Transfer Function Gain and Phase Calculator

What Is a Transfer Function? A transfer function H(s) describes the input-output relationship of a linear time-invariant (LTI) system in the Laplace domain. H(s) = Y(s)/X(s) = numerator polynomial / denominator polynomial. The frequency response is found by substituting s = jω (where j = √−1, ω = 2πf): H(jω) gives the complex gain at frequency ω. Magnitude |H(jω)| = gain in linear scale. Phase ∠H(jω) = phase shift in degrees.

First-Order Systems Low-pass filter: H(s) = K/(τs + 1), where τ = time constant, K = DC gain. At ω = 1/τ (cutoff frequency): gain = K/√2 ≈ 0.707K = −3 dB point. Phase at cutoff: −45°. Roll-off: −20 dB per decade above cutoff. High-pass filter: H(s) = Kτs/(τs + 1). Phase at cutoff: +45°. Lead/Lag compensators are first-order transfer functions with adjustable zero and pole locations.

Second-Order Systems Standard form: H(s) = ωₙ²/(s² + 2ζωₙs + ωₙ²) Where ωₙ = natural frequency (rad/s), ζ = damping ratio. ζ < 1: underdamped — oscillatory step response. Resonance peak in frequency response. ζ = 1: critically damped — fastest non-oscillatory response. ζ > 1: overdamped — two distinct real poles. Resonant frequency: ωᵣ = ωₙ√(1 − 2ζ²) (for ζ < 1/√2). Peak magnitude: |H(jωᵣ)| = 1/(2ζ√(1−ζ²)).

Bode Plot A Bode plot shows gain (in dB) and phase (in degrees) vs. log frequency. Gain in dB: 20·log₁₀(|H(jω)|). 0 dB = unity gain (output = input magnitude). −3 dB corresponds to a gain of 0.707 — the standard bandwidth definition. −20 dB/decade slope = one-pole rolloff. −40 dB/decade = two-pole rolloff. The Bode plot was developed by Hendrik Wade Bode at Bell Labs in the 1930s in the United States.

Gain and Phase Margin Gain margin: how much gain can increase before the closed-loop system becomes unstable. Phase margin: how much phase lag can be added at the 0 dB crossing before instability. Well-designed systems typically have: gain margin > 6 dB, phase margin > 45°. These margins define the robustness of a feedback control system.

Common Transfer Functions Integrator: H(s) = K/s. Gain = K/ω, phase = −90° (constant). Differentiator: H(s) = Ks. Gain = Kω, phase = +90°. PID controller: H(s) = Kp + Ki/s + Kd·s. Combines all three. Notch filter: removes a specific frequency. Used in audio and vibration control.