PLL Noise Model

 Now consider the PLL block schematic in figure 5.3.

 

Figure 5.3:   PLL Block schematic

Assume that purely flicker phase noise is generated at point A at the output of the phase detector and that only white phase noise is generated at point B at the VCO output:

where c and k are constants. Since it will be assumed that the noise is small compared to the useful signal, superposition may be used. Now the transfer functions between the points A and B and the output are

where K_d is the phase-detector gain factor measured in volts/radian, K₀ is the VCO gain factor measured in rad/sec-V and F(s) is the transfer function of the loop filter. Using the relationship (5.2) the Power Spectral Density of S_A(f) at the PLL output is

$$S_{A_{out}}=\frac{c}{f} \cdot \left\vert\frac{F(s)K_0}{s+K_0K_dF(s)}\right\vert^2$$

which can be simplified to  

$$S_{A_{out}}(f)=\frac{c}{f} \cdot \frac{\left(\frac{K_0}{2 \pi} F(s)\right)^2}{f^2 + \left(\frac{K_0K_d}{2 \pi} F(s)\right)^2}$$ (40)