Page 66 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL ANALYSIS Ronald E. Best 47

we get the normalized form

(3.24)

This is identical to the phase-transfer function we obtained for the PLL using a phase detector
with voltage output [cf. Eq. (3.18)].

Transient Response of the PLL in the Locked State

Knowing the phase-transfer function H(s) and the error-transfer function H (s) of the PLL, we
e
can calculate its response on the most important excitation signals. We therefore analyze the
PLLs answer to

■ A phase step
■ A frequency step
■ A frequency ramp

applied to its reference input.

Phase step applied to the reference input
A reference signal performing a phase step at time t = 0 has been shown in Fig. 2.2a. In this
case, the phase signal θ (t) is a step function,
1

(3.25)

where u(t) is the unit step function and ΔΦ is the size of the phase step. For the Laplace
transform Θ (s), we therefore get
1

(3.26)

The phase error θ is obtained from
e

(3.27)

Inserting Eq. (3.20) into Eq. (3.27) yields

(3.28)

Applying the inverse Laplace transform to Eq. (3.28), we get the phase error functions θ (t)
e
shown in Fig. 3.5. The phase error has been normalized to the phase step ΔΦ.