PLL Tracking

 From the previous section, the noise rejection property of a phaselock loop dependent on a narrowband filter. If such a filter had a fixed filter then the signal would rarely be within the passband. Therefore it is essential that a narrowband filter accurately tracks the input signal. This section presents material from [11] analysing this property. This will allow the ability of a PLL to track to be related to its basic properties. From the material in section 3.4.2, it can easily be seen that the phase error defined by $\varphi_e(s)=\varphi_i(s)-\varphi_o(s)$ is given by  

$$\varphi_e(s)=\frac{s\varphi_i(s)}{s+K_oK_dF(s)}$$ (22)

The above equation may be used to analyse steady state errors by use of the Laplace Transform final value theorem  

$$\lim_{t\to\infty}f(t)=\lim_{s\to0}s\mathcal{L}\big(f(t)\big).$$ (23)