FROM -THE- LI BRARY- OF
WILLIAM -A HILLEBRAND
PHYSICS OEPT.
ELECTRIC TRANSIENTS
c MsQra&)'3/ill Book & 1m
PUBLISHERS OF BOOKS F O R^
Electrical World ^ Engineering News-Record
Power v Engineering and Mining Journal-Press
Chemical and Metallurgical Engineering
Electric Railway Journal v Coal Age
American Machinist ^ Ingenieria Internacional
Electrical Merchandising v BusTransportation
Journal of Electricity and Western Industry
Industrial Engineer
ELECTRIC TRANSIENTS
BY
CARL EDWARD MAGNUSSON
AUTHOR OF "ALTERNATING CURRENTS," PROFESSOR OF ELECTRICAL ENGINEERING,
DEAN OF THE COLLEGE OF ENGINEERING, DIRECTOR OF THE ENGINEERING
EXPERIMENT STATION, UNIVERSITY OF WASHINGTON
A. KALIN
INSTRUCTOR IN ELECTRICAL ENGINEERING, UNIVERSITY OF WASHINGTON
J. R, TOLMIE
INSTRUCTOR IN ELECTRICAL ENGINEERING, UNIVERSITY OF WASHINGTON
FIRST EDITION
McGRAW-HILL BOOK COMPANY, INC.
NEW YORK: 370 SEVENTH AVENUE
LONDON: 6& 8 BOUVERIE ST., E. C. 4
1922
COPYRIGHT, 1922, BY THE
MCGRAW-HILL BOOK COMPANY, INC.
THE MAPLE PRESS - YORK PA
PREFACE
Transient electric phenomena generally increase in
commercial importance with the size and complexity of
electric systems, and a knowledge of the fundamental
principles of electric transients and their application to the
solution of quantitative problems is as essential to the
successful operation of large power and communication
systems as a mastery of the basic laws of direct and alter-
nating currents.
This work is an outline of an introductory lecture and
laboratory course given during the past twelve years to
electrical engineering students in the University of Wash-
ington. The purpose of the book is to aid the student in
gaining clear concepts of the fundamental principles of
electric transient phenomena and their application to
quantitative problems. The course as outlined is pro-
fessedly of an elementary character with emphasis placed
on the physical properties of electric transients. The text
is illustrated and supplemented by a large number of
oscillograms of transients that occur in the various types of
machines and electric circuits in common use in electrical
engineering laboratories. The problems are based on
quantitative data obtained from laboratory experiments
under circuit conditions that may easily be reproduced by
the student.
Quantitative laboratory work is essential in order to
readily gain insight into the physical nature of transient
electric phenomena. It is advisable to require the student
to devote at least two-thirds of the time allotted to a course
in electric transients to the taking of oscillograms. Adjust-
ing an oscillograph so as to obtain sharply defined, well
proportioned oscillograms of electric transients is an
effective method for acquiring due appreciation of quanti-
995863
vi PREFACE
tative values, both absolute and relative, of the factors
involved. The quality of the photographic record depends
as much on painstaking care in handling the films and in
developing and printing the oscillograms as on skilful
operation of the oscillograph. Many pitfalls in the photo-
graphic part of the work may be avoided by carefully
following the directions given in the Appendix.
No attempt is made to give references to original investi-
gations or to papers and books dealing with the various
phases of electric transient phenomena, as the principles
discussed are well established and the material is arranged
in text book form. A distinctive feature of the book lies in
the illustrations. All of the oscillograms were taken by
A. Kalin and J. R. Tolmie or by students in the course
under their direction in the electrical engineering
laboratories of the University of Washington.
C. EDWARD MAGNUSSON.
SEATTLE, WASH.,
March, 1922.
CONTENTS
PAGE
PREFACE v
CHAPTER I. INTRODUCTION 1
Magnetic circuit Dielectric circuit Electric circuit.
CHAPTER II. OSCILLOGRAPHS 10
Three element oscillographs Timing wave from oscillator genera-
tor Oscillograms Problems and experiments.
CHAPTER III. SINGLE ENERGY TRANSIENTS. DIRECT CURRENTS . 23
Single energy circuits The exponential law The time constant
Dissipation or attenuation constants The exponential curve
Initial transient values Current, voltage and ^magnetic flux
transients Problems and experiments. ^'4
CHAPTER IV. SINGLE ENERGY TRANSIENTS. ALTERNATING
CURRENTS 40
Single phase, single energy load circuit transients Three phase,
single energy load circuit transients Starting transient of a poly-
phase rotating magnetic field Polyphase short circuits. Alter-
nator armature and field transients -Single phase short circuits.
Alternator armature and field transients Single phase short cir-
cuits on polyphase alternators Problems and experiments.
CHAPTER V. DOUBLE ENERGY TRANSIENTS 75
Double energy circuits Surge or natural impedance and admit-
tance Frequency of oscillations in double energy circuits
Dissipation constant and damping factor in simple double energy
circuits Equations for current and voltage transients Problems
and experiments.
CHAPTER VI. ELECTRIC LINE OSCILLATIONS. SURGES AND TRAVEL-
ING WAVES 101
Artificial transmission lines Time, space and phase angles
Natural period of oscillation Length of line Velocity unit
of length Surge impedance Voltage and current oscillations and
power surges General transmission line equations Traveling
waves Compound circuits Problems and experiments.
CHAPTER VII. VARIABLE CIRCUIT CONSTANTS 130
Variable resistance Variable inductance Variable conductance
Variable condensance Problems and experiments.
vii
viii CONTENTS
PAGE
CHAPTER VIII. RESONANCE 145
Voltage resonance Current resonance Coupled circuits Direct
coupling Inductive coupling Condensive coupling Coupling
coefficient Multiplex resonance Resonance growth and decay
Problems and experiments.
CHAPTER IX. OSCILLOGRAMS 168
Starting transients of a D.C. lifting magnet Opening of D.C. and
A.C. circuit breakers due to overload T.A. regulator operating
transients Short circuits on series generators "Bucking
broncho" transients Current transformer transients Single
phase short circuit on a two phase alternator Undamped oscillo-
graph vibrator oscillations Starting transients on a three phase
induction motor Starting transients on a repulsion-induction
motor and a split phase motor Single phase operation of a three
phase induction motor Transients in three phase induction motor
due to short circuit on stator terminals Short circuits on a rotary
converter Synchronizing a rotary converter from 85 per cent
synchronous speed Synchronous motor falling out of step due to
overload The magnetic flux distribution of a synchronous motor
when slipping a pole Problem and experiments.
APPENDIX 189
Instructions' for developing and printing oscillograms.
INDEX. . 193
ELECTRIC TRANSIENTS
CHAPTER I
INTRODUCTION
The laws for direct currents, as usually -expressed, state
the relations of the several factors involved under continu-
ous or permanent conditions, and cannot be correicyy
applied while the current or voltage is increasing or decreas-
ing. Similarly, alternating currents are expressed as
continuous phenomena by means of effective values and
complex quantities, on the basis that the successive cycles
are of the same magnitude and wave shape. Observations
and test data for both the direct-current and alternating-
current systems are ordinarily taken only during steady
or permanent conditions. The equations derived, and the
data obtained from tests, apply only to permanent or
constant conditions and cannot be correctly applied during
transition periods when the conditions vary. Transient
electric phenomena, as the term implies, are usually of
short duration and relate to what occurs in an electric
circuit between periods of stable conditions. This defini-
tion is, however, not rigidly adhered to in electrical discus-
sions. Frequently other disturbances that militate against
successful operation of electric systems, such as unstable
electric equilibrium, permanent instability, resonance and
cumulative oscillations are included with the true transients
under the caption of transient electric phenomena.
It is important that the student should realize that
electric transients are of very frequent occurrence in all
commercial electric systems. Any change, such as the
starting or stopping of a motor, the turning on of a lamp,
or any change in the operating conditions necessitates a
2 ELECTRIC TRANSIENTS
re-adjustment of the energy content in the whole system
and produces electric transients just as truly as a stroke
of lightning or a short circuit. In the operation of street
car systems the changes in load, and hence the transients
on the system, are so frequent that they overlap and occupy
by far the greater part of the time; hence, for street railway
systems, it might appear simpler to define the permanent or
steady conditions as short periods occurring between succes-
sive series of overlapping transients.
Electrical ^engineering deals with the transmission and
transformation of electric energy. During permanent
conditions- the flow of energy is uniform and continuous;
any change in the power indicates a transient condition.
Changes in the current and voltage factors imply a cor-
responding change in the energy content of the electric
field, since a magnetic field surrounds all electric currents,
and an increase or decrease in the current necessitates a
corresponding change in the stored magnetic energy.
Similarly, any change in voltage between conductors must
be accompanied by a corresponding re-adjustment in the
energy stored in the dielectric field of the system.
Magnetic Circuit. In the study of transient phenomena,
as well as of all phases of the electric field, Faraday's
concept of magnetic and dielectric lines of force is of funda-
mental importance. All magnetic lines are continuous and
closed on themselves. Ohm's law applies to the magnetic
circuits in the same way as to the electric circuit. The
magnetic flux produced is equal to the magneto-motive
force divided by the reluctance.
-,, ,. a magneto-motive force
Magnetic flux = - :
reluctance
cy
$ = - or ff = (R$ (1)
The magnetic field is produced by, and is proportional to,
the electric current.
$ = Li (2)
INTRODUCTION
The proportionality factor L is called the inductance of
the circuit.
The reluctance varies directly as the length and inversely
as the cross section of the magnetic circuit. The specific
reluctance per cm. 3 is the reciprocal of the permeability ju.
If the magneto-motive force is expressed in ampere turns,
the resultant field intensity is given by the equation.
H ---- 4irnl lO^per cm. (3)
This magnetizing force produces a magnetic flux density
of B lines per cm. 2 in materials having /* permeability.
B = [J.H lines per cm. 2 (4)
The permeability is the reciprocal of the specific reluct-
ance in the magnetic circuits and corresponds to the specific
FIG. 1. Magnetic field of single conductor.
Magnetic field of circuit.
conductivity of the conductor in the electric circuits. In
empty space ^ = 1 and for all non-magnetic materials it is
very nearly equal to unity. For magnetic materials the
permeability is greatly increased and may reach several
thousand for soft iron and steel. The factor 4?r comes from
the definition of a unit magnetic pole as having one line
per cm. 2 on the surface of a sphere of unit radius. The
10" 1 factor results from the definition of the ampere.
In building up a magnetic field, lines of force cut the
conductor and thus produce a counter e.m.f., or inductance
4 ELECTRIC TRANSIENTS
voltage, L e, which is equal to the time rate of change of the
interlinked magnetic flux.
d$ T di
* - dt = L dt < 5 >
Necessarily an equal opposite voltage must be impressed
to force the current through the electric circuit. The prod-
uct of the voltage and the current represents the power
required to generate the field. Hence, the energy stored in
a magnetic field by a current, 7, in a circuit having an induc-
tance, L, is given by equations (6) and (7).
C w C 1 C 1 .
I dw = I L eidt = L I idi (6)
Jo Jo Jo
w-%
The energy is stored magnetically in the electric field
surrounding the conductor and is proportional to the square
of the current. When the current decreases the energy is
returned to the circuit, for if i and therefore <f> decrease,
di/dt and hence L e are negative, which means that the energy
is returned to the electric circuit.
The practical unit of inductance, L, is the henry. In
any consistent system of units a circuit possesses one unit
of inductance, if a unit rate of change of current in the
circuit generates or consumes one unit of voltage. If the
current changes at the rate of one ampere per second, and
the voltage generated or consumed is one volt, then the
inductance is one henry.
Dielectric Circuit. For the dielectric field similar rela-
tions exist. All dielectric lines of force are continuous and
end on conductors. Ohm's Law may be applied to the
dielectric circuit in the same manner as to the magnetic
and electric circuits.
Dielectric flux = , - ^ -;
elastance
* = = Ce (8)
INTRODUCTION
The dielectric flux is directly proportional to the voltage
between the conductors and inversely proportional to the
elastance of the dielectric circuit. The elastance, S, is the
reciprocal of the condensance, C, and varies directly as
the length, x, and inversely as the cross section, A, of
the dielectric circuit. It corresponds to resistance of the
electric circuit and to reluctance of the magnetic circuit.
FIG. 3. Dielectric field of single
conductor.
FIG. 4. Dielectric field of circuit.
S = . ; C = - '- in c.g.s. electrostatic units
K.A 4:irX
(9)
S = - ', C = -. -in electromagnetic units (10)
'
i ,
- 1 -a; darafs
, A 1 O 9 * A
C = - = 88.42 *~10- 15 farads
X
C = 88.42 " 10- 9 microfarads
JU
(12)
(13)
The permittivity K is unity for empty space and very
nearly equal to unity for air and many other materials.
In Table I is given the permittivity constants for the more
common dielectrics used in electric apparatus. The con-
stant v = 3-10 10 cm/sec., the velocity of the propagation
of an electric field in space (equivalent to the velocity of
light) , is the ratio of the units used in the electromagnetic
6
ELECTRIC TRANSIENTS
and electrostatic systems. The factor 4w comes from the
definition of a unit line of dielectric force.
TABLE I
Material
Permit-
tivity
Material
Permit-
tivity
Air and other gases ....
1.0
Olive oil
3 . to 3 2
Alcohol, amyl
Alcohol, ethyl
Alcohol, methyl
Asphalt
Bakelite
Benzine
Benzol
15.0
24.3 to 27.4
32.7
4.1
6.6 to 16.0
1.9
2 2 to 24
Paper with turpentine
Paper or jute impreg-
nated
Paraffin
Paraffin oil
Petroleum
Porcelain
2.4
4.3
2.3
1.9
2.0
5 3
Condensite
Glass (easily fusible) . .
Glass (difficult to fuse)
Gutta-percha
Ice
Marble
6.6 to 16.0
2 . to 5.0
5.0 to 10.0
3 . to 5.0
3.0
6.0
Rubber
Rubber vulcanized. . .
Shellac
Silk
Sulphur
Turpentine
2.4
2 . 5 to 3 . 5
2.7 to 4.1
1.6
4.0
2 2
Mica
5.0to 7.0
Varnish
2.0 to 4.1
Micarta
4.1
The charging current, c i, storing energy in the dielectric
circuit is equal to the time rate of change in the dielectric
flux.
^ r<^ e n/n
= dT L dt
Hence the energy stored in the dielectric field by a voltage,
E, in a circuit having a condensance, C, is given by equa-
tions (15) and (16):
JW (*E /
dw = I c iedt = C I
E
ede
CE*
2
(15)
(16)
The energy stored dielectrically in the electric field sur-
rounding a conductor is proportional to the square of the
voltage. When the voltage decreases the energy is
returned to the electric circuit, for if e and therefore ^
INTRODUCTION 7
decreases, then de/dt and hence c i are negative, which
means that the energy is returned to the electric circuit.
The unit of condensance (capacitance), C, is the farad.
In any consistent system of units a circuit possesses one
unit of condensance if a unit rate of change of voltage
produces (or consumes) one unit of current. If the voltage
changes at the rate of one volt per second and the current
produced (or consumed) is one ampere, the condensance
FIG. 5. Electric field of conductor.
FIG. C. Electric field of circuit.
of the circuit is one farad. The farad is too large a unit for
practical purposes and hence in commercial problems the
condensance is usually measured in microfarads.
1 farad = 10 6 microfarads (17)
Electric Circuit. The electric circuit relates specifically
to the conductor carrying the electric current although the
term is frequently made to include the dielectric and mag-
netic fields, since the electric, dielectric and magnetic
circuits are interlinked. Under steady or permanent
conditions in a direct current system the electric circuit
transmits the energy without causing any change in the
energy stored magnetically and dielectrically in the space
surrounding the electric circuit. In starting the system a
transient condition exists until the magnetic and dielectric
fields have been supplied with the required amount of
energy as determined by the magnitude of the current and
voltage and the circuit constants.
8 ELECTRIC TRANSIENTS
If the electric circuit be considered as something separate
and apart from the surrounding magnetic and dielectric
fields, no storage of energy would be involved and hence
no transients could exist, since all the changes would be
instantaneous. But the electric circuit is interlinked with
the dielectric and magnetic circuits. Changes in the cur-
rent and voltage in the electric circuit are accompanied by
changes in the energy stored in the dielectric and magnetic
fields, thus necessitating a readjustment of the energy con-
tent in the whole electric system. The transfer of energy
requires time and thus the transient period is of definite,
although often of extremely short, duration.
The close analogy existing between electric, dielectric
and magnetic circuits may be shown to advantage by
arranging the corresponding quantities in tabular form as
in Table II.
For convenience in solving problems the energy equations
are expressed in the units used in commercial work:
Energy in a Magnetic Field :
= W (joules) = ^ en ^)il(am_peres) (lg)
2i
Energy in a Dielectric Field :
TTT/. i N CYmicr of arads) e 2 (volts) /irvv
w oouies) = -~
Energy in a Moving Body:
= TF(ergs) = M (g rams ) ^ (meters per sec.)
2i
Energy in a Moving Body :
= TF(joules) = M(k S') v 2 (meters per sec.)
Energy in a Moving Body :
= TFfft Ib } = -' v " ( ft -_Per_sec.)
2 X 32.2
1 joule = 1 watt-sec. = 10 7 ergs = 0.7376 ft.-lb.
- 0.2389 g.-cal. ='0.102kg.-m. = 0.0009480 B.t.u. (21)
1 ft.-lb. = 1.356 joules = 0.3239 g. = 0.1383 kg.-m.
= 0.001285 B.t.u. = 0.0003766 watt-hour (22)
1 B.t.u. - 1,055 joules = 778.1 ft.-lb. = 252 g.-cal.
- 0.2930 watt-hour (23)
INTRODUCTION
TABLE II
Electric circuit
Dielectric circuit
Magnetic circuit
Electric current:
Dielectric flux (dielectric
Magnetic flux (magnetic cur-
current):
rent):
i = Ge = electric current.
R
* = Ce = * lines of di-
</> = Li 10 8 lines of mag-
o
netic force.
electric force.
Electromotive force, voltage:
Electromotive force:
Magnetomotive force:
e = volts.
e = volts.
5 = 4-irni ampere-turns.
gilberts.
Conductance:
Condensance, capacitance,
Inductance:
permittance or capacity
n<j> _ <b
G = mhos.
e
C = farads.
5
e
henrys
Resistance:
Elastance:
Reluctance:
R = ohms.
S = Q = ^ darafs.
R = , oersteds.
Electric power:
Dielectric energy:
Magnetic energy:
'P = R?' 2 G^ = ic watts.
Ce 2 tye
T i 2 (hi
w = == joules.
w = - = n 10~ 8 joules.
i
Electric-current density:
Dielectric-flux density:
Magnetic-flux density:
J = = yG amp. per cm. 2
D = = KK lines per cm. -
A.
B = . - = pH lines per cm. 2
Electric gradient:
Dielectric gradient
Magnetic gradient:
G' = volts per cm.
G' = volts per cm.
/ = -j amp. -turns per cm.
Conductivity:
Condensivity, permittivity
Permeability:
j
or specific capacity:
7 = mho. -cm. 3
D
B
~ K
M = H
Resistivity:
Elastivity:
Reluctivity:
1 G ,
P = = ohm-cm. 3
1 K
k D
r = B
Specific electric power:
p = p /2 = 7 G 2 = GI watts
Specific dielectric energy:
kG'* G'D.
Specific magnetic energy-
0.47TU/ 2 fB.
per cm. 3
cm. 3
Condensance, permittance,
capacitance current:
d = ,, = C - - amperes.
at at
Dielectric-field intensity:
K = - 2 lines of dielectric
force per cm. 2
w = ^ = -10-
joules per cm. 3
Inductance voltage:
Magnetic-field intensity.
// = 47T/10- 1 lines of mag-
netic force per cm. 2
CHAPTER II
OSCILLOGRAPHS
The oscillograph is the most important apparatus for
obtaining quantitative data on electric transient phenom-
ena. To gain clear concepts of the relative magnitude of
the physical quantities involved it is highly desirable for
FIG. 7. Magnetic field and vibrating elements.
the student to take oscillograms of a number of typical
transients. For this purpose an oscillograph with a photo-
graphic attachment is necessary.
While several types of oscillographs are in commercial
use all operate on the same basic principle. The essential
10
OSCILLOGRAPHS
11
element of the oscillograph is the galvanometer, an insu-
lated loop of wire, placed in a magnetic field, through which
the electric current flows. The direction and magnitude
of the currents cause a proportional turning movement of
a small mirror attached to both sides of the loop. The
deflection of a beam of light thrown on the mirror indicates
the angular position of the mirror and hence the magnitude
and direction of the current flowing through the loop.
Three Element Oscillographs. The three element, port-
able type oscillograph manufactured by the General Electric
Co. is shown in Figs. 7 to 13. The arrangement of the
Mirror-
FIG. 8. Vibrating element.
FIG. 9. Cross section of vibrating element.
electromagnetic field and the three vibrating elements is
shown in Fig. 7. One of the vibrating elements removed
from its magnetic field is shown in Fig. 8 and its vertical
cross-section in Fig. 9. The three vibrators are indepen-
12
ELECTRIC TRANSIENTS
dent units and insulated so as to carry three separate elec-
tric currents. The vibrating strips and mirrors are of
silver. The vibrating element can be turned around a
vertical axis, passing through the center of the mirror, by
the screw Q. The containing cell for the whole vibrating
element is also movable around a horizontal axis, passing
through the center of the mirror, by means of the screw S.
Hence the beam of light reflected from the vibrating mirror
may be directed to any desired spot and so adjusted as to
pass through the cylindrical lens to the slit in front of the
rotating photographic film.
In Figs. 7 and 8, the letters TT' mark the terminals of
the vibrating strips marked ST in Fig. 9. The mirror
with the vibrating portion of the loop lies between the
supports BB 1 '. The size of the mirror is about 20 by 10 mils
and the vibrating element has a natural period of approxi-
XA
FIG. 10. Optical train horizontal projection.
FIG. 11. Optical train vertical projection.
mately one five-thousandth of a second (0.0002 sec.). By
immersing the vibrating element in oil the instrument is
made dead-beat.
OSCILLOGRAPHS
13
The horizontal projection of the optical train for photo-
graphic work is shown in Fig. 10 and a vertical projection
in Fig. 11. The arc lamp is at A and the arrows indicate
the directions of the beams of light. PI, P 2 , P* are right-
angled prism mirrors; Si, $ 2 , $ 3 , adjustable slits; h, 1%, h
condensing lenses; VMi, VM^, VM Z the vibrating mirrors ;
CL a cylindrical lens for bringing the light beams to a sharp
focus on the photographic film on the surface of the revolv-
ing cylinder in the film holder.
FIG. 12. Oscillograph on operating stand. (Gen. Elec. Co.)
In Fig. 12 the oscillograph is shown mounted on a con-
veniently arranged operating stand. The positions of the
arc lamp, film motor, film holder, controlling rheostats,
time wave oscillator and other accessory appliances for
recording electric transients are clearly indicated. The
corresponding wiring diagram, with quantitative circuit
data, is shown in Fig. 13.
14
ELECTRIC TRANSIENTS
The three-element, portable oscillograph of compact
design, manufactured by the Westinghouse Elec. & Mfg.
Co., is shown in Figs. 14 to 16. The photographic film
drum and driving pulley with rheostats, switches, etc., are
shown on the right side of Fig. 14 a, while on the left are the
FIG. 13. Wiring diagram three element oscillograph. (Gen. Elec. Co.)
three sets of dial resistances, one for each vibrating element,
with switches, binding posts and fuses. In Fig. 146 is
shown the driving motor with control apparatus for opera-
ting the film holder at several speeds. Light for making
the photographic record is obtained from a low voltage
incandescent lamp of special design. For high speed records
an arc lamp is used, in place of the incandescent lamp, to
gain the greatest possible light intensity.
The galvanometer, with one of the three elements re-
OSCILLOGRAPHS
15
FIG. 14a. Front and resistance panel side of portable oscillograph. (Westing-
house Elec. & Mfg. Co.}
FIG. 14&. Front view of portable oscillograph coupled to motor. (Westinghouse
Elec. & Mfg. Co.)
16 ELECTRIC TRANSIENTS
moved, is shown in Fig. 15. The moving element consists
of a single turn or oblong loop of wire forming two parallel
conductors. A tiny mirror is attached to both conductors
and placed in a strong magnetic field. Hence when a
current passes down one conductor and up the other, one
tends to move forward and the other backward. The
mirror bridging these conductors is given an angular deflec-
tion proportional to the current.
FIG. 15.- -Three element galvanometer. (Westinghouse Elec. & Mfg. Co.)
The design of the electromagnetic field circuit is unique.
A direct current passing through a single coil sets up a mag-
netic flux which passes through the three vibrating elements
in series. To insulate the elements from each other and
from the main magnetic core and yokes four insulating
gaps are used, thus placing seven air gaps in series in the
path of the magnetic flux. The three gaps in the galvano-
meter elements are J^ 2 in. long, giving sufficient space for
the vibrators and producing uniform distribution of the
magnetic flux. The four insulating gaps are KG in. long
but of large cross-sectional area so as to give comparatively
low reluctance in the magnetic circuit. The field excitation
requires 6 volts, direct current.
A view of the trip magnet and shutter release mechanism
OSCILLOGRAPHS
17
is shown in Fig. 16, in which the trip magnet holds the long
shutter finger so that the short finger does not quite touch
the shutter tripping arm. The shutter is a tube with two
opposite longitudinal slots. The tube rotates and when
the slots are in a horizontal plane the beams of light, re-
flected from the tiny mirrors of the galvanometer vibrators,
pass through the cylindrical condensing lens and are
focused on the revolving photographic film. This occurs
between the time the short finger falls from the shutter
FIG. 16. Trip magnet and shutter release mechanism. (Westinghouse Elec. &
Mfg. Co.)
tripping arm and the time the variable finger falls from the
arm one revolution later. The shutter is actuated by the
spiral spring seen just beyond the finger hub. A pin on
the shutter shaft strikes an arm on the lamp extinguishing
switch. On the hub are attached laminated copper strips
which complete the lamp circuit when the shutter is set and
which break the circuit when the shutter snaps closed.
The tripping device can be adjusted so as to start exposures
at any desired part of the film.
Timing-waves from the Oscillator Alternator. The
time factor is of special importance in electric transient
phenomena and some means for recording the time elapsed
18
ELECTRIC TRANSIENTS
is necessary. In taking oscillograms in which the transient
current or voltage recorded does not give directly an indica-
tion of the time consumed it is customary to impress an
alternating current timing wave of known frequency on
one of the vibrators.
Current for the timing wave may be takefh directly
from any available power circuit, but the frequencies of
commercial systems are to some extent variable and the
indicating frequency meters may not be sufficiently accu-
: /VWK&&&
1ZOV.D.C
FIG. 17. Oscillator alternator circuit diagram.
rate for this purpose. A convenient source of supply for
timing wave current of constant frequency is found in the
oscillator generator. The circuit diagram of a simple porta-
ble form used in the electrical engineering laboratories of
the University of Washington is shown in Fig. 17. The
alternator consists of an audion tube connected to condens-
ance, resistance and inductance, as shown in the circuit
diagram, of the following quantitative values:
LI = 0.756 henry s R\ = 99 ohms
L 2 = 0.756 henrys E 2 = 99 ohms
L(total) == 2.38 henrys R 3 = 96 ohms
C = 1.06 microfarads R = 50 ohms
The impressed d.c. voltage was 110 volts but other values
may be used by adjusting the resistance, R s . The ampli-
tude of the timing wave may be varied by means of the
OSCILLOGRAPHS
19
20 ELECTRIC TRANSIENTS
resistance, R*. The alternating current produced by the
oscillator is of simple sine wave form and has a constant
frequency of 100 cycles per second.
Oscillograms. Great care must be taken in making
the adjustments on the oscillograph in order to produce
good oscillograms. The speed of the revolving drum carry-
ing the sensitized film and the amplitude of the galvano-
FIG. 19. Starting transient of induction motor with secondary resistance in
circuit. See Fig. 167, Chap. IX.
meter mirror vibrations must be adjusted to meet the
conditions imposed by the transient under investigation.
Thus the relative drum speed for the oscillograms shown
in Figs. 18, 19, and 20 was as 4 : 1 : 29, and the amplitude
adjusted in each case so as to use the film area to good
advantage.
The time lag of tripping devices and shutter operating
mechanism must be determined so as to expose the film at
the instant the transient occurs. The optical train must
be adjusted so as to give a spot of light sharply focused on
the sensitized film.
Instructions for developing the films and for printing the
oscillograms are given in the Appendix. A circuit diagram
OSCILLOGRAPHS
21
22
ELECTRIC TRANSIENTS
with quantitative data should be attached to each film.
It is important to show the circuit position of each vibrator
FIG. 21.-
-Transfer of oscillating energy in inductively coupled circuits.
Chap. VIII.
See
FIG. 22. Same circuit as in Fig. 21. Primary opened at the instant all the oscil-
lating energy was in the secondary circuit.
so that the record will indicate precisely where the transient
appearing on the film was taken.
OSCILLOGRAPHS
23
Problems and Experiments
1. Examine the oscillograph with care; trace all the circuits; operate the
arc lamp; adjust the optical train until the mirror on each vibrator throws
a spot of light through the slit and this is sharply focused on the ground
glass screen. Arrange a circuit with variable inductance and condensance
as indicated in Fig. 23. Connect vibrator V\ by means of a shunt, S, so
(VWWVW
FIG. 23.
as to indicate the current wave and vibrator Vz with a resistance, R\, in
series to show the voltage wave. By means of the small synchronous motor
operate the large oscillating mirror throwing the beams of light on the
mica screen on top of the oscillograph. By varying the resistance, induct-
ance and condensance in the circuit the time phase relations of the voltage
and current may be changed from lag to lead.
2. Reproduce, as nearly as available equipment will permit, the oscillo-
gram in Fig. 18
CHAPTER III
SINGLE ENERGY TRANSIENTS. DIRECT CURRENTS
Transient electric phenomena are produced by changes
in the magnitude, distribution and form of the energy
stored in electric systems. The simplest types of electric
transients are found in electric circuits having only one
kind of energy storage that is, either the magnetic or the
dielectric field, but not both. A condenser discharging
through a non-inductive resistance, as illustrated by the
oscillogram in Fig. 24, gives electric transients of the
simplest type. Since the resistance in the circuit is con-
stant the current is at all instants directly proportional
to the voltage across the terminals of the condenser. The
curve on the oscillogram can therefore be used as represent-
ing either the current-time or the voltage-time relation as
indicated by the two scales in the figure.
The Exponential Law. The energy stored in the con-
denser is at any instant equal to Ce 2 /2. The rate of
discharge is ei, which must be equal to the Ri 2 rate of energy
dissipation into heat in the resistance. The rate of energy
discharge is therefore at any instant proportional to the
energy stored in the condenser and the rate of change in the
current is at any instant directly proportional to the magnitude
of the current.
Let i and i' represent the currents at any two points on
the current-time curve of the oscillogram, in Fig. 24. Then :
di di' . .,
dt : dT ::i:i (24)
Let the line OP be drawn from starting point perpendi-
cular to the X axis. Let the line OQ be drawn tangent to
the curve at and intersecting the X axis at Q. The time
represented by the distance PQ is called the time constant,
24
DIRECT CURRENTS
25
.s
26 ELECTRIC TRANSIENTS
T, of the circuit. Since i' may be any point on the curve,
let it be taken at the starting point, ; then
t'-7 and ^ = -^ (25)
From (24) and (25)
di . IQ . . - . T
df ~T % ' Io (26)
Separating the variables and taking the limits of integra-
tion from the starting point, 0, to any point (i, t) oil the
curve :
r*.
JJ
(27)
(28)
t = Joe-* (29)
Similarly, for the corresponding voltage-time curve:
- F f -'f (30)
e J^Q
Equations (29) and (30) show that the fundamental
relations in simple electric transients are expressed by the
exponential equation. The minus sign is used as di/dt is
negative. The exponential curve represented by equations
(29) and (30) is as fundamental in the study of electric
transients as the sine wave in alternating currents.
If the energy stored in a magnetic field is released by
short circuiting through a resistance and dissipated into
heat, the same relations exist. Oscillograms of the current-
time or voltage-time curves similar to Fig. 24, may be
obtained by discharging a magnetic field through a resis-
tance, Fig. 27. Likewise, the electric transients existing
while a condenser is charged, or while a magnetic field is
established, obey the exponential law. In Fig. 25 is shown
the oscillogram of a current-time transient obtained while
establishing a magnetic field in the circuit shown in the
diagram. Let the line OP be drawn through the starting
point at right angles to the X axis. Let PS be drawn
DIRECT CURRENTS
27
parallel to the X axis and be an asymptote to the current-
time curve. Let the line OQ be drawn tangent to the curve
FIG. 25. Forming a magnetic field.
E= 109 volts;/ - 0.36 amps.; R = 303ohms;L = 5.1 henrys; T = 0.017 seconds.
/v
/
! w
FIG. 26. Showing permanent, transient and instantaneous values for oscillogram
in Fig. 25.
at and intersecting the line PS at Q. The line OP repre-
sents the value of the permanent current / which is equal
28 ELECTRIC TRANSIENTS
to E/R. The time measured by the line PQ is the time
constant, T, of the circuit. The rate of storing the mag-
netic energy at any instant is proportional to the remaining
magnetic storage facilities in the circuit under the given
conditions. Therefore the rate of change in the current is at
any point on the curve proportional to / - - i, and equation
(31) is derived in the same manner as equations (24) and (26)
(33)
(34)
The transient which by definition represents the change
from one permanent condition to another is in equation
_ t
(34) represented by the factor Ie T . Before the circuit
was closed the value of the current was zero, while the
final permanent value is I.
In Fig. 26 the permanent or final value of the current is
represented by OP, the distance of the line PS from the X
axis. The transient values are given by the ordinates to
the broken curve WVX, while the instantaneous current
which must at any instant be equal to the algebraic sum of
the permanent and transient values is given by the ordi-
nates to the curve OYS, which is the curve photographed
on the oscillogram in Fig. 25. It is important to note that
the photographic record of the actual instantaneous values
gives at each point the resultant or the algebraic sum of
the corresponding permanent and transient ordinates.
Thus for any time, t:
RY = RU + (-RV) (35)
The Time Constant. From the starting point of the
current-time curve in the oscillogram, Fig. 27, which shows
the discharge of a magnetic field through a resistance, the
line OP is drawn at right angles to the X axis ; the line OQ tan-
DIRECT CURRENTS
29
gent to the curve at the point and intersecting the X axis at
Q' } the line QN perpendicular and ON parallel to the X axis.
From the principle of the conservation of energy, the
energy stored in the magnetic field must be equal to the
amount dissipated as heat in the resistance of the circuit
when the field is discharged.
/00 /
= R I tfdt = RI Q 2 I
Jo t/o
RIJ-T
(36)
(37)
In circuits having resistance and inductance in series, as
in Fig. 27, the time constant is equal to the inductance
divided by the resistance.
FIG. 27. Discharge of a magnetic field through a constant resistance.
E = 60 volts; I = 0.21 amps.; R = 28.6 ohms; L= 0.89 henrys; T = 0.031
seconds.
In a similar manner the expression for the time constant,
T, in terms of the circuit constants may be found for circuits
having condensance and conductance, Fig. 24. The energy
stored in the condenser when the discharge starts must be
equal to the energy expended as heat in the Ri 2 losses.
30 ELECTRIC TRANSIENTS
CF 2 f ra f ra 2t J?T 2 T
^p_ = R tfdt = RI 2 e~ Tdt = ^ (38)
/o /o
Hence,
T = CR = C G (39)
In circuits having resistance and condensance in series,
as in Fig. 24, the time constant is equal to the condensance
divided by the conductance.
Equations (37) and (39) are of fundamental importance
in the study of transient phenomena. The exponential
equation for the transients in Figs. 24 and 26 may be
rewritten using the value of T as given in (37) and (39) and
the data in the circuit diagrams.
Fig. 24, equations (29), (39)
/ C 1
i = I Q ~T = 7 oc -c' = 4.18 " : '"'amperes ( 40 )
Fig. 24, equations (30), (39)
e = E Q e~ * = E$. = 120.6 e~ 38 - 5 Volts ( 41 )
Fig. 25, equations (34), (37)
i = I - Ie~ T = I - 7e~*' = 0.36 - 0.36e~ 58 ' 7 amperes (42)
The reciprocals of the time constants appearing in the
exponential equations as R/L and G/C or such combinations
of circuit constants as the complexity of the system may
require, are often called the dissipation constants or the
attenuation constants of the circuit.
The expressions for the time constants in equations (37)
and (39) may be derived from the current-time and voltage-
time curves instead of basing the equations directly on the
principle of the conservation of energy. In Fig. 24 the
quantity of electricity (coulombs) in the condenser when
starting the transient must be equal to the total amount
expended when the condenser is discharged, as represented
by the area between the current-time curve and the X axis.
rco /oo
idt = 7 I e~ * dt = I Q T (43)
Jco /
idt = 7 1
*/ o
DIRECT CURRENTS 31
Hence,
T = C E T = CR = C n = 0.026 seconds (44)
-/O (JT
Similarly, in Fig. 27, the magnetic flux in the field when
starting the transient may be equated to the total number
of lines of force cutting the circuit when all the magnetic
energy in the field changes into heat in the resistance.
Hence,
Jf- r~ t
edt = R\ idt = RI Q I e T dt = RI T (50)
t/o /o
T = = 0.31 seconds (51)
H
If the initial rate of discharge in Fig. 24 be continued
unchanged, the current-time curve would coincide with the
line OQ and the condenser would be completely discharged
in the time represented by PQ or T. Hence the area of the
rectangle OPQN must be equal to the area between the
current-time curve and the X axis.
Similarly, in Fig. 27, if the initial rate of discharge
continued unchanged, the current-time curve would coin-
cide with the line OQ and all the energy stored in the mag-
netic field would appear as heat in the resistance in the time
represented by PQ or T. Hence the area of the rectangle
OPQN must be equal to the area between the current-time
curve and the X axis.
Expressions for the transient current and voltage as given
in (40), (41) and (42) are derived without using the time
constant term. The customary differential equations
giving the. basic relations, with expressions for the transient
currents, are given in (52) to (55).
For circuits having resistance and condensance, Fig. 24,
while the condenser is discharged through a constant
resistance :
Ri + f- == 0; hence i == / O e~*c == 1$ (52)
32 ELECTRIC TRANSIENTS
For circuits having resistance and condensance, similar
to Fig. 24, the transient current while charging from to
the voltage E is the same as for discharging through the
resistance.
Ri + ( l ~ ---- E', hence i = 7 e"V (53)
For circuits having resistance and inductance, Fig. 19,
while the magnetic field is formed:
Ri +L- = E-, hence, i = f - ^"^ = / - /e~ (54)
at K H
For circuits having resistance and inductance, Fig. 21,
and a magnetic field supplied by a current 7 , while the
field discharges through a short circuit:
Ri + L~ = 0; hence, i = 7 e~ ^ (55)
Cvt'
The Exponential Curve. Oscillograms of simple electric
transients give a photographic record of the current-time
factors. The amplitude of the curve varies directly with
the magnitude of the current passing through the vibrator
and the strength of the magnetic field in which the vibrator
moves. The length of film used for any given unit of
time depends on the speed of the revolving drum carrying
the film. It is evident that both the amplitude of the
mirror vibrations and the speed of the film may be adjusted
independently of the circuit in which the transient occurs.
By examining exponential equations representing simple
electric transients it is apparent that if the value of the time
constant, T, be used as the unit of length on the X axis and
the initial value of the variable as the unit of measure for
the ordinates, then all exponential transients will have the
same shape and may be represented by the numerical
values of the exponential equation, y = e~ x . The same
space unit need not be used on both axes to represent the
unit values of current and time, but the scale may be
selected so as to secure a convenient shape for the available
DIRECT CURRENTS
33
space. In Fig. 28 is shown a current-time curve in which
the unit representing the initial value of the transient
current is five cm., while the unit used on the X axis, that
is, for the time constant of the circuit, is one cm.
FIG. 28. The exponential curve. Current-time transient.
By using the initial value of the transient as the unit of
ordinates and the time constant of the circuit as the unit
TABLE III
y = e *; = 2.71828
y
0.00 1.000 1.2
0.301
0.05
0.951
1.4
0.247
0.10
0.905
1.6
0.202
0.15
0.860
1.8
0.165
0.20
0.819
2.0
0.135
0.30
0.741
2.5
0.082
0.40 0.670
3.0
0.050
0.50
0.607
3.5
0.030
0.60 0.549
4.0
0.018
0.70
0.497
4.5
0.011
0.80
0.449
5.0
0.007
0.90
0.407
6.0
0.002
1.00
0.368
7.0
0.001
34 ELECTRIC TRANSIENTS
of abscissae, all exponential transients are of the same shape
and if plotted to the same scale would be identical with the
curve in Fig. 28. The numerical relations between y and
x in the exponential equation y = e~ x are given in Table
III. While the plotting of transients may be facilitated
by the selection of the above units, the actual initial values
of the transient quantity, expressed in amperes or volts,
may be of any magnitude as determined by the circuit
conditions.
Initial Transient Values. In simple electric transients
the initial value of the variable quantity depends on both
the permanent value and on the relative magnitude of the
circuit constants. Thus equations (33) and (42) show that
the time constant of a magnetic field depends on the induc-
tance and resistance in the circuit. If the energy stored in
the magnetic field be discharged by short circuiting the
terminals of the field, the initial value of the transient volt-
age will be equal in magnitude but opposite in direction to
the previously permanent value. But if the discharge be
made through an additional resistance, R^ the initial
voltage transient will be greater in magnitude in the ratio
of Ri + Rz : Ri, when Ri represents the resistance of the
field winding. The time constant of the circuit in which
the transients appear would be,
T = D when the field is short circuited,
HI
and TI = - D -~ '=- when the additional resistance # 2 is
Hi T- Hz
inserted in the discharging circuit. With the same amount
of energy stored in the magnetic field, the products of the
initial value and the corresponding time constants must
be equal.
E T = E 'T' (56)
Hence,
E Q :E Q f ::R l :R 1 + R 2 (57)
Eo > = E Q R R * (58)
Hi
DIRECT CURRENTS
35
The initial induced discharge voltage is therefore greater
than the permanent impressed voltage in the proportion
of the resistances in circuit for the two cases. In the
voltage-time curve, Fig. 29, the initial discharge voltage,
Eo", is that part of the induced voltage, E Q ', due to the
Ri 2 drop.
EV/
111 Q =
= E
R
(59)
This relation is of great importance in the design and opera-
tion of electrical machinery. In breaking electric circuits,
as induction coils, motor and generator fields, transmission
wwwwvwww
FIG. 29. Magnetic field discharging through additional resistance, Ri.
lines, etc., in which energy is stored magnetically, the air
or oil gap in the switch introduces a rapidly increasing
resistance. The faster the contact points of the switch or
circuit breaker separate, the more rapidly the resistance is
inserted and the higher the induced voltage.'
In Fig. 30 is shown the voltage-time and current-time
oscillogfams for breaking the field circuit of a direct-
current motor. In opening the switch an arc is formed
by which a resistance of rapidly increasing magnitude is
36 ELECTRIC TRANSIENTS
introduced into the circuit. The oscillograrn shows that in
about >f 5 of a second the induced voltage increased to more
than twenty-eight times the voltage impressed on the termi-
nals of the field before the switch was opened. Although
the voltage applied to the motor field was only 31.5 volts
the opening of the switch in the field circuit produced a
transient stress of over 900 volts on the field insulation.
FIG. 30. Breaking field circuit of direct current motor. Current and voltage
transients.
Since the transient induced voltage on the motor field
winding is directly proportional to the rate of cutting lines
of force the shorter the time used in opening the switch,
or the faster the resistance is inserted in the circuit the
greater the transient voltage-stress tending to puncture the
field insulation. If the circuit breaker operates in steps by
which resistances of known value are introduced into the
circuit in rapid succession the transient induced voltage
will be proportionately lower and the destructive action
of the arc greatly reduced. Since the energy stored in an
DIRECT CURRENTS 37
electromagnetic field depends on the current flowing in the
field windings, it must be converted into some other form
when the current is interrupted.
Current, Voltage and Magnetic Flux Transients. In
electromagnetic circuits having constant permeability the
current, voltage and flux transients have the same shape
and are expressed by the exponential equation. Referring
to Fig. 27
e = Ri (60)
The curve in the oscillogram therefore represents either
the current or voltage transients and the quantitative
values are obtained by applying the corresponding ampere
and volt scales. From the law of electromagnetic induc-
tion the induced voltage is equal to the rate of cutting
lines of force.
e = . 10~ 8 volts (61)
at
Hence,
edt =
= 10 s I e
- R t
= constant e L lines of flux (62)
The flux transient therefore is an exponential curve of
the same form as the current and voltage transients.
In Fig. 31 is shown the corresponding transients for the
current, voltage and flux in forming an electromagnetic
field in a magnetic circuit of constant permeability. The
transients are shown by the dotted lines, the permanent
values by the broken lines and the instantaneous values
by the full line curves.
/ _ / - (63)
/ _t\ _R t
= E + \-Ee T ) = E - Ee L (64)
_* 4
= $ _ $ e L (65)
38
ELECTRIC TRANSIENTS
T
-/
/
L
T
1
'N
T
$
\\
1
f N
FIG. 31. Single energy voltage, current and magnetic flux transients in forming a
magnetic field in a magnetic circuit of constant permeability.
DIRECT CURRENTS 39
Equations (61), (62) and (63) express the instantaneous
values as equal to the algebraic sum of the permanent and
transient quantities. At any instant in time, t, as indicated
in Fig. 31:
PQ = the instantaneous value, i, e or < (66)
PS = the permanent or final value, /, E or < (67)
t t t
PN -- the transient value, I e~ T , E Q e~ T , 3> Q e~ T (68)
In discharging energy stored in a magnetic field, in a
magnetic circuit of constant permeability, through a con-
stant resistance, the transient and the instantaneous values
are equal as the permanent value is zero.
i = 1^^ = I^~ Lt (69)
-' - R t
e = E e T = E e L (70)
-' - R t
= $ oe T = $ oe L (71)
Problems and Experiments
1. A condenser of 115 mfds, charged to 500 volts, is discharged through a
constant resistance of 425 ohms.
(a) Derive the equations for the current-time curve.
(6) Find the time constant of the circuit.
(c) Plot the voltage across the terminals of the condenser-time curve.
Ordinates in volts and abscissae in seconds.
(d) Draw an ampere scale of ordinates so that the curve plotted in (c)
will represent the current transient.
2. The time constant of an inductance coil is found by taking an oscillo-
gram to be 0.04 seconds. The resistance in circuit was 15.8 ohms.
(a) Find the inductance in henrys.
(6) With 110 volts impressed on the coil plot the starting current
transient.
(c) Write the equation for the current-time curve in (6).
3. Take an oscillogram of the starting current transient of the field of a
laboratory motor or generator.
(a) From the oscillogram find the time constant of the field.
(6) Measure the resistance of the circuit and calculate the inductance of
the field.
4. With the vibrators connected as shown in the circuit diagram in Fig.
30 taken an oscillogram showing the current and voltage transients pro-
duced by breaking the field circuit of a motor or generator.
6. By means of oscillograms determine the time required for the opera-
tion of automatic circuit breakers. Arrange the connection for the vibrators
so as to show the time consumed by each step in the operation.
CHAPTER IV
SINGLE ENERGY TRANSIENTS. ALTERNATING
CURRENTS
In direct-current systems the transient electric phenom-
ena described in the preceding chapter, are due to the
storage of energy in magnetic and dielectric fields. If a
constant direct-current voltage is impressed on a circuit
having constant resistance but neither inductance or
condensance the current would instantly reach its perma-
nent value, and any change in the voltage would at the same
time cause a proportional change in the current. With
either condensance or inductance in the circuit a short
period of time is required for the current to reach its perma-
nent value after any change in voltage, and the current-
time curve during the transition period is expressed by the
exponential equation. The value of the variable current
is at any instant equal to the algebraic sum of the per-
manent and transient values; and single energy transients
for direct currents in circuits having constant resistance
and inductance or constant resistance and condensance,
may be expressed by exponential equations similar in form
to (63), (64) and (65).
Single Phase, Single Energy Load Circuit Transients.
The same principle applies to single energy transients in
alternating-current systems. In circuits having constant
resistance but neither inductance nor condensance, no
transients appear. Any change in the voltage produces
instantly a proportionate change in the current. In
circuits having either inductance or condensance a tran-
sient period for the readjustment of the energy content of
the system follows any change in voltage. At any instant
the transient current or voltage is the algebraic sum of the
40
ALTERNATING CURRENTS
41
. > II
42 ELECTRIC TRANSIENTS
corresponding permanent and transient values. The per-
manent term, i' is the alternating current wave assumed
to be of simple sine form as expressed in equation (72) with
71 as the time phase angle at the starting moment.
i' = +Vsin (co - 7l ) (72)
e ' = +-# sin (co - T2 ) (73)
The transient term, i" is expressed by the exponential
equation of the same form as in direct currents with an
initial value equal in magnitude but opposite in time phase
to what would have been the permanent value at the start-
ing point if the circuit has been closed at some previous time.
i" = Vsin 7, e~r (74)
e" -- M E sin 72 ^ T (75)
The instantaneous value of the current or voltage would
therefore be expressed by (76) and (77) :
i = t' + i" a! + V sin (ut - 71) "I sin 71 e~* (76)
e = e ' + e " = M I s in (ut - 72) + "E sin Tl e~ T (77)
The oscillogram in Fig. 32 shows the current-time curve,
i, produced in a circuit having 20 ohms resistance and 0.575
henrys inductance when a 60 cycle alternating-current volt-
age, e, was impressed by closing a switch at the instant in
time presented by the OY axis.
The voltage impressed is represented by the sine wave,
e = M E sin (co - T2 ) (78)
The actual current flowing is represented by the curve
i, which practically coincides, after completing 6 cycles,
with the permanent value i, shown by the dotted sine wave.
The transient, i", is shown as the broken line whose initial
value,
ON = -OP = Vsin Tl (79)
At any instant, t, after the closing of the switch, i is equal
to the algebraic sum of i' and i" .
ALTERNATING CURRENTS
43
I = I + I
= 0.835
= V sin co -
wt 2
+ 0.835 sin
-4
'/ sin Tl e L (80)
-35*
(81)
It is evident that the initial value of the transient may
vary from - M I to + A 7, depending at what point of the
voltage-time curve the circuit is closed. If the switch be
thrown at the instant when the permanent current wave
would be zero, 7 = 0, no transient would appear and the
E
FIG. 33. Single phase, single energy, current transient.
129 volts; M E = 182.5 volts; I = o'.59 amps.; "I .835 amps.;
R = 20 ohms; L = 0.575 henrys; / = 60 cycles; 71 = 0.
permanent and actual current time curves would coincide
throughout as shown in Fig. 33.
i = i' = M I sin (0 (82)
The transient current would have a maximum initial
value if the circuit is closed at the instant the permanent
current wave is at a maximum, that is when sine 71 = 90%
The time constant for the current transient would be the
44 ELECTRIC TRANSIENTS
same at whatever point in the cycle the circuit is closed, as
it depends on the resistance and inductance in the load
circuit.
Three-phase, Single Energy, Load Circuit Transients.
For three-phase circuits similar relations exist. The start-
ing current transients in three-phase systems in which
energy may be stored either magnetically or dielectrically
follow the same laws as discussed for single-phase circuits.
In Figs. 34 and 35 are shown oscillograms of the three
starting load currents in a three-phase system, star con-
nected and having 9.0 ohms resistance and 0.205 henrys
inductance in each phase to neutral. The corresponding
permanent current waves and transient currents were
traced on the oscillogram in Fig. 34. In Fig. 35 the circuits
were closed at the instant the current in v\ was of zero
value.
Phase 1 :
Impressed voltage,
61 = "Ei sin (co 72) (83)
Permanent current,
i'i = M Ii sin (ut 71) (84)
Initial value starting current transient,
OQ 1 = -OP, = "/sin 71 (85)
Transient current,
_Ri
i"i = "I i sin 71 e L I (86)
Actual current, oscillogram,
^ = i\ + i'\ = Vi sin (ut - 71) + Vi sin yie ^ ( 87 )
Time constant,
T = - = 0.023 seconds (88)
Phase 2:
Impressed voltage,
e, = E 2 sin (ut - 72 - 120) (89)
Permanent current,
i\ = / 2 sin (ut - 71 - 120) (90)
ALTERNATING CURRENTS
45
IS
46
ELECTRIC TRANSIENTS
ALTERNATING CURRENTS 47
Initial value starting current transient,
OQz = -OP 2 = "I z sin (71 - 120) (91)
Transient current,
t" 2 = V 2 sin (71 -- 120)~5 (92)
Actual current, oscillogram,
i 2 = i' 2 + i" 2 = / 2 sin (co - 72 -- 120)
+V 2 sin (71 - 12Q)e~ (93)
Time constant,
T 2 = f 2 = 0.023 seconds (94)
-^2
Phase 3:
Impressed voltage,
e, = M Ez sin (* -- 72 - 240) (95)
Permanent current,
i' 8 = "/ 8 sin (* -- 71 - 240) (96)
Initial value starting current transient,
OQ 3 = -OP, = M I Z sin (71 - 240) (97)
Transient current,
t" 8 == "I* sin (71 -- 240)e *' (98)
Actual current, oscillogram,
i 3 = i' 8 + i'% = V 8 sin (co^ - 71 - 240)
+ M / 3 sin (71 - 240) e~^ (99)
Time constant,
T z =^ = 0.023 seconds (100)
/i3
It is of interest to note that the sum of the instantaneous
values of the three currents, ii + i z + i$, is equal to zero
during the starting period as well as after the permanent
state has been reached. This is evidently the case since
under permanent conditions the sum of the currents is at
any instant equal to zero and hence at the instant the
circuit is closed, OPi + OP 2 + OP 3 = 0. Therefore, the
sum of the initial values of the transient currents, OQi +
+ OQz = 0, and as the time constants of the three
48 ELECTRIC TRANSIENTS
transients are equal the sum of the actual currents in the
three phases must at any instant be equal to zero.
Starting Transient of a Polyphase Rotating Magnetic
Field. In the preceding illustrations the single energy
transients are due to changes in the amounts of energy
stored in the given circuits, and the current-time curves
show a continuous decrease of the current as expressed by
the exponential equation. If the permanent condition
relates to interconnected circuits which permit a transfer
of energy from one circuit to another, although the total
amount of energy stored in the magnetic field is constant,
as in the rotating field of a polyphase induction motor,
pulsations will appear during the transition period, that
merit attention.
In Fig. 36 let a vector of constant length, ON, rotating in
a counter clockwise direction represent a constant rotating
magnetic field, as would be produced by three equal mag-
netizing coils, placed 120 deg. apart, and excited by three-
phase currents, as, for example, in a three-phase induction
motor. For simplicity let the rotor be removed and con-
sider the stator circuit and the magnetic flux during the
starting transition period in which the rotating field is
built up to its constant permanent value. Let the switch,
connecting the stator circuit to three-phase mains, be
closed at the instant the rotating magnetic flux vector, ON,
lies along the X axis, ON in Fig. 36 ; as would have been the
case if the switch had been closed at some previous
time. The actual value of the rotating flux at the instant
the circuit is closed is zero. The permanent value is
represented by ON Q and since the initial value of the transi-
ent flux must be equal in magnitude but of opposite time
phase, it is represented by the vector OQ .
OQ Q =-- (-(W ) (101)
From its initial values OQ the transient flux decreases
in magnitude, as indicated by the exponential flux-time
curve in Fig. 37, but continues fixed in space direction
ALTERNATING CURRENTS 49
along the X axis. After the time, ti, has elapsed, repre-
sented by the time angle N ONi, the transient has a value
OQi. The actual value of the flux must be the vector sum
of the permanent value ONi and the transient OQi, or the
resultant OPi. In the time t 2 , the transient has decreased
to OQ 2 and the permanent flux vector reached the position
ON 2 . The actual flux OP 2 is the resultant of OQ 2 and ON 2 .
Similarly OP 3 is the resultant of OQ 3 and ON Z ] OP 4 , of
FIG. 36. Permanent, transient and instantaneous values of the magnetic flux in
starting a rotating magnetic field. Polar coordinates.
OQ 4 and ON^ etc. From the vector diagrams, Figs. 36
and 38, it is evident that the actual starting flux will oscil-
late having values greater and smaller than the permanent
value, the number of oscillations depending on the time
constant of the circuit. The maximum value of the flux
in the starting period will in any case be less than double the
permanent value, as the transient flux continuously
decreases from an initial value equal to the permanent flux
in magnitude and different by 180 deg. in time-phase.
50
ELECTRIC TRANSIENTS
The flux-time curve in Fig. 39 gives in rectangular coordi-
nates the same relation as shown by the flux vector OP in
the polar diagram in Fig. 38.
Polyphase Short Circuits. Alternator Armature and
Field Transients. Consider a three-phase alternator carry-
ing a constant balanced load of constant power factor.
The three-phase currents flowing in the armature produce
a resultant constant armature flux or armature reaction.
FIG. 37. Starting magnetic flux transient from Fig. 36. Rectangular coordi-
nates.
With respect to the field the armature flux is stationary but
with respect to any diameter of the armature taken as a
reference axis, the armature currents produce a constant
rotating field of the same nature as the constant rotating
field of a three-phase induction motor. For a machine in
which the field is on the rotating spider while the armature
is stationary, the resultant flux producing armature
reaction rotates synchronously with the field. For an
alternator with the field stationary and the armature
rotating the resultant constant flux rotates at the same
ALTERNATING CURRENTS
51
speed but in direction opposite to the rotation of the arma-
ture and therefore is stationary with respect to the frame
of the machine. In either case the armature flux or the
armature reaction is stationary, if referred to the alternator
field, but is a synchronously rotating field with respect to
the armature.
FIG. 38. Starting a rotating magnetic field. Polar vector diagram of magnetic
flux for three cycles of Fig. 36.
Due to the close proximity of the armature conductors
to the field poles a large part of the magnetic flux produced
by the armature currents passes through the field magnetic
circuit. This causes a reduction in the field flux and
therefore in the amount of energy stored magnetically by
the exciting current in the field. Hence, although a con-
stant direct-current voltage is impressed on the field circuit
the useful flux is greatly reduced by the armature reaction,
and as a consequence the generated armature voltage
decreases in the same proportion. To effect any change in
the amount of energy stored magnetically takes time and
therefore the interaction of the armature flux with the field
magnetic circuit produces electric transients.
During the transition period following the instant the
short circuit occurs, two distinct causes are therefore
superimposed in producing transient phenomena in the
52
ELECTRIC TRANSIENTS
interlinked electric and magnetic circuits of polyphase
alternators.
(a) The armature transient which is equivalent to the
starting transient of a rotating magnetic field including full
frequency pulsations as illustrated in Figs. 38, 39.
(b) A field transient due to the reduction of the field flux
by the armature reaction.
FIG. 39. Same data as in Fig. 38. Rectangular coordinates.
The transients produced under (a) and (b) differ in
duration, the ratio being in each case determined by the
relative time constants of the armature and field circuits.
In general the time constant in the field circuit is greater
than in the armature circuits. Large turbo-alternators
have very slow field transients as compared to the duration'
of the armature transients.
In Fig. 40 is shown an oscillogram of transients produced
by a short circuit on all three phases of a 7.5 kw., 240 volt,
60 cycle, three-phase, star-connected alternator running
idle and with 40 per cent normal field excitation. A similar
oscillogram of short circuit transients for the same machine
while carrying 50 per cent of full load is shown in Fig. 41.
ALTERNATING CURRENTS 53
As indicated in the circuit diagrams, Figs. 40 and 41,
vibrator Vi records the armature voltage across one pair of
slip rings, e a \ vibrator i> 2 , the current, i a , in one armature
circuit, and vibrator v s the field current, i f . As the short
circuit is directly across v\, the voltage e a instantly drops to
zero. The transient in the field winding is due to the com-
bined action of the starting transient of the rotating field
FIG. 40. Short circuit transients from no load. Three-phase alternator, star-
connected.
E, no load = 109 volts; 7, short circuit = 12.0 amps.; I, field = 1.25 amps.;
/ = 60 cycles.
in the armature, which produces the full frequency pulsa-
tions, and the slower field transient resulting from the
reduction of the field flux by the armature reaction. In
breaking the short circuit the field transient alone will
appear in the field winding, as shown by the oscillograms
in Figs. 42 and 43. Necessarily the transient is reversed
in direction from what is represented in Figs. 40 and 41,
when the short circuit is made. It should be noted that
breaking the armature short circuit was not instantaneous
54
ELECTRIC TRANSIENTS
a
o ^
II
ALTERNATING CURRENTS
55
O ^H
.^ CO
bC 3
a
56
ELECTRIC TRANSIENTS
ALTERNATING CURRENTS
57
as arcs formed at the switch and continued the circuit
during the time, a b, Fig. 42, approximately for % of
a cycle or J^oo f a second. During this period the
energy stored magnetically in the armature circuits was
dissipated. Much more time, over 10 complete cycles,
was required to restore full excitation in the field poles.
FIG. 44. Armature current transients. Short-circuit on three-phase, star-con-
nected alternator, no load.
E = 280 volts; /, short circuit = 28.7 amps.; 7, field = 3.3 amps.; / = 59
cycles.
The direct relation of the voltage generated in the arma-
ture to the variable useful field flux is shown by the voltage
wave, e a , and the field transient, i f , in Fig. 42. When the
short circuit is made the same transients occur, but reversed
in time, as is evident from oscillograms in Figs. 40, 41, 44
and 45.
In the operation of alternators the relative value of the
initial or momentary to the final or permanent short circuit
currents is of great importance. At any instant the short
circuit current obeys Ohm's law, that is in magnitude it
58 ELECTRIC TRANSIENTS
will be directly as the voltage generated and inversely as
the impedance of the armature circuit.
ia = ^ --~^-^-'~ x - ( 102 )
Since the armature resistance, R a , is small compared to
the armature reactance L x a , equation (102) may be written
as in (103).
ia =- ~ (103)
For constant speed the generated voltage, e , is directly
proportional to the useful flux. At the instant the short
circuit occurs and the alternator carries no load, as in Fig.
40, the useful flux depends on the direct current voltage
impressed on the field winding and produces an armature
voltage, O e a , and hence the initial or momentary value of
the short circuit current,
O ia = -?- a - (104)
joX a
If expressed in effective values as if the current sine wave
continued at the initial magnitude,
./. = -- (105)
ifi a
During the transient period following the short circuit
the armature reaction reduces the field flux and as a conse-
quence the voltage generated in the armature decreases
in the same ratio. With the expiration of the field tran-
sient the useful flux, <j> u , is constant and hence the gener-
ated voltage, E a , and the armature current, I a , are constant
or have permanent values.
Ia == (106)
I a E a $ M _ field excitation armature reaction
Ia E a & u field excitation
(107)
Although the decrease in the armature current from its
initial to the permanent value, as shown in Figs. 44 and 45,
A L TERN A TING C URREN TS
59
is due to a reduction in the useful field flux and hence in the
generated armature voltage it is customary to consider the
voltage constant and ascribe the change to a fictitious
increase in the reactance of the armature circuit. The
combined effect of the armature reaction and the true arma-
ture reactance is represented by the so-called synchronous
reactance s x a .
FIG. 45. Armature current transients. Short-circuit on three-phase, star-
connected alternator, 50 per cent, of full load.
The permanent short circuit current may therefore be
expressed by equation (108) and the ratio of the permanent
to the initial or momentary values by (109)
Let : I a = permanent short circuit armature current.
I a = initial short circuit armature current.
L x a = armature reactance.
s x a = synchronous reactance = armature reactance
+ armature reaction.
T a
a ~ r
S'^ffl
J- a L %a
a* a sX a
(108)
(109)
60 ELECTRIC TRANSIENTS
If it be assumed that the permeability of the magnetic
circuits remains constant for the changes in flux density,
the field current-time curve may also be expressed in the
form of an equation in terms of the circuit constants and
the initial value of the transients.
Let, R f = resistance of field circuit.
L f = inductance of field circuit.
R a = resistance of armature circuit.
L a = inductance of armature circuit.
t= time from the instant short circuit occurs.
oo = 27r/; / = frequency in cycles per second.
i/= instantaneous value of field current.
//= permanent value of field exciting current
before short circuit occurs.
i'/= instantaneous value of current in field circuit
due to field transients.
/'/ initial value of i' f .
i' a f= instantaneous value of current in field circuits
due to armature transient.
I' af= initial value of i' ' af
(110)
- R *t
i' af = r af sin (0 (111)
In Figs. 42, 43:
if=I f - i' s = If - r f e J (112)
In Figs. 40, 41:
i af = i f +i' f + i' af -.= J + rr^V/'e'^'sinarf (113)
Short circuit currents, particularly under normal field
excitation, produce so great changes in flux density that
the permeability is not constant and hence L a and L f are
not constant. The purpose of the equation is however,
merely to state in concise form the factors involved without
taking into consideration the complications due to
variations in the permeability of the magnetic circuits.
ALTERNATING CURRENTS
61
While short circuits produce electric transients of greater
magnitude than the changes that occur during normal
operation of alternators, it should be kept in mind that any
modification in the armature currents, as, for example, an
increase or decrease in the load, produces transients having
the same characteristics as those produced by short circuits.
Any change in the amount of energy stored magnetically
in the armature or field circuits requires time and during
the period of readjustment electric transients are produced
in the interlinked electric and magnetic circuits.
FIG. 46. Short circuit transients, single phase alternator. Symmetrical. No
load.
E = 108 volts; 7, short circuit = 10.4 amps.; /, field = 2.6 amps.; / = 60
cycles.
Single -phase Short Circuits. Alternator Armature and
Field Transients. In polyphase alternators the permanent
armature field produced by the balanced armature currents,
and hence the armature reaction, is constant in value and,
with respect to the alternator field poles, fixed in position.
In single-phase alternators the magnetic field produced by
the armature currents, and therefore the armature reaction,
62 ELECTRIC TRANSIENTS
pulsates synchronously with the armature rotation. The
pulsations of the armature reaction necessarily appear in
the field circuit. As the armature rotates 180 electrical
degrees for each half cycle of the armature current, the
pulsations of the armature reaction with respect to the
field poles will have double the frequency of the armature
currents. Therefore, the field current has a permanent
double frequency pulsation as shown in Figs. 46 and 47.
FIG. 47. Short circuit transients, single phase alternator. Symmetrical. Load.
E, load = 106 volts; /, load = 16.83 amps.; /, short circuit = 21.5 amps.;
7, field = 2.6 amps.; / = 60 cycles.
Since the armature reaction is pulsating and not constant,
as in polyphase alternators, the initial value of the starting
transient of the armature flux will depend on the point on
the current wave at which the short circuit occurs. Thus
in Figs. 46 and 47 the short circuiting switch closed nearly
at the instant the armature current was zero and hence
only a very small armature transient was produced. With
the armature transient absent the field current-time oscit-
lograms, as illustrated in Figs. 46 and 47, are symmetrical
ALTERNATING CURRENTS 63
showing the permanent double frequency pulsations of the
armature reaction superimposed on the field transient.
If the short circuit occurs at other than the zero points
on the armature current wave, an armature transient of full
frequency is produced for the same reason as explained for
short circuits in polyphase alternators. The oscillograms
in Figs. 48 and 49 show the asymmetrical field current-
FIG. 48. Short circuit_transients, single phase alternator. Asymmetrical. No
load.
E = 57 volts; /, short circuit = 23.0 amps.; I, field = 1.3 amps.; / = 60
cycles.
time curves on which are superimposed the double fre-
quency permanent armature reaction, the field transient,
and the full frequency pulsation produced by the armature
transient. The combination of the full frequency arma-
ture transient pulsation with the permanent double
frequency armature reaction produces the asymmetry
in the curves. The ordinates for the odd numbers of the
double frequency waves add to the full frequency values,
while for the even number of waves the difference in the
ordinates produces the wave recorded by the oscillograph.
64
ELECTRIC TRANSIENTS
ALTERNATING CURRENTS 65
Hence, during the transition period the peaks of the odd
numbered waves decrease, while the even numbered peaks
increase, and at the expiration of the armature transient
all reach the permanent constant pulsation produced by
the pulsating armature reaction. While the field current
pulsates as a result of the double frequencyarmature reac-
tion and the full frequency armature transients, the voltage
across the field terminals will pulsate to a greater or less
degree depending on the amount of external resistance and
inductance in series with the field circuit. With much ex-
ternal resistance or impedance the voltage at the terminals
of the field winding may reach high values which may
puncture the insulation and cause a short circuit in the
field exciting circuit.
The field transient separated from the armature reaction
may be shown by taking an oscillogram of the field current
when the short circuit on the single phase alternator is
broken, as shown in Figs. 50 and 51. The armature tran-
sient is dissipated during the opening of the switch, indi-
cated by the time a b on the oscillogram, while several
complete cycles are required before the field flux, and as a
consequence the armature voltage, regains its full value.
In the transition period following the closing or opening of
the short circuiting switch the oscillograms of the field
currents show the effects of the energy changes taking
place in both the field and armature circuits.
Under the assumption that the permeability of the mag-
netic circuits is constant the field-current-time curve in
Figs. 46 to 51 may be expressed in terms of the circuit
constants and the initial values of the transients :
Let: R f = resistance of field circuit.
L f = inductance of field circuit.
R a = resistance of armature circuit.
L a = inductance of armature circuit.
t = time from the instant short circuit occurs.
co = 27r/; / = frequency in cycles per second.
if = instantaneous value of field current.
66
ELECTRIC TRANSIENTS
ALTERNATING CURRENTS
67
J3 05
a- 1
83
68 ELECTRIC TRANSIENTS
I f = permanent value of exciting current before
short circuit occurs.
i' f = instantaneous value of current in field circuit
due to field transient.
/'/ = initial value of i f /.
i a f = instantaneous value of current in field circuit
due to armature reaction.
la/ = maximum value of i a /.
i f af = instantaneous value of current in field circuit
due to armature transient.
Iaf = maximum initial value of i' af .
71 = phase angle of i af .
72 = phase angle of i' /.
iaf = M I a /' sin (2 cot 71) (115)
i
In Fig. 50:
_R O(
'af = "I'af6 La sin (ut - 72) (116)
-i
i, = // - 7/e L ' (117)
In Fig. 46:
if ---- If + r f * L ' + "J a/ sin (2^ - 71) (H8)
In Fig. 48:
- Rf t
i f = I f + r f e L! + M I af sin (2wf -- 71)
t
-7a) (119)
- R "t
As indicated by the difference in the upper and lower
halves of the double frequency pulsation the permeability
of the magnetic circuit changed with the flux density.
Under full field excitation the short circuit transients
would produce much greater changes in the flux density
and hence in the permeability of the steel in the armature
and field poles. For this reason the equations are not
directly applicable to commercial problems but state the
ALTERNATING CURRENTS
69
relations of the factors involved provided the permeability
of the iron core is constant.
Single -phase Short Circuit on Polyphase Alternators.
If all phases of polyphase alternators are short circuited
simultaneously the armature transients appear in the field
circuit as full frequency pulsations produced by the rotating
magnetic field, as illustrated for three-phase machines in
Figs. 40 to 43.
FIG. 52. Single phase short circuit on three phase alternator.
If one phase only is short circuited the effect on the field
circuit is essentially the same as illustrated for single phase
alternators in Figs. 46 to 48. In Fig. 52 is shown the
transient of the field current of a three-phase alternator
after short circuiting one phase. The field current-time
curve shows the effects produced by the field and armature
transients and the permanent double frequency pulsations
due to the armature reaction. In Fig. 53 is shown an
oscillogram for a single-phase short circuit on a three-phase
alternator which after 4 cycles is followed by a short circuit
70
ELECTRIC TRANSIENTS
on all three phases. While only one phase is short circuited
the field current shows the double frequency pulsations
combined with both the armature and field transients.
After the three-phase short circuit occurs the field current
shows the full frequency pulsations of the armature tran-
sient combined with the slower field transient.
FIG. 53. Single phase short circuit on three phase alternator followed by a three
phase short circuit.
E = 118 volts; 7, load =10 amps.; /, short circuit = 22.5 amps.; I, field =
2.6 amps.
Oscillograms of transients in polyphase systems produced
by single-phase short circuits necessarily differ with the
type of machine and the way the transient magnetic fluxes
interlink with the electric circuit to which the oscillograph
vibrator is connected. Thus in Fig. 54 the open phase
voltage of a two-phase alternator with short circuit on one
phase shows a triple frequency harmonic, while the field
current shows the double frequency pulsation combined
with the field and armature transients of the same charac-
teristics as for single-phase alternators.
ALTERNATING CURRENTS
71
72
ELECTRIC TRANSIENTS
Problems and Experiments
1. Let the sine wave curve in Fig. 55 represent the 60 cycle alternating
current that would flow in a circuit having 3.0 ohms resistance, 0.05 henrys
inductance for a given voltage.
(a) Let the switch impressing the voltage on the circuit be closed at the
instant marked (a) in the diagram. Draw in rectangular coordinates:
1. The permanent current sine wave as in Fig. 55.
2. The starting transient.
3. The actual current flowing in the circuit during the first ^ second
after the switch is closed.
\
FIG. 55. Single phase current, sine wave, 60 cycle starting transient.
(6) Similar to (a) except the voltage is impressed at the instant marked
(6).
2. In a circuit having 60 ohms resistance and 0.045 henrys inductance a
25 cycle current is flowing, as represented by the sine wave on the left side
in Fig. 56. At the instant marked (a) the impressed voltage is suddenly
changed so that it will produce a permanent 60 cycle current shown by the
dotted line sine wave in the figure.
(a) Draw in rectangular coordinates:
1. The sine current waves as in Fig. 56.
2. The starting transient.
3. The actual 60 cycle current for the first ^lo second after the voltage
was changed.
(6) Same as (a), except the change is made at some other point along the
time axis.
3. Take an oscillogram of the starting current in a circuit of known
resistance and inductance. Calculate the starting transient and draw it
on the oscillogram. Check by combining the ordinates for the actual current
ALTERNATING CURRENTS
73
recorded by the oscillograph with the corresponding values of the calcu-
lated transient and compare the resulting curve with the permanent cur-
rent sine wave.
4. Let the sine waves in Fig. 57 represent the permanent value of the
currents flowing in a balanced three-phase system, whose time constant is
M>,ooo f & second.
FIG. 56. Single phase current, sine wave, 25 cycles to 60 cycles transient.
FIG. 57. Three phase current, sine wave, 60 cycle starting transients.
(a) Let the voltage be impressed at the instant marked (a). Draw in
rectangular coordinates :
1. The permanent current sine waves as in Fig. 57.
2. The starting transients for the three phases.
74 ELECTRIC TRANSIENTS
3. The actual currents as would be recorded by an oscillograph
if a vibrator was connected to each of the three phases so as to
record the current-time curves.
(6) Same as (a), except the voltage is impressed at the instant marked (6).
6. Take an oscillogram of the starting currents in a three-phase system
connecting the vibrators as in the circuit diagram in Fig. 34. From the
oscillogram and the circuit constants plot the starting transients and check
with the permanent current waves as explained in Prob. 3.
6. Make oscillograms similar to Figs. 40 and 42 or 41 and 43. Obtain
the necessary data to draw the scale in amperes or volts for each vibrator.
A circuit diagram showing the position of each vibrator should be attached
to each film.
7. Make oscillograms similar to Figs. 46 and 48 or 47 and 49. Quanti-
tative data should be obtained for each vibrator and for the circuit constants.
8. Make an oscillogram similar to Fig. 53.
CHAPTER V
DOUBLE ENERGY TRANSIENTS
Single energy transients occur in electric circuits or
other apparatus in which energy can be stored in only one
form. Any change in the amount of energy stored produces
transients and whether the stored energy is decreased or
increased the transient itself is a decreasing function with
its maximum value at the first instant. In magnetic,
electric and dielectric circuits in which the resistance,
inductance and condensance are constant during the transi-
tion period, single energy transients may be expressed by
the exponential equation as discussed in Chaps. Ill and IV.
In apparatus having two forms of energy storage as a
pendulum or electric circuits having both inductance and
condensance, a series of oscillations may take place by
which the energy is transferred from one form to the other,
while the dissipation of the stored energy into heat proceeds
in much the same manner as in single energy systems.
Thus a pendulum, freely suspended in air, will swing back
and forth over arcs of decreasing amplitude, with energy
changing from the kinetic to the potential form and back
to the kinetic twice for each cycle. The amplitude of each
swing is less than for the one preceding since part of the
energy has been dissipated into heat by friction during the
intervening time. The pendulum comes to rest when all
the stored energy is dissipated into heat.
In electric circuits having both dielectric and magnetic
storage facilities the energy stored in one form may change
to the other and back and forth in a series of oscillations
of definite frequency. This is illustrated by the oscillogram
in Fig. 58. The energy stored in a condenser is discharged
through a resistance in series with an inductance. In
75
76
ELECTRIC TRANSIENTS
o <
iO ^
DOUBLE ENERGY TRANSIENTS 77
passing from the dielectric field to the magnetic field or
the reverse, the energy goes through the resistance and a
part is dissipated by the Ri 2 losses. Hence the amplitude
of each oscillation is less than for the one preceding. By
referring to the timing wave on the oscillogram, Fig. 58,
it is found that the frequency of oscillation was 1,070 cycles
per second, and that practically all the energy was dissi-
pated into heat by the Ri 2 losses in 50 cycles, or approxi-
mately Ho of a second.
Surge or Natural Impedance and Admittance. If no
energy is dissipated during the transfer the stored energy in
the dielectric field when the voltage is a maximum must be
equal to the quantity stored in the magnetic field when the
current is a maximum. Hence from (7) and (16)
C f L f (120)
Therefore, from (120)
M j = ^\~ = n z< the surge or natural impedance of
the circuit (121)
"I 1C
fY, = A/7 n y, the surge or natural admittance of
rj \L
the circuit (122)
The quantity, \/L/VC, is in the nature of an impedance
and is called the surge or natural impedance of the circuit,
and its reciprocal, \/C/\/L, the natural or surge admit-
tance of the circuit.
Frequency of Oscillation in Simple Double Energy
Circuits. Consider circuits "a" and "6" in Fig. 59. Let
the inductance, L, the condensance, C, and the resistance,
/2,'be constant and of the same value in the two cases. Let
an alternating current voltage be impressed on the ter-
minals and let the frequency be varied until the current is
in phase with the voltage at the terminals. All the energy
absorbed by the Ri 2 losses is supplied from the a.c. mains.
78
ELECTRIC TRANSIENTS
In circuit (a) under the given conditions:
J L % jcX =
Hence,
Likewise for circuit (b)
j c b - j L b =
Hence,
(121)
(122)
(123)
(124)
(125)
(126)
The expressions in equations (123) and (126) are generally
used to determine the "resonance frequency" of the cir-
(a) (b)
FIG. 59. Simple series and parallel double energy circuits.
cuits. As shown in Chap. VIII a strict application of the
definition for resonance gives a different value for the true
resonance frequency unless the resistance is negligible.
If all the resistance were removed from the circuits in
Fig. 59 no energy would be supplied from the bus bars and
the stored energy would be transferred back and forth
between the inductance and the condensance. With no
losses the frequency of the natural or free oscillations
would be the same as the " resonance frequency" given in
equations (123) and (126).
DOUBLE ENERGY TRANSIENTS
79
In circuit a, Fig. 60, and for the oscillograms in Figs.
61 to 65 the condenser is charged from a direct current
supply main after which the switch " S" is thrown to the
right so as to form an independent closed circuit with the
condenser, C, resistance, R, and inductance, L, in series.
The energy stored in the condenser is dissipated into heat
by the Ri 2 losses during a series of oscillations between the
dielectric and magnetic fields.
FIG. 60: Simple oscillatory double energy circuits.
From Kirchoff's Laws the voltage in the closed circuit,
Figs. 60 to 64, while the energy originally stored in the
condenser is dissipated into heat, is expressed by equations
(127) or (128).
= (127)
dt* + R di + (J = (128)
This is a homogeneous differential equation of the second
order and its general solution is given by equation (129),
in which A\ and A 2 are the arbitrary constants.
(129)
(130)
In equation (129)
B
LC
R
LC
(131)
80
ELECTRIC TRANSIENTS
FIG. 61. Double energy transient.
E = 120 volts; R = 40 ohms; G = 0; L = 0.205 henrys; C
farads; timing wave 100 cycles.
0.813 mtero-
FIG. 62. Double energy transient.
E = 120 volts; R = 75 ohms; G = 0; L = 0.205 henrys; C = 0.873 micro-
farads; timing wave 100 cycles.
DOUBLE ENERGY TRANSIENTS
81
FIG. 63. Double energy transient.
E = 120 volts; R = 150 ohms; G = 0; L = 0.205 henrys; C = 0.813 micro-
farads; timing wave 100 cycles.
FIG. 64. Double energy transient.
E = 700 volts; R = 770 ohms; G = 0; L = 0.205 henrys; C
farads; timing wave 100 cycles
6
0.813 micro-
82 ELECTRIC TRANSIENTS
In order to more readily keep the dissipation or damping
factors separate from the parts indicating oscillations,
equations (130), (131) are rewritten in (132), (133):
R
Ul = ~ n r +
R . I I
-- 3 -
From (129), (132), (133):
_ 2" \ orr/ 2 " , _ m _ ;\i L T ~& t
i = An 2L e 3 \LC 4L'+^ ae 2L e 3 \LC 4L f (134)
But from Euler's equation for the sine and cosine:
v/JL _ *!
\LC 4Li = ; " = ^ + j sin co^ (135)
rf - j sin ut (136)
Hence from (134), (135), (136):
_Rt _Rt
i = Aie ^[cos ut +' sin ut] + A 2 e 2L
n (137)
[cos ut j sin at]
From (135), (136):
co == ^f=^- C ~f^ ( 138 )
Hence,
/ = 2W/!c- (139)
In circuits for which the quantity under the radical is
real, oscillations occur at a definite frequency as determined
by equation (139) and as illustrated by the oscillograms in
Figs. 61 to 63.
If the resistance,
. R >2 J5 (140)
DOUBLE ENERGY TRANSIENTS 83
the quantity under the radical sign in (139) becomes imagin-
ary, and hence the circuit is non-oscillatory. All the
energy initially stored in the condenser is dissipated into
heat as the voltage and current decrease to zero. This
condition is illustrated by the oscillograms in Figs. 64 and
65.
For circuits having comparatively little resistance the
naturalfrejjuency of oscillations, as given in equation (139),
is ver$; Dearly the same as the " resonance frequency"
given by equations (123) or (126). Thus for the circuit
data in Fig. 62 the nafuFal frequency of oscillation, using
equation (139), is given in equation (141), while the cor-
responding " resonance frequency" from equation (126),
is given in equation (142).
1 / 1
ir\LC "
4/ 2 cycles per second
f = n /T ^ = 391 cycles per second (142)
For circuits corresponding to the conditions that would
exist if the condenser in Fig. 606 were leaky, similar equa-
tions may be obtained. The voltage equation, based on
KirchofPs Laws for circuits of the type shown in Fig. 606,
and in Figs. 66 to 70, is expressed by equations (143) and
(149).
L % + *$ + c = (143)
Using the notation shown in the circuit diagram, Fig.
606, letting c e be the voltage across the condenser terminals,
and applying Ohm's and KirchofPs Laws.
d = i + O i (144)
d = G L e (145)
dj
c e = Ri'+L (146)
Hence,
di
c i = i + GRi +GL (147)
84
ELECTRIC TRANSIENTS
DOUBLE ENERGY TRANSIENTS 85
From (143) and (147),
LC d jl + (RC + GL)~.+ (1 + GK)i = (148)
at ctt
or,
M ' '*<** '^-M = (149)
Equation (149) is a homogeneous differential equation
of the second order of the same form as equation (133).
Hence, the same general solution applies to both equations,
as expressed by equation (150), in which B } and B 2 are the
two arbitrary constants.
+ B 2 e v * (150)
* - -
Rewriting (151), (152) so as to more clearly indicate the
damping and oscillation factors, equations (153), (154) are
obtained.
From (150), (153), (154),
~\(L + c)' - y "N/Lc - KL - ?)' (155)
From Euler's equation,
_ _
C 4- c< = e ^ = cos ut + sin
V_L _ i/j? _ o\
c LC iU e^ = e wf = cos ut _j s in ^ (157)
Hence,
[cos cot + j sin cotj
^ 2e 2^L c/ r cos ^1 j sin
86
ELECTRIC TRANSIENTS
FIG. 66. Double energy transient.
~E = 625 volts; R = 4.5 ohms; G = 1.67 10~ 4 mhos.; L = 0.205 henrys; C
0.813 microfarads; timing wave 100 ovnles.
FIG. 67. Double energy transient.
E = 640 volts; R = 4,.5 ohms; G =3.33 10" 4 mhos; L = 0.205 henrys; C
0.313 microfarads; timing wave 100 cycles.
DOUBLE ENERGY TRANSIENTS
87
From (156), (157),
' = 2 & -
Zir\LL 4\
fi
L
G
(159)
(160)
(161)
The circuit is non-oscillatory if
R _G _2_
L C > \LC
For circuits in which the quantity under the radical sign
is greater than zero, the energy in the condenser will be
dissipated into heat during a series of oscillations of definite
frequency as determined by equation (160) and as illus-
trated by the oscillograms in Figs. 66, 67 and 68.
FIG. 68. Double energy transient.
E = 625 volts; R = 4.5 ohms; G = 6.66 10~ 4 mhos; L = 0.205 henrys;
C = 0.813 microfarads; timing wave 100 cycles.
If the resistance and the conductance are of such values
relatively to the inductance and the condensance that the
quantity under the radical sign in (160) becomes imaginary,
88
ELECTRIC TRANSIENTS
FIG. 69. Double energy transient.
E = 550 volts; R = 4.5 ohms; G = 1.67 10~ 3 mhos;
C = 0.813 microfarads; timing wave 100 cycles.
0.205 henrys;
FIG. 70. Double energy transient.
E = 550 volts; R = 4.5 ohms; G = 4.4 110~ 3 mhos;
C = 0.813 microfarads; timing wave 100 cycles.
L = 0.205 henrys;
DOUBLE ENERGY TRANSIENTS 89
the circuit would be non-oscillatory. The energy initially
stored in the condenser is dissipated into heat while the
voltage and current decrease to zero, as illustrated by the
oscillograms in Figs. 69 to 70.
The circuits in which the resistance and conductance
are comparatively small the natural frequency of oscilla-
tion is very nearly the same as the resonance frequency or
the natural frequency of circuits in which R and G are
equal to zero. Thus for the circuit in Fig. 67 the natural
oscillation frequency,
f = ~ := 391 CyCl6S PCT S6COnd (162)
Considering R and G as negligible in determining the
frequency of oscillation,
/ = - 7 = 391 cycles per second (163)
2ir\/LC
A very interesting circumstance is revealed by equation
(160). A circuit having resistance greater than the critical
value for oscillatory discharges as given in (161), may be
made oscillatory by increasing the conductance across the
terminals of the condenser without changing the resistance.
This is illustrated by the oscillograms in Figs. 71, 72, 73
and 74.
For the given circuit constants in Fig. 71 the circuit is
non-oscillatory. Letting R, L and C remain constant and
of the same value as in Fig. 71 but increasing the conduc-
tance, G, the circuit is made oscillatory in Fig. 72 although the
damping factor is greater than for the circuit in Fig. 71.
In Fig. 73 the oscillation was greatly reduced and by still
further increasing the conductance while R, L and C remain
constant, the circuit is again made non-oscillatory as shown
by the oscillogram in Fig. 74.
Dissipation Constant and Damping Factor in Simple
Double Energy Circuits. In the solution for the current in
double energy circuits, Fig. 60a and Figs. 61 to 65, as
given in equation (137), the damping factor and the dissi-
90
ELECTRIC TRANSIENTS
FIG. 71. Double energy transients.
E = 700 volts; R = 150 ohms; G = mhos; L = 0.205 henry
microfarads; timing wave 100 cycles.
C = 36
FIG. 72. Double energy transients.
E = 700 volts; R = 150 -ohms; G = 4.35 X 10~ 3 mhos; L
C = 36 microfarads; timing wave 100 cycles.
0.205 henry s;
DOUBLE ENERGY TRANSIENTS
91
FIG. 73. Double energy transients.
E = 400 volts; R = 150 ohms; G = 1.31 10~ 2 mhos; L = 0.205 henrys
C = 18 microfarads; timing wave 100 cycles.
FIG. 74. Double energy transients.
E = 700 volts; R = 150 ohms; G = 2.63 10~ 2 mhos;
C = 36 microfarads; timing wave 100 cycles.
0.205 henrys;
92 ELECTRIC TRANSIENTS
pation constant have already been found. Similarly
for circuits in Fig. 606 and in Figs. 66 and 67, the factors
may be obtained from equation (158)
Dissipation or damping constant = J(T + f )
Damping factor = ~2\L + c) i (1(35)
While the above expressions are obtained mathematic-
ally by the solution of the differential equation of the cir-
cuit, it is important that the student gain a clear concept of
the physical phenomena involved.
In Chap. Ill it was shown that for single energy tran-
sient in circuits having resistance and inductance in series
the time constant is directly proportional to the inductance
and inversely to the resistance.
,T 1 -- ^ (166)
Similarly for circuits having" condensance in parallel
with conductance, the time constant is directly proportional
to the condensance and inversely to the conductance.
C T, -- g (167)
In double energy circuits the energy is alternately stored
in the magnetic and dielectric fields. In circuits having
inductance, resistance, condensance and conductance,
arranged as shown in the circuit diagrams in Figs. 66 to 70,
energy is dissipated into heat both in the resistance and in
the conductance. The rate of dissipation is greatest in
the conductance, (re 2 , when the voltage across the condenser
is a maximum, that is, at the instant all the energy is
stored in the dielectric field. Similarly the rate of dissipa-
tion in the resistance, Ri 2 , is a maximum, when the current
is a maximum, that is, when all the energy is stored in the
magnetic field. It is evident that since the energy is
oscillating it will be in the dielectric field half of the time
and in the magnetic field half of the time. Since the energy
DOUBLE ENERGY TRANSIENTS 93
is in the dielectric form only half the actual time, the
rate of dissipation in the conductance will be equal to half
of what would be the case for the same circuit constants
in the corresponding single energy transient. Hence, the
time constant, C T^ for the dielectric half of the double energy
circuit would be twice the time constant, c Ti, in the corre-
sponding single energy transient.
9C
J\ = 2 o r, = ^T (168)
(jT
Similarly the time constant, ,7%, for the inductance-
resistance part of the double energy circuit would be twice
the time constant, L Ti, for the corresponding single energy
transient.
or
,T 2 = 2 L T 1 = - (169)
K
Under the given circuit conditions with R, L, G and C
constant, the proportionality law applies to double energy
transients on the same basis as for single energy transients.
The transient term is therefore expressed by the exponential
equation and appears as a factor in the current-time and
voltage-time equations and represents the dissipation of
energy into heat by the resistance, Ri 2 , and the conduc-
tance, Ce 2 , in the circuit.
Let u represent the dissipation constant of double energy
circuits. The damping factor is therefore,
1 l - R t --t
ut = ,T> C T 2 = 2L 2G
c (17!)
\/R . G\ /ihr<n\
-2(1 + c) (172)
This is the same value as obtained in equation (158).
For circuits in which G = 0, as illustrated by Figs. 61
to 65, the term G/C would be zero.
94 ELECTRIC TRANSIENTS
Hence,
u' -g- (173)
- R t
e u't = e 2L Q74)
This corresponds to the value of the damping factor in
equation (137).
For circuits similar to Figs. 66 to 70 but in which R = 0,
the term R/L would be zero.
Hence,
u" = - c (175)
_G_ t
e u"t = 2c (175)
As it is not possible to completely eliminate the resis-
tance in circuits having inductance, the conditions for u"
can not be fully realized experimentally.
Equations for Current and Voltage Transients. For
simple double energy circuits, with R, L and C in series,
as in Fig. 60a, the general equation (137) for the current is,
_Rt
i = Aie 2Z/ [cos cot + j sin cofl
_ Rt
+ A 2 e~ 2L [cos cot j sin co(] (177)
In equation (177) A\ and A 2 are the arbitrary constants,
which for any specific case are determined by the perman-
ent circuit conditions preceding and following the transient
period. Equation (177) may be written in a more compact
form as in (178), in which A 3 and A 4 are the arbitrary
constants which for any specific case may be determined
from the given limiting conditions under which the tran-
sient occurred.
_Rt
i = e 2L [A 3 cos ut + A sin co] (178)
The voltage across the terminals of the condenser,
c e = -Ri - L (179)
DOUBLE ENERGY TRANSIENTS 95
From (178), (179),
c e = e 2L I rt~[A 3 cos co + A 4 sin co]
+ wL[A 4 cosco^ A 3 sincofl (180)
For the transients in Figs. 61 to 65 the starting conditions
are:
t = 0; i = 0; c e == # (181)
From (178), (180), (181),
Hence,
p -* 1
L sin w< (183)
CO-L
Rt 7?
2L [cos co^ + ^ sin co] (184)
T>
For the given circuit constants, ^ j is very small and
hence,
_Rt
c e = Ee 2L sin ut (very nearly) (185)
To illustrate the application of equations (183) and (185)
for the solution of numerical problems, equations (186)
and (187) give the value of the current and voltage in
amperes and volts for the oscillograms in Fig. 62.
i = -0.24 ~ 182 "sin (170760*) amperes (186)
e =-. 120. e~ 182 'cos (1707600 volts (187)
For the circuit in Fig. 606 the equations are of a similar
nature. The general equation (158) for the current is
given in (188) and may be written in a more compact
form as in equation (189), in which B s and B 4 are constants
that in each case depend on the permanent circuit condi-
tions preceding and following the transient period.
[cos coZ + j sin <*t]
2L c
[cos ut - j-sin.ut] (188)
96 ELECTRIC TRANSIENTS
- t
i = 2 \ L c > [B 3 cos ut + B 4 sin <*t] (189)
c e = -Ri _ L-- (190)
di
--
From (188) and (190),
cos ut + ^ 4 sin
+ <o[/?4 cos cot J5 3 sin ool (191)
For the transients in Figs. 66 to 70 the initial conditions
are:
t = 0;t = 0; c e = E (192)
From (189), (191) and (192),
3 = 0; B, = Jr (193)
Hence,
i=- 4 '^* + ^'sin (194)
(195)
If in equation (195),
_ ,
c e = Ee 2\L + c) cos ut (197)
and
(198)
As an illustration of the application of equations (194)
and (197) to the solution of a specific problem, let the
numerical values of the circuit constants in Fig. 67 be
used. Equations (199) and (200) give the instantaneous
values of the current and voltage for the oscillogram in
Fig. 67.
_ 9 1 r, i
i=-l.28e' sin (170760*) amperes (199)
c e = 640 e' cos (1707600 volts (200)
DOUBLE ENERGY TRANSIENTS
97
o
o
LO^
II J
^'E
fi
O c3
2 3
o
w> S
03
98
ELECTRIC TRANSIENTS
FIG. 76. Starting current and voltage transients.
E = 125 volts; R = 5.0 ohms;(? = 0.00167 mhos;L = 0.205 henrys; C = 9.0
microfarads; timing wave 100 cycles.
FIG. 77.- Starting current and voltage transients.
E = 125 volts; R = 5.0 ohms; G = 0.0025 mhos; L = 0.205 henrys; C
microfarads; timing wave 100 cycles.
9.0
DOUBLE ENERGY TRANSIENTS
99
FIG. 78. Starting current and voltage transients.
E = 125 volts; R = 5.0 ohms; G = 0.005 mhos;L = 0.205 henrys; C
microfarads; timing wave 100 cycles.
FIG. 79. Starting current and voltage transients.
E = 125 volts; R = 5.0 ohms; G = 0.0132 mhos; L = 0.205 henrys; C = 9.0
microfarads; timing wave 100 cycles.
100 ELECTRIC TRANSIENTS
The oscillogram in Fig. 74 shows the current and voltage
transients in a circuit having a high damping factor but
in which the frequency of oscillation is the same as if both
R and G w r ere zero. The data in Fig. 74 show that the
circuit constants were of such values as to satisfy equation
(196).
For the oscillograms in Figs. 75 to 79 the circuits are of
the same type as in Figs. 66 to 70, but the permanent con-
ditions preceding and following the transition period are
different. The oscillograms show the starting current and
voltage transients at the points in the circuit indicated by
the positions of the vibrators in the circuit diagram and for
the values of R, L, G and C, as given in each figure.
Problems and Experiments
1. Given a circuit similar to Fig. 60 (a) having R, L, and C in series. Let
R = 20 ohms, L = 0.31 henrys, C = 1.2 microfarads and E = 120 volts,
the initial condenser discharge voltage.
(a) Find the natural period of oscillation of the circuit.
(6) Find the time constant, and the damping factors.
(c) Write the equation for the transient condenser discharge current.
(d} For what values of R would the circuit be non-oscillatory.
2. Derive the equations for e e, the transient voltage across the condenser
terminals in Fig. 62. Trace the voltage-time curve for c e on rectangular
coordinates, using the same time scale on the X axis as in the oscillogram.
3. Take a double energy oscillogram similar to Fig. 58. Obtain all the
necessary data and write the equations for the transient current.
4. Write the equations for the voltage and current curves of the oscillo-
gram in Fig. 61 similar to equations (199) and (200) for Fig. 67 in the text.
5. Take a series of oscillograms similar to Figs. 66 to 70. Find the values
of the circuit constants and place on the film ampere and volt scales for the
current and voltage curves.
6. For the oscillogram in Fig. 75 with the given circuit conditions:
(a) Write the expression for c e and i similar to equations (194), (195).
(6) Insert the numerical values of circuit constants and express c e and
i in volts and amperes, similar to equations (199) and (200).
CHAPTER VI
ELECTRIC LINE OSCILLATIONS, SURGES AND
TRAVELING WAVES
Electric lines whether designed for poWer*tVaVL&fnissioTi or
telephone service, may be considered "as^cpnisisjifig^f 8it
infinite series of infinitesimal double energy 'Circuits o*f the
simple types discussed in Chap. V. Each infinitesimal
length of line may be represented by the resistance and
inductance in one of the series circuit elements in Fig. 80
and the corresponding portion of the dielectric between
the conductor and neutral by the conductance and con-
densance in the adjacent parallel circuit. The line con-
stants, R, L, G and (7, depend on the size and spacing of the
conductors and the electrical properties of the dielectric
and conductor materials. To readily gain clear concepts
of transmission line phenomena it is essential for the student
to conduct experiments and obtain quantitative test data.
Commercial transmission lines are seldom available for
experimental work but artificial lines having the electrical
characteristics of actual lines can be readily constructed
of convenient design for operation in the laboratory.
Artificial Electric Lines. Since the operating charac-
teristics of transmission lines are determined by the line
constants, the resistance, inductance, conductance and
condensance and are independent of the space and mass
factors, much of the experimental work can to good advan-
tage be performed on equivalent artificial electric lines. 1
The oscillograms of electric line transients used for illus-
trations in this chapter were obtained from an artificial
transmission line, 2 one section of which is shown in Fig.
1 DR. A. E. KENNELLY, "Artificial Transmission Lines."
2 Trans. A. I. E. E., vol. 31, p. 1137.
101
102
ELECTRIC TRANSIENTS
81. This line is of the lumpy "T" type of design, which
means that each unit has resistance and inductance in
series combined with condensance and conductance in
parallel as shown in Fig. 82. If the insulation is sufficiently
high the conductance factor may be omitted and the section
circuit diagram would be as in Fig. 83, which represents
the circuit diagram for the "T" unit in Fig. 81.
R L
C =tr^G C=t^G C==
R L L R
R L
FIG. 80. Transmission line circuit diagram showing three elements.
In the lumpy types the line constants R, L, G and C, are
massed instead of uniformly distributed as in actual lines.
As the lumpy type only approximates a uniform distribu-
tion of the resistance, inductance, conductance and con-
densance in the line, the size of each unit must be small in
comparison to the total length of the line. In Fig. 81
is shown one of the twenty ten-mile units in the artificial
transmission line in the electrical engineering laboratories
of the University of Washington. In each unit the line con-
stants may be adjusted within the following limits:
Resistance, minimum value, 2.59 ohms.
Inductance, maximum value, 0.021 henry.
Condensance, 0.1 to 1.0 microfarad.
The resistance may be increased to any desired amount by
moving the clamp on the resistance loop or by inserting
resistance elements between the units; the inductance
may be decreased by turning the right hand coil and by
taps in the lower coil; and the condensance may be varied
in steps by using ten or a less number of condensers in
OSCILLATIONS, SURGES AND TRAVELING WAVES 103
series. Adjustments can be made so as to give to the artifi-
cial line the electrical constants equivalent to an actual
transmission line of any size of wire up to No. 0000 A.W.G.
hard-drawn copper and for any spacing up to 120 inches.
FIG. 81. Section of artificial electric line, University of Washington.
The line may also be adjusted so as to be equivalent to
commercial telephone lines.
Time, Space and Phase Angles. In Chap. V the equations
for the current and voltage transients were derived for
104
ELECTRIC TRANSIENTS
simple double energy circuits, Fig. 60, in which the circuit
constants, R, L, G and C are massed. Evidently the
energy transfer between the magnetic and dielectric fields
would be of essentially the same nature if the inductance
and resistance were intermixed with the condensance and
conductance or uniformly distributed as in a transmission
line. However, one important difference must be noted
which necessitates an additional factor in the expression
for the transient current and voltage. In circuits having
massed circuit constants the maximum value of the voltage
f=f /?
FIG. 82. T-circuit with leaky condenser.
FIG. 83. T-circuit.
will be impressed on all of the condensance at the same
instant, and all parts of the magnetic field reach a maximum
at the instant the current is a maximum. On the other
hand, with R, L, G and C distributed, as in long transmission
lines, the time required for the electric wave to travel along
the length of the line enters into the problem. If a constant
direct current voltage is impressed at one end of an electric
line a short but definite time will elapse before the voltage
reaches the other end of the line. If an alternating current
is transmitted over the line the successive waves travel over
the line at definite velocity in the same manner as the
impulse from the direct current voltage. The maximum
point of any wave travels at a definite velocity as deter-
mined by the distribution of the resistance, inductance,
conductance and condensance in the line. In trans-
mission lines with air as the dielectric and with copper or
aluminum conductors the speed at which a wave or impulse
OSCILLATIONS, SURGES AND TRAVELING WAVES 105
travels is approximately the same as the velocity of propa-
gation of an electromagnetic wave in space or the velocity of
light.
v = 3-10 10 cm. per second (205)
In a medium having a permeability ^ and a permittivity /c,
3-10 10
v' = - -=,- cm. per second (206)
v M*
The time required for the voltage wave to travel a
distance x along the line having distributed line constants,
depends on the distance and velocity of propagation.
(207)
In comparing the transient voltage and current conditions
at any two points on an electric line, x distance apart, con-
sideration must be given to the time required for the
propagation of the electric wave over the given distance
and hence the factor t, must be included in the equations.
In double energy circuits having massed R, L, G and C,
as in the oscillograms in Figs. 66 to 69, and for oscillations
produced by the discharge of energy initially stored in the
condensers, the instantaneous values of the voltage and
current, under the stated conditions, are given in equations
(194), (197). Under similar conditions, as illustrated by
the oscillograms in Figs. 84 to 91, and by the introduction
of space angles, the equations may be considered as apply-
ing to circuits having distributed R, L, G and C, as in trans-
mission lines.
To simplify the notations, let
= K? - D
(208)
I = - E (209)
co-L
y = time phase angle for t = (210)
For oscillations in circuits with massed R, L, G and C,
under the stated assumptions:
106
ELECTRIC TRANSIENTS
e = Ee~ ut cos (ut 7)
(211)
(212)
FIG. 84. Electric line oscillations.
E = 500 volts; R = 52.14 ohms; G = 0; L = 0.427 henrys; C = 3.66 micro-
farads; length = 232 miles; timing wave 100 cycles.
& V
1C)
FIG. 85. Circuit and wave diagram for Fig. 84,
OSCILLATIONS, SURGES AND TRAVELING WAVES 107
For oscillations in circuit with distributed R, L, G and C,
under similar conditions:
FIG. 86. Electric line oscillations.
E = 500 volts; R = 26.12 ohms; # 2 = 26.02 ohms; Gi = 0; Gz = 0; Li = 0.2128
henrys; Li = 0.2146 henrys; Ci = 1.831 microfarads; Cz 1.834 microfarads;
timing wave 100 cycles.
FIG. 87. Circuit and wave diagram for Fig. 86. ~ j
i = I e -ut s i n [ w ( t _ tl ) - y] (213)
e = Ee~ ut cos [u(t - ti) -- y] (214)
108
Substituting
ELECTRIC TRANSIENTS
c for coi :
= e-ut sin (o> -- 0x 7) (215)
e = Ee~ ut cos (ut - <$>x - 7) (216)
In equations (215), (216) ut is the time angle, <f)X the
space angle and 7 the phase angle.
FIG. 88. Electric line oscillations.
E = 500 volts; Ri = 31.28 ohms; R 2 = 15.64 ohms; Gi = 0; G-i = 0; Li =
0.2564 henrys; L 2 = 0.1282 henrys; C\ = 2.201 microfarads; C 2 = 1.10 micro-
farads; timing wave 100 cycles.
FIG. 89. Circuit and wave diagram for Fig. 88.
Natural Period of Oscillation. Since the space angle,
4>x, in equations (215), (216) is directly proportional to
OSCILLATIONS, SURGES AND TRAVELING WAVES 109
the distance, x, from the origin, it is evident that the phase
of the current, i, and the voltage, e, changes progressively
along the line. At some distance, 1 Q , the current and volt-
FIG. 90. Electric line oscillations.
E = 500 volts; Ri = 39.10 ohms; #2 = 13.04 ohms; Gi = 0; Gz = 0; Li =
0.3204 henrys; Z/ 2 = 0.1070 henrys; Ci = 2.748 microfarads; (7 2 = 0.917 micro-
farads; timing wave 100 cycles.
VBWWtfWtf^^
FIG. 91. Circuit and wave diagram for Fig. 90.
age are displaced 360 deg. from their starting point values.
The distance, 1 , is called the wave length and is the distance
110 ELECTRIC TRANSIENTS
the electric field travels during the time, t Q , required for the
completion of one cycle or complete wave.
If /is the frequency of oscillations in cycles per second,
t Q = - seconds (217)
J
h = vto (218)
The fundamental frequency or natural period of free
oscillation depends on the length of the line and on the
imposed circuit conditions. For the oscillations recorded
in the oscillogram in Fig. 84, the line is open at the receiver
end and connected through the vibrator circuit at the
generator end. The diagram in Fig. 856 shows that
under these conditions the length of the line is one-fourth
wave length of the fundamental oscillations. In Fig. 85c is
shown the wave diagram for the ninth harmonic which
appears as ripples on the fundamental oscillation.
In Fig. 86 the vibrator is connected at the middle point
leaving both ends open. The corresponding wave diagram
in Fig. 876 shows that the length of the line is two quarter-
wave lengths or one-half wave length, and the frequency of
the fundamental oscillation is twice that in Fig. 84. Simi-
larly in Fig. 88, in which the vibrator is connected at one-
third the distance from one end, each of the two parts
becomes a vibrating element giving fundamental oscilla-
tions. The frequency of the oscillation of the shorter
part is twice as great as for the longer portion. In Fig. 90,
with the vibrator connected at one-fourth the distance from
one end of the line, the short end oscillates at three times
the frequency of the long end. In all cases the voltage
and current vary progressively along the line so that at
any instant the average voltage, instead of the maximum
value, is impressed on the condensers and the average
current, in place of the maximum value, flows through the
inductance.
The same results would be obtained in circuits with
massed R, L, G and C in which the maximum voltage is
OSCILLATIONS, SURGES AND TRAVELING WAVES 111
impressed on all the condensance simultaneously or all of
the magnetic field reaches a maximum at the instant the
current is a maximum, by reducing the condensance and
inductance in the ratio of the maximum to the average
values. This ratio is ir/2 for sine waves.
The frequency for free oscillations in simple circuits
with massed R, L, G and C was derived in Chap. V, equation
(162).
~ 2 cycles per second (219)
The frequency of free oscillations in circuits having
distributed R, L, G and C and a sine wave distribution of
the voltage and current may be obtained by multiplying
L and C in equation (219) by Tr/2, the ratio of the maximum
to the average value.
~ c cycles per second (220)
In commercial electric lines the quantity
negligibly small in comparison with 1/LC. For practical
problems the frequency of the fundamental oscillations or
surges in transmission lines with uniformly distributed
R, L, G and C may therefore be obtained by equation (221).
/ = . . n cycles per second (221)
Thus the fundamental frequency of oscillation for the
transmission line in Fig. 84,
ThuS = /= - 2 - CydeS Per S6COnd (222)
This may be checked by measurements on the oscillogram
in Fig. 84. On the original film (the cut in the text is
reduced in size) 10 cycles of the timing wave measured
14.3 cm., while 10 cycles of the transient oscillations
measured 7.1 cm. Hence the frequency,
/ = -j 4 ^ == 200.1 cycles per second (223)
112 ELECTRIC TRANSIENTS
Since L and C represent the total inductance and conden-
sance of the line the frequency depends on the total length
of the line or the length of time in which the oscillation
occurs, as illustrated by the oscillogram in Figs. 84, 86, 88
and 90. The transmission line, or other circuits of dis-
tributed R, L, G and C, therefore, have a fundamental
frequency at which the whole line oscillates, but as any
fractional part of the line may also oscillate independently
of the whole line, particularly if the oscillating section is
short as compared to the entire line, oscillations of any
frequency may occur. At high frequencies the successive
waves are so close together that a small variation in the
time constants will cause them to overlap. Since R, L,
G and C are not perfectly constant high frequency oscilla-
tions interfere with each other, and on this account reso-
nance phenomena occur only at low or moderate frequencies.
Length of Line. In ordinary transmission lines, with air
as the dielectric and conductors of copper or aluminum, an
electric wave or impulse travels approximately 3 10 10 cm.
per second, the velocity of propagation of an electromag-
netic field in free space, equation (205). This fact is of
much practical importance in transmission line calculations.
If the length of the line is known the frequency of the
fundamental oscillation and of the harmonics can readily
be determined. The length of the line is one quarter wave
length of the fundamental oscillation as illustrated by the
oscillogram in Fig. 84 and corresponding diagrams in Fig.
85.
v =- Wo (224)
Conversely, if the frequency of the oscillation is known
the length of the oscillating section may be determined.
In artificial transmission lines with the frequency of the
fundamental oscillation obtained from oscillograms the
equivalent length of the line can be calculated. Thus from
measurements on the oscillogram in Fig. 84, equation (223),
/ = 200 cycles per second. Hence the length of the line,
OSCILLATIONS, SURGES AND TRAVELING WAVES 113
v 3-10 10
/o = 4 ,- = cm. = 375 km. == 233 miles (225)
From equations (205), (221) and (224), relations are
obtained by which L or C may be calculated if the length
of the line, I in cm., and either C or L are known.
v = 3-10" - 4/7 = ~ (226)
Hence,
For cables or circuits in which the permeability, /*, and
the permittivity, K, are greater than unity the corresponding
relations are obtained from equations (206), (221) and
(224).
3-10 10 I
These equations are useful in the calculation of the con-
densance of circuits in which the inductance can be more
easily determined, as in complex overhead systems and in
calculating inductance in cables or other circuits in which
the condensance may readily be measured.
Velocity Unit of Length. Surge Impedance. In hand-
books and tables the values of R, L, G and C are given for
some unit of length as cm., km., 1,000 ft., mile, etc. In
discussions and calculations of transient phenomena the
velocity unit of length is sometimes used. For overhead
structures the unit of length, I, on this basis would be v t or
3-10 10 cm. Hence from equation (227), and under the
assumptions made in its derivation,
L, = ~ (230)
O v
The natural or surge impedance from equations (121),
(230) :
114 ELECTRIC TRANSIENTS
= J'-=L. ==V (231)
\ U v ^v
By the use of the velocity unit of length investigations
on transmission systems having sections of different con-
stants and hence of different wave length are greatly
simplified. In systems having overhead lines, cables,
coiled windings, as in transformers, arresters, etc., the
wave length becomes the same in the velocity measure of
length.
Voltage and Current Oscillations and Power Surges.
It has been shown that in free or stationary oscillation
transmission lines or other electric circuits having uniformly
distributed R, L, G and C the current and voltage are essen-
tially in time quadrature. From equations (215), (216) :
i = It-ut s i n ( w $ __ X __ T ) (234)
e = Ee~ ut cos (coZ - 4>x - 7) (235)
Hence, the instantaneous power, p, at any point in the
circuit is given by equation (236) :
TjJT
p = ei = - - e~ ut sin 2(J - <f>x - 7) (236)
tU
The direction of the flow of power changes 4/ times each
second since the sine function becomes alternately positive
and negative for successive r time degrees. That is, a
surge of power occurs in the circuit of double the frequency
of the current or voltage oscillations, although the average
flow of power along the line is zero.
Average power, p Q = (237)
General Transmission Line Equations. In the preceding
paragraphs various phases of the electric transients that
occur during the free or natural oscillations of electric
circuits have been discussed. The general problem, in which
transient phenomena occur while continuous power is
supplied to the system and transmitted along the line, is
OSCILLATIONS, SURGES AND TRAVELING WAVES 115
necessarily much more complex. In transmission lines
or other electric circuits having uniformly distributed resis-
tance, inductance, conductance and condensance, with
R, L, G and C the constants per unit length of line, the
voltage and current relations in time may be expressed by
partial differential equations as in (238), (239):
-< + ft (239)
Differentiating (238) with respect to x and (239) with
d^i
respect to / and eliminating equations (240), (241)
(7 JU(J L
may be derived:
^2 P x2 p 3p
LC + (RC + GL) + BGe (240)
+ (RC + GL) | + RGi (241)
A general solution for these equations is given in equation
(242) , one term of which represents the sum of the outgoing
and the other the sum of the incoming waves.
e = Aie at e b * sin (at + 0x + 71)
+ A 2 e at e- b * sin (at + $x + T2 ) (242)
In order to determine the values of A\ t A 2 , a, 6, a, |8, 71,
and 72, the specific conditions under which the line operates
must be given. It is, however, of first importance to
understand the purpose or functions of each term in the
equation. On the basis of energy flow and dissipation in
a line transmitting power the following interpretation of
the symbols in equation (242) may be helpful.
A i, and A 2 are constants whose values are determined by
the limiting conditions of each specific problem.
e~ at may be called the time damping factor and a the time
dissipation constant for the transient oscillations.
116 ELECTRIC TRANSIENTS
This factor represents the same form of energy
dissipation as e~ wi in Chap. V. Ordinarily the trans-
formation of electric energy into heat by the Ri 2 and
Ge 2 losses is non-reversible and therefore the sign of
the dissipation constant must be negative.
e bx may be called the distance damping factor and b the
distance dissipation constant. It relates both to the
losses along the line in the steady flow of energy, as in
transmission lines carrying permanent load, and to the
flow of transient energy in the system as with travel-
ing waves or in the oscillations of compound circuits.
at is the time angle. Under permanent or steady condi-
tions with a simple sine voltage, M E sin ut, impressed
at the generating station a = w and has only one
value. However, if the impressed voltage is a
complex wave or during transition periods between
two permanent conditions while transient currents
and voltages are flowing in the system, a would have
more than one value.
fix is the space or distance angle with x as the distance
along the line from the origin. If waves of more
than one frequency are passing over the line |8
would have more than one value.
71 and 72 are phase angles for t = 0.
Traveling Waves. Traveling waves are in many respects
similar to free oscillations or standing waves as the transfer
of energy between the dielectric and magnetic fields is the
basis for all double energy electric phenomena. The essen-
tial difference is that in traveling waves power flows along
the line while in free oscillations or standing waves the
energy oscillates between the two fields but does not travel
from one line element to another. Oscillograms of the cur-
rent and voltage factors in traveling waves are shown in
Figs. 92 to 97. It should be noted that the current is
in time phase with the voltage for the outgoing waves
and differs by 180 deg. for the returning waves. In both
cases a flow of power occurs along the line.
OSCILLATIONS, SURGES AND TRAVELING WAVES 117
In Fig. 92 the receiver end of the line is short circuited.
The reflected voltage wave is in opposite time phase to the
outgoing wave while the corresponding current waves are
in the same direction.
In Fig. 93 the receiver end of the line is open and as a
consequence the reflected current wave reverses in sign
while the corresponding voltage wave is in the same direc-
tion as the outgoing wave.
FIG. 92. Traveling waves on artificial transmission line. Receiver end short
circuited.
Eo = 120 volts, d.c.; Ei = 5 volts; 7i = 19.5 amps.; R =56.1 ohms; G = 0;
L = 0.418 henrys; C = 3.053 microfarads; timing wave 100 cycles.
For the circuit in Fig. 94 a resistance equal to the surge
impedance of the circuit, VL/VC, is inserted at the
receiver end of the line. All the energy of the traveling
wave was dissipated into heat by the Ri^ losses at the
receiver end of the line and as a consequence there was no
reflected voltage or current waves or return flow of power.
From the timing wave and known length of line it is found
that the velocity of propagation of the impulse is equal to
118
ELECTRIC TRANSIENTS
v or 3-10 10 cm. per second, the velocity of propagation of
an electromagnetic field in free space.
A traveling wave in an electric line is sometimes trans-
formed into a standing wave, as illustrated by the oscillo-
grams in Figs. 95, 96 and 97. In Fig. 95, with the
receiver end of the line open, both the voltage and current
waves show that the traveling wave passed from the genera-
tor to the receiver end of the line and back again four
times before it was changed into a standing wave. During
this period the voltage and current waves are in phase or
FIG. 93. Traveling waves on artificial transmission line. Receiver end open.
Eo = 120 volts d.c.; Ei = 5 volts; /i = 19.5 amps. R =56.1 ohms; G = 0;
L = 0.418 henrys; C = 3.053 microfarads; timing wave 100 cycles.
180 deg. apart, showing a flow of power along the line, but
when the traveling wave is changed to an oscillation the
current leads the voltage (note position of vibrators in
the circuit diagram) by 90 deg. If the current leads or
lags 90 deg. with respect to the voltage, the power in the
circuit is reactive and therefore the average flow of power
along the line is equal to zero.
OSCILLATIONS, SURGES AND TRAVELING WAVES 119
Similarly, the oscillograms in Figs. 96 and 97 show
impulses which after passing over the lines several times
as traveling waves are transformed into standing waves or
oscillations. In each case the impulse starts as a traveling
wave with the current and voltage in phase and a flow of
power along the line. The oscillogram shows that the
traveling wave was converted into an oscillation or stand-
ing wave, in which the current and voltage differ by 90 deg.
in time phase, in less than one hundredth of a second, and
that the energy then oscillated between the magnetic and
dielectric fields without flow of power along the line.
FIG. 94. Traveling waves on artificial transmission line.
Receiver resistance =\/L /\/C; Eo = 120 volts d. c.; Ei = 5 volts; I\ = 19.5
amps.; R = 56.1 ohms;G= 0; L = 0.418 henrys; C = 3.053 microfarads; timing
wave 100 cycles.
In Fig. 97 the vibrator connections for the voltage wave,
t> 3 , were reversed; the voltage and current were in phase
instead of 180 apart as indicated by the oscillogram.
The change in frequency when the traveling wave is
converted into a standing wave should be noted. In the
120 ELECTRIC TRANSIENTS
traveling wave the inductance and condensance of the
line alone determines the velocity of propagation while
for the oscillations or standing waves the line and trans-
former oscillate together as a compound circuit.
In determining the instantaneous values for the current
and voltage at any point on the system the power flow must
be taken into consideration in addition to the dissipation
of the transient electric energy into heat as expressed by
the damping factor e~ ut . It is evident that the flow of
power may be increasing, decreasing or unvarying in the
direction of propagation.
If the power flow is uniform the expressions for the cur-
rent and voltage are in the simplest form (244), (255), as
the power transfer factor does not appear in the equations.
i = Io~ ut cos (ut + </> 7) (244)
e = Eoe~ ut cos (ut + <f>x - 7) (245)
p = # /oe-<[l -- sin 2 (ut + 4>x - 7)] (246)
F 1 J
Average power, p = ~ e~ 2ut (247)
Uniform flow of transient power is infrequent but may
occur in special cases. Thus if a transformer line and load,
as in Fig. 100, are disconnected from the power supply and
left to die down together, uniform flow of power in the line
may be realized provided the dissipation constant of the
line is equal to the average dissipation constant of the
whole system. Consider the transformer as having stored
in the magnetic field a comparatively large quantity of
energy while its resistance and conductance are relatively
small compared to the corresponding value for the line.
Likewise assume that the load part of the circuit has very
little energy stored in its magnetic and dielectric fields and
that its dissipation constant is large as compared to that
of the line. Under these conditions the dissipation of
energy is most rapid in the load part of the circuit and
slowest in the transformer. Hence a 'flow of energy will
occur from the transformer to the load. If the rate of
OSCILLATIONS, SURGES AND TRAVELING WAVES 121 -
energy dissipation of the line is midway between the corre-
sponding rates for the load and transformers the energy
dissipated in the line would be equal to the amount initi-
ally stored in the line while part of the energy originally
stored in the transformer flows through the line and is dissi-
pated in the load part of the circuit. The flow of power in
the line would be uniform as it delivers to the load part of
the circuit all the energy received from the transformer.
FIG. 95. Traveling waves changing to standing waves on artificial transmission
line.
R = 55.32 ohms; G = 0; L = 0.419 hemys; C = 3.05 microfarads; Length =
207 miles; 4/0 copper; 96 in spacing; Transformer L = 37.8 henrys; timing wave
100 cycles. ,
The flow of power decreases along the line in the direction
of propagation, if energy is left in the circuit elements as
the traveling wave passes along the line. That is, the
traveling wave scatters part of its energy along its path
and thus decreases in intensity with the distance traveled.
This decrease is expressed by a power transfer constant, s,
comparable to the power dissipation constant u. If no
energy were supplied to the line by the traveling wave the
122 ELECTRIC TRANSIENTS
voltage and current would decrease by the dissipation
factor t~ ut . With power supplied by the flow of energy
the decrease would be slower and would be expressed
by a combination of the damping and power transfer
factors.
For decreasing flow of power:
Damping factor = e~ ut (248)
Power transfer factor = e +st (249)
Combined damping and power transfer factor
' (250)
Similarly if the flow of power increases along the line in
the direction of propagation the traveling wave receives
FIG. 96. Traveling waves changing to standing waves of artificial transmission
line.
Eo= 110 volts; 7i= 19.8 amps.; R = 52.9 ohms; G = 0; L = 0.412 henrys;
C = 3.03 microfarads; timing wave 60 cycles.
energy from the line elements and the actual decrease in
the voltage and current is greater than indicated by the
dissipation constant. The power transfer would in this
case be negative, and the combined damping and power
transfer factor would be expressed by equation (253).
OSCILLATIONS, SURGES AND TRAVELING WAVES 123
For increasing flow of power :
Damping factor = e- ut (251)
Power transfer factor = e~ st (252)
Combined damping and power transfer factor
= e- (M + s) ' (253)
To express the instantaneous values of the current and
voltage at any point in the circuit a distance factor must
be included. For if the traveling wave either scatters or
gathers in energy as it travels along the line the voltage and
current factors decrease at a lesser or greater rate, as the
case may be, in the direction of propagation than if the flow
of power were uniform. In order to use only one power
transfer constant, s, in the equation, let X = the distance x
expressed in velocity measure (254)
For decreasing flow of power along the line :
the distance damping factor = e~' (255)
For increasing flow of power along the line :
the distance damping factor = e sX (256)
The instantaneous values of the transient current, voltage
and power under conditions producing a flow of power along
the line from the point of reference, in the direction of propa-
gation may be expressed by equations (257), (258), or
(259), (260).
i = I e~ ( ' e + ?X cos (cot + X -- T ) (257)
e = E e "c cos (cot + 0X 7) (258)
i = Le~ ut e cos (cot + 0X - 7) (259)
e = E e ~ ut e~ cos (cot + </>X -- 7) (260)
T -2ut 2s(t - X) .
p = loEoe e [I Sin 2 (cot + 0X 7)!
(261)
Average power, p = Lfj>r** - (262)
2i
The upper sign of 4>X applies to waves traveling in the
direction of increasing values of X and the lower sign for
returning waves, for which X is decreasing. For s =
124
ELECTRIC TRANSIENTS
which represents a constant flow of power, equations
(259) and (260) become identical with equations (244) and
(245). Referring to Fig. 100, already used for illustrating
the flow of constant power, it is evident that if the dissipa-
tion constant for the line is less than the average dissipa-
tion constant for the system the flow of power from the
transformer will be such as to increase the power stored
in the line, while if the line dissipation constant is greater
than the average the reverse would be the case.
FIG. 97. Traveling waves changing to standing waves on artificial transmission
line.
Eo = 120 volts; Length = 200 miles; 4/0 copper; 120 in. spacing; timing wave
100 cycles.
Traveling waves are of very frequent occurrence in elec-
tric power systems. . Not merely such violent disturbances
as direct strokes of lightning or short circuits, but practi-
cally every change in load or circuit conditions produce
transient waves that travel over the system. Simple travel-
ing waves as illustrated by the oscillograms in Figs. 92 to
101 are frequently called impulses. In the first part of the
OSCILLATIONS, SURGES AND TRAVELING WAVES 125
FIG. 98. Oscillation of compound circuit. Starting transient of artificial
transmission line and step-up transformer.
Length of line = 52 miles; 4/0 copper; 96 in. spacing, R = 13.84 ohms ; G =
0; L = 0.105 henrys; C = 0.764 microfarads; transformer L = 37.8 henrys; 60
cycle supply.
FIG. 99. Oscillation of compound circuit. Starting transient of (artificial
transmssion) line and transformers.
Length of line 52 miles; 4/0 copper; 96 in. spacing; R = 13.84 ohms; G = 0;
L = 0.105 henrys; C = 0.764 microfarads; 60 cycle supply.
126 ELECTRIC TRANSIENTS
impulse as it passes along a line the wave energy increases
at a rate depending on the steepness of the wave front, and
after the maximum value is reached the wave energy
decreases. While the wave energy increases the combined
dissipation and power transfer factor is represented by
c~' '* as in equation (253), and during the decreasing
stage by e }t as in equation (250). The steepness of
the wave front which corresponds to the sharpness or
suddenness of a blow is often a more important factor in
causing damage to the electric system than the quantity
of energy involved.
Compound Circuits. In commercial systems the trans-
mission line is not an independent unit but merely a link
between the generator and load circuits. Step-up and
step-down transformers, generators and load circuits,
lightning arresters and regulating devices, and all the
apparatus necessary for the operation of the system are
electrically interconnected into one unit. In the several
parts of the system the circuit constants differ in relative
magnitude and hence the velocity of
propagation of an electric impulse
varies and no two sections may have
the same natural period of oscilla-
tion. While the whole system may
FIG. 100. circuit diagram oscillate as a unit partial oscilla-
of a compound circuit. , . ,. , ,.
tions are of much more frequent
occurrence.
In Figs. 98, 99, 101 and 102 are shown the oscillations of
compound circuits consisting of an artificial transmission line
and transformers. The ripples on the current wave, Vi, indi-
cate a wave traveling over the transmission line alone. From
measurements on the film, Fig. 101, the length of the line is
found to be 207 miles. The line and transformers oscillate
as a compound circuit at a frequency of 10.5 cycles per
second. In Fig. 102 the length of the second half wave is
longer than for the first half wave. This is due to a varia-
tion in the permeability of the iron in the transformer core.
OSCILLATIONS, SURGES AND TRAVELING WAVES 127
o i
60
I
en
sl
&
O iO
B "
03 .0
II
fl ft
S o
128
ELECTRIC TRANSIENTS
^ g
"
^3 O
fl >>
g'a
+3 a
<Jo
to
3 ,y
o o
*-i 03
'3 a
o o
fl"
a
^ o
a
o
OSCILLATIONS, SURGES AND TRAVELING WAVES 129
Problems and Experiments
1. Given a transmission line, 80 miles long, of No. 0000 copper, spaced
12 ft. and with the receiver end open. From handbook tables obtain the line
constants. Find the fundamental oscillation frequency of the line. Check
the results by solving for the frequency from the known velocity of propaga-
tion of an electric wave in space and use the given length of the line.
2. Make a series of oscillograms similar to Figs. 84, 86, 88 and 90, on an
artificial transmission line. From the oscillograms determine the equiva-
lent length of actual line. Check by determining the natural frequency of
oscillation from the line constants.
3. From the oscillogram in Fig. 95 or 97 determine the frequency of oscil-
lation of the transmission line alone and the transmission line combined
with the transformer. Assume the condensance of the transformer equal to
zero. From the data given calculate the inductance of the transformer.
4. In the oscillogram in Fig. 101 the ripples on the voltage wave indicate
reflections of traveling waves in the transmission line with the receiver end
open. Calculate the length of the line.
6. From the oscillograms and data in Fig. 101 calculate the inductance
in the transformer in the compound circuit. Assume the condensance of
the transformer equal to zero. It should be noted that the inductance is
essentially massed while the condensance is distributed and hence for the
combined circuit/ = / -
^ v 2irLC
6. From the data given in Fig. 102 calculate the average inductance of
the transformers during the first half cycle after the current and voltage
wave lines cross; also during the second half cycle.
7. Make oscillograms of the oscillations of compound circuits, similar to
Figs. 99, 100, 101, and 102.
CHAPTER VII
VARIABLE CIRCUIT CONSTANTS
In the preceding chapters the fundamental laws of tran-
sient electric phenomena are derived under the assumption
that in any given circuit the resistance, inductance, con-
ductance and condensance, the so-called circuit constants,
remain constant in value during the transition period under
discussion. The transients are due to changes in circuit
condition or in the impressed voltage, but during the period
required for the dissipation of the stored energy, or the
readjustment of the energy content in the system the values
of R, L, G and C are assumed constant. The oscillograms,
Chaps. Ill to VI inclusive, of electric transients were
obtained from circuits in which the resistance, inductance,
conductance and condensance remained essentially constant.
It is evident that if the circuit constants do not remain
constant during the period the transients occur but vary
rapidly over a wide range of values the nature of the result-
ing electric phenomena must be correspondingly more
complex. The laws for the variations in R, L, G and C are
not always known or are so complex that they can not be
represented in the form of equations. For example, data
for the quantitative ratios between the magnetomotive
force and the resulting magnetic flux in iron clad circuits,
as indicated by the hysterises loop, may readily be obtained
experimentally but it has not been possible to express the
relation in the form of a mathematical equation. The
empirical equations in common use are limited in their
application and give only approximate values.
Variable Resistance. Change in temperature is the
most important factor in producing variations in the resis-
tance of electrical conductors, the R circuit constant. For
130
VARIABLE CIRCUIT CONSTANTS
131
metals the specific resistance is a linear function of the
temperature over a fairly wide range.
---- po + at (270)
= specific resistance at t C.
specific resistance at C.
= temperature coefficient.
Pt
PO
For rapid changes in temperature the rate of change in
the resistance may be large. This is illustrated by the
oscillograms in Figs. 103, 104. For the tungsten incandes-
cent lamp, Fig. 103, a starting transient appears in the
current due to a rapid increase in the resistance of the fila-
FIG. 103. Starting current transient of a 60-watt, 120 volts, tungsten incan-
descent lamp. Resistance variable; timing wave 100 cycles.
ment as the temperature rises. When the switch is closed
the filament is at room temperature and the resistance low.
The current flowing through the lamp rapidly heats the
filament to incandescence with an accompanying increase in
the resistance and a decrease in the current. The timing
132
ELECTRIC TRANSIENTS
wave shows that it required about 0.02 of a second for the
lamp to reach full brilliancy. During this period the resis-
tance of the filament increased by 400 per cent of its initial
value.
For carbon the resistance decreases with an increase in
temperature, or the temperature coefficient is negative, as
illustrated in Fig. 104, showing that the time required for
the resistance to reach a constant value was approximately
0.5 of a second and that the resistance of the incandescent
carbon filament is about 70 per cent of its value at room
temperature.
FIG. 104. Starting current transient of a 50 watt, 120 volts, carbon incandescent
lamp. Resistance variable; timing wave 100 cycles.
The temperature of the lamp filament increases until the
dissipation of heat by radiation from the lamp is equal to
the heat generated by Ri 2 losses. For a direct current sup-
ply with constant impressed voltage the constant tem-
perature condition is quickly reached. For alternating
currents the power supplied to the lamps pulsates with double
VARIABLE CIRCUIT CONSTANTS 133
the current frequency and as the lamp emits or radiates
heat continuously the temperature, and therefore the resis-
tance of the filament, pulsates. This is illustrated by the
oscillogram in Fig. 105. Alternating currents are impressed
on two pairs of tungsten and carbon lamps, arranged as
shown in the circuit diagram, with the vibrator of the oscil-
lograph in the bridge connection. Since the resistance of
FIG. 105. Pulsating resistance of tungsten and carbon lamps, alternating
currents; 60 cycle supply.
the tungsten lamp increases and the carbon lamp decreases
with an increase in temperature, the pulsations in the Ri 2
losses unbalance the bridge as indicated by the pulsations
in the currents flowing through the vibrator.
The resistance of the electric arc depends on many factors
and may vary over a wide range with extreme rapidity.
Since the resistance of the arc decreases with the increase
in temperature the arc alone is unstable and hence must be
provided with a " ballast" to make continuous operation
possible. On alternating currents an inductance placed in
134
ELECTRIC TRANSIENTS
series with the arc serves as the stabilizer and the variations
in the resistance of the arc are counterbalanced by the
induced voltage in the inductance. In direct current arc
lamps a series resistance serves the same purpose.
In commercial systems the electric arcs that affect the
series resistance, the R circuit constant, occur chiefly in
the opening of switches. In breaking the circuit under
load, especially when a large quantity of energy is stored
magnetically in the circuit, arcs form in which the resistance
varies rapidly from zero at start to infinity when the circuit
is open.
^^^^^^^^^^^MM^^^^MMM^^^^^^MMM^M^^^M
FIG. 106. Transformer magnetizing current; no starting transient.
Vi = 106 volts; v-2 = primary current; 03, calibration current = 10.0 amps.; 10
KVA. transformers;/ = 60 cycles.
This is illustrated by the oscillogram in Fig. 110. In
the opening of the switch an arc forms whose resistance
rapidly increases, approaching infinity when the circuit
opens, which occurs at the point of maximum value in the
voltage curve. The increase in the resistance can be deter-
mined quantitatively from the oscillogram by combining
VARIABLE CIRCUIT CONSTANTS 135
data from the rapidly increasing voltage and decreasing
current curves.
Variable Inductance. In iron-clad circuits as in trans-
formers the magnetic flux is not directly proportional to the
ampere turns or magnetizing force. Hence the inductance,
the L circuit constant is not constant but varies with the
permeability of the iron. Moreover, the variation in the
inductance is different for decreasing and increasing flux
FIG. 107. Starting transient of magnetizing current in iron-clad circuit.
vi, primary voltage = 236 volts; vz = primary current; % calibration current
= 13.0 amps.; 10 KVA.. transformers;/ = 60 cycles.
values and depends on the maximum flux density as indi-
cated by the form of the hysteresis loop. As no satisfac-
tory mathematical expression has yet been found for the
hysteresis cycle, solutions of practical problems are obtained
by a series of approximations. As a first step in obtaining
the shape of transients in iron-clad circuits, neglecting the
difference between increasing and decreasing flux values,
Frohlich's formula is generally used.
H B = v + *H (271)
136
ELECTRIC TRANSIENTS
The formula is based on the assumption that the permea-
bility of the iron is proportional to its remaining magnetiza-
bility and states that the reluctivity of an iron-clad circuit
is a linear function of the field intensity.
FIG. 108. Starting transient of magnetizing current in iron-clad circuit.
i, primary voltage = 106 volts; 02, primary current; 03, calibration current
= 10.0 amps.; 10 KVA. transformer; / = 60 cycles.
The effect of variable inductance in iron-clad circuits
may be illustrated by the starting transients of alternating
current transformers. The magnitude of the starting
current transient depends more on conditions affecting the
value of the inductance in the circuit than on what point
on the voltage cycle the switch is closed. The direction
and magnitude of the residual magnetism are important
factors as a combination of much residual flux with an
additional magnetizing force in the same direction may
bring the flux density in the core beyond the saturation
point and hence greatly reduce the inductance in the
circuit.
For the oscillograms in Figs. 106 to 109 a constant alter-
nating current voltage of sine wave shape was impressed on
VARIABLE CIRCUIT CONSTANTS 137
the transformer terminals. In Figs. 106, 107 and 108 the
residual magnetism in the iron core was, in each case,
removed before the oscillogram was taken. The three
oscillograms form a series showing the transient current due
to the closing of the switch at different points of the voltage
cycle. In Fig. 106 the switch was closed at an instant the
magnetizing current would have been zero (maximum point
on the voltage wave), if the circuit had been closed earlier,
FIG. 109. Starting transient in transformer magnetizing current. Residual
magnetism.
vi, primary voltage = 150 volts; 02, primary current; vs, calibration current
= 2.5 amps.; / = 60 cycles.
and hence no starting transient. In Figs. 107 and 108 the
switch was thrown at other than the zero point of the mag-
netizing current cycle. The impressed voltage was less than
normal and the change in the flux density is not large and
hence the inductance at the maximum points of the mag-
netizing current wave is practically constant. The starting
current transients under the given conditions may be expres-
sed by an exponential equation as explained in Chap. IV.
138
ELECTRIC TRANSIENTS
The starting transient in Fig. 109 differs greatly both in
form and magnitude, as compared to Fig. 108, although the
circuits were closed in the two cases at approximately the
same point on the voltage wave. In Fig. 109 the impressed
voltage was higher than the rating of the transformer and
the residual magnetism in the iron core was in the same
direction as the flux produced by the magnetizing current
during the first half cycle. Above saturation of the iron
FIG. 110. Breaking generator field circuit. Field current and voltage
transients.
vi = 100 cycle timing wave; v?, impressed voltage =31.5 volts; va, field cur-
rent = 4.0 amps.
core the transformer inductance is relatively small and
hence the first half cycle shows a correspondingly large
current transient. A smooth curve drawn through the
successive maximum values of the starting transient in
Figs. 107 or 108 could with a fair degree of accuracy be
expressed by the exponential equation ; but the correspond-
ing curve drawn through the successive maximum values
of the current wave in Fig. 109 would have a much steeper
VARIABLE CIRCUIT CONSTANTS
139
gradient due to the variation in the inductance, L, of the
transformer winding.
The same effect, due to variable inductance, may be
obtained in breaking the field circuit of a direct current
generator as illustrated by the oscillogram in Fig. 110.
The change in the voltage and current curves from the
instant the jaws of the switch separate to the peak value
of the voltage is largely due to a change in the arc resistance.
After the arc breaks, at the peak of the voltage curve, the
FIG. 111. Building up generator field. Field current and armature voltage
transients.
vi = generator terminal voltage; vz = field current; vs 100 cycle timing
wave.
vibrator circuit provides a path for the dissipation of the
energy stored in the field. As the resistance in the vibrator
circuit is constant the voltage curve also represents the
transient current. The dotted curve traced on the oscillo-
gram shows the exponential curve conforming with the
latter part of the actual voltage or current curves. The
relative magnitude of the peak value of the voltage to the
140
ELECTRIC TRANSIENTS
corresponding initial value of the dotted curve indicates
the change in magnitude of the inductance in the field
winding.
The corresponding variation in the inductance when the
generator field is formed is evidenced by the starting field
current and armature voltage curves shown in Fig. 111.
FIG. 112. Arcing grounds on transmission line.
Ground at generator end. Impressed voltage = 90 volts;/ = 60 cycles; ^2 =
arc voltage; 03 = arc current.
Variable Conductance. In the calculations on power
transmission lines and in general for constant potential
systems in good condition the leakage through the insulation
is small, so that the conductance is negligible and the G
circuit constant may be taken as equal to zero. The insula-
tion of electric circuits deteriorate with varying rates and
the conductance and leakage increase and may become very
large, as for example, if the insulation completely breaks
down and a short circuit is formed. A rupture of the insula-
tion or any sudden change in the conductance of the electric
circuit will of necessity cause violent disturbances in the
VARIABLE CIRCUIT CONSTANTS
141
FIG. 113. Arcing- grounds on transmission line.
Semi-continuous copper-carbon arc 114 miles from generator end of 207
mile artificial transmission line. 4/0 copper, 96 in. spacing. v\ = arc voltage;
#2 = arc current; vs line current.
FIG. 114. Arcing grounds on transmission line.
Arc at receiver end. vi = 100 cycles timing wave; vz = current receiver end:
= voltage receiver end.
142
ELECTRIC TRANSIENTS
system. Arcing grounds or intermittent arcs, as illustrated
by the oscillograms in Figs. 112 to 115, are prolific sources
of electric transients. It is evident that momentary
short circuits, as would be produced by an intermittent
arcing ground with the conductance varying practically
from zero to infinity at an extremely rapid rate, would give
rise to oscillations of any frequency and produce waves and
impulses that would travel to all parts of the system.
FIG. 115. Arcing grounds on transmission line.
Arc at middle of line. v\ 100 cycle timing wave; vi = arc current; va arc
voltage.
Variable Condensance. Under ordinary conditions and
for low voltages, air is very nearly a perfect insulator. In
other words, the conductivity of air is practically zero, the
permittivity, unity and the energy loss extremely small.
If the voltage is increased until the limit of the insulating
strength of the air is reached important changes occur
in both the electric and dielectric circuit constants. With
the occurrence of visual corona in high voltage circuits the
conductivity of the air in the space filled by the corona is
VARIABLE CIRCUIT CONSTANTS
143
increased. Thus in circuits with parallel wires as high
tension transmission lines a voltage gradient above 29.8 ky.
per cm. will produce corona in the air surrounding the
conductor surface and this space filled by the corona glow
becomes semi-conducting. This produces a change in the
circuit condensance as with the appearance of the corona the
effective size of the conductor, and hence of the condenser
surface, is increased. For alternating currents the visual
FIG. 116. Variable condensance. Corona.
Single phase line 135 ft. long, 10 in. spacing, No. 24 A.W.G. steel wire. Line
voltage = 3400 volts; line current = 0.0008 amps.
corona, is intermittent, appearing only near the peaks of
the successive voltage waves, when the instantaneous volt-
age gradient exceeds 29.8 kv. per cm., the required value
for producing visual corona. As a consequence the con-
densance of the alternating current circuit when corona
occurs is variable, pulsating with double the frequency of
the voltage. This is illustrated by the oscillograms in
Figs. 116, 117. If an alternating current voltage of sine
wave shape is impressed on a circuit having constant con-
144
ELECTRIC TRANSIENTS
densance the charging current would also follow the
sine law. If the condensance, the C circuit constant, varies
during the voltage cycle, a corresponding change is produced
in the wave shape of the charging current.
FIG. 117. Variable condensance. Corona.
Line covered with snow and swaying in the wind. Line constants same as
for Fig. 116.
Problems and Experiments
1. Take oscillograms, similar to Figs. 106, 108 and 109, showing the
starting transients of transformers.
2. Take oscillograms showing the variable condensance of an arcing
ground for direct and alternating currents on a transmission line.
3. Take oscillograms similar to Figs. 116 and 117, showing the change in
condensance produced by corona.
4. Take an oscillogram similar to Fig. 110, showing the voltage across
the terminals. Compare the operating voltage with the maximum value
when the switch is opened.
CHAPTER VIII
RESONANCE
Electric resonance phenomena have essentially perman-
ent or stable characteristics but are closely related to, and
frequently accompanied by, true electric transients. The
conditions required for producing resonance and expres-
sions for the frequency at which resonance occurs, in simple
electric circuits, are referred to in Chap. IV in connection
with the derivation of the equations for the natural fre-
quency of free oscillations. Resonance in an electric circuit
implies a forced oscillation of energy between the magnetic
and dielectric fields, during which the energy dissipated as
heat by the Ri 2 and Ge' 2 losses, is supplied from some outside
source. Distinction is usually made between voltage reson-
ance occurring in series circuits, and current resonance that
may be produced in parallel circuits.
Voltage Resonance. In series circuits voltage resonance
occurs at that frequency of the impressed voltage for which
the impedance of the circuit is a minimum. In series circuits,
as in Fig. 118, the impedance is a minimum when the con-
densive and inductive reactances are equal.
,x = c x; 27T/L = (275)
/= \ (276)
2irVLC
z ---- VR 2 + ( L x - c x) 2 =R (277)
Frequently the assumption is made that a circuit is in
resonance when the current and the impressed voltage are
in phase, as illustrated by the vector diagram in Fig. 119.
For straight series circuits the conditions required for unity
power factor of the power supplied to the circuit are iden-
10 145
146
ELECTRIC TRANSIENTS
tical with the requirements for minimum impedance, but
in complex circuits or for current resonance in parallel
circuits this is not always the case.
FIG. 1 18. Series circuit for voltage resonance.
Equation (276) gives the optimum condition for reson-
ance in series circuits for given values of the R, L and C, the
circuit constants. Resonance phenomena are, however,
Y
FIG. 119. Vector diagram for voltage resonance in series circuit Fig. 118.
not limited to the exact frequency determined by equation
(276), but persist over a range of frequencies, more or less
sharply defined, depending on the relative magnitude of
the resistance and the inductive or condensive reactance.
The voltage-frequency relation for given constant values
of R, L and C, is shown in Fig. 120. The feature of special
RESONANCE
147
interest is the large increase in L E and C E, the voltages
across the inductance and the condensance under resonance
conditions. If the resistance is small ,E arid C E may rise
FIG. 120. Voltage resonance for series circuit as in Fig. 118
to many times the value of the impressed voltage E n .
Voltage resonance in power circuits is undesirable as the
increase in voltage above the normal operating value
endangers the insulation.
The effect of varying the resistance on the sharpness of
resonance is illustrated by Fig. 121. The smaller the resis-
tance the higher and sharper the voltage and current reson-
ance peaks. The sharpness of resonance may be defined
as the ratio of the inductive reactance or the condensive
reactance at resonance frequency to the resistance in the
circuit.
Sharpness of resonance = ^ = C D (278)
ti H
Reactance Curves. Curves in rectangular coordinates
showing graphically the changes in magnitude of the
148
ELECTRIC TRANSIENTS
inductive reactance and the condensive reactance produced
by varying the frequency of the impressed voltage are of
much value for giving a clear insight into resonance phe-
nomena. The ordinates of the curves in Fig. 122 represent
respectively the inductive reactance, L x, the condensive
reactance, c x, and the total reactance, x, with the frequency
of impressed voltage as the other variable represented
/ \
FIG. 121. Resonance curves for series circuit with different resistances.
along the X axis. Since resonance occurs when the
impedance of the series circuit is a minimum, the resonance
frequency is indicated by the intersection of the total
reactance curve, in Fig. 122, with the X axis.
Current Resonance. Forced oscillatory transfer of
energy between dielectric and magnetic fields is the basis of
resonance phenomena in parallel circuits in much the
same manner as in series circuits, but the resultant voltage
and current values are different. In simple parallel
circuits, as illustrated by Figs. 123 and 127, current reson-
ance occurs at that frequency of the impressed voltage for
which the total admittance is a minimum. In discussions of
resonance phenomena it is frequently assumed that the
RESONANCE
149
conditions for current resonance in parallel circuits are
met when the inductive and condensive susceptances
are equal, that is, when the impressed current and voltage
are in phase. That this assumption is not in full accord
with the above definition of current resonance for all
FIG. 122. Reactance curves. Series circuit.
values of R in the circuits shown in Figs. 123 and 127, may
readily be seen from the corresponding vector diagrams in
Figs. 124 and 128. For the circuit in Fig. 123 current
resonance occurs when ,b = ( b under the condition that
R = 0. From the vector diagram in Fig. 124:
R
J == E Q (g - jjb) ---- E ^
J = j c bE Q = juCE
I == J + J == E [g +j( c b - L b)]
I = 1
(279)
(280)
(281)
(282)
150
ELECTRIC TRANSIENTS
The total current, 7, will be in phase with the impressed
voltage, E, if
* = *><*<> =*** (284)
Hence for unity power factor supply, the frequency for
the circuit in Fig. 123,
~
(285)
'
FIG. 123. Parallel circuit for current resonance.
FIG. 124. Vector diagram for circuit in Fig. 123.
For maximum current resonance the total admittance
of the circuit must be a minimum and hence for constant
impressed voltage, E Q , the total current must be a minimum.
Therefore, the resonance frequency may be obtained by
RESONANCE
151
equating the first derivative of / to co, L, or C, as the case
may be, in equation (283) to zero. Taking co as the variable
factor with R, L, C, and E constants for the circuit in Fig.
123:
Letting C be the variable factor with R, L, co, and E
constant:
1 /I "722
/ =
(287)
FIG. 125. Current resonance. Variable u>. For Fig. 123, Equation (286).
Letting L be variable with R, C, co, and E constants:
(288)
f = l - 1 -
J o \ or r<
UY
V C /
In a similar manner expressions may be obtained for
unity power factor frequency and maximum current resonance
frequency for co, C or L respectively as the variable with the
other factor constants for the circuit in Fig. 127.
152
ELECTRIC TRANSIENTS
j = E (g ~ jjb)
J == E (G + job)
t --= J+ J ---- E Q [(g + G) + j( c b - L b)}
(289)
(290)
(291)
FIG. 126. Vector diagram. Variable C. For Fig. 123, equation (287).
The total current, /, will be in phase with the impressed
voltage, EQ if
E>9 | 97" 2
it -J- CO JLJ
(293)
Hence, the frequency required to give unity powerfactor
for the circuit in Fig. 127 is the same as for Fig. 123.
L 2
(294)
The frequency for maximum current resonance if w is
variable while R, L, (7, G and E'o are constant, Figs. 127, 128:
RESONANCE 153
If C be the variable, while R, L, G, u and E Q are constant:
(296)
1 / 1 ~ R' 2
L' 2
If L be the variable, while R, C, G, co and E^ are constant:
= 27r\ 2LC + [r 4 + CL 3 4L 2 C 2
(297)
o
o
L
G O
FIG. 127.- -Parallel circuit with leaky condenser.
In tuning ratio receiver sets resonance is obtained by
varying C or L as expressed by equations (296) (297).
FIG. 128. Vector diagram for circuit in Fig. 127.
Changes in the inductance by varying the number of turns,
also changes the ohmic resistance but the conditions
154
ELECTRIC TRANSIENTS
required for equation (297) may be obtained experiment-
ally for circuits in which the change in L may be produced
by varying the mutual or self-induction between parts of
the inductance in circuit.
The smaller the resistance in the resonating circuit the
greater the increase in the resonance current and voltage.
Resonance phenomena are of commercial importance only
when 'the resistance in circuit is small as compared to the
inductance and condensance.
FIG. 129. Susceptance curves for parallel circuit.
In most cases and particularly those of greatest impor-
tance, the resistance is negligibly small. If R and G are
taken equal to zero all the resonance frequency equations
(295) to (297) become identical in form.
Resonance frequency, massed circuit constants (approxi-
mate value) :
RESONANCE 155
> = 2.VLC (298)
In commercial work equation (298) is in general use, giv-
ing with sufficient accuracy the resonance frequency for
simple circuits having massed condensance, inductance and
resistance.
For distributed circuit constants, as in long transmission
lines, the space distribution of the voltage and current
waves must be taken into consideration, the approximate
resonance frequency is given by equation (299), as explained
in Chap. VI on Transmission Line Oscillations.
Resonance frequency, uniformly distributed circuit con-
stants (approximate value)
f - 4VLC (299)
In power circuits resonance conditions must be avoided
or the resistance in circuit be sufficiently large to prevent
any marked increase due to resonance in the current and
voltage.
Coupled Circuits. Resonance phenomena are of funda-
mental importance in the operation of radio communica-
tion apparatus. The circuits in commercial use are more
complex than the forms discussed above but may be con-
sidered as combinations of simple circuits. In general
the component simple circuits have certain parts in
common.
The couplings or connections may be made in a number of
ways. For two circuit apparatus the coupling is generally
made in one of the following ways:
1. By direct connection across an inductance coil.
Direct coupling as in Fig. 130.
2. By magnetic induction. Inductive or magnetic coup-
ling as in Fig. 131.
3. By dielectric induction. Condensive, capacitative
or dielectric coupling as in Fig. 132.
loG
ELEC TRIG TEA NSIEN TS
The inductive interaction of the voltages and currents
in tAvo resonating coupled circuits and the transfer of the
PTXRP 1 nRHT^-lf
M
FIG. 130. Direct coupling.
oscillating energy between the primary and secondary
circuits are illustrated by the oscillograms in Figs. 133 to
FIG. 131. Inductive or magnetic coupling.
138. The oscillations of the energy between the dielectric
and magnetic fields of each circuit are combined with a
FIG. 132. Condensive or dielectric coupling.
rapid to and fro transfer of the energy between the mag-
netically or dielectrically coupled circuits. In Fig. 133
the energy was initially stored in the condenser in the pri-
RESONANCE 157
mary circuit. By closing the switch oscillations are set up
between the dielectric and magnetic fields in both the
primary and secondary circuits, and these are combined
with a rapid to and fro transfer of the energy between the
two circuits. The oscillogram shows that the frequency
of oscillation between the magnetic and dielectric fields in
both the primary and secondary was 790 cycles per second,
while the frequency of transfer between the circuits was
approximately 99 cycles per second. That is, the time
required for the transfer of the energy from the primary to
the secondary through the magnetic coupling and back
again was approximately equal to eight complete oscilla-
tions between the magnetic and dielectric fields of either the
primary or the secondary circuits. The oscillations
decrease in magnitude due to the Ri 2 losses and practically
all of the energy was dissipated into heat in ^ of a second.
For the oscillogram in Fig. 134 the primary circuit was
opened at the instant all the energy had been transferred
from the primary to the secondary circuit, thus preventing
its return to the primary circuit. Hence the secondary
continues to oscillate until all the energy has been dissi-
pated as heat by the Ri 2 losses.
The oscillogram in Fig. 135 shows the starting oscillatory
transient of two inductively coupled circuits when an
alternating current of resonance frequency is impressed on
the primary. Similar oscillograms showing the oscillatory
transfer of energy between the primary and secondary of
dielectrically coupled circuits are shown in Figs. 136, 137
and 138. The difference in form in the three oscillograms
is due to change in the degree of coupling as indicated by
the quantitative data in each case.
Coupling Coefficient. In coupled circuits as in Figs.
130 and 131, the interaction will depend on what part of
the total magnetic flux interlinks both circuits. The degree
of coupling which is often termed "loose" or " close, "
depending on whether a small or large fraction of the flux
interlinks both circuits, is quantitatively expressed [by
158
ELECTRIC TRANSIENTS
RESONANCE
159
160
ELECTRIC TRANSIENTS
the coupling coefficient. This is defined as the ratio of
the mutual reactance to the square root of the product of
the primary and secondary circuit reactances.
FIG. 135. Transient oscillations. Inductive or magnetic coupling. Resonant
charge.
Impressed frequency = 750 cycles; R = 6.5 ohms; L = 0.205 henrys; C =
0.2 microfarads; coefficient of coupling = 11 percent; timing wave 100 cycles;
natural frequency 790 cycles when K = 0.
Inductive coupling coefficient, Fig. 131:
a m M
(300)
M = mutual inductance
L^ = inductance of primary with the secondary open
or removed
L 2 = inductance of secondary with the primary open
or removed.
Condensive coupling coefficient, Fig. 132:
cX,
vc
c,
C 1 I
= V(C.
c r
\j a\* /
(C. + C.) (C. + C
C m = condensance in common condenser
-, (301)
RESONANCE
161
,
II
0>
I!
o3 o
11
162
ELECTRIC TRANSIENTS
O
d
I
O
S b
I"
II
g 3
2 2
fl
RESONANCE
163
s
-i*
"Eg
s
-^ o
i s
^
1 64 ELECTRIC TRANSIENTS
C a = condensance in primary circuit
Ci = condensance in secondary circuit
C C
d = -^j ~ m r - = total condensance in primary
C o ~\~ C m
C C
Cz = ~/V = total condensance in secondary.
Cb ~\- (j m
Multiplex Resonance. In complex circuits or series of
double energy loops the conditions for resonance may be
satisfied for more than one frequency of the impressed
voltage. The degrees of freedom, or the number of fre-
quencies at which resonance may occur, depends on the
number and interconnection of the elemental double
energy circuits in the system. Thus, a transmission line
having uniformly distributed R, L, G and C, and hence to
be considered as consisting of an infinite series of infinitesi-
mal double energy circuits, would resonate for the funda-
mental frequency of the line as a unit and for any multiple
or harmonic of the fundamental frequency. As the line
constants are not perfectly constant and the distribution
of R, L, G and C not quite uniform, resonance is limited to
the fundamental and a few of the lower harmonics.
Resonance Growth and Decay. As stated in the begin-
ning of this chapter resonance in electric circuits implies a
forced oscillation of energy between magnetic and dielectric
fields, at such frequencies of the impressed voltage as to make
the total impedance or admittance a minimum. To supply
the resonating circuit with the oscillatory energy necessitates
a transient starting period during which the amplitude of
each oscillation is greater than the one preceding. For
systems having constant finite circuit constants in which
the resonance phenomena reach permanent values, the
growth of the transient follows the exponential law. This
increase in the magnitude of the oscillations during the
starting period is illustrated by the oscillograms in Figs.
139 and 140. In these oscillograms the power supply was
cut off when the resonance had reached the permanent
stage. The decay parts of the oscillograms in Figs. 139 and
RESONANCE
165
B^
03 O)
a a
a T2
c3 to
I
166
ELECTRIC TRANSIENTS
FIG. 140. Resonance in high speed signaling.
R = 10 ohms; L = 89 millihenrys; C = 0.25 microfarads; timing wave 100
cycles; frequency = 1070 cycles; decrement = 0.052.
FIG. 141. Resonance limited by spark gap discharge.
R = 15 ohms; L = 89 millihenrys; C = 0.25 microfarads; timing wave 100
cycles; frequency = 1070 cycles; decrement = 0.079.
RESONANCE 167
140, represent, therefore, free oscillations with a decrease in
amplitude as the electric energy is dissipated into heat.
In Fig. 141 the starting period is of the same form as in
Fig. 139 or 140, but not the decay stage. It is evident from
the circuit connections that the decay of the resonating
currents or voltages will differ in shape depending at what
instant in the cycle the short circuit occurs. The oscillo-
gram in Fig. 141, for which the short circuit was produced
by spark-over, occurred near the maximum point of the
voltage wave with practically all of the oscillating energy
initially stored in the dielectric field of the condenser.
Problems and Experiments
1. Take oscillograms showing the transients accompanying the growth
and decay of cumulative resonance in circuits similar to Figs. 139, 140
and 141.
2. Take oscillograms of the transient oscillations of two inductively
coupled circuits similar to Figs. 133, 134 and 135.
3. Take oscillograms of the transient oscillations in two dielectrically
coupled circuits similar to Figs. 136, 137 and 138.
CHAPTER IX
OSCILLOGRAMS
In the preceding chapters the fundamental principles
of electric transient phenomena are illustrated by a number
of oscillograms, many of which the student should repro-
duce in order to gain the necessary appreciation of the
quantitative value of the factors involved. However, the
laboratory work in the course should not be restricted to
the reproduction of oscillograms appearing in the text for
which quantitative data are provided, or to the taking of
other oscillograms that merely illustrate the fundamental
principles. For while the gaining of clear concepts of the
basic laws of transient electric phenomena is of primary
importance, training in applying the principles to practical
engineering problems is likewise an essential part of the
work. Ample material for this purpose is available in all
electrical engineering laboratories. The oscillograms in this
chapter, Figs. 142 to 161, which were selected from the labo-
ratory reports of students in the introductory course in
electric transients, may be taken as typical examples. The
students were required to outline the problem, to select the
necessary apparatus and instruments, to make preliminary
calculations and to predict the form and shape of the
transients to be recorded. They made all the adjustments
on the oscillograph, obtained experimentally the recorded
quantitative data, took the oscillograms, developed the
films and prepared a report on the transients photographic-
ally recorded by the oscillograph. Each oscillogram repre-
sents a separate problem to be analyzed on the basis of the
principles discussed in the preceding chapters.
168
OSCILLOGRAMS
169
.
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170
ELECTRIC TRANSIENTS
C
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CSCILLCGRAMS
171
172
ELECTRIC -TRANSIENTS
FIG. 145. T. A. regulator operating transients.
Fi = exciter field current; V* = alternator field current; Va = alternator
terminals.
FIG. 146. Undamped oscillograph vibrator oscillations.
Vi = timing wave, 100 cycles; Vz = Oscillations of undamped oscillograph
vibrator superimposed on tungsten lamp starting transient. Vz = starting
transient (vibrator damped) of tungsten lamp, imperfect contact.
OSCILLOGRAMS
173
174
ELECTRIC TRANSIENTS
OSCILLOGRAMS
175
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176
ELECTRIC TRANSIENTS
FIG. 150. Current transformer transients.
Vi = secondary current; Vz = secondary voltages; Va = primary current;
primary / = 60 amps.; secondary / = 3.5 amps.; core undersaturated before
transient.
FIG. 151. Single phase short circuit on a two-phase alternator.
Open phase voltage = 605 volts; short circuit current = 23 amps.; E, field =
500 volts; I, field = 3.25 amps.; frequency = 60 cycles; Vi = open phase voltage;
Vz = short circuit current; V = field current; brushes sparking.
OSCILLOGRAMS
477
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ELECTRIC TRANSIENTS
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OSCILLOGRAMS
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ELECTRIC TRANSIENTS
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OSCILLOGRAMS
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ELECTRIC TRANSIENTS
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OSCILLOGRAMS
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ELECTRIC TRANSIENTS
OSCILLOGRAMS
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ELECTRIC TRANSIENTS
12
OSCILLOGRAMS 187
Problems and Experiments
1. Take oscillograms of a number of transients in circuits of the types
shown in this chapter. In each case obtain quantitative data and pre-
pare'a report giving an explanation of the transients appearing in the oscillo-
gram based on the fundamental principles of transient electric phenomena.
2. Find several electric transients in the laboratory under different
circuit conditions from those described in the book. For each case draw
diagrams of the proposed circuit connections showing the location of the
vibrators; make preliminary calculations as to the amount of resistance
required in each vibrator circuit; the most desirable speed of the film drum,
etc., to give a well proportioned oscillogram; take the oscillogram; record
the quantitative, data; develop the film and make prints. Compare the
predicted forms of the curves with the photographic record and check the
preliminary calculations with the final circuit data. Prepare a report on
the transients recorded on the oscillogram.
APPENDIX
Developing and Printing Oscillograms. The finished
oscillogram, even if perfect electrically, is often disappoint-
ing photographically. Care and cleanliness in the manipu-
lation of the photographic film and printing paper will reduce
these failures to a negligible quantity.
Starting with the unexposed film, the photographic proc-
ess will be traced to the completed print, ready for the files.
Cleanliness is essential. During no part of the process
should the hands come in contact with the sensitized side
of the negative. In order to accomplish this, the film and
its black protecting paper should be placed on the drum as
a unit, with the black paper on the outside. After the film
and paper have been adjusted to the proper position, the
paper may be removed from the drum. In this way the
hands have not touched the surface of the film.
Unlike most photographic work, the permissible time of
exposure for oscillograms is limited, especially in high speed
work. Stray light of any nature is injurious. For this
reason it is highly desirable to load the film-holders in
complete darkness and to develop for the first two or three
minutes without even the ruby light. After a little practice
the student will have no trouble in working without the
darkroom light.
Any metol-hydrochinon film developer may be used with
varying degrees of success. Where only a few T negatives
are made at odd times, Eastman's " Special" developer is
satisfactory. This developer will give better results if
some of the used developer be added to the fresh solution.
In our laboratories the following stock solution is used:
water 64 oz., metol one drachm, hydrochinon one-half oz.,
sodium sulphite 2 oz., sodium carbonate 3 oz., potassium
bromide 30 grains. This stock solution is diluted in the
proportion of two parts stock solution to one part water.
188
APPENDIX 189
It is very important that the developer be used at a tem-
perature of 65 deg. F. The hydrochinon is inactive at lower
temperatures, resulting in slow development and a flat
negative which lacks density and contrast. If used at a
higher temperature, the negative will gain density rapidly
but will be lacking in contrast and show a decided tendency
to fog in the unexposed portions.
The exposed negative should be given maximum develop-
ment possible without fogging the unexposed portions.
The image should be allowed to develop until it appears
quite definite on the reverse side of the negative. A good
rule to follow is to develop until by comparison with the
back of the negative, the sensitized side appears quite
gray. The gray tone will disappear in the fixing bath and
further development is detrimental.
Care should be taken to fix and wash the negatives prop-
erly. The film should be left in the standard fixing bath
at least five minutes longer than is necessary to dissolve the
last visible trace of un-reduced silver salts. After careful
fixing, the film should be washed for at least twenty minutes
in running water. It is desirable to rinse off the surface
with a tuft of cotton before hanging up to dry. The hurry
which often comes in the completion of the day's work in
the laboratory, results in haste in the darkroom. If the
fixing and washing processes are slighted, the film, though
apparently good at the time, becomes worthless in a few
months on account of staining.
The same developer may be used for the printing paper,
except that it should be always mixed fresh just before
using. The best results are obtained by following the
printed instructions accompanying the photographic paper.
In order to get the maximum contrast in the finished
print, it is necessary to use the most contrasting photo-
graphic paper. The paper which has proven the best is
the Eastman "Azo," grade No. 4, glossy, although others
may satisfy the individual user. If this is purchased in ten
yard rolls, twenty inches wide and cut on a circular saw or
190 APPENDIX
band-saw to four and one-half inch widths, four small rolls
result with a two inch roll left over for use in testing
exposure.
Prints should be given normal exposure so that with
normal or full development the background reaches good
density without appreciable reduction of the silver in the
highlights. As usual, prints should be fixed fifteen minutes
in a standard hypo bath and washed for at least twenty-five
minutes in running water. The best finish is obtained by
drying the prints on ferro-type plates, which imparts high
gloss to the surface.
INDEX
Alternating current transients, 40
Alternator field transients, 50, 61
Alternators, single-phase, 61-69
three-phase, 53-60, 69
two-phase, 71, 176
Arcing grounds, 141
Armature reactance, 59
reaction, 58
transients, 50, 61
Artificial electric lines, 101
Asymmetrical field transients, 63
Attenuation constant, 30
B
Breaking field circuit, 138
"Bucking broncho," 175
Capacitance, 7, 9
Carbon lamps, 132
Circuit breakers, 170
constants, 101, 130
Compound circuits, 126
Condensance, 5, 7, 9
variable, 142
Condensive coupling, 156
Conductance, 9
Corona, 143
Coulombs, 9
Coupled circuits, 154
Coupling coefficient, 155
Current resonance, 148
D
Damping factor, 92, 115
Developing oscillograms, 188
Dielectric circuit, 4
coupling, 156
Dielectric field intensity, 9
flux, 4, 9
gradient, 9
Direct coupling, 156
current transients, 24
Dissipation constant, 30, 92, 115
Distance angle, 116
Double energy transients, 21. 75-100
Elastance, 4, 9
Electric circuit, 7
line oscillations, 101
Energy, 8, 9
Exponential curve, 32
law, 24
Farad, 7, 9
Faraday's lines of force, 2
Forming magnetic field, 27, 139
Frequency, distributed R, L, G
and C, 110
massed R, L, G and C, 111
Frohlich's formula, 135
Galvanometers, 14
Generator field transient, 138
Gilbert, 9
II
Henry, 4, 9
High frequency signalling, 166
Impedance, 9
Impulses, 124
191
192
INDEX .
Inductance, 3, 9
variable, 135
Induction motors, 20, 48, 177, 181
Initial values, 33
Intermittent arcs, 141
Joule, 8, 9
Permittivity, 4, 6
Phase angle, 103
Polyphase short circuit, 50
Power surges, 114
transfer factor, 123
Printing oscillograms, 189
Pulsating condensance, 142
inductance, 135
resistance, 133
K
Kirchoff's Laws, 79, 83
L
Leaky condenser, 83, 104
Length of line, 112
Lifting magnet transient, 169
Line constants, 101
oscillations, 101
Lumpy line, 101
R
Reactance curves, 149, 154
Reluctance, 3, 9
Repulsion induction motor, 178
Resonance, 145
"Resonance" frequency, 78
growth and decay, 164
Resistance, 9, 130
Resistivity, 9
Rotary converter, 182-187
Rotating magnetic field, 48
M
Magnetic circuit, 2
coupling, 156
field intensity, 3, 9
flux, 2, 9
Microfarads, 7
Multiplex resonance, 164
N
Natural admittance, 77
impedance, 77
period of oscillation, 108
Oersted, 9
Ohm's law, 2, 4, 9
Oscillator alternator, 17
Oscillatory circuits, 79
Oscillograms, 20
Oscillographs, 10
Permeability, 3, 9
Permittance, 9
Series generator, 173
Sharpness of resonance, 147
Short circuits, polyphase, 50
single-phase, 61
Single energy transients, a.-c., 40
d.-c., 23
Space angles, 103, 116
Split-phase motor, 178
Standing waves, 118
Surge admittance, 77
impedance, 77, 113
Susceptance curves, 154
Synchronous reactance, 58
T. A. regulator, 172
"T" circuits, 104
Three-phase transients, 44
Time angles, 103, 116
constant, 28
Timing waves, 17
"T" line, 102
Transformers, 135, 176
INDEX
Transmission line, artificial, 101 Units, 9
constants, 101 V
equations, 114
oscillations, 101 Variable circuit constants, 130
Traveling waves, 116 Velocity unit of length, 113
Tungsten lamp, 1'31 Voltage resonance, 145
U W
Undamped vibrator, 172 Watt, 8
193
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