# Motor Inductance & PWM Frequency

This paper discusses the effect of servo drive Pulse Width Modulation (PWM) frequency on heat generating current ripple in a permanent magnet motor. The basic idea of PWM is introduced and the RL circuit, which models the motor winding is reviewed. The concept of PWM average voltage and current is discussed. Techniques to reduce current ripple are presented.

## PWM Concept

A linear drive can be modelled as a variable resistor in series with the motor winding. Energy (I^{2}R) is dissipated in the effective resistance resulting in very low efficiency. A PWM drive delivers current to the motor in pulses via electronically controlled switches – transistors in the drive power stage. No energy is lost in a perfect switch. When the switch is open, no current flows, so I^{2}R is zero. When the switch is closed, current flows but there is zero resistance so again I^{2}R is zero. The effective current applied to the motor winding depends on the duty cycle of switching – time_{closed} relative to time_{open}.

### Inductance Refresher – RL Circuit

Inductance in an electrical system is analogous to inertia in a mechanical system. Inertia resist change to the motion of an object. Inductance resists current change. Energy is stored in the magnetic field of the inductor in the same way kinetic energy is stored in a rotating mass. In Fig. 2, when switch A is closed the inductor opposes current flow. When A is opened and B closed, the inductor maintains current flow until the energy stored in the magnetic field is depleted. The time taken to reach maximum current is approximately 5 x L/R. L/R is known as the time constant of the resistor and inductor combination.

## Average Voltage and Current

Fig. 3 shows modulation of the average voltage as the PWM duty cycle changes. Starting at a 50% duty cycle the motor winding sees half the bus voltage. As the PWM t_{on} is reduced (t_{off} increased) the voltage seen by the motor is reduced accordingly.

The motor winding can be modeled as an RL circuit. During t_{off}, the energy stored in the inductor magnetic field maintains current flow through “freewheeling” diodes in the drive power stage. This effectively smoothes the current as the voltage applied to the motor winding is turned on and off.

## Current Ripple

If we look more closely at the average current, we see a ripple as shown in Fig. 4. The magnitude of the ripple depends on the time constant L/R. A lower inductance motor winding has a faster risetime and can reach a larger value, in a given amount of time, than a higher inductance winding.

The motor can only respond to the average current to produce useful torque. The current ripple, however, has an RMS component which causes heating in the motor winding. To understand the RMS value of a waveform it is instructive to look at a more familiar example. Fig. 5 shows two alternating current sinewaves of different magnitudes centered around zero average current. The higher magnitude waveform clearly has a higher RMS current which would generate more heat (I^{2}_{RMS} x R) in a resistive heating element.

Ripple current in the motor winding also generates a changing magnetic field which induces eddy currents in the motor iron which further add to thermal losses.

### Current Ripple and PWM Frequency

The pk-pk magnitude of the current ripple (CR) is given by the following formula:

${{C}{R}}_{{p}{k}{\u2013}{p}{k}}{=}\frac{{{(}{V}}_{{B}{U}{S}}{\u2013}{{V}}_{{B}{E}{M}{F}}{\u2013}{{V}}_{{I}{R}}{)}{*}{{P}{e}{r}{i}{o}{d}}_{{P}{W}{M}}}{{2}{*}{L}}$

As an example, assume V_{BUS} = 350 V, V_{BEMF} ≈ 0 (low speed) V_{IR} ≈ 0 (low resistance), Period_{PWM} = 0.1 msec (10 kHz PWM frequency). Also assume the current ripple approximates a triangular wave with an RMS value of 0.6 x CR_{pk}.

A moderate to high inductance winding of 10 mH would have a CR_{RMS} of 0.5 A. A low inductance winding of 0.5 mH would have a CR_{RMS} of 10 A. Considering that heat produced in the motor winding is proportional to I^{2}_{RMS}, the losses generated in low inductance motors can be very significant.

From the equation, if we increase the PWM frequency (shorter period) CR_{RMS} reduces proportionally. High performance drives have higher PWM frequencies (20-100 kHz) to reduce the current ripple in low inductance motors. As shown in Fig. 6., for the same duty cycle, doubling the PWM frequency reduces the amount of time for the current to rise relative to the L/R time constant. In this way current ripple magnitude is reduced.

An additional benefit of high (>20 kHz) PWM frequency is operation beyond the audible frequency range. Audible noise from PWM drives can be a problem in the quiet environments typically found in the medical and semiconductor industries.

## PWM Frequency and Switching Losses

The transistors in the drive power stage are not ideal switches. A finite on-resistance causes conduction losses. The on-off/off-on transitions are not infinitely fast, so the transistors conduct current during the PWM edges generating switching losses. Conduction losses are not affected by PWM frequency but as PWM frequency increases, switching losses increase as there are more edge transitions in a given time.

Switching losses can be significant. The magnitude of the losses depends on the transistor technology and whether PWM edge shaping is used to soften the transitions for reduced radiated emissions. If a drive has a programmable PWM switching frequency, the datasheet will incorporate power loss derating curves so that an appropriate heatsink can be selected.

### External Inductors

If a drive does not support higher PWM rates, adding inductors in series with the motor winding is an option. Added inductance can significantly reduce ripple current but it increases resistance with a consequent lowering of efficiency. External inductors increase wiring complexity, and they are very bulky in higher power applications. Note that the servo drive current loop must be tuned after the inductors are installed.

## Conclusion

Motor heating due to current ripple can be significant in low inductance motors. Increasing PWM switching frequency reduces current ripple but increases switching losses in the drive. A larger heatsink may be required to compensate for switching losses. External inductors can be employed but this is considered a last resort.