Motor NotesPWM Driving of Motors: Relationship between PWM Period and Electrical Time Constant of the Motor

Where PWM driving of brushed DC motors is concerned, it must be born in mind that the PWM period should be set to be sufficiently shorter than the electrical time constant of the motor. In this article, we consider the actual meaning of this “sufficiently shorter period”.

About PWM Periods that are “Sufficiently Shorter” than the Electrical Time Constant of a Motor During PWM Driving

We begin by presenting numerical relationships. Current ripple values are largest when the duty ratio is m=0.5; the following table indicates the relationship between the ratio [motor electrical time constant τ/PWM period tpwm] and the ratio [current ripple/average current].

From the table, we see that in order to hold ripple to 5% or less, the value of τ/tpwm must be 10 or higher; in actuality, however, the value of τ/tpwm must be determined according to the required characteristics. Mathematically, for the PWM period tpwm to be sufficiently short compared with the motor electrical time constant τ means that tpwm/ τ ≈ 0, and so one may consider that τ/tpwm should be greater than about 100.

We derive a numerical relationship expressing this. We begin by reviewing the equivalent circuit when a power supply voltage is applied to a brushed DC motor.

Ea：Supply voltage
Ia：Motor current
R：Motor equivalent resistance
L：Motor equivalent inductance
Ec：Motor generated voltage

The electrical time constant of the motor is a value indicating the current rise characteristic when a voltage is input, and more specifically, is the time that has elapsed when 63.2% of the peak value is reached. As indicated in the equivalent circuit, electrically the motor is a circuit by adding the generated voltage Ec of the motor to a series-connected resistance R and inductance L. The electrical time constant of the motor τ is expressed by L/R. Smaller values of this quantity means that the current waveform rises more rapidly.

With the motor generated voltage at Ec = 0 V, the equation for the transient current i when a voltage Ea is applied in steps to the inductance L and resistance R of the motor equivalent circuit is

 L・(di/dt)＋R・i＝Ea ……（1）

The general solution of this differential equation is

 i＝Ea/R＋A・exp(－R・t/L)     A：Initial value ……（2）

If we assume that at time t = 0 the initial current is i = i_0, then we have

 A＝i_0－Ea/R ……（3）

And therefore

 i＝(Ea/R)・(1－exp(－R・t/L))＋i_0・exp(－R・t/L) ……（4）

Next, the equivalent circuit for when the motor terminals are shorted and current regeneration occurs is shown.

Ia：Motor current
R：Motor equivalent resistance
L：Motor equivalent inductance
Ec：Motor generated voltage

We derive the transient current i for this case. If in equation (2) Ea is set to 0 V, then i becomes

 i＝A・exp(－R・t/L) ……（5）

If it is assumed that an initial current i_0 is flowing at time t = 0, then we have

 A＝i_0 ……（6）

and so

 i＝i_0・exp(－R・t/L) ……（7）

From these equations, if the current at the time of voltage application is i_1, then the transient current flowing in the motor coil during PWM operation is

 i_1＝(Ea/R)・(1－exp(－m・tpwm/ τ ))＋i_01・exp(－m・tpwm/ τ ) ……（8）

Moreover, if the current flowing when the motor terminals are shorted and regeneration occurs is i_2, then we obtain the exponential function

 i_2＝i_02・exp(－(1－m)・tpwm/ τ ) ……（9）

Here Ea is the applied voltage, R is the equivalent resistance of the motor, m is the on-duty ratio (between 0 and 1), tpwm is the PWM period, τ is the electrical time constant of the motor (= L/R), i_01 and i_02 are initial current values, and the motor generated voltage Ec = 0 V.

From the above transient current equations, upon considering a PWM period that is sufficiently short with respect to the electrical time constant of the motor, we find that

 －m・tpwm/ τ ≈ 0 and －(1－m)・tpwm/ τ ≈0

so that

 exp(－m・tpwm/ τ )≈1 and exp(－(1－m)・tpwm/ τ )≈1

and consequently we obtain

 i_1≈i_01, i_2≈i_02

and a constant current always flows. In order for the relationships

 －m・tpwm/ τ ≈0 and －(1－m)・tpwm/ τ ≈0

to obtain, mathematically it is necessary that

 τ /(m・tpwm)>100, τ /((1－m)・tpwm)>100

In terms of the PWM period tpwm, it is thought that approximately

 τ /tpwm>100

be satisfied.

Where current ripple is concerned, when the current is stable, the i_2 initial value is i_1, and because the result for i_2 is the initial value i_0 of i_1, the following three equations obtain.

 i_1＝(Ea/R)・(1－exp(－m・tpwm/ τ ))＋i_0・exp(－m・tpwm/ τ ) ……（10）
 i_2＝i_1・exp(－(1－m)・tpwm/ τ ) ……（11）
 i_2＝i_0 ……（12）

By arranging the equations such that i_1 and i_2 cancel, the relationships of i_0, m, tpwm, and τ can be found, and upon inserting parameters and calculating, the several current values can be determined.

Using these equations, two examples of transient current waveforms in PWM driving when the motor current starts from 0 A are shown. The first is a graph for when Ea = 12 V and R = 6 Ω, τ/tpwm = 10, and tpwm = 100 µs and m is varied; the ripple is largest when m = 0.5.

The next graph is an example in which, once again, Ea = 12 V and R = 6 Ω, but with m = 0.5, tpwm is changed to cause τ/tpwm to change; a larger value of τ/tpwm results in less ripple.

In actuality, when connecting to a power supply and when shorting the motor terminals, one must also take into account the on resistance of the driving circuit output MOSFET, the fact that the regenerated current flows through the MOSFET parasitic diode, and the like.