PLL is an electronic circuit which causes a output signal to keep track of the input
reference signal applied to it and the output signal keeps synchronization with the input one.
Three basic functional blocks, a phase detector (PD), a loop filter (LF) and a voltage
controlled oscillator (VCO), are contained in a PLL depicted in Fig. 6.1.
Assume the reference input ( )v ti =A sin(ωit+θi) and the VCO output
( ) Bsin( o )
o o
v t = ω t+θ , where ωi and ωo are angular frequencies, θi and θo are phases,
A and B are amplitudes of v t and i( ) v t , respectively. o( )
In a PLL, the function of the PD is to measure the phase difference between v t and i( )
0( )
v t and produces an output voltage v t proportional to the phase error of ( )d( ) v t and i
0( )
v t . Assume the PD be a linear multiplier in linear PLL (LPLL) through this chapter.
The LF is a low-pass filter and is used to suppress noise and high-frequency signal
components which are unwanted signals. The lower frequency and dc parts are passed
through the LF and delivered to control the frequency of the VCO output.
The VCO is an oscillator which produces a periodic signal with the frequency that is
proportional to the dc voltage from the LF.
When the PLL is locked, the frequencies ωi and ωo are identical. The PD is linear
and the LF output voltage is proportional to the phase error. The linearized mathematical model of the PLL is shown in Fig. 6.2 if the phase difference θ θi− o is very small. As seen
in this figure, the PLL structure is in fact a feedback control mechanism. The phase transfer function ( )Γ s that relates the phase θi of the reference input to the phase θo of the VCO
transfer function of the LF.
6.3 Stability Boundary Analysis
Consider a gain-phase tester ke−jθ included in series with the original control system
as in Fig. 6.3, and its transfer function is given by
F s q is the transfer function of the LF with the perturbed parameter vector q .
1 2
[ , ,...., r]
m= m m m is a designed parameter vector of PLL in the m1−m2−....−mr coordinate
space under user-defined specifications.
The closed-loop characteristic polynomial is P s q m k( , , , , )θ and
6.3.1 Gain Boundary Curves
Let 0θ = Ο and q be a specific perturbed parameter. Equation (6.4) is rewritten into
and a region of the designed parameters in m−space is to be found out so that the whole
PLL system with the designed parameters chosen from the above determined region will
satisfy the GM condition. The GM of the control system at a point on one side of a specific
gain boundary curve is greater than that at a point on the boundary curve. But it is less at the
points on the other side.
6.3.2 Phase Boundary Curves
Given k=1 and a specific q , (6.4) is written into the form ( , , ) 0
( , , ) 0
U m
V m
ω θ ω θ
⎧ =
⎨ =
⎩ . (6.6) Phase boundary curves are developed under the PM specification in a similar way. They
are generated in m−space from the solutions m of (6.6) by varying ω for every θ .
Given a specific θ , a phase boundary curve will be generated and a region of the designed
parameters in m−space is to be found out so that the whole PLL system with the designed
parameters chosen from the above determined region will satisfy the PM condition. The PM
of the control system at a point on one side of a specific phase boundary curve is greater than
that at a point on the boundary curve. But it is less at a point on the other side.
6.3.3 PLL Robust Design
In physical systems, uncertainties usually exist in system parameters. The LF is usually
connected to a PLL IC externally and implemented by the designer under the specified
constraints. The system parameters are separated into the designed parameters, which are the
parameters of PD and VCO, and the perturbed parameters, which are the ones of the LF. The
designed parameters are the parameters the range of which is to be determined so that the
performance of the whole PLL system can meet the specified conditions under the perturbed
parameters varying in a region as long as the designed parameters are within the determined
range.
In this chapter, the LFs with different order are used as examples to demonstrate the
proposed design method of PLL robust design. The robust design is based on gain and phase
boundary curves with respect to k and θ of a gain-phase margin tester and they are drawn
in m−space from the locations of the roots of (6.5) and (6.6) with respect to different k
and θ , respectively. The range of the designed parameters is going to be found out in
space
m− under the constraints of specified GM and PM.
Based on the discussions mentioned above, the design algorithm is as the followings:
Step (1) Set up user-defined specifications on GM and PM.
Step (2) For every system parameter q at the vertices of the perturbed system parameter
region in q -plane, draw the gain boundary curves corresponding to the specified
GM in m-plane by solving (6.5).
Step (3) For every q at the vertices of the perturbed system parameter region in q -plane,
draw the phase boundary curves corresponding to the specified PM in m-plane by
solving (6.6).
Step (4) Determine a gain region in m−space with the help of the gain boundary curves as
in step (2)
so that the designed parameters with the coefficients in that region satisfy the
specified GM constraints.
Step (5) Determine a phase-region in m−space with the help of the phase boundary curves
as in step (3) so that the designed parameters with the coefficients in that region
satisfy the specified PM constraints.
Step (6) Find out the common region of the determined gain and phase ones as in steps (4)
and (5). The perturbed PLL system with the designed parameters in that region is the
desired one satisfying the specified GM and PM conditions.