Transfer Functions and Root Locus Plots of PLLs

Key Takeaways

• The basic building blocks of a PLL are a phase-frequency detector (PFD),  a loop filter (LF), a voltage-controlled oscillator (VCO), and a frequency divider (FD).

• The open-loop transfer function of the second-order PLL can be given by 

• The open-loop  transfer function of a third-order PLL can be given by 

Electrical smart grids, electronic instrumentation, communication networks, and computer clusters are engineering applications that have been revolutionized by primary-secondary clock distribution systems. In all these applications, phase-locked loops (PLLs) are responsible for process synchronization and recovery of the correct time basis. Various clock distribution topologies are possible using PLLs, but master-slave time distribution is the most frequently used. PLLs are closed-loop systems that can generate the desired frequency with respect to a reference. Since they are a closed-loop system, system performance can be analyzed using root locus plots.

Transfer Functions and Root Locus Plots of PLLs

The basic building blocks of a PLL are a phase-frequency detector (PFD), a loop filter (LF), a voltage-controlled 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 closed-loop PLL system. It also smooths the PFD output and makes it a slow-changing analog input to VCO. Usually, first-order or second-order RC filters are used as the LF. The closed-loop 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 closed-loop 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 open-loop 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 first-order system. 

Root Locus of Second-Order PLLs

The introduction of a first-order filter makes the order of the PLL transfer function equal two. The LF consists of a single resistor (R) and a (C). The first-order filter adds one more pole at origin and a zero at 1RC. The open-loop transfer function  of the second-order PLL can be given by

where K=K_dK_o/N. 

The RC leg connected across to the PFD output is a first-order 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 second-order PLL is given by the below figure. There exist two poles at origin and a zero at -10⁸.

Root Locus of Third-Order PLLs

Consider adding a capacitor C₂ across the RC filter in second-order PLLs. The LF is transformed into a second-order system with two passive elements. The second-order LF turns the PLL into a third-order system and the open-loop  transfer function can be given by:

There are three poles and one zero in the transfer function of third-order PLLs. The two poles are at origin and the third pole is at . The zero of the third-order transfer function lies at .  Let the value of capacitor C2 be ten times lesser than C ( 0.001µ F). 

The root locus plot of second-order PLLs is entirely different from that of third-order 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 master-slave 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.