--- title: "Double field theory; not self-starting" subject: "Electrical Machine" chapter: 7 key: "semester_4/electrical_machine:dfrt-not-self-starting-proof" source: https://www.bctengineer.com/q/semester_4/electrical_machine/dfrt-not-self-starting-proof site: BCT Engineering --- # Double field theory; not self-starting Electrical Machine · Chapter 7: Fractional Kilowatt Motors - Asked in: 2073 Bhadra, 8 marks; 2069 Bhadra (asked 2 times) - Video: [1. DOUBLE FIELD REVOLVING THEORY || WHY SINGLE PHASE INDUCTION MOTOR ISN'T SELF STARTING?](https://www.youtube.com/watch?v=_3MSZwP9fdg) — ELECTRICAL NEST - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/electrical_machine/dfrt-not-self-starting-proof - Whole chapter: https://www.bctengineer.com/md/semester_4/electrical_machine/chapter-7 ## Question Explain double field revolving theory with reference to a single phase induction motor and prove that a single phase induction motor is not self starting. ## Short answer Pulsating field resolves into two equal counter rotating fields. At rest, both have slip 1, so rotor impedances and torques are equal and opposite; net torque zero proves no self-starting. ## Answer A single-phase winding carrying a.c. produces a pulsating (alternating) field, φ = φ_m cos ωt, along a fixed axis. It does not rotate. The double revolving field theory says that such a pulsating field is equal to two rotating fields of half the magnitude, turning in opposite directions at synchronous speed. #### Resolution of the pulsating flux $$\phi_m\cos\omega t = \frac{\phi_m}{2}e^{j\omega t} + \frac{\phi_m}{2}e^{-j\omega t}$$ i.e. the sum of two vectors, each of length φ_m/2: - forward field φ_f = φ_m/2, rotating at +N_s (say anticlockwise) - backward field φ_b = φ_m/2, rotating at −N_s (clockwise) ```text ωt = 0 ωt = 90° ωt = 180° ff and fb ff and fb point ff and fb both both point opposite ways, at point the other along +axis: 90° to the axis: way along the axis: sum = φm sum = 0 sum = -φm ff = fb = φm/2, turning in opposite directions; resultant = φm cos ωt along the fixed axis ``` The components at right angles to the axis cancel at every instant, and the components along the axis add to give φ_m cos ωt. #### Working of the motor Let the rotor turn in the forward direction at speed N. The slip with respect to the forward field is s_f = (N_s − N)/N_s = s. The slip with respect to the backward field is s_b = (N_s + N)/N_s = 2 − s Each field induces rotor currents and develops its own torque, T_f (forward) and T_b (backward). The net torque is T = T_f − T_b. | Condition | Slip f / b | Result | |---|---|---| | Standstill (s = 1) | 1 / 1 | T_f = T_b, so net torque is zero | | Rotor started forward | s / 2 − s | T_f > T_b, net forward torque | | Rotor near N_s (s → 0) | ~0 / ~2 | T_f large, T_b small | ```text Torque | | ffffff | fffff f | ffffff ********f | fffffffffff **** *f | fffffffffffffffffffffff **** *f |f ****** --+b*------------------------*****------------------------*f> N | ****** b | b* **** bbbbbbbbbbbbbbbbbbbbbbb | b* **** bbbbbbbbbbb | b******** bbbbbb | b bbbbb | bbbbbb | -Ns 0 +Ns f = Tf (forward field), b = Tb (backward field, acting in reverse) * = net torque T = Tf - Tb: zero at N = 0, positive for forward running, and zero again just below +Ns ``` The net torque is zero at standstill, so the single-phase induction motor has no starting torque. If the rotor is given a push in either direction, it keeps rotating in that direction, since the torque of the field rotating that way (T_f or T_b) becomes larger than the other. Near synchronous speed the backward torque is small, because the slip 2 − s for the backward field is near 2: the rotor current it causes is at nearly twice the supply frequency, lags heavily and gives little torque. The net torque falls to zero at a speed slightly below N_s, so the motor never reaches N_s. #### Proof that the motor is not self-starting Let the rotor be at rest, N = 0. 1. Slip with respect to the forward field: s_f = (N_s − 0)/N_s = 1. 2. Slip with respect to the backward field: s_b = (N_s + 0)/N_s = 1. 3. The two fields are equal, φ_f = φ_b = φ_m/2, and the rotor impedance is the same for both (same slip, same frequency). So the rotor currents and torques are equal: T_f = k (φ_m/2)² · R₂ /(R₂² + X₂²) = T_b 4. The two torques act in opposite directions, so the net starting torque is **T_st = T_f − T_b = 0** The rotor is acted on by two equal and opposite torques and stays at rest. Hence the single-phase induction motor is not self-starting. It needs an auxiliary starting arrangement (a second winding, a capacitor or shading rings) that produces a phase difference between two fluxes, and so a rotating field, at starting. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A single-phase winding on a.c. produces a field that is equivalent to: A. a single field rotating at N_s B. two equal half-magnitude fields rotating in opposite directions at N_s C. two fields of full magnitude rotating at N_s/2 D. a constant d.c. field Answer: B. The pulsating field resolves into a forward and a backward field of phi_m/2 each. (Key point: Equivalent rotating fields) 2. At standstill (N = 0), what are the slips s_f and s_b? A. s_f = 0 and s_b = 2 B. s_f = 1 and s_b = 2 C. s_f = 2 and s_b = 0 D. s_f = 1 and s_b = 1 Answer: D. s_f = (N_s - 0)/N_s = 1 and s_b = (N_s + 0)/N_s = 1. (Key point: Slips at standstill) 3. Why is T_st = T_f - T_b = 0 at standstill? A. The rotor impedance is zero B. The fields are equal and the rotor impedance is the same for both, so the torques are equal and opposite C. The forward field is absent at standstill D. The backward torque is larger than the forward torque Answer: B. Equal flux phi_m/2 and equal slip give the same rotor current and torque magnitude, acting in opposite directions. (Key point: Proof of zero starting torque) 4. What is needed to make the motor start? A. Increase the supply voltage B. A starting arrangement giving a phase difference between two fluxes to create a rotating field C. Lower the supply frequency D. Remove the rotor squirrel cage Answer: B. An auxiliary winding, capacitor or shading ring produces a rotating field at starting, so a net starting torque appears. (Key point: Remedy for starting) ## Related in Electrical Machine - [Single phase motor not self-starting; DFRT](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Aspim-not-self-starting-dfrt) — Electrical Machines, chapter 7, asked 3× - [Single phase motor not self-starting; two starting methods](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Aspim-not-self-starting-starting-methods) — Electrical Machines, chapter 7 - [DFRT and making single phase motor self-starting](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Adfrt-self-starting-explain) — Electrical Machines, chapter 7, asked 3× - [Reason: single phase motor not self-starting](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Aspim-not-self-starting-reason) — Electrical Machines, chapter 7 - [Double revolving field theory](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machine%3Adouble-field-revolving-theory) — Electrical Machine, chapter 7, asked 3× - [Double field theory; starting methods](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machine%3Adfrt-two-starting-methods) — Electrical Machine, chapter 7 ## Related in other subjects - [Double field revolving theory and a starting method](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Adfrt-one-starting-method) — Electric Machines II, chapter 3, asked 5× - [Why single-phase IM is not self starting; methods](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Asingle-phase-not-self-starting-methods) — Electric Machines II, chapter 3, asked 2× - [Double field revolving theory](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Adouble-field-revolving-theory) — Electric Machines II, chapter 3, asked 5× - [Not self starting and permanent split capacitor motor](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Anot-self-starting-psc-motor) — Electric Machines II, chapter 3