Transfer Function

A transfer function G(s) is defined for a linear time-invariant (LTI) system as the ratio of the Laplace transform of the output Y(s) to the Laplace transform of the input U(s), with all initial conditions set to zero:

The variable s = \sigma + j\omega is the complex Laplace variable. Along the imaginary axis (s = j\omega), the transfer function becomes the frequency response G(j\omega), which describes how the system amplifies and phase-shifts sinusoidal inputs at each frequency \omega.

Rational form and system order

For systems described by linear ordinary differential equations with constant coefficients, the transfer function is a rational function — a ratio of polynomials in s:

The roots of the numerator polynomial are the zeros z_i; the roots of the denominator polynomial are the poles p_j. The order of the system is n — the degree of the denominator. A physical system must be proper (n \geq m), meaning it cannot have more zeros than poles.

Poles, stability, and transient response

The poles of the transfer function are the eigenvalues of the system and determine its natural modes of response. A pole at s = p = \sigma_p + j\omega_p contributes a term of the form e^{p t} = e^{\sigma_p t} e^{j\omega_p t} to the impulse response. For the system to be stable, all poles must lie in the left half of the complex plane (\sigma_p < 0), so that all natural modes decay with time. A pole on the imaginary axis produces sustained oscillation; a pole in the right half-plane produces a growing, unstable response.

The location of poles in the complex plane encodes the character of the transient response: poles close to the imaginary axis correspond to slowly decaying modes; poles with a large imaginary part correspond to oscillatory modes with frequency \omega_p; the damping ratio \zeta of a complex conjugate pair is related to the angle from the negative real axis.

Closed-loop transfer function

In a feedback control system with plant transfer function G(s) and controller transfer function C(s), the open-loop transfer function is L(s) = C(s) G(s). The closed-loop transfer function from reference R(s) to output Y(s) is:

The denominator 1 + L(s) is the characteristic polynomial of the closed loop. Its roots — the closed-loop poles — determine stability and transient performance. The Routh–Hurwitz criterion, the Nyquist criterion, and root locus methods all use the transfer function to analyse how the closed-loop pole locations depend on the controller parameters.

Frequency response and Bode plots

Evaluating the transfer function on the imaginary axis gives the frequency response:

The magnitude |G(j\omega)| (in decibels: 20 \log_{10} |G|) and phase \angle G(j\omega) as functions of frequency \omega are displayed on a Bode plot. The Bode plot reveals bandwidth, resonance peaks, and the gain and phase margins that quantify how close the system is to instability — essential information for controller design and performance assessment.

Common mistakes

A common mistake is using a transfer function for a system that is strongly nonlinear, time-varying, saturated, or operating far from the linearization point. Another is forgetting that the definition assumes zero initial conditions, so stored energy and initial state can affect time response even when the input-output transfer function is unchanged. A strong transfer-function review states input and output variables, units, operating point, model order, poles and zeros, delay treatment, validity range, and whether the model has been validated in time and frequency domains.