Part 2 · Control Loops

Pole/Zero Primer

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.

Along the way

1
Write factors, not a blob of parts. Same low-entropy idea as Part 1: \(H(s)=K\cdot(\text{zeros})/(\text{poles})\).
2
A pole is −20 dB/dec above \(f_p\). Phase rolls toward −90°. Toggle the LF pole and watch the dashed asymptote kink.
3
A zero is +20 dB/dec above \(f_z\). Phase leads. A mid-band zero can cancel one pole’s slope.
4
Origin pole tilts; it does not raise a single number. \(K/s\) equals K only at \(\omega=1\) rad/s (~0.16 Hz). At 10 Hz (this plot’s left edge) it is below K. Toward DC it heads to infinity.
5
Stack them. Origin + LF pole + zero + HF pole is the Type II shape you will meet later.

Try it: leave K at 60 dB. Add only “Pole at origin.” Read 10 Hz vs the old flat 60 dB. Then add the mid-band zero and HF pole.

What we ignored: a real circuit, loading, and closing the loop. Next page: why we close it.

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 · ∞

Gain K (dB) Pole at origin (1/s) LF pole kHz 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

Dig deeper — 60 dB and a pole at the origin

Short answer: toward DC, yes — ideal loop gain → ∞ (the integrator’s job). At 10 Hz on this plot, \(H=K/s\) is usually lower than 60 dB, not higher. Both are true once gain is a function of frequency.

\[ |H(j\omega)| = \frac{K}{\omega} \quad\text{(origin pole only)} \]

\(|H|=K\) exactly when \(\omega=1\) rad/s (~0.16 Hz). That is the anchor of the −20 dB/decade slope. Below ~0.16 Hz, |H| exceeds K and heads to infinity. Above it (including this plot’s 10 Hz left edge), |H| is below K. The integrator tilts; it does not paste +X dB on a flat line.

Toggle Pole at origin with K = 60 dB and read the 10 Hz line. Then stack the other factors — that stacked shape is what you will recognize in the compensator pages.