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

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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 1st 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 1st 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.

This page titled 4.5: Derivation of the Complex Frequency-Response Function - Easy derivation of the complex frequency-response function for standard stable first order systems. is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by William L. Hallauer Jr. (Virginia Tech Libraries' Open Education Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.