5.2.1 Transfer Function

The steady-state characteristic in the frequency domain after phase synchronization is called a transfer function. It shows the amplitude changes of frequency components passing through the PLL. In general, the Laplace transform and its special case, the Fourier transform, are used for analysis in the frequency domain over continuous time. In particular, a differential equation can be converted into an algebraic one by using the Laplace transform because of its linearity and the conversion form for derivative functions. The first-order differential equation shown in (5.6) is converted by Laplace transformation into

where ? _o( s) and ? _i( s) represent the Laplace transform of ? _o( t) and ? _i( t), respectively, and it is assumed that the free-running frequency of the controlled oscillator always equals the frequency of the input signal, and that w _free( t) = 0.

The ratio of the output phase change to the input phase change can be calculated as shown in (5.8), which is derived from that based on (5.7).

The response in the real frequency domain can be found by the Laplace transform (Fourier transform) along the jw axis on the s plane by setting s = jw. After this, the response in (5.8) is becomes transfer function H( jw) of angular frequency ? and the amplitude or absolute value of H( jw) showing the frequency response of...