In the previous blog post of this series, we looked at the principles behind phase lock loops (PLLs). In this post, we look at how the following transfer function is used to characterize a PLL performance:
We can also express the transfer function relative to the phase error:
Two specific PLL inputs are of special interest in characterizing a PLL:
· Simple phase difference
· Frequency shift radians/second
The response of the loop function for these two inputs is called, respectively, the step response and ramp response.
The goal of a PLL is to be able to get perfectly in phase with its input. In other words, the phase error between the input (either a step or ramp input) and the VCO needs to be zero at some point in time. The only loop filter that can provide such characteristics to a PLL is the proportional-plus-integrator loop filter with the following transfer function:
By replacing the above transfer function in the PLL transfer function, we get the following transfer function:
This is very convenient as it’s a second order system that can be expressed as:
This form is interesting as we can now relate the damping factor and the natural frequency to the different gains of our system:
The damping factor is most interesting as it controls the transient response of our PLL. The damping factor can be lower than 1, equal to 1 or higher to 1 and define the rise-time, the overshoot and oscillations created at the output of our PLL in response to a disturbance (a step input for example):
Another way to characterize a PLL is to compute the bandwidth of the frequency response or more importantly, the equivalent noise bandwidth Bn. The equivalent noise bandwidth depends on the loop filter used. For a proportional-plus-integrator loop filter, the noise bandwidth is:
The noise bandwidth is also useful to determine the range of frequency offsets for which the loop can acquire lock (called the pull-in range ). This is well approximated by:
That being said, the last step is to select the loop constants. The results above are showing that the designer has two degrees of freedom: the damping constant and the natural frequency (which can be related to the noise bandwidth and pull-in range). Following the above equations, we can solve for the loop constants:
Since is a parameter chosen by the designer, all that remains is to find an expression for :
From this point, it’s possible to express the loop constants in terms of the damping constant and the equivalent noise bandwidth.
In part 3 of this blog, we will discuss how we can use the same concepts for digital PLLs.