Page 247 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL APPLICATIONS PART 1: INTEGER-N FREQUENCY
SYNTHESIZERS Ronald E. Best 147
Let’s assume for the moment that u is a sine wave with radian frequency ω m and
n
amplitude

(6.21)

Inserting Eq. (6.21) into Eq. (6.20) yields

(6.22)

with ω = 2π f and ω = 2π f . ω is the radian frequency of a modulated carrier with carrier
0 0

radian frequency ω . The peak frequency deviation is , so we can state that the signal u (t)
0 n
modulates the frequency of the carrier. Knowing the radian frequency of the frequency
modulated signal, we can compute its phase φ(t) by integrating ω over time t. This leads to

(6.23)

Note that φ(t) is the phase of a frequency modulated triangular wave. Because the
harmonics of the triangular signal are of no importance since they are suppressed almost
entirely by the loop filter, it is sufficient to account for the fundamental only. The fundamental
of the capacitor voltage (denoted U , ) is a sine wave with amplitude U whose phase is
cap 1 0
given by φ(t), thus

(6.24)

When computing the Fourier series of a symmetrical triangular signal, the fundamental U 0

can be shown to be .

Applying the addition theorem of trigonometric functions to Eq. (6.24) gives

(6.25)

Now we observe that the argument of the cosine function in square brackets is much less