# Induction machine drive example with a V / f scalar control

This example shows a induction machine supplied with a PWM inverter and driven with a $V / f$ scalar control, with:

• a dc bus voltage of 600 V,
• a switching frequency of 10 kHz,
• a variable load torque with a square-law characteristic (such as booster pumps and centrifugal fans), with $T_{load} = 0.1 + 0.003 \times \Omega^2$.

## Control structure

The diagram below shows the closed-loop $V / f$ control structure.

This $V / f$ control makes possible to drive the induction machine with a constant flux (considering the stator resistance can be neglected).

## Principle of the control

The electromagnetical torque of an induction machine can be given by:

T_{em} = 3 N_{pp} \left(\frac{V_s}{\omega_s}\right)^2 \frac{R_r / \omega_r}{(R_r / \omega_r)^2 + L_r^2}

The maximum torque - when $\omega_r = R_r / L_r$ - is then:

(T_{em})_{max} = \frac{3 N_{pp}}{2 L_r} \left(\frac{V_s}{\omega_s}\right)^2
• $N_{pp}$ number of pole pairs
• $V_s$ equivalent single-phase voltage
• $\omega_s$ pulsation of the stator currents
• $\omega_r$ pulsation of the rotor currents
• $R_r$ rotor resistance at the stator side
• $L_r$ rotor leakage inductor at the stator side

If the ratio $V_s / \omega_s$ can be kept constant, the maximum value of the torque will be constant for different synchronous speeds $\Omega_s$. This principle is illustrated in the figure below:

For low speeds, the ohmic voltage drop due to the stator resistance cannot be neglected. For high speeds, when the voltage reaches this maximum, this ratio decreases leading to a flux-weakening zone and a decreasing of the torque. This makes possible to reach higher speeds than the nominal speed.

Industrial variable speed drives propose modified $V / f$ control profiles with:

• an adding term for low frequencies to keep a minimum torque,
• a saturation when the voltage reaches its maximum value.

This control profile is illustrated in the global control diagram above in the $V / f$ block. Such a control profile has not been implemented in the demo example as it was not necessary.

## Torque control

The torque can then be controlled with $\omega_r$ according to the first expression of the electromagnetical torque. For small values of $\omega_r$, the torque expression can even be considered linear with $\omega_r$:

T_{em} \approx 3 N_{pp} \left(\frac{V_s}{\omega_s}\right)^2 \frac{\omega_r}{R_r}

## Speed loop

The speed loop involves a classic PI regulator.

## Reference

This circuit example is issued from an example proposed by Christophe Haouy.