Classical Feedback Primer
Why feedback?
A regulator compares what you want (\(V_\text{ref}\)) to what you have (\(V_\text{out}\)) and drives the error toward zero. Without feedback, the output drifts with load, line, and temperature. With feedback, the loop forces \(V_\text{out} \to V_\text{ref}\) — but only if the loop is stable.
You already know how to write \(H(s)\) for a single block (see the Transfer Function Primer). A power-supply control loop is several such blocks wired in a circle. This page stays abstract: no circuits, only transfer functions.
The negative-feedback loop
Every classical loop has the same skeleton: a reference, a summer, a forward path, and a return path.
The error signal at the summer is:
\[ e(s) = V_\text{ref}(s) - F(s)\,V_\text{out}(s) \]Plant \(P(s)\) is the power stage and output filter. Compensator \(C(s)\) shapes the loop. Feedback \(F(s)\) is usually a resistor divider — often just a constant gain. We draw boxes, not schematics.
Closed-loop transfer function
Trace the algebra in three steps. The forward chain from error to output is \(L(s) = C(s)\,P(s)\):
\[ V_\text{out}(s) = L(s)\,e(s) \] \[ e(s) = V_\text{ref}(s) - F(s)\,V_\text{out}(s) \]Substitute and solve for \(V_\text{out}/V_\text{ref}\):
\[ T_\text{cl}(s) = \frac{V_\text{out}(s)}{V_\text{ref}(s)} = \frac{L(s)}{1 + L(s)\,F(s)} \]
When the denominator goes to zero, the loop blows up.
Instability means closed-loop poles where
In plain language: at some frequency the loop has unity gain and −180° phase — the returned signal reinforces the error instead of canceling it.
Closed-loop step preview
2nd-order approximation from BW and PM — same engine as the transient panel below.
Break the loop → loop gain
To analyze stability we break the loop at the summer: inject a test signal and ask what comes back around. The open-loop transfer function — usually called loop gain — is the product of every block in the circle:
\[ T(s) = P(s)\,C(s)\,F(s) \]On a Bode plot we draw \(|T(j\omega)|\) in dB and \(\angle T(j\omega)\) in degrees. Loop gain is not the same as the closed-loop response \(T_\text{cl}(s)\) — it is what we measure and shape during design.
In many textbooks the loop is written \(T(s) = K\,P(s)\,C(s)\,F(s)\), with a scalar \(K\) collecting DC gains (modulator gain, sense resistor, PWM gain, and so on). In practice \(K\) is not one physical part — it is distributed across the plant and compensator. Sliding \(K\) moves the entire Bode curve up or down in dB, which shifts the 0 dB crossover and changes phase margin without changing pole/zero frequencies. The Feedback Network Analyzer bakes those gains into \(P\) and \(C\); the Root Locus Tool sweeps \(K\) explicitly and shows where closed-loop poles land.
Loop bandwidth
Loop bandwidth \(f_\text{bw}\) is the frequency where the loop gain crosses unity:
\[ |T(j\omega_\text{bw})| = 1 \quad (0\,\text{dB}) \]Below \(f_\text{bw}\) the loop has gain and rejects load and line disturbances. Above \(f_\text{bw}\) the loop cannot correct fast enough. Higher bandwidth generally means faster response — but only if phase margin stays healthy.
Phase margin
Phase margin (PM) is the phase headroom at the 0 dB crossing — how far \(\angle T(j\omega)\) sits above the −180° stability line:
\[ \text{PM} = 180° + \angle T(j\omega_\text{bw}) \]Example: if \(\angle T = -120°\) at crossover, then PM \(= 180° + (-120°) = 60°\). At −180° exactly, the loop would oscillate.
Loop-gain Bode demo
Plant \(P(s)=1/(1+s/\omega_p)\), Type-II compensator \(C(s)=(1+s/\omega_z)/[s(1+s/\omega_c)]\), unity feedback. Pick PM, slide bandwidth — \(K\) and pole/zero locations are solved behind the scenes. Green ○ = zero, red × = pole (origin pole at left edge).
Phase margin on the scope
PM is a frequency-domain number — but it shows up as how the output recovers from a unit load step. Fixed axes (Bode to 1 MHz, step to 5 ms) so traces slide as you change BW and PM.
Load-step response
- ~10° — heavy ringing, near instability
- ~30° — visible ringing, slow settle
- ~45° — moderate overshoot
- ~60° — clean step; industry default
- ~75° — sluggish, almost no overshoot
- ~120° — model tops out near 93°; very overdamped if achievable
Green ○ = zero, red × = pole (origin pole at left edge of plot). Full-featured version: Control System Transient Analyzer
Bandwidth vs settling time
Holding PM fixed, a wider loop bandwidth means the loop corrects faster — shorter rise and settling time. Push bandwidth up without adding phase (e.g. with a compensator zero), and PM falls — ringing returns.
\[ \zeta \approx \frac{\text{PM}}{100} \quad\text{(rule of thumb)} \] \[ t_s \approx \frac{4}{\zeta\,\omega_n} \]These links are approximate. Bode analysis gets you close; confirm with a load step in simulation or on the bench. Try BW and PM yourself in the Transient Analyzer.
What the compensator does
The plant \(P(s)\) is usually low-pass — it eats phase before crossover. Feedback \(F(s)\) is often a constant divider. The compensator \(C(s)\) exists to buy phase back: a zero below crossover, a pole above. That is why power supplies use Type II or Type III networks — not magic, just pole-zero placement on \(T(s)\).
Shape \(P\), \(C\), and \(F\) live in the Feedback Network Analyzer or Type II Compensator tool.
What Bode does not tell you
Bode analysis is fast and indispensable, but it focuses on one frequency (crossover) and assumes linear, small-signal behavior. It does not show saturation, slew limits, or large-signal effects.
Root locus complements Bode: instead of magnitude and phase vs frequency, it shows where closed-loop poles move in the \(s\)-plane as gain \(K\) changes. Those poles set oscillation frequency, damping, and settling — directly in the time domain.
Many engineers use Bode to pick a compensator, then root locus to verify the closed-loop poles landed where they expected.
Practical targets (power supplies)
- Phase margin: 45–60° is common; below ~30° expect ringing; very high PM often means a slow loop.
- Bandwidth: often 1/10 to 1/5 of switching frequency (topology-dependent rule of thumb).
- Always verify with a load step — on the bench or in simulation.