Page 162 - Phase-Locked Loops Design, Simulation, and Applications

PLL PERFORMANCE IN THE PRESENCE OF NOISE Ronald E. Best 100
In Sec. 3.4, we showed that the transient response of the PLL is best at ζ = 0.7. Because the
function B (ζ) is fairly flat in the neighborhood of ζ = 0.5, the choice of ζ = 0.7 does not
L
noticeably worsen the noise performance. For ζ = 0.7, B is 0.53 ω instead of the minimum
n
L
value 0.5 ω . For this reason, ζ = 0.7 is chosen for most applications.
n
For the output phase jitter , we can now write

(4.11)

where Φ is already known from Eq. (4.4). Combining Eqs. (4.1), (4.3), and (4.10), we obtain

(4.12)

We saw in Eq. (4.3) that the phase jitter at the input of the PLL is inversely proportional to
the (SNR) . By analogy, we can also define a signal-to-noise ratio at the output, which will be
i
denoted by (SNR) (SNR of the loop). We define this analogy in Eq. (4.13):
L

(4.13)

Comparing Eqs. (4.12) and (4.13), we get

(4.14)

Equation (4.14) says that the PLL improves the SNR of the input signal by a factor of
B /2B . The narrower the noise bandwidth B of the PLL, the greater the improvement.
i
L
L
All this sounds very theoretical. Let us therefore look at some numerical data. In radio and
television, the SNR is used to specify the quality of information transmission. For a stereo
receiver, a minimum SNR of 20 dB is considered a fair design goal. The same holds true for
PLLs. Practical experiments performed with second-order PLLs have demonstrated some very
useful results. 1

1. For (SNR) = 1 (0 dB), a lock-in process will not occur because the output phase noise
L
is excessive.
2. At (SNR) = 2 (3 dB), lock-in is eventually possible.
L
3. For (SNR) = 4 (6 dB), stable operation is generally possible.
L
In quantitative terms according to Eq. (4.13), for (SNR) = 4 the output phase noise
L
becomes