IM equivalent circuit; starting/running torque

A three-phase induction motor is like a transformer with a rotating secondary. Its per-phase equivalent circuit is drawn with all rotor quantities referred to the stator.

Rotor circuit at standstill (s = 1) and running (slip s)

Rotor emf per phase at standstill = E₂, reactance X₂. At slip s: emf = sE₂ and reactance = sX₂ (rotor frequency f₂ = sf).

 Standstill (s = 1)            Running (slip s)

   R2      X2                   R2      sX2
 --/\/\/--(XXX)--+            --/\/\/--(XXX)--+
   E2 ->         |               sE2 ->        |
                 (short)                       (short)
 I2 = E2/√(R2²+X2²)           I2 = sE2/√(R2²+s²X2²)

Dividing the running circuit by s gives the same current with a constant-frequency circuit: I₂ = E₂ / √((R₂/s)² + X₂²). Here R₂/s = R₂ + R₂(1−s)/s, where R₂(1−s)/s is the load resistance that stands for the mechanical output.

Complete per-phase equivalent circuit (referred to stator)

 I1    R1     X1                 X2'      R2'
 -->--/\/\/--(XXX)--+-------+--(XXX)--/\/\/--+
                    |       |                |
  V1 (phase)      Rc|      Xm|               | R2'(1-s)/s
                    |       |                | (load)
 --------------------+-------+---------------+

Rc and Xm form the exciting branch (core loss and magnetising reactance). At standstill the load resistance R₂'(1−s)/s = 0 (s = 1). The motor behaves like a short-circuited transformer.

Starting torque

At start s = 1. Rotor current I₂ = E₂ / √(R₂² + X₂²), rotor power factor = R₂ / √(R₂² + X₂²).

Rotor input (air-gap power) P₂ = 3 I₂² R₂ / s. Torque T = P₂ / ω_s. At s = 1:

T_st = (3/ω_s) · E₂² R₂ / (R₂² + X₂²)

(Neglecting stator impedance, E₂ is proportional to the supply voltage, so T_st ∝ V².)

Running torque

For slip s, I₂ = sE₂ / √(R₂² + s²X₂²). Rotor input P₂ = 3 I₂² R₂ / s. Hence

T = P₂/ω_s = (3/ω_s) · s E₂² R₂ / (R₂² + s²X₂²)

Mechanical power P_m = (1−s)P₂ and T = P_m/ω_r = P₂/ω_s. The maximum running torque occurs at s_m = R₂/X₂ and equals T_max = 3E₂² / (2ω_s X₂).

Relations

• Rotor copper loss = s · P₂
• Mechanical power P_m = (1 − s) · P₂
• P₂ : P_cu : P_m = 1 : s : (1 − s)