Part 4 · Control Loops

Loop gain \(T = P\,C\,F\)

Key point: break the loop at the summer and ask what comes back. That product is loop gain \(T(s)=P(s)\,C(s)\,F(s)\) — what we measure and shape. It is not the closed-loop map \(V_{\mathrm{out}}/V_{\mathrm{ref}}\).

Along the way

1
Open the circle at Σ. Inject \(V_{\mathrm{test}}\). \(T\) is what returns: \(V_{\mathrm{return}}/V_{\mathrm{test}}\).
2
\(T\) is the product of the blocks. Compensator \(C\), plant \(P\), feedback \(F\). On a Bode plot, dB add.
3
\(K\) is not a third part. It is the leftover DC gain sitting in \(C\) or \(P\). Sliding \(K\) moves the whole \(|T|\) curve up or down.
4
Crossover is where \(|T|=0\,\mathrm{dB}\). That frequency is \(f_c\). Next page reads phase there — that is phase margin.
5
\(T_{\mathrm{cl}}\) is a different fraction. Closed-loop \(V_{\mathrm{out}}/V_{\mathrm{ref}} = L/(1+LF)\). We design by shaping \(T\), not by staring at \(T_{\mathrm{cl}}\).

Try it: Low K — \(f_c\) sits near the plant pole. Raise K: the blue curve lifts and \(f_c\) walks right. Phase (the plant’s lag) does not move with \(K\).

What we ignored: phase margin, Type II zeros, and whether that \(f_c\) is still stable. Next page reads PM at the 0 dB crossing.

Loop-gain Bode

\(T = K\cdot F/(1+s/\omega_p)\) · plant is one pole · 0 dB is crossover

K (loop DC in C)

\(f_c\) → —

\(|T|\) at DC → —

Plant \(f_p\) (kHz) Feedback F

T(s) = K · F / (1 + s/ωp). dB add on the Bode. K lifts |T| without moving the plant pole.

Loop opened at the summer: Vtest into Comp then Plant, Feedback returns Vreturn, no connection back to Vtest — Opened at Σ. \(T = V_{\mathrm{return}}/V_{\mathrm{test}} = P\,C\,F\).

Bode of T · 10 Hz–1 MHz

|T| (dB) Phase 0 dB fc

Dig deeper — \(T\) vs \(T_{\mathrm{cl}}\)

Forward path \(L=CP\). Closed-loop from reference to output:

\[ T_{\mathrm{cl}} = \frac{V_{\mathrm{out}}}{V_{\mathrm{ref}}} = \frac{L}{1+LF} \qquad T = L F = P\,C\,F \]

We shape \(T\) because it is what a network analyzer measures when you break the loop. \(K\) in textbooks is \(T(s)=K\,P\,C\,F\) — a bag for modulator gain, sense gain, and so on. It is not a third capacitor. Sliding \(K\) does not move pole/zero frequencies; it only moves crossover. Instability is \(1+T=0\): unity gain and −180° together. That reading is the next page.