Hello all, I'm trying to better understand PLL systems. As far as modeling goes, I think I have and understand all the puzzle pieces (in terms of blocks and their associated transfer functions) except for the VCO block. When the VCO's output frequency is directly related to the input voltage (freq_out(t) = K*Vin(t)), For this case, I understand the math perfectly and I get phase_out(s)/Vin(s) = K/s. Great.
However, I have a hard time understanding the math if the VCO's output frequency is offset so that the VCO has the relationship freq_out(t) = a + K*Vin(t). With this equation, if you go through the math to get a relationship between the phase and Vin(t), you get that phase_out(s)=a/(s^2)+K*Vin(s)/s. In this case, it is impossible to write the transfer function because it is impossible to solve for phase_out(s)/Vin(s)... The inability for this relationship to have a transfer function is also obvious from the fact that the initial input voltage to output frequency is a nonlinear relationship from a systems point of view. Herein lies the problem.
Online, I have been able to find many references that claim that the transfer function for a VCO with a frequency offset as I have described it is still just K/s. (one example of such a document is [http://www.ti.com/lit/an/scha003b/scha003b.pdf page 33 dealing with Ko)
My question is this: What happened to the a/(s^2) term / what do I do with it???
My reference for figuring out how to do the math in general was this video:
The attached image called pg1 shows a sketch of the system as I understand it, as well as the math that I understand for the non offset VCO
The attached image called pg2 shows my attempt at the math for the frequency offset VCO. All the equations scattered in this post are better written on these two pages. dotted lines separate the three 'sections'
At the bottom of pg2, I drew my best idea of what a solution might be for modeling purposes, but I'm pretty sure it still wrong and it still has a question mark.
Any help is appreciated, thank you!