--- title: "IM equivalent circuit; starting/running torque" subject: "Electrical Machine" chapter: 5 key: "semester_4/electrical_machine:im-equivalent-circuit-starting-running-torque" source: https://www.bctengineer.com/q/semester_4/electrical_machine/im-equivalent-circuit-starting-running-torque site: BCT Engineering --- # IM equivalent circuit; starting/running torque Electrical Machine · Chapter 5: Three Phase Induction Machines - Asked in: 2072 Magh - Video: [Rotor Equivalent circuit | Induction Motor | Lec-44](https://www.youtube.com/watch?v=tcrztOFXXvQ) — Education 4u - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/electrical_machine/im-equivalent-circuit-starting-running-torque - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machine/chapter-5 ## Question Draw equivalent circuit of 3 phase induction motor at stand still and running conditions. Derive the expression for starting torque and running torque. ## Short answer At standstill (s=1), rotor circuit has E2, R2, X2. Starting torque: T_st = (3/ω_s) E2² R2/(R2²+X2²). At any slip s, rotor emf is sE2, reactance sX2; running torque: T = (3/ω_s) s E2² R2/(R2²+s²X2²). Maximum torque occurs at s_m = R2/X2, T_max = 3E2²/(2ω_s X2). ## Answer 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). ```text 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) ```text 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) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. At slip s, what are the rotor emf and reactance? A. sE2 and sX2 B. E2/s and X2/s C. sE2 and X2 D. E2 and X2 Answer: A. Rotor frequency is s f. (Key point: Rotor quantities at slip s) 2. What does R2'(1-s)/s represent? A. Stator loss B. Mechanical output C. Core loss D. Leakage Answer: B. It is the load resistance for mechanical power. (Key point: Load resistance) 3. What is T_st? A. (3/ω_s) E2 R2 B. 3 E2^2/(2 ω_s X2) C. (3/ω_s) E2^2 R2/(R2^2 + X2^2) D. (3/ω_s) E2^2/R2 Answer: C. Put s = 1 into the torque equation. (Key point: Starting torque) 4. What is P2 : P_cu : P_m? A. 1 : (1-s) : s B. s : 1 : (1-s) C. 1 : 1 : 1 D. 1 : s : (1-s) Answer: D. Rotor copper loss is s·P2 and mechanical power (1-s)·P2. 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