📈 Bode Plot: Frequency Response & Stability Margins

A Bode plot is a pair of graphs — magnitude in decibels and phase in degrees, both against frequency on a logarithmic axis — that show how a linear system responds to sinusoidal inputs of every frequency. It is produced by substituting s = jω into the system's transfer function G(s) and evaluating the resulting complex number G(jω) at many frequencies ω; this simulation does exactly that using real complex-number arithmetic, not an approximation.

Gain margin is how much the open-loop gain could increase, in decibels, before the closed-loop system becomes unstable; it is measured at the phase-crossover frequency ω_pc, the frequency where the phase first reaches −180°, as GM = −20·log₁₀|G(jω_pc)|. Phase margin is how much additional phase lag the loop could tolerate before instability; it is measured at the gain-crossover frequency ω_gc, where |G(jω_gc)| = 0 dB (unity gain), as PM = 180° + ∠G(jω_gc). Both are read directly off the plotted curves in this simulation, by interpolating between the two nearest sample points that bracket the crossover.

Under classical Bode/Nyquist stability theory, for a large class of open-loop transfer functions closing the loop with unity negative feedback remains stable as long as, at the frequency where the gain equals 0 dB, the phase has not yet reached −180° (positive phase margin), and at the frequency where the phase reaches −180°, the gain is still below 0 dB (positive gain margin). If either condition fails, the loop gain and phase align at −180° with gain ≥ 1, so a small disturbance is fed back in phase and amplified every cycle — the definition of instability. This simulation flags "marginal" rather than simply "stable" when a margin is positive but small, because real systems have model uncertainty and small margins leave little robustness headroom.

Yes. Every pole and zero you place is stored as a complex number; the simulation multiplies out G(s) = K·Π(s−zᵢ)/Π(s−pᵢ), substitutes s = jω for 400 log-spaced frequencies between the chosen limits, and performs genuine complex multiplication and division (using the standard a+bi arithmetic rules) at every point. Magnitude is 20·log₁₀ of the resulting complex number's modulus and phase is its argument, unwrapped across the sweep so it doesn't jump by ±360° — the same computation a textbook or MATLAB's bode() function performs.

Each pole of a transfer function contributes an angle that approaches −90° as frequency rises past that pole's break frequency, and each zero contributes +90° in the same way; a system with three poles and no zeros therefore ends up near −270° at high frequency, not somewhere between −180° and +180°. This simulation "unwraps" the raw arctangent output so the phase curve is continuous instead of snapping back every time the raw angle would jump across ±180°, which is essential for correctly finding the −180° phase-crossover frequency.

If the magnitude curve never crosses 0 dB anywhere in the chosen frequency range, there is no gain-crossover frequency and phase margin cannot be located that way — the loop gain is either always below or always above unity across the swept band. Likewise, if the phase never reaches −180° in range, there is no finite gain margin. Widening the frequency range with the ω min / ω max sliders will usually reveal the crossover if one exists at all.

A pole at s = 0 makes G(s) a "type-1" system, meaning it contains a free integrator; type-1 systems are common in practice (motor position control, level control) because the integrator drives steady-state error to zero for a step input. They are also a standard teaching example for Bode analysis because their phase starts at −90° at low frequency and their gain margin is finite and gain-dependent, making the stability boundary easy to see by sliding K.