Purpose

The purpose of the loop filter is to take the error current from the charge pump, i_err, and convert it to a voltage that the VCO can use.  The loop filter also integrates the i_err signal, providing a voltage signal as shown below.
 
 

Design

After some preliminary calculations, it was decided that a bandwidth of around 5 kHz and a Q value of 0.707 would be needed for the loop filter to track the error current.  Since on chip resistors and capacitors tend to use up considerable chip space, the goal was to keep them as small as possible.  Calculations for the loop filter are shown below.
 

                            B = 5 kHz
                            Q = 0.707
                            K_V= VCO sensitivity = (2MHz/volt)*(2p)
                            K_F = (i_err / 2p) = (8mA / 2p)
                                   N = (average division from divide by N counter) = 126
                                   w₀= (2p * B) = 31,416 radians/second
 
 

                                w₀² = (1/N)K_VK_F(1/C)                  =>             C = ((1/N)K_VK_F) / (w₀² )

                                (w₀/Q) = (1/N)K_VK_FR                   =>             R = (w₀N) / (QK_VK_F)
 
 

                          C = ((1/126)(2x10⁶)(2p)(8x10^-6 / 2p)) / (31,416)² = 1.28x10^-10=128pF

                    R = ((31,416)(126)) / ((0.707)(2x10⁶)(2p)(8x10^-6 / 2p)) = 349,930 = 350 kW
 
 

Problems

The capacitor and resistor values found in the calculations above ended up being very large.  Even worse, the I*R product ends up being 350kW*8mA= 2.8V, which is too high for the current sources to supply.  Calculations were performed with many different values for error current, B, and Q, but the results were not any better.  Clearly, further study needs to be done to try to get the R, C, and I*R values down to more reasonable levels.