Page 14 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL BUILDING BLOCKS Ronald E. Best 12
the simplest case: the phase θ (t) performs a step change at time t = 0; hence, it is given by
1

(2.3)

where u(t) is the unit step function. This case is an example for phase modulation.
Let us consider next an example of frequency modulation (Fig. 2.2b). Assume the angular
frequency of the reference signal is ω ′ for t < 0. At t = 0, the angular frequency is abruptly
0
changed by the increment Δω. For t ≥ 0, the reference signal is consequently given by

(2.4)

In this case, the phase θ (t) can be written as
1

(2.5)

Consequently, the phase θ (t) is a ramp signal.
1
As a last example, consider a reference signal whose angular frequency is ω ′ for t < 0
0
and increases linearly with time for t ≥ 0 (Fig. 2.2c). For t ≥ 0, its angular frequency is
therefore

(2.6)

where denotes the rate of change of angular frequency. Remember that the angular
frequency of a signal is defined as the first derivative of its phase with respect to time:

(2.7)

Hence the phase of a signal at time t is the integral of its angular frequency over the time
interval 0 < τ < t, where τ denotes elapsed time. The reference signal can be written as

(2.8)

Consequently, the corresponding phase signal θ (t) is given by
1

(2.9)

in other words, it is a quadratic function of time.