Phase-lead and Phase-lag Controllers

Imagine adjusting the volume equalizer on a stereo. When the bass is cranked up, a phase-lead controller is at work. When the treble is increased, a phase-lag controller comes into play.

PD controllers act as high-pass filters, PI as low-pass filters, and PID as either band-pass or band-stop filters.

The phase-lead controller, or high-pass filter, introduces a positive phase to the system over a specific frequency range. Conversely, the phase-lag controller, or low-pass filter, introduces a negative phase.

A single transfer function can represent both controllers, where if the first parameter is greater than the second, it's high-pass; if less, it's low-pass.

The function can be realized using an op-amp circuit. Its transfer function relates output and input voltages through the product of two capacitors' ratio and the sum of resistance and capacitance product for each path.

Equating the capacitors reduces the design parameters from four to three, simplifying the circuit equation.

Constants can replace the ratio of resistances and the product of resistance and capacitance.