4.5: Derivation of the Complex Frequency-Response Function - Easy derivation of the complex frequency-response function for standard stable first order systems.

This section is an example of a much easier method (than that of Sections 4.2 and 4.3) for deriving the frequency-response function of a system. Let us find the frequency response of standard stable 1^st order systems. From Equation 3.4.8 the standard stable ODE with sinusoidal excitation is

\[\dot{x}+\left(1 / \tau_{1}\right) x=b u(t)=b U \cos \omega t \nonumber \]

We seek steady-state sinusoidal response \(x_{s s}(t)=X(\omega) \cos (\omega t+\phi(\omega))\), in which \(X(\omega)\) and \(\phi(\omega)\) are functions to be found. The first step in the method is to take the general [for arbitrary \(u(t)\)] Laplace transform, setting the IC to zero:

\[\left.\left(s+1 / \tau_{1}\right) L[x(t)]\right|_{x_{0}=0}=b L[u(t)] \nonumber \]

Next, we form the system general transfer function, \(T F(s)\), defined as the ratio of the output transform to the input transform, with zero IC:

\[T F(s) \equiv \frac{\left.L[x(t)]\right|_{x_{0}=0}}{L[u(t)]}=\frac{b}{s+1 / \tau_{1}} \nonumber \]

The Laplace independent variable \(s\) is complex in general. However, in order to analyze frequency response, we let \(s\) in \(TF(s)\) be purely imaginary, \(S=j \omega\) (\(\omega\) being the real circular frequency), producing the complex frequency-response function \(FRF(\omega)\):

\[ \left.T F(s)\right|_{s=j \omega} \equiv T F(j \omega) \equiv F R F(\omega)=\frac{b}{1 / \tau_{1}+j \omega}=b \tau_{1} \frac{1}{1+j \omega \tau_{1}}\label{eqn:4.18a} \]

Next, with use of Equations 2.1.6, 2.1.7, 2.1.8, and 2.1.11, we convert \(T F(j \omega) \equiv F R F(\omega)\) algebraically into polar form:

\[\begin{align} FRF(\omega) &=b \tau_{1} \frac{1}{1+j \omega \tau_{1}} \times \frac{1-j \omega \tau_{1}}{1-j \omega \tau_{1}} \\[4pt] &=b \tau_{1} \frac{\sqrt{1+\left(\omega \tau_{1}\right)^{2}}}{1+\left(\omega \tau_{1}\right)^{2}} \exp \left[j \tan ^{-1}\left(\frac{-\omega \tau_{1}}{1}\right)\right] \\[4pt] &\equiv |F R F(\omega)| e^{j \angle F R F(\omega)} \\[4pt] &=\frac{b \tau_{1}}{\sqrt{1+\left(\omega \tau_{1}\right)^{2}}} e^{j \phi(\omega)}\label{eqn:4.18b} \end{align} \]

in which phase angle \(\phi(\omega)=\tan ^{-1}\left(-\omega \tau_{1} / 1\right)=\tan ^{-1}\left(-\omega \tau_{1}\right)\).

Equations \(\ref{eqn:4.18a}\) and \(\ref{eqn:4.18b}\) define the complex frequency-response function, \(F R F(\omega)\), of standard stable 1^st order systems. It is proved in Sections 4.6 and 4.7 for LTI systems in general that the real magnitude \(|F R F(\omega)|\) of function \(F R F(\omega)\) is the magnitude ratio of system frequency response, and the phase angle \(\phi(\omega)\) of function \(F R F(\omega)\) is the phase angle of system frequency response. For example, let us adapt standard solution Equation \(\ref{eqn:4.18b}\) to the damper-spring system, for which [from Equation 4.2.5] \(b=1 / c\) and \(\tau_{1}=c / k\). Thus, the magnitude of \(F R F(\omega)\) from Equation \(\ref{eqn:4.18b}\) is

\[|F R F(\omega)|=\frac{X(\omega)}{U}=\frac{b \tau_{1}}{\sqrt{1+\left(\omega \tau_{1}\right)^{2}}}=\frac{1}{k} \frac{1}{\sqrt{1+\left(\omega \tau_{1}\right)^{2}}}\label{eqn:4.19} \]

which is identical to damper-spring system \(FRF\) magnitude ratio \(X(\omega) / F\) of Equation 4.3.8. Also, the phase angle of \(FRF(\omega)\) from Equation \(\ref{eqn:4.18b}\) is

\[\angle F R F(\omega)=\phi(\omega)=\tan ^{-1}\left(-\omega \tau_{1}\right)\label{eqn:4.20} \]

which is identical to damper-spring system \(FRF\) phase \(\phi(\omega)\) of Equation 4.3.9. Thus, with \(F R F(\omega)\) of Equation \(\ref{eqn:4.18b}\), we have obtained here the same final results as before for the damper-spring system, but much more easily.