--- title: "Variable frequency control of induction motor" subject: "Utilization of Electrical Energy" chapter: 3 key: "semester_7/utilization_of_electrical_energy:induction-motor-variable-frequency-control" source: https://www.bctengineer.com/q/semester_7/utilization_of_electrical_energy/induction-motor-variable-frequency-control site: BCT Engineering --- # Variable frequency control of induction motor Utilization of Electrical Energy · Chapter 3: Control of Electric Drive - Asked in: 2073 Shrawan - Video: [Variable Frequency Drives Explained - VFD Basics IGBT inverter](https://www.youtube.com/watch?v=yEPe7RDtkgo) — The Engineering Mindset - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/utilization_of_electrical_energy/induction-motor-variable-frequency-control - Whole chapter: https://www.bctengineer.com/md/semester_7/utilization_of_electrical_energy/chapter-3 ## Question How three phase induction motor has controlled by variable frequency method? Explain with necessary mathematical relation and figures. ## Short answer Variable frequency control changes synchronous speed by changing stator frequency; below base speed V/f is kept constant to maintain flux and constant maximum torque. Above base speed voltage is fixed, flux weakens, giving constant power operation, and voltage boost compensates stator resistance drop at low frequency. ## Answer In **variable frequency control**, the stator supply frequency is changed by an inverter so that the synchronous speed $N_s = 120f/P$ changes, and the motor speed changes with it at small slip. To keep the flux constant, the voltage is changed in proportion to frequency (**V/f control**). #### Why voltage must change with frequency Stator induced emf: $E_1 = 4.44\, f\, N_1 \phi\, k_w$, so $$ \phi \propto \frac{E_1}{f} \approx \frac{V}{f} $$ - If $f$ is reduced at constant $V$, the flux rises, the core saturates and the magnetising current becomes very large. - If $f$ is raised at constant $V$, the flux falls and the torque capability falls. Hence, below base frequency, $V/f$ is kept constant. #### Torque relations Neglecting the stator impedance, the torque at slip $s$ (slip speed $\omega_{sl} = s\omega_s$) is $$ T = \frac{3}{\omega_s}\frac{V^2 (R_2'/s)}{(R_2'/s)^2 + X_2'^2} $$ With $X_2' \propto f$, the maximum torque is $$ T_{max} = \frac{3}{2\omega_s}\frac{V^2}{X_2'} \propto \left(\frac{V}{f}\right)^2 $$ So with constant $V/f$, $T_{max}$ is constant at all frequencies, and the slip speed at $T_{max}$, $s_m\omega_s = R_2'\omega_s/X_2'$, is also constant. The speed–torque curves simply shift parallel to each other. #### Operating regions ```text V, T, P | ____________ V (rated) | /| | / | T falls ~1/f | / |____________ P constant |/______|____________ T constant +-------+----------> f base f (50 Hz) constant torque | constant power (V/f const) | (V fixed, field weak) ``` - **Below base speed:** $V/f$ constant; constant torque region. - **Above base speed:** the voltage cannot exceed rated, so $V$ is fixed and $f$ is increased; the flux falls ($\phi \propto 1/f$), giving the constant power (field weakening) region with $T_{max} \propto 1/f^2$. - At very low frequency the stator $I_1R_1$ drop is significant, so a **voltage boost** is added to keep the flux. #### Converter ```text ac -> [rectifier] -> [dc link] -> [PWM inverter] -> IM V/f reference and firing logic ``` A PWM voltage source inverter (or a cycloconverter for very low speeds) supplies variable voltage and frequency. **Advantages:** wide speed range above and below base speed, high efficiency (low slip), smooth starting with full torque at low current, regenerative braking possible; suitable for cage motors. **Drawbacks:** costly power electronics and harmonics. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is voltage changed in proportion to frequency? A. To keep slip constant B. To keep flux (proportional to V/f) constant C. To keep rotor resistance constant D. To increase speed Answer: B. Reducing f at constant V saturates the core and raises magnetising current. (Key point: Why V/f) 2. With V/f constant, what happens to Tmax? A. Remains constant, curves shift parallel B. Falls with f C. Rises with f D. Proportional to 1/f^2 Answer: A. Tmax is proportional to (V/f)^2, so constant. (Key point: Tmax) 3. Above base frequency the motor runs in: A. Constant torque B. Constant power region as flux falls and Tmax is proportional to 1/f^2 C. Regeneration only D. Constant V/f Answer: B. V is limited to rated, so flux falls with 1/f. (Key point: Above base) 4. Why is voltage boost added at very low frequency? A. Because core losses vanish B. Because the stator I1R1 drop becomes significant and would weaken flux C. To reduce harmonics D. To raise frequency Answer: B. At low f the applied V is small compared to I1R1. 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