Compensate a one-pole plant — Control Loops

Along the way

1
Green is given. \(P=K_P/(1+s/\omega_p)\) on the top plot. You wrap it; you do not redesign it.
2
Blue is what you add. \(C\) is an integrator plus a zero. Same axes as \(P\), so you can see the two pieces.
3
Purple below is the sum. \(F=1\), so \(|T|_{\mathrm{dB}}=|P|_{\mathrm{dB}}+|C|_{\mathrm{dB}}\). Read \(f_c\) and PM there — same skill as Part 5.
4
\(C\) gain walks \(f_c\). Lift blue, purple lifts with it. Green stays put. Same lesson as Part 4.
5
The zero buys PM. Drop \(f_z\) below crossover so the orange phase of \(T\) sits farther above −180°.

Try it: drag the blue handles on the top Bode — |C| up/down for gain, \(f_z\) left/right — or use the sliders. Land purple \(f_c\) on the gold target, then the PM.

What we ignored: a Type II high-frequency pole, and \(F\neq 1\). The time plot is an output bump (disturbance), not a \(V_{\mathrm{ref}}\) step — a reference step of PI overshoots extra because the zero is in that path. Next page puts real \(F\) in the sum.

Add C on top of P · F = 1

Target \(f_c\) and PM — plant stays put, you tune C

C gain (dB) Zero fz (kHz)

fc —

PM —

Given plant — not the exercise

Plant gain (dB) Plant fp (kHz)

T = P · C. Raise C gain to walk fc. Place the zero below fc to buy phase margin.

Loop with compensator you tune, plant given, feedback equal to 1 — \(C\) is the exercise. \(P\) is given. \(F=1\).

Output bump · loop pulls Vout back to 0

Vout idle = 0

Parts · |P| given + |C| you tune · drag handles

|P| given |C| · drag ↕ fp fz · drag ↔

Loop gain T = P · C

|T| phase of T 0 dB fc target fc −180°

Dig deeper — why an integrator plus a zero

A one-pole plant plus a number \(K\) already has PM ≥ 90°. That is too easy, and it has finite DC loop gain. Power stages need the extra pole at the origin so leftover DC error goes to zero. Then at high \(f_c\) you have roughly −90° from \(C\) and −90° from \(P\): PM sits near 0°. The zero buys some of that back:

\[ C(s)=G\Bigl(1+\frac{\omega_z}{s}\Bigr) \qquad \mathrm{PM}=180°-\arctan(f_c/f_p)-\arctan(f_z/f_c) \]

\(G\) (the C-gain slider) sets how high \(|C|\) sits after the zero — that walks \(f_c\). \(f_z\) sets how much phase is back by crossover. A Type II network (later) adds a high-frequency pole so \(|C|\) does not stay flat forever. A real \(F\) (next lesson) is another dB and phase term in the same sum.

A \(V_{\mathrm{ref}}\) step of this PI would overshoot more than the PM rule of thumb: the zero is in the reference path. The lab’s time plot is an output bump instead, so leftover ringing tracks the closed-loop poles (the PM you bought) without that extra kick.