The transfer function is the Laplace-domain ratio of output to input, assuming zero initial conditions. It completely characterizes LTI (Linear Time-Invariant) system behavior.

$G(s) = \frac{Y(s)}{U(s)}$

Deriving Transfer Functions

From differential equations: Take Laplace transform, assume zero ICs, solve for Y(s)/U(s).

Example — RC lowpass filter:

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$G(s) = \frac{Y(s)}{U(s)} = \frac{1}{\tau s + 1}$

Poles and Zeros

Transfer function in factored form:

$G(s) = K \frac{(s-z_1)(s-z_2)\cdots}{(s-p_1)(s-p_2)\cdots}$

Zeros ($z_i$): Values where $G(s) = 0$ Poles ($p_i$): Values where $G(s) \to \infty$

Poles determine stability and natural response. Zeros affect transient shape and frequency response.

Closed-Loop Transfer Function

For unity feedback system with forward path $G(s)$ :

$T(s) = \frac{G(s)}{1 + G(s)}$

With feedback path $H(s)$ :

$T(s) = \frac{G(s)}{1 + G(s)H(s)}$

The denominator $1 + G(s)H(s) = 0$ is the characteristic equation — its roots are the closed-loop poles.

Block Diagram Algebra

Series: $G_1(s) \cdot G_2(s)$ Parallel: $G_1(s) + G_2(s)$ Feedback: $\frac{G(s)}{1 \$

Advanced

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Power Systems