A Design Tip

A Note on Frequency Domain Design

When designing control systems using frequency response techniques there are some general "rules of thumb" that you might want to be aware of and use. One of those rules is the following:

In any closed loop system, the frequency response must cross zero db at a slope less than -40 db/dec. In other words - since slope tends to be at -20 db/dec, -40 db/dec, etc., this means that the slope at the zero db crossing should be -20 db/dec, or that it should tend toward that value.

One implication of this rule is the following.

In any system in which the DC gain must be at a high level (because of SSE requirements,etc.) the gain must roll off at -20 db/dec to beyond the zero db crossing.

Let's look at a few examples.

Consider a sysem with this block diagram.

If we apply the Nyquist stability criterion using Bode' plots we will look at the zero db crossing and check the phase. Here's the Nyquist plot for a system with G(s) = Kp/(s + 1)². This plot is drawn for Kp = 100.

Note that the zero db crossing is at f = 1.6 Hz, and that the phase is around -165^o. Also, note the following conclusion that can be drawn.

The longer the -40 db/dec slope persists - as frequency is increased - the closer the phase gets to -180^o.

As you examine higher frequencies, the slope stays at -40 db/dec and the phase gets closer to -180^o.

Now, consider a slightly different system. This one has G(s) = 1/(s + 1)(.01s + 1).

This Bode' plot has a segment that drops off at -20 db/dec. That's between about f = 0.2 to f = 10. Then from around f = 20 on up, the Bode' plot drops off at -40 db/dec. Note that when the drop off is -20 db/dec the phase tends to be -90^o.

Now, here is a question for you. If you are designing a system, is it better to have the zero db crossing when the magnitude slope is -20 db/dec or when the magnitude slope is -40 db/dec? Your answer should be clear. You want to choose the option that gives the largest phase margin, and that means you want the magnitude slope at the zero db crossing to be -20 db/dec.

Now, can you use this? The answer is that you can use this concept whenever you find the following combination of circumstances.

There is a reason for the DC gain to be large. That is usually related to a SSE/Step Input specification.
You - the designer - have an opportunity to shape the frequency response at higher frequencies.

One case where this combination of circumstances occurs is in IC operational amplifier design. Click here to go to the lesson where op-amp circuit stability is discussed, and watch how the op-amp frequency response has been shaped by the chip designer.