Variable frequency control of induction motor

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

 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

 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.