCUITS
tion, and transients, may impose conflicting conditions on rectifier
design. To save time in what otherwise would be a laborious cut-and-
try process, charts are used. In Fig. 100 the more usual filter prop-
erties are presented on a single chart to assist in arriving at the best
filter directly. This chart primarily satisfies ripple and regulation
equations 46 and 53 for a choke-input filter.
Abscissa values of the right-hand scale are bleeder conductance in
milliamperes per volt, and of the left-hand scale, filter capacitance
in microfarads. Ordinates of the lower vertical scale are inductance in
henrys. Lines representing various amounts of ripple in the load are
plotted in quadrant I, labeled both in db and rms per cent ripple.
In quadrant II, lines are drawn representing different types of recti-
fiers and supply line frequencies. A similar set of lines is shown in
quadrant IV.
Two orthogonal sets of lines are drawn in quadrant III. Those
sloping downward to the right represent resonant frequency of the
filter L and C, and also load resistance Rl- The other set of lines is
labeled ^L/C, which may be regarded as the filter impedance. It can
be shown that the transient properties of the filter are dependent upon
the ratio of ^/L/C to Rt-
The L scale requires a correction to compensate for the fact that
ripple is not exactly a linear function of L but rather of X^ — Xq.
The curves in the lower part of quadrant IV give the amount of
correction to be added when the correction is greater than 1 per cent.
Instructions for Using Chart
1. Assume suitable value of bleeder resistance or bleeder current /i
in millamperes per volt of E^c- This is also steady-state peak ripple
current in milliamperes.
2. Trace upward on assumed bleeder ordinate to intersect desired
value of load ripple, and from here trace horizontally to the left to
diagonal line for rectifier and supply frequency used. Directly under,
read value of C.
3. Trace downward on same assumed bleeder ordinate to intersect
diagonal line below for rectifier and supply frequency, and read
value of L.
4. From desired ripple value, determine correction for L on graph
at lower right, and add indicated correction to value of L.
5. Using corrected value of L and next standard value of C, find
intersection in third quadrant, and read maximum resonant fre-
quency /r.
RECTIFIER PERFORMAXCE 135
6. Using same values of L and C as in 5, read value of ratio y/L/C.
7. Under intersection of ^L/C with load resistance R/, read values
of the four transients illustrated in Fig. 101 (in per cent).
BxampZe (shown dotted) . Three-phase full- wave 60-cycle rectifier; Edc =
3,000 v; h = 1 amp; 7i = 96 ma; load ripple = — 50 db; balanced line.
Solution:
Bleeder ma/volt = 0.032.
C = 4.5 Mf (use 5 /if).
Scale value of L = 0.78 h; corrected value = 0.82 h.
Resonant frequency = 75 cycles.
Load resistance Rl = 3,000 ohms.
im = 7/2 = 7 amp; AEd = 12 per cent; AEu = 15 per cent; AEs = 80
per cent.
In polyphase rectifiers the possibility exists of enough phase unbal-
ance to impress a voltage on the filter having a frequency lower than
the normal fundamental ripple frequency. If the filter L and C
resonate near the unbalance frequency, then excessive ripple may be
expected. Conversely, the L and C should have a resonant frequency
lower than the unbalance frequency to avoid this trouble. Quadrant
III of the chart has a series of lines labeled /r, and the intersection of
L and C thereon indicates this resonant frequency. It should be no
higher than the value given in the small table on the chart if excessive
ripple is to be avoided. This table is based on 2 per cent maximum
unbalance in the phase voltages.
For most practical rectifier filters, transient conditions fall within
the left-hand portion of the third quadrant. The other conditions
sometimes help in the solution of problems in which L and C are inci-
dental, e.g., the leakage inductance and distributed capacitance of a
plate transformer.
Although the chart applies directly to single-stage, untuned filters
with constant choke inductance, it can be used for other types with
modifications:
(a) Shunt-Tuned Choke per Fig. 93. Figure 100 can be used di-
rectly for capacitance C, but, for a given amount of ripple, divide the
chart values of inductance by 3 in order to obtain the actual henrys
needed in the choke.
(6) Swinging Choke. If at light load the filter choke swings to S
times the full-load value of henrys, multiply the capacitance obtained
from the chart by the ratio S to find the capacitance needed (C„). The
136
ELECTRONIC TRANSFORMERS AND CIRCUITS
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RECTIFIER PERFORMANCE
137
RIPPLE =RMS A-C VOLTAGE ACROSS Ci-E^^
ZERO CHOKE AND RECTIFIER REGULATION ASSUMED
L ASSUMED CONSTANT WITH VARYING DIRECT CURRENT
RMS A-C VOLTAGE ACROSS C
' 0.70/ E(,r
Ob RIPPLE = 20 LOG.c
Fig. 100. (Continued)
138
ELECTRONIC TRANSFORMERS AND CIRCUITS
value of L obtained by projecting the bleeder current downwards is
the maximum or swinging value. It must be divided by S to obtain
the full-load value. Transient conditions then may be approximated
by using capacitance C„ and the full-load value of henrys.
Edc
TRANSIENT CONDITIONS WHEN RECTIFIER IS STARTED
VOLTAGE DIP AEq
VOLTAGE RISE AEr
TRANSIENT CONDITIONS WITH VARIABLE LOAD
Fig. lOL Four transient conditions in choke-input filter circuit and curves.
(c) Two-Stage Filters. In a filter with two identical stages, Fig.
84 (fo), the chart can be used if it is recognized that the ripple is that on
the load side of the first choke. For example, if the filter consists of
two stages both equal to that in the example given for the single-stage
filter, the ripple would not be —100 db but —75 db, because of the
fact that the rectifier output has (per Table VII) only 4 per cent
ripple, which is —25 db.
The regulation in a two-stage filter, as far as capacitor effect is con-
cerned, depends upon the inductance of the first choke as in the single-
stage filter. Therefore the chart applies directly to the inductance
and capacitance of one stage. The peak ripple current likewise de-
pends upon the inductance of the first choke, regardless of the loca-
tion of the bleeder resistor. Transients, however, are more compli-
cated, owing to the fact that the two stages interact under transient
conditions.^
57. Rectifier Efficiency. Losses in a rectifier consist of transformer,
tube, and filter losses. Filament power should be counted as loss,
especially when a tube rectifier is compared with a rotating machine
1 See Proc. I.R.E., 22, 213 (February, 1934).
RECTIFIER PERFORMANCE 139
or metal disk rectifier. In spite of this loss, a high-voltage polyphase
rectifier of the mercury-vapor or pool type may have 95 per cent
efficiency at full load. In contrast, the rectifier for a radio receiver
rarely has more than 60 per cent efficiency. Reasons for this low figure
are the high tube and reactor IR drops and low transformer efficiency.
The filament power, too, is a greater portion of the total.
58. Rectifier Tests. Even though the transformers, chokes, tubes,
and capacitors have been tested before assembly of the rectifier, per-
formance tests of the rectifier are desirable. These generally include
tests of output, regulation, efficiency, ripple, and input kilovolt-
amperes or power factor. Accurate meters should be used, and poly-
phase rectifiers should have balanced supply voltages. Wiring is
tested at some voltage higher than normal, preferably with trans-
formers, tubes, and capacitors disconnected to avoid damage during
the test. Ordinary care in testing is sufficient except for regulation
tests. If the regulation is low, the difference in meter readings at no
load and full load may be inaccurate. Differential measurements are
sometimes used, such as a voltmeter connected between the rectifier
and a fixed source of the same polarity and voltage. Artificial loading
of a high-voltage rectifier is often a problem. Water rheostats have
been used for this purpose. Load tests, preferably in combination
with the transmitter or other apparatus which the rectifier is to supply,
are safeguards against field troubles. Operating tests are essential
when the load is keyed or modulated, so that overheating or inadequate
transformer operation may be detected.
Ripple is measured either with a special hum-measuring instru-
ment or with a capacitance-resistance network arranged to block the
direct current from the measuring circuit. Capacitance and resistance
values in the measuring circuit should be so chosen as to avoid influ-
encing the ripple or loading the rectifier transformer. Sometimes
capacitance dividers are used for this purpose. The problem of proper
values becomes particularly critical with high-voltage low-current
rectifiers. The effect of stray capacitance is especially important.
5. AMPLIFIER TRANSFORMERS
An amplifier is a device for increasing voltage, current, or power
in a circuit. The original wave form may or may not be maintained;
the frequency usually is. An amplifier may be mechanical, electro-
mechanical, electromagnetic, or electronic in form, or it may be a com-
bination of these. In this chapter the transformer-coupled electronic
amplifier is considered. The amplifier consists of a vacuum tube, or
similar device, with transformers, capacitors, and resistors. Input
voltage or current is impressed on some element of the tube ; this causes
higher voltage or current to appear in the output circuit.
59. Amplifier Potentials. Electronic amplifiers are characterized by
the use of tubes having three or more elements. In triodes or three-
element tubes, the addition of the third element, the grid, alters the
CATHODE
ANODE
CATHODE GRID
ANODE
POSITIVE GRID
VOLTAGE
NEGATIVE GRID
VOLTAGE
VOLTAGE GRADIENT
IN DIODE
VOLTAGE GRADIENTS
IN TRIODE
Fig. 102. Diode and triode voltage gradient.
voltage gradient between cathode and anode as shown in Fig. 102. The
grid either aids or opposes the flow of electrons from cathode to anode,
depending on whether the grid voltage is positive or negative respec-
tively, compared to the cathode, which is shown at zero voltage in
Fig. 102.
As the grid voltage is made more and more negative, electron flow
is diminished and finally stops. At this point the anode current is
zero; the condition is called anode current cut-off.
If the grid voltage is made more and more positive, eventually
140
AMPLIFIER TRANSFORMERS
141
further increase in grid voltage causes no additional anode current
increase. This condition is called grid saturation.
Tetrodes and pentodes have respectively two and three grids. The
voltage gradient between cathode and anode is more complex than
that indicated in Fig. 102. The advantages to be gained from the addi-
tional grids are mentioned below.
60. Transformer-Coupled Amplifiers. Amplifier circuits in which
transformers are used can be represented by a circuit similar to that
of Fig. 103(a). Here a triode is shown with a voltage e^ impressed
upon the grid, which comprises the grid bias (a constant negative
direct voltage) and a superimposed alternating voltage eg. Anode
Zg
i L
(0) (b)
Fig. 103. (a) Transformer-coupled amplifier; (b) equivalent circuit.
voltage Eb is supplied from some source through the primary of the
transformer, across which appears an alternating voltage e.p. The
secondary of the transformer is connected to a load Z^. Under certain
conditions, which will be defined below, this circuit may be simplified
to that of Fig. 103(6). A fictitious alternating voltage fie^ is impressed
on the circuit, where ,/x is the tube amplification factor. Internal tube
resistance Zq is in series with the load Zr„ which is reflected by the
transformer to the proper value in the primary circuit for tube opera-
tion. That is, Zt, in Fig. 103(6) is equal to that in Fig. 103(a) only if
the transformer has a 1 : 1 ratio. For any turns ratio, the quotient of
two Z's is equal to the (turns ratio)- as in equation 7 (p. 8). Note
that the winding resistances are regarded as zero, so that, in the ab-
sence of a grid signal, full voltage Eb appears on the plate of the
vacuum tube.
Alternating voltage jxCg causes voltage ep to appear across the load
Zx,. The voltage e^ is not /j, times e^ but is related by the following
equation:
142 ELECTRONIC TRANSFORMERS AND CIRCUITS
(58)
Zg + Zr.
Although transformer-coupled amplifiers are used sometimes for volt-
age amplification, they are used mostly where power output is required
of the amplifier and where a good reproduction of the grid voltage is
required in the plate circuit.
61. Tuned Amplifiers. Figure 104 shows the circuit for an amplifier
in which the output voltage appears across a parallel-tuned circuit.
Fig. 104. Tuned amplifier.
This circuit is shown coupled to a load Zr,. This type of amplifier may
be used where large outputs are required, but the voltage e^ is not neces-
sarily a reproduction of e^, and they are not related as in equation 58.
62. Amplifier Classification. Amplifiers can be divided into classes,
depending upon the mode of operation. A class A amplifier is one in
which the grid bias and alternating grid voltage are such that anode
current flows continuously. In a class B amplifier the grid bias is
almost equal to the cut-off value, so that plate current is nearly zero
when no exciting grid voltage is applied. When full alternating grid
voltage is applied, plate current flows for approximately one-half of
each cycle. A class C amplifier has a grid bias greater than the cut-off
value, so that the plate current is zero when no alternating grid voltage
is applied and it flows for appreciably less than one-half of each cycle
when an alternating grid voltage is applied. These classes are illus-
trated in Fig. 105, in which the alternating plate current, plate voltage,
grid voltage, and grid current are shown with the steady or average
values which are, respectively. Is, Eb, Ec, and Iq. Relative plate and
grid voltage amplitudes for these three types of amplifiers are shown
in Fig. 105, and other properties are summarized in Table XI.
Class A amplifiers are characterized by comparatively high no-signal
anode current. Usually the grid never swings positive. Anode cur-
AMPLIFIER TRANSFORMERS
143
CLASS A
CLASS B
CLASS C
r
\
r
w
PLATE
f
\
T
\
Y
B
CURRENT
\
J
'b
\
\
\
\
'b
\
\
/
1
\
1b
^
N.
PLATE
i\
'\
V
B —
VOLTAGE
Eb
/
■\
\
I
\
\^
/
\
\
1
\
1
V
i
+
/
\
/
\-
— Emin 1
N^
f
\
/
\
GRID
"v
'-^
'-o
i\
/
to
l\
|\
Ec
VOLTAGE
W-
/
1 \
•J
LnL
/
GRID
\l
\
^
y
CURRENT
^
j_i
q
h
^
'9-
Fig. 105. Amplifier voltages and currents.
Table XI. Amplifier Classes
Amplifier Class A B
Anode efficiency
a. Theoretical maximum 50% 78.5% *
b. Practical value for
low distortion Up to 30% 40-67% *
Output proportional to e^" eg
Grid current 7g None Small
Anode current In Fairly eg — 0, 1b low
100%
70-85% t
Eb^ (grid saturated)
Large (may ~ Ib)
eg = 0, /b =
constant Bg = max, Ib high eg = max, Ib high
* These values are for push-pull amplifiers.
t With a high-Q tank circuit, the efficiency depends on excitation power.
144
ELECTRONIC TRANSFORMERS AND CIRCUITS
1
rent remains comparatively constant, when averaged over a whole a-c
cycle. In class B amplifiers, the grid is biased at a greater negative
potential so that current is nearly cut off in the absence of a signal.
Positive swings of grid voltage result in anode current being drawn;
this causes a dip in the residual voltage on the plate of the amplifier.
Negative grid swings cause no plate current to flow but do cause a
positive plate voltage swing. In class C amplifiers, the grid is biased
more negatively still, with the re-
sult that plate current flows for less
than half a cycle, and mostly when
the plate voltage on the tube is at
a relatively low value. Grid cur-
rent in this class of amplifier
reaches values comparable to the
plate current. Output voltage
wave form is maintained by a
tuned plate circuit.
Operation may sometimes be im-
proved by the use of two tubes
connected push-pull, as shown in
Fig. 106. This is the most common
connection for class B amplifiers;
also, it is frequently used in class A amplifiers. Intermediate between
class A and class B amplifiers are those known as class AB with grid
bias and efficiency intermediate between class A and class B amplifiers.
Such amplifiers are further subdivided into class ABi and class ABo.
Class ABi amplifiers draw no grid current, but the bias voltage is
somewhat higher than the class A value and the plate current may be
discontinuous during the cycle when grid signal is applied. Class AB2
amplifiers draw grid current but are not biased as close to cut-off as
class B amplifiers. Both class ABi and AB2 amplifiers are commonly
used with the push-pull connection.
Tube properties such as plate resistance r^, amplification factor /n,
and mutual conductance g,„ may be calculated from data published for
each tube in the form of characteristic curves. Operating conditions
such as plate- and grid-voltage swings, power output, plate dissipation,
and efficiency also are found from these curves. Theoretical discus-
sions of such data may be found in books on amplifiers.
63. Decibels; Impedance Matching. In amplifier work, the ratio of
two voltages E^ and E2 at the same impedance level is often stated in
decibels (db) according to the definition
Fig. 106. Push-pull amplifier.
AMPLIFIEE TRANSFORMERS 145
db = 20 logio {EJE2) (59)
Amplifier voltage gain, transformer ratio, frequency response, and
noise levels all may be expressed in decibels. Volume, voltage, or
power in decibels must be compared to a reference level; otherwise the
term is meaningless. A standard reference level is 1 milliwatt. This
is expressed as zero dbm. Across 600 ohms, the voltage for zero dbm
is VO.OOl X 600 = 0.775 volt; for 20 dbm the voltage is 7.75 volts.
Transmission lines at audio and higher frequencies exhibit properties
commonly ignored at 60 cycles. Line wavelength, characteristic im-
pedance, and attenuation are important at audio frequencies; so is
the matter of matching impedance. If a long transmission line has no
attenuation, its characteristic impedance is given by
Zo = ^l (60)
where L and C are the inductance and capacitance per unit length. If
such a line terminates in a pure resistance load equal in ohmic value
to Zo, all the power fed into the line appears in the load without
attenuation or reflection. This is called matching the impedance of
the line. It is very desirable to save audio power and avoid reflec-
tions; therefore impedance matching of lines is the usual practice
wherever possible. The notion has been extended to include the load-
ing of vacuum tubes, but this is stretching the meaning of the term
matching. A vacuum tube has its optimum load impedance, but the
value depends upon the conditions of tube operation and is not neces-
sarily the same as the tube internal impedance.
Power transmission lines operating at 60 cycles are rarely long
enough to act as appreciable source impedances. When a short-circuit
or low-impedance fault occurs on the load side of a power transformer,
the load current is limited mainly by the transformer short-circuit
impedance. In a vacuum-tube amplifier, the load current delivered
into a short-circuited load is limited mainly by the vacuum-tube in-
ternal resistance rather than by the transformer. At certain fre-
quencies the transformer itself may contribute to low load impedance.
But the greatest difference between power and amplifier transformers
is the difference in source impedance. Even the use of the word im-
pedance in the two fields of application reflects this difference. In
power work, transformer impedance denotes tlie short-circuit im-
146 ELECTRONIC TRANSFORMERS AND CIRCUITS
pedance; in amplifier work, the same term refers to the load or source
impedance.
64. Amplifier Transformers. The major problem of amplifier trans-
former design is obtaining proper output when the transformer is
operated in conjunction with the apparatus for which it is intended.
Several factors external to the transformer affect its performance,
namely, (1) impedance of the source; (2) linearity of this impedance;
(3) impedance of the load; and (4) frequency.
The simplest method of dealing with amplifier transformers is an
adaptation of the so-called equivalent network which has long been
used for power transformers. The transformer that connects the
source to its load in Fig. 103 (a) may be represented more fully by the
diagram of Fig. 107(a).
65. Low-Frequency Response. At low frequencies, the leakage re-
actances are negligibly small. Resistance Rp may then be combined
with Zg to form Ri for a pure resistance source, and Rs with Zl to form
i?2 for a resistance load. At low frequencies both source and load are
pure resistance, and the circuit may be simplified to that of Fig. 107(6) .
Here the a^ has been dropped; in other words, a transformer with a
1:1 ratio is shown, referred to the primary side. Xj{ is the primary
open-circuit reactance, or 27r/ times the primary open-circuit inductance
(OCL) as measured at low frequencies.
If shunt resistance R^' is included in load resistance R2, the circuit
becomes like that of Fig. 107(c). Winding resistances are small com-
pared with source and load resistances in well-designed transformers.
Likewise, Rs is high compared with load resistance, especially if core
material of good quality is used.
Therefore, to a good approximation, in Fig. 107(c) , Ri may represent
the source impedance and R2 the load impedance. On a 1:1 turns-
ratio basis, the voltages E2 and Ei are proportional to the impedances
across which they appear or
jX^R2
E2 jXiY + R2 , ^^
— = : (61)
El jXnR2
R\ +
jX^r + R2
The scalar value of this ratio is found by taking the square root of the
sum of quadrature terms:
AMPLIFIER TRANSFORMERS
147
(62)
Equation 62 holds for any values of Ri, R^, and Xn "whatsoever, but
there are three cases that deserve particular attention: (a) R2 = Ri;
(b) i?2 = 2i?i; and (c) R^ = x. Of these, (a) corresponds to the
usual line-matching transformer with the source and load impedances
equal; (6) is often recommended for maximum undistorted output of
triodes; (c) is realized practically when the load is the grid of a class A
SYMBOLS
a = RATIO OF SEC. TO PRI. TURNS
Cp=PRI. WINDING CAPACITANCE
Cs= SEC. WINDING CAPACITANCE
Ct=Cp + o2Cs
f = ANY AUDIO FREQUENCY
1,= RESONANCE FREQ. OF X^ a X^
Rp = PRI. WINDING RESISTANCE
Rs = SEC. WINDING RESISTANCE
■PRI. NO LOAD (CORE LOSS)
EQUIVALENT RESISTANCE
■ PRI. OPEN CIRCUIT REACTANCE
"PRI. LEAKAGE REACTANCE
■ SEC. LEAKAGE REACTANCE
Xl-Xp + Xj/o^
Xc= TOTAL CAPACITY REACTANCE
I
2Trf Ct
= SOURCE IMPEDANCE
■LOAD IMPEDANCE
Fig. 107. (a) Transformer equivalent circuit; (b) low-frequency equivalent cir-
cuit; (c) simplified low-frequency circuit; (d) high-frequency equivalent circuit;
(e) simplified high-frequency circuit.
148 ELECTRONIC TRANSFORMERS AND CIRCUITS
amplifier. For these cases, equation 62 becomes
E2
E2
eI
1
4 +
2.25 +
\xJ
1 +
\xj
(62a)
(626)
(62c)
These three equations are plotted in Fig. 108 to show low-frequency
response as "db down" from median. The median frequency in an
audio transformer is the geometric mean of the audio range; for other
transformers it is a frequency at which the ratio Xjj/Ri is very large.
At median frequency the circuit is properly represented by Fig. 103(6).
1
f
-'-''''^--'
:===
y-"
^ /
^
/
/
/
/
'/
/
/
/
/
/
/
*
/}.
/
/
/
J
'V
g
/
/
/
{
J
/
/
f
l\ f
R«
/
/
7 \
/
f
/
12
/
/
/
0.2
0.4 0.6 0.8 1.0
2.0
4.0 6.0 8.0 10.0
Fig. 108. Transformer characteristics at low frequencies.
AMPLIFIER TRANSFORMERS 149
The equivalent voltage ratio E2/E1 has maxima of 0.5, 0.667, and 1.0
for cases (a) , {b) , and (c) , respectively, at the median frequency, or for
Xx/Ri = 00 in Fig. 108. The higher OCL, the nearer the transformer
voltage ratio approaches median-frequency value. The lower the
value of loading resistance R2, the lower the equivalent voltage ratio
is. The factors 0.5, 0.667, and 1.0 multiplied by the turns ratio, a,
give the actual voltage ratio at median frequency. At lower fre-
quencies, the factors diminish.
The transformer loaded by the lowest resistance has the best low-
frequency characteristic. A transfoiTner having an open-circuit sec-
ondary has twice the voltage ratio and gives the same response at
twice the "low end" frequency of a line-matching transformer of the
same turns ratio.
Figure 108 is of direct use in determining the proper value of primary
OCL. Permissible response deviation at the lowest operating fre-
quency fixes Xif/Ri and therefore X^. At the corresponding fre-
quency, this represents a certain value of primary OCL. As this in-
ductance determines the size and weight of the transformer, the impor-
tance of Fig. 108 is evident.
If the primary and equivalent (1:1) secondary winding resistance
each are 5 per cent of Ri, the total effect will be a decrease of 10 per
cent in the median-frequency voltage ratio, in the case of the line-
matching transformer, with corresponding decreases at lower fre-
quencies. On the other hand, the primary resistance of an open
secondary transformer has no effect upon the median- frequency voltage
ratio but has some effect at lower frequencies, whereas the secondary
resistance has no effect either at median or at lower frequencies.
Hence it is important in the open secondary case, for the sake of low-
frequency response, to keep the primary winding resistance low, but
the secondary winding resistance may be any value. The maximum
number of secondary turns may be determined by the smallest practi-
cable wire size rather than by winding resistance.
As the frequency increases, the primary inductive reactance Xn also
increases until it has almost no effect upon frequency response. This is
true for median frequency in Fig. 108. It is also true for higher fre-
quencies; in other words, the OCL has an influence only on the low-
frequency end of the frequency response curve. The ratio of R2 to Bi
still limits the voltage ratio, however. If the amplifier works at one
frequency only, OCL is determined by the deficiency in voltage gain
that can be tolerated in the amplifier design. This can be found in
Fig. 108.
150 ELECTRONIC TRANSFORMERS AND CIRCUITS
In an amplifier with a band of operating frequencies, e.g., the audio
band, a well-designed transformer has uniform voltage ratio for a
frequency range extending from the frequency at which X^ ceases to
exert any appreciable influence, upward to a zone designated as the
high-frequency end of the transformer frequency range.
66. High-Frequency Response. The factors that influence the high-
frequency response of a transformer are leakage inductance, winding
capacitance, source impedance, and load impedance. Hence a new
equivalent diagram, Fig. 107(d), is necessary for the high-frequency
end. Winding resistances are omitted or combined as in Fig. 107(6).
Winding capacitances are shown across the windings. If primary and
secondary leakage inductances and capacitances are combined, X^ is
omitted as if it were infinitely large, and a^ is dropped as before, the
circuit becomes that shown in Fig. 107(e). X^ is the leakage reactance
of both windings, Xc the capacitive reactance of both windings, and
B2 the load resistance, all referred to the primary side on a 1 : 1 turns-
ratio basis.
At any frequency, the equivalent voltage ratio in the circuit of
Fig. 107(e) can be found by the ratio of impedances, as for the low-
frequency response. The scalar value is
E2 1
— = , (63)
El /Ih XlV /£l _ ^ _ ^
\ \Xc R2 ' \Xc R2
In equation 63 the term X^/Xc may be written Aw^fLC = f^/fr^,
where l/(2ir\/LC) = fr, the resonance frequency of the leakage in-
ductance and winding capacitance, considered as lumped and without
resistance. Also Xl/R2 = Xcf^/R2fr^. Assign to the ratio Xc/R\ a
value B at frequency fr- Then at any frequency /, Xc/Ri = Bfr/J. In
the three cases considered at the low frequencies.
R2 = Ri,
R2 = 2Si,
i?2
E
' M-dH^-)^
E2
e[
Ci40-(S-)
(63a)
(63M
AMPLIFIER TRANSFORMERS
151
Ro = =0,
E2
1
El
^
l( ' V+ C^ -
-;
(63c)
Equations 63a, h, and c are plotted in Figs. 109, 110, and 111. If
Xc/Ri has certain values at frequency /r, the frequency characteristic
is relatively flat up to frequencies approaching jr- In particular, per-
formance is good at fi = 1.0 in all three figures.
z - I
<
Q-2
UJ
s
o
0: -4
< -6
5-7
m
e
-9
-10
-II
O
X
^■"'■t F
,„
.i
R
r
2=R
T ^
Xl=Xc at FREQ. fr
B= -5- AT FREQ. f.'
"=a
~!
-^
^
=:
■~>
■>
S
■^
^
N,
"V
N
s
s
\
\
S
.\
\
s
\\^
.
\
S.
\
\\
\
S
s
»\
s
\l
B=I.O
\
\\
B = b.8,' B- r.25
B=0.67, B= 1.5-
B=0.5, B=2.0
\
\
\
L
^8 = 0.25
B =
4.0
0.2
0.4 0.6 0.8 1.0
f/fr
2.0
Fig. 109. Transformer characteristics at high frequencies (line matching).
When leakage inductance and winding capacitance are regarded as
"lumped" quantities, current distribution in the windings is assumed
to be uniform throughout the range of frequencies considered. As
shown in Chapter 7 (Section 97), this assumption is valid up to the
resonance frequency jr- At frequencies higher than jr, there may
be appreciable error in Figs. 109, 110, and 111. But good frequency
characteristics lie mainly below the frequency jr, where the curves are
correct within the assumed limits.
To use these curves in design work, choose the most desirable
152
ELECTRONIC TRANSFORMERS AND CIRCUITS
characteristic curve and, from a knowledge of the source impedance,
find the proper value of capacitive reactance Xc at frequency /r. The
value of /r should be such that the highest frequency to be covered lies
on the fiat part of the curve. Xo and jr determine the values of wind-
o
o
z
UJ
ID
o
UJ
£ +1
1
o
s -I
-2
'^ -3
z
<
S -5
<l
>-6
ui
en
z -7
o
Q-
tf) -8
liJ
Q
-10
-I I
0.1
Yv-y
1 A
;
1 "U J,
Rj
= 2R|
r
— Xl=X(; at FREO.fr
Ri
Lm
'r
SS
M=
—
^
^,
--
^
N
^^.
^^
s
N
\
v\\
V
N
\
s
\\
\
\
s
s
•A\
\
\
s
\\
\\
\
\^
\\\
\
\
\\\
\
\\1
B=I.4I
\
M
B=I.O, B=2.0
\
\
8 = 0.7, B=2.8
B =
0.25, B =
3-
\
\
B = 0.5,B=4
1 \ 1
0.2
0.6 0.8 1.0
f/fr
2.0
Fig. 110. Transformer characteristics at high frequencies (triode output).
ing capacitance and leakage inductance that must not be exceeded
in order to give the required performance.
In Fig. 107(e) the capacitance is shown across the load. This is
correct if the main body of capacitance is greater on the secondary
than on the primary side. Normally this is true if the secondary wind-
ing has the greater number of turns. Figures 109, 110, and 111 are thus
plotted specifically for step-up transformers. Modifications are neces-
sary for step-down transformers, the equivalent circuit for which is
shown in Fig. 113. Analysis shows the scalar voltage ratio to be
E2
1
Ri Xl\
Xc R2'
\R2
Xl Ri
Xc R2
(64)
AMPLIFIER TRANSFORMERS
153
Notice the similarity to equation 63. In fact, if Ri = R2, equation 64
reduces to equation 63 ; for this case the response is the same for step-
down and step-up transformers, and is given by Fig. 109.
+ 8
+7
+ 6
UJ +4
o
S + '
s
o
c -I
-7
1
n
R
n
n
n
_
'
T
L j
R
i0^
x.a=
2=<D
1
/
1 M-
/
/
E
=1.'
\
- Xl = Xc AT FREQ. f
MM 1
/
/
u
1 III
/
9
/'
\\
"1
1 1 .1.^
^
/>
y
^
6
•^
\'<r'
w
^
i^
^
"
s
*n
^ \\
-■
-
■
-^B
~"
—
-
S-e
hs
e
\\\
X
■V
s
<s
to
F
\
AW
\
N
\
\
a\v
\
\
V
'aWi
\
\
\
\\v\
\
\
\
Av
\
\
\\\
\
i
\ \\\\\
0.1
0.2 0.4 0.6 0.8 1.0
f/fr
Fig. 111. Transformer characteristics at high frequencies (class A grid).
For jR2 = 2iJi, equation 64 becomes, after substitution in terms of
frequency,
B2 1
(65)
which is plotted in Fig. 113. Non-uniform response comes at somewhat
lower frequency than in Fig. 110.
The case of JK2 = 00 for step-down transformers is not important.
By inspection it can be seen to be the response of R^ and Xc in series,
because Xl carries no current. This case rarely occurs in practice.
67. Harmonic Distortion. Audio response may be good according to
Figs. 109, 110, 111, and 113, but at the same time the output may be
badly distorted because of changes in load impedance or phase angle.
154
ELECTRONIC TRANSFORMERS AND CIRCUITS
■ i
Fig. 112. Audio amplifier. Audio transformers are inverted on chassis at left.
Power supply is at right.
1
Xl
1
T^ ^
Rg E2-
— ~^
■■~^==^
*^
T
*^
^Xo
^,
^
*<:
^. l'
\
\
\
sj^^
r 1 1
-N
s.
NJ
\\, «
Xc
= 5^ AT FRE
0. f.
\,
\\\v
R|
L=Xc AT FREQ.fr -
!> = 2R.
^
s.
\\v
B=I.4I R
\
\n
M ' ^
v
V
B=l,2
B= 0.7, 2.8
\
\ \
8=0.5,4
\
\
\
\
8=0.25,8
1.0
f/fr
Fig. 113. High-frequency response of step-down transformers.
AMPLIFIER TRANSFORMERS
155
This possibility is considered here for the case in which the load im-
pedance is twice the source impedance.
The phase angle of the equivalent circuits of Figs. 107(a) and (e)
is found by taking the angle whose tangent is the ratio of imaginary
to real components of the total circuit impedance in each case. This
40
30
20
10
uj
-J
o
z
<
<
X
Q.
-10
-20
^
^
^
^
\
1 1 1
'2
JXn f
'
k 1
R2 = 2R|
0.1 0.2 0.4 0.6 0.8 1.0
Fig. 114. Variation of amplifier phase angle at low frequencies.
angle is plotted in Figs. 114 and 115 for the low- and high-frequency
ranges, respectively, with the same abscissas as in Figs. 108 and 110.
It is the angle between the voltage H-i and the current entering the
equivalent circuits of Figs. 107(c) and (e) and therefore represents the
angle between a-c grid voltage and plate current. Positive angle indi-
cates lagging plate current.
The phase angle exhibited by a transformer over the range con-
sidered in Figs. 114 and 115 does not exceed 30°, whereas for the most
favorable curve in Fig. 115 {B = 1.0) it does not exceed 15°. To study
the effect of phase angle alone upon distortion, the light load of 8,800
ohms is plotted upon the plate characteristics of triode type 851 in Fig.
116. The result is a sine wave of plate voltage. If the phase angle
156
ELECTRONIC TRANSFORMERS AND CIRCUITS
30
20
rr
o
10
III
o
z
UJ
-1
v>
z
<
UJ
-10
to
<
I
-30
1
1 —
1 — 1
-I l-^Y
R|
X,
1
"I
Ic
T
Rg
/
B = 2.0
1
/
/
B = l.5
Xl=Xc at FREQUEN(
R2 = 2R,
=Yfr
^
/
i
B= 1.0
B = 0.8
B = 0.5
^
^
4
f
^-^
r:/
7
^—
-^
0.1
0.6 0.8 1.0
2.0
Fig. 115. Variation of amplifier phase angle at high frequencies.
between grid voltage and plate current waves is then arbitrarily made
30°, as in Table XII, the elliptical load curve obtains. The wave of
Table XII. 851 Triode Operation
WITH 8,800-Ohm 30° Ph
e (deg)
Be
iB
es
- 60
0.245
1850
30
- 33
0.300
1400
60
- 13
0.365
1080
90
- 6
0.395
960
120
- 13
0.410
1160
150
- 33
0.395
1520
180
- 60
0.355
2000
210
- 87
0.300
2460
240
-107
0.245
2790
270
-114
0.205
2880
300
-107
0.190
2720
330
- 87
0.205
2350
360
- 60
0.245
1850
Phase Angle Load
plate voltage is plotted for both zero and 30° phase angle in Fig. 117.
These wave forms indicate that the phase angle encountered in audio
AMPLIFIER TRANSFORMERS
157
\
-"
—
—
\
(-20
---
\
\
-40
—
' \
(\
^
<
-60
-
\
\/
/
-80
Er,
\
V\
/
-100
/
•^
■~"
^
^
/
/
-120
/
K
/
^
V
^
-^
f
/
J
7^
^
r
^
.y/
\,
s/
y
WITH 30° PHASE
/
/
f\
'
'^
^
^
Y
/i^
^
/
GRID VOLTAGE 8
/
/
/
)(
V^\
7
-88O0/0HM LOAD LINE
/
1
/
1
1
/
/
\\
A?
/
/
/
/
/
y
V
/ (RECOMMENDED)
/
/
/
y
y
X
' ^
>
1550 OHM LOAD LINE
2000
PLATE VOLTAGE
Fio. 116. Triode type 851 with reactive load.
3 000
o
>
UJ
a
o
z
<
2000
1000
//^
X
/
\;
V
/
\
1
\
1
1
j
\
I
\\
r
DOTTED CURVE IS PLATE VOLT- "
AGE Bp WITH A-C COMPONENT OF
\
L
1
7
PLATE CURRENT Ip DISPLACED '
30° FROM eg
\
1/
\
r~
SOLID CURVE IS Bp WITH Ip IN
\
"Vj
y
/
PHASE WITH Bg
e, AND ip ARE SINUSOIDAL IN
BOTt
H CAS
ES
90° 180° 270°
GRID VOLTAGE PHASE ANGLE
360°
Fig. 117. Plate voltage wave forms with zero and 30° phase angles.
158
ELECTRONIC TRANSFORMERS AND CIRCUITS
transformers does not of itself introduce much distortion in a lightly
loaded triode.
The influence of load impedance on distortion will be considered next.
In Fig. 107(c) the load impedance, to the right of the dotted line, is
Z =
Hence
Z
/?2
1 +
R2
R2 Xff
Xif R2
(66)
Equation 66 is plotted in Fig. 118. It shows the change in load Z from
its median-frequency value R2, as the frequency is lowered. Abscissas
are X^/Ri instead of X^/Ri as in Fig. 108.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
- —
1
1-*-
^
r ]
1
R2 -
J -
/
X
/
/
A
i
^^
1
/
!
1
-J 1 —
_
0.4 0.6 0.8 1.0
R?
Fig. 118. Variation of load impedance with transformer characteristics at low
frequencies.
For the higher audio frequencies, the load impedance at the right of
the dotted line in Fig. 107(e) is
AMPLIFIER TRANSFORMERS
JXlR2 + XlXc - jXcR2
159
z
i?2
«2 - jXc
V\fi2/ \ R2' Xc
- 1
/
i?2 Xc
Xc i?2
(67)
If we let Xc/R-z = -D at frequency Jr, then, at any frequency /, Xc/R2
= Dfr/f. If this substitution is made in equation 67 and also if
Xl/Xc = f/fr",
(67a)
z
,/T^(-4-
-y
R2
Dfr I
Equation 67a is plotted in Fig. 119 for several values of D. The
impedance varies widely from its median-frequency value, especially
at lower values of D.
1.1
1.0
0.9
0,8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
— n
n
' — 1
— 1
""'
— 1
n
Yf /
D= 1.0
^
5=5
55,
"2
1 1 1 1
<U^
N,
^:
"■^
s
■v
s.
/
S
s
^^
\
S
"s
s
/ J
D'0.75
s
k
S
\
/
\
\
\
S
S
\
V
/
z
L
R2
r
\
s
\^
\.
V
V.
' 1
D^O.'S
Xr,:
\
\-
S
//
0=0.4
s
K
\
S
</
= 25
■* Xi. = Xc AT FREQ. f
r
\
s
^/
-/
O.l
0.2
0.4 0.6 0.8 1.0
f/fr
2.0
Fig. 119. Variation of load impedance with transformer characteristics at high
frequencies.
160 ELECTRONIC TRANSFORMERS AND CIRCUITS
From Figs. 118 and 119 it is possible to compare the change in im-
pedance with the frequency response curves in Figs. 108 and 110.
When this comparison is made it should be remembered that B = 2D
for the triode conditions assumed here. If the amplifier response is
allowed to fall off 1.0 db at the lowest frequency, the corresponding
value of Xif/Ri from Fig. 108 is 1.3. This means that X^/R.^ is 0.65.
The corresponding load impedance in Fig. 118 is only 0.55 of its me-
dian-frequency value. Likewise, for 0.5-db droop of the frequency
characteristic, the load impedance falls to Q.7R2, whereas for a good
load impedance of 0.9i?2 the frequency characteristic can fall off only
0.1 db. It is thus evident that load impedance may vary widely even
with comparatively flat frequency characteristics.
At high audio frequencies the divergences are still greater. Suppose,
for example, that a transformer has been designed so that Xo/Ri is
1.0 at fr (that is, B = 1.0 in Fig. 110) . Suppose further that the highest
audio frequency at which the transformer operates is 0.75/^. The
amplifier then has a relatively flat characteristic, with a slight rise
near its upper limit of frequency. In Fig. 119, the curve corresponding
to B = 1.0 is marked D = 0.5, for which at 0.75/r the load impedance
has dropped to 32 per cent of R2, an extremely poor match for the tube.
It might be thought that, since 0.75/r is the upper frequency limit,
the harmonics resulting from the low value of load impedance would
not be amplified, and no harm would be done. But at the frequency
0.375/r, whose second harmonic would be amplified, the load imped-
ance is only 0.69i?2-
Between 0.375/,- and 0.75/r (over half of the amplifier frequency
range) the load impedance gradually drops from O.69E2 to 0. 327^2.
Thus distortion is large over a wide frequency range. It would be
much better to design the transformer so that B = 2.0; the change in
impedance is much less, and the rise in response is slight.
To ascertain how much distortion these low load impedances pro-
duce, a series of loads was plotted in Fig. 116 on 851 plate character-
istics: 100, 70, and 50 per cent of the class A UPO value of twice the
plate resistance (3,100, 2,200, and 1,550 ohms, respectively). The
distortion is tabulated below for 54 volts grid swing.
Percentage Percentage Plate Voltage
of 2nd of 3rd Swing (Peak
Load Harmonic Harmonic to Peak)
3100 ohms 4 1 1600
2200 ohms 10 4 1270
1550 ohms 19 6 1100
AMPLIFIER TRANSFORMERS 161
The plate voltage amplitude decrease with low impedance loads
means that the combination of tube and transformer has a character-
istic which droops instead of remaining flat as indicated by the curve
B = 1.0 in Fig. 110.
This droop modifies the upper ends of the curves of Fig. 110.
Although these curves were intended specifically for vacuum tubes,
they were derived on the basis of a constant sinusoidal voltage in the
source. Figure 119 demonstrates one important fact: For vacuum
tubes operating into loads of twice the tube plate resistance, it is
better to design transformers so that B = 2 or more. Then the out-
put voltage and distortion are less affected by impedance variations
at high frequencies. The actual frequency characteristics for triodes
lie somewhere between the curves of Fig. 110 and the corresponding
curves of Fig. 119.
Designing transformers for B ^ 2.0 means keeping the effective
capacitance lower, but the leakage inductance may be proportionately
greater than for transformers having B = 1.0.
Variations of load impedance at high frequency shown in Fig. 119
are for step-up transformers. Similar variations for step-down trans-
formers may be found from equation 68.
Z 1 + (DY/fr^)
R2 Uj_ ^^_^
.3 ^.. - (68)
Equation 68 is plotted in Fig. 120. Impedance rises to peaks in the
vicinity of fr, in contrast to the valleys in Fig. 119. For the same
variation of impedance, the frequency range is greater for step-down
transformers, especially with values oi D = 0.5 and 0.7.
Besides the harmonic distortion caused by variations in load im-
pedance, at low frequencies additional distortion is caused by non-
linear magnetizing current. If a transformer is connected to a 60-cycle
supply line, the no-load current contains large harmonics, but the
voltage wave form remains sinusoidal because the line impedance is
low. But if distorted magnetizing current is drawn from an amplifier
tube, the plate resistance is high enough to produce a distorted voltage
wave form across the transformer primary winding, caused mainly by
the third harmonic. If the harmonic current amplitude In in the
magnetizing current is found by connecting the transformer across a
low-impedance source, the amplitude of harmonic voltage appearing
in the output with a higher-impedance source is
162
ELECTRONIC TRANSFORMERS AND CIRCUITS
Ejj IhR / R
IfX
f^N
1 -
4Xa.
(69)
where Eh = harmonic voltage ampHtude
Ef = fundamental voltage amplitude
/// = harmonic current amplitude
If = fundamental current amplitude
R = RiR2/{R\ + /?2)- ^1, R2, and Xv are as shown in Fig.
107(c).i
3.5
2.5
2.0
1.5
!\
Xl
I
'
1
J
\
/.
V
V
/
\
\
XL=Xcatfr
D=2.0
^
y
'\'
^
D--I.4I
-. \
D = I.O
"^
b=u.c
D=0.5
0.4 0.6 0.8 1.0 2.0
Fig. 120. Step-down transformer impedance at high frequencies.
If flux density is below the knee of the saturation curve, and if
Xif = 3R2 at the lowest operating frequency, the harmonic amplitude
is less than 5 per cent. An air gap in the core reduces this figure.
Table XIII gives typical harmonic currents for silicon steel.
Output voltage distortion may be analyzed to find harmonic content
by the usual Fourier method. Several simplifications have been de-
1 For a discussion of equation 69 and magnetizing currents in general, see "Har-
monic Distortion in Audio-Frequency Transformers," by N. Partridge, Wireless
Engr., 19 (September, October, and November, 1942).
Percentage of
Percentage of
3rd Harmonic
5th Harmonic
4
1
7
1.5
9
2.0
15
2.5
20
3.0
30
5.0
AMPLIFIER TRANSFORMERS 163
Table XIII. Typical Silicon-Steel Magnetizing Current Harmonic Com-
ponents WITH Zero Impedance Source
Gauss
100
500
1,000
3,000
5,000
10,000
vised to reduce the labor and increase accuracy. "^ In general, if the
recommended tube load impedances are maintained, harmonic per-
centages will be as given in the tube manuals. If other load imped-
ances obtain at some frequencies, to predict the harmonic output re-
quires harmonic analysis.
68. Push-PuU Amplifier Transformers. The analysis of single-side
amplifiers in Section 67 applies to class A push-pull, except that the
second-harmonic components in the amplifier output are due to unlike
tubes rather than to low-impedance distortion.
The internal tube resistance of a class B amplifier varies so much
with the instantaneous signal voltage on the grids, power output, and
plate voltage that it is not practicable to draw curves similar to Figs.
110 and 113 for class B operation. Qualitatively, the characteristic
curves may be expected to follow the same general trend as for class A
amplifiers. A basis for class B amplifier design is to make the trans-
former constants such that the load impedance does not fall below a
given percentage of the load resistance R2. This is discussed below.
Usually the decline with frequency response is greater for class B
than for class A amplifiers, because the effect of internal plate im-
pedance is greater. In the extreme, frequency response falls off pro-
portionately with load impedance.
A change in mode of operation occurs in a class B amplifier as the
output passes from one tube to the other in the region of cut-off. This
change-over may cause transient voltages in the amplifier which dis-
tort the output voltage wave form. If the two halves of the trans-
former primary winding are not tightly coupled, primary-to-primary
leakage inductance causes nicks in the output voltage wave, in some-
what the same way as leakage inductance in a rectifier plate trans-
1 For example, "Graphical Harmonic Analysis," by J. A. Hutcheson, Electronics,
9, 16 (January, 1936).
164 ELECTRONIC TRANSFORMERS AND CIRCUITS
former. In a class B amplifier, the change from one tube to the other
is less abrupt than in a rectifier, but in triode amplifiers perceptible
nicks in the voltage wave occur if the ratio of primary-to-primary
leakage reactance to average plate resistance is 4 or more.^
CORE-
Pl/2 -
S/2 -
P2/2-
fx
no leakage flux in
"space between coils
- + B
^
r
-Pl/2
■S/2
-P2/2
Fig. 121. Core-type push-pull balanced windings.
Balanced operation in a push-pull amplifier, i.e., equal plate current
and voltage swing on both sides, is possible only if the tubes are alike
and if transformer winding turns and resistances per side are equal.
Shell-type concentric windings do not fulfill this condition because the
half of the primary nearer to the core tongue has lower resistance than
the other half. Balance is easier to achieve in the core type of arrange-
ment shown in Fig. 121. In class A am-
plifiers close primary-primary coupling
is not essential, and balance may be
attained by arranging part coils as in
Fig. 122.
Because only half of the primary
winding of a class B amplifier carries
current during a half-cycle, the leak-
age flux and therefore the primary-to-
secondary leakage inductance have approximately half the values
with both windings active all the time. With capacitive currents,
both windings are active, at least partially. Transformers with
D > 1.0 have low capacitive currents, low leakage inductance, high
resonance frequency, and extended frequency range, in addition to the
1 See "Quasi Transients in Class B Audio-Frequency Push-Pull Amplifiers," by
A. Pen-Tung Sah, Proc. I.R.E., U, 1522 (November, 1936).
Fro.
122. Shell-type push-pull
balanced windings.
AMPLIFIER TRANSFORMERS
165
load-impedance advantages given in Section 67. At higli frequencies a
class B amplifier transformer presents a circuit to the tubes like that
in Fig. 123. Let Li be leakage inductance between the halves of the
\.z
'
D =
i&-
—
~N
l\\
-^
1.4
^J
\
IIWV
1.0
~
■~^
1.2
\
1
'
,\\
N^
\
if
\,
\
] ' ^
\\
0.8
\
V
\
\
\
1
\ \
\,
\
\
\
\
\\
0.6
N,
«v
N
^
y
\
1
\
EQUIVALENT CIRCUIT \.
\
\
\
^ \
Ja^ ^J^
-^N
as
\
N
\
1
\^
0.4
CL5 1
\
\
\
J
\
~
-~H
i
J
-~'^z
\
N
\
7
\
0.2
T
"2
\
\
\
i>/
\
J
1
1.4
1.2
0.8
0.5
0.4
f/fr
0.8 1.0
2.0
Fig. 123. High-frequency load impedance of class B amplifiers.
primary winding, and L2 between each half of the primary and the
secondary. Li is the inductance of one half of the primary winding,
measured with the other half-primary short-circuited and the sec-
ondary open. L2 is the inductance of one half of the primary winding,
measured with the other half-primary open and the secondary short-
circuited. In Fig. 123, Li = 2Lp and L2 = Lp -\- Lg. Resonant fre-
quency /r is determined by Xc and X^i — 27r/Li. In this figure D =
Xo/R-i at jr-
First one tube delivers power into the equivalent circuit at one end ;
then, during the next half-cycle, this tube is cut off and the other tube
delivers power into the circuit at the other end. Thus the transformer
equivalent impedance Z seen looking into the circuit, first at the end
166 ELECTRONIC TRANSFORMERS AND CIRCUITS
shown and then at the other end, is fed by one of the tubes at all times.
Impedance ratio Z/R2 varies with frequency as in Fig. 123. For some
values of parameter D, impedance falls more rapidly than for class A
amplifiers (Fig. 119), but frequency fr in Fig. 123 is determined by L
and C having approximately half the values of these elements in class
A amplifiers. Hence class B impedance stays flat at higher frequencies,
although response may droop at lower frequencies, than for class A.
Figure 123 is drawn for a ratio of L1/L2 = 1-5, which is a practical
design ratio. Lower ratio I/1/Z/2 results in deeper valleys in the im-
pedance curve; higher L1/L2 is more likely to cause nicks in the volt-
age wave. Good practice consists in designing class B amplifier trans-
formers so that the highest operating frequency is less than /r/2 and
L1/L2 ^ 1.5. Then harmonic distortion at high frequencies should not
exceed 5 per cent.^ Class B modulation transformer impedance is in-
fluenced by circuit elements, so that maintenance of constant imped-
ance over a wide frequency band becomes an overall amplifier problem.
This is discussed further in Chapter 6.
Capacitive currents also cause unbalance at high frequencies, even
with winding arrangements like Figs. 121 and 122. This is evident if
the secondary winding in these figures is grounded at one end; the
efi^ective capacitances to the two primary windings are then unequal.
This problem may be solved by keeping the capacitances small with
liberal spacing, but this practice increases leakage inductance and
cannot be carried very far. Coil mean turn length should be kept as
small as possible by the use of the most suitable core steel. Core-type
designs have smaller mean turns than shell-type. Also, the two
outer coil sections have low capacitance to each other and to the case
if liberal spacing is used, without an increase in leakage inductance.
Flux in the space between the outer sections links all the windings on
one leg and hence is not leakage flux. Consequently, this space is not
part of the term a in equation 33 (p. 76). In push-pull amplifiers the
winding arrangement of Fig. 121 is advantageous because of the low
capacitance between the points of greatest potential difference, A and C.
69. Plate Current Increase. In a lightly loaded amplifier the fre-
quency characteristic stays flat at high frequencies, even with a droop
in load impedance, but the plate current rises in inverse proportion to
the impedance. If the plate current can rise enough to maintain con-
stant output voltage, this plate current rise may be objectionable from
1 See "The Design of Broad-Band Transformers for Linear Electronic Circuits,"
by H. W. Lord, Trans. AIEE, 69, 1005.
AMPLIFIER TRANSFORMERS
167
the standpoint of tube heating or plate supply regulation. Values of
plate current rise calculated on the basis of constant output for low
and high frequencies are shown in Figs. 124 and 125. Many satisfac-
tory audio amplifiers have plate currents which would be excessive at
the extremes of the range if high or low notes were amplified continu-
1
f
T
O
'2
2
U.
O
z
UJ
o
UJ
^ 300
UJ
Q: 200
a:
z>
o
Ul
5 100
-1
--
— 1
0.1
0.2
0.4 0.6 0.8 1.0
X^/ Rg
2.0
4.0
Fio. 124. Rise in plate current due to transformer impedance change at low fre-
quencies.
ously. They are not damaged because these tones are of short dura-
tion.
70. Pentode Amplifiers. Tetrode tubes have an additional grid be-
tween anode and control grid to reduce the grid-to-anode capacitance.
This additional grid is known as the screen grid and is operated at a
positive potential with a-c bypass to reduce the grid-to-anode capaci-
tance. The chief drawback to this type of tube is that the anode volt-
age swing is limited to the difference between the anode voltage and
screen voltage. This disadvantage is overcome by the addition of a
third grid known as the suppressor, which removes this limitation and
allows large anode voltage swings down to the diode line of the tube.
Sometimes the third electrode is connected internally to the cathode.
168
ELECTRONIC TKANSFORMERS AND CIRCUITS
Similar characteristics are obtained with the so-called beam tubes,
which are tetrodes with special screen-grid spacings. Figure 126
shows 6L6 beam tube plate characteristics, with a typical load line of
2,500 ohms. As a single-side amplifier, such a tube is likely to have
large distortion because of the uneven spacing of constant-grid-voltage
lines. Distortion is reduced in a push-pull amplifier, especially for
800
700
o 600
300
200
0.1
^
1
Xl
ll >.
±v. Ho'
= 0.25-
1
T"^ U"
/
Xr
-^ AT FREQ. fr
= Xo AT FREQ. fr
1 1 1 1 1 1
1
D
I
Xl
1
/
/
0=0.4
/
/
//
D = 0,5
/
/
/
V
^
y
/'
/
/
'y
D = 0.75
^
X
r-*
^
y
y'
^
^
y^
D= 1.0
D= 1.41-
D = 2.0
^ -
----:
^
=
^
^
L-;
i
s
!===i
0.2 0.3 0.4 0.6 0.8 1.0
f/fr
2.0
Fig. 125. Rise in plate current due to transformer impedance change at high
frequencies.
high power output. Plate resistance r^, is very high in pentodes and
beam tubes, of the order of 10 times the load resistance.
Pentodes are essentially constant-current devices. The value of
load impedance is thus an indication of the output voltage, at least for
low frequencies. Response of a low-frequency transformer-coupled
pentode amplifier can be taken from Fig. 118.
At high frequencies, leakage inductance of the transformer inter-
venes between the pentode and its load, so that the primary voltage
and secondary or load voltage are not identical. In Fig. 127 the
change of output voltage for a constant grid voltage at high frequencies
is shown. In this figure, the equivalent circuit is a pi-filter, which is
AMPLIFIER TRANSFORMERS
169
T-'
O in O
1 1 I
.?
in
tf>
1
rl
a>
in
1
r
/
1
^>^'
c
)
c
f
(
5
1
$.
1
¥1
'ftl
/'
1
>
•^
'
f
o
+
i
J
"l ,
1
\
k
W
vO
\
\
%
°
+V
\
ol
o
, c
^
\
1
3\
\
\
\ ii
\
\
'^ \
b
A
\j
V
\
if
"^
^
X
■\
^ v\
k>
--;
3
f-5
O
>
<
oof
o
03
f^
O
o
o o
o o
ro CJ
saaadwvmiw aivnd
o
o
170
ELECTRONIC TRANSFORMERS AND CIRCUITS
desirable for pentode transformers, and is approximated when the
transformer ratio is 1:1. Harmonic content of pentodes is high, espe-
cially in single-side amplifiers. Large phase angle and low load im-
pedance produce undesirable distortion. It is best to use values of
X}//R2 greater than 2 in Fig. 118 at the lowest frequency to avoid
distortion.
3.2
3.0
2.8
>_ 2.6
o
^ 24
z>
2.2
LlJ
Z 2.0
1 1.8
* 1,6
|l»j 1.4
S 1.2
o
R,
1 Xl
— f — '
En
F.^X
X
.4: F*.
R2
y
i I " "1 r
'*<; ^ . -]
D=-5- AT FREQ. fr
R2
Xc,
= Xc2=Xl^
Mf,
/
i\
/
O
/
f^
r
5?
V
y
-y
^
--'
,-'
.,
p / \
*~'
■=^
^ —
—
^
^,
D = C
i
^'Ai
--.
'y ^
m
%
0.1
f/fr
1.0
Fig. 127. Pentode frequency response with pi-filter output circuit.
Semiconductor amplifiers known as transistors have emitter, collec-
tor, and base electrodes; these are analogous, respectively, to grid,
plate, and cathode in a vacuum tube. Emitter and collector currents
are of the same order of magnitude in grounded-base transistors, but
collector impedance is much larger than emitter impedance. To match
impedances, transformer coupling is often used between stages of tran-
sistor amplifiers. Junction transistors resemble pentode amplifiers
in having nearly constant collector current over a large range of col-
lector voltage. Hence junction transistor transformer operation closely
resembles that of pentode vacuum-tube transformers, and the fore-
going discussion is generally applicable to both.
71. Calculation of Inductance and Capacitance. Transformer-
coupled amplifier performance is dependent at low frequencies upon
transformer OCL, and at high frequencies upon leakage inductance
AMPLIFIER TRANSFORMERS 171
and winding capacitance. Calculation of these quantities is essential
in design and useful in tests for proper operation. Inductance formulas
are repeated here for convenience, along with capacitance calculations.
3 2N^A
OCL = '■ '— (henrys) (38)
10*
{'■-'{)
where N = turns in winding
Ac = core area in square inches
Ig = total length of air gap in inches
Ic = core length in inches
M = permeability of core (if there is unbalanced direct current
in the winding, this is the incremental permeability).
For concentric shell- or core-type windings the total leakage induct-
ance referred to any winding is
l0.6N^MT(2nc + a)
no X 10
where N = turns in that winding
MT = mean length of turn for whole coil
a = total winding height
b — winding width
c = insulation space
n = number of insulation spaces
= number of primary-secondary interleavings (see Fig. 57,
p. 75).
Winding capacitance is not expressible in terms of a single formula.
The effective value of winding capacitance is almost never measurable,
because it depends upon the voltages at the various points of the wind-
ing. The capacitance current at any point is equal to the voltage
across the capacitance divided by the capacitive reactance. Since
many capacitances occur at different voltages, in even the simplest
transformer, no one general formula can suffice. The major com-
ponents of capacitance are from
1. Turn to turn.
2. Layer to layer.
3. Winding to winding.
4. Windings to core.
172 ELECTRONIC TRANSFORMERS AND CIRCUITS
5. Stray (including terminals, leads, and case) .
6. External capacitors.
7. Vacuum-tube electrode capacitance.
These components have different relative values in different types of
windings. Turn-to-turn capacitance is seldom preponderant because
the capacitances are in series when referred to the whole winding.
Layer-to-layer capacitance may be the major portion in high-voltage
single-section windings, where thick winding insulation keeps the
winding-to-winding and winding-to-core components small. Items 5,
6, and 7 need to be watched carefully lest they spoil otherwise low-
capacitance transformers and circuits.
If a capacitance C with E-^ volts across it is to be referred to some
other voltage E2, the effective value at reference voltage E2 is
C, = C(.BiV-E2^) (70)
By use of equation 70 all capacitances in the transformer may be
referred to the primary or secondary winding; the sum of these capaci-
tances is then the transformer capacitance which is used in the various
formulas and curves of preceding sections.
In an element of winding across which voltage is substantially
uniform throughout, capacitance to a surface beneath is
C = (0.225^ e/0 (MMf) (71)
where A — area of winding element in square inches
€ = dielectric constant of insulation under winding = 3 to 4 for
organic materials
t = thickness in inches of insulation under winding. This in-
cludes wire insulation and space factor.
If the winding element has uniformly varying voltage across it, as in
Fig. 128, the effective capacitance is the sum of all the incremental
WINDING
-c
CORE
/minininliiniin/in
Fig. 128. Transformer winding with
uniform voltage distribution.
AMPLIFIER TRANSFORMERS 173
effective capacitances. This summation is
C^ = \l, (72)
where C = capacitance of winding element as found by equation 71
El = minimum voltage across C
E2 = maximum voltage across C
E = reference voltage for Ce-
If El is zero and £'2 = E, equation 72 becomes
Ce = C/3 (73)
or the capacitance, say, to ground of a single-layer winding with its
low-voltage end grounded is one-third of the measured capacitance of
the winding to ground. Measurement should be made with the wind-
ing ungrounded and both ends short-circuited together, to form one
electrode, and ground to form the other.
In a multilayer winding, Ei is zero at one end of each layer and
E2 = 2E/N1, at the other, where E is the winding voltage and A^'l
is the number of layers. The effective layer-to-layer capacitance of
the whole winding is
Ce = (1 (74)
3iVi\ Nzl
where Ci is the measurable capacitance of one layer to another.
The first and last layers have capacitance to other layers on one side
only, and this is accounted for by the term in parentheses in equa-
tion 74.
Because the turns per layer and volts per layer are greater in wind-
ings with many turns of small wire, such windings have higher effective
capacitance than windings with few turns. In a transformer with
large turns ratio, whether step-up or step-down, this effective capaci-
tance is often the barrier to further increase of turns ratio. With a
given load impedance across the low impedance winding, there is a
maximum effective capacitance (7,„ which can be tolerated for a given
frequency response. If layer and winding capacitances have been
reduced to the lowest practicable figure Ci, the maximum turns ratio
is y/Cm/Ci. Appreciable amounts of capacitance across which large
voltages exist must be eliminated by careful design.
174 ELECTRONIC TRANSFORMERS AND CIRCUITS
Since effective capacitance is greater at higher voltages, in step-down
transformers the capacitance may be regarded as existing mainly across
the primary winding, in step-up transformers across the secondary
winding. The effect of this on frequency response has been discussed
in Section 66.
The input capacitance of a triode amplifier is given by ^
Cinput = Cg-f +(a+ 1)Cg-P (75)
where Cq-f = grid-to-cathode capacitance
Cg-p = grid-to-anode capacitance
a = voltage gain of the stage.
Cg~f and Cg-p are given for many tubes in the tube handbooks. They
can be measured in any tube by means of a capacitance bridge.
72. Amplifier Transformer Design. In amplifiers which operate at a
single frequency, transformers are similar in design to rectifier plate
transformers. Size of core is determined by the required value of OCL.
See Section 65. If the winding carries unbalanced direct current, an
air gap must be provided to keep Bm within the limits discussed in
Section 38 (Chapter 3). Winding resistances are limited by per-
missible loss in output, or in larger units by heating.
If the amplifier operates over a frequency range, the start of the
design is with OCL to insure proper low-frequency performance. After
ample core area and turns have been chosen, attention must be given
to the winding configuration. Leakage inductance and winding capaci-
tance are calculated and, from them, /r and B. If the high-frequency
response does not meet the requirements, measures must be taken to
increase fr or change B to a value nearer optimum. Sometimes these
considerations increase size appreciably.
Below frequency /,, the leakage inductance per turn is constant and
equal to the total coil inductance divided by the number of turns.
Capacitance per turn is constant and may be large because of the close
turn-to-turn spacing. But the LC product per turn is smaller than the
LC product per layer, because the layer effective capacitance is greater.
Therefore the frequency at which the turns become resonant is higher
than that at which the layers become resonant. Likewise, if there is
appreciable coil-to-coil capacitance, the layer resonant frequency is
higher than the coil resonant frequency /,-. If the coil design is such
1 See Principles of Radio Communications, by J. H. Morecroft, John Wiley &
Sons, 2nd ed., New York, 1927, p. 511.
AMPLIFIER TRANSFORMERS
175
that resonance of part of a coil occurs at a lower frequency than fr,
the transformer frequency response is limited by the partial resonance.
This condition is especially undesirable in
wide range designs, but with reasonable
care it can be avoided.
It is helpful in amplifier transformer
design work to use a reactance chart,
especially at the higher frequencies where
resonance frequency fr must be known in
order to determine high-frequency proper-
ties. Several reactance charts have ap-
peared in the literature.^
Two examples of audio transformer de-
sign are given here to illustrate low- and
high-frequency response calculations.
Fig. 129. Input transformer
driving push-pull grids.
Example (a). Input Transformer. To terminate a 500-ohm line and apply
input to push-pull class A grids as in Fig. 129. Primary voltage is 2.0 volts.
Frequency range 100 to 5,000 cycles. Step-up ratio 1:20. No direct current
flows in either primary or secondary winding. Nickel-iron laminations are used.
Refer to Fig. 57 for dimension symbols and winding arrangement.
Ac = 0.5 sq in.
Ic = 4.5 in.
Permeability (initial value) = 5,000.
Coil mean turn 4.5 in.
Window 0.578 in.
Primary 400 turns No. 30 single enamel wire.
Secondary 8,000 turns No. 40 single enamel wire (total).
Primary layers 7; layer paper 0.0015 in.
Secondary layers 44; layer paper 0.0007 in.
Vertical space factor 0.9.
b = 0.75 in.
c = 0.008 in.
3.2 X (8,000)2 X 0.5 X 10^
Secondary OCL
0.0005 -t-
4.5
5,000
Secondary leakage inductance
10.6 X (8,000)2 X 4.5 x (4 X 0.008 -|- 0.578)
= 730 henrys with small-
est possible air gap
(per Table IX, p. 99).
= 540 henrys with aver-
age gap = 0.001 in.
= 0.62 henry
4 X 0.75 X 10''
1 See "Reactance Chart," by H. A. Wheeler, Proc. I.R.E., 38, 1395 (December,
1950) .
176 ELECTRONIC TRANSFORMERS AND CIRCUITS
Capacitances :
^ , , , , 0.225 X 4.5 X 0.75 X 3X0.9
Secondary layer-to-layer = _^-^^^^_^ = 1,700
Ditto referred to whole secondary = ^^ X 1.33 X 0.977 = 51 titii
Primary layer-to-layer, referred to secondaiy < 1
Tube input capacitance = 25
Winding-to-core capacitance = 40
Stray capacitance = 10
Total secondary capacitance = 127 MMf
With the primary winding located at audio ground on the secondary, there
is virtually zero winding-to- winding capacitance. Secondary-to-core
capacitance is based on a coil form >i6 in. thick.
Total secondary load resistance = 500 X (20)^ = 200,000 ohms.
Based on inductance with 0.001-in. gap, Xn = 339,000 ohms and X}f/Ri
= 1.7.
Response is 0.3 db down, in Fig. 108.
Resonance frequency of 0.62 henry and 127 ju^uf is 18,000 cycles and Xc =
70,000 ohms. B = XJRx = 0.35. ///, = 5,000/18,000 = 0.28. Re-
sponse is 0.6 db down at 5,000 cycles (from Fig. 109).
Example (b). Interstage Transformer. In interstage coupling the impedance
level is high, to maintain both high load impedance and high grid excitation in
the following stage. The limit on the secondary side is the highest resistance
which affords grid circuit stability. There is no impedance limit on the primary
side except that imposed by transformer design. Usually a 1 : 1 ratio is about
optimum. A step-down ratio gives less voltage on the following grid. A step-up
ratio reflects the secondary load into the plate circuit as a lower impedance.
This reduces the voltage gain, especially with pentodes which have high plate
resistance compared to load resistance. Under this condition, equation 58
becomes
ep/eg = ixZJrp = g,J^L (76)
or the voltage gain is proportional to load impedance. Interstage transformers
commonly have many turns and high OCL.
Suppose that a transformer is required to connect a 6SK7 tube to a 6L6 oper-
ating class A with 10 volts rms on the grid over a frequency range of 300 to
3,000 cycles. 6L6 grid resistance is 90,000 ohms, which is to be reflected into
the 6SK7 plate circuit as a 90,000-olim load. Hence a 1:1 turns ratio is used.
6SK7 plate current is 10 ma. The same core as in Example (a) is used, except
that here it is made of silicon steel, and the stacking is reduced so that 4 c is
0.32 sq in. Primary and secondary windings are single sections; with the pri-
mary start lead connected to 6SK7 plate and secondary finish lead connected
to 6L6 grid. This leaves adjacent turns in both these windings at zero audio
potential, and effective primary-secondary capacitance is zero.
AMPLIFIER TRANSFORMERS 177
Primary turns = secondary turns = 6,600 turns No. 40 enamel wire.
Primary and secondary layers = 37.
Primary mean turn = 3.3 in.
Secondary mean turn = 4.2 in., Ig = 0.005 in.
0.6 X 6,600 X 0.010
Bdc = Q-^ = 7,750 gauss.
_ 3.49 X 10 X 10'^ _
^'"' ~ 300 X 0.32 X 6,600 ~ ^^ ^''''''-
Bm = 7,805 gauss.
From Fig. 70, ma = 1,100.
_^-. 3.2 X (6,600)2 -^ 0.36 X 10"^ ^^ ,
UCL = — = 55 henrj's.
0-°°^+ 1:100
^ , ^ 10.6 X (6,600)2 X 3_8(-2 x 0.008 + 0.578) , , ,
Leakage L = q yg ^ 10" " " '""'■^^•
Capacitances :
, , , 0.225X3.3X0.75X3 , ,^^ .
r rimary layer-to-layer = .,.,,- = 1,120 M^f.
U.UUlO
a , , , , 0.225X4.2X0.75X3 , ^^^ ,
secondary layer-to-layer = ^tt^tttt = 1,415 j"y"i-
0.0015
Referred to the whole winding, the capacitances are
1,120 X 4 X 0.973
and
37 X 3
1,415 X 4 X 0.973
37 X 3
4:0 nix
50
Primary — core C =44
Secondary — core C =14
Tube capacitances = 17
Stray capacitances = 10
Total capacitance = 175 /xfii
fr = 10 kc. X = 90,000, = 1, and response is 1 db down at 3,000 cycles
(from Fig. 127). Xn/R2 = 6.28 X 300 X (55/90,000) = 1.04. Figure 118
shows a Z/R2 of 0.72; therefore the response is 3 db down at 300 cycles.
6. AMPLIFIER CIRCUITS
Amplifier applications may require control of hum, distortion, or
frequency response beyond the limits of practical transformer design.
Sometimes the additional performance is obtained by designing extra
large transformers; this is usually an expensive procedure. Sometimes
extra features can be incorporated into the transformers without
marked increase in size. At other times additional circuits are used
either preceding or in conjunction with the amplifier.
In this chapter several devices for obtaining special performance
are considered. Transformer and amplifier design both are affected
by them to a marked degree.
73. Inverse Feedback. If part of the output of an amplifier is fed
back to the input in such a way as to oppose it, the ripple, distortion,
and frequency response deviations in output are reduced. The ampli-
fier gain is reduced also, but with the availability of high-gain tubes
an extra stage or two compensates for the reduction in gain caused by
inverse feedback, and the improvement in performance usually justifies
it. In the amplifier of Fig. 130, a network is shown connected to out-
t
<l
T
E2
AMPLIFIER
a
. 1
Eo
1
FEEDBACK
NETWORK
IB
4r
Fig. 130. Voltage feedback.
put voltage Eo ; part of this output is fed back so that the input to the
amplifier is
E2 = El - fiE„ (77)
Here /J is the portion of Eq which is fed back. If a is the voltage ampli-
178
AMPLIFIER CIRCUITS 179
fication of the amplifier and Er and Eh are the ripple and harmonic
distortion in the output without feedback, and a!, E'r, and E'h are the
same properties with feedback, the following equations hold, if a, Er,
and Eff are assumed to be independent :
Without feedback,
^0. = aE2 + Eb + Eh (78)
With feedback,
Eo = a'Ei + E'r + E'h (79)
From these equations it can be shown that
a 1
1 + afi
(80)
Er Er
E'r = -— ^ « -^ (81)
Eh Eh
E'h = — ^ - — (82)
l + afi al3
With high-gain amplifiers and large amounts of feedback, the out-
put ripple and harmonic distortion can be made astonishingly small.
Likewise the frequency response can be made flat, even with mediocre
transformers. Inverse feedback is not used in class C amplifiers, be-
cause the output and input are not linearly related.
Incidental effects in the amplifier, like distributed capacitance
and leakage inductance, have to be carefully matched in the inverse
feedback network so that the phase shift around the loop does not be-
come too large. If it reaches 180°, feedback is regenerative, so that
the amplifier may become an oscillator with a frequency determined
by the circuit constants. Nyquist has shown ^ that oscillation does not
take place so long as the gain X feedback product ac^ is less than unity
at the frequencies for which the phase shift is 180°. In a plot of ap
made on the complex plane, the requirement for stability is that the
curve of afi must not enclose the point 1, 0, with the sign of 13 con-
sidered opposite to that of a. Both gain a and feedback /? are ratios
of voltages. Therefore, both may be expressed in decibels and both are
complex quantities at some frequencies. Proper care in application is
required so that amplifiers with 180° or more phase shift do not oscil-
iSee "Regeneration Theory," by H. Nyquist, Bell System Tech. J., 11, 126
(January, 1932).
180
ELECTRONIC TRANSFORMERS AND CIRCUITS
late at some frequency outside the pass band. If it is desired to cor-
rect for distortion or hum over a frequency range of 30 to 10,000
cycles, the amplifier should have low phase shift over a much wider
range, say 10 to 30,000 cycles. In the frequency intervals of 10 to
30 cycles and 10,000 to 30,000 cycles, both the amplification and the
feedback should taper off gradually to prevent oscillations.
Low phase-shift amplifiers benefit most from inverse feedback.
Feedback in such amplifiers reduces size or improves performance, in-
cluding phase shift. Transformer phase shift, therefore, is a vital
(/>
uj
,
• ■
o
'
ij:^
o
.^^
y'
; ?^R
X
^J>-\^V(
■f\- R|
\ \
R2 Eg
S60
*
y
^^
1' 3
J
■^
<i
Q.
.-iS^"
CJ
—
— '
Ep LEADS E
1
liJ
1 1 III
1
11
Fig. 131. Transformer-coupled amplifier low-frequency phase shift.
property in feedback amplifiers and may take precedence over fre-
quency response in some instances.
Phase shift at low and high frequencies is shown in Figs. 131 and
132 for transformer-coupled stages. At high frequencies, 180° phase
shift is possible whereas at low frequencies but 90° is possible. In a
resistance-coupled amplifier, only 90° phase shift occurs at either low
or high frequencies. Partly for this reason, partly because less capaci-
tance is incidental to resistors than to transformers and good response
is maintained up to higher frequencies, it is in resistance-coupled am-
plifiers that inverse feedback is generally employed. But if the dis-
tortion of a final stage is to be reduced, transformer coupling is in-
volved. It is preferable to derive the feedback voltage from the pri-
mary side of the output transformer. This is equivalent to tapping
between J?i and Xn in Fig. 132, where the phase shift is much less.
The transformer must still present a fairly high impedance load to the
output tube throughout the marginal frequency intervals to permit
gradual decrease of both amplification and feedback.
AMPLIFIER CIRCUITS
181
Current feedback is effected in the circuit of Fig. 133 by removing
capacitor C. This introduces degeneration in the cathode resistor
circuit, which accomplishes the same thing as the bucking action of
20
40
60
80
J 100 -
120 -
140^
1 1 1 1
XfXcAT FREQ fr
— '
—
-.
B=^ATFREOfr
R2=2R|
B -0 7 TO 1 41
—
k
\
F
1
Xl
1 — '—\ ^
YYV> r
*
s
- E| ^
1 1
-Xc R2 E
1 1
\
\
\
1 1 1 1 t
\
Ej LAGS E
\
1
\
III
t/tr
.03 0.1
FREQUENCY
Fig. 132. Transformer-coupled amplifier high-frequency phase shift.
voltage feedback. It is less affected by phase shift and consequently
is used with transformer-coupled amplifiers.
74. Cathode Follower, The circuit of Fig. 134 is known as a cathode
follower. Here the anode is connected to the high-voltage supply Eg
without any intervening impedance, so that for alternating currents it
Fig. 133. Cathode bias.
Fig. 134. Cathode follower.
is essentially grounded. Grid voltage eg must be great enough to
include the output Eq in addition to the normal grid-to-cathode volt-
age at E^ia. However, the grid power is still the same as it would be
182 ELECTEONIC TRANSFORMERS AND CIRCUITS
were the cathode grounded. This circuit is used when the output
impedance Z^ is variable or of low power factor so that normally it
would be diflBcult to produce in it full output from the tube. The cir-
cuit has a low internal effective impedance as far as the output is con-
cerned. It is approximately equal to the normal plate resistance rp
divided by the amplification factor /» of the tube. This is equivalent
to saying that the effective internal impedance is approximately the
reciprocal of the mutual conductance Qm ^ for class A or B amplifiers.
Cathode followers have been used to drive grids of class B modulator
tubes, which are highly variable loads. The circuit produces nearly
constant output voltage but at the expense of increased grid swing.
If the tube feeds a low impedance load, output may be increased by
coupling the load through a transformer. Frequency response in
cathode output transformers is usually flat over a very wide range
because of the low effective source impedance.
75. Wave Filter Principles. In preceding sections dealing with
transformer frequency response, means for extending frequency range
have been considered. In broadcast transmitters this is a vital prob-
lem. But in other applications amplifiers are used over a limited fre-
quency range. It is sometimes desirable to allow certain frequencies
which are present to pass through the amplifier at full amplitude but
to suppress as nearly as possible certain other frequencies. The means
usually employed to accomplish this result is a wave filter. In any
such filter, the band of frequencies which it is desired to transmit is
known as the transmission band, and that which it is desired to sup-
press is known as the attenuation band. At some frequency, known
as the cut-off frequency, the filter starts to attenuate. Transition be-
tween attenuation and transmission bands may be gradual or sharp;
the filter is said to have gradual or sharp cut-off accordingly. When
a filter is used in conjunction with a transformer-coupled amplifier,
the frequency response of both filter and amplifier must be coordi-
nated. In a later section it will be shown how transformer response
may be improved through the use of wave filter principles.
To avoid introducing losses and attenuation in the transmission
bands, reactances as nearly pure as practicable are used in the elements
of a wave filter. For example, in the "low-pass" filter T-section of
Fig. 135, the inductance arms shown as L/2 and the capacitance C are
made with losses as low as possible. Capacitors ordinarily used in
filters have low losses, but it is a problem to make inductors which
1 See "Feedback," by E. K. Sandeman, Wireless Engr., 17, 350 (August, 1940).
AMPLIFIER CIRCUITS
183
have low losses. Values of inductor Q ranging from 10 to 200 are
common, depending upon the value of inductance and the frequency
of transmission. Therefore in wave filters the loss is mostly in the
L/2
L/2
T SECTION TT section
Fig. 135. Low-pass filter sections.
inductors. It can be shown ^ that for pure reactance arms the values
of reactance are such that in the transmission band
0>^>
4Z2
■1
(83)
where Zi is the reactance of the series arm and Z2 is the reactance of
the shunt arm. In the T-section of Fig. 135, Zi is 27r/[ (L/2) + (L/2) ]
Ob
60
55
50
945
o
uj 40
tf)
Q: 35
UJ
d.
o
P 25
<
i 20
UJ
t 15
<
10
5
100
Zi
4Z,
5000
Db
90
85
80
75
70
65
60
55
50
Fig. 136. Attenuation per section with pure reactance arms.
= 27r/L and Z^ is the reactance of C. The attenuation for sections of
filter like Fig. 135 is shown in Fig. 136, for a pure reactance network
1 See Transmission Networks and Wave Filters, by T. E. Shea, D. Van Nostrand
Co., New Yorli, 1929, p. 187.
184 ELECTRONIC TRANSFORMERS AND CIRCUITS
starting at the cut-off frequency. The attenuation is shown in decibels,
and the abscissas are one-fourth of the ratio of series to shunt reactance
in a full section.
It is important, in the transmission band, to terminate the sections
of filter in the proper impedance. Like a transmission line, a wave
filter will deliver its full energy only into an impedance which is equal
to its characteristic impedance. Many wave filters are composed of
several sections which simulate transmission lines. A properly con-
structed filter exhibits the same impedance at either end when termi-
nated at the opposite end with an impedance equal to its characteristic
impedance. The impedance seen at any one point in the filter is
called its image impedance; it will be the same in either direction pro-
vided that the source and terminating impedances are equal. In gen-
eral, however, the image impedance will not be the same for all points
in the filter. For example, the impedance looking into the left or T-sec-
tion of Fig. 135 (if it is assumed to be terminated properly) will not be
the same as that seen across the capacitor C. For that reason, another
half-series arm is added between C and the termination to keep equal
input and output impedances. The terminating sections at both the
sending and receiving ends of a filter network are half-sections, whereas
the intermediate sections are full sections. A full T-section of the
type shown in Fig. 135 includes an inductance L equal to L/2 -f- L/2.
The image impedance seen at the input terminals of the T-section of
Fig. 135 is known as the mid-series impedance, and that seen across
capacitor C is known as the mid-shunt impedance.
Likewise, in the pi-section shown at the right in Fig. 135, the mid-
shunt image impedance is seen at the input or output terminals. The
mid-series impedance is seen at a point in the middle of coil L. This
section terminates properly in its characteristic impedance at either
end. Note that adjacent sections have C/2 for the shunt arm, so that
a full section would again be composed of a capacitor C and an induct-
ance L. The choice of T- or pi-sections is determined by convenience
in termination, or by the kind of image impedance variation with fre-
quency that is desired.
If these precautions are not observed, wave reflections are likely to
cause a loss of power transfer in the transmission band.
76. Limitations of Wave Filters. Several factors modify the per-
formance of wave filters, shown in Fig. 136, especially in the cut-off
region. One is the reflection due to mismatch of the characteristic
AMPLIFIER CIRCUITS
185
impedance.^ The load resistor is usually of constant value, whereas
the image impedance changes to zero or infinity at cut-off for lossless
filters. The resulting reflections cause rounding of the attenuation
curve in the cut-off region instead of the sharp cut-off of Fig. 136.
Another cause of gradual slope at cut-off is the Q of the filter chokes,
or ratio of reactance to resistance. Figure 137 gives the attenuation
db
10
/
/
V
'-I '-I
2 2
A
>H M^
1
p ^
r
^
f^
/^
i
Q.
«-^
YA
f
.
—
—
^
_—
^
^y
'
■ — "!
;
[ -
Q^
0.8
0.9 1.0
f/fc=FOR LOW PASS FILTER
fc/f=FOR HIGH PASS FILTER
I.I
Fig. 137. Insertion loss near cut-off of a constant-i? filter section.
at cut-off in terms of Q for a section of the so-called constant-^ filter
(e.g., Fig. 135).
Still another cause of the gradual slope of cut-off is the practice of
inserting a resistor to simulate the source impedance in attenuation
tests. In typical cases the source and terminating resistances are equal.
The correct prediction of filter response near cut-off requires a good
deal of care. It cannot be taken directly from the usual attenuation
charts.
Phase shift is nearly linear with frequency up to approximately 50
1 See "An Analysis of Constant-if Low- and High-Pass Filters," by O. S. Meixell,
RCA Rev., 5, 337 (January, 1941) ; also "Single-Section m-Derived Filters," by
C. W. Miller, Wireless Engr., 21, 4 (January, 1944).
186 ELECTRONIC TRANSFORMERS AND CIRCUITS
per cent of cut-off frequency for constant-X filters in the transmission
band. This fact is important in connection with networks used for the
transmission of steep wave fronts, as in video amplifiers. It is proved
in books on network theory ^ that, when a non-sinusoidal voltage
wave is applied to the input of a network, it appears at the output
without distortion of its original shape if the phase shift of the net-
work is proportional to frequency and if the amplitude response is
flat for all frequencies. In no actual network are these conditions
fulfilled completely, but the closer a network approximates them the
smaller the distortion it causes in irregular wave forms. Linearity
of phase shift is usually more essential to good wave form than flatness
of response. For this reason, when a non-sinusoidal wave passes
through a filter, distortion is minimized if the major frequency com-
ponents of the wave all lie in the linear region of the phase shift
curve. Considerable judgment must be exercised in the choice of
cut-off frequencies. Higher-order harmonics are usually of smaller
amplitude, and the natural tendency is to include too few of them in
the pass band; then the output wave form is a poor reproduction of
the input.
In band-pass filters, the effects just noticed are present, with the
additional complication of band width. The filter designer must choose
a band width of transmission such that high attenuation is afforded
at unwanted frequencies and low attenuation at desired frequencies.
This is often not a simple choice. For a given frequency separation
from the mid-frequency, attenuation decreases as the filter band width
is made wider. Impedance variation is much less with a wider band
width. Therefore, choosing a narrow band width attenuates fre-
quencies in the transmission band because of reflections.
77. Artificial Lines. Sometimes a certain amount of time delay must
be interposed between one circuit and another. Or, if the length of a
transmission line is not an exact multiple of 90°, some means must
be found to increase its length to the next higher multiple of 90°. For
either of these purposes, artificial lines are used. They may operate
at a single frequency or over a range of frequencies. They may be
tapped for adjustment to suit any frequency in a given range, so that
impedance and line length are correct. The configuration may be
either T or w, high- or low-pass. Figure 138 shows these four
combinations for any electrical length 6 of line section in degrees. It
1 See Communication Networks, by E. A. Guillemin, John Wiley & Sons, New
York, 1935, Vol. II, p. 474.
AMPLIFIER CIRCUITS
187
is assumed in this figure that the line operates at a single frequency
and is terminated in a pure resistance equal in value to the line charac-
teristic impedance Zq. Figure 139 is the vector diagram for a leading
Xxc Xx
Xo Xc
Hl-HH>
Xxo
Xl.
Xl = ZoSIN9
Zo
TAN e.
2
Xl-ZoTAn|
Zo
. Zq
TANe.
2
Xl =
Zq
SJNe
Xc"
SIN e
SHIFT 1
Xc-ZoSiNe
LEADING
Xc= ZoTAn|-
LEAOINC
Zq-characteristic impedance of line
e -electrical length of line
Fig. 138. Artificial line relations.
phase shift pi-section line of 90° electrical length. Proportions of L
and C are somewhat different in these line sections than in wave
filters.
It — *■ Xc
Fig. 139. Vector diagram for 90-degree line length.
To obtain approximately constant time delay over a range of fre-
quencies, several constant-X low-pass filter sections may be used, each
having a cut-off frequency high enough so that the phase shift is pro-
188 ELECTRONIC TRANSFORMERS AND CIRCUITS
portional to frequency. The time delay per section is then Q/l-w] at
any frequency in the range, and B = 2Trj\/LC, where B is the phase
shift in radians, L is the inductance per section, and C is the capaci-
tance per section. In Fig. 139, Er = Eg. If the section were termi-
nated in impedance higher than Zq, Er > Eg. The line section is then
a kind of transformer, although the ratio Er/Es varies with frequency.
Ninety-degree line sections are often used at high frequencies to obtain
transformation of voltage.
FREQ. IN CYCLES FOR .014" THICKNESS -
1 1 1 I I
MULTIPLY CORE LOSS * 4T 60 CY
AND 10,000 GAUSS BY CORE
LOSS FACTOR
* FOR SILICON STEEL, 0.6 W/LB
FOR NI-FE ALLOY 0.25 W./LB.
.FOR GRAIN ORIENTED STEEL 035W./IB,
APPROXIMATE CORE LOSS FACTOR
Fig. 140. Core loss in laminations 0.014 and 0.005 in. thick.
78. Filter Inductor Design. In Sections 75 and 76, it was pointed
out that inductors for wave filters must have Q great enough to pro-
vide low attenuation in the pass band. In design, attention must be
given as much to Q as to inductance.
Low-loss core material is essential for high Q. Nickel-iron alloys
are widely used; the lamination thickness depends on frequency. At
frequencies up to 400 cycles, 0.014-in.-thick laminations are used, and
at frequencies higher than 400 cycles, 0.005 in. thick. This is an ap-
proximate practical guide. Figure 140 shows how loss varies with
thickness, frequency, and flux density. At frequencies higher than
1,000 cycles, flux density must be quite small for low core loss. In
the majority of audio applications, low flux density conditions prevail.
Under such conditions, core loss is largely eddy-current loss and may
be treated as a linear resistance.
AMPLIFIER CIRCUITS 189
Core gaps are used in filter reactors to obtain better Q. For any
core, inductance per turn, and frequency, there is a maximum value
of Q. The reason for this is that the
a-c resistance is composed of at least
two elements: the winding resistance
and the equivalent core loss. In pre-
vious chapters the core loss has been
regarded as an equivalent resistance
across a winding. But it can also be Rsh
regarded as an equivalent resistance x "ser
in series with the winding. Figure 141 — /nrvw> 1 I
shows this equivalence, which may be Fig. 141. Shunt and series equiva-
stated: lent core-loss resistance.
Rser + jX
For values of Q > 5,
jR^hX
iJsh + jX
X^
i?sh « -- (84)
■H/ser
where Rsh = equivalent shunt resistance
^ser = equivalent series resistance
X = winding reactance = 27r/L.
The equivalence depends upon frequency. The formula for large Q may
then be changed to
Q = ^ = ^ (85)
X 2tjL
or Q is proportional to shunt resistance, the winding resistance being
neglected. Thus Q can be increased by lowering L, and L is lowered
by increasing core gap, but there are limits on the increase of Q that
can be obtained in this way.
First, the winding resistance is not negligible. With small gaps,
maximum Q is obtained when winding resistance and equivalent series
core-loss resistance are equal. For a given air gap there is a certain
frequency /„ at which this maximum Q holds. At higher and lower
frequencies, the manner in which Q falls below the maximum is found
as follows: Let R^ be the coil winding resistance. Then
X
Q =
190
ELECTRONIC TRANSFORMERS AND CIRCUITS
If for Rser we substitute the value obtained from equation 84, we have,
approximately,
X 1
Q = — = T^ (86)
Rc +
X^
Rsh
Re X
X i?sh
Equation 86 therefore gives the relation of Q to frequency. When it
is plotted on log-log coordinates with frequency as the independent
variable,^ it is symmetrical about the frequency fm for which Q is a
100
50
10
■
W''~
;^*^
^5'q
-.^^ -
>
9j
i /
V,
>:n
^ ■•>
— c
— y
y(/
4
^ S
e;:'^
:^^.
-^
^
i 1 >>1 i 1 nj
10
50 100 500 1000 5000 10000
FREQUENCY IN CYCLES PER SECOND
Fig. 142. Frequency variation of Q for an iron-core coil with air gaps.
maximum. If the core gap is changed, frequency /« changes. Figure
142 shows how the Q of a small inductor varies with frequency for
several values of air gap in the core. All these curves have the same
shape, a fact which suggests the use
of a template for interpolating such
curves.
Another phenomenon that limits Q
is the ^vx fringing at the core gap, the
influence of which on inductance was
discussed in Chapter 3. As the air
gap increases, the flux across it fringes
more and more, like that shown in
Fig. 143, and L ceases to be inversely
proportional to the gap. Some of the
fringing flux strikes the core perpen-
dicular to the laminations and sets up
eddy currents which cause additional loss. Accurate prediction of gap
loss depends on the amount of fringing flux. For laminated cores it
can be estimated from
iSee "How Good Is an Iron-Cored Coil?" by P. K. McElroy and R. F. Field,
General Radio Experimenter, XVI (March, 1942). This article also discusses
choke design from the standpoint of simihtude.
GAP
Fig. 143.
Magnetic flux fringe at
core gap.
AMPLIFIER CIRCUITS 191
Wg = Glgd/xteVfBj watts (87)
where G = a constant (17 X 10^^° for silicon steel)
d = lamination tongue width in inches
fi — permeability
te = lamination tks. in inches
/ = frequency in cycles
Bm = peak core induction in gauss
Ig = gap length in inches.
In Section 33, Chapter 3, it was shown that under certain conditions
maximum transformer rating for a given size is obtained when core
and winding losses are equal. The same would be true for inductors
with zero core gap. Similarly it may be shown that, if the core gap is
large enough to cause appreciable gap loss, maximum Q is obtained
with core, winding, and gap losses equal. In a given design, if this
triple equality does not result in the required Q, size must be in-
creased. Losses may be compared by finding either the equivalent
series resistances or the equivalent shunt resistances.
79. Powdered Iron Cores. As frequency increases above a few
thousand cycles, gap loss becomes predominantly large. At such fre-
quencies, cores of powdered iron are preferable for large Q. Powdered-
iron cores are made from several grades of iron and nickel-iron alloys.
Proportions of insulating bond and iron powder are varied to obtain
permeabilities ranging from 10 to 125. Permeability in such cores is
only apparent; it is far less than the inherent permeability of the iron
used because of the many small gaps throughout the core structure.
Finely divided iron has low eddy-current loss and virtually zero gap
loss. Equation 85 indicates how Q varies with frequency ; that is, low-
permeability cores should be used to reduce inductance and maintain
large Q at high frequencies. At frequencies higher than audio, coil
eddy-current losses make stranded wire necessary. This is discussed
further in Chapter 7.
One of the problems of filter design is the maintenance of cut-off
and attenuation frequencies under conditions of varying temperature.
This may be so important as to dictate the choice of core material.
Powdered cores are available which have very low temperature coeffi-
cients. Usually these cores have less than the maximum Q for a given
kind of iron powder. With low-temperature-coefficient cores, attention
also must be paid to filter capacitors in order to obtain the requisite
overall frequency stability.
192 ELECTRONIC TRANSFORMERS AND CIRCUITS
Table XIV. Shapes of Powdehed-Iron Cores
Core Shape Use
Shell Low-voltage r-f transformers and inductors.
Cup Adjustable low and medium r-f inductors.
Slug Adjustable r-f inductors. Also used to adjust cup-core inductance.
Toroid Audio and low r-f inductors.
C High- voltage audio and r-f transformers and inductors.
Powdered cores are made in several forms. Table XIV indicates the
main areas of usefulness of such forms. Figure 144 illustrates the core
shapes in this table. A study of available molds and materials is
worthy of the designer's time.
Fig. 144. Powdered iron-core shapes.
80. Modulation Transformers. Wave-filter principles are applied in
many circuits other than filters. An example is the design of modula-
tion transformers for high-level amplitude-modulated radio transmit-
ters. Some of these transformers are large. In plate-modulated power
amplifiers, the modulator power required to produce 100 per cent
modulation is half of the power amplifier input. Improved audio
quality and reduction in size of components are achieved through the
use of what may be called the pi-filter method.
For low modulation frequencies, this method may be illustrated by
means of the circuit diagram of Fig. 145(a). The modulator usually
is a class B amplifier. Output transformer OCL, coupling capacitor,
and modulation reactor combine to form a pi-section high-pass filter,
Fig. 145(b). The elements are proportioned for characteristic imped-
ance equal to the equivalent plate input resistance Eb/Ib of the modu-
lated power amplifier.
Formerly these components were made as large as was considered
practical. Transformer secondary and reactor reactances were each
AMPLIFIER CIRCUITS
193
3 to 4 times the power amplifier plate input resistance, and the
coupling capacitor reactance was a fraction of this resistance, at the
lowest modulation frequency. Advantages of the pi-filter are a reduc-
MODULATION
TRANSFORMER
MODULATION
INDUCTOR
TO R.F. POWER
AMPLIFIER
■^h
Xl= TRANSF. sec. OPEN CIRCUIT REACTANCE
Xl= REACTANCE OF MODULATION INDUCTOR
Xt;= REACTANCE OF COUPLING CAPACITOR
R = P.A. INPUT RESISTANCE = 4^
lb
NOTE: BYPASS ON Et AS€UMED INFINITELY LARGE
(6)
Fig. 145. (o) Circuit diagram of anode modulation system; (b) equivalent pi-
filter modulator tube load.
tion of the two inductive reactances to 1.41 times the terminating
load resistance, and increase in capacitive reactance to the same value,
at a low frequency /i, which is 1.41 times cut-off frequency of the filter.
Down to /i the filter maintains a tube load of almost 100 per cent
194 ELECTRONIC TRANSFORMERS AND CIRCUITS
power factor, although the ohmic value rises to 190 per cent of the
terminating load resistance at /i. The voltage required for constant
output rises to 138 per cent of normal. Partly compensating for this
defect is the tendency of class B amplifier tubes to deliver higher volt-
ages with higher tube load impedances. Thus, in a certain radio trans-
mitter, the type 805 modulator tubes deliver 1,035 peak volts per side
into a normal tube load of 1,860 ohms. At 30 cycles, the lowest audio
frequency, the load impedance rises to 3,600 ohms, and the voltage re-
quired for full output is 1,440 volts. Plotting a 3,600-ohm load line on
the tube curves shows that 1,275 volts will be delivered at 30 cycles,
which is 1 db down from normal.
To obtain the same frequency response with the older "brute force"
method, at least twice the values of transformer and reactor inductance
and much more coupling capacitance would have been necessary. The
voltage across the capacitor increases as the capacitance decreases,
but surge voltages often exist across the coupling capacitor in service.
The voltage rating was formerly determined more by these surges
than by normal voltage. With the pi-filter method, the normal
voltage at low frequencies cannot be greatly exceeded in service, owing
to the limitation in voltage output of the tubes. Hence, the reduction
in coupling capacitance is a real one and is offset very little by in-
creases in voltage rating.
These points are made clearer by reference to Fig. 146. Phase shift
between transformer and load voltages is 90° at /i. At cut-off ( = 70
per cent of /i) the tube voltage for constant output is 224 per cent of
the load voltage. A tube would not deliver so much voltage with this
type of load, especially when the power factor is so low. The corre-
sponding capacitor voltage at cut-off frequency is 284 per cent of Er]
it would not be delivered either, for the same reason. The useful fre-
quency range is higher than /i.
Another advantage of the pi-filter over the older method is the high
power factor load down to frequency /i. The maximum tube load
phase angle above /i is 8°, and at frequencies above 3/i the phase angle
is zero. At 3/i the tube load impedance is equal to R. Hence the tube
works into a unity power-factor load of constant value at frequencies
above 3/i. For the same size of inductive components, the "brute
force" system would have been very much poorer. If these elements
had been made twice as large in order to give the same frequency re-
sponse at 30 cycles, the load phase angle at 30 cycles would be 35° and
the load impedance 80 per cent of R. Hence the possibilities of low-fre-
quency distortion and low efficiency are reduced by the pi-filter method.
AMPLIFIER CIRCUITS
195
1 1
1
R = RESISTANCE OF LOAD
Es = TRANSFORMER SECONDARY VOLTAGE
Zs = TUBE LOAD IMPEDANCE (REFO. TO SEC
IsX(;=VOLTAGE ACROSS COUPLING CAPACITO
e 'PHASE SHIFT BETWEEN Ej a E„
a
o
c
1
<1
o
03
<fi
H
o
O
UJ
1
4
UJ
U
o
z
<
/
!:
O
>
in
<
I
0.
/
UJ
«•
/
/
/
/
^ 1
/
/
/
(
/
/
f
/
/
/
/
«
M
/
/
<
J
w
^
;^
/
/
i
o
0>
M
>
^
111
X
/
/
\
II
in
N
^
^
^
^
{>■■■
^
^
^
)
^^^
,^
^.
,-^
v>
UJ
/
.
'
r ^
-^
3 dO % Nl 'X^I-
"3 dO % Nl *3-
1
1
^
o -
o -^
O u-
o
3
o
I
o
o
o
o
o
u>
o
o
T
9Niav3~i s33aQ3a e
9NI99WT S33a93a-<|)
196 ELECTRONIC TRANSFORMERS AND CIRCUITS
In very high fidelity modulators the lowest frequency is 2 to 3
times /i to reduce phase shift in connection with the inverse feedback.
The pi-filter just discussed conforms with the usual high-pass filter
in regard to values of L and C. These elements can be propor-
tioned on another basis such that the ohmic values of all the reactive
elements are made equal to the load resistance at frequency /i. These
values are those of a 90° artificial line at frequency /i. They give a
unity power-factor tube load equal to the modulated power amplifier
plate input resistance at frequency /i, and thus represent 41 per cent
increase in capacitor and 41 per cent reduction in transformer and
reactor size. The disadvantages of the artificial line are that the tube
load impedance drops to a minimum of 74 per cent B at the frequency
1.5/i, the maximum tube load phase angle is 16°, and it persists up to
frequencies much higher than 3/i, which was the frequency where the
response of the pi-filter became virtually perfect. This appreciable
phase angle spread over a portion of the audio frequency range,
together with the lower load impedance, causes distortion. The arti-
ficial line basis of design is used where larger amounts of distortion
are not objectionable, or in fixed-frequency modulators.
No matter what method of design is used, it is important that the
modulator be loaded properly. If the power amplifier input should be
interrupted while the modulator is fully excited, the voltages on the
various elements are likely to rise to dangerous values, because the
load impedance becomes high and causes indefinitely high voltages
in the positive Bb direction. The transformer and reactor in high-
voltage modulators are equipped with spark gaps like those in Fig. 147
to prevent insulation breakdown due to accidental removal of load.
The pi-filter or artificial line method of design can be applied also at
the higher modulation frequencies. Figure 148 shows how the usual
properties of winding capacitance and leakage inductance are arranged
to give a low-pass filter circuit. Preferably the internal capacitance
and inductance should be so low that external L and C can be added
to the transformer terminals to produce the required proportions at
the highest modulator frequency.^ Figure 149 shows how both high-
and low-pass pi-filter performance can be combined in a modulator to
obtain wide-range high-fidelity performance. Although these methods
apply chiefly to modulation transformers, they may be used in the
design of loaded interstage transformers.
1 See "An Analysis of Distortion in Class B Audio Amplifiers," by True McLean,
Proc. I.R.E., U, 487 (March, 1936) ; also see Section 81.
AMPLIFIER CIRCUITS
197
',""■»--
:-^.-
%..
,^»> ;-
«'
"?'•''.
* ^ ^
j ' '' * ^ * '
1
Fig. 147. Broadcast station modulation transformer.
LEAKAGE
INDUCTANCE
LOAD
RESISTANCE
C|= PRIMARY CAPACITANCE, INCLUDING
TUBE AND WIRING CAPACITANCE.
C2= SECONDARY CAPACITANCE, INCLUDING
BY-PASS AND LOAD CAPACITANCE.
Fig. 148. Equivalent transformer diagram at high audio frequencies.
198
ELECTRONIC TRANSFORMERS AND CIRCUITS
In high power modulation transformers, it is necessary to make the
core larger in order to reduce the number of turns and obtain good
high-frequency response. But, as the core becomes larger, so do the
leakage inductance and winding capacitance. The core must grow
very large to overcome this difficulty, and high power audio trans-
formers are much larger than commercial power transformers of the
same rating. One advantage of grain-oriented steel is that it permits
this process to be reversed. High induction at low frequencies means
Ko»5-
pl.8—
—
~
1
tlo S
Zs/R/ /
540— lod^
\
/
/ /
<30 75
-^ 20—50—
'■^\^
Zs/R
/ J
^■v.^^
Q
/
0/
— —
^ ^
^
^
j
f-i.o—
■■
.
-~-^
J
^^
\
^° ^^T
^
\
r rn y
10
100 1,000
FREQUENCY-CYCLES PER SECOND
40
KG
e • PHASE SHIFT BETWEEN TRANSFORMER PRIMARY VOLTAGE AND LOAD VOLTAGE
• PHASE ANGLE BETWEEN TRANSFORMER PRIMARY VOLTAGE AND CURRENT (TUBE LOAD)
Fig. 149. High- and low-pass pi-filter performance in a modulator.
a smaller core area, smaller mean turn, and better high-frequency per-
formance, or, for the same high-frequency performance, more turns
and a still smaller transformer. Weight savings of 60 to 75 per cent
have been made in this way.
Advantages of the pi-filter and artificial line methods are realized in
transformers for 30 to 10,000 cycles ; some advantage can be gained in
the 100- to 5,000-cycle range, but not below 100 watts. Below this size,
or for a narrower frequency range, the transformer becomes so small
that combination with the modulation reactor into one unit is feasible
and economical. The secondary current wave shape is like the first
wave of Table I (p. 16) . In such a transformer the core gap must be
large enough to prevent saturation by the power amplifier plate
current.
AMPLIFIER CIRCUITS
199
81. Driver Tansformers. Requirements for class B modulator
driver transformers are unusually difficult to satisfy. The transformer
load is non-linear, for grid current is far from sinusoidal. Although
the average load is low, the driver tube must deliver instantaneous
current peaks ; otherwise distortion will appear in the modulator audio
MODULATOR
TUBES
Fig. 150. Cathode follower driver circuit.
output and therefore in the r-f envelope. The grid current peaks con-
tain harmonic currents of higher order, and to insure their appearance
in the modulator grid current an extension of the driver transformer
frequency range at both ends is required: on the high-frequency end
because of the decreased leakage inductance necessary to allow the
higher currents to flow, and on the low-frequency end to prevent
transformer magnetizing current, itself non-linear, from loading the
driver tube so that it does not deliver the peak grid power. If the
driver tube is a pentode or beam tube, it is usually loaded with resist-
ance to minimize current variations. Driver transformers are usually
step-down because the grid potentials are relatively low.
200 ELECTEONIC TRANSFORMERS AND CIRCUITS
These conditions require transformers of exceptionally large size.
For low (1 to 2 per cent) overall harmonic distortion, driver trans-
former design becomes impractical, and it is advantageous to dispense
with driver transformers entirely. This is accomplished by the cathode
follower circuit (Fig. 150), which for a push-pull amplifier takes the
form of a symmetrical pi-filter. The two input chokes connect the
driver tube cathodes to ground and carry their plate current. Coupling
capacitors connect these chokes to the modulator tube grid chokes,
which carry modulator grid current. Sizes of chokes and coupling
capacitors are chosen to give approximately constant impedance from
the lowest modulation frequency up to the higher harmonics of the
highest frequency, and choke capacitance is reduced to preclude pro-
nounced resonance effects throughout the frequency range. In Fig.
151, the filter components are mounted in the exciter cubicle; a trans-
former for this purpose would be too large to locate internally.
The cathode follower circuit is advantageous in another way. Leak-
age inductance in a driver transformer causes high-frequency phase
shift between driver and grid voltage, which does not exist in the
coupling capacitor scheme. Since inverse feedback is often applied to
audio amplifiers to reduce distortion, the absence of phase shift is a
great advantage. The low frequency at which phase shift appears
must be kept below the audio band, but this can be done without ex-
cessively large components.
82. Transformer-Coupled Oscillators. These have circuits similar
to that of Fig. 152. Transformer OCL and capacitor Ci form a tank
circuit, to which are coupled sufficient turns to drive the grid in the
lower left-hand winding. The output circuit is coupled by a separate
winding. For good wave shape in such an oscillator, triodes and class
B operation are preferable. The ratio of turns between anode and
grid circuits is determined by the voltage required for class B operation
of the tube as if it were driven by a separate amplifier stage. Single
tubes may be used, because the tank circuit maintains sinusoidal wave
shape over the half-cycle during which the tube is not operating. Grid
bias is obtained from the RC2 circuit connected to the grid.
In such an oscillator, tube load equals transformer loss plus grid
load plus output. In small oscillators, transformer loss may be an
appreciable part of the total output. This loss consists of core, gap,
and copper loss. Copper loss is large because of the relatively large
tank current, and the wire size in the anode winding is larger than
would be normally used for an ordinary amplifier. The gap is neces-
AMPLIFIER CIRCUITS
201
PI- FILTER CH-'KtS
COyPLIN.-, C-AfriCiTORS
I
i
%
V
JljiS'"'^
«w**
ila
S
f
J*
V
' *
Fig. 151. Rear view of exciter cubicle for 50-kw broadcast transmitter.
202
ELECTRONIC TRANSFORMERS AND CIRCUITS
sary to keep the inductance down to a value determined by the tank
circuit Q or volt-amperes. This in turn is dictated by the required
harmonic content. The use and selection of core materials are approxi-
mately the same as those indicated in Sections 78 and 79.
Class C oscillators are less desirable for very low harmonic require-
ments, because of the difficulty of designing tank circuits with suffi-
ciently high Q. Where large harmonic values can be tolerated, the
transformer can be designed for low Q, but the wave form becomes non-
TO OUTPUT
CIRCUIT
Fig. 152. Transformer-coupled audio oscillator.
sinusoidal. Transformer grid circuit turns are large, approximately
the same as plate turns, and the grid voltage would be high if grid
current did not limit the positive voltage swing. During the half-cycle
when the tube is operating, the voltage wave has a roughly rectangular
shape, and during the rest of the cycle it peaks sharply to a high ampli-
tude in the opposite direction. Core losses are difficult to predict be-
cause loss data are not normally available for such wave forms. Con-
sequently, designs of this type are usually cut-and-try. The fre-
quency of oscillation varies with changes in load ; hence low Q class C
oscillators are to be avoided if good frequency stability is required.
83. Shielding. Gain of 80 to 100 db is often reached in high-gain
amplifiers. It is important in these amplifiers that only the signal be
amplified. Small amounts of extraneous voltage introduced at the
amplifier input may spoil the quality or even make the received signal
unintelligible. One source of extraneous voltage or hum is in stray
magnetic fields emanating from power transformers in or near the
amplifier. The stray fields enter the magnetic cores of input trans-
AMPLIFIER CIRCUITS
203
formers and induce small voltages in the windings, which may be
amplified to objectionably high values by the amplifier. Several de-
vices are used to reduce this hum pick-up:
1. The input transformer is located away from the power trans-
former.
2. The coil is oriented for minimum pick-up.
3. Magnetic shielding is employed.
4. Core-type construction is used.
The first expedient is limited by the space available for the amplifier,
but, since the field varies as the inverse cube of the distance from the
Fig. 153. Refraction of magnetic field by iron shield.
source, it is obviously helpful to locate the input transformer as far
away from the power transformer as possible. The second method is
to orient the coil so that its axis is perpendicular to the field. It re-
quires extra care in testing. Magnetic shielding is the "brute force"
method of keeping out stray fields; core-type construction is effective
and does not materially increase the size. Of course, any of these
methods increases manufacturing difficulties to a certain extent.
Magnetic shielding is ordinarily accomplished by a thick wall of
ferrous metal or a series of thin, nesting boxes of high permeability
material encasing the windings and core of the input transformer.
Neither type of shield is applied to the power transformer because the
flux lines originate at the power transformer and fan out in all direc-
tions from it. A large percentage of the flux would strike the shield at
right angles and pass through it. On the other hand, the stray field
near the input transformer is relatively uniform, and very few flux
lines strike the shield at right angles. Thus more flux is bypassed by
it. The action of a thick shield in keeping stray flux out of its interior
is roughly illustrated in Fig. 153.
204
ELECTRONIC TRANSFORMERS AND CIRCUITS
FIN.
ST.
a
ST.
1
Fig. 154. Flux direc-
tions in a core-type
transformer.
Multiple shields increase the action just mentioned because eddy
currents induced in the shields set up fluxes opposing the stray field.
Sometimes alternate layers of copper and magnetic material are used
for this purpose, when hum pick-up 50 or 60 db below the no-shield
value is required.^
In core-type transformers the flux normally is in opposite directions
in the two core legs, as shown in Fig. 154. A uniform external field,
however, travels in the same direction in both legs, and induced volt-
ages caused by it cancel each other in the two coils.
The relative effectiveness of these expedients
is shown in Fig. 155. Hum pick-up is given in
decibels with zero decibel equal to 1.7 volts
across 500 ohms, and distance from a typical
small power transformer as abscissas. All curves
are for 500- ohm windings working into their
proper impedances, and with no orientation for
minimizing hum. Using impedances much less
than 500 ohms reduces the hum picked up. Ori-
entation of coil position also reduces hum. For
all types of units there is a position of minimum
hum. With the unshielded shell type the angle
between the transformer coil and the field is almost 90° and is ex-
tremely critical. With shielding, this angle is less critical, but the
minimum amount of hum picked up in this position is not noticeably
reduced. The core type is less critical, especially with a shield. The
minimum amount of hum picked up is from 10 to 20 db less than the
shielded shell type in its minimum position. Removing the shields
from the core type may change its position of minimum pick-up.
This is because the shields reduce hum by a process different from that
of the two bucking coils.
It is advantageous to have power transformers of the core type.
Leakage fluxes from like coils on the two legs approximately cancel
at a distant point. The U-and-I shape of lamination is better than the
L shape because of its symmetry. A type C core has the advantage
that gaps are inside the coils. Thus fringing is reduced, and stray
flux from the core gap is minimized.
Static shielding does not prevent normal voltage on a primary wind-
ing from being transferred inductively into a secondary winding. It is
1 See S. L. Gokhale, J. AIEE, 48, 770 (October, 1929) ; also, "Magnetic Shielding
c(f Transformers at Audio Frequencies," by W. G. Gustafson, Bell System Tech. J.,
17, 416 (July, 1938).
AMPLIFIER CIRCUITS
205
effective only against voltage transfer by interwinding capacitance.
High-frequency currents from vacuum tube circuits are thus prevented
from flowing back into the 60-cycle power circuits via filament and
-10
-15
-20
-25
-30
-35
-40
-45
-50
-55
-60
-65
-70
-75
-80
-85
-90
-95
-100
-105
-110
-115
-120
-125
-130
-135
4
DISTANCE FROM
5 6 8 10
HUM SOURCE-INCHES
20 30 50
10
N
c
URGE COIL
'JI =
200
s
N
s
\
\
\
\,
^
\
K^-.
V
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h
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s
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s
h'^n
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ft
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Fig. 155. Hum pick-up in input transformers.
plate transformers. Without shielding, such currents may interfere
with operation of nearby receivers. Likewise, voltages to ground on
telephone lines are kept from interfering with normal voice frequency
voltages between lines. The extent of static shielding depends upon
206
ELECTRONIC TRANSFORMERS AND CIRCUITS
the amount of discrimination required. Usually a single, thin, grounded
strip of metal between windings is sufficient, with ends insulated to
prevent a short circuit. Magnetic flux in the interwinding space
causes eddy currents to flow in such shields, and even shields with
insulated ends indicate a partial short circuit on test. This effect
reduces the OCL of the transformer. If volts per layer are small com-
pared to total winding voltage, a layer of wire is an effective shield.
(Q) NO RESPONSE IN R4 FROM "^
Eq. response in RgSRj
INVERTED HYBRID
R3
(C) RESPONSE IN R2aR3, BUT
NOT IN R] FROM Es
CISJ
_A
<b) response IN R I, Rg.a R4 —
FROM E3
(d) EQUIVALENT CIRCUIT,
R4 REPRESENTS
SECONDARY LOAD
IT
\1
Fig. 156. Hybrid coil operation.
The start turn is grounded and the finish left free, or vice versa. A
wire shield has none of the short-circuit effect of a wide strip shield.
Usually a transformer that requires static shielding has a low -voltage
winding; the shield can be placed close to this winding, needs little
additional insulation, and occupies but a small fraction of the total
coil space. If shields are placed between high-voltage windings, as in
modulation transformers, the shields must be insulated from each
winding with thick insulation. This materially increases the coil mean
turn length, transformer size, and difficulty in obtaining good high-
frequency response. Shields have questionable value in such trans-
formers and usually are omitted.
84. Hybrid Coils. Hybrid coils are used to isolate an unwanted
signal from certain parts of a circuit, and allow the signal to be used in
other parts of the circuit. In the hybrid coil shown in Fig. 156(a) the
lower windings or primary sections are balanced with respect to each
AMPLIFIER CIRCUITS 207
other, and the two resistors R2 and R3 are equah Voltage Eq applied
between the primary center tap and ground causes equal currents to
flow in opposite directions through the two halves of the primary wind-
ing, and therefore produces zero voltage in the secondary winding. By
this means, signal Eq arrives at resistors R2 and R3 undiminished, but
there is no voltage in R4, connected across the secondary coil. Fig-
ure 156(6) shows what happens in this circuit if the voltage is applied
across R3 instead of across Ri. In this case, the voltage E3 appears
across resistors Ri, R2, and R^, that is, in all parts of the circuit.
An inverted hybrid coil is shown in Fig. 156(c). Here voltage Eg is
applied across the upper coil, which is now the primary. The second-
ary sections are assumed to be balanced. Therefore, there is zero
voltage between the center point of the secondary winding and ground,
and though a signal appears at R2 or R3 there is no signal across Ri.
Thus a hybrid coil works in both directions.
It has been assumed that R2 and R3 are equal and that the two
primary half-windings are of equal number of turns. This is not
necessarily true, for, if the resistance of R2 is twice that of R3, the
number of turns connected to R2 should be twice those connected to
R3. However, it is important that, through the range of frequency
in which the hybrid coil is desired to function, the balance between the
two halves be maintained closely. The most exact balance is achieved
for i?2 = R3 by winding the two halves simultaneously with two dif-
ferent wires. This method gives good isolation of the undesired signal.
Other methods introduce some ratio error which reduces the isolation.
For the same reason, it is necessary to balance the circuit with regard
to capacitance and leakage inductance. That is, if a capacitance
exists across R3, such as line capacitance for example, an additional
equivalent amount should be added across R2 in order to achieve the
balance desired. Likewise, any inductive apparatus, adding either
series or parallel inductance in one circuit, should be compensated for
by inductance of like character in the other circuit. Adding series in-
ductance, for example, in series with R3 will not compensate for shunt
inductance across R2, or vice versa, as the two have opposite effects
with regard to frequency and therefore balance is attained only at one
frequency.
Assume a perfect transformer having no exciting current and no
leakage inductance between the two halves, and a transformer with
equal turns in the two halves of the primary winding. Assume cur-
rents in the directions shown in Fig. 156(d). Then
208 ELECTRONIC TRANSFORMERS AND CIRCUITS
h=h+ h (88)
El = I2R2 + hRi (89)
Ez = hRx + E2 (90)
On the assumption of equal turns in the two half -windings, Ei = E2.
If the magnetizing current is assumed to be zero, the ampere-turns and
hence the volt-amperes in the two primary halves are equal. The sec-
ondary load can be considered as reflected into the primary winding as
resistor R^.
h = (El + E2)/Ri (91)
ih - h)E2 = {h + h)Ei (92)
If equations 88 to 92 are combined, an expression for Z3 can be found:
Ez iRiRz + 4:RiRi + R2R4
Z3 = — = — !-^ — -^ (93)
h iRi + 4^2 + ^4
If the secondary circuit is open, ^4 = =0, and equation 93 becomes
Zs = 4:Ri + R2 (94)
85. Amplifier Tests. Tests for hum, distortion, linearity, and fre-
quency response can be made with meters in the output circuit when
voltage of a known frequency and wave form is applied to the input.
Hum and distortion are conveniently measured by instruments spe-
cially made for the purpose. Linearity is measured by varying the
input voltage and measuring corresponding output voltage. Frequency
response is measured at a fixed input or output voltage, but frequency
is varied. Normal production testing of amplifiers requires no more
than such overall tests. But, in the development of the amplifier,
excessive hum, distortion, or other defects may be indicated, and tests
must be applied stage by stage to locate the trouble. Voltage is
usually measured by a tube voltmeter, one terminal of which is
grounded. In a push-pull amplifier, it is therefore necessary to block
the direct voltage and measure the alternating voltage on each side.
A cathode-ray oscilloscope is helpful in checking phase shift and wave
form at various points.
Before being assembled in the amplifier, transformers are tested for
turns ratio, balance, polarity, OCL, winding resistance, core loss, and
insulation strength. Although with new designs it is desirable to
check leakage inductance, winding capacitance, and shielding, these
properties vary less in a given design than the others. Methods for
AMPLIFIER CIRCUITS 209
making most of these tests are the same as those described in Chapter
3. Tests for capacitance are limited to winding-to-winding and wind-
ing-to-core capacitance. These tests are made on a capacitance bridge.
Evaluation of capacitance measurements is made as in Section 71.
Balance (an important property of push-pull amplifiers) and shield-
ing tests are described in Standard TR-121 of the Radio-Electronics-
Television Manufacturers Association.
Reactor Q is measured either on an inductance bridge (which also
measures a-c resistance as in Fig. 75) or on a special Q meter. In
either method, rated voltage and frequency should be used. Modula-
tion reactors are usually measured for inductance with full direct cur-
rent in the winding; great care should be exercised to prevent sudden
interruption of this current and consequent dangerous high voltage.
Such reactors are often surge-tested to guard against breakdown in
service under conditions of overmodulation.
In the diagram of Fig. 157, the first two stages have current feed-
back, and so initial tests were made with the circuit shown. But over-
all voltage feedback from the modulator plates back to the 6J7 grids
was not applied until the amplifier was first tested without it. Then
resistors from which feedback is derived were adjusted to produce the
feedback voltage necessary to give the required performance. The
carrier power amplifier was completely adjusted before modulation was
applied. Percentage of modulation was measured by the increase of
carrier output current when modulation was applied. Inductance RFC
and capacitor Ci maintain the modulator load constant at high fre-
quencies. C2 in this circuit is the audio coupling capacitor. Separate
meters are provided to measure the plate current of each driver and
modulator tube, so that bias may be adjusted for the same plate cur-
rent on each side.
Proper operation is predicated on amplifier stability, which often
is not obtained when power is first turned on. Local or parasitic oscilla-
tions may easily occur as a result of natural resonance of circuit ele-
ments or even in connections and tube electrodes. These must be de-
tected and eliminated by corrective measures which apply to the
trouble. Some of these troubles may be caused by long leads, espe-
cially in the grid circuit. Tubes may require resistors in the plate and
grid leads to damp out parasitic oscillations. Resistors are used in this
manner in the amplifier shown in Fig. 157. Coils in circuits with
widely different voltages should not be coupled closely, because re-
generation may result. In circuits with high voltage, and therefore
large capacitive currents, it may be necessary to add shielding to
210
ELECTRONIC TRANSFORMERS AND CIRCUITS
a
fi*
AMPLIFIER CIRCUITS 211
prevent stray pick-up from one stage to another. In push-pull ampli-
fiers, if some circuit element is unbalanced, it may give rise to a push-
push oscillation which can be eliminated by better balance, or by de-
coupling the tube plates at the unwanted frequency. If insufficient
bypass capacity is used on plate or bias supplies, interstage coupling
may occur at low frequencies. The frequency may be less than 1 cycle
per second. This kind of instability is known as "motor-boating."
Operating tubes so that some electrode becomes a negative resistance
during a portion of the cycle may give rise to oscillations which cannot
be prevented except by avoidance of the cause, or by some power-
absorbing circuit which does not affect normal operation. The elim-
ination of such trouble requires much testing time and skill, but it
must be done before performance tests are made.
86. Design Examples.
Example (a). Transformer for Pi-Filter Modulator.
Frequency range 100 to 5,000 cycles.
Audio output 400 watts.
Power amplifier Eb/Ib = 10,000 ohms.
Voltage ratio primary/secondary (1,180 + l,180)/2,000.
/i = 60 cycles.
Core: 4-in. stack of silicon-steel lamination B, Fig. 44 (p. 55).
Turns primary/secondary (800 + 800)/l,380 No. 26 wire.
Ac = 7.2 sq in. net; 8.0 sq in. gross.
h = 12.75 in.
Ig = 0.012 in.
Possible tube current unbalance = 0.032 amp.
0.6 X 800 X 0.032
Bdc = Q^ = 1,260 gauss.
3.49 X 2,000 X 108
^- = 100 X 7.2 X 1,380 =^g"""-
Bm = 8,260 gauss. From Fig. 70 (p. 98), jua = 9,000+
, . _^ 3.2 X (1,380)2 X 8 X W „„ . ,
Secondary OCL = o.012 + (12.75/9,000) = ^''^ ^''''^'■
Xl at /i = 6.28 X 60 X 36.5 = 13,800 ohms. Z, = 115 per cent R at 100
cycles from Fig. 146. Winding arrangement as in Fig. 158, to reduce layer volt-
age and capacitance.
Winding resistances: Total primary 90 ohms.
Secondary 80 ohms.
Secondary leakage inductance = 53 millihenrys.
Capacitances (referred to secondary) :
212
ELECTRONIC TRANSFORMERS AND CIRCUITS
Pi - Siiat A) = 292 nnf
Pi - S2(at B) =0
P2 - Si(a.t C) =112
P2 - -SaCat D) = 58
Secondary layer to layer = 140
Primary layer to layer = 170
Stray and tube = 50
Power amplifier r-f bypass = 500
Total = 1,322 MMf
TO PLATE
TO MOD. COUPLING
CAPACITOR
\
\
Sz
%%^%%%;:m%;^^^^:<%^?^^
TO PLATE
COIL FORM'
Fig. 158. Section of transformer coil wound for low layer voltage.
^l"
■TO Eb
At high audio frequencies /r = 19,000, D = 0.6, and Z/R2 from Fig. 119
(p. 159) is 87 per cent at 5,000 cycles.
Example (6). Audio Oscillator.
Circuit of Fig. 152, with 6C5 tube, Eb = 150 volts, Ec = -10 volts.
Frequency 800 cycles.
Plate load impedance = 20,000 ohms.
Class B operation; grid swings to +2 volts.
Aeg = — 12 volts
Ae„ = 150 - 35 = 115 volts
Aip = 5.6 ma
■ during positive half-cycle.
Average power output = (115 X 5.6)/4 = 160 mw.
Transformer voltage ratio P/G = 115/12 = 81/8.5 rms.
For low harmonic distortion, volt-amperes = 10 X tube output =1.6 v-a.
^ (115X0.707)2 ,i.n 1,
Xc = ^ r^ = 4,140 ohms.
C
1.6
1
6.28 X 800 X 4,140
= 0.048 ixL
AMPLIFIER CIRCUITS 213
Current in plate winding = 4! |o = 0.02 amp.
Core is the same as in Example (b), Section 72.
Primary 2,100 turns No. 32 enamel. Winding resistance = 125 ohms.
Grid 250 turns No. 42 enamel. Winding resistance = 180 ohms.
Ig = 0.060 in., Ic = 4.5 in., Ac = 0.32 sq in., core weight = 0.4 lb.
0.6 X 2,100 X 0.0056
^"'^ 00601^"^,; ^ ^^^^"''•
„ 3.49 X 81 XIO" .„_
^- = 800X0.32X2,100 = ^^ gauss.
B^ = 560 gauss. (From Fig. 70, ma = 2,000.)
P- nnr 3.2 X (2,100)^ X 0.36 X 10"^
Primary OCL = 0.O6O + (4.5/2,000) = ^■^'' ^''"^^•
From Fig. 140 core loss = 0.2 X 0.6 X 0.4 = 0.048 watt.
Gap loss = 0.030 watt.
Primary copper loss = (0.02) ^ X 125 = 0.05 watt.
This leaves 32 mw available for secondary output.
Example (c). Cathode Follower. Assume that tubes 828 and 849 in Fig. 157
operate at Eb = 1,700 volts. With Ec = -75 and 700 watts output, 849
plate swing is 1,200, E^iin = 500, peak grid voltage = +105, and peak grid
current is 0.090 amp. This is peak load for the 828 tube which can operate
class ABl. The 828 tube plate swing is 1,200 + 180 = 1,380, or E^in = 1,700
- 1,380 = 320 volts, and the 828 peak load is 105/0.090 = 1,170 ohms. The
cathode choke of 828 and the grid chokes and capacitors of 849 should have
1.41 X 1,170 = 1,650 ohms reactance for 100 per cent modulation at frequency
/i. At 10 cycles this would be 27 henrys. Peak voltage across these chokes is
the 828 plate swing.
7. HIGHER-FREQUENCY TRANSFORMERS
In Chapter 6 the influence of low-frequency performance on size
was mentioned. If high transformer OCL is required to maintain
good low-frequency response, many turns or a large core are necessary,
either of which limits the high-frequency response. But if the amplifier
frequency range is wholly composed of high frequencies, this limitation
is in large part removed. For example, in a power-line carrier ampli-
fier, the frequency range is 40 to 200 kc. It is then only necessary that
OCL be high enough to effect good response at 40 kc. This is a great
help in designing for proper response at 200 kc, and makes possible
the use of laminated iron-core transformers for these and higher fre-
quencies.
87. Iron-Core Transformers. At power-line carrier frequencies, the
principles discussed in preceding sections for lower-frequency trans-
formers apply. In terms of the mean frequency, the band is narrow.
Fig. 159. Two-layer bank winding.
but at 40 kc the curves for low-frequency operation portray amplifier
performance just as they do at 30 cycles. Likewise at 200 kc care
must be used that the limiting factors of leakage inductance and wind-
ing capacitance do not interfere with proper operation.
In carrier frequency transmitters, transformers are normally used
for coupling between stages and for coupling the output stage to the
line. They sometimes transform a large amount of carrier power.
Coils are usually wound in single layers, spaced well apart to reduce
capacitance, and have but few turns. If the necessary turns cannot
be wound in a single layer, a bank winding like that shown in Fig. 159
may be used. This winding has more capacitance than a single layer
214
HIGHER-FREQUENCY TRANSFORMERS
215
but much less than two layers wound in the ordinary way. Since
intrawinding layer-to-layer capacitance is zero in these transformers,
the resonance frequency fr is usually determined by winding-to-wind-
ing capacitance.
In high impedance circuits, the winding-to-winding capacitance
may be reduced by winding "pies" or self-supporting vertical sections
side by side. Pies are wound with one or more throws per turn and
may be several turns wide. They have the general appearance of Fig.
160.^ High, narrow core windows or several pies are desirable to re-
COIL FORM
Pig. 160. Pie-section windings.
duce leakage inductance. Transformer loss is mostly core loss. Two-
mil grain-oriented steel can be used advantageously in such transform-
ers, because of its low losses and high permeability. In transmitter
operation, class AB or class B amplifiers are commonly used, with or
without modulation, which may be as high as 100 per cent. In a
receiver, input and interstage transformers also are employed, mainly
to obtain voltage gain or for isolation purposes. Similar transformers
are used for line matching, especially where overhead lines are con-
nected to underground cables. Line impedance changes abruptly, and
transformers may be necessary for good power transfer.
Core data at these frequencies are usually not available except for a
limited choice of materials and gages. Approximate loss for 2-mil
oriented steel is given in Fig. 161. Interpolation or extrapolation from
known data may be necessary to estimate core losses. In spite of
this limitation, carrier frequency transformers are widely used. Some
of the transformers in Fig. 16 operate in the carrier band. Core
steel permeability decreases at high frequencies, depending on the
1 See "Theory and Design of Progressive and Ordinary Universal Windings,"
by Myron Kantor, Proc. I.R.E., December, 1947, p. 1563.
216 ELECTRONIC TRANSFORMERS AND CIRCUITS
lamination thickness. Oriented steel and nickel alloys have high
permeability at low frequencies, but, unless thin laminations are used,
this advantage disappears at frequencies of 20 to 50 kc. The approxi-
mate decline for low induction is shown in Fig. 162. Decrease of per-
meability may be so rapid that OCL nearly decreases inversely as
frequency with 0.014-in. and even 0.005-in. material.^ Grain-oriented
steel 0.002 in. thick is well suited to these frequencies.
lOOJD
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FREQUENCY -CYCLES PER SECOND
Fig. 161. Approximate loss for 2-mil steel at higher frequencies.
Transformers are used at still higher frequencies. Capacitance
limits the upper frequency at which amplifier transformers may be
operated. In a tuned circuit amplifier, the tuning includes the inci-
dental and tube capacitance as well as the tank circuit capacitance.
A transformer has no tuning to compensate for such capacitances.
Even with zero winding capacitance there would be a frequency limit
at which any tube could operate into an untuned transformer without
1 For additional core-loss and permeability data at higher frequencies, see "The
Variation of the Magnetic Properties of Ferromagnetic Laminae with Frequency,"
by C. Dannatt, J.I.E.E., 79, 667 (December, 1936).
HIGHER-FREQUENCY TRANSFORMERS
217
3500
3000
2500
2000
<
S
ir 1500
UJ
iOOO
500
V \
1>J
\ V
x^
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I03 |04 |o5
FREQUENCY- CYCLES PER SECOND
I0«
Fig. 162. Approximate permeabilities of core steels at higher frequencies.
spoiling its efficiency or other characteristics. The most favorable
condition for the use of transformers at higher frequencies is low circuit
impedance. With low leakage inductance and low impedance circuits,
transformer operation is possible in the high- and very-high-frequency
bands.
88. Other Core Materials. In the high radio-frequency bands, fer-
rite cores have the advantage of high resistivity and practically no
eddy-current component of core loss. Several grades are manufac-
tured commercially, usually mixtures of manganese, nickel, and zinc
ferrites. Figure 163 is a set of normal permeability curves for differ-
ent grades of ferrites, and Fig. 164 gives initial permeability. Usually
the lower-permeability materials have lower loss at higher frequencies,
so that permeability is an inverse indication of the relative frequencies
at which ferrites are useful.
218
ELECTRONIC TRANSFORMERS AND CIRCUITS
PERMEABILITY (p) ^0°^°^ /' ,4°0°o°o°0°^o° .
0.04 0.06 0.08
0.4 0.6 0.8 1.0 2 4
MAGNETIZING FORCE (H)-OERSTEDS
Fig. 163. Magnetic ferrite normal permeability.
Fig. 164. Magnetic ferrite initial
permeability.
10,000
o
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0.01
HI 0.001
o
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tjL
.,/
Ij
CD
<
UJ
^ ion
V
\
- >
w~
-\^
D -
—^
~i.
/
V/
1 ,
/"—
C-
^
^V
/
'
B-
__
u
^
"^
-J
<
10
^
7>^
■-ll
A-
0.
1 10 20 40
FREQUENCY, MEGACYCLES
.1
FREQUE
1
NCY, MEGAO
40
rCLES
Fig. 165. Loss factor of ferrite cores.
HIGHER-FREQUENCY TRANSFORMERS
Losses in ferrites are often related to the product fioQ-
165.) This relation is approximately as follows:
Core loss 0.41 X lO-*^^^
in.'
MoQ
219
(See Fig.
(95)
Instead of ixqQ, the quantity Rsei/i^fL is sometimes plotted, where
Rser is the equivalent series resistance corresponding to core loss. Equa-
tion 95 then becomes
Core loss
m:
= 0.065 X 10-75
(96)
SECONDARY COIL
P^T-> >T^ .-T"
PRIMARY COIL
CORE
XTX^ 1 HI"
^i^ nJ.^ -^i^
sj.^ vl^ vix
At the lower radio frequencies, finely divided powdered iron has
loss lower than some ferrites. Owens ^ gives 1.0 mc as the highest
frequency for which this holds. Both ferrites and powdered iron have
temperature limits far below that of strip-wound cores: ferrites because
of low Curie temperature, and pow-
dered iron because of possible damage
to the bonding material. Powdered
iron with certain bonds has the better
temperature coefficient of permeabil-
ity. Both materials are available in
the forms shown in Fig. 144.
89. Capacitance Evaluation. In
high-frequency transformers, the ca-
pacitances differ from those in audio
transformers in that the windings are
usually single layers, whose tum-to-
turn capacitance is negligible com-
pared to capacitance between wind-
ings and to the core. For example, in the transformer in Fig. 166, the
primary and secondary are each wound in a single layer concentrically,
in the same rotational direction, and in the same traverse direction
(right to left) . It will be assumed that the right ends of both windings
are connected to ground (or core) through large capacitances, as shown
dotted, so that the right ends are at substantially the same a-c poten-
tial. Primary capacitance Ci is composed of many small incremental
capacitances Cp, and secondary C2 of many small incremental capaci-
tances Cs, each of which has a different voltage across it. Likewise,
^ See "Analysis of Measurements on Magnetic Ferrites," by C. D. Owens,
Proc. I.R.E., 41, 360 (March, 1953).
Fig. 166. Single-layer windings.
220
ELECTRONIC TRANSFORMERS AND CIRCUITS
many small incremental capacitances Co exist between primary and
secondary, which have different potentials across them. If the trans-
former is step-up,
Ci
1
2C„ and C. = ~
3 ' "3
If the transformer is step-down,
Ci
iV,,
and C2 = - ^Cs
o
If the ratio is 1:1,
N, = Np, Ci = ^2Cp and C2 = ^SC,
(97)
(98)
(99)
For transformers with opposite angular rotations of primary and
secondary windings, or with opposite traverse directions (but not
both), minus signs in the factors [N^ - N,)^/Nj,^ and (iV, - iVp)2/iV/
in equations 97 and 98 become positive ; there is no other change. For
transformers with both angular rotations and traverse directions op-
posite there is no change at all in these equations. If there is a shield
between primary and secondary, omit terms containing Ca in these
equations, and make %Cs and SCj, include the capacitance of secondary
and primary to shield, respectively. 5Cs is the measurable capacitance
of the short-circuited secondary to core, and %Cp that of the short-
circuited primary to core.
In push-pull amplifier transformers, the secondary winding is inter-
leaved between two primary halves. The rotational directions of wind-
ing and traverse are important, as they affect not only effective capaci-
tance but also plate-to-plate coupling. It is usually best to have all
windings with the same rotational direction and traverse, and to con-
nect the primary halves externally.
Winding resistance increases with frequency because of eddy cur-
rents in the larger wire sizes, and copper loss increases proportion-
ately. Formulas for single-layer coils are given in handbooks.^ Eddy-
current resistance of layer-wound coils in deep open slots is plotted
in Fig. 167 as a function of conductor thickness, frequency, and number
of coil layers; it approximates the increase of winding resistance in a
transformer.^
1 See Natl. Bur. Standards Circ. 74, p. 304.
2 See "Eddy-Current Resistance of Multilayer Coils," by T. H. Long, Trans.
AIEE, 64, 716 (October, 1945).
HIGHER-FREQUENCY TRANSFORMERS
221
Rdc
^///^/AcORE
INCREASE OF COIL RESISTANCE
VS.
CONDUCTOR t'kS.(INCHES[ xV
100 50 .20
FREQUENCY (CYCLESI
IO-«-LAYERS
/ y
'
/
/
/ /
/
1
/
//
/
/
"
7
/
/
/
/5
J
//
/ /
/
/
^5
1
/
/
1 j
j
/
/
11
/ /
' /
' /
yZ
\\
//
\
/y
y
jk
^
^
^
^
2 3 4 5 6 7
Fig. 167. Increase in coil resistance at high frequencies.
90. Example. . Line Matching Transformer 50 to 500 ohms.
Frequency range 50 to 150 kc.
Power output 100 watts.
Primary voltage = \/ ZW = VsO X 100 = 70.7 volts.
Secondary voltage = VsOO X 100 = 224 volts.
Core 2-mil oriented silicon steel.
Ac = 0.45 sq in., h = Q in., Ig = 0.002 in. (incidental), core weight M lb.
Window % in. X 1 3^ in.
Primary 31 turns No. 22 wire. Mean turn 3.8 in.
Secondary 100 turns No. 30 wire. Mean turn 4.6 in.
Windings arranged as in Fig. 166.
Insulation between primary and core, and between secondary and primary
K in. of organic material.
Secondary effective capacitance 40 /iyuf .
Bm = 350 gauss.
Secondary OCL = 20 mh.
Secondary leakage inductance = 260 nh.
222
ELECTRONIC TRANSFORMERS AND CIRCUITS
Core loss = 8 watts per lb X 0.75 = 6 watts. This is practically the only
loss.
Xn/Ri at 50 kc = (6.28 X 50,000 X 0.020)/500 = 12.56.
fr = 1,560 kc. B = 5.00.
According to Figs. 108 and 109, this transformer has nearly flat response over
the entire range.
91. Leakage Inductance at High Frequencies. Provided that a
transformer is operated at frequencies below resonance, the leakage
inductance measured at low frequencies governs response at high fre-
V^'°"V
(a) (b)
Fig. 168. (a) Symmetrical and (b) asymmetrical spacing of concentric windings.
quencies. Leakage inductance in concentric windings is lowest if
the windings are symmetrically spaced in the traverse direction, as in
Fig. 168(a). For a given number of turns, the leakage flux is least
in Fig. 166, somewhat greater in Fig. 168(a), and much greater in Fig.
168(b). The increase in leakage flux is a function of core dimensions,
winding-to-winding spacing, and margin inequality. Figure 169 shows
typical increase of leakage inductance when one secondary margin is
increased with respect to the other, as in Fig. 168(5).
Leakage inductance is very low in toroids with windings which cover
the whole magnetic path. Toroids are wound on special machines
which thread wire in and out of the core. Carefully wound toroidal
transformers function at very high frequencies.^ If part of the core is
not covered by the windings, as indicated by dimension G in Fig. 170,
leakage flux sprays out of the ends of the coils and reduces the fre-
quency range.
92. Wide-Band Transformers. Untuned transformers operate in all
frequency ranges from to VHF. The lowest operating frequency is a
1 See "Very-Wide Band Radio-Frequency Transformers," by D. Maurice and
R. H. Minns, Wireless Engr., 24, 168 (June, 1947).
HIGHER-FREQUENCY TRANSFORMERS
223
fraction of 1 cycle. The highest frequency is in the VHF band, some-
where around 150 megacycles. No known transformer covers this
whole range at present. Television coaxial-line terminating trans-
1
/
>
f
/
/
/
00%
/
^s :
r
/
/
/
/
L' =
SINGLt LAYER WINDINGS
LEAKAGE INDUCTANCE WITH
TAPPED SECONDARY SHORT
CIRCUITED
/
/
u
LEAKAGE INDUCTANCE WITH
FULL SECONDARY SHORT
CIRCIITFD
/
/
b =
c=
b/
WINDI
INSUL
C = 40
1-
\IG WIDTH
ATION THICKNESS
100 80 60 40 20
% SECONDARY TURNS SHORT-CIRCUITED
Fig. 169. Leakage inductance of asymmetrical windings.
formers have been made to cover the frequency range of 50 cycles to
6,000,000 cycles, or a ratio of highest to lowest frequency of over
100,000:1. This is an exceptionally wide band. More common wide-
band transformers are those in the audio band
of 20 to 20,000 cycles, or 10 to 30,000 cycles,
that is, with about a 3,000:1 frequency ratio.
Often, transformers used at frequencies on the
order of 100 megacycles are for relatively nar-
row bands, say 20 to 60 megacycles wide.
In low-impedance circuits, it is leakage in-
ductance that determines transformer behavior,
whereas at high impedance it is winding capaci-
tance. In most audio transformers the coupling
WINDINGS
Fig.
170. Toroidal core
and coil.
224 ELECTRONIC TRANSFORMERS AND CIRCUITS
coefficient is 0.9995 or higher. With bifilar windings, this figure may
increase to 0.999995.^ Such a high coefficient of coupling requires the
use of good core materials. For a given source impedance and trans-
former core material, the product of turns ratio and band width is a
rough indication of size. Quite generally, for low power the widest-
band transformers are made of Permalloy or Supermalloy.
In high-impedance circuits the matter of size is not merely one of
space for mounting; it also has a direct bearing on the upper frequency
limit, since transformer capacitance is roughly proportional to size.
If capacitance is low, the band-width ratio (highest/lowest frequency)
is approximately equal to the ratio of OCL/leakage inductance. This
may be verified by comparing Figs. 108 and 109. It is most nearly
true for low-impedance transformers. With given primary impedance,
core size, and material, there is a limit to the step-up turns ratio pos-
sible for any specified frequency response.
93. Air-Core Transformers. Transformers considered hitherto have
had iron or ferrite cores. A class of transformers is widely used in
radio-frequency circuits without cores or with small slugs of powdered
iron. In a transformer with an iron
core, the exciting current required
for inducing the secondary voltage
is a small percentage of the load
component of current. In an air-
core transformer all the current is
Fio. 171. General case of inductive exciting current and induces a sec-
coupling, ondary voltage proportional to the
mutual inductance.
Consider the circuit of Fig. 171 in which Zi is complex and includes
the self-inductance of the primary coil. Likewise, secondary imped-
ance Z2 is complex and includes the self-inductance of the secondary
coil. With a sinusoidal voltage applied, Kirchhoff's laws give the
following:
^1 = Zi7i + jccL^l2 (100)
= Z2I2 + ia,L^/i (101)
where co = 27r times operating frequency, and L^ is the mutual induct-
ance between the primary and secondary coils.
From equation 101 we see that the voltage in the secondary coil is
iSee "New 50- Watt Amplifier Circuit," by F. H. Mcintosh and Gordon J.
Gow, Audio Engineering, December, 1949, p. 9.
HIGHER-FKEQUENCY TRANSFORMERS 225
numerically equal to wLmli, the product of primary current and mutual
reactance at the frequency of applied voltage Ei. The equivalent
impedance of the circuit of Fig. 171 when referred to the primary side
is given by
Z' = Zi + (XmVZ^) (102)
where Xm = j^L^.
In the above formulas, the impedances Zi, Z2, and Z' are complex
quantities whose real and imaginary terms depend upon the values of
resistance, inductance, and capacitance in the circuit. One common
practical case arises when the primary resistance is zero, or virtually
zero, and the secondary coil is tuned to resonance so that Z2 is a pure
resistance R2. Under these conditions, equation 102 reduces to
R' = XmVR2 (103)
where R' is the equivalent resistance in the primary.
Equation 103 gives the value of mutual inductance required for
coupling a resistance R2 so that it will appear like resistance R' with a
maximum power transfer between the two coils, and states that the
mutual reactance Xm is the geometric mean between the two values of
resistance.
The ratio of mutual inductance to the geometric mean of the pri-
mary and secondary self -inductances is the coupling coefficient:
k = LjVl^2 (104)
The value of fc is never greater than unity, even when coils are inter-
leaved to the maximum possible extent. Values of fc down to 0.01 or
lower are common at high frequencies.
Coupling coefficient is reiated to untuned transformer open- and
short-circuit reactance by means of the transformer equivalent circuit
shown in Fig. 107(a), p. 147. Assume that the transformer has a 1:1
ratio, and lealiage inductance is equally divided between primary and
secondary windings. Then if Li and L^ are the self-inductances of pri-
mary and secondary, respectively, L^ is the total leakage inductance
(measured in the primary with secondary short-circuited), and Lm
the mutual inductance,
Xp -|- Xtf _^^ . J
Xs -\- Xji Ls
226 ELECTRONIC TRANSFORMERS AND CIRCUITS
From equation 104,
k
VL1L2
j(--iy
1
1 + iLJ2LJ
If Ljn 5i> Ls,
k
1 -
2L„
(104a)
(1046)
Equations 104(a) and (b) are useful in estimating approximate trans-
former band width.
A tuned air-core transformer often used in receivers is shown in
Fig. 172. Here a sinusoidal voltage Ei may be impressed on the pri-
R| R2
Fig. 172. Tuned air-core transformer.
mary circuit by a vacuum tube amplifier. Resistances Ri and R2 are
usually the inevitable resistance of coils, but occasionally resistance
is added to change the circuit response. The value of voltage E2 ob-
tained from this circuit depends on the impressed frequency; in Fig. 173
it is shown for resonance at three different values of coupling. If the
value of coupling is such that
X
M
VR1R2
we obtain a condition similar to that of equation 103, in which the max-
imum power or current is produced in the secondary circuit. Maximum
current through condenser C2 gives maximum voltage E2. This value
of coupling is known as the critical value. Smaller coefficient of
coupling gives a smaller maximum value of E^. Greater coefficient
of coupling results in a "double hump" as shown in Fig. 173. The
heights of resonant peaks and frequency distance between peaks de-
HIGHER-FREQUENCY TRANSFORMERS
227
pend upon circuit Q and coefficient of coupling k. The double hump
curve of Fig. 173 is desirable because, with modulated waves, fre-
quencies in adjacent channels are rejected; yet very little attenuation
is offered to audio frequencies which effectively add or subtract from
the carrier frequency normally corresponding to resonance. Close
tuning control and high Q are essential to good response and selectivity.
u)Lm<VRiRa
- +
CYCLES OFF RESONANCE
Fig. 173. Response curves for circuit of Fig. 172.
If the primary circuit is made to resonate at a different frequency
from the secondary, audio response is much worse, and considerable
distortion is likely. Moreover, the response at mean frequency is less
than it would be if the circuits were properly tuned. Air-core trans-
formers are usually made adjustable for tuning and coupling.
94. Multiple-Tuned Circuits. Double hump resonance was obtained
with higher-than-critical coupling in the circuit of Fig. 172. Frequency
response with more humps is obtainable if there are more than two
coupled loops. Such circuits are more difficult to tune and adjust than
the circuit of Fig. 172 because of the reaction of each coupled loop on
the others. Easier adjustment can be made with successive "stagger-
tuned" band-pass amplifiers. Each amplifier stage is tuned to a
slightly different frequency. Because of the isolation of the stages by
the associated tubes, tuning one stage does not influence the tuning of
another.
Frequency response similar to that of multiple-tuned coupled cir-
228
ELECTRONIC TRANSFORMERS ANX) CIRCUITS
cuits may be obtained by iterative ladder filter sections. In other
words, it does not matter whether the coupling is inductive or capaci-
tive ; the same shape of response is obtained from the same number of
circuits tuned in the same manner. Since similar results are obtained
from coupled circuits and filters, the choice between them may be
made on the basis of convenience or cost. Considerable literature
has accumulated concerning the design and adjustment of multiple-
tuned circuits, and special techniques have been developed for tuning
them.^
95. Mutual Inductance. It is evident from equation 101 that the
secondary voltage depends upon the mutual inductance between the
coils. Mutual inductance can be calculated by formulas which depend
upon the geometric configuration of the coils. If the coils are arranged
(a) (b)
Fig. 174. Concentric coaxial coupled coils.
concentrically, as shown in either (a) or (b) of Fig. 174, the mutual in-
ductance of the coils can be found from
0.05a^NiN2
(-^d:
juh
(105)
where A'"! = primary turns and A'^2 = secondary turns. All dimen-
sions are in inches. For most purposes, the bracketed portion of this
formula is approximately unity, and it has been plotted in Fig. 175
for a single-turn secondary. To find the mutual inductance for any
given number of secondary turns, multiply the mutual inductance
found from this curve by the number of secondary turns. The range
of ordinates and abscissas can be extended indefinitely.
If the coils are arranged coaxially as in Fig. 176, approximate
values of mutual inductance are found as follows:
1 See "Alignment and Adjustment of Synchronously Tuned Multiple Resonant
Circuit Filters," by M. Dishal, Proc. I.R.E., 39, 1448 (November, 1951).
HIGHER-FREQUENCY TRANSFORMERS
229
iij
I
DOT
,^-PER SECONDARY TURN (N2 = l) IN MICROHENRIES
Fig. 175. Mutual inductance of coils in Fig. 174.
<1
A
1
•{"if*
2 . n2
',j.ar -1-2
i + aJ +a2
Fig. 176. Coaxial non-concentric coiLs with rectangular sections.
230
ELECTRONIC TRANSFORMERS AND CIRCUITS
.0001
V
\
\
S
\
\
\
\
\
\
1
i
\
\
s
\
\
0.8
n \
9 r
1
_j
r
r
1
F
0.1
.03
.01
.001
.7
.8
.0003
.1 .2 .3 .4 .5
Fig. 177. Factor F in equation 106 as a function of r2/r-^ (see Fig. 176).
L^ = FNiNzVAa (106)
In this formula all dimensions are in inches and the mutual inductance
is in microhenrys. The factor F can be conveniently found in Fig. 177.
Self-inductance of single-layer coils is
O.la^N^K
L = ^ (107)
where a = mean coil radius in inches
N = number of turns
I — length of coil in inches
L = inductance in microhenrys
K may be found from Fig. 178.
Equations 105, 106, and 107 are based on equations 192, 187, and 153 in
Natl. Bur. Standards Circ. 74.
HIGHER-FEEQUENCY TRANSFORMERS 231
Receiver intermediate-frequency tuned transformers generally have
coaxial coils. If the wire is more than 0.005 in. in diameter, it is com-
monly subdivided into several strands in the type of cable known as
Litzendraht, to reduce losses and increase Q. In transmitters, the size
of the coils becomes much larger, and concentric coils are employed.
The wire used is Litzendraht at 600 kc or lower frequency, and may
9
8
7
6
5
4
3
2
4 5
DIAMETER
LENGTH
Fig. 178. Factor K in equation 107.
contain many strands for carrying heavy currents. At higher fre-
quencies the wire is of solid or tubular section.
96. Powdered-Iron Slugs. Both self-inductances and mutual in-
ductances of a coil may be increased by inserting a slug of powdered
iron inside the coil tube. Tuning a coil to a given frequency is often
effected in this manner with fixed capacitors instead of tuning with
variable capacitors. Such a coil is shown in Fig. 179, with the pow-
dered-iron core hidden by the coil form. At the left end is the screw
and lock by which the inductance can be adjusted and maintained at a
given value. The mutual inductance of a pair of coils can be changed
similarly. This is preferable to attempting to vary the distance be-
tween the coils, since it requires no flexible connections. Powdered
iron is available in several grades, from ordinary powdered iron to
232 ELECTRONIC TRANSFORMERS AND CIRCUITS
powdered nickel alloy. Insulating compound reduces the permeability
of the core to values ranging from 10 to 125, depending on the grade of
iron and the frequency. In a given coil, the insertion of a powdered-
iron slug raises the inductance from 2 to 3 times the value which it
would have if no iron were present. Circuit Q increases similarly.
Higher Q results from a powdered iron or ferrite magnetic path, closed
except for small air gaps. For an untuned transformer, where high Q
is not essential, the air gap may be zero to reduce magnetizing current.
~4^ J^
Fig. 179. Coil inductance is varied by powdered-iron slug.
97. R-F Chokes. When a choke is used to pass direct current and
present high impedance to radio frequencies, it may have high r-f
voltage across it. High choke impedance at operating frequency is
necessary to avoid loss of r-f current which reduces the useful power
and overheats the choke. If a single-layer choke is connected to an
r-f generator at a given voltage, and if its current is measured as in
Fig. 180, the choke impedance is the ratio of voltage to current meas-
ured.
By disconnecting the choke from the circuit, the tuning reaction
may be noted, and from this whether the reactance of the choke is
inductive or capacitive. The difference in watts input to the generator,
when the coil is removed and the tank condenser is retuned for mini-
mum plate current, is readily observable. This difference times the
generator efficiency is the loss in the coil at a particular voltage and
frequency.
The impedance of a typical coil, found as described above, is plotted
in Fig. 180 against frequency. At low frequencies (a) , the curve follows
straight reactance line Xl{= 2ttJL). At a frequency somewhat below
natural frequency b (determined by the choke inductance and effec-
tive capacitance), the slope starts to increase and reaches a maxi-
mum point at a frequency c of 1.26 to 1.76. Above this frequency, the
HIGHER-FREQUENCY TRANSFORMERS
233
impedance decreases until a minimum value is reached at d, which is
from 2.2 to 3.0 times 6. At higher frequencies, the increase and de-
crease are repeated in a series of peaks and valleys at approximately
equal frequency intervals. The second, fourth, and sixth peaks are of
Fig. 180. R-f choke impedance.
lower value than the first, third, and fifth, respectively. The seventh
peak is followed by a flattened slope which suggests a submerged eighth
peak. The points of minimum impedance rise in value, so that at
higher frequencies the valleys appear to be partly filled in and the
peaks to be level off. The watts loss are high at points of low imped-
ance, and they rise sharply at the frequency d.
The change in reactance is shown in Fig. 180. The coil is inductive
up to frequency b. From b to c it has no noticeable effect on the tuning
and hence is pure resistance, or nearly so. Above c it is capacitive up
to a frequency slightly below d, where it again becomes of indefinite
234 ELECTRONIC TRANSFORMERS AND CIRCUITS
reactance. Thereafter, it is capacitive, except for brief frequency in-
tervals, where it is resistive, or only slightly inductive. At all fre-
quencies higher than the fifth peak, the coil is capacitive.
Since a coil has distributed constants it is subject to standing waves
at the higher frequencies. The character of these waves may be found
by tapping the coil at various points and inserting thermogalvanom-
eters in series with the coil at these points. The current distribution is
plotted in Fig. 180 against coil length. These diagrams show the kind
of standing waves as the frequency increases.
Current distribution is uniform at all frequencies below b. Most
chokes are used within the first impedance peak. The useful range for
choke impedance of 20,000 ohms in Fig. 180 is 1,700 to 2,800 kc. This
choke could be operated at 5,500 kc safely also, but the frequency range
is narrower. Also, the safe loss dissipation is less because it takes place
over half of the coil surface. Pie-section chokes have similar imped-
ance curves, but impedance peaks following the first are less pro-
nounced.
98. Large Power Coupling Coils. In the tank circuits of large power
amplifiers, the coupling coils are arranged to couple the antenna to the
power amplifier, and the equivalent circuit is like that of Fig. 172.
Optimum coupling between tank and antenna circuits is given by equa-
tion 103. The construction of the coupling coil itself is usually similar
to that of Fig. 174(6), with the coupling coil on the outside and spaced
from the main tank coil to reduce capacitive coupling. Taps are pro-
vided on the coupling coil for frequency and antenna resistance ad-
justments.
When the secondary circuit is untuned, and the secondary load is re-
active, all the secondary volt-amperes (which may exceed the sec-
ondary watts many times) flow through the transformer windings. It
is then necessary that tight coupling be used between primary and sec-
ondary in order to prevent loss of power, due to current circulating in
the primary without corresponding current flowing in the load. If the
load power factor is less than 20 per cent, currents and volt-amperes
in the circuit may be considered independent of the winding and load
resistances. In Fig. 171, let the load Z2 be inductive, comprising L3
and Rl. Also let
«i =
Li = primary self-inductance
-L2 = secondary self-inductance
Lm = mutual inductance
k = coefficient of coupling = L„j/VLiL2
HIGHER-FREQUENCY TRANSFORMERS
235
Then the secondary volt-amperes = I^Z2, and primary volt-amperes
= Ell I. The ratio of maximum secondary volt-amperes transformed
to the primary volt-amperes is related to k as follows:
\ £^1/1 /max
fc2
2(1 + Vl -fc2)- I?
(108)
This equation is plotted, together with values of ratio L^/L^ for maxi-
mum power transfer, in Fig. 181. If perfect coupling could be attained,
1.0
.9
.8
.7
X .5
<
S
^-
-^
Sfc.
^
N
_:
-U iA iz^
t ir—^^
k
N
1 : C r
\
/
1
i >— 1 H
\
/
I*"- 1
k=Lm/VL|L2
RL«2TrfL3
\
/
/
\
2
/
/
\
I2Z2
7max
^
/
\
^
^''
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
k
Fig. 181. Effect of coupling on maximum volt-amperes in untuned load.
all the primary volt-amperes could be transferred to the load. Iron-
core transformers operate at the extreme right of Fig. 181. With air-
core transformers it is often difficult to approach this condition, espe-
cially if voltages are high.
99. High-Frequency Power Supplies. In cathode-ray tubes for
oscilloscopes or television receivers, high-voltage low-current sources
of d-c power are needed for the accelerating anodes. Voltages range
from 1 to 30 kv, and currents are on the order of 1 ma down to 1 /xa.
Because of the many turns of wire required in high-voltage 60-
cycle transformers, high-frequency power supplies are often used. A
tetrode or pentode tube is used in a double-tuned circuit as shown in
236
ELECTRONIC TRANSFORMERS AND CIRCUITS
Fig. 182. Here the plate circuit is tuned by means of elements Li — Ci,
and the load circuit by means of L2 — C2. The grid winding is coupled
from Li. The coupling between Li and L2 is usually higher than criti-
cal, so that changes in load do not affect the resonant frequencies, but
instead vary the relative height of the two humps.
In general, the transformer represented by inductances L^ and I/2
:c2 ^Rl
Fig. 182. R-f power supply.
is step-up. Load R^ may be a voltage doubler, tripler, or quadrupler,
depending on the voltage needed from the oscillator.^
Frequencies used in such power supplies range from audio to medium
high radio frequency. The oscillator is usually operated class B or C ;
loaded Q (or ratio of volt-amperes to watts) ranges from 10 to 20.
Lower values of Q result in oscillator instability with load changes.
With ferrite cores, departures from the overcoupled case are feasible.
Frequency is usually lower than with air-core transformers. The core
itself affords a certain amount of load to the circuit and therefore
results in better voltage regulation from load-on to load-off conditions.
The single-tuned oscillator of Fig. 152 then becomes practical.
1 See "Radio-Frequency-Operated High-Voltage Supplies for Cathode-Ray
Tubes," by O. H. Schade, Proc. I.R.E., 31, 158 (April, 1943).
8. ELECTRONIC CONTROL TRANSFORMERS
Electronic devices are used to control speed, voltage, and current, or
may require control of these quantities. Most of the circuits can be
grouped into a few basic types. This chapter comprises typical cir-
cuits which use transformers and reactors.
100. Electronic Control Circuits. Vacuum-tube control circuits are
used for amplification of the input voltage, not always at a single
LOAD
INPUT
60 CYCLE
SUPPLY
ANODE
VOLTAGE
FiQ. 183. Basic thyratron circuit.
frequency. With thyratrons the input voltage triggers the tube, which
then allows current to flow into the controlled circuit, but the output
wave may not resemble the input wave, as is described below.
A simple circuit for thyratrons
operated with alternating anode
supply and resistive load is
shown in Fig. 183.^ During that
part of each cycle when the
anode is positive with respect to
the cathode, the tube conducts
current which passes through the
load, provided that the grid is at
the right potential. In Fig. 184
is shown the positive anode volt-
age for a half-cycle, with the
Fig.
A-G CRITICAL
GRID VOLTAGE
184. Anode and critical voltages in
basic thyratron circuit.
1 See Industrial Electronics, by F. H. Gulliksen and E. H. Vedder, John Wiley
& Sons, New York, 1935, p. 45.
237
238
ELECTRONIC TRANSFORMERS AND CIRCUITS
corresponding critical grid voltage. Any value of grid voltage higher
than this critical value will permit the tube to conduct. Once tube
conduction is started, change of grid voltage to a value less than criti-
D-C GRID
INPUT VOLTAGE
CRITICAL
GRID VOLTAGE
FiQ. 185. Grid control of a thyratron tube with (a) variable d-c grid voltage,
(b) variable d-c grid voltage with superposed a-c grid voltage, (c) fixed d-c grid
voltage with superposed a-c grid voltage of variable phase position.
cal will not stop conduction. Conduction does stop, however, at the
end of the half-cycle, or when the anode voltage falls to zero. Three
methods of controlling the load current are shown in Fig. 185. Figure
185 (a) shows how direct voltage applied to the grid permits conduction
through the tube over the shaded portion of the cycle. Minimum con-
ELECTRONIC CONTROL TRANSFORMERS 239
trollable current is half that which would flow if the tube were free to
conduct over the entire positive half-cycle. This method of control is
not precise, especially near the half -power point, because a small dif-
ference in d-c input control voltage produces a comparatively large
change in conduction angle or may cause the tube to fail to fire alto-
gether.
Figure 185(6) shows a more satisfactory method of amplitude con-
trol. The grid is maintained at a positive d-c potential, and an alter-
nating voltage is superposed on it which lags the anode voltage by
90°. Varying the magnitude of the d-c grid voltage shifts the zero
axis of the a-c wave up or down, and intersects the critical a-c grid
voltage at different points of the cycle. Tube current can then be
varied from zero to maximum. Close control of the tube current can
be obtained because the grid voltage wave intersects the critical curve
at a large angle.
In Fig. 185(c) another method is shown. The phase of a superposed
alternating voltage is shifted upon a negative d-c bias which is more
negative than the critical characteristic. Changing the phase position
of the a-c grid voltage varies the tube current from zero to maximum.
The phase position of the grid voltage can be shifted by several
methods, one of which is discussed in Section 101.
The anode supply transformer carries load direct current. Core
saturation may be prevented by an air gap; heating and regulation in
the primary winding due to excitation current govern the length of
air gap. Ordinarily, permissible maximum induction may be higher
than in a single-side amplifier transformer because impedance or fre-
quency response considerations are irrelevant with a 60-cycle supply
line. Excitation current may be comparable in magnitude to load
current. However, there is this difference: with a resistive load, cur-
rent flows only during the positive half-cycle, whereas magnetizing
current flows during the whole cycle. Secondary current is a series of
pulses, the maximum width of which is 180°. The rms value of these
pulses is half the peak amplitude, and this is the current which
governs secondary wire size. Rms secondary voltage is 2.22 times
maximum d-c load voltage, as in a single-phase half-wave rectifier.
Design of the transformer is similar to the anode transformers in
Chapter 3, except for the higher induction and current wave form.
Full-wave circuits ^ operate with two thyratrons and a center-tapped
transformer in which the net d-c fiux is zero. The design of the anode
transformer is described in Section 102.
1 See Gulliksen and Vedder, op. cit., p. 64.
240 ELECTRONIC TRANSFORMERS AND CIRCUITS
101. Grid-Controlled Rectifiers. The basic a-c grid control circuit
described in the last section may be extended to more than one tube
and may control large amounts of power. Any of the rectifier circuits
of Table VII (p. 62) may be used with grid control of output voltage,
which supplants the older practice of using induction regulators in
the supply lines.
Smooth control of rectifier d-c voltage under load conditions is
possible through the use of thyratrons or ignitrons with phase-shift
control of the grid or igniter. Stable control of filtered output is possi-
ble only with choke input filters. In Chapter 4 the regulation of a
rectifier is shown to be lowest if the input choke has inductance greater
than critical value. With grid control, if the filter choke inductance is
great enough, the tube conducts even after the anode reaches zero.
The tendency of current to stop at voltage zero builds up voltage
across the filter choke in such a direction that cathode potential is less
than zero after the anode reaches zero. Thus conduction in the tube
is maintained until the next tube fires. If the choke inductance is less
than critical, tube current wave is discontinuous, regulation is poor,
transient surges and oscillations in the output voltage occur, and con-
trol is unstable.
For a single-phase full-wave rectifier with grid control, the direct
voltage output decreases as shown by curve I in Fig. 186. Critical
value of inductance increases with firing angle and so does ripple volt-
age as shown by curves II and III in this figure. For a three-phase
full-wave rectifier, the direct output voltage is approximately 41 per
cent greater than the single-phase values shown in Fig. 186, and the
critical value of choke reactance less filter capacitor reactance is ap-
proximately one-tenth of the single-phase values over the range of
20° to 90° firing angle. At 90° firing angle, the d-c output always is
zero. Voltage across the choke reverses in sign but does not increase
in magnitude even with the maximum angle of 90°. Therefore the
maximum voltage from choke to ground is not changed, and the de-
sign of a reactor for this type of rectifier is the same as for a rectifier
without grid control, except for the value of inductance.
Choke-input filters can be used to maintain continuous current flow
in single-phase half-wave rectifiers. Although the output voltage is
reduced, as mentioned in Chapter 3, this combination is occasionally
useful.^
1 For general calculation of discontinuous waves, see "Voltage and Current
Relations for Controlled Rectification witii Inductive and Generative Loads," by
K. P. Puchlowski, Trans. AIEE, 64, 255 (May, 1945). For theory of controlled
ELECTRONIC CONTROL TRANSFORMERS
241
Grid-controlled rectifiers have more irregular current wave forms and
therefore more pronounced a-c line harmonics than ordinary rectifiers.'
Two methods of providing phase shift control of a constant amplitude
a-c grid voltage for grid-controlled rectifiers are shown in Fig. 187. In
ii 1.0
a.
Id 09
^^
o
o08
Li
07
0,6
0,5
0.4
03
02
0,1
O,
X
1 1 1 1 1 1 1 1 '
,_= FIRST INDUCTOR REACTANCE AT RIPPLE FREQUENCY
- "l
— M
-FIR
■MAX
JLTIF
;t 0/
LOA
)
VPACI
3 RES
TOR
ISTA
3Y0F
?EAC
■(CE t
OINA
rANC
"dC'
EOF
E AT
DCF
CUR
RIPPLE Ffi
OR CONTIN
1
VEm TO
EQU
uous
■IND
Epk
FIED
:ncy
CUR
RIP
E
RENT
'LE-
30
/
\
/
\ RECT
\
/
\
\ 1 VOLTAGE
\
i
\
-F
IRINC
ANS
-E
"^
s
\
\
\
\'
1
\
S
r\
/
\
/
j
\
/y
y\
A
y
V
\
-=
m
\
10 20 30 40 50 60 70 80 90
FIRING ANGLE IN DEGREES
20O ji
S 5
18 O 3
I 5
16 '- ^
'° z _j
O Q.
U Q.
» a: 2
o
12,"
X
loi
6
6
4
2
Fig. 186. Output voltage, ripple, and current continuity in single-phase full-wave
grid-controlled rectifier.
(a) a small value of resistance R effectively connects the upper grid
circuit terminal to the left-hand terminal of the supply transformer,
and a large value of R shifts it nearer to the right-hand terminal of the
rectifiers, see "Critical Inductance and Control Rectifiers," by W. P. Overbeck,
Proc. I.R.E., Z}, 655 (October, 1939).
1 See "Harmonics in A-C Circuits of Grid-Controlled Rectifiers and Inverters,"
by R. D. Evans and H. N. Muller, ,lr.. Trans. AIEE, 58. S61 (1939).
242
ELECTRONIC TRANSFORMERS AND CIRCUITS
supply transformer. If the supply transformer is center-tapped, the
vector diagram of Fig. 188 shows the phase position of the grid voltage
Eg in solid lines for a small value of i?, and in dotted lines for a large
value of R. Varying the rheostat R thus varies the rectifier output
from full voltage to a low voltage.
17
TO GRID
CIRCUIT
■VX>.>«.A>O^A>^
TO 60
CYCLE SUPPLY
(a)
TO D-C CONTROL
VOLTAGE
17
TO GRID
CIRCUIT
■vaa>JLjuui^>^
TO 60
CYCLE SUPPLY
(b)
Fig. 187. Resistance-inductance phase shift circuits.
In Fig. 187(6) resistor R is fixed and inductance L is varied by means
of direct current flowing in one of its windings. The vector diagram of
Fig. 188 still applies; the solid lines are for high inductance and the
dotted lines for low inductance. Direct current for varying the induct-
ance may be obtained through a thyratron or a vacuum tube, espe-
cially when rectifier output voltage is automatically controlled. The
reactor is usually of the saturable type.
ELECTRONIC CONTROL TRANSFORMERS 243
The widest range of inductance is obtained with zero direct current
at the higher inductance. In some vacuum-tube circuits, the minimum
direct current is not zero, and a bias winding is added to the center
leg to cancel the d-c ampere-turns with minimum current in the main
d-c winding. Saturable reactors have many uses besides that described
here, and are discussed more fully in Chapter 9.
Fig. 188. Vector diagram for Fig. 187.
102. Thyratron Transformers. Anode transformers used for supply-
ing thyratrons resemble rectifier anode transformers but generally
have higher rms current for a given direct current in the load, and are
more subject to voltage surges. With resistive loads, anode current
has the same wave shape as the shaded portion of the anode voltage in
Fig. 185. The relation of peak, rms, and average currents is shown
in Fig. 189 as a function of firing angle 6 for single-phase full-wave
circuits. Voltage reduction as a function of 6 is shown in Fig. 190.
If a transformer is designed for operation with zero firing angle, maxi-
mum current flows in any given load; the transformer is then capable
of carrying the current with greater firing angle, so long as the load
impedance remains the same. If the load impedance is changed with
6 > to keep the load current as high as possible, the limiting value
may be found from Fig. 189. The average load current which may
flow without overheating the transformer decreases as increases.
High-voltage surges occur when capacitance input filters are used
with grid-controlled rectifiers. To a degree these surges are likely to
occur even when the load is nominally resistive, because of incidental
capacitance in the transformer, wiring, and other components. If the
load is a radio-frequency generator, the r-f bypass capacitor adds to
this effect. In Fig. 191 (a) the total amount of external capacitance is
designated Ci. A half-wave anode transformer is shown for simplicity,
but each half of a full-wave transformer, or each phase of a three-phase
transformer, behaves similarly. When the thyratron firing angle is
244
ELECTRONIC TRANSFORMERS AND CIRCUITS
greater than zero, a steep voltage wave front occurs at the instant of
firing t^, Fig. 191(6), as follows:
Normal voltage induced at point A in the secondary winding is ei
volts above ground, just prior to tg. As soon as the thyratron fires,
? a:
ft
06
5g
/
0.8
0.7
' N
\
\
-*
4
/
f
e -
-TT-
\
/
1
HK
■^
•V,
N
\
^RMS
0.6
IPK
\
\
0.5
1
AV
PK
\,
\
1
\
\
\
0.4
\
\
\
\
\
0.3
0.2
\
k
\
\
Ir
N
\.y
\
/
"'"-■
\
\
Ia\
/
^
--
\
\
\
s.
0.1
^
\
\
\
15 30 45 60 75 90 105 120 135 150 165 180
FIRING ANGLE 9 (DEGREES)
Fig. 189. Single-phase thyratron currents.
the external wiring and circuit capacitance Cx momentarily forms an
effective short circuit from A to ground. A large surge current flows
into this short circuit, but initially this current cannot be drawn from
the primary because of the inevitable inductance of the windings. The
initial current is therefore supplied by the secondary winding capaci-
tance. Since point A is momentarily short-circuited, a surge voltage,
equal and opposite to ei, is developed in the secondary winding. This
voltage surge appears across the turns or layers of winding nearest
to A. Unless precautions are taken in the design of the anode trans-
ELECTRONIC CONTROL TRANSFORMERS
245
former the voltage may be high enough to damage the winding insu-
lation.
As is shown in Chapter 10, with steep wave fronts in single-layer
windings initial voltage distributes most equally between turns when
ratio a = y/Cg/C\„ is small, Cg, being the capacitance of the winding to
ground and C„ the series capacitance across the winding. If the sec-
ondary of Fig. 191 were a single-
layer winding of n turns, C„,
would be Cs/n. In multilayer
coils, ratio a is not so readily
defined. In general, small effec-
tive layer-to-layer capacitance
means small effective Cg in rela-
tion to C„, small a, and more
linear initial distribution of volt-
age. Many layers are better
than few layers in keeping ca-
pacitance Cg small. In the limit,
a one-turn-per-layer coil would
have small « and good initial
voltage distribution. In practice
this extreme is not necessary to
avoid layer insulation break-
down. It is usually sufficient to
split the secondary into part
coils, like S, and & in Fig. 59.
This reduces C, to a quarter of
the corresponding capacitance of
full-width coils. Ratio a is reduced, and voltage distribution im-
proved.
Even with part coils there is some non-linearity of voltage distribu-
tion, especially in the top layer. This non-linearity may be minimized
by providing a static shield over the top layer and connecting it to
point A, Fig. 191. The momentary voltage described above appears
within the winding, and unless there are taps it may not be observable.
If a surge suppressor circuit (usually a capacitor-resistor network
across the secondary) is used, it does not appreciably diminish the
internal winding voltage surge, but such a surge suppressor may be
necessary to damp out oscillations in the external circuit due to firing
of the thyratrons.
Fig. 190.
Relation of firing angle to
voltage output.
246
ELECTRONIC TRANSFORMERS AND CIRCUITS
Secondary windings of control transformers in thyratron grid cir-
cuits, like those of Fig. 187, should be insulated for the anode voltage.
When thyratrons arc-back the grids may be subjected to full anode
potential, which would damage lesser amounts of insulation.
(o) SCHEMATIC CONNECTIONS FOR EACH PHASE
\a
L^.
(b) SECONDARY VOLTAGE 6$ AND CURRENT Is
WITH PHASE- BACK
Fig. 191. Thyratron plate transformer operation.
103. Peaking Transformers. It is stated in Section 100 that a large
angle of intersection between the grid fii'ing voltage wave and critical
grid voltage is desirable for accurate control. A grid wave form with
vertical front edge would be ideal. To produce a steep peaked wave
form for firing thyratron tubes, sometimes special transformers are
used. Usually the design depends on the non-linearity of the mag-
netizing current. Figure 192 shows a peaking transformer in which the
magnetic core is made of special laminations. The primary is wound
on the full-width left leg, and the secondary on the right leg which is
made of a few laminations of small width. In the space between pri-
mary and secondary is a laminated shunt path with an air gap. In
Fig. 193 are shown the core fluxes <^,„ and <^s, linking the primary and
secondary coils, respectively. At low inductions, the same flux links
both coils. As the flux rises from zero in each cycle, at first all the
ELECTRONIC CONTROL TRANSFORMERS
247
PRIMARY
MAIN CORE
AIR GAP ,
■=;
SECONDARY
CORE LEG
SECONDARY
OUTPUT VOLTAGE
—n 6c
Fig. 192. Peaking transformer.
flux links the secondary coil, but because of the smaller cross section of
the right leg it saturates at the value </>s and the main flux (l>m flows
through the shunt path. Thus there is a long interval in each cycle
during which the flux change is substantially zero, and no voltage is
induced in the secondary coil. During the short period dg, a voltage
is induced in the secondary coil
which has a very peaked wave form.
This happens twice in each cycle.
Because of the shorter time for the
change in (f>^, d(f>/dt would remain
nearly constant over the angle 6s
if there were no leakage flux, and for
1 : 1 turns ratio there would be ap-
proximately equal volts in the pri-
mary and secondary coils. The sec-
ondary flux change takes place over
a much shorter period of time, and
the flux rises to only a fraction of its maximum value (j>„,. Therefore
less core area is needed in the secondary leg to obtain the desired volt-
age Bg. This leads to the following approximate ratio.
Fw. 193. Fluxes and secondary volt-
age in peaking transformer.
As
A„
— sm —
TT 2
(109)
where A^ = core area in the secondary leg
Ap = core area in the primary leg.
Peaked secondary voltage may be made steeper by the use of nickel-
iron laminations in the secondary leg, because these alloys have sharp
saturation. The air gap in the center leg prevents it from shunting all
the primary flux, which would reduce the secondary voltage to zero.
248 ELECTRONIC TRANSFORMERS AND CIRCUITS
This air gap should be no more than 5 to 10 per cent of the window
height, to keep leakage flux from threading through the secondary coil
and giving a less peaked wave form. With this length of air gap and a
total window length of twice the window height, the secondary turns
are, for a 1:1 voltage ratio,
N, « 2Nj, (110)
where A^p is the number of primary turns.
Transformers may be made to peak by the use of special circuits
instead of special cores. Voltage wave forms like Fig. 193 are obtain-
able if the primary winding is operated at a voltage exceeding satura-
tion but is connected in series with a large linear inductance or other
high impedance. Grain-oriented core material, with its rectangular
hysteresis loop, is well suited to peaking transformers. When primary
and secondary windings are wound over the same magnetic path, the
same volts per turn are induced in both windings, and equation 110
no longer applies. A peaking reactor circuit is shown in Chapter 11,
Fig. 262.
104. Current-Limiting Transformers. Filaments of large vacuum
tubes sometimes nmst be protected against the high initial current they
draw at rated filament voltage. This is done by reducing the starting
voltage automatically through the use of a current-limiting trans-
former, with magnetic shunts between primary and secondary windings.
The shunts carry very little flux at no load; as the load increases, the
secondary ampere-turns force more of the flux into the shunts until at
current I^c, Fig. 194(B), the output voltage is zero. The same principle
is used to limit current in transformers for oil-burner ignition, precipi-
trons, and neon or other gas-filled tube signs.
Cross-sectional area through each shunt path is the same as that
of the upper or lower leg of the shell laminations; then flux in the
shunts does not exceed that in the core, shunt iron loss is not abnormal,
and secondary voltage is sinusoidal. At short-circuit current Igc, half
the total flux flows through each set of shunts. The air-gap length in
each shunt path can be found from equation 35:
0.6A^/,,
Ig = (inches)
Bm
where A^ = secondary turns
B,n = allowable induction in the shunts (in gauss).
ELECTRONIC CONTROL TRANSFORMERS
249
The constant 0.6 is generally too small because of the flux fringing
around the gap. The increase of gap made necessarj- by fringing may
be found from Fig. 72 (p. 102). If the shunts are too short, the trans-
former does not limit the current properly. It is best to have slightly
less air gap than necessary, and find by trial the right length of
MAGNETIC SHUNTS
AIR GAP
-MAIN CORE
P= PRIMARY WINDING SECTION
S= SECONDARY WINDING SECTION
(A)
OPEN-CIRCUIT
VOLTAGE
SECONDARY
VOLTAGE
SECONDARY LOAD
CURRENT
(B)
Fig. 194. (A) Current-limiting transformer; (B) output voltage versus current
curve.
shunt. Fringing flux heats the coils and core somewhat more than
in an ordinary transformer. If the secondary current is heavy, coils
are wound pancake fashion and connected in parallel; they may
have to be cross-connected for the coils to divide the load equally.
If the ordinate for open-circuit voltage and abscissa for short-circuit
current in Fig. 194(B) are equal, the curve is a quarter-circle for a
perfect transformer because the secondary current at short-circuit is
all reactive. With core, shunt, and winding losses the curve for an
actual transformer falls some 10 to 15 per cent less than the quarter-
circle at currents 0.5 to 0.75 times /„..
250
ELECTRONIC TRANSFORMERS AND CIRCUITS
Fig. 195. Autotransformer voltages and
currents.
105. Autotransformers. An autotransformer has a single winding
which is tapped as shown in Fig. 195 to provide a fraction of the pri-
mary voltage across the secondary load. The connections may be
reversed so that a step-up voltage is obtained. The regulation, leak-
age inductance, and size of an
autotransformer for a given rat-
ing are all less than for a two-
winding transformer handling
the same power. Where the volt-
age difference is slight, the gain
is large. Where the voltage dif-
ference is great, there is not
much advantage in using an
autotransformer, nor can it be
used where isolation of the two
circuits is required.
Autotransformers are used in
electronic applications chiefly for the adjustment of line voltage, either
to change it or to keep it constant. Examples are the reduction of
plate voltage for tuning an amplifier and the maintenance of constant
filament voltage. Taps may be chosen by means of a tap switch to
adjust the load voltage. The load voltage may be adjusted to within
half the voltage increment between taps.
If the voltage is adjusted while load remains connected, bad switch-
ing arcs occur, either from breaking the circuit or from short-circuiting
turns. To provide for adjustment under load conditions, a resistor
may be momentarily connected in the circuit as the tap switch bridges
from one tap to the next, and current is limited to full-load value. In
large power tap changers, a reactor replaces the resistor to avoid heat-
ing and losses.
The v-a rating of an autotransformer depends on the ratio of input
to output voltage. In Fig. 195 the output current I2 = h + I3. Let
p = per cent tap/100 = E2/E1. Neglecting losses, I2 = Ii/p and
h = n/p - l)/i. Then
Volt-amperes (in the upper portion) = (1 — p)EiTi
Volt-amperes (in the lower portion) = pE^I^ = (1 — p)EiIx
which satisfies equality of volt-amperes in each section. For ratio p
close to unity, the v-a rating and hence size for a given output can be
made very small; for small values of p the size is not much less than
ELECTRONIC CONTROL TRANSFORMERS
251
that of a two-winding transformer, but the autotransformer has much
less regulation. Its effective winding reactance and resistance de-
crease as (1 — p)^; that is, for a given unit,
X (or R) as autotransformer
X (or R) as two-winding transformer
(1 - pf
(111)
Appreciably less regulation is obtained in an autotransformer, even
when size is not reduced much, because the right-hand term in equa-
tion 111 is squared.
»
Fig. 196. Adjustable primary anode transformer.
When the power for electronic equipment is supplied by a 230-volt
line, but auxiliary items such as relays and small motors are used at
115 volts, a convenient way of obtaining the latter voltage is to center-
tap the primary of a large plate transformer, and use it as a 2 : 1 step-
down autotransformer. The larger primary winding copper requires
little extra space, and an additional transformer is thereby saved.
To improve the closeness of voltage control, a variable autotrans-
former has been developed in which the moving tap is a carbon brush
252 ELECTRONIC TRANSFORMERS AND CIRCUITS
which slides over exposed turns of the winding. Brush resistance pre-
vents excessive transition current and permits smooth voltage control;
yet it offers little additional series resistance to the load. The same
idea can be applied to two-winding transformers for secondary voltage
adjustment. A typical unit of this kind is shown in Fig. 196.
When autotransformers are used on three-phase supply lines, they
may be connected the same as two-winding transformers in star, delta,
open-delta, or Scott connections. The last two connections are less
subject to objectionably high regulation in autotransformers and, if
they supply three-phase anode transformers, cause no serious primary
voltage unbalance for voltage ratio p close to unity.
106. Static Voltage Regulators. Automatic regulators of various
kinds have been devised for keeping comparatively small amounts of
power at a constant voltage. Figure 197 shows one circuit for a res-
K^^JZkJ^
7-^^T^
A-C SUPPLY
LINE
VOLTAGE
I
}'
OUTPUT
VOLTAGE
Fig. 197. Resonant-circuit voltage regulator.
onant-reactor voltage regulator. Inductance Li is linear. Inductance
L2 and capacitor C2 are parallel-resonant at the supply line frequency
and rated voltage. The pair draws very little current, so that the reac-
tive voltage in Li is low. Output current flows through its secondary
winding which is of such polarity as to maintain rated voltage. In-
ductance L2 is partially saturated at this voltage. If line voltage falls
below rated value, less current is drawn by La, and the L2C2 combina-
tion becomes untuned. Total current to the parallel circuit is then
capacitive, and this capacitive current, drawn through Li, raises the
output voltage. Conversely, if line voltage rises above rated value, the
L2C2 combination becomes untuned on the inductive side, and the out-
put voltage falls below the line value. Output voltage variations of
±1 per cent are obtained with ±10 per cent line voltage variations
in this manner, and with load changes from zero to full load.
Constant supply frequency is a condition for resonance at rated
voltage; with the good frequency control of modern power systems
this condition is generally fulfilled. Load power factor variations
ELECTRONIC CONTROL TRANSFORMERS
253
cause output voltage to change. Some regulators are provided with
taps to minimize this effect. Output wave form contains a noticeable
third harmonic, because the large magnetizing current of L2 must flow
through appreciable impedance in Lj. Owing to the partial saturation
of reactor L2, it tends to operate at a high temperature and requires
good ventilation. Practical regulators are in use with ratings from
25 v-a to several kva.
UNREGULATED
D. C. SUPPLY
_i
SERIES R
—'VsAAj-
ANODE
VOLTAGE
V-R
TUBE
REGULATED
D. C. OUTPUT
L
80
60
40
20
5 10 15 20 25
ANODE MILLIAMPERES
Fig. 198. Voltage regulator characteristic.
30
Electronic voltage regulators make use of a gas-filled regulator
tube, which has a v-a characteristic such as that shown in Fig. 198.
Current drawn by this tube changes between wide limits with virtually
no change in voltage. A series resistor is ordinarily used to limit the
current. When output current in excess of the "V-R" tube rating is
desired, it may be used as a voltage reference for a current amplifier.
Some voltage regulators amplify the difference between a voltage
reference and the output voltage of a rectifier or generator. This
difference is called the error voltage. The amplifier output reduces the
generator field if generator output voltage is high and increases the
field if the output voltage is low. Likewise, motor speed may be
regulated by the difference between tachometer output and a voltage
reference. Or the angular position of a motor may be controlled elec-
trically as desired by remote means. These means are discussed in
254 ELECTRONIC TRANSFORMERS AND CIRCUITS
books on servomechanisms.^ If a thyratron amplifier is used as part
of a servo system, one thyratron may produce an effect opposite to
tiiat of the other, such as reversing current in the load. This amplifies
the power controlled by error voltage.
In many modern regulator and servo systems, magnetic amplifiers
are used. These devices are described in Chapter 9.
107. Demodulators. Demodulators or detectors measure phase or
amplitude variations which are used to convey intelligence or control
^^XL-I-
4 — ! — •— • ■ — » — »♦
Fig. 199. Phase-difference demodulator.
another device. A circuit often used for phase detection is shown in
Fig. 199. Transformer Ti has a balanced secondary with ei volts per
side. In frequency or phase demodulators, the circuit is called a dis-
criminator, and uses air-core transformers. This circuit is used in
phase demodulation to produce a d-c voltage proportional to phase
shift. In this case e^ leads ei by 90° for zero output. This condition
is shown in the upper vector diagram. If 62 should lead the upper ei
by less than 90°, e^ would increase and Cb decrease, causing a net d-c
voltage to appear across the output. If 62 should lead the lower Ci
by less than 90° the d-c output voltage would change polarity. The
plate voltage of one diode is the vector sum of ei and 62, and is 90°
out of phase with the plate voltage of the other diode (which is the
vector difference of ei and 62). If the phase angle <f> between ei and 62
changes either positively or negatively, output voltage Eg^ is very
nearly proportional to c^, up to </> = ±60°. For positive phase shift,
the output Efc is negative; for negative phase shift, Eac is positive.
For good linearity, load resistances Rl should be large compared to the
diode resistances.
Modifications of this circuit made to eliminate the vector difference
1 See Principles of Servomechanisms, by G. S. Brown and D. P. Campbell,
John Wiley & Sons, New York, 1948.
ELECTRONIC CONTROL TRANSFORMERS
255
voltage or to reduce the degree of amplitude modulation are known as
balanced and ratio detectors.^
As used in servo control, the demodulator produces d-c output which
changes polarity when one voltage reverses with respect to the other;
in this case 62 adds directly to one voltage ej and subtracts from the
other. The lower vector diagram shows how a voltage 62 — ei causes
a low difference voltage on diode A and a large sum voltage on B to
produce a net d-c output voltage. Transformers in phase-reversal de-
modulators are usually iron-cored. Both transformers Ti and T^
/RECTIFIED
N / LOOPS
CARRIER
MODULATION
ENVELOPE
Fig. 200. Rectified amplitude-modulated wave.
should have low exciting current so that the phase angle between the
voltages to be measured is not appreciably increased. For the same
reason, the source impedances should be small; when this is not pos-
sible, the transformers should be matched. Figure 131 shows how
necessary this is. With equal source and load resistances, several de-
grees of phase shift are introduced even if the ratio of transformer re-
actance to source resistance is 10:1. Variations in this reactance cause
errors in the phase-demodulator output.
In a-m receivers, the received signal is modulated radio frequency.
If 100 per cent amplitude modulation is used, the r-f amplitude is varied
from to 200 per cent at an audio-frequency rate. The detector, or
demodulator, of the receiver first rectifies the r-f signal and then elimi-
nates the r-f component, leaving only audio frequencies in the output.
A rectified signal, before the r-f is eliminated, is shown in Fig. 200.
Demodulation is accomplished in the circuit of Fig. 201 by means
of diode 6H6-2. Each half of the r-f cycle is rectified and the d-c
1 See "Diode Phase-Discriminators," by R. H. Dishington, Proc. I.R.E., 37, 1401
(December, 1949) ; also, Radiotron Designer's Handbook, F. Langford-Smith,
RCA Victor Division, Harrison, N. J., p. 1088.
256
ELECTRONIC TRANSFORMERS AND CIRCUITS
power is absorbed in resistor R^. Audio power is bypassed around i?3
by capacitor C2, and the voltage is impressed upon the primary of trans-
former 7*2. If an amplitude-modulated wave is used in this amplifier,
the output voltage of winding Si on transformer Ti has the form shown
in Fig. 200. The first few cycles are shown as full-wave rectified loops
with constant amplitude, that is, with no modulation. The audio out-
put for this section of the wave is zero. A sine-wave envelope of 100
~; -^^ AUDIO O
Fig. 201. Demodulator and automatic gain control circuits.
per cent modulation is shown in the rest of the figure. Average volt-
age left after the carrier frequency half-loops have been absorbed by
the r-f filter LiCi is the audio voltage impressed on transformer T2.
The method of demodulation just described is known as diode de-
modulation. It is often accomplished by means of a single diode, and
then every other lobe of the wave in Fig. 200 is omitted. Methods
are in use also for demodulation with a triode, in which some amplifica-
tion of the demodulated wave is obtained.
108. Automatic Gain Control. Vacuum-tube amplification factor is
constant under certain conditions of operation. With high current
operation the amplification factor in the region of high anode current
and low anode voltage is no longer constant.
Some tubes are designed to have large variations in amplification
factor. These are known as variable-mu, remote cut-off, or super-
ELECTRONIC CONTROL TRANSFORMERS
257
control tubes. The mutual conductance of such tubes is highly vari-
able with grid bias. Figure 202 is the curve of mutual conductance
for a tube of this kind. Such a characteristic can be used to reduce
gain at high amplitudes and thus prevent overmodulation in audio
Ef
1 1 — r—
-6.3 VOLTS
=o'
1
SUPPRESSOR VOLTS
SCREEN VC
LTS
" iO
Im
1
r
V
/"/
1
//
/
^
^
y
2500
2000
o
X
>
o
iE
o
z
15 I 1500 -g
■* o
" r
O H
10 S 1000 g
UJ
o
(D
5 - 500
UJ
-50
-40 -30 -20 -10
CONTROL-GRID VOLTS
Fig. 202. Variable-mu tube (6SK7) mutual conductance curve.
systems. In Fig. 201 the circuit shown automatically reduces gain for
excessive values of applied grid voltage eg on the grid of the 6SK7
tube. This tube drives a 6L6 output tube through transformer Ti.
On this transformer there is an auxiliary winding So which is con-
nected to rectifier tube 6H6-1 and produces the rectified output
across resistance 2?2 having a negative potential at the point shown.
With large signals, the voltage rectified across J?2 is large and reduces
the mutual conductance and plate voltage swing of the 6SK7 tube.
Nearly constant output voltage is maintained in the 6L6 output.
If the power output of the 6L6 tube is delivered mainly into a linear
258 ELECTRONIC TRANSFORMERS AND CIRCUITS
a-c impedance, the slight additional load imposed by the gain control
makes little difference. But if all the output is delivered to rectifier
loads, as it is in Fig. 201, the non-linearity of both tube and load
causes output distortion. This is true particularly of beam or pentode
output tubes. The normal class A output of a 6L6 beam tube is 6
watts but, if the output power is all rectified, only 50 mw can be drawn
without excessive distortion. Half-wave rectifiers and capacitor-input
filter outputs are worst in this respect, because of the current dis-
continuities. If the automatic gain control rectifier input is taken
from a tuned amplifier, these difficulties decrease. The tuned circuit
capacitor readily supplies irregular current wave forms, provided the
amplifier has sufficient power output available.
Automatic volume control (AVC) is applied in receivers to either
the r-f or audio stages, to maintain approximately constant volume in
spite of fading or other causes of input voltage variations. It is ap-
plied in audio amplifiers to maintain better output volume with differ-
ing voice levels.
If the input grid resistor Ri in Fig. 201 is connected to a fixed
negative bias the AVC is inoperative below the value of bias voltage.
This is called delayed AVC; with it, no AVC is applied until a certain
output level is reached. In some receivers, more than one stage may
be controlled, and the AVC action is amplified.
Circuits similar to this are used in power-line carrier receivers. The
carrier frequency is 40 to 200 kc, and audio frequencies are employed
for modulation. In transformer Ti some special problems are encoun-
tered because the transformer operates over a range of 40 to 200 kc,
delivers the correct amount of voltage to the automatic gain control
tube 6H6-1 for proper AVC action, delivers the proper output to the
audio load without distortion, and obtains these voltages from a nearly
constant current source. The transformer ratio is obtained by estimat-
ing the r-f voltage swing obtained with a square primary input current
wave, and dividing this by the voltage required to produce the neces-
sary audio output after choke Li smooths the rectified lobes to the
average value shown by the heavy dotted lines in Fig. 200. Trans-
former voltages and currents are calculated as in Table VII (p. 62)
for a single-phase full-wave rectifier, but with peak audio current and
voltage taking the place of d-c output.
9. MAGNETIC AMPLIFIERS
Amplifiers with saturable reactive elements are known as magnetic
amplifiers. Such amplifiers have been built with power gains of over
1,000,000. Compared with electronic amplifiers, magnetic amplifiers
have the advantage of long life. It is the purpose of this chapter to
describe the operation and design of elementary magnetic amplifiers.
109. Saturable Reactors. From the fundamental theory of trans-
formers, it will be recalled that the voltage induced in a winding
usually far exceeds the resistance drop in that winding. In other
words, winding open-circuit reactance usually is much greater than
winding d-c resistance. Further, it will be recalled that a relatively
small amount of direct current flowing into the winding of a trans-
former, in the core of which there is no air gap, causes the core to
saturate. Thus, the reactance of the transformer may be varied by
a small amount of d-c power. Now, if one winding of a transformer
is connected between an a-c supply and a load, the amount of power
delivered to the load may be controlled by a small amount of d-c
power flowing in another winding. Because of the fact that open-
circuit reactance ordinarily exceeds d-c resistance, the possibility of
power amplification is inherent in a transformer. When one winding
of a transformer is used for d-c control power and another for a-c
output power, the transformer is called a saturable reactor.
110. Simple Magnetic Amplifiers. A single reactor, with a battery-
fed d-c source controlling one winding and a-c power fed through
the other winding, would have a-c voltage induced in the d-c winding.
If this d-c winding were closed only on the battery, it would effectively
short-circuit the a-c voltage in the power winding. This difficulty
might be overcome by using a high impedance in the d-c control cir-
cuit. A more common solution is to use two reactors, one of the d-c
windings of which is reversed, while the a-c windings add normally.
Connections of this sort are shown in Fig. 203 (a) , with the a-c wind-
ings in series; it is possible to connect them in parallel as in Fig. 203(6)
in order to allow more load current to flow at lower a-c voltage.
259
260
ELECTRONIC TRANSFORMERS AND CIRCUITS
When there is zero direct current in the control windings of Fig.
203, both reactor impedances are large and prevent any load current
except exciting current from flowing throughout the a-c voltage cycle.
AVhen direct current is applied to the control windings, impedance
remains large for the first part of a cycle, until saturation flux density
is reached. Then reactor impedance is reduced and a large load cur-
rent may flow. With rectangular B-H loop core material, such as
that shown in Fig. 22 (p. 26) , the change from high to low impedance
LOAD I — I
oA-CO
SUPPLY
oA-C6
SUPPLY
(a) (b)
Fig. 203. (a) Series and (6) parallel-connected simple magnetic amplifiers.
is abrupt. If the loop were a true rectangle, the load current wave
form would be as shown by II in Fig. 204(a). Only the exciting cur-
rent flows in the load during the interval O-^i. Then saturation is
reached and load current suddenly rises to a large value. From (9i
to T, ir, has sinusoidal shape. During the next half-cycle, this load
current shape is repeated but in the reverse direction.
For a 1 : 1 turns ratio in each reactor, current 4 in each control
winding equals zV, minus the exciting current. In one reactor, because
of the reverse connection, current v flows in the opposite direction.
Total current in the control circuit is as shown by the lower trace of
Fig. 204(a), the average value of which is the input direct current /<..
Thus load current contains fundamental and odd harmonics, whereas
control current contains only even harmonics. If sufficient control
current flows to saturate the cores over the full cycle, load current
also flows over the full cycle and is sinusoidal in wave form. For
turns ratios other than unity, load and control currents are inversely
proportional to turns ratio.
In the foregoing it was assumed that control current was free to
assume the shape shown in Fig. 204(a). This is true, on a 1:1 turns-
ratio basis only if the control circuit impedance is small. If total
control circuit resistance is denoted by Re and load resistance by Rt,,
MAGNETIC AMPLIFIERS
261
for Ru <<C Rl, load and control currents are sine waves, or portions
thereof. If the opposite is true, namely Re y> Rl, control current wave
shape is determined by Re- For very large Re, control current is con-
tinuous, and the current wave shapes approach those in Fig. 204(6).
In this figure, d-c source impedance is large, even harmonics cannot
flow, magnetization is "constrained," load current is flat-topped, and
voltage across the reactor is distorted considerably. This distortion
can be overcome by the use of a capacitor across the control coils as
(b)
Fig. 204. Simple magnetic amplifier voltage and currents with (a) R^<<:^^Rj^, and
(b) RcyyRj,.
shown dotted in Fig. 203(a). When the reactors are parallel-con-
nected, as in Fig. 203(6), even harmonics may flow in the load coils,
and capacitors are unnecessary for Re » Rl-
Sometimes the two cores are combined into one, in the manner
shown in Fig. 205. This is called a three-legged reactor, with one
d-c coil and two a-c coils. Figure 205 shows the relative paths for
the a-c and d-c fluxes. Equal turns in the a-c coils set up equal a-c
magnetomotive forces which cancel in the center leg, and cause flux
to flow as indicated by the solid line. No fundamental alternating
voltage is induced in the d-c coil, but d-c flux flows in both outer legs
as indicated by the dotted lines. A change of current in the d-c coil
causes a change in total flux linking the a-c coils and hence a change
of inductance. A-c coils may be connected in parallel instead of
series, provided that equal turns in each coil and the flux polarity of
Fig. 205 are maintained; for the same total number of turns the in-
262
ELECTRONIC TRANSFORMERS AND CIRCUITS
Fig. 205. Windings and core flux paths in a saturable reactor.
«»•>—
s=
^
^
^=^
--5=
^
^
y
■ ^
^
-^
/
^
^
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/
y
y
"/
y
/
D-C
AM
p.
^
^
-^
y
/
'/
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TURNS
PER INC
H*0
A
y
W:
Y
5.55.
4.
4
/ii
A
3
e/
^
<n
//
/
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28
/''^
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ml
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^
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r^
It
//
%.
<y
^
m
y^
•y
==
\
==
A-C AMPERE-TURNS PER INCH
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Fig. 206. Magnetization curves for 4% silicon steel.
MAGNETIC AMPLIFIERS
263
ductance is halved and the alternating current doubled. The middle
core leg shunts the even harmonics of a-c flux.
Rectangular B-H loops are obtained in grain-oriented core mate-
rials. It is only in these materials that the wave shapes of Fig. 204(a)
are even approximated. In unoriented core steel, wave shape is much
more rounded, and control current bears less resemblance to load cur-
rent. Figures 206 and 207 indicate the contrast in saturation control
afforded by unoriented silicon steel and oriented nickel steel. In
Nl/lN BASED ON TURNS AND 1 (AMPS) FOR ONE TOROID
Fig.
30 40 50
LOAD A-C NI/IN
80
207. Typical magnetization curves for 0.002-in. grain-oriented nicliel-steel
toroidal cores.
grain-oriented steel cores there is an approximately linear relationship
between d-c ampere-turns per inch and a-c ampere-turns per inch over
a large range of flux density. Moreover, the a-c Nl/m. for a given
d-c A^7/in. change but little with a-c flux density over this range. In
Fig. 207, each of the lines for a given value of control magnetizing
force is nearly vertical. For a given value of control A'"7/in., load
current is almost independent of flux density and therefore of a-c
supply voltage. The sections following are based on the use of grain-
oriented core steel.
111. Graphical Performance of Simple Magnetic Amplifiers. Since
with a given core and supply frequency there corresponds a definite
voltage for every flux density, and since for a given number of turns the
ampere-turns per inch are proportional to current, the curves of Fig.
207 may be replotted in terms of voltage and current. We may then
plot load lines on these curves in a manner similar to those in electronic
264
ELECTRONIC TRANSFORMERS AND CIRCUITS
amplifiers, so that the operation, efficiency, control power, etc., may
all be determined from a study of these load lines.
In Chapter 5 (p. 142) equation 58 indicates that the a-c voltage in
a vacuum-tube circuit is divided between the load and the tube. If
a resistive load is used, a straight line can be drawn on the charac-
teristics of a vacuum tube which will form the locus of plate current
and plate voltage for any given load and supply voltage. This line is
called the load line, and by use of it the gain and power output of the
amplifier can be determined.
A-c MAGNETIZING FORCE AMPERE TURNS PER INCH
2 4 6
1.6
1.4
1.2
1.0
.8
.6
.4
.2
8
10
k
- ^
___,
7
T
s
/
CO
~ N
2
3-C
NTRO
l/lN"
L
ZER
\
4
6
8
\
\
z
\
\
\.
Q
2^
\,
10
1
40
J.
___
__-
/_
-^
-4-
->
20
30
40
NI= AMPERE TURNS
oc
Fig. 208. Generalized magnetic amplifier characteristics and load line.
A similar method can be used with magnetic amplifiers. If a linear
reactor were connected in series with the load, the voltage across the
load and the voltage across the reactor would add at right angles.
With rectangular loop core materials, currents are not reactive in the
linear sense, so that the actual load line is neither a straight line nor
an ellipse. For practical calculations the straight line is used, and the
results obtained are correct within small percentages if the reactor
voltage and current are measured on an average-reading voltmeter
and ammeter.
Figure 208 shows similar information to Fig. 207, except that it is
for a given core. The scale of abscissas is ampere-turns, and the
scale of ordinates is volts per turn. These characteristics can be used
for any amplifier which uses the same cores and the same supply
voltage and frequency as the amplifier on which the measurements
MAGNETIC AMPLIFIERS 265
were made to obtain these characteristics. These characteristics can
be derived for a given core from the parameters of a-c flux density,
a-c magnetizing force, and d-c magnetizing force as shown to the right
and top of Fig. 208. These curves then give a set of characteristics
for a given core rather than for a given core material. Some error is
involved if these curves are used with a different supply frequency
from the one used in making the original curves. Over a narrow
range of frequency the curves of Fig. 208 using the scale at the top
and to the right can be used to determine the operation of a magnetic
amplifier for different loads. The curves of Fig. 207 may be used for
magnetic amplifier calculations in this manner. For convenience of
calculation, it is usually preferable to make a set of characteristic
curves for several core sizes and for each supply frequency.
An example will show how these curves can be used in the design
of magnetic amplifiers. Assume that it is necessary to design an ampli-
fier with 30 watts output using cores with E/N and NI of Fig. 208.
The supply voltage is 100 volts, the load is 200 ohms, and 0.01 amp is
available for use in the control winding. The characteristics show
the E/N can be varied from about 1.4 to 0.2 and still stay on the
linear part of the characteristic curves. The power output is equal to
a£ X a/ which is also equal to A {E/N) X aA7, where E is the alter-
nating voltage across the reactors, / is the alternating current through
the reactors, and N is the number of turns in the load windings of the
reactor. A A/ needed for 30 watts = 30/(1.4 — 0.2) = 25 ampere-
turns. Load impedance is AE/Al. A load line on Fig. 208 is {aE/N^)
-^ aNI = (aE/aI) (1/AV), where Nl = turns in load winding. For
200 ohms, the load line passes through the points E/Nl = 1.4', NIac =
2.5, and through E/Nl = 0.2, A7„<, = 27.5. When this line is extended
to the ordinate it intersects at 1.54. This is the point of zero alternat-
ing current or of infinite reactor inductance. At this point the total
supply voltage would be across the reactor. Since the supply voltage
is 100 volts, 100/A = 1.54 and Nl = 65 turns. By interpolation of
the d-c NI curves, we see that, for E/N = 0.2 and Nhc = 27.5, 25
ampere-turns are necessary in the control winding. The turns in the
control windings are Nclc/Ic = 25/0.01 = 2,500 turns. Here I^ is the
current in the control winding, and N^ are the turns in the control
winding. Control winding resistance is determined by the wire size.
For the purpose of this example, assume that the resistance of the con-
trol winding is 500 ohms. Then the power in the control winding is
500 X J^<? = 0.05 watt. Power gain of the amplifier is power out/
power in = 30/0.05 = 600. The impedance of either the input cir-
266
ELECTRONIC TRANSFORMERS AND CIRCUITS
cuit or output circuit can be cliangcd by ciianging the number of turns
in the respective windings. Either impedance varies with the square
of the number of turns used in the winding. For example, the load line
which was used for 200 ohms in the preceding example could be used
for 800 ohms. The load winding would then have V 800/200 X 65 =
130 turns, and the supply voltage would be 200 volts instead of 100
for E/N = 1.54 at zero current.
Power output is proportional to the area of the rectangle of which
the load line forms a diagonal. More power output can be obtained
by using a load line with less slope, but gain may increase or decrease,
depending upon the winding resistances and core material. In the
60
50
1 1 1
100% FEEDBACK—
, T
vo
^^
^
z
40
/
y^ia
^100
-l2
N
^
s
z
o
30
/
-^
s
<
3
20
/y
/
1
-%
10 /J
/
cc
NTRC
L NI
/IN
120
-60 -50 -40 -30 -20 -10 10 ZO 30 40 50 60
Fio. 209. Simple magnetic amplifier transfer curves with line-voltage variations.
preceding example, the load windings were assumed to be in series
with the load, as in Fig. 203. This is the connection commonly used
when the source is a 60-cycle a-c line. AVith a high impedance source,
it is preferable to connect the load windings in shunt with the load.
Then the ordinates of Figs. 207 and 208 correspond to load voltage at
all times.
If we choose three line voltages corresponding to flux densities
within the linear portions of Fig. 207, and plot the d-c control versus
a-c load ampere-turns per inch, the curves of Fig. 209 result. If,
instead of A''//in., average load current is plotted, Fig. 209 gives the
transfer curves for a simple magnetic amplifier. The curves are sym-
metrical about zero ampere-turns. The difference between the trans-
fer curve and a straight line indicates the degree of non-linearity in
the amplifier for any load current. With grain-oriented core mate-
rial the a-c load current is nearly independent of supply voltage for
a-c inductions less than saturation. Provided that appropriate changes
MAGNETIC AMPLIFIERS 267
in scale are made, transfer curves may be plotted between load volt-
age and control current, or between load ampere-turns and control
ampere-turns, or between combinations of these.
Load current is the result of flux excursions beyond the knee of the
normal magnetization curve. In Fig. 207 the curve for zero control
A^7/in. is normal magnetization for the material. When direct current
flows in the control windings, it sets up a constant magnetizing force
in the core. Then superposed a-c magnetizing force readily causes a
flux excursion beyond the knee of the curve, permeability suddenly
drops, and a large current flows through the load winding. The point
in the voltage cycle at which this sudden increase in current occurs
depends upon the amount of direct current in the control winding.
Magnetic amplifiers with steep current curves like those of Fig. 207
can be used as control relays.
Load current is usually measured with an average-reading am-
meter, such as a rectifier-type instrument. This kind of ammeter
is generally marked to read the rms value of sinusoidal current but
actually measures the average value. Thus the ammeter reading is
0.707/0.636 = Lll times the average current over a half-cycle. When
the meter is used to measure non-sinusoidal current, it still reads 1.11
times the average.
Except for the slight amount of non-linearity noted in Fig. 209,
the average value of ampere-turns in the load winding of each re-
actor equals the d-c ampere-turns in the control winding. But since
the a-c ammeter reads 1.11 times this value, the load a-c Nl/in. are
1.11 times the control d-c iV//in., plus the differential due to core
magnetizing current. Thus, if a core had infinite permeability up to
the knee of the magnetization curve and zero permeability beyond the
knee, the transfer curve would be exactly linear. Oriented nickel-
iron alloy cores approach this ideal and therefore are more nearly linear
than other materials.
112. Response Time. Because of the inductance of the reactor
coils, when a change is made in the control winding direct current,
load current does not change immediately to its final value. An inter-
val of time, called response time, elapses between the change in con-
trol current and the establishment of a new steady value of load cur-
rent. If the inductance were constant during the change, the response
time constant would be the time required for a load current increase
to rise to 63 per cent of the final value after a sudden control current
increase. Magnetic amplifier response time cannot be evaluated as an
268 ELECTRONIC TRANSFORMERS AND CIRCUITS
ordinary linear L/R time constant. Storm ^ shows that the time of
response of simple magnetic amplifiers is independent of core permea-
bility. An average or equivalent control circuit inductance may be
found from the relation
r,.^£ = ^(*-5Y (112)
Re ilRc\Ni,/
where Ta = time for load current increment to reach 63 per cent of
final value
Lc = equivalent total control coil inductance (henrys)
Re = total control circuit resistance (ohms)
Rl = load resistance
/ = line frequency
Nc = turns in control winding
Nl = turns in load winding.
An obvious method of decreasing magnetic amplifier response time is
by increasing Re, but this has the disadvantage of reducing overall
power gain. Gain and response time are so related that the ratio of
gain to time constant in a magnetic amplifier is usually given as a
figure of merit.
113. Feedback in Magnetic Amplifiers. If a rectifier is interposed
between the reactor and load, and a separate winding on the reactor
is connected to this rectifier as in Fig. 210, it is possible to obtain
LOAD WDG
• \»
CONTROL
D-C WDG.—
INPUT
« RECTIFIER-J lC^
Fig. 210. Magnetic amplifier with external feedback.
sufficient power from the rectifier to supply most of the control power.
If the control power from the rectifier furnishes the ampere-turns
represented by the straight line in Fig. 209, the amplifier is said to
^ "Transient Response of Saturable Reactors with Resistive Load," by H. F.
Storm, Trans. AIEE, 70, Part I, p. 99 (1951).
MAGNETIC AMPLIFIERS
269
have 100 per cent "feedback." It is then necessary for the control
winding to supply only the amount represented by the horizontal dif-
ference between the transfer curve and the straight line. This greatly
increases the amplification of a pair of reactors.
Typical transfer curves for a simple magnetic amplifier are plotted
in Fig. 209 for three a-c supply voltages: 100, 110, and 120 volts. A
100 per cent feedback line intersects the transfer curves at 7i, I^, and
li, respectively. The control iV7/in. are furnished by the feedback,
except for the control current difference between the feedback line and
+ 10
+20
CONTROL Nl/lN
Fig. 211. Transfer curves for magnetic amplifier with feedback.
the transfer curve. Positive control current is required when the trans-
fer curve is at the right, and negative current when it is at the left, of
the feedback line. Net control jV7/in. for the three voltages are plotted
in Fig. 211 with expanded abscissa scale. Now the transfer curve is
asymmetrical. Most of the amplifier gain occurs with negative con-
trol current changes. On the steep parts of the transfer curves, gain is
fairly linear and greatly exceeds the gain of simple amplifiers. Be-
low the steep parts, output current reaches a minimum but remains
small with relatively large excursions of negative control current.
These current minima are called cvt-off points. Reference to Fig. 207
shows that cut-ofT current is I^T, the normal exciting current at supply
voltage E. With positive control, current output levels off to a nearly
constant value, depending on the voltage. Feedback causes output
current to be quite dependent on variations in a-c supply voltage,
because Is has a greater effect than in simple amplifiers.
Computing control current for transfer curves with feedback as
described in the preceding paragraph involves a small difference be-
tween two large quantities. Minor measurement errors in the original
270 ELECTRONIC TRANSFORMERS AND CIRCUITS
data cause large inaccuracies in the feedback transfer curves of Fig.
211. A more accurate derivation of 100 per cent feedback transfer
curves is given in Section 117.
To the left of the cut-off points, the transfer curve rises slowly to-
ward the left along a straight line, as in Fig. 212(a). This line cor-
responds to 100 per cent negative feedback; it is practically linear,
but gain is much reduced. The transfer curves of Fig. 211 would, if
continued to the left, merge into such a line.
Polarities in Fig. 210 are for positive feedback with positive direct
current entering the control winding at the top. Negative feedback is
obtained if the control current is reversed. If series feedback is de-
rived as shown in Fig. 210, the feedback current is Ei,/Ri,. It is pos-
sible to connect the feedback circuit across the load to obtain voltage
feedback. To conserve power, the feedback resistance should be large
relative to J?j,.
114. Bistable Amplifiers. Positive feedback in a magnetic amplifier
can be increased to more than 100 per cent by increasing turns in the
feedback winding. Transfer curves may then become double-valued
and give rise to abrupt load current changes with changing control
current. Such amplifiers are called bistable. In Fig. 209, the effect
of increasing feedback would be to decrease the slope of the feedback
line. If the feedback were increased gradually, operation would re-
main stable until the feedback line had the same slope as the transfer
curve. Then the load current would become some indefinite value
along the transfer curve. If the feedback were increased further stable
operation would be had at only one of two values of load current.
Bistability is illustrated in Fig. 212(a). Here a transfer curve similar
to those of Fig. 211 is shown except that it is with load voltage ordi-
nates and expanded Ncic abscissas. The amount of feedback in excess
of 100 per cent is drawn as line AB with slope less than that of the
main part of the 100 per cent feedback transfer curve. Another line,
CD, is drawn parallel to the line AB. These lines are tangent to the
transfer curve at points A and C. With feedback > 100 per cent, let
d-c control current be decreased from some negative value toward zero.
Load voltage or current follows the transfer curve until it reaches
point A ; then it jumps to point B, and further increase of control cur-
rent results in very little load voltage increase beyond point B. If
control current is subsequently reduced, load voltage follows the top
of the transfer curve until it reaches point C; then it drops abruptly
to point D.
MAGNETIC AMPLIFIERS
271
Bistable action is shown in Fig. 212(6) as a function of control
Nl/in., with points A, B, C, and D corresponding to those in Fig.
212(a). Line AB in Fig. 212(a) represents feedback ampere-turns
Nflf in excess of 100 per cent, which are proportional to E^. Line AB
extended intersects the axis of abscissas at F\ and CD extended inter-
sects at G. Vertical lines erected at A' and F' intersect the transfer
-L
,^
>^^
•^B
HSFER CURVE
FOR
100% —
yf
FEEDBACK
A
/
'W/
/^
(a)
D-
■
_ F
T^
A
«1
F'
A'
NA
t
Nclc
Fig. 212. (a) Typical transfer curve, and (b) bistable magnetic amplifier.
curve at A and F, respectively. F'A' represents ampere-turns A'';//
when control ampere-turns iVc^c are at point F . When decreasing
negative NqIo reach value F , the load voltage jumps from A to B.
Points F' and G are projected downward to Fig. 212(5). In this fig-
ure the output jumps to final value B, but the increase actually takes
place along the dotted line. Decreasing additional feedback 'N^li re-
duces the differential amount F'G of control NqIq and reduces the
width of the bistable loop. Conversely, increasing A'//; widens the
loop and provides a greater margin for variations in A^/7/ due to volt-
age, temperature, etc. Bistable amplifiers are used in protective and
control circuits to turn relays or indicators on or off when control
272
ELECTRONIC TRANSFORMERS AND CIRCUITS
power varies between narrow limits and the inherent lock-in action is
desirable.
115. Current Transductors. In some countries the term transductor
is used to denote any magnetic amplifier. Here it denotes a saturable
reactor circuit for measuring direct current. A current transductor is
hardly an amplifier; it is a metering device. A transductor circuit
is shown in Fig. 213. It is similar to that of Fig. 210 but with feed-
TO 0-C '
• o
" A-
<;iipp
SUPPLY ,
1 {
^ p ■
[©J
Fig. 213. Current transductor circuit.
back windings and a-c load removed. Operation is entirely differ-
ent. Cores are circular or square, and are wound in-and-out toroid-
ally in a manner resembling through-type current transformers. The
heavy d-c bus then may be inserted through the toroid to form a single
turn on each core. In Fig. 213 the d-c load windings are shown aid-
ing, and the a-c windings bucking; this accomplishes the same core
flux polarities as for Fig. 205. Load direct current is determined by
the load resistance, which is large compared to the reactance of the
transductor. Control circuit impedance multiplied by the turns ratio
is large; magnetization is constrained. It will be recalled from Sec-
tion 110 that, under this condition, even current harmonics cannot
flow. Therefore a-c winding current is flat-topped. After this flat-
topped current is rectified, it flows through the ammeter as smooth
direct current.^
At any instant one reactor of the pair is saturated, and the other
unsaturated. On each a-c half-cycle the unsaturated reactor main-
tains the output current constant. Total output d-c ampere-turns of
course must equal twice the load direct current at all times. Trans-
ductors are like simple magnetic amplifiers as far as the relations of
load and output currents are concerned. They have been built to
measure currents of 10,000 amp or more, with good linearity.
• For a description of the current and flux conditions, see "IVLiKnotic Amplifier.^,"
by S. E. Tweedy, Electronic Eng., February, 1948, p. 38.
MAGNETIC AMPLIFIERS 273
116. Self-Saturated Magnetic Amplifiers. In Section 113 it was
seen that tlie use of feedback windings greatly increases the gain of a
magnetic amplifier. Several circuits have been devised to provide
the feedback by means of the load circuit and thus eliminate the extra
feedback winding. Such circuits are termed self-saturating. A
"building-block" or elementary self-saturating component is the half-
wave circuit of Fig. 214, from which several magnetic amplifiers may
D-C
CONTROL
INPUT
-^
— -I LOAD I— I-
+
-o
A-C
SUPPLY
k-e,-^
Fig. 214. (a) Half-wave self-saturated magnetic amplifier circuit and (6) load
and rectifier voltage wave shapes.
be formed. Impedance Z in the control circuit prevents short-circuit-
ing the reactor. It may be the control winding of another reactor in a
practical amplifier. Rectifier RX prevents current flow into the load
in one direction, so that the core tends to remain in a continually
saturated condition. This condition is modified by negative control
winding AV/in., which opposes the load winding A7/in. and permits
the core to become unsaturated during the portion of the cycle when
there is no load current flowing. The greater the control A^7/in., the
less the average output current. Transfer characteristics are similar
to those of Fig. 211. Ideally the circuit has 100 per cent feedback.
Assuming the core to be saturated at all times with zero control
current, current flows into the load throughout the whole positive half-
cycle and is zero for the whole negative half-cycle. With a given
274 ELECTRONIC TRANSFORMERS AND CIRCUITS
value of negative control current, reactor inductance is high at the
start of the positive half-cycle and load current does not build up
appreciably until an angle 9i is reached when the core saturates.
Then it climbs rapidly and causes most of the supply voltage to ap-
pear across the load as shown by the curve marked e^ in Fig. 214(6) for
the remainder of the positive half-cycle. As negative control current
increases, so does angle di. In the limit di = 180°; that is, with large
negative control current, virtually no load current flows. The simi-
larity of load voltage wave shape to thyratron action is at once evi-
dent. It has led to the use of the same terminology. Angle Oi is often
called the firing angle of a magnetic amplifier. Load voltage is re-
duced as di increases, approximately as in Fig. 190. There are some
important differences, too:
(a) Reactor inductance is never infinite, and magnetizing current
is therefore not zero. This means that during the interval O-^i a
small current flows into the load. The change in reactor inductance at
the firing instant is not instantaneous; the time required for the in-
ductance to change limits the sharpness of load current rise.
(b) Even with tight coupling between control and load windings,
the saturated reactor inductance is measurable. This saturated in-
ductance causes the load current to rise with finite slope.
(c) After load voltage reaches its peak and starts to drop along with
the alternating supply voltage e, core flux continues at saturation
density. An instant a is reached when the load voltage exceeds the
supply voltage. Beyond a, the reactor inductance increases and mag-
netizing current decreases, but at a rate slower than the supply volt-
age because of eddy currents in the core.
(d) After supply voltage e in Fig. 214(6) reaches zero, the reactor
continues to absorb the voltage until the core flux is reset to a value
dependent on the control current, that is, until angle 62 is reached.
Then part of the negative supply voltage rises suddenly across recti-
fier RX as shown by the wave form of eu-
During the interval 0-5i the reactor inductance is high and virtually
all the supply voltage appears across it. The voltage time integral
j e dt represented by the reactor flux increase during this interval is
equal to j edt during ir-92- That is, the energy stored in the core
before the firing instant is given up during the negative half-cycle of
supply voltage.
MAGNETIC AMPLIFIERS
275
Self-saturated magnetic amplifiers have transfer curves similar to
that of Fig. 212(a). A small amount of additional positive feedback
makes them bistable. Negative feedback makes the transfer curve
more linear but reduces the gain. Ordinates and abscissas may be cur-
rent, ampere-turns, or oersteds, as for simple magnetic amplifiers.
117. Hysteresis Loops and Transfer Curves. Several workers have
observed ^ that the transfer curves of Fig. 211 are similar in shape to
the left-hand or return trace of the hysteresis loop. There is a con-
(3)
Fig. 215. Minor loops in rectangular hysteresis loop core material.
nection between the two. In Fig. 21, p. 25, it was shown that in a
core with both a-c and d-c magnetization the minor hysteresis loop
follows the back trace of the major loop in the negative or decreasing
B direction, and proceeds along a line with less slope in the positive di-
rection until it joins the normal permeability curve at B,„. Also, it
was pointed out in connection with Fig. 69, p. 94, that, if AB has the
maximum value B,„, the result is the banana-shaped figure OB,nD'.
Here again the loop representing flux excursion 0-Bm follows the left-
hand side of the hysteresis loop in the downward or negative direction.
In a rectangular hysteresis loop material with B-H loop shown in
Fig. 215 (a) , the path traced over a fiux excursion BoB^ is more irregu-
lar in shape but still follows the left-hand trace of the loop. If mag-
netic amplifier cores are biased to a series of reset flux positions Bo
to B:i the corresponding flux excursions and minor loops are those
shown in Fig. 215(b). Usually, the load current far exceeds the con-
trol current necessary to reset the cores, so that these loops actually
have a much longer region over which the loop width is practically
^ See "Self-Saturation in Magnetic Amplifiers," by W. J.
AIEE, 68, 835 (1949).
Doi'nhoefer, Tranfi
276 ELECTRONIC TRANSFORMERS AND CIRCUITS
zero, as shown in Fig. 215(c). This is true of all the loops regardless
of flux excursion.
The foregoing is true of a slowly varying flux excursion, so that the
locus of the lower end points of the minor loops is the left-hand trace
of the d-c hysteresis loop. Most magnetic materials, including rectan-
gular loop materials, have a wider loop when the hysteresis loop is
taken under a-c conditions, because of eddy currents. The difference
between loops is as shown in Fig. 216. The locus of the end points
of the minor loops under a-c flux excursions is neither the a-c nor the
d-c loop but an intermediate line such as that drawn dot-dash in
Fig. 216. The slope of this line is less than that of either the a-c or
the d-c loop, and the gain of the magnetic amplifier is accordingly
reduced.
An analysis for the self-saturated magnetic amplifier of Fig. 214(a)
is given below. Load current is assumed to have the same shape as
ex, in Fig. 214(6), and the following assumptions are made:
1. Sinusoidal supply voltage and negligible a-c source impedance.
2. Negligible reactor and rectifier forward voltage iR drops.
3. Negligible rectifier back leakage current.
4. Negligible magnetizing current compared to load current.
5. Negligible saturated inductance.
6. High control circuit impedance.
7. E = 4:.UfN<i>s X 10-^ (113)
This will be recognized as equation 4 (p. 6) with peak flux at saturation
value <t>s- Other terms are listed as follows:
di = firing angle as in Fig. 214(6).
h = ^i/co.
w = 2ir X supply frequency /.
E = rms supply voltage.
^s = saturation flux = BgAc (for Bg see Fig. 215).
Ac = core section in cm^.
00 = reset core flux = BqAc (for Bq see Fig. 215).
Rl = load resistance.
7av = average load current.
i = instantaneous load current.
N — turns in load winding.
Under the assumptions, equation 1 becomes
f- N d(j>
V 2 Esinoit = -——- for < cat < wh (114)
10* dt
MAGNETIC AMPLIFIERS
277
o
=1 -4
° -8
-12
-16
~~\ r
DC LOOP
A-C LOOP-.I
?r=
LOCUS OF Brr
-X
JH;
1.0
ih
Fig. 216. D-c and a-c B-H loops for grain-oriented nickel steel.
Integrating equation 114 gives
and
Nd4, = j V 2 E sin coi <
CoiV(0S - 0O)
= 1 — COS Olti
V2E X 10*
During the interval di < wt < t, load voltage is
V2 E sin ut = zi?L
where Rl is the load resistance. This may be integrated to give
o>Rl
V2E
1 + cos wti
(115)
(116)
(117)
(118)
„ -- (119)
'tl tiL
The left side of equation 119 is the average load current over the con-
ducting interval tt/oj — ^i . Average load current over the whole cycle is
JN^^ (120)
Rl X 10*
Equation 120 has two flux terms: <l>s, which is a fixed quantity for a
given core material; and <^o- The relation between ^o and control
idt =
Combining equations 116 and 118 and substituting equation 113,
X
7r/a> jq
idt = — {4>s + 4>a) X 10"
278
ELECTRONIC TRANSFORMERS AND CIRCUITS
current is, as indicated in Fig. 215, the return trace of the major
hysteresis loop. Thus equation 120 states that the average load cur-
rent is the sum of a constant term and a term which has the same shape
as the return trace of the hysteresis loop. Quantitatively, a self-
saturated half-wave magnetic amplifier has a current transfer curve
the same as the return trace of the core hysteresis loop, except that
ordinates are multiplied by fAcN/W^Bi, and are displaced vertically
by an amount fBgAcN/lO^Ri,.
Comparison with equation 113 reveals that the ordinate multiplier
and vertical displacement are E/4:A4:RlBs and E/4:A^Rr„ respectively.
As noted above, the return trace should be modified to mean the dot-
dash line of Fig. 216.
118. Self-Saturated Magnetic Amplifier Circuits. In Fig. 217 three
single-phase circuits are diagrammed which comprise two of the half-
D-C
INPUT
-o E o C
(3) DOUBLER CIRCUIT
I
(C) CENTER-TAP D-C OUTPUT CIRCUIT
Fig. 217. Self-saturated magnetic amplifier circuits.
MAGNETIC AMPLIFIERS
279
O)
wave elements described in the preceding sections. These circuits are
discussed briefly below.
(a) Doubler Circuit. This is really two half-wave circuits working
into a common load. Rectifier polarities are such as to cause a-c
voltage to appear across the load, as in Fig. 218(a). The wave shape
departs somewhat from alternately reversed half-waves. In the
doubler, the reactor which is carrying load current during a given half-
cycle causes a reduction in the resetting voltage, and therefore in the
time rate of resetting flux change of the other reactor. This increases
the output and gain for a given control cur-
rent compared to the half-wave circuit but
has no effect on current minima at the cut-
off points (see Fig. 211).
When control circuit resistance Re is
large, the control current and associated
magnetizing force are fixed, but, when Re
is small, even harmonic currents flow freely
in the control circuit and influence the
wave shape for a given control current
further. Generally, low values of resist-
ance Re cause a slight increase in the con-
trol oersteds for a given output but vir-
tually no change in slope. In other words, the whole transfer curve
is displaced slightly to the right.
(b) Single-Phase Bridge Circuit. Here two extra rectifiers isolate
the two reactors at all times, and the wave form is like that of the
half-wave rectifier, except that it occurs twice each cycle. Load cur-
rent is d-c; that is, both reactors produce load current of the same
polarity, as in Fig. 218 (fe). Because of the isolation of the two re-
actors, the transfer curve closely follows a dot-dash line like that in
Fig. 216 if the core is grain-oriented nickel steel, or a similar line be-
tween a-c and d-c loops for other core material. Control resistance
Ri; affects output in a manner similar to that mentioned for the
doubler.
(c) Center-Tap D-C Circuit. Although the reactors are not isolated
in this circuit, load and resetting currents are still the same as for the
bridge circuit, and hence the transfer curve has the same shape, unless
the rectifier reverse currents are appreciable. Then gain is appreciably
reduced.
In all these single-phase circuits, the load current is twice that of
(b)
Fio. 218. Single-phase mag-
netic amplifier output; (a)
a-o voltage across load; (b)
d-c voltage across load.
280
ELECTRONIC TRANSFORMERS AND CIRCUITS
the half-wave circuit. Therefore transfer curves may be predicted
from B-H loops as in Section 117, but ordinates are multiplied by
E/2.22RlBs, and the vertical displacement is E/2.22Rl. From these
multipliers it can be seen that output current is proportional to supply
voltage E, and therefore power gain is proportional to E^. In this re-
spect, a self-saturated amplifier contrasts with a simple magnetic
amplifier, the output current of which is nearly independent of E, for
rectangular B-H loop core material. At least this is true below maxi-
mum current, or current flow over a complete half-cycle.
500
. 400
<
s
— 300
UJ
>
<
- 200
-I
H
100
M
_J
<
Si
o
f
f
/
J
^
-OER
STEDS
1
-0.5
-I
5
I
-0.5
O 0.5
H-OERSTEDS
(a)
ib)
Fig. 219. Self-saturated magnetic amplifier output; (a) calculated for 500 ohms
from Fig, 216; (6) in actual amplifier.
As an example of the manner in which a transfer curve is found
from the B-H loop, suppose that, in a given self-saturated amplifier,
Fig. 216 is the B-H loop, supply voltage E = 230 v, Rr^ = 500 ohms,
B, = 14.7 kilogauss. The ordinate multiplier is 230/(2.22 X 500 X
14.7) = 0.0141, and displacement is 230/(2.22 X 500) = 0.207.
Table XV indicates the change in ordinates. The last two columns
of the table are plotted in Fig. 219(a) as load current in milliamperes.
Also indicated is the "normalized" value of unity for maximum output
current. For any load impedance the same calculated transfer curve
can be used, and all ordinates multiplied by E/l.llRt. Abscissas may
be normalized likewise, with cut-off H = —1.0.
Normalized output current at cut-off is A^, = I^Rl/E. Cut-off con-
ti'ol current is most accurately found from H corresponding to —B,.
This is H = —0.5 in Fig. 216. These relations are, of course, idealized,
but they are still very useful in practical work. For example, winding
MAGNETIC AMPLIFIERS 281
Table XV.
Derivation
OF Transfer Curve from
B-H Loop (Fig. 216
Load Current,
Vertical
Fig. 219(a) (av)
H
B
0.0141B
Displacement
(oersted.s)
(kilogauss)
amp (av)
(amp)
Amp
Normalized
-0.5
-14.3
-0.202
+0.207
0.005
0.012
-0.4
-14.0
-0.197
0.207
0.010
0.024
-0.3
0.207
0.207
0.500
-0.15
13.0
0.183
0.207
0.390
0.943
14.0
0.197
0.207
0.404
0.975
0.5
14.7
0.207
0.207
0.414
1.000
resistance Ro and rectifier forward resistance Rp reduce load cur-
rent and output power, but these resistances may be added to the actual
Rtj arithmetically to obtain total resistance Rt = Rg + Rf + Rl-
Then the transfer curve ordinates are
E(B-H loop ordinates)
/av = -^ (121)
2.22R'['Bs
displaced vertically by
E/2.22Rt (122)
Output current and power are reduced somewhat by these inevitable
resistances. This can be verified in Fig. 219(6) which is a plot of
transfer curves for an actual doubler amplifier with E = 230 v, with
350-, 500-, and 1,000-ohm load resistances, and with average load
current X 1-1 1 as read directly on the output meter. The 500-ohm load
resistance curve is approximately the same as Fig. 219(a) ; this means
that Rj,' + Rg == 0.11i?L in this particular amplifier. The accuracy of
Fig. 219(a) is evidently poorest at cut-off. Upward slope at control
currents more negative than cut-off is not shown at all. For the most
practical region, i.e., to the right of cut-off, the calculated curve is
eminently useful.
Additional windings are often used on the reactors for control pur-
poses. One common winding, called a bias winding, carries negative
control current. The function of this winding is to maintain low out-
put in the absence of control current. Thus in Fig. 211, with E = 120,
~5NI/m. of bias magnetizing force keeps the amplifier load jV//in. at
5. Then positive control current raises the load current to the desired
value. Most of the gain is obtained with less than +5A^//in. control
magnetizing force.
282 ELECTRONIC TRANSFORMERS AND CIRCUITS
Additional control windings are used for adding or subtracting input
signals. This provides a simple means of combining several control
functions in one magnetic amplifier.
Response time in a self-saturated amplifier is longer than in a simple
amplifier, but the gain per second is much greater. The time con-
stant is
T4 = 0Lj2f (seconds) (123)
where T^ = time for 63 per cent response to step input
a„ = amplifier voltage gain for 1 : 1 turns ratio
= (AEl/AEc) X (Nc/Nl) for any turns ratio
/ = supply frequency
AEl = change in load voltage
AEc = change in control voltage
Nc = turns in control winding
Nl = turns in load winding.
Equation 123 is valid for T^ down to approximately 4 cycles minimum.
Although smaller T^ may be obtained, it does not follow equation 123.
Push-pull amplifiers are used to provide a-c or d-c output, with the
output polarity dependent on input polarity. A d-c push-pull circuit
which senses input polarity is shown in Fig. 220. Bias windings on
each reactor carry current in such directions that amplifier outputs
cancel for Ec = 0. For positive Ec, amplifier 1 produces positive Er„
and for negative Ec, amplifier 2 produces negative El. This circuit has
low efficiency, owing to the power dissipated in the balance resistances
Rr and R2 but has linear output.
It is important, wherever two or more reactors are used together in
magnetic amplifiers, that the reactors be alike in turns and in cores.
Cores are generally selected to "match," with closely duplicated B-H
characteristics. It is not feasible to compensate core differences in bal-
anced amplifiers by bias adjustments and still obtain linear output.
119. Half- Wave Control of Magnetic Amplifiers. Through atten-
tion to a-c voltages present in the control circuit, Dr. R. A. Ramcy
analyzes magnetic amplifiers in a manner which gives rise to new cir-
cuits with desirable properties.^ A half-wave building block of such
circuits is shown in Fig. 221 (a) for a 1 : 1 turns-ratio reactor. The load
circuit is the same as in the half-wave amplifier of Fig. 214. The con-
1 See "On the Mechanics of Magnetic Amplifier Operation" and "On the Con-
trol of Magnetic Amplifiers," by R. A. Ramoy, Trans. AIEE, 70, 1214 and 2124,
respectively.
MAGNETIC AMPLIFIERS
283
trol circuit comprises a-c voltage E and rectifier RXc in addition to
variable rectified control voltage ec of polarity indicated. A-c volt-
age polarities are for the positive or conducting half-cycle in the load
TO
A-C
SUPPLY .
LTS ACROSS Rl
THRl»R|
R|=R2
AS
^ Ncic
Fig. 220. D-c push-pull magnetic amplifier.
circuit. During this half-cycle, RXc blocks and the control volt-
age is zero. During the next half-cycle, a-c line voltage E — ec ap-
pears across the reactor control coil. If ec is zero, the core is not mag-
netized by control current flowing during the positive half-cycle, and
284 ELECTRONIC TRANSFORMERS AND CIRCUITS
^1 1 I— ^1—
— OEO— I ' OEO 1
'—OEO— II l—OEO—i
* — k-
-K-*
(a) (b)
Fig. 221. Half-wave controlled magnetic amplifior.s.
the core is completely reset by E during the negative half-cycle. If
the peak value of ec is equal to ■\/2E, it appears across the reactor in
opposite phase to the line voltage and completely cancels it during
the resetting half-cycle. This is shown dotted in Fig. 222, with both
voltage waves designated by capital let-
ters. This cancellation results in zero
resetting; therefore full output current
flows over 180° of the positive half-cycle.
If Ec = E/2 it subtracts from E, result-
ing in the lower dot-dash line of Fig. 222.
The area under E — Ec (shown hatched)
is just half of the area under E and there-
fore equals the hatched area under E dur-
ing the interval to ir/2 of the positive
half-cycle. That is, the reactor absorbs
voltage E during the interval to -n-/2 and allows current to flow from
ir/2 to TT. But this is half of full or maximum output. Thus the out-
put current is:
Fig. 222. Resetting voltages
with half-wave control.
zero for Er
H max for Ec = E/2
max for Ec = E
Several advantages accrue from this type of control :
1. Output is proportional to control voltage.
2. Output depends only on control voltage and is independent of
variations in line voltage or frequency.
MAGNETIC AMPLIFIERS 285
3. Time of response is short (2 cycles or less).
4. Filtered d-c source of control power is not necessary.
Proportionality of output to input voltage is strictly true only for
zero control circuit resistance or zero reactor exciting current. The
lower the control circuit resistance and reactor exciting current, the
more nearly is output proportional to input. Rectangular B-H loop
core material is necessary for linearity. Control circuit resistance can
be made small without causing slow response in this circuit. Exciting
current and control circuit resistance give rise to load voltage output
with zero control voltage. Raising control voltage ec restores linearity
of output. With half-wave control, voltage gain is more important
than power gain; voltage gain is approximately equal to turns ratio.
Mixing is not so readily accomplished in half-wave control circuits.
Figure 221 (b) shows how two half- wave sections are combined to form
a full-wave bridge circuit with d-c output. This circuit differs from
the circuit of Fig. 217(5) in that the control windings are isolated
from each other by the control circuit rectifiers. Voltage E in the con-
trol circuit is an a-c bias voltage, and e© is rectified a-c signal voltage.
Zero output voltage appears across Rz, with ec = 0. When ec is in-
creased, full-wave rectified voltage appears across Rl- The funda-
mental a-c component of this voltage is zero.
120. Magnetic Amplifier Design. Of first concern in design is the
reactor core material. Supermalloy or other high-percentage nickel
alloys are best suited as core material for the low power input stage.
Grain-oriented nickel steel is used in the stages where output power is
appreciable, and grain-oriented silicon steel where power is large.
Figure 223 shows the d-c loops of two grain-oriented core materials,
Hipersil and Orthonik. Although both materials have approximately
rectangular B-H loops, the difference in rectangularity is marked.
Grain-oriented nickel-steel strip such as Orthonik is usually wound
into toroids, to insure that the flux flows in the preferred direction.
The toroidal cores are protected from mechanical damage and strain
by encasing them, as in Fig. 224, after the core material is annealed
to preserve the magnetic properties. Grain-oriented silicon-steel cores
are much less sensitive to damage; type C cores may be used, with
coils wound as described in Chapter 2.
In either type of core there is a small inevitable air gap. In a
toroid, the flux must change from one lamination to the next as it
flows around the core. If the insulating space between laminations
is 0.0005 in. and the average core length is 5 in., the effective core gap
286
ELECTRONIC TRANSFORMERS AND CIRCUITS
/
^Z
-■
— '
^ '
•
/
^>
(
/
/
/
3 in
/
1
/
< '0
/
y
1
^ ^
m
1
, /
.5
.5 1.0 1.5
H-OERSTEDS
2.0 2.5
Fig. 223. Typical d-c magnetization curves and hysteresis loops for 2-miI Hipersil
and 2-mil Orthonik toroidal cores.
Fig. 224. Toroidal core of grain-oriented nickel steel in case, and with top of
case removed.
MAGNETIC AMPLIFIERS 287
is 0.0005/5 = 0.0001 in. This gap is not negligible in high-permeability
core material, but it is about one-tenth of the gap that manufacturers
allow in type C cores. Effective core gap requires more control Nl/in.
and reduces gain because the gap causes a more sloping B-H loop. Sec
Fig. 242 (p. 310). Special U-shaped punchings of grain-oriented steel
are sometimes used with alternate stacking to reduce the efTective core
gap.
Another effect that reduces gain is rectifier "back" resistance, or cur-
rent flow during the part of the negative half-cycle when inverse volt-
age exists across the diode. The peak value of inverse voltage divided
by the corresponding reverse current is the rectifier back resistance.
For a given peak source voltage y/'IE, the inverse peak rectifier volt-
age is 2 ■\/2 E in the center-tap d-c circuit, and it is -\/2 E in the
bridge circuit, for zero winding and rectifier forward IR drops. In a
doubler amplifier with zero forward drop, inverse peak voltage is zero,
and increases with forward drop up to a maximum of -\/2 E. The re-
verse current corresponding to these voltages resets the core more than
control circuit current with no rectifier reverse current. This causes
transfer characteristic slope to decrease; the unity ordinate of the
normalized transfer curve is displaced to the right by the ratio of re-
verse current to cut-off control current 7c. Normal cut-off control
current /,.. and cut-off output current Zy are not affected, because Ic
operates to reduce load current during the positive half-cycle. Good-
quality rectifiers are as important as good core material. This applies
equally well to leakage current and forward current IR drop. Losses
may limit output in rectifiers as well as in reactors. Most of the PR
loss in windings of self-saturated amplifiers is usually in the load
windings. This loss occurs during the part of the cycle in which load
current flows, or while the core is saturated and core loss is zero. PR
loss is a maximum when 6ii = in Fig. 214(6). When (9i = 180°, PR,
loss is negligible and core loss is a maximum.
When the supply frequency is high, choice of rectifiers is limited to
those with good high-frequency properties. At radio frequencies this
may mean that suitable rectifiers are not available; simple magnetic
amplifiers must then be used. To reduce core loss at high frequencies,
ferritcs are used.
Insulation of toroidal coils is difficult to apply. Insulation between
concentric windings is taped in and out like the wire. If voltage is low,
the wire enamel is sufficient insulation. For 115- or 230-volt circuits,
windings are laid on the core progressively, that is, with turns bunched
288 ELECTRONIC TRANSFORMERS AND CIRCUITS
SO that adjacent turns have but a small a-c voltage difference. Insula-
tion difficulties increase with voltage, and high-voltage reactors are
preferably layer wound, with type C or stacked cores.
Induced voltage in control windings requires careful attention, espe-
cially when control current is limited and many control turns are re-
quired. Although fundamental a-c voltage cancels in the control cir-
cuit, the full magnitude of this voltage is induced in the control wind-
ings. In the example of simple magnetic amplifier given in Section
111, the voltage induced in the control windings is 2,500/65 X 100 =
3,850 volts. With layer-wound coils and solventless resin coil impreg-
nation the insulation is readily provided, but it would be difficult with
toroidal coils.
Winding space in a toroid is limited by the minimum practicable hole
size in the finished coil. This varies with the kind of winding ma-
chine and also with the size of toroid. If
di = hole diameter
d2 = core case inside diameter
^3 = core case outside diameter
^u, = total winding area,
then
A„ = (7r/4)(d2' -rfi') (124)
On the outside of the toroid, the winding builds to a smaller height
than on the inside. Since ^^ is fixed by the minimum hole size, the
coil outside diameter is
di = Vd^^ + (4A„/7r) (125)
Coil axial length = Core case height + 2Au,/lc (126)
Mean turn of first winding = Case periphery -|- irA^i/lc (127)
where Au,i is area occupied by first winding. Equation 127 is approxi-
mate because wire turns tend to become circular after several layers are
wound on the core.
Example. Control Reactors for Single-Phase Rectifier. Assume the following
conditions :
Power supply 400 cycles.
Center-tap d-c circuit per Fig. 217(c).
Control current available = 40 ma d-c.
Plate transformer E = 125 volts per side.
MAGNETIC AMPLIFIERS 289
At full output Idc = 2 amp in Rl-
Per cent reduction in Edc = 33 per cent at minimum output.
Assume grain-oriented nickel-steel core with A^ = 0.1 sq in., l^ ^ 5.5 in.,
and B, = 14,700 gauss.
Core-case dimensions 134 in. I.D., 2^6 in. O.D., '5li2 in. high.
Each reactor must be capable of absorbing the voltage-time integral corre-
sponding to 33 per cent voltage reduction, or 0.33 X 125 = 41 volts. From
equation 34 (p. 83),
3.49X41X10'' ^,,^
''^ = 400 X^TxT4-J00 = 244 turns
With full output, load winding current = 2x7(2 X 2) = 1.57 amp rms. From
Fig. 219(a) this can be controlled with H = 0.5 oersted
0.5 = 0.5NcIc/lc
Nc = Ic/Ic = 5.5/0.04 = 138 turns
This will be increased to 276 turns to allow for rectifier reverse current, varia-
tions in slope of the core B-H loop, and effective core gap. Using 650 cir mils
per ampere, and single enameled wire, yields 1.57 X 650 = 1,020 cir mils or
No. 20 wire for N l, and 0.040 X 650 = 26 cir mils or No. 35 wire for Nc- With
an average winding area space factor of 60 per cent, the coil winding areas
required are, from Table V (p. 37), 244/(860 X 0.60) = 0.48 in.^ for Nl and
276/(24,500 X 0.60) = 0.019 in.^ for Nc. If Nc turns are wound concentrically
over N L, the load winding inside diameter is, from equation 124,
di = Vd-I' - (4A,,/7r)
= ^(1-25)2 - (4 X 0.48/ir) = 0.975 in.
Nc turns occupy but a single layer. Then, for Nc, di = 0.955 - 2(0.0064)
= 0.94 in. With 10-mil insulation over Nc, the hole diameter becomes
0.94 — 0.02 = 0.92 in. Space required to insulate the ends of the windings and
space for additional control windings reduce the hole diameter further.
Winding mean turn lengths are, for a core-case periphery of 1.88 in.,
M7V= 1.88 + -^^-J-€ 2.16 in.
0.5
MTc = 2.16 + 7r([0.48/5.51 + 0.0064 + 0.029) = 2.44 in.
T) ■ , fi 1 ■ r 244 X 2.16 X 10.3 ^ ,. ,
resistance or load wmdmg = ToTwi ^ ohms
T? • f f f 1 • .^- 276 X 2.44 X 338 ^^ ,
Kssistance of control wmduig = o^nnA ^ ^^ ohms
290 ELECTRONIC TRANSFORMERS AND CIRCUITS
Load winding IR = 0.71 volt. PR = 1.12 watt.
Control winding IR = 0.76 volt. PR = 0.0305 watt.
^ X [1 - (0.67)2]
P""-^'- ^'"" = o:0305 = 2'°^°
41 X 276
Time constant = -^---_^^^_^ = 0.07 sec
with no external series resistance in the control circuit. With feedback applied
to the control winding, this rectifier can be made self-regulating. If the feed-
back is further refined by comparison with a voltage reference, a stable voltage
regulator results.
121. Magnetic- Amplifier Limitations. Several limitations may affect
the practical usefulness of magnetic amplifiers. Some of these limita-
tions are beneficial in certain applications:
1. Residual output with zero input.
2. When more than one reactor is used in a circuit, reactor cores must
often be matched.
3. Zero drift. At low input levels (of the order of 10^^* watt for
toroids of rectangular loop core material) magnetic amplifiers do not
track because of hysteresis.
4. Amplifiers with feedback or high-gain self-saturated amplifiers
are subject to instability when biased to cut-off and may change linear
amplifiers into bistable amplifiers.
5. When the amplifier operates over a wide range of ambient tem-
perature, variations in resistance of the reactors and rectifiers, and
hysteresis loop width, cause changes in gain, output, and balance.
6. Response time of a magnetic amplifier is a limitation in compari-
son with an electronic amplifier.
7. Variations in supply frequency and voltage cause variations in
gain and output, especially with self-saturated amplifiers.
8. "Whereas the vacuum tube is a relatively high-impedance device,
the magnetic amplifier is better adapted to low impedances, where the
turns arc fewer.
9. Saturation inductance is greater than the leakage inductance of
the reactor, measured as in a transformer. The B-H curve slope at
Ss, even with rectangular loop core materials, always gives fn greater
than unity at the top. This effect reduces output and gain, and causes
a sloping wave front at the instant of firing.
MAGNETIC AMPLIFIERS 291
Many ingenious circuits have been devised to overcome one or more
of these limitations. For descriptions of these circuits, for refinements
of operation, and for fields of application, the reader is referred to the
bibliography on magnetic amplifiers.
10. PULSE AND VIDEO TRANSFORMERS
122. Square Waves. Square waves or pulses differ from sine waves
in that the front and back sides of the wave are very steep and the top
fiat. Such pulses are used in the television and allied techniques to
produce sharp definition of images or signals. A square wave can be
thought of as made up of sine waves of a large number of frequencies
starting with, say, audio frequencies and extending into the medium
r-f range. The response of a transformer to these frequencies deter-
mines the fidelity with which
J i P I a .
(a)
(c)
Fig. 225. Square waves differentiated and
clipped.
the square wave is reproduced
by the transformer. Some
pulses are not square, but have
sloping sides and a round top,
ni K I K ''^^ '"'^ half-wave rectifier volt-
/ ]/ I * ' age. Such pulses will not be
' ' discussed here, because if a
transformer or circuit is capable
of reproducing a square wave,
it will reproduce a rounded wave
at least as well.
Square waves can be gener-
ated in several ways: sometimes
from sine waves by driving a tube into the diode region each cycle. The
plate circuit voltage wave form is then different from that of the grid
voltage because the round top of the sine wave has been removed. By
repeating this process (called clipping) in several stages, it is possible
to produce very square waves, alternatelj^ plus and minus, like those of
Fig. 225(a). If a voltage having such a wave form is applied across a
capacitor, it causes current to flow in the capacitor only at the vertical
edges, as in Fig. 225(b). If a voltage proportional to this current is
then successively amplified and clipped at the base, it results in the
wave form of Fig. 225(c). Here the negative loops are assumed to be
missing, as they could be after single-side amplification. The wave is
292
PULSE AND VIDEO TRANSFORMERS 293
again square but of much shorter duration than in Fig. 225(a), and
the interval between pulses greatly exceeds the pulse duration. The
pulse duration is usually referred to as the pulse width, and the fre-
quency at which the pulses occur is called the repetition rate and is
expressed as pulses per second (pps). Common pulse widths lie be-
tween 0.5 and 10 microseconds; the intervals between pulses may be
between 10 and 1,000 times as long as the pulse width. These values are
representative only, and in special cases may be exceeded several-fold.
The general wave shape of Fig. 225(c), with short pulse duration com-
pared to the interval between pulses, is the main subject of this chap-
ter. The ratio of peak to average voltage or current may be very
high, and the rms values appreciably exceed average in such pulses.
There are two ways in which the response of any circuit to a square
wave can be found. The first of these consists in resolving the pulse
into a large number of sine waves of different frequencies, finding the
response of the circuit to each frequency and summing up these re-
sponses to obtain the total response. This can be formulated by a
Fourier integral, but for most circuits the formulation is easier than
the solution. An approximation to this method is the arbitrary omis-
sion of frequency components having negligible amplitude, and calcula-
tion of the circuit response to the relevant frequencies. This approxi-
mation has two subjective criteria: the number of frequencies to be
retained, and the evaluation of the frequency components for which the
circuit has poor response.
The second method, which will be used here, consists in finding the
transient circuit response to the discontinuities at the front and trailing
edges of the square wave. It is possible to reduce the transformer to a
circuit amenable to transient analysis, without making any more as-
sumptions than would be necessary for practical design work with
the Fourier method. The transient method has the advantage of
giving the total response directly, and it can be plotted as a set of
curves which are of great convenience to the designer. The major
assumption is that one transient disappears before another begins. If
good square wave response is obtained, this assumption is justified.
Analyses are given of the influence of iron-core transformer and
choke characteristics on pulse wave forms. In all these analyses the
transformer or choke is reduced to an equivalent circuit; this circuit
changes for different wave forms, portions of a wave form, and modes
of operation.
Initial conditions, resulting formulas, and plots of the formulas for
294
ELECTRONIC TRANSFORMERS AND CIRCUITS
design convenience are given for each case. Formulas may be verified
by the methods of operational calculus.
123. Transformer-Coupled Pulse Amplifiers. The analysis here
given is for a square- or flat -topped pulse impressed upon the trans-
former by some source such as a vacuum tube, a transmission line, or
a b
I
I
1
E
Fig. 226. Flat-topped pulse.
Fig. 227. Transformer coupling.
even a switch and battery. Such a pulse is shown in Fig. 226, and a
generalized circuit for the amplifier is shown in Fig. 227. The equiva-
lent circuit for such an amplifier is given in Fig. 228. At least this is
the circuit which applies to the front edge a of the pulse shown in Fig.
226 as rising abruptly from zero to some steady value E. This change
is sudden, so that the transformer OCL can be considered as presenting
infinite impedance to such a change, and is omitted in Fig. 228. Trans-
former leakage inductance, though, has an appreciable influence and
is shown as inductance Lg in Fig. 228. Resistor Ri of Fig. 228 repre-
I
^S R|
n.
Fig. 228. Equivalent circuit.
sents the source impedance; transformer winding resistances are gen-
erally negligible compared to the source impedance. Winding capaci-
tances are shown as Ci and C2 for the primary and secondary windings,
respectively. The transformer load resistance, or the load resistance
into which the amplifier works, is shown as 7?2- All these values are
PULSE AND VIDEO TRANSFORMERS
295
referred to the same side of the transformer. Since there are two
capacitance terms Ci and Co, it follows that, for any deviation of the
transformer turns ratio from unity, one or the other of these becomes
predominant. Turns ratio and therefore voltage ratio affect these
capacitances, as discussed in Chapter 5 ; for a step-up transformer, Ci
may be neglected, and, for a step-down transformer, C2 may be neg-
lected. The discussion here will be confined first to the step-up case.
124. Front-Edge Response. The step-up transformer is illustrated
by Fig. 229. When the front of the wave (Fig. 226) is suddenly im-
J R
Fig. 229. Circuit for step-up tran.sformer.
pressed on the transformer, it is simulated by the closing of switch S.
At this initial instant, voltage e across R2 is zero, and the current from
battery E is also zero. This furnishes two initial conditions for equa-
tion 128, which expresses the rate of rise of voltage e from zero to its
final steady value Ea = ER^/iRi -f- R2) :
ERo
Ri + R2
1 +
mae'"''
- 1
mi — ni2
TO2
nil
(128)
where mi, m2 = — m(l =t v 1 — l/fci^).
Figure 230 shows the rate of rise of the transformed pulse for R^ =
and Fig. 231 for Ri = R2. In hard-tube modulators, source resist-
ance is comparatively small and approaches R^ = 0. Line-type modu-
lators are usually designed so that R^ = R^-
The scale of abscissas for these curves is not time but percentage
of the time constant T of the transformer. The equation for this time
constant is given in Figs. 230 and 231. It is a function of leakage
inductance and of capacitance C.,. The rate of voltage rise is governed
by another factor fci, which is a measure of the extent to which the
circuit is damped. The relation of this factor fci and the various con-
296
ELECTRONIC TRANSFORMERS AND CIRCUITS
slants of the transformer is given directly in Figs. 230 and 231. The
greater the transformer leakage inductance and distributed capaci-
tance, the slower is the rate of rise, although the effect of Ri and R2 is
important, for they affect the damping factor fci. If a slight amount of
1.2
# 0.8
0.6
0.4
0.2
I'
0.
25
R| =
/
\
h
XA 1
\l
\
\
";?•
/
\
^
^
■^
c
1
/"=,
f^
/
n.5
^
^
//
/.
2
y"
^
L
/
/
4'
.
^
k
="10/
IRCF
RF'^I
1TAN
CF IK
DHK
f«;
//
/
/
y
4
Rz'LOAD RESISTANCE IN OHUS
/
/■
/
/
- Lg-LEAKAGE INDUCTANCE IN HENur^^
Cz'EOUIVALENT CAPACITANCE IN FARADS
If/
/
/
T=Z1T*i/LsCz
/
/
k-rn^LCz
1
/
"'=2R,C,
1 1
51
2
2T
2
3T
Fig. 230. Influence of transformer constants on front edge of pulse (iJj = 0) .
equivalent circuit see Fig. 229.
For
oscillation can be tolerated, the wave rises up faster than if no oscilla-
tion is present. Yet, if the circuit is damped very little, the oscilla-
tion may reach a maximum initial voltage of twice steady-state voltage
Ea, and usually such high peaks are objectionable. The values for fci
given in these figures are those which fall within the most practicable
range. Time required for pulse voltage to reach 90 per cent of Ea is
given in Fig. 232.
PULSE AND VIDEO TRANSFORMERS
297
EqO.B
0.4
0.2
0.25
I
r\
\
\
OA
\
"l
"2
//
'
^
/
°^
ro7
V
^
^
\k
/^
J^
^
k
/
^
■
\
1
'/^
/
)>
\
>
/
"^
_^___
^
K
—
1
\i
/
y
^
^
^
1
'/
/
/
-b,
^
y
7
^
^
-^
III
,
/
/
f)
y
^
III
/
/
/
/
y
1 /
f
/
/
/
Rl = SOURCE RESISTANCE (OHMSI
R^=LOAD RESISTANCE (OHMS)
/
/
l-g-LEA
Cg^EQL
T =2T
KAGE INDUCTANCE (HENRYS)
IIV.CAPACITANCE (FARADS)
Iv
/
Wi-s Cz
R, 1
y
k, m^/LsCz.m- ^^^ +
2RZC2
3T
2
Fig. 231. Influence of transformer constants on front edge of pulse {R^ = R,,).
125. Response at the Top of the Pulse. Once the pulse top is
reached, Ea is dependent on the transformer OCL for its maintenance
at this value. If the pulse stayed on indefinitely at the value Ea, an
infinite inductance would be required to maintain it so, and of course
this is not practical. There is always a droop at the top of such a pulse.
The equivalent circuit during this time is shown in Fig. 233. Here the
inductance L is the OCL of the transformer, and R^ and i?2 remain the
same as before. Since the rate of voltage change is relatively small
during this period, capacitances Ci and €■> disappear. Also, since
leakage inductance usually is small compared with the OCL, it is- neg-
lected. At the beginning of the pulse, the voltage e across R2 is assumed
to be at steady value Ea which is true if the voltage rise is rapid.
Curves for the top of the wave are shown by Fig. 234. Several curves
298
ELECTRONIC TRANSFORMERS AND CIRCUITS
are given; they represent several types of pulse amplifiers ranging from
a pentode in which R2 is one-tenth of Ri, to an amplifier in which load
resistance is infinite and output power is zero. In the latter curve, the
1 2
//
'-0
/
/
--■^^i^^^
^
^^
/J
1
Fig. 232. Time required to reach 90 per cent of final voHage.
voltage e has for its initial value the voltage of the source. All the
curves are exponential, having a common point at 0, 1. Abscissas
are the product of time t and Ri/Le, time t being the duration of the
pulse between points a and b in
Fig. 226. The greater the in-
ductance Le the less the devia-
tion from constant voltage dur-
ing the pulse.
126. Trailing-Edge Response.
At instant b in Fig. 226, it is as-
sumed that the switch S in Fig.
233 is opened suddenly. The
equivalent circuit now reverts
to that shown in Fig. 235, in which Le is the OCL, and Co is total
capacitance referred to the primary. Figure 235 illustrates the de-
cline of pulse voltage after instant b (Fig. 226), the equation for
which is:
e = [(mi + 2Am)e"'"i' - (mj + 2Am)t'"^'] (129)
mi — m2
where mi,m2 = — m(l ± VI
r
t
Fig. 233. Circuit for top of pulse.
l/fcs ),m = }^RiCr>, and other terms
are defined in Fig. 235. Abscissas are time in terms of the time constant
determined by OCL and capacitance Cd- Ratio k^ on these curves has
PULSE AND VIDEO TRANSFORMERS
299
a different meaning, and time constant T is greater than in Fig. 230,
but with low capacitance /cs is high and the curves with higher values
of fcs drop rapidly. The slope of the trailing edge can be kept within
tolerable limits, provided that the capacitance of the transformer is
small enough. Accurate knowledge of this capacitance is therefore im-
portant. Measurement and evaluation of transformer capacitance
should be made as in Chapters 5 and 7.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
£ -
'^
— ■
1
:J;
^~~~'
Pg =0.1 Ri-
\
^
V.
\1
^
■--
'^
N
■^
^>^
R2 = 1?
(
r"^
R|
R
2 = to
"2=2"/
1 1
i
«2
JL
T
t
Le'O.C.L. IN MICROHENRYS
-
=•
\ R, 'SOURCE RESISTANCE IN OHMS
0.2
0.1
"u-
0.6
1. 2
TR,
Fig. 234. Droop at top of pulse transformer output voltage.
If the transformer has appreciable magnetizing current, the shape
of the trailing edge is changed. The greater the magnetizing current,
the more pronounced the negative voltage backswing. The ordinates
at the left of Fig. 235 are given in terms of the voltage E„, at instant
a, as if there were no droop at the top of the pulse. These curves apply
when there is droop, but then the ordinates should be multiplied by the
fraction of £"„ to which the voltage has fallen at the end of the pulse.
Magnetizing current at the end of the pulse is
iM = (Ea/mL)(l - e-n (130)
where m = RiR2/{Ri + R2)L (see Fig. 233)
T = pulse duration in seconds
L = primary OCL in henrys
Magnetizing current can be expressed as a fraction A of the primary
load current /, or A = im/I- For any R1/R2 ratio, A = [{Ri + R2)/Ri\
300
ELECTRONIC TRANSFORMERS AND CIRCUITS
Q.I 0.2 0.3 0.4 0.5 0.6 07 0.8 0.9 U)
To find e/Ea at any time t/T, ks and A:
(a) Take initial e/Ea for the appropriate t/T and k from left-hand
chart: project this point to the right to obtain intersection with
A = line.
1,6) Take second e/Ea at the same t/T and ^3 from right-hand
chart; project to the left to obtain intersection with A — 3 line.
(c) Through these intersections draw a straight line.
(d) Drop given value of A to intersect this line; project horizon-
tally to obtain actual e/Ea-
Example shown dotted is for ks = 3.84, t/T = 0.5, and A = 0.256.
Answer e/Ea = -0.21.
Ea = Volts at end of pulse
he = Primary OCL
Cd ^ Primary equivalent capacitance
Rl = Primary equivalent resistance
, VLe/Cl)
T - 2irVLeCD
Magnetizing current
3
Cd fit
•T
e
J.
EQUIV. CIRCUIT
A = '-
Load current
-.2
-.4
-.6
-.8
-1.0
Fig. 235. Interpolation chart for
X voltage droop at point h (Fig. 226), or
W/E, = 1 - Rx^|(Rr + n<,) (131)
where E^ = voltage at point a (Fig. 226), and E' = voltage at point b.
This equation gives the multiplier for finding the actual trailing-edge
voltage from the backswing curve parameters in Fig. 235. With in-
creasing values of A the backswing is increased, especially for the
damped circuits corresponding to values of kg ^ 1.0. The same is
also true for lower values of fcg, but with diminishing emphasis, so
that in Fig. 235 exciting current has less influence on the oscillatorv
PULSE AND VIDEO TRANSFORMERS
01 0.2 0.3 0.4 0.5 6 0.7 0.8 9 1.0
301
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
pulse transformer backswing.
backswings. These afford poor reproduction of the original pulse
shape, but occasionally large backswing amplitudes are useful, as men-
tioned in Section 137.
Equation 129 is plotted at the left of Fig. 235 for A = 0, and at the
right of Fig. 235 for A = 3. Instructions are given under Fig. 235
for finding the backswing in terms of E^ by interpolation for < A < 3,
and for given values of fca and t/T. This chart eliminates the labor of
solving equation 129 foi' a given set of circuit conditions. Elements Le,
Cu, and Ri in Fig. 235 sometimes include circuit components in addi-
302 ELECTRONIC TRANSFORMERS AND CIRCUITS
tion to the transformer, as will be explained later. For linear resistive
loads, the terms are interchangeable with L and i?2 of Fig. 233, and
with C2 of Fig. 229, all referred to the primary winding.
In transformers with oscillatory constants the backswing becomes
positive again on the first oscillation. In some applications this
would appear as a false and undesirable indication of another pulse.
The conditions for no oscillations arc all included in the real values of
the equivalent circuit angular frequency, i.e., by the inequality
1 1
>
4/2/ Cd LeC£
or
VlJCd > 2Ri (132)
Terms are defined in Fig. 235.
The quantity -y/LJCi) niay be regarded as the open-circuit imped-
ance of the transformer. Its value must be more than twice the load
resistance (on a 1 : 1 ratio basis) to prevent oscillations after the trailing
edge. This requires low distributed capacitance.
Likewise the negative backswing may prove objectionable in certain
apparatus. Certain conditions for avoiding all backswing are those
represented in Fig. 235 by fc = 5 and A = 0; these require good core
material, low exciting current, low distributed capacitance, and a
loaded transformer.
127. Total Response. By means of the curves we can now construct
the pulse shape delivered to load i?2. Suppose that a transformer with
the following properties is required to deliver a flat top pulse of 15
microseconds' duration.
Primary leakage inductance (secondary
short-circuited) = 1 .89 X 10~* henry
Primary open-circuit inductance =0.1 henry
Primary/secondary turns ratio Nj,/Ns =1:3
Source resistance R\ = 800 ohms
Load resistance (primary equivalent) R2 = 5,000 ohms
Primary effective capacitance C2 = 448 ///xf
From the expressions given in Fig. 230
m = 2.34 X W
r = 1.8 microseconds
ki = 0.68
PULSE AXD VIDEO TRANSFORMERS
303
The front of the wave follows a curve between those marked fci = 0.4
and ki = 0.8 in Fig. 230. Value Ea is reached in 0.57" or 0.85 micro-
second, and an overshoot of about 10 per cent occurs in 1.2 micro-
seconds.
The top of the wave slopes down to a voltage determined by the
product tRi/Lc = 0.12, and by a curve between those for R-, = oc and
R. = 2Ri in Fig. 234. Voltage E' at b is evidently 0.9Ea.
The trailing edge is found from Fig. 235. Here
T = 42.2 X 10^''
Vo. 1/448 X 10-12
h = = 1.5
2 X 5,000
5,800
A = X 0.09 = 0.65
800
Load voltage reaches zero in 0.05T or 2.11 microseconds. The
negative loop has maximum amplitude of 33 per cent £" at 0.2T or 8.44
microseconds beyond the pulse edge b. The pulse delivered to load
i?2 is shown in Fig. 236, in terms of E instead of i?„.
Fio. 236. Output voltage of pulse transformer.
So far we have assumed that the pulse source is disconnected at the
end of the pulse. In some applications the source remains connected.
304
ELECTRONIC TRANSFORMERS AND CIRCUITS
This would result if switch S (Fig. 233) were left closed, and battery E
were short-circuited. Under these conditions the leakage inductance
remains in the circuit, and an additional transient occurs. The tran-
sient has a shape similar to one of the curves of Fig. 231 but is in-
verted and superposed on the backswing voltage due to OCL. In the
example just given, this superposed oscillation has an amplitude of 10
per cent of E, with a total result similar to the oscillogram of Fig. 237.
Superposed backswing oscillations are discussed more fully in Section
134. Because of the distributed nature of leakage inductance and
Fig. 237. Oscillogram of voltage pulse.
capacitance, higher-frequency superposed oscillations may sometimes
occur even when the load is disconnected at the end of a pulse. By
their very nature, the conditions for these oscillations are difficult to
state with certainty, but if oscillations occur on the front edge they are
likely to appear on the trailing edge, superposed on the voltage back-
swing.
128. Step-Down Transformers. The circuits of Figs. 229 and 233
for step-up pulse transformers arc essentially the same as those of
Figs. 107(e) and 107(c), respectively, for audio transformers. Low-
frequency response corresponds to the top of the pidse and high-fre-
quency response to the front edge. In step-down pulse transformers
the top is unchanged, but the front edge corresponds to Fig. 113. Step-
down transformer analysis shows that the form of equation is simi-
lar to that for step-up transformers, except that the damping factor
for the sine term is greater by the quantity R-z/L^li. Also, the decre-
ment, although still composed of two terms, has the resistances R^ and
R2 in these two terms reversed with respect to the corresponding terms
for the step-up transformer. Except for this, the front-edge curves are
little different in shape from those of step-up transformers. Where
PULSE AND VIDEO TRANSFORMERS
305
Ri = R'l the curves are virtually the same as in Fig. 231. Pentode
amplifiers, with their constant-current characteristics, can be repre-
sented by the circuit of Fig. 238. Here 7 is the current entering the
primary winding from the tube, and is constant over most of the
voltage range. The transformer is usually step-down for the reasons
of impedance mentioned in Section 70 (Chapter 5). Front-edge re-
sponse of these transformers is the same as in Fig. 230 if the rise in
load cm-rent is expressed as a fraction of final current 7, and the de-
crement is changed to R2/2Lg. It is reproduced in Fig. 239 with this
change in constants. Flatness of top is approximately that of the
curve 7^2 = 0.17?, in Fig. 234. Trailing edge is the same as in Fig. 235.
Fig. 238. Step-down transformer equivalent circuit.
It is evident that many practical cases are represented by the
figures. If transformer constants are outside the curve values, the
pertinent equation should be plotted to obtain the response.
129. Frequency Response and Wave Shape. Because of the preva-
lent thinking of engineers in terms of frequency response rather than
wave shape, it is sometimes necessary to correlate the two concepts.
The matter of phase shift enters, for the reason that the relative phase
of the different frequency components affects wave shape. It is some-
times convenient to know whether a transformer, whose frequency
response is known, can deliver a given wave shape. Starting with the
low-frequency response, assume equal source and load resistances; the
upper curve of Fig. 108 (p. 148) applies. This curve shows 90 per cent
of maximum response at the frequency for which X}//R^ = 1. How
does this frequency compare with the reciprocal of the pulse width at
the end of which there is 10 per cent droop in the top of the pulse?
Xs/R-[ can be written
2vfL/Ri = 1 or / = Ri/2TrL (133)
likewise, from the proper curve of Fig. 234, for 10 per cent droop,
tRi/L = 0.2 (134)
306
ELECTRONIC TRANSFORMERS AND CIRCUITS
Combining equations 133 and 134 gives / = 0.0318 (1/t), or the trans-
former should be not over I db down at a frequency about %o of the
reciprocal of the pulse width. For example, if a maximum of 10 per
cent droop is desired at 2 microseconds the response should be not
1.4
1.2
1.0
0.8
0.6
0.4
0.2
k,
/
r
/
\
/j
0.4 '
/
\,
(/
0.8
\
jC.
^'?
L^
^
//
/
.y(
U
j/
^
//
/
2.0,
/
/
r
Y
//
/
/
3.0,
/
4
"r
1
/ /
f
/
/
/
\
1
/
/
/
R2=RESISTANCE IN OHMS
Ls=INOUCTANCE IN HENRYS
C =CAPACITANCE IN FARADS
1
//
/
f
/
/
r =2TTVLsC
k,=mVLsC J
1
m
/
-=».
r
3T
2
2T
5T
2
3T
Fig. 239. Pentode amplifier front-edge response.
more than -1 db at 0.0318 X 0.5 X 10® = 16 kc. Maximum phase
shift is 27 degrees (from Fig. 131, p. 180), but this is taken into ac-
count in Fig. 234.
Similarly, front-edge steepness can be related to transformer high-
frequency response, which for the case of i^i = i?2 is found in Fig. 109.
The corresponding front-edge curves are found in Fig. 231. Parameter
fci of these curves is related to B in Fig. 109 as follows.
PULSE AND VIDEO TRANSFORMERS 307
(for Ri = R2)
2RiC
B = — = — at frequency fr
Ri Ri
2lTjrL L
Ri RiVlC
From equation 135 we can prepare Table XVI.
Table XVI. Pabametebs foe Fbequency Response and Wave Shape
B ki
1.0 1.0
0.8, 1.25 1.025
0.67, 1.5 1.08
0.5, 2 1.25
0.25, 4 2.125
If a transformer has frequency response according to the curve for
B = y,, 2 in Fig. 109, its front edge will rise somewhere between
curves for fci = 1 and fci = 1.4 in Fig. 231.
Transformer OCL, leakage inductance, and effective capacitance
must be known to make this comparison, but these quantities are
already known if it is established that the frequency response is given
by Figs. 108 and 109, or the wave shape by Figs. 231 and 234. If con-
ditions other than Ri = R2 prevail, another set of response curves can
be used, and corresponding approximate relations can be found in the
manner here outlined.
Pulse transformer windings are similar to those in the high-frequency
transformers described in Section 87 (Chapter 7). Resonance fre-
quency jr is determined largely by leakage inductance and winding-to-
winding capacitance. With pulse operation, partial resonances of sec-
tions of a coil, and even turn-to-turn resonance, may appear because
of the steep front edge of voltage impressed on the transformer. If
these resonances cause pronounced oscillations in the output wave
form, larger coil or turn spacings or fewer turns may be necessary to
reduce them.
308
ELECTRONIC TRANSFORMERS AND CIRCUITS
130. Core Material. In Chapter 7 it was shown that core per-
meability decreases with frequency, especially at frequencies higher
than audio. This decrease also occurs with short pulse widths. When
a pulse is first applied to the transformer, there is initially very little
penetration of flux into the core laminations because of eddy currents.
Hence initially only a fraction of the total core is effective, and the
apparent permeability is less than later in the pulse, or after the flux
density becomes uniform throughout the laminations.
A typical B-H curve for pulse transformers is shown in Fig. 240.
Flux density builds up in the core in the direction shown by the arrows.
/
^
^V
B
/
\
/
^
— -,
^
NORMAL
PERMEABILITY
1
/
/'
N
/
1
/d
^b
/
o
f 1
.-^
^
//
/
h-
j
1 ^
/
\
/
/
/
3
1
f
}
,/
/
Z
//
'
/
'1
/
/
Ih
^
^,
> A
/
J
^
/
-
A
V
/
y
—
/
A
A
V'
MAGNETIZING FORCE H
Fig. 240. Pulso B-H loops.
For a typical loop such as ohcd, the slope of the loop (and hence the
permeability) rises gradually to the end of the pidse h which corre-
sponds to point h' in Fig. 236. Since magnetizing current starts de-
creasing at this point, H also starts decreasing. Current in the wind-
ings does not decay to zero immediately but persists because of wind-
ing capacitance, and sufficient time elapses for permeability to increase.
Therefore, flux density B may also increase during a short interval
after point b. The trailing edge of the pulse voltage soon reaches
zero, and this corresponds to point c on the loop. At some interval
later, the maximum backswing amplitude is reached, which corre-
sponds to point d on the loop. At this point the slope or permeability
is several times as great as at point h.
For any number of pulses of varying amplitudes but of the same
width, there are corresponding loops having respective amplitudes c.
PULSE AND VIDEO TRANSFORMERS
309
A curve drawn through point b of each loop is called the normal per-
meability curve, and this is ordinarily given as the permeability curve
for the material. The permeability fi for a short pulse width is less
than the 60-cycle or d-c permeability for the same material. Values of
pulse permeability for 2-mil grain-oriented steel are given in Fig. 241.
The permeability values include the irreducible small gap which exists
in type C cores; the cores on which the measurements were made had
a ratio of gap to core length Ig/k. =« 0.0003, but the data are not criti-
■1400
1
oX^o
A
^
y
//
>-
1-
/ /
,00° J
—
-J
000 S
/^
UJ
5
/ V
^,ooO^
- —
"^"
UJ
<<V^
So^-
UJ
^
^
-1
3
^
^
/
^
y
/
^
P^
.1 .2 .4 .6 .8 1.0 2 4 6 8 10
PULSE WIDTH IN MICROSECONDS
Fig. 241. Effective permeability versus pulse width.
cally dependent on this ratio. The effect of penetration time is clear.
Flux densities attained in pulse transformers may be low for small
units where very little source power is available, or they may be high
(several thousand gauss) in high power units. This is true whether
the pulse width is a few microseconds or 1,000 microseconds.
The nickel-iron alloys in general have lower saturation densities, but
higher permeabilities below saturation than either grain-oriented or
ordinary silicon steel. Depending on the flux density chosen, the in-
crease of permeability with the use of a nickel-iron alloy may vary
from zero to 300 per cent. This increase holds also for long-time
pulses, during which permeability may approach the 60-cycle value.
In order to overcome the net d-c pulse magnetization which is in the
same direction throughout each pulse, an air gap may be inserted in
the core to prevent it from returning only to the residual magnetism
310
ELECTRONIC TRANSFORMERS AND CIRCUITS
value Br at the end of each pulse, and thereby limiting its useful pulse
flux density range AS to the difference between maximum flux density
Bm and residual Br (see Fig. 242). This gap increases the effective
length of the magnetic path and reduces OCL from the value it has
with symmetrical magnetization. The reduction is less with core mate-
rials of low permeability. To maintain the advantage of high per-
meability in nickel-iron alloj's, the core is "reset." This is done by
iB WITHOUT
GAP OR RESETTING
Fig. 242. Flux density range in pulse transformer cores.
arranging the circuit so that, during the period of backswing, sufficient
negative current flows through the windings to overcome coercive force
He and drop the flux density to the negative value of residual mag-
netism. Then nearly twice the previous maximum flux density (AB'
in Fig. 242) is available for the pulse. Where resetting is possible,
it is advantageous to use nickel-iron alloys; where resetting is not
practicable, grain-oriented silicon steel is preferable.
131. Windings and Insulation. Pulse transformers generally have
single-layer concentric windings with solid insulation between sections.
For high load impedance, a single section each for primary and second-
ary as in Fig. 166 is favorable, as the effective capacitance is lowest.
For low load impedance, more interleaving is used to reduce leakage
inductance. To reduce capacitance to a minimum, pie-section co-
PULSE AND VIDEO TRANSFORMERS 311
axial windings may be used. In these, coil capacitance is kept low by
the use of universal windings, and intersection capacitance between
windings is low because the dielectric is air. Such coils are more diffi-
cult to wind, require more space, and therefore are used only when
necessary.
Coil sections can be wound with the same polarity as in Fig. 166
(p. 219) or with one winding reversed. Effective capacitance between
P and iS is given below for three turns ratios. Capacitance is. based on
100 /x/xf measurable capacitance.
Turns Ratio Effective Capacitance Referred to Primary
N1/N2 Same Polarity Reversed Polarity
1:5 533 1200
1:1 133
6:1 21 48
From this it can be seen that the polarity exemplified in Fig. 166 is
preferable for reducing effective capacitance, but that the percentage
difference is greatest for turns ratios near unity and less as the ratio
increases.
Attention to the insulation so far has centered around capacitance.
The insulation also must withstand the voltage stress to which it is
subjected. It can be graded to reduce the space required. Low-fre-
quency practice is adequate for both insulation thickness and end-turn
clearances.
Small size is achieved by the use of solventless varnish. Small size
with consequent low capacitance and low loss results in higher prac-
ticable impedance values and shorter pulses.
In order to utilize space as much as possible, or to reduce space for
a given rating, core-type construction is often used. Low capacitance
between high-voltage coils is possible in such designs. It is advan-
tageous in reducing space to split the secondary winding into non-
symmetrical sections. Although the leakage inductance is higher with
non-symmetrical windings, there is less distributed capacitance when
the high-voltage winding has the smaller length. Lower capacitance
obtains with two coils than with a shell-type transformer of the same
interleaving. In core-type transformers high-voltage windings are the
outer sections. It is preferable to locate terminals or leads in the coil
directly over the windings in order to maintain margins. Insulating
barriers may be located at the ends of the windings to increase creep-
age paths.
312 ELECTRONIC TRANSFORMERS AND CIRCUITS
Autotransformers, when they can be used, afford opportunity for
space saving, because there are fewer total turns and less winding
space is needed. Less leakage inductance results, hut not necessarily
less capacitance; this always depends on the voltage gradients.
Initial distribution of voltage at the front edge of a pulse is not uni-
form because of turn-to-turn and winding-to-winding capacitance. In
a single-layer coil the total turn-to-turn capacitance is small compared
to the winding-to-ground capacitance, because the turn capacitances
add in series but the ground or core capacitances add in shunt. There-
fore a steep wave of voltage impressed across the winding sends cur-
rent to ground from the first few turns, leaving less voltage and less
current for the remaining turns. Initially, most of the pulse voltage
appears across the first few turns.
After a short interval of time, some of the current flows into the
remaining turns inductively. Before long the capacitive voltage dis-
tribution disappears, all the current flows through all the turns, and
the voltage per turn becomes uniform. This condition applies to most
of the top of a pulse. Between initial and final current distribution,
oscillations due to leakage inductance and winding capacitance may
appear which extend the initially high voltage per turn from the first
few turns into some of the remaining turns.
Winding capacitance to ground is evenly distributed along the wind-
ing of a single coil, and so is the turn-to-turn capacitance. If a rec-
tangular pulse E is applied to one end of such a winding, and the other
end is grounded, the maximum initial voltage gradient is ^
aE
— coth a
N
where N = number of turns in winding
Cg = capacitance of winding to ground
Cu, = capacitance across winding
= turn-to-turn capacitance/A''.
Practical values of a are large, and coth a approaches unity. Then
Maximum gradient ~ aE/N (136)
1 For the development of this expression see "Surge Phenomena," British Elec-
trioal and Allied Industries Research Association, 1941, pp. 223-226.
PULSE AND VIDEO TRANSFORMERS 313
or the maximum initial voltage per turn is approximately a times the
final or average voltage per turn.
If the other end of the winding is open instead of grounded, equation
136 still governs. This means that maximum gradient is independent
of load. If there is a winding Ni between the pulsed winding iV2 and
ground, a depends on Ci„2 and Ci in series. The initial voltage in
winding iVi is ^
EC i_2
El = (137)
C,_2 + C\
where Ei = initial voltage in winding A^i
E = pulse voltage applied across N2
Ci_2 = capacitance between Ni and N2
Ci = capacitance between A^i and core.
Thus the initial voltage in winding A^i is independent of the transformer
turns ratio. It is higher than the voltage which would appear in N2 if
A^i were pulsed, because then current would flow from Ni to ground
without any intervening winding. If winding A^i is the low-voltage
winding (usually true), applying pulses to it stresses turn insulation
less than if iV2 is pulsed.
Reinforcing the end turns of a pulsed winding to withstand better
the pulse voltages is of doubtful value, because the additional insula-
tion increases a and the initial gradient in the end turns. Increasing
insulation throughout the winding is more beneficial, for although a is
increased the remaining turns can withstand the oscillations better as
inductance bect)mes effective. Decreasing winding-to-corc capacitance
is better yet, for then a decreases and initial voltage gradient is more
uniform.
132. EfRciency. Circuit efficiency should be distinguished from
transformer efficiency. Magnetization current represents a loss in
efficiency, but it may be returned to the circuit after the pulse. Circuit
efficiency may be estimated by comparing the area of the actual wave
shape across the load to that impressed upon the transformer; it in-
cludes the loss in source resistor Ri (Fig. 233). Except for this loss,
the circuit and transformer efficiency are the same when the source is
cut off at the end of the pulse. It is important in testing for losses to
use the proper circuit.
Core loss can be expressed in watt-seconds per pound per pulse. A
convenient way to measure core loss is to use a calorimeter. The trans-
iSee "Surge Phenomena," pp. 227-281.
314 ELECTRONIC TRANSFORMERS AND CIRCUITS
former is located in the calorimeter, and the necessary connections are
made by through-type insulators. Dielectric loss is included in such a
measurement. It is appreciable only in high-voltage transformers, and
may be separated from the iron loss by first measuring the loss of the
complete transformer and then repeating the test with the high-volt-
age winding removed. At 6,000 gauss and 2 microseconds pulse width,
the loss for 2-mil grain-oriented steel is approximately 6,000 watts per
pound, or 0.012 watt-second per pound per pulse. For square pulses,
core loss varies (a) as B^ or E^ for constant pulse width and (6) as
pulse width, for constant voltage and duty t/, where r is the pulse
width and / is the repetition rate. Dielectric loss is independent of
pulse width and varies (a) as the repetition rate, for constant voltage,
and (b) as E^ for constant repetition rate.
Copper loss is usually negligible because of the comparatively few
turns required for a given rating if a wire size somewhere near normal
for the rms current is used. If the windings are used to carry other
current, such as magnetron filament current, the copper loss may be
appreciable but this is a circuit loss.
Efficiencies of over 90 per cent are common in pulse transformers,
and with high-permeability materials over 95 per cent may be ob-
tained. These figures are for pulse power of 100 kw or more. Maxi-
mum efficiency occurs when the iron and dielectric losses are equal.
133. Non-Linear Loads. The role played by leakage inductance and
distributed capacitance in determining pulse shape has been mentioned
in Sections 124 and 126. It has been shown that the first effect is a
more or less gradual slope on the front edge of the pulse, and that the
second effect consists of oscillations superposed upon the voltage back-
swing following the cessation of the pulse. Consider the additional
influence of non-linear loads upon the first effect, that is, upon the
pulse front edge.
Figure 230 is based on the following assumptions:
(a) Load and source impedances are linear.
(6) Leakage inductance can be regarded as "lumped."
(c) Winding capacitance can be regarded as "lumped."
Assumptions (b) and (c) are approximately justified. Pulses effec-
tively cause the coils to operate beyond natural resonance, like the
higher-frequency operation of r-f coils in Section 97 (Chapter 7).
The distributed nature of capacitance and leakage inductance, as well
as partial coil resonance, may cause superposed oscillations which re-
PULSE AND VIDEO TRANSFORMERS
315
quire correction. But the general outline of output pulse shape is
determined by low-frequency leakage inductance and capacitance.
Assumption (a) may be a serious source of error, for load impedances
are often non-linear. Examples are triodes, magnetrons, or grid cir-
cuits driven by pulse transformers. In a non-linear load with current
flowing into the load at a compara-
tively constant voltage, the problem
is chiefly that of current pulse shape.
First assume that no current flows
into this load for such time as it
takes to reach steady voltage E.
During this first interval, the trans-
former is unloaded except for its
own losses, and is oscillatory. After
voltage E is reached, the current
rises rapidly at first and then more
slowly, as determined by the new
load R2- The sudden application of
load at voltage E damps out the os-
cillations which would exist without
this load, and furnishes two condi-
tions for finding the initial current.
A rigorous solution of the problem
involves overlapping transients and
is complicated.
The problem can be simplified by
assuming that the voltage pulse
has a flat top E. When the pulse
voltage reaches E, capacitance C-z
ceases to draw current. At the in-
stant tr (Fig. 243) when voltage E
is first reached, the current in L^ which was drawn by capacitance d
flows immediately into the load. Also since the voltage was rising
rapidly at instant t,-, the energy which would have resulted in the first
positive voltage loop (shown shaded in Fig. 243) must be dissipated
in the load. The remaining oscillations also are damped. Prior to the
time tr, all the current through L^ flowed into €2- The value of this
current is C2 de/dt. Therefore we may find the slope of the appro-
priate front-edge voltage curve and multiply by the transformer ca-
pacitance to obtain the initial current. Unloaded transformer front
edge means small fci in Figs. 230 and 231. The front-edge slope at
12- -f--
FiG. 243. Non-linear load voltage
and current pulse shapes.
316
ELECTRONIC TRANSFORMERS AND CIK(^UITS
voltage E is given in Fig. 244, the ordinates of which ure {T de/E)/dt,
with E corresponding to the Ea of Fig. 230. Ordinates of Fig. 244 are
multiplied by C2E/T to find the initial load current.
Few non-linear loads have absolutely zero current up to the time
that voltage E is reached, and the foregoing assumptions are thus
10
1.0
0.1
^ — ^
"^
^
^ —
"•^^
'^--
\
^
K
\
N
\
\
\
s
'
T^cti-I.C
)
\
\
ki
\
0.1 0.2 0.3 0.4 0.5 06 0.7 08
Fig. 244. Front-edge slope of pulse tran-sfonuer.
0.9
approximate. In spite of this, the following procedure gives fair
accuracy.
(a) Find the initial capacitance current as just outlined.
(b) Estimate the current at which the load e-i cui'\'e departs from a
straight line {ii, in Fig. 243).
(c) Add currents (a) and (b) . This gives j'a (Fi^' -43), as the total
initial current.
Pulse current continues to rise beyond the value (;. if the initial cur-
rent value just found is less than the final operating cuirent correspond-
ing to the voltage E; it will droop if the initial cuiient is higher than
PULSE AND VIDEO TRANSFORMERS 317
the load current at voltage E. To obtain constant current over the
greater part of the pulse width, t2 should equal the load current at volt-
age E. When this equality does not exist, the rate of rise or droop is
determined by transformer leakage inductance, source impedance, and
load resistance. Where the mode of operation depends upon the rate
of voltage rise, as it does in some magnetrons, the initial current may
drop off to nearly zero before the main current pulse starts. When
there is negligible initial current I'l, the condition for a good current
pulse is E/i2 ^ ^/Lg/Ci, where Lg is the leakage inductance.
At the end of the pulse, when the source voltage is reduced to zero
(point h, Fig. 226), the circuit reverts to that shown on Fig. 235, but
the transformer loses all the load except its own losses. Since by this
time it has drawn exciting current, the higher values of A in the back-
swing curves apply. Backswing amplitudes with non-linear loads are
complicated and can be predicted only approximately. A procedure
for line-type pulsers is given in Chapter 11.
134. Design of Pulse Transformers. (A) Requirements. The per-
formance of a pulse transformer is usually specified by the following:
(a) Pulse voltage. (/) Slope of front.
(6) Voltage ratio. ig) Droop on top.
(c) Pulse duration. (h) Amount of backswing
(d) Repetition rate. permissible.
(e) Power or impedance level. ((') Type of load.
Design data for insuring that these requirements are met are pro-
vided in the foregoing sections, in several sets of curves. Below are
outlined the steps followed in utilizing these curves for design purposes.
{B) Start of Design. The first step in beginning a design is to choose
a core. It is helpful if some previous design exists which is close in
rating to the transformer about to be designed.
After choosing the core to be used, the designer must next figure the
number of turns. In pulse transformers intended for high voltages, the
limiting factor is usually flux density. If so, the number of turns may
be derived as follows, for unidirectional pulses:
Nd(t> , dB
e = X 10-** = A^^, — X 10"**
di dt
fe dt = NA, fdB X 10"** (138)
318 ELECTRONIC TRANSFORMERS AND CIRCUITS
For a square wave, e = E and
Et = NA,B X 10"^
or
Et X 10^
N = (139)
QA5BAe
where E = pulse voltage
T = pulse duration in seconds
B = allowable flux density in gauss
Ac = core section in square inches
A^ = number of turns.
In many designs, the amount of droop or the backswing which can be
tolerated at the end of the pulse determines the number of turns, be-
cause of their relation to the OCL of the transformer.
After the turns are determined, appropriate winding interleaving
should be estimated and the leakage inductance and capacitance cal-
culated.
With the leakage inductance and winding capacitance estimated,
the front-end performance for linear loads can be found from Figs. 230
and 231. Likewise, from OCL and winding capacitance, the shapes of
the top and trailing edge are found in Figs. 234 and 235, If perform-
ance from these curves is satisfactory and the coil fits the core, the
design is completed.
(C) Final Calculations. Preliminary calculations may show too
much slope on the front edge of the pulse (as often happens with new
designs). Two damping factors Ri/2Ls and I/2R2C2 contribute to the
front-edge slope, and the preliminary calculations show which one is
preponderant. Sometimes it is possible to increase leakage inductance
or capacitance without increasing time constant 7' greatly, and this
may be utilized in decreasing the slope.
If the front-edge slope is still too much after these adjustments, the
core chosen is probably inadequate. Small core dimensions are desir-
able for low leakage inductance and winding capacitance. Small core
area Ac may require too many turns to fit the core. These two con-
siderations work against each other, so that the right choice of core is
a problem in any design.
If the calculated front-edge slope is nearly good enough it may be
improved by one of the following means:
PULSE AND VIDEO TRANSFORMERS 319
(a) Change number of turns. (d) Increase insulation thickness.
(6) Reduce core size. (e) Reduce insulation dielectric
(c) Change interleaving. constant.
High capacitance is a common cause of poor performance and items
[b) to (e) may often be changed to decrease the capacitance. It is
sometimes possible to rearrange the circuit to better advantage and
thereby make a deficient transformer acceptable. One illustration of
this is the termination of a transmission line. Line termination re-
sistance may be placed either on the primary or secondary side. If it
is placed on the primary side there is usually a much improved front
edge. Figure 231 does not show this improvement inasmuch as it was
plotted for Fig. 229. For resistance on the primary side, the damping
factor reduces to the single term
R1R2
a = — (140)
2(i?i + R2)L,
Improvement of the trailing-edge performance usually accompanies
improvement of the front edge.
Core permeability is important because it requires fewer turns to
obtain the necessary OCL with high-permeability core material. Per-
meability at the beginning of the trailing edge (point b', Fig. 236) is
most important, for two reasons: the droop at this point depends on
the OCL, so that for a given amount of droop the turns on the core are
fixed; also, the normal permeability data apply to such points as b'.
Flux density is chosen with two aims: it should be as high as possible
for small size, but not so high as to result in excessive magnetizing cur-
rent and backswing voltage.
(D) Example. Assume that the performance requirements are:
Pulse voltage ratio 2,000:10,000 volts.
Pulse duration 2 microseconds.
Pulse repetition rate 1,000 per second.
Impedance ratio 50:1,250 ohms (linear).
To rise to 90 per cent of final voltage in }4 microsecond or less.
Droop not to exceed 10 per cent in 2 microseconds.
Backswing amplitude not to exceed 60 per cent of pulse voltage.
50-ohm source.
The final design has the following :
Primary turns = 20.
Secondary turns = 100.
320 ELECTRONIC TRANSFORMERS AND CIRCUITS
Core: 2-mil silicon steel with I -mil gap per leg.
Core area = 0.55 sq in.
Core length = 6.3 in. {IJh = 0.0003).
Core weight 0.75 lb, window % in. X I9i6 in.
Primary leakage inductance = 2 miorohenrys.
Effective primary capacitance = 1,800 /i/uf.
No-load loss equivalent to 400 ohms (referred to primai'y)-
^, , .^ 10,000 X 2 X 102
Flux density = eXs-^OO^^irSS = '■'*^" ^""^"
At 2 microseconds and B = 5,600 ai ~ 600.
nrj 3.2 X 400 X 0.55 X 10^^
Primary OCL = -^oo^T-TfWeW^^ = ""'''''■
Front-edge performance is figured as follows:
R, 1 50X10» 10" ,.^,,1
"' = 2Z; + Yr,c, = ^4~- + oTi-^i^g = ' '^ X 1 "
r = 6.28 X I0-''V'2 X 0.00r8 = 0.375 miciospcond
k = 1.08
According to Fig. 231, this value of k gives 90 per cent of £"„ in 0.35T' or 0.131
microsecond.
The top is figured at
tRi _ 2 X 50 X 10-" ^
L, 550 X I0-«
and from the curve Ri = Ri in Fig. 234, the top droops 9 per cent.
The magnetizing current is
{Ri -\- R'i)/Ri X 9 per cent or 18 per cent of the load current
For the backswing
10«
T = 6.28 X 10-= V550 X 0.0018 = 6 micro.seconds
k = 5.2
P'rom Fig. 235, the backswing is 20 per cent of Ea- If tlie load resistance is
connected to the transformer when the pulse voltage is removed, the backswing
superposed oscillation has the same k (1.08) as the front edge, that is, there is
no oscillation and the total backswing voltage is 20 per cent of Ea-
Suppose the load were non-linear; the voltage would rise up to E within }4T
or 0.094 microsecond. The front edge
PULSE AND VIDEO TRANSFORMERS
321
50 X 10«
+
106
0.8 X 1.8
13.2 X 10''
and
From Fig. 244,
k = 0.8
Tde
Edt
0.44
The secondary effective capacitance is 1,800/25 = 72 /i/if and the initial load
current is
,de 72 X 0.44 X 10,000
dt
0.375 X 10"
= 0.84 amp
Final load current is 10,000/1,250 = 8 amp, and current is non-uniform dur-
ing the pulse. The backswing is calculated in Chapter 11,
Secondary current is 8 amp. The rms value of this current is, from Table I
(p. 16),
/rms = 8-\/2^0^^xT000 = 0.36 amp
and the primary current is 5 X 0.36 = 1.8 amp. The wire insulation must
withstand 10,000 -^ 100 = 100 volts per turn, and with single-layer windings
this normally requires at least 0.0014 in. of covering insulation. Heavy enamel
wire. No. 28, has a margin of insulation over this value. This is further modi-
fied by the initial non-uniform voltage distribution as figured below. A sectional
view of a two-coil design is shown in Fig. 245, with No. 28 heavy enamel wire
TO SECONDARY
LEAD HV
>MICA
Fig. 245. Section of pulse transformer.
322 ELECTRONIC TRANSFORMERS AND CIRCJUITS
in the secondary and No. 22, wound two in parallel to occupy the form fully,
in the primary. The core section is % in. by % in. The primary turn length
is 3.75 in. and that of the secondary is 4.13 in. Primary and secondary d-c
resistances are 0.05 and 2.3 ohms, and the respective coi)per losses are
(1.8)2 X 0.05 = 0.162
(0.36)2 X 2.3 = 0.3
Total = 0.462
The no-load loss is [(2,000)7400] X 1,000 X 2 X lO"'' = 20 watts. Copper
loss is therefore of little significance.
From the coil dimensions and insulation thicknesses we can figure the capac-
itances. The total winding traverse for both coils is 1.875 in. The primary-to-
core capacitance is
0.225 X 3.75 X 1.875 X 5
0.030
and the secondary-to-primary capacitance is
0.225 X 4.13 X 1.875 X 5
0.060
264 tiixi
= 145 MM'
so that these two capacitances in series are 94 /i/xf. Secondary turn-to-turn
capacitance is, approximately,
0.225 X 4.13 X 0.0126 X 3 _
oM9 " ^^-^
or
C„ = 0.184
a is therefore V 94/0.184 = 22.5, and the wire enamel initially must withstand
2,250 volts per turn.
Figure 246 is a photograph of the transformer with Fosteiite-treated coils.
135. Testing Technique. Tests for open circuits, short circuits, turns
ratio, and d-c resistance are made on pulse transformers in the same
manner as in other transformers. The instruments used must be suit-
able for the low values of inductance encountered, but otherwise no
special precautions are necessary. Usually the d-c resistance is some-
what lower than the winding resistance during most of the pulse, but
even the latter value is so low that it causes no significant part of the
transformer loss. Losses are measured as described in Section 132.
Various methods have been used to check effective pulse OCL.
These may involve substitution of known inductances, or current
build-up, or decay, depending on the time constant of the transformer
PULSE AND VIDEO TRANSFORMERS
323
inductance and an external known resistance. When such measure-
ments are attempted under pulse conditions, there is usually a certain
amount of error due to reflections, incidental capacitance, and the like.
A method involving the measurement of pulse permeability and cal-
culation by the OCL formula is given here.
If the air gap and pulse permeability are known, the OCL for a given
core area and number of turns can be calculated. If the gap used is
.^"^
-$^^p
^,
"'^W^Sf-'-'-'--^,
t
t
yj ■
.»-. s;
"'■ i.\:
..
V
w>^-
Fig. 246. Pulse transformer with coils of Fig. 245.
purposely made large to reduce saturation, proper allowance for it can
be made in equation 38 (p. 97). If the gap is the minimum obtainable,
it is necessarily included in the permeability measurement, but this is
often done in taking pulse permeability data, as it was in the data of
Fig. 241. With this definition of permeability equation 38 reduces to
OCL = (3.2/iA'2A X 10~^)/lc
(141)
Equation 141 is valid only when Ig/ix )$> Ig.
B-H data for a pulse transformer are taken by means of a circuit
similar to that of Fig. 247. Primary current flowing through small
resistor Ri gives a horizontal deflection on the oscilloscope propor-
tional to / and therefore H for a given core. R-^ should be low enough
in ohmic value not to influence the magnetizing current wave form
324
ELECTRONIC TRANSFORMERS AND CIRCUITS
appreciably. If the voltage drop across a higli-rcsistance load R2
(«= 50 times normal pulse load) is almost the entire secondary volt-
age e-2, then voltage e^ applied to the vertical plates is the time in-
tegral of 62 and is therefore proportional to flux density at any instant.
(See equation 138.)
Short leads and the reduction of incidental capacitance arc essential
to obtain acciu'ate measiu'ements. Distributed capacitance of the
winding, shown dotted, should be minimized, as it introduces extrane-
_rL
PULSE
TRANSFORMER
Fig. 247. B-H test for piilsc transfonii('i>
ous current into the measurement of H. One way to minimize this
capacitance is to omit the high-voltage winding, and make all meas-
urements from the low-voltage low-capacitance coil only.
The pulse source should be the kind for which tlic transformer is to
be used. If it must be loaded to obtain proper ])ulse shape, a diode
may be used to prevent backswing discharge thi-oiigh this load and
therefore a reset core, unless reset core data are desired. Difficulty
may be experienced in seeing the B-H loops of pulses having a low
ratio of time on to time off because of the poor spot brilliance, unless
an intensifier is used to brighten the trace.
With a calibrated oscilloscope it is possible to detcimine the slope of
the dotted line ob in Fig. 247, drawn between the origin and the end of
the pulse, and representing effective permeability ^ at instant b, Fig.
226. This value of /x can be inserted in equation 141 to find OCL.
Cores failing to meet the OCL should first be examined for air gap.
Effective values of leakage inductance and capacitance are difficult
PULSE AND VIDEO TRANSFORMERS
325
to measure. The calculations of capacitance and leakage inductance
are based on the assumption of "lumped" values, the validity of which
can be checked by observing the oscillations in an unloaded trans-
former when pulse voltage is applied. The frequency and amplitude
of these oscillations should agree with those calculated from the leak-
age inductance and effective capacitance. The pulse source should be
Fig. 248. Transformer constants may be found from pulse shape.
chosen for the squareness of its output pulse. Because of the light
load, the transformer usually will be oscillatory, and produce a second-
ary pulse shape of the kind shown in Fig. 248. In this figure, the dot-
dash line is that of the impressed pulse and the solid curve is the
resulting transformer output voltage. This curve is observed by con-
necting the vertical plates of a synchroscope (oscilloscope with syn-
chronized swec])) across the transformer output winding.
The first check of leakage Lg and Co is made by finding the time con-
stant T from
This time constant can be related to the time interval to-tr in Fig.
326 ELECTRONIC TRANSFORMERS AND CIRCUITS
248 by consulting Fig. 230. Formulas in this figure can be used for
finding values of parameter fci using L^, C^, source resistance i?i and
load resistance R^. This load resistance will be that corresponding to
transformer losses only; hence R'> Ri for a pulse source with plenty
of power, and
VlJc2
fci =
2i?2
AVith this value of fci, the increase or overshoot of the first voltage
oscillation over the flat top value E may be found from Fig. 230, and
may be compared with that observed in the test. When the load is
resistive, or when the voltage pulse is the criterion of pulse shape,
these are the only checks that need to be made on leakage inductance
and distributed capacitance.
When the load is a magnetron, triode, blocking oscillator, grid cir-
cuit, or other non-linear load, the shape of the current pulse is impor-
tant. Ordinarily the current will not build up appreciably before
time tr in Fig. 248. The shape of this current pulse and sometimes the
operation of the load are determined to a large extent by slope AB of
the no-load voltage at time t,-. This time is the instant when the first
oscillation crosses the horizontal line E in Fig. 248. As indicated in
Fig. 244, there is a relationship between this slope and the parameter
fci. If the slope AB is confirmed, correct current pulse shape is also
assured.
Insulation can be tested in one of two ways, depending on whether
the insulation and margins are the same throughout the winding or
whether the insulation is graded to suit the voltage. In the former
case an equivalent 60-cycle peak voltage, applied from winding to
ground at the regular 60-cycle insulation level, is sufficient. But, if
the winding is graded, this cannot be done because the voltage must
be applied across the winding and there is not sufficient OCL to support
low-frequency induced voltage; hence a pulse voltage of greater than
normal magnitude must be applied across the winding. Adequate
margins support a voltage of the order of twice normal without insula-
tion failure.
Such pulse testing also stresses the windings as in regular operation,
including the non-uniform distribution of voltage gradient throughout
the winding. The higher-voltage test ought to be done at a shorter
pulse width so that saturation does not set in. In cases of saturation,
PULSE AND VIDEO TRANSFORMERS
327
the voltage backswing is likely to exceed the pulse voltage of normal
polarity and thus subject the insulation to an excessive test. This
backswing may be purposely used to obtain higher voltage than the
equipment can provide, but it must be carefully controlled. Corona
tests are sometimes used in place of insulation tests, and this can be
done, where the insulation is not graded, by using a 60-cycle voltage
and a sensitive receiver to pick up the corona noise. With graded
CUT OFF BY GRIDS
POLARITIES + AND - OCCUR DURING PULSE
TERMINALS © INDICATE PULSE VIEWING CONNECTIONS
TO
OSCILLOSCOPE
Fig. 249. Pulse amplifier with oscilloscope connections.
insulation a high frequency must be used. The method becomes too
difficult to use because the receiver may pick up part of the high-fre-
quency power emitted from the transmitter, or the transmitter parts
may generate a certain amount of corona which is more troublesome
than at 60 cycles.
In pulse amplifiers, the mode of operation of the tubes and circuit
elements is important. A round irregular pulse may be changed by
grid saturation, or by non-linear loading of some other sort, into a
practically square wave pulse. It may take several stages of amplifi-
cation to do this in certain instances, and a transformer may be used
at each stage. Often the function of the transformer is to invert the
pulse for each stage; that is, the transformer changes it from a nega-
tive pulse at the plate of one stage to a positive pulse on the grid of
the next. Polarity is therefore important and should be checked dur-
328 EI.ECTRONIC TRANSFORMERS AND CIRCUITS
ing the turns-ratio test. If the transformer fails to deliver the proper
shape of pulse, it may he deficient in one of the properties for which
tests are mentioned above. Figure 249 shows a i)ulse amplifier with
normal pulse shapes for each stage. Checking each stage at the points
indicated, without spoiling the pulse shape by the measuring appara-
tus, requires attention to circuit impedance, stray capacitance, cable
termination, and lead length.
11. PULSE CIRCUITS
Pulse generation and utilization require the use of various non-
sinusoidal devices. Formation of pulses from sine waves was men-
tioned briefly in Chapter 10. In the following sections, other meth-
ods of forming and using pulses are described.
136. Blocking Oscillators. Blocking oscillators are used to obtain
pulses at certain repetition rates. The pulse may be used to drive a
OSCILLATOR
TUBE
Fig. 250. Blocking oscillator.
pulse amplifier, or it may be used to modulate a UHF oscillator.
A typical blocking oscillator circuit is shown in Fig. 250. The grid
is driven hard, and grid current usually is comparable in magnitude to
anode current. Grid and anode winding turns are approximately
equal. The oscillator operates as follows.
If the grid is only slightly negative, the tube draws plate current
and because of the large number of grid turns the transformer drives
the grid positive, increases the plate current, and starts a regenerative
329
330
ELECTRONIC TRANSFORMERS AND CIRCITITS
action. During this period, the grid draws current, charging the bias
capacitor to a voltage depending on the grid current flowing into the
bias resistor-capacitor circuit. The negative plate voltage swdng is
determined by grid saturation, so that large positive swings of grid
voltage result in virtually constant plate voltage. This continues for
a length of time determined by the constants of the transformer, after
which the regenerative action is reversed. Because of lowered plate
voltage swing, the plate circuit can no longer dri^■e the low impedance
INSTANT a -PULSE STARTS. ANODE CURRENT RISES.
INSTANT b- SUDDEN RISE IN ANODE CURRENT INDUCES
ANODE WINDING VOLTAGE PEAK,
PROBABLY AT SATURATION VALUE.
INSTANT c-DUE TO COUPLING K < I, GRID VOLTAGE
PEAKS LATER. GRID CURRENT PEAKS
SIMULTANEOUSLY.
INSTANT d- ANODE CURRENT MAXIMUM
ANODE VOLTAGE STARTS TO FALL
INSTANT e-GRID VOLTAGE AND CURRENT ZERO
INSTANT f- ANODE VOLTAGE AND CURRENT ZERO.
INSTANT g- VOLTAGE BACKSWING REACHES PEAK.
CAUSES PEAK LOAD CURRENT i^-
Fig. 251. Blocking oscillator voltages and ciinents.
reflected from the grid, and the charge accumulated on the bias capaci-
tor becomes great enough to decrease plate current rapidly in a de-
generative action. Plate current soon cuts off, and then the plate
voltage overshoots to a high positive value and tlie grid voltage to a
high negative value. Grid voltage decays slowly because of the dis-
charge of the bias capacitor into the grid leak. The next pulse occurs
when the negative grid voltage decreases sufficiently so that regenera-
tive action starts again. Hence the repetition rate depends on the
grid bias R and C.
Either the negative or the positive pulse voltage may be utilized.
Instantaneous voltages and currents are shown in Fig. 251 for a load
which operates only on the positive pulse. The general shapes of these
currents and voltages approximate those in a pi'actical oscillator,
except for superposed ripples and oscillations which often occur.
The negative pulse has a much squarer wave shape than the positive
PULSE CIRCUITS 331
pulse, and consequently it is used where good wave shape is required.
No matter how hard the grid is driven, plate resistance cannot be
lowered below a certain value; so a limit to the negative amplitude is
formed. There is no such limit to the positive pulse, and this charac-
teristic may be used for a voltage multiplier.
If the transformer has low OCL, the leakage inductance may be high
enough to perform like an air-core transformer. That is, there are
optimum values of coupling for maximum power transfer, grid drive,
and negative pulse shape, but they are not critical. Comparison of
peak voltages in Fig. 251 shows that the usual 180° phase relationship
between grid and plate swings do not hold for such a blocking oscil-
lator, if the term "phase" has any meaning in this case.
The front-edge slope of the negative pulse is determined by leakage
inductance and capacitance as in Fig. 230, with two exceptions: the
pulse is negative and the load is non-linear ; hence there are no oscilla-
tions on the inverted top. The slope of this portion can be computed
from Fig. 234, provided tube and load resistances are accurately
known. The positive pulse can be found from Fig. 235 if these curves
are inverted.
Pulse width, shape, and amplitude also are affected by the ratio of
grid turns to plate turns in the transformer. Voltage rise is steeper as
this ratio is greater, with the qualification that grid capacitance in-
creases as the square of the grid turns ; the ratio is seldom greater than
unity. The exact ratio for close control depends on tube data which
may not be available and must be determined experimentally. The
situation parallels that of the class C low-Q oscillators mentioned in
Chapter 6.
The circuit of Fig. 250 is called a free-running blocking oscillator.
When it is desired to synchronize or otherwise control the pulse repe-
tition rate, an external "trigger" pulse is applied to the blocking
oscillator grid or cathode.
137. Line-Type Radar Pulsars. Figure 252 is the schematic dia-
gram of a line-type pulscr or modulator. This pulser is of the variety
known as d-c resonant charging, with hold-off diode. The operation
of the pulser is as follows:
During the charging period of each cycle, the diode permits direct
current to flow through the charging reactor to the pulse-forming net-
work and through the primary of the pulse transformer to ground.
The rate of charging is governed largely by the inductance of the
charging reactor and pulse-forming network capacitance. The in-
ductances in the pulse-forming network and the leakage inductance
332
ELECTRONIC TRANSFORMERS AND CIRCUITS
of the pulse transformer are so small as to be negligible during this
period.
The pulse transformer load is the magnetron, which is operated
with high negative potential on the cathode. A double winding is
provided on the pulse transformer secondary to peiiiiit filament cur-
rent to flow into the magnetron filament. The reactances of the pulse-
forming network and the charging reactor at their resonant frequency
are high compared to the circuit equivalent series resistance. There-
fore, the pulse-forming network charges up to a voltage 2E, where E
PULSE -FORMING
NETWORK
PULSE
TRANSFORMER MAGNETRON
^ J- TRANSFORMER
Fig. 252. Simplified schematic of modulator.
is the d-c supply voltage. Negligibly small voltage appears on the
pulse transformer and magnetron during the charging interval.
After the pulse-forming network has charged up fully, it is pre-
vented from discharging back through the d-c source by the hold-off
diode. At some subsequent instant, a trigger voltage on the grid of
the hydrogen thyratron causes the thyratron to conduct and permits
the pulse-forming network to discharge through the very low internal
resistance of the thyratron.
The sudden discharge of current through the thyratron causes a
voltage wave to start down the pulse-forming netwoi'k as in Fig. 253.
This voltage is an inverted step function with a value {2E — E) = E.
Initial voltage 2E on this network is divided equally between the net-
work and the pulse transformer primary, and produces pulse voltage
E, of duration t. The pulse width t is the length of time that the
pulse takes to travel down the pulse- forming network (PEN) and
back. After time ti the circuit is ready to charge slowly again through
the charging reactor. Magnetron current and voltage rise at the
PULSE CIRCUITS
333
pulse leading edge in general accordance with the explanation of
Section 133, but sometimes "despiking" network R'C is included to
reduce current oscillations. In this pulser, the magnetron equivalent
resistance Rm (referred to the pulse transformer primary) is equal to
Zjf, the pulse-forming network impedance.
2E —
t, t
(o) VOLTAGE ON SENDING- END OF
PULSE-FORMING NETWORK.
(b) VOLTAGE ON PULSE- TRANSFORMER
PRIMARY WINDING
Fig. 253. Modulator voltages.
If the magnetron circuit opens during pulser operation, voltage ap-
plied to the pulse transformer primary is doubled. This may cause
insulation failure if the open circuit continues. For this reason, spark
gaps are often provided on pulse transformers in line-type pulsers.
138. False Echoes after Main Pulse. Trailing-edge oscillations are
of two general types: (1) a long-term or low-frequency oscillation
(see Fig. 254) dependent on capacitance Cn and pulse transformer
open-circuit inductance Le, and (2) a superposed high-frequency
oscillation dependent on capacitance Co and LV ( = PFN inductance
L^r plus pulse transformer leakage inductance Lg) . If these oscilla-
tions exceed zero in the positive or main pulse direction, false "echoes"
of two kinds may occur: close echoes, adjacent to the main pulse, and
distant echoes which appear later at a comparatively longer time in-
terval. Either of these cause the magnetron to give a false indication.
The distant echo corresponds to oscillation (1), and the close echo
superposed on this long-term backswing to oscillation (2). These are
represented in Fig. 254 as oscillations /g and /2, respectively. With
proper attention to the circuit constants, it is possible to eliminate
both types of echoes.
The low-frequency backswing oscillation or axis is affected by Cs
only while it is still in the circuit. When thyratron conduction ceases
(soon after the trailing edge becomes negative) Ck is cut out of the
circuit. Once this happens, the presence or absence of distant echoes
is determined only by Co in combination with L^ and Rg. More-
334
ELECTRONIC TRANSFORMERS AND CIRCUITS
over, if Ch^Cd, close echoes are also determined by Co and L'}i,
regardless of whether or not the thyratron is conducting during the
close-echo interval.
Both low- and high-frequency oscillation amplitudes depend on the
amount of resistance in the circuit. At instant t-i this resistance is
Rm, the magnetron resistance E/i2 (Fig. 243), in comparison with
which the transformer core-loss equivalent resistance R^ is negligibly
high. After the magnetron ceases conducting, only Rr remains. Dur-
RISE TIME
'DEPENDS ON k,
DISTANT ECHO
OSCILLATION f.
CLOSE ECHO
OSCILLATION fj
(K2)
CLOSE BACKSWING AXIS (kj)
Fia. 254. Oscillations on pulse baokswing.
TIME-*
ing the trailing-edge interval, circuit resistance varies from Rm to Re-
Resistances Rm and Re may be replaced by their geometric mean Ri
during the trailing-edge interval and the part of the backswing im-
mediately following. This applies to both low- and high-frequency
backswing oscillations during the interval ^2-^1 < Fig- 254). The
low-frequency or long-term axis of backswing may then be found from
values of parameter fcs determined by L^, Co, and R[, as indicated in
Fig. 235. If the oscillations are damped out, the impedance ratio still
determines wave shape. This ratio is designated A'l. /c2, fcs, or fc^, de-
pending on the portion of the pulse as indicated in Fig. 254.
If PFN produces an essentially square wave, front-edge wave shape
at the magnetron is determined by impedance ratio
h =
i?A
2VlJCd
It may be shown ^ that if fci = 0.5 the magnetron voltage and current
1 See Pulse Generators, by Glasoe and Lebacqz, M.I.T. Radiation Laboratory
Series, Vol. 5, McGraw-Hill Book Co., New York, 1948, p. 567.
PULSE CIRCUITS 335
rise without oscillations to final value at U = 1.6 VLsCb. This ki has
little influence on the trailing edge because L., is usually small compared
to Le.
Oscillations occurring close to the trailing edge of the pulse are of
frequency /2 (determined by L'^C'd, where C'd = CdCn-/{Cd + Cm),
L'm is the sum of the transformer leakage and PFN inductance), and of
amplitude determined by
2Ri\C'd
This amplitude is superposed on the backswing as an axis, which, if
oscillatory, has frequency /a determined by LeCo- Since one purpose of
good pulser design is the elimination of false echoes, the backswing axis
considered here is always non-oscillatory. Assuming the thyratron is
non-conducting for most of the backswing interval, the condition for
non-oscillatory backswing is
C'd
^ 2Ri for the close part of the backswing
and
JLe
^ 2Re for the distant part
Cd
where L^ is the OCL at time ti and J is the ratio of low-frequency coi-e
permeability to pulse permeability. If
and
then
because generally
fc.s
h^ =
2Ri \C'd
1 /jZ^
2Re v Cd
fcs > fc4
Re jJC J
— >
Ri \ Cd
So, if the pulser is designed to prevent distant echoes, ki^ 1 and fca
is several times the value of k^. Time intervals influenced by these
336
ELECTRONIC TRANSFORMERS AND CIRCUITS
impedance ratios are illustrated in Fig. 254. In general, for a good
pulse, the close part of the backswing axis follows a non-oscillatory
pattern of relatively high-impedance ratio, such as those shown on
Fig. 235 for fcs > 1.
The general effect of pulse transformer magnetizing current is to
depress the backswing axis. Magnetizing current does not affect the
criterion for absence of false echoes ^4 > 1 ; hence a high ratio J is
helpful in eliminating close echoes. The ratio A of magnetizing cur-
rent to load current is much less for the close echo than for the distant
echo because Rj is less than Rg. Close echo A is aiJ|)roxiraated by
tRi
~l7
(142)
To prevent close echo, the first positive peak of oscillation should
be no greater than the negative backswing axis voltage at instant t.,.
Backswing axis voltage may be equated to the ani[)litude of the first
positive oscillation peak at time (2 in Fig. 254. This equation leads to
a transcendental relation between fc2, fc.s, and A, wliich is plotted in
0.7
K3
6
-
IF
F
RC
PULSER kj EXCEEDS VAL
M CURVE , THERE IS NO
UE
4
1
*^
~
-
-
-^^
FALSE ECHO,
1
'^-^-A
ji_
2
~
■~
■"
-
=:^
^
^
"3 oi>\lrr-
1
~-
^5
^
S
-■'i
D
"~
' ■
~
^
*v.
■--;
^
:ij
^
>
•^
A = iji
1
.01 0.1
Fig. 255. Borderline of close false eclioc
Fig. 255.^ It will be noted that all values of k2 in Fig. 255 are less
than unity. Hence, under the conditions here assumed, there is always
1 This equation is developed in tlie author's "False Echoes in Line-Type Radar
Pulsers," Pruc. I.R.E., 42, 1288 (August, 1954).
PULSE CIRCITITS
337
a certain amount of high-frequency oscillation superposed on the back-
swing axis. But, if fc2 is greater than the value given by Fig. 255,
there is no false close echo. Because of the approximation represented
by equivalent resistance Ri, to prevent false echoes it is best if fc2
exceeds the curve value substantially. For convenience, the various
impedance ratios are tabulated in Table XVII.
Table XVII. Pulser Impedanxe Ratios
Part of Pulse
Affected
Front edge
Close echo
Backswing axis
close to pulse
Distant echo
Value of Load
Resistance
Ri = \/RmRc
Ri = \/RmR,
Re
Impedance
Ratio Defined
R,r
h =
2-\/l,/C/-;
'''-2R,MC'u
1
2Ri \ Co
u =
1
2Re V Cd
Condition for (jood
Pulse Shape
ki = 0.5 for min. tr with
flat-top current pulse
I \L'n k2 g value in Fig. 255
for no close echo
ki > ki by definition
./ ki ^ 1 for no distant
echo
Example. Suppose that the transformer of Section 134 were used in a line-
type modulator with Lat = 50iJ.h,CK = 0.02/xf, Kj = VSO X 400 = 141 ohms.
/ = 2.0. Using equation 142, for close echo,
0.02 X 1,800 X 10-i« , ^,,, ^
^^ = (0:02-+0T0018yx"l0"- ^ ^''""^^^
L'y = 50 -I- 2 = 52 yuh
Impedance ratios:
*' " 2X 141 A/ ",650 X 10-
52 X 10-
0.628
550 X 10-
^' ~ 2 X 141^1,650 X 10-12
2.04
^4 =
2 X 550 X 10-
2 X 400 \ 1,800 X 10-
0.98
338 ELECTRONIC TRANSFORMERS AND CIRCTJITS
From ratio ki we see that the long-term backs wing is barely oscillatory. Hence
some distant false echo is possible. In Fig. 255, mininunu ki for close false
echoes is 0.17. Since k2 calculated above is 0.628, there arc no close false echoes
with this modulator. According to Fig. 235, the transfoinier has 56 per cent
backswing. It can be improved by using 0.002 in. grain-oriented silicon steel
having ju = 1,000 at 5,600 gauss. Le then becomes 1,000 nh, A = 0.283, ks =
2.75, and ^4 = 1.39. Now there are no false echoes, distant or close, and the
backswing is 30 per cent.
139. Charging Reactors. In the modulator of Fiu;. 252, a reactor
is used to charge the pulse-forming network to nearly double d-c volt-
age E. The inductance of this reactor is not critical but should not
be so great as to resonate with PFN at a frequency /,■ less than half
the pulse repetition frequency (PRF) . That is, the voltage across
PFN would not reach 2E if the resonant-charging frequency /,■ were
less than PRF/2, and the hold-off diode would be useless. It is the
function of this diode to prevent discharge of PFX through the d-c
source. This makes it possible to use reactors with normal manu-
facturing tolerances, or with variations in PRF, within the limit set
by U > PRF/2.
When no hold-off diode is used, both reactor inductance and PRF
must be held to close tolerances, to maintain high voltage across PFN,
with resonant frequency /,■ exactly equal to PRF/2. Linearity of in-
ductance with change in direct current is also necessary for accurate
control. Reactors designed by Fig. 71, p. 100, arc usually linear
enough for the purpose.
If fr < PRF/2 the hydrogen thyratron fires before voltage across
PFA^ reaches its peak, but after steady-state conditions obtain this
voltage nearly equals 2E. If /,■ is low enough, the increase of voltage
is approximately linear with time. This result is ('ailed linear charg-
ing. Reactor inductance is large in this case, and charging current
flows continuously. Here also a range of PRF'i^ may be used, all
greater than 2/r.
Reactor voltages and currents for the three methods of charging
are illustrated by Fig. 256. In all cases, Q should be high (more than
10) for efficient pulser operation. Voltages in Fig. 25G are for infinite
Q. Voltage is shown for the reactor terminal that connects to PFN.
With a hold-off diode, this voltage appears on both terminals during
the intervals wdicn current is zero. Without the diode, or if diode and
reactor in Fig. 252 are interchanged, the voltage at the left-hand
terminal, Fig. 252, is E. Insulation may then be gi'aded accordingly.
In all cases, peak voltage across the reactor is E. A-v flux density may
PULSE CIRCUITS
339
be calculated as in a filter reactor, but the effective frequency is 2/r
because the flux excursion is in one direction only at frequency /r.
In a-c resonant charging, reactor voltage would increase to QE,
where E is the a-c input voltage, if charging continued for a sufficient
number of cycles. If reactor Q > 10 and the thyratron is fired at the
end of one full charging cycle, maximum PEN voltage is wEpTc, where
Ep!c is the peak value of applied a-e voltage. In a-c charging the sup-
reactor
AND PFN
VOLTAGE
THYRATRON
FIRES
REACTOR
CURRENT
REACTOR
LINEARITY
D-C RESONANT CHARGING
WITH HOLD-OFF
DIODE
PRF< 2fr
NOT IMPORTANT
WITHOUT HOLD-OFF
DIODE
iYf\
PRF= 2fr
SHOULD BE LINEAR
FOR ACCURATE
CONTROL
D-C LINEAR
CHARGING
PRF>2ff
PRODUCES CHARGING
LINEARITY
Fig. 256. Pulser charging reactor voltage and current.
ply transformer inductance may be used instead of a separate reactor
and results in somewhat less total weight. A high reactance trans-
former is useful for this purpose, with shunts as in Section 104. If a
hold-off diode is used with a-c charging, an ordinary plate transformer
may be used and weight reduced still further.^
140. Sweep Generators. Successive pulses, which are separated in
time and cause vertical deflection, can be displayed on an oscilloscope
by a horizontal sweep. This is the term applied to deflecting means
which move the oscilloscope beam horizontally from left to right, usu-
ally at a uniform time rate. If the sweep rate is non-uniform, picture
distortion results. In magnetic deflection uniform sweep is produced
by a sawtooth current which varies linearly with time during the
sweep interval. A transformer for magnetic deflection is used in the
1 For a detailed description of a-c charging, see Glasoe and Lebacqz, op. cit.,
p. 380.
340
ELECTRONIC TRANSFORMERS AND CIRCUITS
circuit of Fig. 257. Pulses of sweep duration t,, ajiplied to the grid of
a beam tetrode cause plate current to increase throughout the pulse.
If the transformer had no losses, a pulse of constant amplitude Ea
Fig. 257. Tetrode sweep generator.
would cause current to increase linearly with time until the pulse ended,
in accordance with equation 40:
di
di
(40a)
Losses in a practical transformer are equivalent to a resistance in
series with L, and the rise in current is exponential. If the losses are
small, current rise may be confined to the part of the exponential curve
which is nearly linear, as indicated at the right in Fig. 257. The trans-
former load is usually the deflection coil of a picture tube. If this coil
also has sufficiently low loss, the deflection coil current has the same
wave form as the transformer primary, and a linear sweep results.
At the end of the pulse, current i does not stop flowing immediately,
because of the transformer and deflection coil inductance. A large
voltage backswing amplitude results, corresponding to large values of
A in Fig. 235. Values of L, C, and Re are such that the backswing is
oscillatory with a period 2Tr which is small compared to sweep dura-
tion Ts. During t,-, the first or negative half of the backswing oscilla-
tion, the oscilloscope beam is usually "blanked" or cut off quickly,
to allow the beam to retrace to the extreme left, ready for the next
sweep period. For a bright picture, the retrace i)eriod t^ should be
short compared to jg, so that scanning occurs during a large percentage
of the time. In television receivers, the sweep frequency is 15,750
cycles and the backswing frequency approximately 77,000 cycles.
Thus the retrace time is about 10 per cent of the sweep period. Posi-
tive voltage backswing is used in starting the next sweep trace, as will
be described later. Magnetic deflection sweep transformers are made
PULSE CIRCUITS
341
De-
with low-loss cores. Manganese-zinc ferrites find wide use here
sign of sweep transformers is closely integrated with the sweep circuit,
as will be shown in the next section.
Electrostatic deflection is accomplished by application of sawtooth
voltages to the horizontal plates of an oscilloscope. Such a voltage is
shown in Fig. 258. Sawtooth voltages may be formed in several ways
Fig. 258. Sawtooth wave.
Fig. 259. Sawtooth
transformer circuit.
from repetitive pulses.' If the pulse requires amplification before
being applied to the tube plates, a sweep amplifier is necessary. Here
again linearity is important. The spot is moved at a uniform rate
across the screen and quickly returned to repeat the trace. In such
a circuit, the load on the transformer can be regarded as negligible.
Assume a linearly increasing voltage as shown in Fig. 258 to be ap-
plied to the equivalent circuit of Fig. 259.
ei = Kt
Then
Lpci = LK
where p ^ d{ )/dt, and the voltage across the transformer primary is
Lpei
Lp + Ri
LK
Slope of e is
Ri
de/dt = Ke
(1
_ ^—R^tlL\
-Bit/L
(143)
(144)
1 See "Response of Circuits to Steady-State Pulses," by D. L. Waidehch, Proc
I.R.E., 37, 1396 (December, 1949).
342 ELECTRONIC TRANSFORMERS AND CIRCUITS
Thus voltage e has the same slope as the applied voltage times an
exponential term which is determined by the resistance Ri of the
amplifier, the OCL of the transformer, and the time between the be-
ginning and the end of the linear sweep. Under the conditions as-
sumed, the value of the exponential for any interval of time can be
taken from the curve marked R2 = 00 in Fig. 234. For example, sup-
pose that the sweep lasts for 500 microseconds, that the sweep amplifier
tube plate resistance is 800 ohms, and that the transformer inductance
is 10 henrys. The abscissa of Fig. 234 is 0.04, and, since the slope of
this exponential curve equals its ordinate, the slope of the voltage ap-
plied to the plates of the oscilloscope will be, at the end of the saw-
tooth interval, 96 per cent of the slope which it had at the beginning
of the interval.
Assume that at the end of the time interval ( (Fig. 258) the amplifier
tube is cut off. Then the sweep circuit transformer reverts to that of
Fig. 235, in which Cd has the same meaning as before but Ri includes
only the losses of the transformer, which were neglected in the analysis
for linearity of sweep. That is to say, the voltage does not immediately
disappear but follows the curves of Fig. 235 very closely, as in mag-
netic deflection. Backswing voltage may be kept from affecting the
screen by suitable spacing of the applied wave forms or biasing the
cathode-ray tube grid.
Vertical sweep transformers are used in television receivers to dis-
place the horizontal sweep lines at a 60-cycle rate, in order to produce
a picture. The vertical displacement is fairly linear, retrace rapid, and
sawtooth wave form is necessary here also. Because of the relatively
slow vertical displacement, yoke inductance is negligible, so that
vertical sweep amplifiers effectively operate into resistive loads dur-
ing trace periods. The transformers present no ])articularly difficult
problem beyond that of high OCL at low cost.
141. Magnetic Sweep Circuits. A simple television receiver sweep
circuit is shown in Fig. 260. Pulse voltage applied to tlic tetrode
grid appears across the transformer primary winding inverted. Cur-
rent in the lower part of the transformer primary has the shape shown
in Fig. 257. This is the current wave shape in tlie transformer sec-
ondary and deflection coil (termed the yoke). An autotransformer
extension of the transformer primary winding is used to transform the
pulse voltage backswing shown in Fig. 257 to a high value. This
voltage is actually much larger than Fig. 257 indicates, and needs only
3 : 1 step-up to furnish 7 to 14 kv. It is then rectified and applied to
the accelerating anode of the oscilloscope. In this way, a separate
PULSE CIRCUITS
313
high-voltage supply is avoided. A damper diode is used to convert
the backswing current into useful current during the next sweep in-
»-T0 SCOPE 2ND ANODE
YOKE
CURRENT
t
Fig, 260. Television sweep circuit.
terval. Backswing current reaches its negative peak at the end of
retrace period t,- As indicated by the dotted oscillation at the left
of Fig. 261, this current would continue
to oscillate for some time if left un-
damped. With the damper diode cir-
cuit, this current never oscillates but
instead charges the diode R-C net-
work, which slowly discharges into the
yoke. Before damper current reaches
zero, the tetrode starts to conduct.
Because of winding capacitance, the
tetrode tube current is not initially lin-
ear. It is offset by exponential decay
of damper tube current. Yoke current then proceeds in a linear man-
ner, following the dotted line in the transition from damper to tetrode
current, as in Fig. 261.
With the large consumer demand for television receivers, there has
been an incentive for improved efficiency of the basic sweep circuit.
Partly this has been accomplished by ingenious schemes for improved
Fk;
261. Deflection yoke
rent and voltage.
344 ELECTRONIC TRANSFORMERS AND CIRCUITS
linearity with lower transformer Q, and partly by using the plate
input resistance of the tetrode for the damper diode bias resistance.
Thus otherwise wasted power is put to a useful purpose.^
142. Magnetic Pulse Generators. In Chapter 9 it was seen that
thyratron operation can be approximated by self-saturating magnetic
amplifiers. This fact points to a saturable reactor to replace the
hydrogen thyratron in the pulser of Fig. 252. Sevei'al factors militate
against the direct substitution of saturable reactors for thyratrons:
1. Departure of core material from sharp rectangularity interferes
with steep pulse voltage rise.
2. Saturated value of inductance interferes with large current flow
needed during pulse.
3. Reactors are a-c devices; hence a-c charging must be used. This
limits the choice of PRF.
Despite these difficulties, saturable reactors have been used success-
fully in pulsers. Low power pulses may be formed by use of the cir-
cuit of Fig. 202. Reactor Li is
_rvnno_
i —
C2
He-
TO
A-C
SUPPLY
Fig. 262. Magnetic; pulse generator.
linear and re-onates with Ci at
the supply frequency. Reso-
nance therefore tends to main-
tain current i sinusoidal in wave
form. Current i is large enough
to saturate reactor L>, which
has rectangular B-H loop core
material. Twice each cycle cur-
rent i passes through zero, and near these current zeros L2 inductance
becomes large. This large inductance forces most of current i instan-
taneously into C2 and Rl, and builds up a comparatively large jjcak
voltage across 7?^. Such pulses are peaked in wave shape and alternate
in polarity twice each cycle. Pulse durations of less than 0.1 micro-
second have been obtained with this circuit. Owing to the large interval
of time during which i is large, and not producing pulses, the power
output is limited to small values.
In Fig. 262 the voltage across Lo, at a given line frequency / is nearly
proportional to the saturation flux density Bs of the core for rectangular
loop material. If the core area is A^ and turns A" in L2, then this
voltage is, neglecting losses,
iSee "Television Deflection Circuits," by A. W. Friend, RCA Rev., March, 1947,
p. 98; also "Magnetic Deflection Circuits for Cathode-Ray Tubes," Ijy O. H.
Schade, RCA Rev., September, 1947, p. 506.
PULSE CIRCUITS 345
O.SBJAcN
lO^ds
(145)
where 6^/2 is the angle, starting from zero, at which saturation is reached,
as in Fig. 193 (p. 247). If ds is very small, and 2ir/w » RlC2 » djw,
substantially all of eg appears across Rl-
Higher power may be obtained from cascaded stages as in Fig. 263.
C| Cz
H(— I— If-
PFN
TO
SUPPLY
"n
Fiu. 263. Cascaded stages in magnetic pulse generatoi'.
Reactor Li is linear and resonates with Ci at the supply frequency.
Reactors Lo, L-.i, and L^ are saturable; bias windings are used, but not
shown. Suppose that L3 and L4 are initially unsaturated, and L3 is
saturated. Ci charges in series with Li and L3. As the voltage across
L2 reaches maximum, L-2 saturates and discharges Ci. Discharge cur-
rent causes L^ to become unsaturated and L4 saturated; then C2
charges until L.^ saturates again. As the wave proceeds towards Rl
both charge and discharge times become successively shorter. Pulse
duration in each stage is determined by the saturated value of induc-
tance and associated C. In the last stage, the pulse is shaped by PFN
to the desired duration and flatness at the top. As this sequence is
repeated once each cycle, the line current is not sinusoidal, so a line
capacitor is useful for supplying the current harmonics. One modifi-
cation of this circuit is the use of saturating transformers instead of
reactors in order to provide the stepped-up voltage necessary for
magnetron operation. With a magnetron, Rl is replaced by a pulse
transformer primary winding. In another modification saturable re-
actors are the series elements and capacitors the shunt. ^
1 See "The Use of Saturable Reactors as Dis{^harge Devices for Pulse Gener-
ators," by W. S. Melville, J.I.E.E. (London), 08, Part III, p. 185.
BIBLIOGRAPHY
TRANSFORMER THEORY
1. E. G. Reed, Essentials of Transformer Practice, McGraw-Hill Book Co.,
2nd ed., New York, 1927.
2. L. F. Blume, Editor, Transformer Engineering, John Wiley & Sons, 2nd ed.,
New York, 1951.
3. M.I.T. Electrical Engineering Staff, Magnetic Circuits and Transformers,
John Wiley & Sons, New York, 1943 (contains extensive bibliography).
CORE MATERIALS
4. Magnetic Core Materials Practice, Allegheny Steel Co., Brackenridge, Pa.,
1937.
5. J. K. Hodnette and C. C. Horstman, "Hipersil, a New Magnetic Steel and Its
Use in Transformers," Westinghouse Engineer, 1, 52 (August, 1941).
6. A. G. Ganz, "Applications of Thin Permalloy Tape in Wide-Band Telephone
and Pulse Transformers," Trans. AIEE, April, 1946, p. 177.
7. ASTM Standards on Magnetic Materials A34-49 to A345-49 inclusive, Ameri-
can Society for Testing Materials, Philadelphia, Pa.
8. R. M. Bozorth, Ferromagnetism, D. Van Nostrand Co., New York, 1951.
9. C. C. Horstman, "Progress in Core Material for Small Transformers," West-
inghouse Engineer, 12, 160 (September, 1952).
10. S. R. Hoh, "Evaluation of High Performance Magnetic Core Materials," Tele-
Tech, 12, 86 (October, 1953).
11. James R. Wait, "Complex Magnetic Permeability of Spherical Particles,"
Proc. I.R.E., 41, 1664 (November, 1953).
RECTIFIERS
12. D. C. Prince and P. B. Vogdes, Mercury Arc Rectifiers, McGraw-Hill Book
Co., New York, 1927.
13. A. J. Maslin, "Three-Phasc Rectifier Circuits," Electronics, 9, 28 (December,
1936).
14. D. L. Waidelich and H. A. Taskin, "Analyses of Voltage-Tripling and -Quad-
rupling Circuits," Proc. I.R.E., 33, 449 (July, 1945).
15. R. S. Mautner and O. H. Schade, "Television High-Voltage R-F Supplies,"
RCA Rev., 8, 43 (March, 1947).
16. E. V. Bheux, "High- Voltage Rectifier Circuits," General Electric Rev., 61, 22
(February, 1948).
347
348 ELECTRONIC TRANSFORMERS AND CIRCUITS
17. T. Spooner, "Effect, of a Superposed Alternating Fie'd on Apparent Magnetic
Permeability and Hysteresis Loss," Pliys. Rev., S6 (2nd series), 527 (January-
June, 1925).
18. L. B. Arguimbau, P. K. McElroy, and R. F. Field, Iron-Cored Coils jar Use
at Audio Frequencies, General Radio Co., Cambridge, Mass.
19. V. E. Legg, "Optimum Air Gap for Various Magnetic Materials in Cores
of Coils Subject to Superposed Direct Current," Trans. AlEE, 64, 709 (1945).
20. George Katz, "Effect of Temperature on Iron Powder Cores," Elec. Mfg., 53,
135 (Februarj', 1954).
AMPLIFIERS
21. H. S. Black, "Stabilized Feedback Amplifiers," Bell !<i/stem Tech. J., 13, 1
(January, 1934).
22. R. Riidenberg, "Electric Oscillations and Surges in Subdivided Windings,"
,/. Awl- Phys., 11, 665 (October, 1940).
23. F. E. Terman, Radio Engineers' Handbook, McGraw-Hill Book Co., New
York, 1943, Sections 2 and 3.
24. K. Schlosinger, "Cathode Follower Circuits," Proc. I.R.K., S3, 843 (December,
1945).
25. H. W. Bode, Network Analysis and Feedback Amplifier Design, D. Van
Nostrand Co., New York, 1945, Chapters XVI to XIX.
26. D. T. N. Williamson, "Design for a High-Quality Amplifier," Wirelcis World,
April, 1947, p. 118.
27. H. W. Lamson, "Advantages of Toroidal Transformers in Communication
Engineering," Tele-Tech, 9 (May, 1950).
28. T. Halabi, "Audio Transformer Design Charts," Electronics, October, 1953,
p. 193.
NON-SINUSOID-AL WAVES
29. J. CJ. Braincrd, G. Koehler, H. J. Reich, and L. F. Woodruff, lUtra-High Fre-
quency Techniques, D. Van Nostrand Co., New York, 1942, pp. 36-47.
30. H. E. Kallman, R. E. Spencer, and C. P. Singer, "Transient Response," Proc.
I.R.E., 33, 169 (March, 1945).
31. C. E. Torsch, "A Universal-Application Cathode-Ray Sweep Transformer
with Ceramic Iron Core," Proc. Natl. Electronics Conf.. .'->, 130 (1949).
32. L. W. Husscy, "Non-Linear Coil Generators of Short Pulses," Proc. I.R.E., 38,
40 (January, 1950).
33. H. W. Lord, "A Turns Index for Pulse Transformer Design," Trans. AlEE, 71,
Part 1, pp. 165-168 (1952).
34. M. Chodorow, E. L. Ginzton, I. R. Nielsen, and S. Sonkin, "Design and Per-
formance of a High-Power Pulsed Klystron," Proc. I.R.E.. Jfl, 1595 (November,
1953).
35. M. B. Knight, "Practical Analysis of Vertical Deflection Circuits," Tele-Tech,
12, 62 (July, 1953).
MAGNETIC AMPLIFIERS
36. A. V. Lamm, "Some Fundamentals of a Theory of the Transductor or Mag-
netic Amplifier," Trans. AlEE, 66, 1078 (1947).
BIBLIOGRAPHY 349
37. E. L. Harder and W. F. Horton, "Response Time of Magnetic Amplifiers,"
Trans. AIEE, 69, 1130 (1950).
38. A. I. Pressman and J. P. Blewett, "A 300 to 4000 Kilocycle Electrically Tuned
Oscillator," Proc. I.R.E., 39, 74 (Januai-y, 1951).
39. James G. Miles, "Bibliography of Magnetic Devices and the Saturable Reactor
Art," Trans. AIEE, 70, 2104 (1951) (containing a list of 901 references).
40. H. L. Durand, L. A. Finzi, and G. F. Pittman, Jr., "The Effective Ratio of
Magnetic Amplifiers," Trans. AIEE, 71, 157 (April, 1952).
41. H. W. Lord, "Dynamic Hysteresis Loops of Several Core Materials Employed
in Magnetic Amplifiers," Trans. AIEE, 72, 85 (1953).
42. S. B. Batdorf and W. N. Johnson, "An Instability of Self-Saturating Magnetic
Amplifiers Using Rectangular Loop Core Materials," Trans. AIEE, 72, 223
(1953).
43. W. A. Geyger, Magnetic Amplifier Circuits, McGraw-Hill Book Co., New
York, 1954.
44. R. W. Roberts, "Magnetic Characteristics Pertinent to the Operation of Cores
in Self-Saturating Magnetic Amplifiers," Trans. AIEE, 73, 682 (1954).
45. H. F. Storm, Magnetic Amplifiers, John Wiley & Sons, New York, 1955 (con-
tains extensive bibliography).
TB.'VNSFORMEB STANDARDS
46. ASA, American Standards for Transformers, Regulators, and Reactors C57.22-
1948.
47. NEMA, Standards for Transformers, No. 48-132.
48. AIEE Standards: No. 1. General Principles Upon Which Temperature
Limits Are Based. No. 551. Master Test Code for Temperature Measure-
ment. "Progress Re])ort of AIEE Magnetic Amplifier Subcommittee," Trans.
AIEE, 70, 445 (1951).
49. RETMA Standards: Power Transformers for Radio Transmitters TR-102-B.
Power Filter Reactors for Radio Transmitters TR-llO-B. Audio Transform-
ers foi' Radio Transmitters TR-121. Audio reactors TR-122. Iron-Core
Charging Reactors TR-127. Pulse Transformers for Radar Equipment TR-
129.
50. D. S. Stephens, "Lightweight Aircraft Transformers," Trans. AIEE, 68, 1073
(1949).
51. P. G. Sulzer, "Stable Electronic Voltage Regidator," Electronics, 23, 162
(December, 1950).
52. W. D. Cockrell, Industrial Electronic Control, McGraw-Hill Book Co., New
York, 2nd ed., 1950.
53. R. E. Collin, "Theory and Design of Wide-Band Multisection Quarter-Wave
Transformers," Proc. I.R.E.., 43, 179 (February, 1955).
INDEX
Air gap, see Core gap
Air-core transformer, 224
resonant peaks, 227
tuned, 226
Aircraft power supplies, 30, 80
Alternate stacking, 98
A-c grid voltage, thyratron, 239
A-c resistance, 106, 220
Ambient temperature, 53, 107, 290
Ampere-turns, 13, 208, 248
per inch, 90, 262, 269
Amplification factor (m), 141, 144, 182
variations in, 256
Amplifiers, 140
bistable, 270
classes, 142, 163, 182, 200, 212, 215,
236
constant output, 257
efficiency, 143
equivalent circuit, 141
frequency response, 147, 175, 179, 194,
214,' 222, 305
load line, 157, 264
magnetic, 259
phase angle, 155, 179, 194
plate resistance, 141, 160, 168, 182, 298
potentials, 140
power output, 143, 264
pulse, 3, 294
push-pull, 143, 163, 175, 193, 200, 209,
282
reactive load, 155, 194
sawtooth, 339
self-saturated, 273
stability, 179, 209, 270, 290
tests, 208
transformer-coupled, 141, 170, 176,
214, 294
tuned, 142, 216, 226, 235
turns ratio, 141, 147, 152, 170, 176,
258, 266, 268, 282, 295, 302
voltage gain, 145, 176, 178, 202
voltage ratio, 149, 152, 211, 285
voltages, 141, 157, 194
Amplitude-modulated wave, 255
Angular frequency, 6, 67, 114, 118, 276,
302
Anode, see also entries beginning with
Plate
Anode characteristics, 155, 169
Anode current cut-off, 140, 256
in class B and C amplifiers, 142
Anode transformer, center tap, 75
currents, 74
design, 77, 82
800-cycle, 82
leakage inductance, 74
secondary voltage, 75
Anode windings, rectifier, 63
Apparent permeability, 308
Artificial fine, 186, 196
Attenuation, 117, 145, 227
wave filter, 182
Audio transformers, see Amplifiers
Automatic gain control, 256
Automatic volume control (AVC), 258
Autotransformer, 250
pulse, 312
variable, 251
Average current, 15
rectifier, 64
Backswing voltage, 299, 320, 330, 333,
340, 342
Balance in 3-phase transformers, 118, 126
Band width, 186, 224, 226
Bank winding, 214
Beam tube, 168, 199
Bias, magnetic amplifier, 275
Bleeder load, 125, 134
Blocking oscillator, 329
Breakdown voltage, 43
Bridge-type rectifiers, 62, 75, 113, 278
Butt stacking, 99
Bypass capacitance, 197, 212, 219, 243
Calculation form, anode transformers, 80
filament transformers, 71
Calorimeter, 3)3
Capacitance, air-core transformer, 226
351
352
INDEX
Capacitance, amplifier transformer, 147,
150, 164, 166, 196, 211, 216
calculation, 170, 176, 211, 219, 311
distributed, 104, 179, 232, 312, 324
effective, 171, 220, 302, 321
filament transformer, 67
filter, 64, 117, 125, 134
layer-to-layer, 171, 176, 215
measurement, 173, 324
pulse transformer, 294, 298, 311, 320
reactor, 103, 233
rectifier anode transformer, 118
vacuum tube, 174, 216
Capacitive current, 166, 244, 252, 308,
313
Capacitive reactance (Xc), 116, 123
transformer, 150, 159
(Capacitor charging current, 63, 112
Capacitor effect, 123
(yapacitor-input filters, 63, 126
Carrier frequency, 214, 255
Cathode follower, 181, 199, 213
Onter tap, anode transformer, 75
Characteristic (curves, amplifier anode,
155, 169
i-ectifier, 61
Charact(!ristic impedance, 145, 184
("hoke, see Reactor
Class (of amplifier), 142, 163
Class A grid load, 147, 150, 174
Clearance, coil, 38, 71, 77, 311
Clipping, 292
("oaxial coils, 215, 228
Coeffi(dent of coupling, 224, 234, 330
0)ercive force, 23, 310
Coil l)ulging, 38
Coil clearance, 38, 73
Coil form, 38, 73, 79
C!oil insulation, 35
Coil interleaving, 75, 164, 171, 212, 219,
321
Coil orientation, 203
Coil resonant frequency, 151, 174, 232, 314
Coil section, anode transformer, 75
balanced windings, 164
filament transformer, 72
low layer voltage, 212
pulse transformei-, 321
symmetry, 219, 222
Coil taping, 46
Coils, treatment of, 49
Combined anode and filament trans-
former, 79
Commutation reactance, 120, 128
Concentric windings, 38, 72, 75, 79, 84,
164, 171, 212, 219, 321
Conpernik, 34
Control circuit, thvratron, 241
Copper loss, 55, 73, 78, 84, 109, 213, 290,
322
Copper weight, calculation, 73
Core area, reactor, 91, 289
transformer, 10, 70, 75, 171, 175, 211,
317
Core dimensions, 72, 83, 85, 91, 175, 289,
320
Core gap, 19, 88, 175, 189, 211, 285, 309
Core gap loss, 190, 200, 213
Core loss, 55, 73, 81, 313
curve, 27, 29, 32, 81, 188, 216, 218
400 and 800 cvclcs, 81
high frequency, 216, 218
oscillator, 202, 212
(^ore-loss current, 9
Core saturation, 92, 198, 247, 259, 309
Core tongue, 17, 72, 164
Core-type transformer, 18, 164, 204, 311,
321
Core window, 17
Core window height, 18, 101, 248
Cores, type C, assemblies, 20
description, 1 8
Corona, .audible. It
effects, 50
measurement, 11
tests, 44, 1 10
voltage, 73
Coupling, critical, 226
Coupling capacitor, 192, 200, 209
Coupling coefficient (k), 224, 234, 330
Couj)ling, inductive, 224, 234
Creejiage curves, oil, 52
Oeepage distance, air, 46, 48
Cross-connected coils, 249
Current, magnetizing, 9, 26, 130, 161,
199, 253, 299, 336
Current density, curve for, 35, 86
Current distribution, coil, 151, 234, 312
Current feedback, 181, 209
t!urrent inrush, 1 30
Current interruption, 103, 132, 196, 252
Current-limiting transformer, 248
Current transductors, 272
Current wave form, 10, 16, 63, 120, 244,
261, 315, 339, 343
Cut-off, amplifier, 140, 269
frequency, 182, 193
Decibels, 144
attenuation, 183
frequency response, 145
hum reduction, 205
INDEX
353
Deflection coil, 340, 342
Degeneration, 330
Demodulation, 254
Design, amplifier transformer, 174
anode transformer, 77
audio oscillator, 212
audio transformer, 175, 211
carrier-frequency transformer, 221
cathode follower, 213
800-cycle transi'ormer, 82
filament transformer, 71
magnetic amplifier, 265, 285, 288
modulation transformer, 211
pulse transformer, 317
reactor, 97
Design charts, 3, 84, 99, 136
Dielectric constant, 45, 172
Dielectric loss, 314
Dielectric strength, 43
Dimensions, aircraft apparatus, 81
coil, 38, 72
core, 72, 83, 91, 175, 289, 320
lamination, 55
Diode, 254, 331, 343
D-c flux, ti'ansformer core, 105
Distortion, see Harmonic distoi'tion
Driver transformer, 199
Droop, pulse top, 297, 317
Eddy current, 23
Eddy-current loss, 9, 220
Efficiency, 14, 143
pulse transformer, 313
vs. rating, 54
Enamel wire, 2, 36
Enclosure, degree of, 22
End cases, 22, 54
Equivalent air gap, alternate stac'king, 99
E(juivalent circuit, amplifier, 141, 147,
170, 197
high magnetizing current, 11
plate modulation, 193, 197
pulse amplifier, 294
resistive load, 7
Ecjuivalent impedance, 9, 192, 225
load, 8
Equivalent resistance, core-loss, 7, 9, 104,
147, 334
secondary load, 225
shunt and series, 189
Equivalent sphere, radius of, 57
Excitation volt-amperes, 28, 32
Exciter chokes, 200
lOxciting current, 6, 9, 260, 269
Extension, insulation, 52
lead wire, 42
False echoes, 333
Feedback, inverse, 178, 209
magnetic amplifier, 268, 270
Ferrite, 33, 34, 217
Filament transformers, current limiting,
69, 248
design, 70
insulation, 67
low capacitance, 67
multiwinding, 70, 88
regulation, 72
Filament voltage, 73
Filter, a-c line, 129
attenuation. 111, 183
band-pass, 186
capacitor. 111, 182
capaidtor effect, 123
capacitor-input, 63, 118, 243
characteristic impedance, 184
constant-K, 185
cut-off frequency, 182
design charts, 136
half-sections, 184
high-pass, 192
image impedance, 184
inductor design, 78
insertion loss, 185
key click, 131
keying, 96, 130
limitations, 184
load impedance, 184
low-pass, 182, 196
multistage, 114
oscillations, 128, 130
phase shift, 185
pi-section, 183, 192, 196
reactor-input, 62, 112, 240
rectifier. 111
rectifier, tuned, 127
series- tuned, 127
shunt-tuned, 127
source impedance, 185
T-section, 183
termination, 184
voltage drops, 118, 124
wave, 182
Filter charts, 133
Filter current, 125, 134
Fine wire corrosion, 22
Firing angle, 238, 274
Flat-top pulse, 294, 303, 325, 333
F\\ix, time variation of, 5, 247, 276
Flux density, see Induction
Flux fringing, 97, 101, 190, 249
Flux linkage, 1, 4, 76, 97
354
INDEX
Flux path, current-limiting transformer,
248
divided, 17
peaking transformer, 246
pulse transformer, 310
reactor, 88, 262, 276
Flux penetration, 309
Fourier integral, 293
Fourier series, 114, 162
Frequency, power supply, 67, 80, 84
range, 3, 180, 222
r-t choke, 232
response, air-core transformer, 227
high-, 150, 196, 214, 306
low-, 146, 194, 305
zones, gage for various, 30
Full-wave rectifier, 63, 113, 240
Functional evaluation, 51
Fundamental transformer equation, 5
Gap, core, 88, 101, 188, 211, 285, 309
Gap loss, 191, 200, 213
Gauss, 34
Glass-covered wire, 84
Graded insulation, 75
Grain-oriented steel, core loss, 29, 77, 81,
188, 216, 309
in audio transformer, 198
in saturable reactors, 33, 263
magnetization curves, 263, 286
maximum induction, 81
permeability, 29, 30, 34
saturation curve, 29, 263
thickness, 29, 81
thin-gage, 216, 286, 309, 320, 338
Grid bias voltage, 141, 238, 257, 330
Grid-controlled rectifier, 240
Grid current, 143, 199, 329
Grid excitation, 176
Grid saturation, blocking oscillator, 330
class C oscillator, 202, 331
definition, 140
pulse amplifier, 292, 327
Grid voltage swing, 182, 212, 329
amplifier, 142
Half-wave rec'tifier, 63, 256, 282
Harmonic distortion, 153, 168, 178, 212
in non-linear loads, 258
measurement, 208
Harmonics, a-c line current, 128, 345
bridge, 106
magnetizing current, 161
pentode amplifier, 168
ripple, 114
Heat dissipation, 54
Heat dissipation, area, 54
coil, 234
Heat run, 107
Henry, 91, 171, 211, 296
Hermetical sealing, 22
High-fidelity modulator, 198
High-frequency current, power line, 205
High-frequency response, 150, 196, 214,
306
High voltage, 3, 46, 67, 133, 243, 313
pulse transformer, 311, 319
Hipernik properties, 34
Hipcrsil, core, 34, 188, 285
eddy-current loss, 29
frequency zones, 30
hysteresis loss, 29
properties, 34
Hum measurement, 139
Hum reduction, 179, 204
Hum voltage, see Ripple
Hybrid coil, 206
Hysteresis loop, 23, 24, 275, 277, 286
pulse, 308, 310, 324
reactor, 89, 94
Hysteresis loss, 9, 23
Ideal transformei', 12
Ignitron, 240
Impedance, characteristic, 145, 184
complex, 224
e(}uivalent, 9, ] 02, 225
high, 2, 161, 215, 223
image, 184
level, 2, 145, 317
load, 2, 145, 155, 161, 310
low, 2, 217, 223
matching, 144
mid-series, 184
mid-shunt, 184
non-linear, 314
r-f choke, 232
ratio, 141, 166, 298, 319, 337
source, 2, 145, 294
Impregnation, coil, 49, 77, 84
Impulse ratio, 110, 133
Incremental pernufability, 25, 89
Induced voltage, 5, 75
test, 109
Inductance, bridge, 95, 107
critical, 126, 241
decrease of, 216, 260
formulas, 76, 97, 171
increase of, 94, 274
leakage, 3, 74, 76, 121, 164, 196, 222,
224, 290, 311, 330, 335
non-linear, 89, 242, 260, 274, 344
INDEX
356
Inductance, open-circuit (OCL), 11, 149,
174, 192, 224, 297, 300, 318, 322,
333, 342
r-f choke, 232
reactor, 61, 209, 338, 344
saturated, 274, 344
Induction (B), curve, 23, 24, 29
filter choke, 97, 188
400- and 800-cycle, 80
high-frequency, 216
Hipersil, 29, 34
maximum, 24, 89, 275, 310
modulation transformer, 198
pulse, 308
reactor, 88, 97, 188, 263, 388
residual, 23, 130, 310
shunt, 247
silicon steel, 29, 34, 97
Inductive reactance (Xl), 117, 241
Inductor, see Reactor
Initial conditions, 295, 312
Initial current, 315
Initial voltage, 295
distribution, 245
Input transformer, 175, 202
Instability, grid-controlled rectifier, 240
Insulating channel, 46
Insulation, class, 41, 50, 81
extension (margin), 48, 52
graded, 75
high-voltage, 46, 52, 246
layer, 37, 38, 77
leads, 40
life, 43
margin, 48
materials, 41
oil, 51, 68, 133
operating temperature, 43
pulse transformer, 310
reactor, 96, 103, 133, 200, 287, 338
test, 109, 326
thickness, 45, 77, 101, 172, 321
Interleaving coil, 75, 310, 321
I-f transformer, 231
Interstage transformer, 176, 196
Inverse feedback, 178, 209
Key-click filter, 131
Keyed wave, 131
Keying filter, 96, 131
Lamination, 175, 211
shape, 17
size, 55
space factor, 31, 83
stack, 55
Lamination, thickness, 17, 31, 83
Large air gaps, 97
Law of cooling, 57
Layer insulation, 39, 77
Layer voltage, 75, 245
L-C filter, see Filter, reactor-input
Lead, anchoring, 40
location, 40, 311
Leakage flux, 6, 76, 164
Leakage inductance, see Inductance
Leakage reactance, 11, 120, 147
drop, IX, 6, 108, 120
Line-matching transformer, 147, 175, 221
Line- voltage adjustment, 250
Linear sweep, 340
Litzendraht cable, 231
Load current, 6
rise, 315
Load line, amplifier, 157, 264
Load resistance, 134, 157, 197, 264
Load voltage, 7
Losses, 14, 55, 60, 81, 84; see also Core
loss and Copper loss
Low-frequency response, 146, 194, 305
Magnetic amplifier, bias, 275
bistable, 270
circuits, 278
current, 261
design, 285
feedback, 268
graphical performance, 262
half-wave control, 282
limitations, 290
response time, 267, 282
self-saturated, 273, 278
simple, 259
transfer curves, 266, 269, 271, 280
voltage, 261
Magnetic field, 202
Magnetic path, see Flux path
Magnetic shunts, 247
Magnetic terms, 23
Magnetization curves, 262
Magnetizing current, 9, 26, 130, 161, 199,
253, 299, 336
large, 11
non-linear, 199, 246
pulse transformer, 300, 303, 313, 336
Magnetizing force (H), 10, 23, 89, 264
Magnetron, 315, 326, 332
Margin, coil, 46, 72, 80, 101, 222, 311
Maximum power transfer, 226, 235, 331
Mean length of turn, 38, 71, 77, 166
Median frequency, 148
Mica, 321
356
INDEX
Microhenry, 228, 320
Microsecond, 293, 302, 319
Mid-series impedan<;e, 184
Mid-shunt impedance, 184
Modulation, 192, 329
filter, 96, 130
reactor, 192
transformer, 192, 211
Modulator, 192, 332
Moisture, sealing against, 22
Mountings, 22
Multilayer winding, 173, 220, 245
Multiple-coil winding, 35
Multiple-tuned circuits, 227
Multiple-wound coil, 77
Multiwinding transformers, 53, 72
Mumetal, 34
Mutual conductance ((/„), 144, 182
variable, 257
Mutual inductance, 224, 228
Mutual reactance, 225
NEMA radio influence test, 44
Natural frequency, see Resonance
Nicaloi, 34
Nickel-iron alloy, 30, 33, 34, 175, 263,
286, 309
core loss, 188
maximum induction, 34, 97
No-load loss, 320
Non-linear load, 199, 315, 331
Non-symmetrical windings, 222, 311
Normal magnetization curve, 24, 286, 308
Normal permeability, 24, 308
Normalized transfer curve, 280
Oersted, 29, 34, 275
Oil insulation, 51, 68, 133
Open-circuit impedance, 302
Open-circuit inductance (OCL), see In-
ductance
Open-circuit reactance, 11, 147, 259
Open-circuit secondary, 147, 208, 249,
298
Open-delta connection, 84, 252
Operating temperature, 42
Oscillation, 179, 312
conditions for, 302
parasitic, 209
rectifier, 131, 240
superposed, 304, 314, 330, 333
Oscillator, transformer-coupled, 200, 212,
236
blocking, 329
Oscillatory circuit, pulse, 296, 300, 315,
325, 334
Oscillogram, kevcd wave, 132
pulse, 304
Oscilloscope, 324, 327
Output power, 14, 139, 265
Overshoot voltage, 296, 303, 325
Pancake coils, 249
Parasitic oscillations, 209
Part coils, 79, 245
Partial resonance, 174, 307
Peak currcuit, 244
rectifier, 66, 82, 96, 127
Peak voltage, 24 1
Peaking transformer, 246
Pentode, 141, 167
amplifier translormer, 168
characteristics, 169
distortion in, 170, 199
pulse amplifiei', 305
Permalloy, 34
Permeability (m), 24, 88
a-c, 26
d-c, 88
high-frequency, 217
incremental, 25, 89
pulse, 309, 323
various steels, 34
Phase angle, load, 194
transformer, 153, 255
Phase control, njctifier, 240
thj'ratron, 238
Phase-shift, amplifier, 180, 195
artificial line, 186
filter, 194
inverse feedback, 180
linear, 186
wave shape, 305, 331
Phase voltage, 121
Phases, effect on ripple, 114
power supply, ()2
Pi-filter, 183
Pic-section coils, 215
modulator, 194
Plate, see also entries beginning with Anode
Plate current, am|)lifier, 143, 166, 199
balance, 164
increase of, 166
Plate resistance, amplifier, 141, 160, 168,
182, 298
blocking oscillator, 329
cathode followci', 181
pentode, 169
sweep amplifiei', 342
Plate transformer, center tap, 77, 113
thyratron, 237, 243
voltage drop, 78, 120
INDEX
357
Plate voltage swing, 143, 160
Plate voltage wave form, 143
Polarity, voltage, 254
Polarity, winding, 13, 106, 260, 270, 327
Polyphase rectifiers, 135
Polyphase voltage unbalance, 84, 135,
252
Powdered iron rxne, 33, 191, 219
Power factor, 14, 194
Power-line carrier, amplifier, 214
receiver, 258
Power line sm'ges, 110, 132
Power supply frequency, 80, 118
Primary-primary coupling, 164
Pulse, blocking oscillator, 329
current, 306, 315
flat-top, 294, 297
front-edge, 295, 314, 318, 325
trailing-edge, 298, 330
Pulse duration, see Pulse width
Pulse forming network (PFN), 331, 345
Pulse generator, magnetic, 344
Pulse inversion, 327
Pulse shape, see Pulse
Pulse voltage, 317, 344
Pulse width, 294, 317, 344
Pulser, line-type, 331, 338
Push-pull amplifiers, 144, 163, 283
Q, 3, 106, 183, 188, 209, 338
air-core transformer, 227
filter reactor, 188
i-f coils, 231
R-f (ihoke, 233
R-f generator, 233
R-f power supply, 235
Radio influence, 44, 205; see al.to Corona
Random winding, 35
Ratings, continuous, 53
intermittent, 56
Ratios, air gap/core length, 91
amplifier voltage, 144, 145, 149
current, 6
extension/thickness, of insulation, 52
filter reactances, 1 16
impedance, 102, 134, 300, 319, 337
reactance/resistance, B = Xc/Iii, 150
reactance/resistance, D = Xc/R'i, 159
reactance/resistance, Xhj/Ri, 148, 255
reactance/resistance, Xm/R^, 158
ripple amplitude, 117, 241
source/load resistance, 147, 175, 295
turns, 1, 5, 7, 105, 141, 147, 152, 170,
176, 248, 258, 266, 268, 282, 295,
302, 311, 331, 342
Ratios, voltage, 5, 6, 145, 149, 317, 319
Reactance (Xl), 115, 147
choke, critical, 126, 241
R-C filter, 118
Reactive voltage drop (IX), 3, 6, 7, 120
Reactor, aii' gap, 88, 101
a-(^ flux density, 89, 262
a-c volts, 88, 265, 338
capacitance, 103, 200
charging, 338
core length, 88
core size, 90
design, 91, 97, 188, 265, 288
dimensions, 99
direct current, 88, 261
d-c flux densitj', 88
energy, 90
flux swing, 94
frequency range, 190
hysteresis loop, 89, 94, 275
impedance, 233
in high-voltage lead, 118
incremental permeability, 89
inductance, 90, 137, 182', 188, 230, 274,
338
IR drop, 77
input-filter, 61, 112, 241
insulation, 96, 103, 133, 200, 287, 338
linear, 97, 99, 338, 344
losses, 182, 188, 233
magnetizing force (H), 89, 275
maximum flux density, 94, 99, 262
modulation, 192, 209
Q, 3, 183, 188, 209, 338
r-f, 233
saturable, 242, 259, 344
saturation, 88, 96, 260, 274, 344
single-layer, 233
surges in, 131
swinging, 96, 135
tuned, 127, 135
turns, 88, 99, 265, 289
winding resistance, 90, 100, 259, 289
Receiver rectifier, 79, 254
Rectangular jjulse, 294
Rectifier, capacitor-input filters, 63, 82
characteristic curve, 61
circuits, 62
current, 63, 127
current wave, 125, 128
efficiency, 138
inverse; peak voltage, 62, 287
load current, 124
load resistance, 126
losses, 138
output voltage, 77, 111
358
INDEX
Rectifier, peak current, 127, 241
peak plate current, 66
polyphase, 113
connections, 62
phase voltage balance, 84
power supply current, 128
re;j;ulation, 78, 120, 124
resistance, 61, 281, 287
ripple voltage, 114, 119
amplitude, 115
attenuation. 111
series resistance, 120, 126, 128
single-phase full-wave, 63, 77, 82, 113,
240, 255
single-phase half-wave, 63, 79, 111, 239
voltage, 62, 77
drop, 60, 120
full- wave, 113
halt-wave. 111
polyphase, 113
Reflections, wave, 184
Regeneration, 179
Regulation, 3, 13, 78, 85
autotransformer, 250
rectifier, 77, 118
voltage, 252
Reinforced end-turns, 313
Reliability, importantie of, 2
Remote cut-off, 256
Repetition rate (PRF), 293, 314, 317,
338
Reset core, 274, 310
Residual induction, 23, 130, 310
Resistance, bridge, 106
load, 199, 264
Resistance vs. temperature, copper, 108
Resonance, choke, 200
r-f, 233
frequency, air-core transformer, 226
frequency, filter, 134
pulser, 338, 344
transformer, 150, 159, 165, 174, 307
partial, 174, 314
Resonant circuit, 252, 331, 344
Resonant peaks, 233, 236, 338
RKTMA, 110
Ripple, HI
current, 64, 124
voltage, 61, 115, 139, 241
Rms current, 15, 83
pulse, 321
rectifier, 62
Saturable reactor, 243, 344
Saturation, reactor, 88, 96, 260, 274, 344
Sawtooth transformer, 340
Scott connection, 252
Self-inductance, air-core coils, 224, 230
powdered-iron (;ore, 231
single-layer coil, 230
Shell-type lamination, 17, 55
Shell-type transformer, 17, 72, 164, 175,
205, 211, 219, 311
Shield, magnetic, 202
static, 204, 209, 220, 245
wire, 206
Short-circuit current, limiting, 248
rectifier, 120
test, 108
Short-circuit reactance, 11, 108
Short-circuit turns, 250
Silicon steel, 27, 176, 211
core-loss, 27, 81, 188
maximum induction, 97
permeability, 30, 98
properties, 27, 34
saturation cuive, 29, 262
Silicone, 43
Similitude, 102
Single-layer winding, 172, 219, 310
Single-phase rectifier, 63, 77, 111, 239,
255
Sinusoidal voltage, 6, 224, 276
Size, 3, 53, 149, 192, 224
autotransformer, 250
core, 85, 88, 198
400- and 800-cycle transformer, 80
Slope, front-edge, 316
pulse top, 297
sawtooth wave, 339
trailing-edge, 298, 329, 334
Solventless varnish, 50, 85, 311
Source impedance, 145, 294
Source resistance, 295, 313
Space (volume), 2, 22
Space factor, 31, 35
Spark gap, 133
Square wave, blocking oscillator, 330
front, 294
test, 322
voltage, 294, 303, 325
Stacking dimension, 55, 72, 21 1
Standard test voltage, 109
Static shield, 204, 220, 245
Step-down transformer, 1, 152, 304
Step-up transformer, 1, 152, 221, 295
Surge voltage, 196, 243
pulse, 312
reactor, 103, 131, 209
Sweep circuit, 342
Sweep generator, 339
Symmetry, winding, 222, 311
INDEX
359
Table, amplifier classes, 143
phase angle, 156
core steel properties, 34
current wave forms, 16
distortion, 160
800-cycle cores and insulation, 81
equivalent core gaps, 99
frequency response and wave shape,
307
harmonic magnetizing currents, 163
Hipersil core data, 31
paper-insulated coil data, 39
pulse winding capacitance, 31 1
radio influence voltage, 110
rectifier circuit data, 62
transfer curve, 281
transformer sizes, 85
wire sizes, 36
wire turns per square inch, 37
Tap-changing, 250
Taping, 46, 49
Telephone interference, 128, 205
Temperature, operating, 42
Temperature gradient, 55
Temperature rise, 22, 55, 59, 84
Terminal voltage, 9
Termination, filter, 184, 193
Test, amplifier, 208
corona, 44, 110, 327
d-c resistance, 105, 322
insulation, 43, 109, 326
losses, 109, 313
OCL, 106, 322
polarity, 106, 327
pulse, 322
rectifier, 139
regulation, 107
temperature rise, 107
turns-ratio, 105, 322
Tetrode, 141
Thermal time constant, 56
Three-phase rectifier, 62
Three-phase transformer, 84
Thyratron, amplitude control, 239
critical grid voltage, 237
firing, 243, 332, 338
Thyratron transformer, 243
Time constant, front edge, 296, 303
thermal, 56
trailing edge, 298
Time delay, line, 186
Toroid, 192, 222, 285
Trailing-edge pulse, 298, 303, 320, 330,
334
Transient, amplifier, 164
backswing, 298
Transient, circuit response, 293
current, 96, 132
cyclic, 130
keying, 131
rectifier, 129, 133, 240
starting, 130, 134
voltage, 96, 131
Transistor, 170
Transmission band, 183
Transmission line, 133, 145, 294
Triode, 140
amplifier, 155
characteristics, 157
demodulation, 256
voltage gradients, 140
T-section filter, 182
Tuned amplifiei', 142, 216
Tuning capacitors, 231
Turns per layer, 18, 77, 173
Turns-ratio bridge, 107
Two-dielectric effect, 45
Two-jjliase transformer, 84
Type C core, 18, 29, 75, 85, 99, 192,
321
Unbalanced direct current, 63, 166, 174
Undistorted power output (UPO), 147,
160
Unloaded transformer, 133, 147, 315,
325
Variable-mu, 256
Varnish, insulating, 50
VARS/lb, 26
Vector diagram, artificial line, 187
high magnetizing current, 11
rectifier phase control, 243, 254
resistive load, 7
Video frequencies, 223, 292
Voltage, high, 3, 46, 67, 133, 243, 313
operating, 46, 84
regulator, 252
rise, blocking oscillator, 330
pulse, 296, 303
reactor, 103, 131, 209
rectifier, 123, 287
stress, 311
Voltage change, rate of, 5, 276, 297
Voltage doublers, 67
Voltage droop, pulse, 299, 317
Voltage gradient, 140, 312
initial, 244, 312
Volt-ampere ratings, 53, 82, 85, 250
Volt-amperes, 14, 63, 80, 83, 234, 250
Volts per layer, 75, 245
Volts per turn, 5, 244, 264, 312
360
IXDKX
Wave filter, see Filter
Wave form, current, 14, 198, 244, 261,
306, 315, 339
distortion, 105
voltage, 243, 255, 261, 273, 279, 303,
315, 339
Wave reflections, 184
Wave shape and frequency response, 2,
305
Wide-band transformers, 222
Winding, balance, 164, 209
bulge, 72
capacitance, 147, 171, 219, 245
lieight, 38, 72, 77
interleaving, 75, 319
IR drop, 3, 6, 7, 73, 78, 120, 290
polarity, 13, 106, 260, 270, 327
Winding, primary, 4
reactance, 6, 189, 251, 259
resistance, 6, 7, 73, 146, 149, 189, 220,
251, 259, 281, 289, 322
air-core tiansformer, 225
iiigh-frec|uency, 221
pulse transformer, 294
rotation, 219
secondary, 4
traverse, 219, 222
Wire, 34, 85
insulation, 35, 37, 41
tables, 36, 37, 39
tolerance, 105
Wound core, 18
Zigzag conned ions, 62, 118
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