Bode Plot Generator

Transfer Function Evaluated at s = jω

G(s) = N(s) / D(s) | N(s) = Σ a_i · s^(m−i) (i = 0…m) | D(s) = Σ b_i · s^(n−i) (i = 0…n) | s = jω | (jω)^p = ω^p · [ cos(p·π/2) + j · sin(p·π/2) ]

Variables: G = transfer function (dimensionless gain ratio), s = Laplace variable (rad/s), ω = angular frequency (rad/s), a_i = numerator coefficient multiplying s^(m−i), entered highest power first (units of the numerator's common unit × (rad/s)^(i−m), so every term shares one unit), b_i = denominator coefficient multiplying s^(n−i), entered highest power first (units of the denominator's common unit × (rad/s)^(i−n)), m = numerator order (dimensionless integer), n = denominator order (dimensionless integer), p = power of jω in a term (dimensionless integer), j = imaginary unit (dimensionless). Source: Ogata, Modern Control Engineering, 5th ed., Ch. 7 (Control Systems Analysis and Design by the Frequency-Response Method), sinusoidal transfer function G(jω)

Magnitude in Decibels and Frequency Conversion

|G(jω)| = √( Re[G(jω)]² + Im[G(jω)]² ) = |N(jω)| / |D(jω)| | |G(jω)|_dB = 20 · log₁₀ |G(jω)| | |G(jω)| = 10^(|G(jω)|_dB / 20) | f = ω / (2π)

Variables: |G(jω)| = linear magnitude of the frequency response (dimensionless gain ratio), Re[G(jω)], Im[G(jω)] = real and imaginary parts of G(jω) (dimensionless), |N(jω)|, |D(jω)| = magnitudes of the numerator and denominator evaluated at s = jω (units of the respective polynomials; their ratio is dimensionless), |G(jω)|_dB = magnitude in decibels (dB), ω = angular frequency (rad/s), f = cyclic frequency (Hz). Source: Ogata, Modern Control Engineering, 5th ed., Ch. 7 (Control Systems Analysis and Design by the Frequency-Response Method), logarithmic (Bode) plots

Phase Angle (Unwrapped, Anchored at Low Frequency)

φ(ω) = ∠G(jω), continuous in ω | φ(0⁺) = 90° · (z₀ − p₀) − (180° if the DC gain is negative) | atan2( Im[G(jω)], Re[G(jω)] ) fixes φ only modulo 360°

Variables: φ = phase of the frequency response (degrees), NOT confined to (−180°, +180°]: it is unwrapped along ω, so it can run to −270° or below; z₀ = number of zeros at s = 0 (dimensionless count); p₀ = number of poles at s = 0 (dimensionless count); DC gain = ratio of the lowest-order non-zero coefficients of N(s) and D(s) (dimensionless); atan2 = two-argument arctangent, whose principal value on (−π, +π] gives the phase only modulo 360°, so the calculator fixes the branch by matching the exact low-frequency limit φ(0⁺) — e.g. −270° for G = 1/s³, where the principal value would read +90°. A pole exactly on the jω axis steps the phase by −180° and a zero by +180°. Source: Ogata, Modern Control Engineering, 5th ed., Ch. 7 (logarithmic plots: phase of integral and derivative factors); Nise, Control Systems Engineering, 7th ed., Ch. 10 (Frequency Response Techniques)

Gain and Phase Stability Margins

at every gain crossover: |G(jω_gc)| = 1 (0 dB) | PM = 180° + φ(ω_gc), wrapped to (−180°, +180°] | at every phase crossover: φ(ω_pc) ≡ −180° (mod 360°) | GM_dB = −20 · log₁₀ |G(jω_pc)| | reported: the smallest |PM| and the smallest |GM_dB|

Variables: PM = phase margin (degrees), GM_dB = gain margin (dB), ω_gc = a gain-crossover frequency where the magnitude passes through 0 dB in either direction (rad/s), ω_pc = a phase-crossover frequency where the unwrapped phase passes through −180° + k·360° (rad/s; ω = 0 counts when the DC gain is negative), φ(ω_gc) = open-loop phase there (degrees), |G(jω_pc)| = linear open-loop magnitude there (dimensionless). Every crossover is located by bisection on the exact response, not read off the plotted grid, and all of them are listed; the cards show the worst case, following MATLAB margin and python-control stability_margins. Margins alone decide stability only for an open-loop-stable, minimum-phase loop, so the calculator also states closed-loop stability from the roots of D(s) + N(s). Source: Ogata, Modern Control Engineering, 5th ed., Ch. 7 (Control Systems Analysis and Design by the Frequency-Response Method), gain margin and phase margin definitions

Example 1: Third-order loop K/(s(s+1)(s+2)) at K = 2: read both margins

A unity-feedback loop with open-loop transfer function G(s) = 2/(s(s + 1)(s + 2)). Find the phase and gain crossover frequencies and the gain and phase margins.

Example 2: Same loop at K = 10: negative margins mean an unstable loop

Raise the gain of the same loop to K = 10: G(s) = 10/(s(s + 1)(s + 2)). What do the margins say?

Example 3: Lead compensation: K_v = 20 s⁻¹ with about 50° of phase margin

The plant G(s) = 4/(s(s + 2)) has velocity error constant K_v = lim(s→0) s·G(s) = 4/2 = 2 s⁻¹. The design asks for K_v = 20 s⁻¹ and a phase margin near 50°. Compare raising the gain alone with adding the lead compensator C(s) = 41.7(s + 4.41)/(s + 18.4).