Transfer Function Primer

Part 1 · Control Loops

Key point: a transfer function is how a box scales and delays a signal as frequency changes: \(V_{\mathrm{out}}(s)=H(s)\,V_{\mathrm{in}}(s)\). A resistor divider is a real number. An RC is not.

Click a step, or Next, to set the lab. The sliders wait until you have walked it.

1
Black box \(H(s)\).
In → \(H(s)\) → out. Magnitude and phase vs frequency, not a single gain.
2
Divider is a ratio.
\(H=R_{\mathrm{bot}}/(R_{\mathrm{top}}+R_{\mathrm{bot}})\), 0° 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.
Sweep \(R_s\). High-\(f\) gain floors and a green \(f_z\) walks in. The scope RC trace moves with it.

Default 90 Ω / 10 Ω is 0.1 (−20 dB) at 0°. Match that on the RC at 20 kHz and the phase lag appears. After the walk, drag \(R_s\).

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 —

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

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 \(R_s\); 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.