Page 86 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL ANALYSIS Ronald E. Best 60
A third way can be employed to cause a PLL to unlock from an initially stable operating
point. If the frequency of the reference signal is increased linearly with time, we have

where is the rate of change of the angular frequency. In the mechanical analogy, this
corresponds to a weight G that also builds up linearly with time. This can be realized by
feeding a material onto the platform at an appropriate feed rate. It becomes evident that too
rapid a feed rate acts on the pendulum like the impulse generated in the last example by
dropping a weight onto the platform. As has been shown in Sec. 3.4, the rate of change of the
reference frequency must always be smaller than to keep the system locked:

These examples have demonstrated that three conditions are necessary if a PLL system is to
maintain phase tracking:

■ The angular frequency of the reference signal must be within the hold range.
■ The maximum frequency step applied to the reference input of a PLL must be smaller than
the pull-out range.
■ The rate of change of the reference frequency must be lower than .

Whenever a PLL has lost tracking because one of these conditions has not been fulfilled, the
question arises as to whether it will return to stable operation when all the necessary
conditions are met again. The answer is clearly no. When the reference frequency exceeded
the hold range, the pendulum in our analogy tipped over and started rotating around its axis. If
the weight on the platform were reduced to a value slightly less than the critical limit that
caused instability, the pendulum would nevertheless continue to rotate because there is enough
kinetic energy stored in the mass to maintain the oscillation. If there were no friction at all, the
pendulum would continue to rotate even if the weight G were reduced to zero. Fortunately
friction exists, so the pendulum will decelerate if the weight is decreased to an appropriate
level. For a PLL, this means that build-up of the phase error will decelerate if the reference-
frequency offset is decreased below another critical value, the pull-in frequency. If the slope of
the average phase error becomes smaller, the (down-scaled) frequency ω ′ of the VCO more
2
and more approaches the frequency of the reference signal, and the system will finally lock.
The pull-in frequency Δω is markedly smaller than the hold range, as can be expected from
P
the mechanical analogy. The pull-in process is a relatively slow one, as will be demonstrated
in Sec. 3.9.3.
In most practical applications, it is desired that the locked state be obtained within a short
time period. Suppose again that the weight put on the platform of the mechanical model is
large enough to cause sustained rotation of the pendulum. It is easily shown that the pendulum
can be brought to rest within one single revolution if the weight is suddenly decreased below
another critical

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