The phase lock loop (PLL) was first proposed in 1932 by the French scientist de Bellescize, who is also credited with being the inventor of coherent demodulation. PLLs come from a heritage of control and vibration theory where they are used to describe the freebody behaviour of mechanical systems. In communication systems, PLLs are commonly used for the following applications:

- Carrier synchronization
- Carrier recovery
- Frequency division and multiplication
- Demodulation

A PLL can be analog or digital. The same concepts apply to both and they also share the same standard parameters like loop bandwidth, damping factor, etc.

A PLL consists of three basic components:

- Phase detector
- Loop filter
- Voltage-controlled oscillator (VCO)

The three components have the following relationship:

The phase detector is a device whose output is a function of the phase difference between the two inputs. This difference is called phase error. The phase error is then filtered by the loop filter to produce a control voltage that is used to adjust the frequency of the VCO. Ideally, the PLL should produce a phase estimate that has zero phase error. This characteristic of the phase error transfer function will be used to determine the desirable properties of the loop filter and the loop transfer function is used to characterize the performance of a PLL.

A common method of analyzing a PLL (which is a non-linear feedback control system) is to linearize the system at a desired operating point. Once this is done, the loop can be analyzed using standard linear system techniques. Here, our desired operating point is at *error = 0*. Let’s illustrate a linearized phase equivalent PLL:

In the above diagram, the phase detector was replaced by a subtractor and a gain *k _{p}*. Also, the VCO was replaced by an integral term multiplied by a gain

*k*. The integral comes from the fact that the frequency of a periodic signal is equal to the rate change of the phase in 2π segments. Conversely, the phase is the integral of the frequency over a certain period of time. Which leads to:

_{o}since the frequency of a VCO is directly linked to its input voltage multiplied by a certain gain.

However, analyzing a control system with integrals and derivatives is tedious. Since the phase equivalent loop is a linear system, it can be analyzed using Laplace transform techniques. Laplace transform is out of the scope of this blog post but basically, it’s a more general form of the Fourier transform. Without going into the mechanics, let’s state the most common transform pairs:

Knowing these three relationships, we can now draw the frequency domain equivalent of the linearized phase equivalent PLL block diagram:

where *F(s)* is the transfer function of the loop filter. This last representation is very useful as we can write the transfer function of PLL as follows:

We can also express the transfer function relative to the phase error:

In the next part of this series, we will look at how these two transfer functions are used to characterize a PLL's performance.