Pole/Zero Primer

Part 2 · Control Loops

Key point: each factor \(1+s/\omega\) is a Bode breakpoint. A pole tilts magnitude down 20 dB/decade and lags phase; a zero does the opposite. An origin pole does not “add 60 dB” — it tilts the whole story.

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

1
K only — \(g_m\).
Only \(g_m\) is lit. \(H(s)\) is just \(K\). Bode is flat 60 dB, 0°. Slope does not change until you add a factor.
2
Add the plant — LF pole.
Series \(R\), shunt \(C\). Left box lights. A denominator factor sits at 1 kHz. Dashed Bode kinks −20 dB/dec; phase toward −90°.
3
Add the origin pole.
\(C_{\mathrm{TH}}\) lights; \(s\) in the denominator. Whole Bode tilts −20 dB/dec, −90° at every \(f\). At 10 Hz you are below \(K\). Toward DC, \(K/s\to\infty\).
4
Add a mid-band zero.
\(R_{\mathrm{TH}}\) lights — the zero lives on that same \(C_{\mathrm{TH}}\). Above \(f_z\) the slope recovers +20 dB/dec and phase comes back.
5
Add the HF pole.
\(C_{\mathrm{THP}}\) lights. Origin + zero + HF on the right box is Type II. The plant is still the left box.

Unused factors stay in the fraction as ∞. After the walk, uncheck boxes or move the kHz numbers.

Build H(s) · watch the Bode

Solid mag + phase · dashed mag asymptote · unused factors stay in the fraction as ∞

H(s) = 60.0 dB × 1 + s 2π · ∞ 1 · 1 + s 2π · ∞ · 1 + s 2π · ∞

Lights follow the walk (and the toggles). Ghosted parts stay in the kit. \(C_{\mathrm{TH}}\) is the origin pole; \(R_{\mathrm{TH}}\) on that same C is the zero; \(C_{\mathrm{THP}}\) is the high pole. The extra LF pole is the plant, not the ITH pin.

Gain K (dB) LF pole kHz Pole at origin (1/s) Mid-band zero kHz HF pole kHz

Bode · 10 Hz–1 MHz

Magnitude Phase Mag asymptote

Hover for coordinates

At 10 Hz: — HF slope: 0 dB/dec · gain only

Infinite at 0 Hz, we plot 10 Hz

A true integrator \(K/s\) has infinite gain at 0 Hz. This Bode does not go there — the left edge is 10 Hz. That 10 Hz number is the one that matters, because you cannot build infinite DC gain in a real circuit anyway.

Turn on Pole at origin. Leave LF pole, Mid-band zero, and HF pole off, and \(K=60\,\mathrm{dB}\). Then

\[ |H(j\omega)| = \frac{K}{\omega} \qquad K = 10^{60/20} = 1000 \qquad \omega = 2\pi f \]

At the left edge, \(f=10\,\mathrm{Hz}\):

\[ |H| = \frac{1000}{2\pi\cdot 10} \approx 15.9 \qquad\Rightarrow\qquad 20\log_{10}|H| \approx 24\,\mathrm{dB} \]

That is the At 10 Hz readout. 60 dB is the gain constant \(K\), not a value that sits on the plot. Toward 0 Hz the ideal curve would keep climbing at −20 dB/decade and never stop. Off the left of this chart — and off any real op-amp or OTA — DC gain is finite. A low-frequency pole (the plant here; \(R_o\) on the Type II page) is how that shows up: same slope in-band, a floor at DC instead of \(\infty\).