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For K-12 Students · Pole–Zero Response

Pole–Zero Response for a phase-locked loop

Built for k-12 students learning it in middle or high school. Watch the idea come alive with plain-language steps and everyday examples — perfect for projects and homework. Simulate a phase-locked loop live below — adjust the inputs and watch it respond, right in your browser.

Pole–Zero & Frequency ResponseLive

poles peak, zeros notch

Controls

Presets

Pole radius

Pole angle (°)

Zero angle (°)

A digital filter is defined by its poles and zeros in the z-plane. A pole near the unit circle creates a resonant peak at its angle; a zero on the circle carves a notch. Move them and watch the frequency response reshape. Educational tool.

▶ Run in Python

Data Inspector

Pole radius0.85

Resonance at45° (ω)

Stabilitystable ✓

Governing equation

H (z) = \frac{( z - e ^{j 12 0^{\circ}} ) ( z - e ^{- j 12 0^{\circ}} )}{( z - 0.85 e ^{j 4 5^{\circ}} ) ( z - 0.85 e ^{- j 4 5^{\circ}} )}, stable ⟺ ∣ p ∣ = 0.85 < 1

Reading this result: A pole at r=0.85, 45° gives a resonant peak while the zero at 120° carves a separate notch — a classic two-feature filter.

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Frequently asked questions

Is this good for k-12 students?: Yes — this version of "Pole–Zero Response for a phase-locked loop" is framed for k-12 students learning it in middle or high school. Watch the idea come alive with plain-language steps and everyday examples — perfect for projects and homework.
Do I need to install anything?: No. It runs in any modern browser, free, with no account required.