Why can the reported margins disagree with my textbook example?

Real loop transfers can cross **0 dB** or **−180°** multiple times. Different textbooks pick different crossings (for example, the **lowest** gain crossover or the **last** one before roll-off). This simulator labels its rule explicitly—**first crossings while ω increases**—and takes the **minimum** phase margin and **minimum** gain margin among detected crossings as a simple “worst case among the obvious crossings” teaching aid.

What happens if I stack poles at the origin?

Each integrator raises low-frequency gain and adds **−90°** of phase per pole. The Bode plot can blow up near **ω = 0**, and the Nyquist trace may need a large **indent** near the origin that this sketch does not draw—so the polar plot is qualitative for **type ≥ 2** loops. Margins may read **undefined** if the searched band never exhibits the expected crossings.

Does placing a zero in the right half-plane break the math?

The evaluation **G(jω)** remains well defined, but the **non-minimum-phase** zero adds extra lag at mid frequencies, often shrinking **phase margin** even when gain looks benign. Nyquist encirclement arguments for closed-loop stability must then be interpreted carefully; this page focuses on the **shape** of **G(jω)** and the **−1** probe rather than completing a full Nyquist counting proof.

How is the s-plane click mapped to actual roots?

Pointer coordinates are inverted from pixel space into **(σ, ω)** with a coarse snap grid so conjugate pairs stay aligned. A click on the **real axis** (within a small tolerance) adds a **single** real root; a click off the axis adds **two** conjugate locations. **Shift-click** deletes whichever listed root (pole or zero) is closest in **(σ, ω)** space.

Bode & Nyquist from Pole–Zero Map

Click the s-plane to place real or complex-conjugate poles (●) and zeros (×); G(s)=K∏(s−z)/∏(s−p). Live Bode magnitude/phase and Nyquist plot with −1 point; phase and gain margins from principal crossings (teaching heuristics).

About this model

This page builds a scalar rational transfer function G(s) = K ∏(s − z_i) / ∏(s − p_j) with real coefficients, so every non-real pole or zero you place is automatically mirrored as a complex-conjugate pair. You edit the map directly on a compact s-plane (σ vs jω), then see the same model three ways: Bode magnitude and phase versus ω on log axes, a Nyquist trace of G(jω) with the −1 reference for unity-feedback stories, and numeric phase margin and gain margin readouts. Margins are computed on a dense logarithmic frequency grid by unwrapping phase to count the first 0 dB crossing of |G| as ω increases (for phase margin) and the first −180° crossing of the unwrapped phase (for gain margin). When several crossings exist—as in conditionally stable or non-minimum-phase loops—the textbook recipe is ambiguous; the page therefore documents a pedagogical heuristic (worst among detected crossings) rather than a full robustness audit. Pair it with the first-order electricity/bode-diagram and preset electricity/nyquist-diagram simulators for layered homework.

Who it's for: Junior or senior undergraduates in classical control or introductory mechatronics who already know poles and zeros algebraically but want a fast, visual link to frequency-domain plots and margin vocabulary.

Key terms

Transfer function
Pole
Zero
Bode plot
Nyquist plot
Phase margin
Gain margin
Unity feedback
Non-minimum phase

How it works

Build a proper rational open-loop model G(s) = K ∏(s−zᵢ)/∏(s−pⱼ) with conjugate-symmetric roots (click the s-plane: off-axis points add a complex pair). The Bode view plots |G| and ∠G versus ω (log scale); the Nyquist view draws G(jω) from ω→0⁺ to ∞ with the −1 point for unity-feedback intuition. Phase margin is 180° + ∠G at the first 0 dB gain crossover when |G| falls through unity as ω increases; gain margin is −|G|_dB at the first −180° crossing of the unwrapped phase (teaching heuristic when multiple crossings exist). Compare with the first-order electricity/bode-diagram and preset electricity/nyquist-diagram pages.

Key equations

G(jω) = K · ∏(jω − zᵢ) / ∏(jω − pⱼ)

PM = 180° + ∠G(jω_gc) at |G(jω_gc)| = 1

GM_dB = −20 log₁₀|G(jω_pc)| at ∠G(jω_pc) = −180° (first crossings, ω increasing)

Frequently asked questions

Why can the reported margins disagree with my textbook example?: Real loop transfers can cross 0 dB or −180° multiple times. Different textbooks pick different crossings (for example, the lowest gain crossover or the last one before roll-off). This simulator labels its rule explicitly—first crossings while ω increases—and takes the minimum phase margin and minimum gain margin among detected crossings as a simple “worst case among the obvious crossings” teaching aid.
What happens if I stack poles at the origin?: Each integrator raises low-frequency gain and adds −90° of phase per pole. The Bode plot can blow up near ω = 0, and the Nyquist trace may need a large indent near the origin that this sketch does not draw—so the polar plot is qualitative for type ≥ 2 loops. Margins may read undefined if the searched band never exhibits the expected crossings.
Does placing a zero in the right half-plane break the math?: The evaluation G(jω) remains well defined, but the non-minimum-phase zero adds extra lag at mid frequencies, often shrinking phase margin even when gain looks benign. Nyquist encirclement arguments for closed-loop stability must then be interpreted carefully; this page focuses on the shape of G(jω) and the −1 probe rather than completing a full Nyquist counting proof.
How is the s-plane click mapped to actual roots?: Pointer coordinates are inverted from pixel space into (σ, ω) with a coarse snap grid so conjugate pairs stay aligned. A click on the real axis (within a small tolerance) adds a single real root; a click off the axis adds two conjugate locations. Shift-click deletes whichever listed root (pole or zero) is closest in (σ, ω) space.

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About this model

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Key equations

Frequently asked questions

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Key equations

Frequently asked questions