Frequency Analysis of sine test data

Frequency response of a system means its steady-state response to a sinusoidal input. For obtaining a frequency response of a system, we vary the frequency of the input signal over a spectrum of interest. The analysis is useful and simple because it can be carried out with the available signal generators and measuring devices. Let us see the theory and procedure. Please note that this procedure is common for data obtained using both local and virtual experiments.

Consider a sinusoidal input

$\displaystyle U(t)$

$\displaystyle = Asin \omega t$

(4.1)

The Laplace transform of the above equation yields

$\displaystyle U(s)$

$\displaystyle = \frac{A\omega}{s^2 + \omega^2}$

(4.2)

Consider the standard first order transfer function given below

$\displaystyle G(s)$

$\displaystyle = \frac {Y(s)}{U(s)} = \frac K{s + 1}$

(4.3)

Replacing the value of U(s) from equation 4.2, we get

$\displaystyle Y(s)$	$\displaystyle = \frac{KA\omega}{(\tau s + 1)(s^2 + \omega ^2)}$	(4.4)
	$\displaystyle =\frac{KA}{\omega ^2\tau ^2 + 1}\left[\frac{\omega \tau ^2}{\tau s +1}- \frac{\tau s \omega}{s^2 + \omega^2}+\frac{\omega} {s^2 + \omega^2}\right]$	(4.5)

Taking Laplace Inverse, we get

$\displaystyle y(t)$

$\displaystyle = \left[\frac {KA}{\omega^2\tau^2+ 1}\right]\left[\omega \tau e^{\frac {-t}{\tau}}-\omega \tau cos(\omega t)+ sin(\omega t)\right]$

(4.6)

The above equation has an exponential term $e^\frac{-t}{\tau}$ . Hence, for large value of time, its value will approach to zero and the equation will yield a pure sine wave. One can also use trigonometric identities to make the equation look more simple.

$\displaystyle y(t)$

$\displaystyle = \left[\frac{KA}{\sqrt{\omega^2 \tau^2 + 1}}\right]\left[sin (\omega t) + \phi \right]$

(4.7)

where,

$\displaystyle \phi$

$\displaystyle = -tan^{-1}(\omega \tau)$

(4.8)

By observing the above equation, one can easily make out that for a sinusoidal input the output is also sinusoidal but has some phase difference. Also, the amplitude of the output signal, $\hat{A}$ , has become a function of the input signal frequency, $\omega$ .