Editing Phase-locked loop (section)

===Linearized phase domain model===
Phase locked loops can also be analyzed as control systems by applying the [[Laplace transform]]. The loop response can be written as

:<math>\frac{\theta_o}{\theta_i} = \frac{K_p K_v F(s)} {s + K_p K_v F(s)}</math>

Where
* <math>\theta_o</math> is the output phase in [[radian]]s
* <math>\theta_i</math> is the input phase in radians
* <math>K_p</math> is the phase detector gain in [[volt]]s per radian
* <math>K_v</math> is the VCO gain in radians per volt-[[second]]
* <math>F(s)</math> is the loop filter transfer function (dimensionless)

The loop characteristics can be controlled by inserting different types of loop filters. The simplest filter is a one-pole [[RC circuit]]. The loop transfer function in this case is

:<math>F(s) = \frac{1}{1 + s R C}</math>

The loop response becomes:

:<math>\frac{\theta_o}{\theta_i} = \frac{\frac{K_p K_v}{R C}}{s^2 + \frac{s}{R C} + \frac{K_p K_v}{R C}}</math>

This is the form of a classic [[harmonic oscillator]]. The denominator can be related to that of a second order system:

:<math>s^2 + 2 s \zeta \omega_n + \omega_n^2</math>

where <math>\zeta</math> is the damping factor and <math>\omega_n</math> is the natural frequency of the loop.

For the one-pole RC filter,

:<math>\omega_n = \sqrt{\frac{K_p K_v}{R C}}</math>
:<math>\zeta = \frac{1}{2 \sqrt{K_p K_v R C}}</math>

The loop natural frequency is a measure of the response time of the loop, and the damping factor is a measure of the overshoot and ringing. Ideally, the natural frequency should be high and the damping factor should be near 0.707 (critical damping). With a single pole filter, it is not possible to control the loop frequency and damping factor independently. For the case of critical damping,

:<math>R C = \frac{1}{2 K_p K_v}</math>
:<math>\omega_c = K_p K_v \sqrt{2}</math>

A slightly more effective filter, the lag-lead filter includes one pole and one zero. This can be realized with two resistors and one capacitor. The transfer function for this filter is

:<math>F(s) = \frac{1+s C R_2}{1+s C (R_1+R_2)}</math>

This filter has two time constants

:<math>\tau_1 = C (R_1 + R_2)</math>
:<math>\tau_2 = C R_2</math>

Substituting above yields the following natural frequency and damping factor

:<math>\omega_n = \sqrt{\frac{K_p K_v}{\tau_1}}</math>
:<math>\zeta = \frac{1}{2 \omega_n \tau_1} + \frac{\omega_n \tau_2}{2}</math>

The loop filter components can be calculated independently for a given natural frequency and damping factor

:<math>\tau_1 = \frac{K_p K_v}{\omega_n^2}</math>
:<math>\tau_2 = \frac{2 \zeta}{\omega_n} - \frac{1}{K_p K_v}</math>

Real world loop filter design can be much more complex e.g. using higher order filters to reduce various types or source of phase noise. (See the D Banerjee ref below)