MATH319 Slides 103 Transfer function of a Laplace transformed system 105 Example of gain and phase

104 Frequency Response

Suppose that we have a SISO with Laplace transform $\hat{Y}(s)=T(s)\hat{U}(s).$ We change variable to $s=i\omega$ so $\hat{Y}(i\omega)=T(i\omega)\hat{U}(i\omega).$ (Consider input $e^{i\omega t}$ , with $i\omega$ on imaginary axis in $s$ plane.)

Definition (Gain and Phase)

Define the frequency response to be $T(i\omega)$ . Then define the gain (or amplitude gain) of the system to be $\Gamma(\omega)=|T(i\omega)|$ at angular frequency $\omega\in{\bf R}$ ; define the phase (shift) to be $\phi(\omega)=\arg T(i\omega)$ . Then $T(i\omega)=\Gamma(\omega)e^{i\phi(\omega)}$ .

The gain $\Gamma(\omega)$ is the factor that changes the amplitude (height) of a signal at angular frequency $\omega.$ The phase (or phase shift) $\phi(\omega)$ is the change in phase of the signal. When $\phi(\omega)>0$ , one talks of a phase gain, so the output is running ahead of the input. When $\phi(\omega)<0$ , the output is running behind the input and the phase lag is $-\phi(\omega)>0$ . In engineering, the frequency response is relatively easy to measure.