Page 80 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL ANALYSIS Ronald E. Best 56
Equation (3.46) is converted into

(3.48)

This nonlinear differential equation for the deflection angle φ looks very much like the
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nonlinear differential equation of the PLL according to Eq. (3.44). A slight difference in the
second term exists, however. In the case of the PLL, the second term contains the factor cos
θ , whereas for the pendulum the coefficient of the second term is the constant 2ζ′ω ′.
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Strictly speaking, the pendulum of Fig. 3.7 would only be an accurate analogy of the PLL if
the friction varied with the cosine of the deflection angle. This would be true if the damping
factor ζ′ were not a constant but instead varied with cos φ . As a consequence, the
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momentary friction torque would be positive for

that is, when the position of the pendulum is in the lower half of a circle around the cylinder
shaft. On the other hand, the momentary friction torque would be negative for

that is, when the position of the pendulum is in the upper half of this circle. A negative friction
is hard to imagine, of course, but let us assume for the moment that is valid.
Imagine further that the weight G is large enough to make the pendulum tip over and continue
to rotate forever around its axis (provided the rope is long enough). Because of the
nonconstant torque generated by the mass M of the pendulum, this oscillation will be
nonharmonic. During the time when the pendulum swings through the lower half of the circle
(−π/2 < φ < π/2), its average angular velocity is greater than its velocity during the time when
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it swings through the upper half (π/2 < φ < 3π/2). The positive friction torque averaged over
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the lower semicircle is therefore greater in magnitude than the negative friction torque
averaged over the upper semicircle. This means that the friction torque averaged over one full
revolution stays positive; hence, it is acceptable to state that the coefficient of friction ζ′
varies with the cosine of φ . The mathematical pendulum is therefore a reasonable
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approximation for the PLL.
Comparing the PLL with this mathematical pendulum, we find the following analogies:
■ The phase error θ of the PLL corresponds to the angle of deflection φ of the pendulum.
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■ The natural frequency ω of the PLL corresponds to the natural (or resonant) frequency of
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the pendulum ω ′.
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