Pole-zero compensators are used to modify the magnitude and phase of feedback amplifier loops. This article goes beyond the textbook level of explanation to consider some of the less obvious design aspects of their use, and even their design using transistors.

Pole-zero compensators can be either *lead-lag* or *lag-lead* compensators. They are often found in circuits and control theory textbooks. The most common passive-circuit compensators have three components, two resistors and a capacitor. The circuit shown below is placed somewhere that is convenient in an amplifier, especially in a feedback loop, to provide additional phase or high-frequency magnitude emphasis.

The circuit is essentially an RC differentiator with the addition of *R _{1}* shunting

Then the transfer function is the voltage-divider formula,

The first factor is the *quasistatic* (0+ Hz) gain, which is the voltage divider without *C*. The dynamic or frequency-dependent factor has one zero at *
ω _{z}* = 1/

Because the paralleled resistors have a lower resistance than *R _{1}*, they also have a lower time constant and thus a higher frequency; the pole frequency is at a higher frequency than the zero. As frequency increases from a low value, the zero begins to contribute positive (leading) phase at about a decade below

If *
ω _{p}* >

Ideally, a compensator would contribute only zeros and no poles, but real analog circuits always have as many or more poles than zeros. (There are ways of circumventing this at a system level by paralleling circuits, such as in the PID compensator. And in the platonic world of DSP, zeros can be programmed without poles - but not in real-time!) The lead-lag compensator has an undesired pole that accompanies the desired zero. The design challenge in using this circuit for compensation is to place the zero where the additional (positive) phase is needed while placing the pole at a higher frequency that is not critical to the loop dynamics, preferably a decade above *f _{T}*.

To separate the pole and zero, the time constants must be separated by making *R _{2}* <<

On the other hand, if the loop has too many poles but the loop gain must be kept what it is to meet precision (quasistatic error) requirements, then the lead-lag compensator is an inadequate remedy. The zero can be used to cancel an unwanted pole, but the compensator pole at a somewhat higher frequency remains. In effect, the compensator merely shifts a loop pole to a higher frequency, though not by more than a decade. And this is sometimes too little to matter. Therefore, the lead-lag compensator is best used when both an additional zero and reduced quasistatic gain will benefit the control situation.

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