Page 74 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL ANALYSIS Ronald E. Best 52

where Δω is the size of the (angular) frequency step. The steady-state error is given by

This only becomes 0 if M is greater or equal to 1—in other words, if the loop filter has at
least one pole at s = 0. Thus, the open-loop transfer function of the PLL must have at least 2
poles at s = 0 if the steady-state error caused by a frequency step must become zero.

Case study 3. Frequency ramp applied to the reference input. As shown in Sec. 3.4, for a
frequency ramp the Laplace transform of the phase signal θ (t) becomes
1

where is the rate of change of angular frequency. For the steady-state error, we get

Here the steady-state error only approaches zero if the loop filter has at least two poles at s =
0. For M = 2 and Q(s) = 1, the order of the PLL becomes 3. Because each pole of the transfer
function causes a phase shift of nearly 90° at higher frequencies, the overall phase shift can
approach 270°, which means the system can become unstable. To achieve a stable loop, the
poles must be compensated for by zeros—that is, the loop filter must have at least one zero,
3
which must be correctly placed, of course. (Appropriate placement of poles and zeros will be
discussed in Chap.9.)

The Order of the PLL System

The number of poles
Most PLLs considered hitherto were second-order PLLs. The loop filter had one pole, and the
VCO had a pole at s = 0, so the whole system had two poles. Generally, the order of a PLL is
always higher by 1 than the order of the loop filter. If the loop filter is omitted—in other
words, if the output of the phase detector directly controls the VCO—we obtain a first-order
PLL. (We will discuss the first-order PLL in Sec. 3.6.2.) In PLL applications, first-order
systems are rarely used, because they offer little noise suppression (more about noise in Chap.
4).
Because higher-order loop filters offer better noise cancellation, loop filters of order 2 and
higher are used in critical applications. As mentioned, it is much

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