--- title: "Max torque condition and torque-slip curve" subject: "Basic Electrical and Electronics Engineering" chapter: 2 key: "semester_1/basic_electrical_and_electronics_engineering:max-torque-induction-motor" source: https://www.bctengineer.com/q/semester_1/basic_electrical_and_electronics_engineering/max-torque-induction-motor site: BCT Engineering --- # Max torque condition and torque-slip curve Basic Electrical and Electronics Engineering · Chapter 2: Electrical Machines - Asked in: 2081 Kartik (new course), 3+3 marks - Video: [torque slip characteristics of 3 phase induction motor | torque speed characteristics |motor control](https://www.youtube.com/watch?v=g4jfjVvgFrs) — Mind of Engineer - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/basic_electrical_and_electronics_engineering/max-torque-induction-motor - Whole chapter: https://www.bctengineer.com/md/semester_1/basic_electrical_and_electronics_engineering/chapter-2 ## Question Derive the condition for maximum torque in an induction motor. Describe the torque slip characteristics of a three-phase induction motor. ## Short answer Maximum torque occurs when rotor resistance equals rotor reactance, i.e., slip s_m = R2/X2. Tmax is independent of rotor resistance. The torque-slip curve shows starting torque, peak at s_m, and a stable linear region for s < s_m where torque is proportional to slip. ## Answer #### Condition for maximum torque The torque of a three-phase induction motor is $$ T=\frac{3}{\omega_s}\cdot\frac{sE_2^2R_2}{R_2^2+(sX_2)^2} $$ where $E_2$, $R_2$, $X_2$ are the rotor emf, resistance and reactance per phase at standstill, and $s$ is the slip. For a constant supply, $E_2$ is constant. Differentiate with respect to $s$ and equate to zero (equivalently, maximise $\frac{sR_2}{R_2^2+s^2X_2^2}$): $$ \frac{d}{ds}\left[\frac{sR_2}{R_2^2+s^2X_2^2}\right]=0 \Rightarrow R_2^2+s^2X_2^2-2s^2X_2^2=0 $$ $$ R_2^2=s^2X_2^2 \Rightarrow s_m=\frac{R_2}{X_2} $$ **Result:** torque is maximum when the rotor resistance equals the rotor reactance at that slip ($R_2=s_mX_2$). Substituting, $$ T_{max}=\frac{3}{\omega_s}\cdot\frac{E_2^2}{2X_2} $$ $T_{max}$ is independent of the rotor resistance, but the slip at which it occurs ($s_m=R_2/X_2$) depends on it. #### Torque-slip characteristic ```text Torque | Tmax | .--*--. | _.-'' \ | _.-'' \ | _.-'' \ *--''' \ Tst \ +--------------------------------*---> slip s=1 sm s=0 (start) (sync) ``` - At $s=1$ (standstill) the torque is the starting torque $T_{st}$, which is lower than $T_{max}$. - From $s=1$ to $s_m$, torque rises to $T_{max}$ (unstable region; here $R_2\ll sX_2$, so $T\propto 1/s$). - Between $s_m$ and zero (the normal working region), $T\propto s$ and the curve is nearly straight and steep; a small slip (2-5%) gives full-load torque, so the speed is nearly constant. - At $s=0$ (synchronous speed) the torque is zero. - Increasing the rotor resistance shifts $s_m$ towards 1 and improves the starting torque; when $R_2=X_2$, $T_{st}=T_{max}$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the slip at maximum torque? A. s_m = X2/R2 B. s_m = R2 X2 C. s_m = R2/X2 D. s_m = 1 always Answer: C. Setting dT/ds = 0 gives R2 = s X2, so s_m = R2/X2. (Key point: Slip at max torque) 2. What happens to Tmax if rotor resistance R2 is increased? A. Tmax increases B. Tmax is unchanged but occurs at a larger slip C. Tmax decreases D. Tmax occurs at a smaller slip Answer: B. Tmax = 3E2^2/(2 ws X2) is independent of R2, while s_m = R2/X2 shifts towards 1. (Key point: Effect of R2 on Tmax) 3. When does the starting torque equal the maximum torque? A. When R2 = 0 B. When R2 is much smaller than X2 C. When s = 0 D. When R2 = X2 (s_m = 1) Answer: D. If R2 = X2 the peak occurs at s = 1, i.e. at standstill. (Key point: Starting torque) 4. In the normal region between s_m and zero slip, how does the motor behave? A. Torque is nearly proportional to slip and speed stays nearly constant B. Torque falls as slip increases C. Torque is constant at all slip D. The region is unstable Answer: A. There T is proportional to s, the curve is steep and small slip (2 to 5%) gives full-load torque; the region from s = 1 to s_m is the unstable one. 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