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Power Electronics

Electric drives: DC motors, induction motors and field-oriented control

The DC motor and equality of the constants proved, the induction motor and slip, constant V/f control proved, permanent-magnet synchronous motors, Clarke and Park transforms, regenerative braking.

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About half of the electrical energy consumed in the world ends up in electric motors: pumps, fans, compressors, conveyors, vehicles. A drive is the combination of a motor, the converter supplying it and the control, and it makes it possible to regulate speed and torque while saving energy. In this lesson we look at the DC motor with the proof that its two constants are equal, the induction motor with slip and the torque-speed characteristic, constant voltage-over-frequency control with its justification, and field-oriented control of permanent-magnet synchronous motors with the Clarke and Park transforms.

The DC motor

The permanent-magnet DC motor has two fundamental relations: the torque is proportional to the armature current, T = k_t\,i, and the back-EMF is proportional to speed, e = k_e\,\omega. The armature circuit is v = R\,i + L\,di/dt + e.

Theorem. In SI units the two constants coincide: k_t = k_e.

Proof. The electrical power absorbed by the back-EMF, e\,i, is what the motor converts into mechanical power, T\,\omega (the resistive losses R\,i^2 are separate). Equating: k_e\,\omega\,i = k_t\,i\,\omega, hence k_e = k_t (in V·s/rad and in N·m/A, which are the same unit). ∎

In steady state, at no load, the speed is about v/k_e: speed is regulated by regulating the voltage, with a bridge DC-DC converter (lesson on converters), which with four quadrants also allows braking with energy recovery. The DC motor is easy to control but has brushes and a commutator, which wear out: today it has almost been replaced by AC motors driven by inverters.

The induction motor

The induction (asynchronous) motor, invented by Galileo Ferraris and Nikola Tesla, is the most widespread in industry: rugged, cheap, with no sliding contacts. Three windings on the stator, fed by three-phase currents, create a rotating magnetic field at the synchronous speed \omega_s, proportional to the frequency. The rotor, a “cage” of short-circuited bars, turns slightly slower: the field sliding past the bars induces currents in them, and currents in the field produce torque. The relative speed difference is the slip

    \[ s = \frac{\omega_s - \omega}{\omega_s} \]

typically a few per cent at full load. Without slip there is no induction, and hence no torque.

Torque-speed characteristic of the induction motor
Starting from standstill (slip 1) the torque grows with speed up to its maximum, then collapses towards zero at synchronous speed. The operating point is where the curve meets the load torque, on the steep part near synchronism, where the speed changes little with load.

Fed directly from the mains the induction motor runs at almost fixed speed, and at start-up it draws currents 5-7 times the rated one.

Constant V/f control

With an inverter (lesson on inverters) the frequency, and therefore the synchronous speed, can be varied. But the voltage must be varied with it.

Theorem. Neglecting the stator resistive drop, to keep the motor flux constant the voltage must be proportional to the frequency.

Proof. The stator voltage equals the derivative of the flux linkage, v = d\lambda/dt. For a sinusoidal flux \lambda = \Lambda\sin\omega t one gets v = \omega\Lambda\cos\omega t: the voltage amplitude is V = \omega\Lambda, so \Lambda = V/\omega. The flux stays constant if V/\omega, i.e. V/f, is constant. ∎

Lowering the frequency without lowering the voltage the flux would grow and the iron would saturate, with enormous currents; raising it without raising the voltage the motor would weaken. With constant V/f the torque-speed curve shifts rigidly as the frequency changes, and the motor delivers rated torque at every speed. It is the simplest control, used in pumps and fans, where regulating speed instead of throttling the flow with a valve often saves 30-50 % of the energy.

Synchronous motors and field-oriented control

In permanent-magnet synchronous motors (PMSMs, and their relatives called brushless) the rotor carries the magnets and turns exactly at the speed of the field. They have very high efficiency and power density, and they are the motors of almost all electric cars, drones and robots. To control their torque well, field-oriented control (FOC) is used.

The idea is to turn the three sinusoidal currents into two constant quantities. The Clarke transform combines the three currents into two components i_\alpha, i_\beta on fixed axes at 90°; the Park transform brings them onto axes d, q that rotate with the rotor, measured by a position sensor:

    \[ i_d = i_\alpha\cos\theta + i_\beta\sin\theta,\qquad i_q = -i_\alpha\sin\theta + i_\beta\cos\theta \]

For balanced three-phase currents of amplitude I and phase \varphi relative to the rotor, i_\alpha = I\cos(\theta + \varphi) and i_\beta = I\sin(\theta + \varphi), and substituting one obtains i_d = I\cos\varphi, i_q = I\sin\varphi: two constants, like the currents of a DC motor. The torque is T = \frac{3}{2}p\,\lambda_m\,i_q, with p the pole pairs and \lambda_m the magnet flux: it is controlled directly by i_q, while i_d is kept at zero (or negative, to weaken the flux and exceed base speed). Two PI controllers on i_d and i_q, the inverse transforms and space-vector modulation (lesson on inverters) complete the scheme, executed by a microcontroller tens of thousands of times a second.

The result is precise, very fast torque, from zero speed to maximum. Reversing the sign of i_q the motor brakes and becomes a generator, returning energy to the battery: this is the regenerative braking of electric cars.

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