---
title: "Torque equation of 3-phase induction motor"
subject: "Electrical Machines"
chapter: 5
key: "semester_4/electrical_machines:torque-equation-induction-max-condition"
source: https://www.bctengineer.com/q/semester_4/electrical_machines/torque-equation-induction-max-condition
site: BCT Engineering
---

# Torque equation of 3-phase induction motor

Electrical Machines · Chapter 5: Three Phase Induction Machines

- Asked in: 2074 Bhadra (old course)
- Video: [torque equation of 3 phase induction motor | torque equation | Motor control | electrical machines](https://www.youtube.com/watch?v=QwSwB8rqn60) — Mind of Engineer 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/electrical_machines/torque-equation-induction-max-condition
- Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines/chapter-5

## Question

Derive the relationship for torque developed by a 3-phase induction motor. Draw a typical torque-slip characteristic and deduce the condition for maximum torque.

## Short answer

Rotor current at slip s is sE2/sqrt(R2^2+(sX2)^2); torque is T = (3/omega_s) * sE2^2R2/(R2^2+(sX2)^2). Maximum torque occurs at sm=R2/X2 and equals 3E2^2/(2 omega_s X2); the curve rises from s=1 to maximum then falls to zero at s=0.

## Answer

#### Derivation of torque
Let $E_2$ = rotor emf per phase at standstill, $R_2$, $X_2$ = rotor resistance and standstill reactance per phase, and $s$ = slip.

At slip $s$, the rotor emf is $sE_2$ and the reactance is $sX_2$, so the rotor current per phase is
$$
I_2 = \frac{sE_2}{\sqrt{R_2^2 + (sX_2)^2}}
$$
Rotor power factor: $\cos\phi_2 = \frac{R_2}{\sqrt{R_2^2+(sX_2)^2}}$.

The torque equals the rotor power input (air-gap power) $P_g$ divided by the synchronous angular speed:
$$
T = \frac{P_g}{\omega_s} = \frac{3I_2^2 R_2/s}{\omega_s}
$$
Substituting $I_2$:
$$
T = \frac{3}{\omega_s}\cdot\frac{s E_2^2 R_2}{R_2^2 + (sX_2)^2}\ \text{N-m}, \qquad \omega_s = \frac{2\pi N_s}{60}
$$
Since $E_2 \propto V_1$, $T \propto V_1^2$.

- Starting torque ($s=1$): $T_{st} = \frac{3}{\omega_s}\cdot\frac{E_2^2R_2}{R_2^2+X_2^2}$.

#### Torque-slip characteristic
```text
 T
 |          Tmax
 |        _.-*-._
 |  Tst _-'      `-._
 |   *-'              `-.
 +--+--------+-----------*--> s
  s=1       s_m         s=0
```

#### Condition for maximum torque
For maximum $T$, $\frac{dT}{ds} = 0$. Writing $T \propto \frac{s R_2}{R_2^2 + s^2X_2^2}$:
$$
\frac{(R_2^2 + s^2X_2^2)R_2 - sR_2(2sX_2^2)}{(R_2^2+s^2X_2^2)^2} = 0 \;\Rightarrow\; R_2^2 = s^2X_2^2
$$
$$
s_m = \frac{R_2}{X_2}
$$
Maximum torque occurs when the rotor resistance equals the rotor reactance at that slip ($R_2 = s_mX_2$). Substituting:
$$
T_{max} = \frac{3E_2^2}{2\omega_s X_2}
$$
The ratio $\frac{T}{T_{max}} = \frac{2 s s_m}{s^2 + s_m^2}$ is useful in numericals.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Torque is obtained from air-gap power as:
   A. T = P_g omega_s
   B. T = P_g/omega_s
   C. T = P_out/omega_s only
   D. T = P_g s
   Answer: B. T = P_g/omega_s with P_g = 3 I_2^2 R_2/s. (Key point: Torque from air-gap power)

2. Rotor current per phase at slip s is:
   A. E_2/sqrt(R_2^2 + X_2^2) always
   B. sE_2/sqrt(R_2^2 + (sX_2)^2)
   C. E_2/(sR_2)
   D. sE_2/R_2
   Answer: B. Rotor emf is sE_2 and reactance sX_2. (Key point: Rotor current)

3. dT/ds = 0 leads to:
   A. R_2 = 0
   B. R_2^2 = s^2 X_2^2, so s_m = R_2/X_2
   C. s = 1
   D. s = X_2
   Answer: B. The numerator reduces to R_2^2 - s^2X_2^2. (Key point: Max torque condition)

4. T/T_max for numerical problems is:
   A. 2 s s_m/(s^2 + s_m^2)
   B. s/s_m
   C. s_m/s
   D. (s + s_m)/2
   Answer: A. This ratio follows from the torque equation. (Key point: Ratio formula)

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