Page 4 - Phase-Locked Loops Design, Simulation, and Applications

INTRODUCTION TO PLLS Ronald E. Best 3
Let us now look at the operation of the three functional blocks in Fig. 1.1a. The VCO
oscillates at an angular frequency ω , which is determined by the output signal u of the loop
f
2
filter. The angular frequency ω is given by
2

(1.1)

−1
−1
where ω is the center (angular) frequency of the VCO and K is the VCO gain in rad s V .
0 0
Equation (1.1) is plotted graphically in Fig. 1.1b. Because rad (radian) is a dimensionless
quantity, we will drop it mostly in this text. (Note, however, that any phase variables used in
this book will have to be measured in radians and not in degrees!) Therefore, in the equations
a phase shift of 180° must always be specified as a value of π.
The PD (also referred to as a phase comparator) compares the phase of the output signal
with the phase of the reference signal and develops an output signal u (t), which is
d
approximately proportional to the phase error θ , at least within a limited range of the latter
e

(1.2)

Here, K represents the gain of the PD. The physical unit of K is V/rad. Figure 1.1c is a
d d
graphical representation of Eq. (1.2).
The output signal u (t) of the PD consists of a DC component and a superimposed AC
d
component. The latter is undesired; hence, it is canceled by the loop filter. In most cases, a
first-order low-pass filter is used. Let us now see how the three building blocks work together.
First, we assume the angular frequency of the input signal u (t) is equal to the center
1
frequency ω . The VCO then operates at its center frequency ω . As we see, the phase error θ e
0
0
is zero. If θ is zero, the output signal u of the PD must also be zero. Consequently, the output
e d
signal of the loop filter u will also be zero. This is the condition that permits the VCO to
f
operate at its center frequency.
If the phase error θ were not zero initially, the PD would develop a nonzero output signal
e
u . After some delay, the loop filter would also produce a finite signal u . This would cause the
d
f
VCO to change its operating frequency in such a way that the phase error finally vanishes.
Assume now that the frequency of the input signal is changed suddenly at time t by the
0
amount Δω. As shown in Fig. 1.2, the phase of the input signal then starts leading the phase of
the output signal. A phase error is built up and increases with time. The PD develops a signal
u (t), which also increases with time. With a delay given by the loop filter, u (t) will also rise.
f
d
This causes the VCO to increase its frequency. The phase error becomes smaller now, and
after some settling time the VCO will oscillate at a frequency that is exactly the frequency of
the input signal. Depending on the type of loop filter used, the final phase error will have been
reduced to zero or to a finite value.
The VCO now operates at a frequency which is greater than its center frequency ω by an
0
amount Δω. This will force the signal u (t) to settle at a final
f