PWM frequency and inductance effects on motor current control

Motor inductance, or more appropriately “electrical time constant”, value affects servo drives in many ways. While high inductance values may limit the system bandwidth, low inductance values can lead to control loop instabilities, inaccuracies in current readings, increased power losses and other problems. These issues are especially critical in high speed brushed motors with very low friction and fast dynamics. 

Attention with low inductance motors

Low inductance motors may be uncontrollable with most off-the-shelf servo drives or may require hardware / firmware modifications for proper operation. Always check with the manufacturer if the driver is appropriate. The problem is more notorious with brushed DC motors.

This page highlights the principal effects of high and low motor inductance values on the whole servo drive system and how to deal with these effects.

Understanding the effects of low inductance motors

Simplified motor electrical model

An electric motor can be modeled with an equivalent electric circuit that can be used to calculate required voltages (for multiphase motors this circuit would be a single phase representation):

The resistance R is due to the copper wire used to create the windings, the inductance L is due to the magnetic circuit formed by the primary and secondary circuit and is greatly affected by winding and core construction. The voltage E (the back EMF voltage) is due to the induced voltage from the secondary circuit and is proportional to the change in magnetic field (linked to motor speed). In the case of stepper motors and brushless permanent magnet motors the back EMF (Electro Motive Force) is due to the magnets on the rotor. In the case of brushed DC motors it is due to the permanent magnets on the stator or the field winding. 

[latex]v(t)=R\times i(t) + L\frac{di(i)}{d(t)}+e(t)[/latex]

This a simplified model based on well understood electrical components such as inductors and resistors. The reality is more complex and can only be understood using advanced electromagnetic simulations.

By the way, depending on the rotor construction method (surface mounted permanent magnet or interior permanent magnet), the inductance will not be always constant, it varies depending on rotor position and the construction of stator and rotor.

Because the magnetic material has a much less relative permeability than the surrounding iron, the reluctance difference for flux flowing through the magnet is greater than reluctance of the iron path. As the rotor’s angle advances, the reluctance has a periodic variation. If the inductance is measured on a coil of the stator, it will look something like below.

Effects of the motor inductance on the servo drive

Current ripple may be big, specially at low speeds

The electrical equation of a motor is given by:

[latex]v=L \frac{di}{dt}+I R+K_{e}\omega[/latex]

Where:

• v is the motor applied voltage (generated by the driver)
• L is the phase inductance of the motor (H)
• [latex]\frac{di}{dt}[/latex] is the rate of change of the current (A/s) 
•  R is the phase resistance of the motor (Ω) 
•  I is the current through the motor (A) 
• [latex]K_{e}[/latex] is the motor voltage constant (v/RPM) 
• ω is the speed of the motor (RPM)

The current variation with time can be expressed:

[latex]\frac{di}{dt}=(v - IR - K_{e}\omega)/L[/latex]

The maximum ripple in a PWM modulated driver is obtained with minimum I·R product and zero motor speed. Motor voltage will be assumed to be equal to bus voltage. Therefore, solving the differential equation and using system values.    

[latex]\Delta i_{max} \approx \frac{VBUS}{R} \bigg( 1-e^{\displaystyle-\frac{R}{2Lf_{PWM}}} \bigg)[/latex]