Pole/Zero Primer — Analog Intuition

Build H(s)

Check factors in order (origin pole → LF pole → mid zero → HF pole) to see slopes and phase stack. Each pole adds −20 dB/dec and up to −90° lag; each zero does the opposite above its breakpoint.

Gain K (dB)

Pole at origin Factor 1/s — constant −90° lag, −20 dB/decade at all frequencies Low-frequency pole kHz Adds another −20 dB/dec above breakpoint; phase rolls −90° around f_p Mid-band zero kHz +20 dB/dec above breakpoint; phase leads up to +90° — cancels one pole’s effect High-frequency pole kHz Rolls gain off again at high frequency (HF noise rejection)

Transfer function (factored)

Hover for coordinates

Tutorial: 60 dB gain + pole at origin — does LF gain go up?

Short answer: Toward DC, yes — loop gain ideally goes to infinity (that is the integrator’s job). At 10 Hz on this plot, H = K/s is usually lower than 60 dB, not higher. Both statements are true once you see gain as a function of frequency, not a single number.

The factored form

Write the error-amplifier path as H(s) = K × (1/s). The 60 dB is the scalar K (linear gain ≈ 1000). The 1/s is an integrator — capacitor feedback in a Type-II compensator — and it is a pole at the origin.

Magnitude at any angular frequency ω (rad/s):

|H(jω)| = K / ω (times any other pole/zero factors you add later)

Where is “60 dB” on the plot?

K is not “the gain at every frequency.” For K/s alone, by definition

|H| = K exactly when ω = 1 rad/s (~0.16 Hz)

Toggle pole at origin in the tool and check the metrics box: at ω = 1 rad/s the magnitude equals K (60 dB). That frequency is the anchor for the −20 dB/decade slope.

Above vs below 0.16 Hz

Below ~0.16 Hz (ω < 1 rad/s): |H| exceeds K — magnitude rises 20 dB/decade as you move left toward DC. Ideal DC gain is infinite → zero steady-state error in a feedback loop.
Above ~0.16 Hz (ω > 1 rad/s): |H| is below K — magnitude falls 20 dB/decade. At 10 Hz (where our axis starts), |H| ≈ 24 dB for K = 60 dB — less than the flat 60 dB you had before adding 1/s.

Intuition for the error amplifier

Think of K as the op amp’s midband gain set by resistors. The integrator does not add a fixed +X dB on top at one frequency — it tilts the response: more gain at lower frequencies, less at higher frequencies. In a closed loop, that rising LF gain is what drives DC error to zero while you still roll off gain at high frequency for stability.

On the Bode plot you never evaluate true DC (ω = 0). You infer infinite LF gain from the −20 dB/dec slope pointing upward as f → 0, and you read a finite number at 10 Hz. Phase sits at −90° from the integrator alone.

Next: stack more poles and zeros

Real compensators add a low-frequency pole, a mid-band zero, and a high-frequency pole on top of K/s. Each adds another ±20 dB/decade segment and ±90° phase — use the toggles above to watch slopes cancel and reappear. That is the shape you will recognize in the Classical Feedback Primer.

How poles and zeros shape the Bode plot