Q16Electrical Machines and Drives

Torque Equation of a 3-Phase Induction Motor

Consider a 3-phase induction motor with rotor standstill EMF per phase E2, rotor resistance per phase R2, and rotor standstill leakage reactance per phase X2, operating at slip s. At slip s, the rotor frequency becomes sf (where f is the supply frequency), so the induced rotor EMF becomes sE2 and the rotor reactance becomes s*X2 (since reactance is directly proportional to frequency), while rotor resistance R2 remains unchanged.

The rotor input power per phase (which, since the rotor circuit is purely resistive-inductive with the EMF as source, all goes into the air gap as the mechanical-equivalent gross power via the torque, apart from rotor copper loss) is proportional to E2I2cos(phi2), and the developed torque T is this air-gap power divided by the synchronous angular speed ws (torque is more conveniently computed via gap power divided by ws rather than mechanical power divided by rotor speed, since gap power = mechanical power + rotor copper loss = T*ws exactly, independent of slip, a standard and important simplification in induction motor torque analysis).

This is the standard torque-slip expression, showing torque as a function of slip s for constant E2, R2, X2, and ws. At standstill (s=1) it gives the starting torque; near synchronous speed (s to 0), the R2^2 term dominates the denominator and torque rises nearly linearly with s (torque proportional to s/R2, a stable, monotonically increasing region); at large slip, the (sX2)^2 term dominates and torque falls with increasing s (an unstable, monotonically decreasing region), giving the overall torque-slip curve its characteristic rise-then-fall bell shape.

To find the condition for maximum torque, differentiate T with respect to s and set dT/ds = 0, holding E2, R2, X2, ws constant (equivalent to treating sX2 as the variable and finding where R2 equals it, by the standard maximum-power-transfer-type argument applicable to this expression's algebraic form, since T is proportional to sR2/(R2^2+(sX2)^2), and writing sX2 = y, T is proportional to (y/X2)R2/(R2^2+y^2), maximized over y when y=R2, i.e., dT/dy=0 gives R2^2+y^2 - y2y =0 => R2^2 = y^2 => y=R2):

Substituting sm = R2/X2 back into the torque expression gives the maximum (breakdown/pull-out) torque:

Notably, Tmax is completely independent of rotor resistance R2 - increasing R2 only shifts the slip sm at which maximum torque occurs (moving it to a higher slip, i.e., lower speed) without changing the peak torque value itself, which is the fundamental principle exploited in wound-rotor motor starting: by inserting external rotor resistance, the maximum-torque point can be deliberately shifted all the way to s=1 (standstill), so the motor develops its maximum possible torque right at starting, without any increase in starting current beyond what is needed for that torque, giving wound-rotor motors with rotor resistance starters their characteristically favorable high-starting-torque, moderate-starting-current performance compared to squirrel-cage motors of comparable rating.