--- title: "Double field revolving theory and a starting method" subject: "Electric Machines II" chapter: 3 key: "semester_5/electric_machines_ii:dfrt-one-starting-method" source: https://www.bctengineer.com/q/semester_5/electric_machines_ii/dfrt-one-starting-method site: BCT Engineering --- # Double field revolving theory and a starting method Electric Machines II · Chapter 3: Fractional Kilowatt Motors - Asked in: 2081 Bhadra, 8 marks - Video: [Double Field Revolving Theory - Single Phase Induction Motor - Electrical Machines 3](https://www.youtube.com/watch?v=z0x1DM4RZRo) — Ekeeda - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/electric_machines_ii/dfrt-one-starting-method - Whole chapter: https://www.bctengineer.com/md/semester_5/electric_machines_ii/chapter-3 ## Question State and explain double field revolving theory of single phase induction motor with detailed diagram and explain any one starting method. ## Short answer The double field revolving theory resolves a pulsating field into two equal counter-rotating fields at synchronous speed, yielding zero net torque at standstill. Once started, forward torque dominates. A capacitor-start method uses an auxiliary winding with a series capacitor and centrifugal switch to create phase shift and produce starting torque. ## Answer **Double field revolving theory** states that a pulsating (alternating) magnetic field of maximum value $\phi_m$ can be replaced by **two fields, each of constant magnitude $\phi_m/2$, rotating in opposite directions at synchronous speed** $N_s = 120f/P$. #### Pulsating field resolved The single-phase stator winding produces a flux that only alternates along its axis: $$ \phi = \phi_m \cos\omega t $$ This is equal to the sum of two rotating phasors $\phi_f$ (forward) and $\phi_b$ (backward), each $\phi_m/2$, turning at $\omega$ in opposite directions: $$ \phi = \frac{\phi_m}{2}e^{j\omega t} + \frac{\phi_m}{2}e^{-j\omega t} = \phi_m\cos\omega t $$ ```text wt = 0 wt = 90 deg wt = 180 deg ^ ^ <-- --> v v | | phi_f phi_b | | sum = phi_m sum = 0 sum = -phi_m (both along (cancel) (both reversed) the axis) phi_f turns anticlockwise, phi_b clockwise ``` At every instant the vertical components add and the horizontal components cancel, so the resultant is the pulsating field. #### Torque at standstill - not self-starting - At standstill each rotating field induces equal rotor currents (slip = 1 for both), so the forward torque $T_f$ and backward torque $T_b$ are equal and opposite. - Net starting torque $T = T_f - T_b = 0$. Hence a single-phase induction motor is **not self-starting**. #### Torque when running If the rotor turns at speed $N$ in the forward direction: $$ s_f = \frac{N_s - N}{N_s} = s, \qquad s_b = \frac{N_s + N}{N_s} = 2 - s $$ Using the induction motor torque expression for each field: $$ T_f = \frac{K s R_2}{R_2^2 + (sX_2)^2}, \qquad T_b = -\frac{K (2-s) R_2}{R_2^2 + \left((2-s)X_2\right)^2} $$ $$ T = T_f + T_b $$ - Near normal speed $s$ is small, $2 - s \approx 2$: the backward rotor current sees a large frequency $(2-s)f$ and high reactance, so $T_b$ is small and $T_f \gg |T_b|$. The motor keeps running in whichever direction it was started. ```text T | _ | .-' \ forward run | .-' | (Tf > Tb) -Ns ---+----.'--------+---> speed | | .' 0 Ns \_|.' reverse run net T = 0 at N = 0 (mirror image) ``` The resultant torque-speed curve passes through **zero at standstill** and is symmetrical: positive in the forward direction, negative in the reverse direction. #### Equivalent circuit (based on DFRT) The rotor is split into two halves, one for each field: forward half with $0.5R_2'/s$ and $0.5X_2'$, backward half with $0.5R_2'/(2-s)$ and $0.5X_2'$, each with $0.5X_m$ in parallel, in series with the stator $R_1 + jX_1$. ```text o--R1--X1--+--0.5X2'--0.5R2'/s ------+ V | (0.5Xm parallel) forward +--0.5X2'--0.5R2'/(2-s)---+ | (0.5Xm parallel) backward o----------+-------------------------+ ``` Hence, an auxiliary means (split phase, capacitor, shaded pole) is needed only to start the motor; once running, the forward field dominates. #### One starting method: capacitor-start motor Since net starting torque is zero, a second (auxiliary) winding is placed $90^\circ$ (electrical) from the main winding, with a capacitor $C$ in series and a **centrifugal switch** S. ```text 1-ph AC o----+-----------+ | | Main wdg Aux wdg (Im) | | C | | | S (opens at ~75% Ns) o------------+-----------+ ``` 1. The capacitor makes the auxiliary current $I_a$ **lead** $V$, while the main current $I_m$ lags $V$; the phase difference $\alpha$ approaches $90^\circ$. 2. Two space-displaced windings carrying time-displaced currents produce a **rotating field**, giving starting torque $T_{st} = K I_m I_a \sin\alpha$, which is high (about 3-4.5 times full-load torque). 3. At about 70-80 % of synchronous speed the centrifugal switch disconnects the auxiliary winding; the motor then runs on the main winding alone, as explained by DFRT. It is used for compressors, pumps, refrigerators and air-conditioners. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. DFRT replaces a pulsating field phi_m with: A. Two fields phi_m/2 rotating in opposite directions at Ns B. One field phi_m at Ns C. Two fields phi_m rotating the same way D. Two fields of phi_m/2 at Ns/2 Answer: A. The sum of forward and backward phasors reproduces phi_m cos(wt). (Key point: Statement of DFRT) 2. At standstill the net torque is zero because: A. Tf and Tb are equal and opposite with slip 1 for both fields B. Rotor current is zero C. The forward field vanishes D. The motor has a capacitor Answer: A. Each field gives equal rotor current and torque in opposite directions. (Key point: Zero starting torque) 3. In the capacitor-start method, the auxiliary current Ia relative to V is: A. Leading V, while Im lags V, so alpha approaches 90 degrees B. Lagging V more than Im C. In phase with Im D. Zero Answer: A. The capacitor makes Ia lead V. (Key point: Capacitor-start phase relation) 4. After the centrifugal switch opens at 70-80 % Ns, the motor: A. Runs on the main winding alone with the forward field dominating B. Stops C. Runs at double speed D. Runs on the capacitor alone Answer: A. Starting needs the auxiliary winding; running does not. (Key point: Switch and running) ## Close variants set in other exams - [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× ## Related in Electric Machines II - [DFRT, starting methods and capacitor start-run motor](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Adfrt-starting-methods-csr) — Electric Machines II, chapter 3 - [Single phase IM principle, torque-slip and starting](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Asingle-phase-principle-torque-slip) — Electric Machines II, chapter 3 - [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× - [DFRT and equivalent circuit of single phase IM](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Adfrt-equivalent-circuit) — Electric Machines II, chapter 3 - [Split-phase motor winding currents at starting](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Asplit-phase-starting-currents) — Electric Machines II, chapter 3 - [DFRT explanation of no starting torque](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machines_ii%3Adfrt-not-self-starting) — Electric Machines II, chapter 3 ## Related in other subjects - [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; not self-starting](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machine%3Adfrt-not-self-starting-proof) — Electrical Machine, chapter 7, asked 2× - [Double field theory; starting methods](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machine%3Adfrt-two-starting-methods) — Electrical Machine, 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×