Transfer Functions and Root Locus Plots of PLLs
Key Takeaways

The basic building blocks of a PLL are a phasefrequency detector (PFD), a loop filter (LF), a voltagecontrolled oscillator (VCO), and a frequency divider (FD).

The openloop transfer function of the secondorder PLL can be given by

The openloop transfer function of a thirdorder PLL can be given by
Electrical smart grids, electronic instrumentation, communication networks, and computer clusters are engineering applications that have been revolutionized by primarysecondary clock distribution systems. In all these applications, phaselocked loops (PLLs) are responsible for process synchronization and recovery of the correct time basis. Various clock distribution topologies are possible using PLLs, but masterslave time distribution is the most frequently used. PLLs are closedloop systems that can generate the desired frequency with respect to a reference. Since they are a closedloop system, system performance can be analyzed using root locus plots.
The basic building blocks of a PLL are a phasefrequency detector (PFD), a loop filter (LF), a voltagecontrolled oscillator (VCO), and a frequency divider (FD)
Transfer Functions and Root Locus Plots of PLLs
The basic building blocks of a PLL are a phasefrequency detector (PFD), a loop filter (LF), a voltagecontrolled oscillator (VCO), and a frequency divider (FD). The figure above shows the block diagram of a basic PLL. The PFD compares the input signal and feedback signal from the FD and produces an output proportional to the phase difference between the two. If the phase of the input signal lags behind the phase of the feedback signal, then PFD produces an inhibiting signal to lower the VCO frequency.
Otherwise, the PFD output to LF is an excitatory signal to increase the frequency of the VCO. The LF controls the bandwidth and lock time of the closedloop PLL system. It also smooths the PFD output and makes it a slowchanging analog input to VCO. Usually, firstorder or secondorder RC filters are used as the LF. The closedloop is formed by the frequency divider, which divides the VCO output frequency by a factor of N and allows the input frequency to be lower than the VCO frequency. The PLL output frequency is programmable using the variable ‘N’.
Transfer Function of PLLs
Assuming the closedloop bandwidth of a PLL is less than the reference frequency and under a nearly locked condition, a PLL can be considered to be a continuous linear system. In such a case, the openloop transfer function of the basic PLL can be given by the ratio of Laplace transform of VCO output to the Laplace transform of the input signal, at zero initial conditions.
Where K_{o} is the VCO gain, K_{d} is the phase detector gain, N is the frequency divider factor, and F(s) is the transfer function of LF. Equation 1 defines the transfer function of a basic PLL.
Depending on the type of LF, the order of the PLL transfer function changes. From equation 1, it is clear that there exists a pole at origin and a zero at infinity and it is a firstorder system.
Root Locus of SecondOrder PLLs
The introduction of a firstorder filter makes the order of the PLL transfer function equal two. The LF consists of a single resistor (R) and a (C). The firstorder filter adds one more pole at origin and a zero at 1RC. The openloop transfer function of the secondorder PLL can be given by
where K=K_{d}K_{o}/N.
The RC leg connected across to the PFD output is a firstorder filter, as there is only one capacitor in the LF. The order of the PLL transfer function will be one greater than the order of the LF transfer function, on the basis of equation 1. Consider a PLL system with an RC filter, where R=1 Ω and capacitor equal to 0.01µF—the root locus plot of the secondorder PLL is given by the below figure. There exist two poles at origin and a zero at 10^{8}.
A secondorder PLL system consists of two poles and a zero
Root Locus of ThirdOrder PLLs
The insertion of an extra capacitor makes an RC filter a secondorder system
Consider adding a capacitor C_{2}, as shown in the figure above, across the RC filter in secondorder PLLs. The LF is transformed into a secondorder system with two passive elements. The secondorder LF turns the PLL into a thirdorder system and the openloop transfer function can be given by:
There are three poles and one zero in the transfer function of thirdorder PLLs. The two poles are at origin and the third pole is at . The zero of the thirdorder transfer function lies at . Let the value of capacitor C2 be ten times lesser than C ( 0.001µ F). The root locus plot of the PLL can be given by the figure below.
Thirdorder PLL systems consist of three poles and a zero
As we have seen, the root locus plot of secondorder PLLs is entirely different from that of thirdorder PLLs. The LF plays a significant role in PLL performance and stability. The bandwidth and settling time of the PLL can be adjusted by changing the order of the LF transfer function. When you are working on masterslave time distribution systems, analyze the root locus plots of your PLL with a different LF before choosing—this will make the PLL locking fast and stable.