Evaluating continuous transfer functions

\( \require{AMSsymbols}\def\lfz#1{\overset{\Large#1}{\,\circ\kern-6mu-\kern-7mu-\kern-7mu-\kern-6mu\bullet\,}} \def\laplace{\lfz{\mathscr{L}}} \def\fourier{\lfz{\mathcal{F}}} \def\ztransform{\lfz{\mathcal{Z}}} \require{cancel} \newcommand\ccancel[2][black] {\color{#1}{\cancel{\color{black}{#2}}}} \) The transfer function can be evaluated using different inputs. We commonly use the impulse, step and sinusoidal input functions.

Let \(H\) be a stable system with transfer function \(H(s)\), input signal \(x(t)\), and output \(y(t)\). In this, “stable” implies that the poles are in left half of \(s\)-plane.

Transfer Function in s-domain

Impulse Response

Earlier, we derived the Laplace transform for the impulse function as $$ \delta(t)\laplace\Delta(s) = 1 $$

Substituting this input function $$ Y(s) = H(s)\,\Delta(s) $$

The response to an impulse input function, is the transfer function \(H(s)\) itself $$ \shaded{ Y(s)=H(s) } $$

Unit Step Response

The page Laplace Transforms gives the Laplace transform for the unit step function as $$ \gamma(t) \laplace \Gamma(s)\frac{1}{s} $$

Substitute the input function $$ Y(s) = H(s)\,\Gamma(s) $$

The response to an unit step input function follows as $$ \shaded{ Y(s) = \frac{H(s)}{s} } $$

Frequency Response

The frequency response is defined as the steady state response of system to a sinusoidal input.

Given sinusoidal input and transfer function $$ \left\{\begin{align} x(t) &= \sin(\omega t)\gamma(t) &\mathrm{input\ signal} \nonumber \\ H(s) &= |H(s)|\,e^{j\angle H(s)} &\mathrm{transfer\ function} \nonumber \end{align}\right. $$

Find the output signal \(y(t)\), using the Laplace transform of the sinusoidal input function $$ \left.\begin{align} X(s) &= \frac{\omega}{s^2+\omega^2} \nonumber \\[8mu] Y(s) &= H(s)\,X(s)\nonumber \end{align}\right\} $$

Substitute \(X(s)\) in \(Y(s)\) $$ \begin{align} Y(s) &= H(s)\,\frac{\omega}{s^2+\omega^2} \nonumber \\ &= H(s)\,\frac{\omega}{(s+j\omega)(s-j\omega)} \end{align} $$

According to Heaviside, this can be expressed as partial fractions [swarthmore, MIT-cu], where the term \(C_h(s)\) represents the transient response resulting from \(H(s)\). This term is independent of \(j\omega\), and dies out for \(t\to\infty\). $$ Y(s) = \frac{\omega}{(s+j\omega)(s-j\omega)}H(s)=\underbrace{\frac{c_0}{s+j\omega}+\frac{c_1}{s-j\omega}}_{Y_{ss}(s)=\mathrm{steady\ state\ response}}+\underbrace{C_h(s)}_\mathrm{transient\ response} \label{eq:partialfractions} $$

Find \(c_0\) $$ \begin{align} H(s)\,\frac{\omega\cancel{(s+j\omega)}}{\cancel{(s+j\omega)}(s-j\omega)} &= \frac{c_0\cancel{(s+j\omega)}}{\cancel{s+j\omega}}+\frac{c_1(s+j\omega)}{s-j\omega}+C_h(s)(s+j\omega) \nonumber \\ \Rightarrow\, H(s)\,\frac{\omega}{s-j\omega} &= \left. c_0 + c_1\frac{s+j\omega}{s-j\omega} + C_h(s)(s+j\omega)\right|_{s=-j\omega} \nonumber \\ \Rightarrow\, H(-j\omega)\,\frac{\omega}{-j\omega-j\omega} &= c_0+c_1\frac{\cancelto{0}{-j\omega+j\omega}}{s-j\omega}+C_h(j\omega)(\cancelto{0}{-j\omega+j\omega}) \nonumber \\ \Rightarrow\, c_0 &= H(-j\omega)\,\frac{\cancel{\omega}}{-2j\cancel{\omega}} = \frac{H(-j\omega)}{-2j} \end{align} $$

Similarly, find \(c_1\) $$ \begin{align} H(s)\,\frac{\omega\cancel{(s-j\omega)}}{(s+j\omega)\cancel{(s-j\omega)}} &= \frac{c_0(s-j\omega)}{s+j\omega}+\frac{c_1\cancel{(s-j\omega)}}{\cancel{s-j\omega}}+C_h(s)(s-j\omega) \nonumber \\ \Rightarrow\, H(s)\,\frac{\omega}{s+j\omega} &= \left. c_0\frac{s-j\omega}{s-j\omega}+c_1+C_h(s)(s-j\omega)\right|_{s=j\omega} \nonumber \\ \Rightarrow\, H(j\omega)\,\frac{\omega}{j\omega+j\omega} &= c_0\frac{\cancelto{0}{j\omega-j\omega}}{s-j\omega}+c_1+C_h(j\omega)(\cancelto{0}{j\omega-j\omega}) \nonumber \\ \Rightarrow\, c_1 &= H(j\omega)\,\frac{\cancel{\omega}}{2j\cancel{\omega}} = \frac{H(j\omega)}{2j} \end{align} $$

Inverse Laplace transform of \(\eqref{eq:partialfractions}\) back to the time domain $$ y(t) = \underbrace{c_0e^{-j\omega t}+c_1e^{j\omega t}}_{y_{ss}(t)=\mathrm{steady\ state\ response}}+\underbrace{\cancelto{0\mathrm{\ as\ }t \to\infty}{\mathfrak{L}^{-1}\left\{C_h(s)\right\}}}_{\mathrm{transient\ response}} $$

Substitute \(c_0\) and \(c_1\) in \(y_{ss}(t)\) $$ \begin{align} y_{ss}(t) &= \frac{H(-j\omega)}{-2j}e^{-j\omega t}+\frac{H(j\omega)}{2j}e^{j\omega t} \nonumber \\ &= \frac{H(j\omega)e^{j\omega t}-H(-j\omega)e^{-j\omega t}}{2j}\label{eq:yss} \end{align} $$

Based on Euler’s formula we can express \(H(s)\) in polar coordinates

$$ \left\{ \begin{align} H(s) &= |H(s)|\,e^{j\angle H(s)} \nonumber \\ |H(j\omega)| &= K \frac{\prod_{i=1}^m \sqrt{\left(\Re\{{z_i}\}\right)^2+\left(\omega+\Im\{z_i\}\right)^2}}{\prod_{i=1}^n \sqrt{\left(\Re\{{p_i}\}\right)^2+\left(\omega+\Im\{p_i\}\right)^2}} \nonumber \\ \angle{H(s)} &= \sum_{i=1}^m\mathrm{atan2}\left(\omega+\Im\{z_i\}, \Re\{{z_i}\}\right) -\sum_{i=1}^n\mathrm{atan2}\left(\omega+\Im\{p_i\}, \Re\{{p_i}\}\right) \nonumber \\ \end{align} \right. \label{eq:euler} $$

Substitute the polar representation of \(H(s)\), in \(y_{ss}\) $$ \begin{align} y_{ss}(t) &= |H(j\omega)|\left( \frac{e^{j\angle H(j\omega)}e^{j\omega t}-e^{j\angle H(-j\omega)}e^{-j\omega t}}{2j} \right) \nonumber \\ &= |H(j\omega)|\left( \frac{e^{j\left(\angle H(j\omega)+\omega t\right)}-e^{j\left(\angle H(-j\omega)-\omega t\right)}}{2j} \right) \nonumber \\ &= |H(j\omega)|\, \underbrace{\frac{e^{j(\omega t + \angle{H(j\omega)})}-e^{-j(\omega t + \angle H(j\omega))}}{2j}}_{\sin(\omega t+\angle H(j\omega))} \end{align} $$

In this, we recognize the Laplace transfer for a sinusoidal function

$$ \sin(\omega t)\gamma(t) \laplace \frac{\omega}{s^2+\omega^2} \nonumber $$

The frequency response follows as $$ \shaded{ y_{ss}(t) = |H(j\omega)|\,\sin(\omega t+\angle H(j\omega))\,\gamma(t) } $$

In other words, for a linear system a sinusoidal input generates a sinusoidal output with the same frequency, but different amplitude and phase.


Suggested next reading is Impedance.

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