Question 19.19: A VCO with a phase noise given by SΦ, vco(f) = h2/f² is plac......
A VCO with a phase noise given by S_{\phi ,~ vco}(f) = h_{2}/f^{2} is placed within an ideal second-order PLL. What is the resulting phase noise at the output?
For a second-order PLL, the open-loop gain L(s) is specified in (19.33).
L(s) = \frac{\omega _{pll}^{2}}{s^{2}} . \left\lgroup1 + \frac{s}{\omega _{z}} \right\rgroup (19.33)
Substitution into (19.93)
H_{osc}(s) = \frac{1}{1 + L(s)} (19.93)
yields the transfer function from VCO phase noise to the PLL output.
H_{osc}(s) = \frac{s^{2}}{s^{2}+ \frac{\omega _{pll}}{Q}s + \omega _{pll}^{2} } (19.94)
The output phase noise is given by,
S_{\phi }(f) = S_{\phi ,~osc}(f)\left|H_{osc}(f)\right|^{2}
= \frac{h_{2}}{f^{2}} . \frac{f^{4}}{\left\lgroup f^{2} + \frac{\omega _{pll}}{2\pi Q}f + \frac{\omega _{pll}^{2}}{4\pi ^{2}} \right\rgroup ^{2} } (19.95)
= \frac{h_{2}f^{2}}{\left\lgroup f^{2} + \frac{\omega _{pll}}{2\pi Q}f + \frac{\omega _{pll}^{2}}{4\pi ^{2}} \right\rgroup ^{2}}
This is a bandpass spectrum with a peak around \omega _{pll} . If (19.95) is substituted into (19.60)
\sigma _{\tau }^{2} = \left\lgroup\frac{T_{0}}{2\pi } \right\rgroup ^{2} \int\limits_{0}^{1 / 2T_{0}}{S_{\phi }(f)df} (19.60)
with the upper limit of integration approximated as infinite (1 / 2T_{0} \approx ∞) a very simple expression is obtained for absolute jitter.
\sigma _{\tau }^{2} = \frac{h_{2}QT_{0}^{2}}{2\omega _{pll}} (19.96)
This expression reveals that the PLL loop bandwidth (approximately equal to \omega _{pll}/Q ) should be maximized in order to minimize the contribution of VCO phase noise to a PLL’s output jitter.
