Lead Compensators - Some Practical Considerations

When you determine that you want to use a lead compensator and you are designing in the frequency domain there is question that you need to know the answer to. Here's the question.

Given a lead compensator,

Determine the largest amount of phase lead that can be produced using the compensator,
Determine where the frequency where the largest amount of phase lead occurs.

In this lesson we will examine that problem.

Imagine that you have a compensator with the transfer function:

Gc(s) = K(s/w_z + 1)/(s/w_p + 1)

In this compensator transfer function:

There is a zero at s = -w_z.
There is a pole at s = -w_p.
There is a DC gain of K. We recognize that when this transfer function is used as a compensator there will need to be an adjustable gain to set zero-db crossing, etc.

The adjustable gain, K, will not affect the phase of the system. If s = jw, the phase is given by:

f = tan^-1(w/w_z) - tan^-1(w/w_p)

Recognize that the phase is always positive (It is a lead!) since the zero frequency, w_z, is less than the pole frequency, w_p.

Here is a little movie you can play. In this movie, the frequency response of a lead compensator is plotted. Note the following:

The ratio of the pole to the zero is a, and a can be varied from 1 to 11 in the movie.
The solid red trace is the magnitude - in db - and the scale for the magnitude is on the right side of the movie.
The dotted blue trace is the phase. This is really a plot of the function, f(w), above.

The scale for phase is on the left side of the movie.
Note how the phase plot changes as you vary the pole-to-zero ratio.
Notice that there is always a single maximum to the phase plot.

To find the frequency at which maximum phase lead occurs, differentiate the phase, f. The result for df/dw is:

To find the point of maximum phase lead, set this expression to zero and solve for the frequency. (We are leaving the details to you.) The result is:

Now, we also need to know how much phase lead we get at that frequency. We get that by computing the phase at the frequency for maximum phase lead. Computing and plotting that, we get the graph below.

Using this plot, you can compute the pole to zero ratio for a compensator. Here is the algorithm for designing a compensator.

Determine how much phase lead you need to add.
Determine the frequency at which that phase must be added.
Choose a pole-to-zero ratio that will give the phase lead you need using the plot above. You may want to add a little safety margin.
Using the expression above for w_max, determine the pole and zero location.