Question 19.20: Consider the charge-pump PLL designed in Example 19.8 and co......

The contribution of the reference input to the PLL output phase noise is given by

S_{\phi ,~ in}(f) = \left\lgroup 8 · 10^{-16}+ \frac{7 · 10^{-11}}{f^{2}} \right\rgroup rad^{2}/Hz

filtered by \left|H(f)\right|^{2} . If C_{2} is neglected, H(s) is given by (19.35) with the values of \omega _{pll} and \omega _{z'}

H(s)=\frac{\phi (s)}{\phi _{in}(s) } =\frac{N(1+s/\omega _{z} )}{1+\frac{s}{\omega _{z} } +\frac{s^{2} }{\omega ^{2}_{pll} } }                       (19.35)

being those chosen in Example 19.8: 2π · 800 KHz and 2π · 400 KHz respectively.

The loop filter resistor R results in thermal noise at the control voltage node, V_{cntl} . Its noise spectral density may be determined by a noise circuit analysis of Fig. 19.35 resulting in

V_{n}^{2}(f) = 4kTR \left\lgroup\frac{C_{eq}}{C_{2}} \right\rgroup ^{2} \left\lgroup\frac{1}{1+ f^{2}R^{2}C_{eq}^{2}} \right\rgroup                  (19.98)

Where C_{eq} = C_{1}C_{2}/(C_{1}+C_{2}) . Maintaining the 2nd-order model, this noise is then bandpass filtered by

H_{n}(s) = \frac{K_{osc}s}{s^{2}+ \frac{\omega _{pll}}{Q}s + w_{pll}^{2} }                  (19.99)

Where Q = 0.5 from Example 19.8.

Finally, from Example 19.18 the VCO phase noise modeled by (19.91)

S_{\phi ,~ osc}(f) = \frac{1350~ rad^{2} · Hz}{f^{2}} + \frac{2.7 · 10^{9}~ rad^{2}· Hz^{2}}{f^{3}}               (19.91)

is filtered by \left|H_{osc}(f)\right|^{2} with H_{osc} as in (19.94).

H_{osc}(s) = \frac{s^{2}}{s^{2}+ \frac{\omega _{pll}}{Q}s + \omega _{pll}^{2} }                      (19.94)

The total output phase noise is the superposition of all contributors.

S_{\phi }(f) = S_{\phi ,~in}(f)\left|H(f)\right|^{2}+ V_{n}^{2}(f)\left|H_{n}(f)\right|^{2} + S_{\phi ,~ osc}(f)\left|H_{osc}(f)\right|^{2}                    (19.100)

A plot of all three phase noise contributors and their sum is shown in Fig. 19.36. Note that at low frequencies, the total output phase noise is equal to the input reference phase noise because the other noise contributors are filtered out by the PLL. Similarly, at high frequencies only the VCO phase noise contributes. In this example, thermal noise in the loop filter plays an insignificant role in the total output phase nois —a typical result. The jitter of the PLL output will be dominated by the high-frequency VCO noise contributions which span a much wider range of frequencies than the input-dominated portion of the phase noise spectrum and, hence, contribute much more to the integral in (19.60).

\sigma ^{2}_{\tau } = \left\lgroup\frac{T_{0}}{2\pi } \right\rgroup ^{2} \int\limits_{0}^{1/2T_{0}}{s_{\phi }(f)df}        (19.60)