Transfer Function Analyzer & LTI System Simulator
Enter numerator and denominator coefficients of a 1st or 2nd-order continuous-time LTI transfer function to visualize its Bode plot, step response, and pole-zero map on the s-plane.
Transfer Function Coefficients
Numerator Coefficients
Denominator Coefficients
Transient Response Metrics
Step Response
Pole-Zero Map (s-Plane)
Bode Plot
Transfer Functions and the Significance of Poles
The transfer function H(s) expresses the relationship between a system's input and output in the Laplace domain as an algebraic ratio of polynomials. The frequency response and transient behavior of the system are entirely determined by the roots of the numerator and denominator polynomials.
Poles (✕) and System Stability
Poles are the roots of the denominator polynomial and govern the system's natural (unforced) response. On the s-plane (s = σ + jω), the real part of each pole (σ) determines whether the system decays or diverges.
- Stable: All poles lie in the left-half plane (LHP, σ < 0). Any excitation decays and the system reaches steady state.
- Unstable: Even a single pole in the right-half plane (RHP, σ > 0) causes the output to diverge exponentially.
- Oscillatory response: Complex poles (non-zero jω component) introduce natural-frequency oscillation. Poles closer to the imaginary axis result in slower decay and longer settling time.
Zeros (◯) and Their Effect
Zeros are the roots of the numerator polynomial — the complex frequencies at which the system output goes to zero. They significantly influence the magnitude of overshoot and the phase profile in the transient response.