*— Lee Stephens*

**Edited by Leland Teschler**

One important aspect of how motors and loads interact is the amount of compliance between them. A compliant or soft system is one that lets the load move while the motor is stationary. A compliant system has a low bandwidth, a low-frequency response. Conversely, a stiff system has a high bandwidth. High-speed, precision mechanical systems need high stiffness — and thus, high bandwidth.

For this reason the bandwidth is a major figure of merit in a servosystem. And the motor inertia ratio affects the bandwidth of the system. I'll show how with a simple mathematical model.

Without regard to specific motors or amplifiers, I've graphed a transfer function of a load and compared its behavior with two different motors. Holding all other parameters of the motors constant lets us see what the inertia ratio of the motor does to the system. I've assumed the rotor inertia of Motor 1 is 0.0002 oz-in.-sec^{2 }while the rotor inertia of Motor 2 is 0.001 oz-in.-sec^{2}. This equates to a change from a 5:1 ratio, to a 1:1 ratio.

A simultaneous plot of the two transfer functions shows the frequency response of the system and, thus, the difference in their bandwidth. Also note the gain of the system is significantly less with the higher inertia. The –3-dB point is 133 Hz for Motor 1 while Motor 2 has a -3-dB point at 80 Hz.

Suppose now that you need a system responsive to 100 Hz. This would not be possible with the servosystem and parameters selected in Motor 2. The general rule of thumb is that at –3 dB and 45° phase shift, you have lost control of the system. The plot shows that we have lost gain here. Can we also conclude we have lost phase? It would seem so. Here is the phase data applied in graphical format for the two cases.

There is not enough room here to show the equations and values assumed for this example. So I've posted them in the online version of this column.

*Lee Stephens is a systems engineer with ***Danaher Motion Corp. ***Got a question about motion control or mechatronics? Ask Lee via e-mail at [email protected].*