Transfer Function Primer — Control Loops

Along the way

1
Black box. In → \(H(s)\) → out. Magnitude and phase vs frequency, not a single gain.
2
Divider is enough when it is only a ratio. \(H=R_{\mathrm{bot}}/(R_{\mathrm{top}}+R_{\mathrm{bot}})\), 0° phase at every \(f\).
3
RC adds a pole. Same \(|H|\) as the divider at one frequency can still lag on the scope. That lag is why we use \(s\).
4
Scope vs Bode. Scope is one \(f\). Bode is the whole story. Violet line on the Bode is the scope frequency.
5
ESR adds a zero. Ideal cap: high-\(f\) gain keeps falling. Real \(R_s\): a floor and a zero. Drag ESR and watch the green marker.

Default 90 Ω / 10 Ω divider is 0.1 (−20 dB) with 0° phase. Tune the RC to match that level at 20 kHz and the phase lag appears.

What we ignored: loading, source impedance, op-amp \(H(s)\). Poles and zeros as a construction kit is Part 2.

\(V_{\mathrm{out}}(s)=H(s)\,V_{\mathrm{in}}(s)\)

Divider vs RC

Scope at one \(f\) · Bode 10 Hz–10 MHz · solid = with \(R_s\) · dashed = ideal

Rs (mΩ)

fp —

fz —

H(s) = 1 + s 2π · ∞ 1 + s 2π · —

With ESR = 0 the zero is at infinity — H(s) is a single pole. Increase ESR to add a zero. Whether you see it depends on fz vs the test frequency.

Resistor divider

\(H = R_{\mathrm{bot}}/(R_{\mathrm{top}}+R_{\mathrm{bot}})\) — magnitude only, 0° phase.

RC low-pass

Series \(R\), then shunt \(R_s\) in series with \(C\).

Scope — 20 kHz sine

V(in) divider RC phase lag outputs ×5 vs 1 V in

Bode · fp red · fz green · f_scope violet

Magnitude Phase Mag · Rs=0 Phase · Rs=0 f_scope

f_scope (Hz) Rtop (Ω) Rbot (Ω) R (Ω) C (µF)

f — div — RC — Δ|H| — ESR Δφ — Hdiv —

Try it: match the divider’s −20 dB on the RC at 20 kHz (default \(R\), \(C\) are close). Then raise ESR and watch \(f_z\) walk in from infinity and the high-frequency Bode flatten.

Writes the schematic with these R, C, Rs, and sine frequency. Run is .tran (scope). Uncomment .ac on the sheet for the Bode.

Dig deeper — writing \(H(s)\) the right way

The schematic is a divider. That algebra is correct — and it hides the corners. Factor so each \(1+s/\omega\) is a Bode breakpoint you can point at (Basso’s low-entropy form). Drag \(R_s\) on the sticky bar; the live fractions below track it.

\[ Z_C(s)=R_s+\frac{1}{sC},\qquad H(s)=\frac{Z_C}{R+Z_C} =\frac{1+s R_s C}{1+s C(R+R_s)} =\frac{1+s/\omega_z}{1+s/\omega_p} \] \[ \omega_z=\frac{1}{R_s C},\qquad \omega_p=\frac{1}{C(R+R_s)} \]

If \(R_s=0\), \(\omega_z\to\infty\) and \(\omega_p=1/RC\) — a single pole. Finite ESR adds the zero and moves \(f_p\) a little, because the time constant is \(C(R+R_s)\), not \(RC\).

High entropy — parts hide the corners

H(s) = 1 —

Low entropy — each factor is a breakpoint

H(s) = 1 + s 2π · ∞ 1 + s 2π · —

\(f_p=\) — · \(f_z=\) ∞ · if \(R_s=0\), \(f_p=\) —

A pole \((1+s/\omega_p)\) in the denominator: −20 dB/decade, phase toward −90°.
A zero \((1+s/\omega_z)\) in the numerator: +20 dB/decade, phase toward +90°.
That is why we factor: the equation and the Bode plot read the same way. Part 2 stacks more of these.