HIGHER HARMONICS 145

to the successive components. The secondary currents induced
by these component fluxes, and the torque produced by the
secondary currents, thus show the same components.

Thus tlie motor torque consists of the sum of a series of
components:

The main or fundamental torque of the motor, given by the
usual sine-wave theory of the induction motor, and due to the
fundamental voltage

is shown as T\ in Fig. 55, of the usual shape, increasing from
standstill, with increasing speed, up to a maximum torque, and
then decreasing again to zero at synchronism.
The third harmonics of the voltage waves are:

cos (3 </> — as),
cos 3 <t> — as +

(4)

c/3 VA.JQ i u *p ut<j ~~p ^ i

f \ ^/

i

| ' As seen, these also constitute EL quarter-phase system of

I voltage, but the second wave, which is lagging in the funda-

| mental, is 90° leading in the third harmonic, or in other words,

the third harmonic gives a backward rotation of the poles with
triple frequency. It thus produces a torque in opposite direc-
tion to the fundamental, and would reach its synchronism, that
is, zero torque, at one-third of synchronism in negative direction, I

or at the speed S, = — J£> given in fraction of synchronous speed.
For backward rotation above one-third synchronism, this triple
harmonic then gives an induction generator torque, and the
complete torque curve given by the third harmonics thus is as
shown by curve T$ of Fig. 55.
The fifth harmonics:

65 COS (5<£ — CL^y

e?3cos ^5 05 — a6 — |

(5)

give again phase rotation in the same direction as the funda-
mental, that is, motor torque, and assist the fundamental. But
synchronism is reached at one-fifth of the synchronous speed of
tfie fundamental, or &t: & = +% and above this speed, the
10