Gain and Phase

The graphing window shows two sinusoids that could be the input and output of a linear time invariant system. The input and output sinusoids are color coded, so the graph of each is the same as their text color. Also, the input sinusoid is always a cosine curve with no phase lag. The input sinusoid is the graph of \(f(t) = B\cos(\omega t)\). That is, it has amplitude \(B\) and phase lag 0. The output sinusoid is \(x(t) = gB\cos(\omega t - \phi)\). That is, the system has gain \(g\) and phase lag \(\phi\).

The values of input amplitude \(B\), gain \(g\), phase lag \(\phi\) and input angular frequency \(\omega\) are all settable using the corresponding sliders.

The values of the time lag \(t_0\), period \(P\) and frequency \(\nu\) are shown at the right below the graphing window. Time lag is the time between the first maximum of the reference sinusoid and the first maximum of the transformed sinusoid. This is also indicated by the horizontal line and vertical strut from the first maximum of the input to the first maximum of the output. The period of both input and output is indicated by the heavy horizontal line and vertical strut along the horizontal axis starting at the origin. Being careful with terminology: the angular frequency \(\omega\) has units of radians per unit time and the frequency \(\nu\) has units of cycles per unit time. Of course, the relationship is \(\omega = 2\pi\nu\).

Note: For an actual linear time invariant system the gain and phase lag would depend on the input frequency, while in this mathlet gain and phase are independent of input frequency.