--- title: "T-S characteristics and effect of rotor resistance" subject: "Electrical Machines" chapter: 5 key: "semester_4/electrical_machines:ts-characteristics-rotor-resistance-effect" source: https://www.bctengineer.com/q/semester_4/electrical_machines/ts-characteristics-rotor-resistance-effect site: BCT Engineering --- # T-S characteristics and effect of rotor resistance Electrical Machines · Chapter 5: Three Phase Induction Machines - Asked in: 2072 Asoj; 2068 Bhadra, 8 marks (asked 2 times) - Video: [Electrical Machines | Lec 6C | Three-Phase Induction Motor | Rotor Quantities: Slip, Torque, etc](https://www.youtube.com/watch?v=s-wCI5jDu3g) — Engineering Devotion - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/electrical_machines/ts-characteristics-rotor-resistance-effect - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machines/chapter-5 ## Question Explain the torque-slip characteristics of a 3-phase induction motor, indicating the starting torque, maximum torque and the operating region. How does rotor resistance affect the torque-slip characteristics? ## Short answer Torque-slip curve rises linearly in low-slip stable region, peaks at sm = R2/X2, then falls. Starting torque occurs at s=1; maximum torque is independent of rotor resistance. Increasing rotor resistance shifts peak toward s=1 and increases starting torque up to Tmax. ## Answer The torque-slip characteristic shows how the torque developed by a 3-phase induction motor varies with slip (from $s = 1$ at standstill to $s = 0$ at synchronous speed), at rated voltage and frequency. Torque of a 3-phase induction motor at slip $s$ (per-phase equivalent): $$ T = \frac{3}{\omega_s}\cdot\frac{sE_2^2R_2}{R_2^2 + (sX_2)^2}\ \text{N·m} $$ where $E_2$ and $X_2$ are the rotor emf and reactance per phase at standstill and $\omega_s$ is the synchronous speed in rad/s. ```text Torque T | Tmax | .--*--. | Tst / | \ Operating | *--' | \ region | | \___ ___ (stable) | | +------------+-----------+----> slip s 1 (standstill) sm 0 (sync) ``` #### Regions of the curve 1. **Starting point ($s = 1$):** the torque is the **starting torque** $T_{st}$. Here the rotor reactance $sX_2$ is large, so the rotor power factor is low and the torque is less than the maximum. 2. **Low-slip (operating) region ($0 < s < s_m$):** the rotor reactance is small, so $sX_2\ll R_2$ and $T\approx \dfrac{3sE_2^2}{\omega_sR_2}\propto s$. The curve is almost a **straight line**. The motor works **stably** here: if the load rises, the speed falls slightly, slip increases and torque increases to meet it. The full-load slip is only about 2–5%. 3. **Maximum torque point ($s = s_m$):** the pull-out or breakdown torque $T_{max}$ occurs at $s_m = R_2/X_2$ and has the value $$ T_{max} = \frac{3E_2^2}{2\omega_sX_2} $$ 4. **High-slip region ($s_m < s < 1$):** torque decreases as slip increases because $sX_2$ dominates ($T\propto 1/s$). This region is **unstable** for normal loads; if the load torque exceeds $T_{max}$, the motor stalls. #### Effect of rotor resistance $T_{max}$ does **not** depend on $R_2$, but the slip at which it occurs does: $s_m = R_2/X_2$. ```text T | R2 low R2 med R2 high | .-*-. .-*-. .-*-. | / \ / \ / \ | / \ / \ / \ +---+--------+---------+--+--------+--> s ``` - Increasing $R_2$ (wound-rotor motor with external resistance) moves the peak toward higher slip (toward $s = 1$). When $R_2 = X_2$, $s_m = 1$ and **maximum torque occurs at starting**. - The **starting torque increases** (up to $T_{max}$ when $R_2 = X_2$; more resistance beyond that reduces it again) and the **starting current falls** (better rotor power factor). - The slope in the operating region becomes less steep, so the speed regulation worsens (more slip for the same torque), and the running efficiency falls due to $I_2^2R_2$ loss. - Hence external rotor resistance is used for starting and speed control in slip-ring motors, and then cut out for running. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. At what slip does maximum torque occur? A. s = X2/R2 B. s = R2/X2 C. s = 1 always D. s = 0 Answer: B. Setting dT/ds = 0 gives s_m = R2/X2. (Key point: Max torque slip) 2. Increasing rotor resistance R2 does what to Tmax? A. Increases it B. Decreases it C. Does not change it, only shifts the slip at which it occurs D. Makes it zero Answer: C. Tmax = 3E2^2/(2 ws X2) is independent of R2. (Key point: Rotor resistance and Tmax) 3. Why is the low-slip region stable? A. Torque falls with slip B. If load rises, speed falls, slip and torque increase to match C. Torque is constant D. Rotor reactance dominates Answer: B. There T is proportional to s, so torque rises with slip and balances the load. (Key point: Operating region) 4. Adding external rotor resistance at start gives: A. Higher starting torque and lower starting current B. Lower starting torque and higher current C. Higher current only D. No change Answer: A. Better rotor power factor raises starting torque; higher impedance cuts current. At R2 = X2, Tmax occurs at start. 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