# How are Loop Filters derived?

It is understood that loop filters are an essential component in PLLs, however, it is not clear to me how they arrived at the required Loop filter transfer function.

I'm not sure if the application affects the design of loop filters, but my application is generating chirps for FMCW radar.

• That's mostly classical control loop theory! Your filter fulfills a specific role – keeping the control loop stable, and suppressing system noise – while still trying to achieve minimum delay. Commented Apr 21, 2022 at 21:34

This is a very broad question where a book would be required to give a full answer. Let me try to give a very high level introduction into what a "Loop Filter" is and how it works using an integer-N PLL as an example:

An integer N PLL can tune a voltage controlled oscillator (VCO) to frequencies that would be $$N$$ times a reference frequency, where $$N$$ is an integer, assuming the VCO itself can produce all those frequencies.

Following the block diagram, the output of the VCO is divided with a frequency divider and this lower frequency out of the divider is compared to the reference oscillator with a "Phase Detector". The phase detector will put out an error voltage that is proportional to the phase error between the two inputs. Often a "Phase/Frequency Detector" is used here, which is also easier to explain. The Phase/Frequency Detector" has an additional feature to put out a maximum positive or negative value if the frequency fed back is higher or lower than the target frequency of the reference. The output of this detector is called the "Error Signal".

This is where the Loop Filter comes in. The Loop Filter typically includes one or more integrators, and for stability reasons I won't be able to get fully into, it is even more complicated than that (including such details that the VCO itself in a PLL is a phase integrator). One observation with having an integrator is that the output of the Loop Filter can be independent of the input. Notably the output can float to any level and stay there while the input (the Error Signal) is 0! If the Error Signal for the Phase Detector (or Phase/Frequency Detector) is 0, then the integrator component in the Loop Filter will do nothing further (integrating 0 is 0), which is what we want as the fact that we have a 0 Error Signal means the loop is locked; the VCO control voltage has set the VCO to exactly the frequency we want it to be.

Consider other cases if the VCO was slightly higher in frequency (and we use a Phase/Frequency Detector) the Error Signal will go negative, the Loop Filter will integrate the negative error and grow in the negative direction, pushing the tuning voltage lower and (assuming a positive frequency vs control voltage slope) pushing the output frequency lower. Once the two frequencies (reference and divided output) match, the Phase/Frequency detector will act as a Phase Detector and produce errors appropriately to adjust the VCO until the phase matches that of the reference: Phase Locked.

The loop filter provides this functionality of control, but the complexity and deeper consideration is in setting it's bandwidth to optimize things like how fast it can track a changing input, how the phase noise of the output is balanced with the phase noise of the reference (often the PLL is used to improve the phase noise of the VCO by using a reference that has better spectral purity over certain ranges of the desired phase noise), and doing this all while maintaining stability over all operating conditions.

As for tuning control for a wideband FMCW application, there are two opportunities to tune the output frequency: summing a tuning control with the output of the Loop Filter or tuning the frequency of the reference. The main point is the control path from VCO input to the output (the first case) has a high-pass filter transfer function: changes slower than the loop bandwidth will be corrected and cancelled out by the loop. In contrast, the control path from the reference oscillator to the output has a low-pass filter transfer function so changes faster than the loop BW will be attenuated at the output.

• Dan, I don't normally modify your posts... so: //The Loop Filter typically includes one or more integrators// --- here you should point out that if the Phase discriminator is as titled and the output error signal is proportional (at least in small deviations) to phase, then the VCO inherently has an integrator in the loop. If it's a Frequency discriminator or a combination Frequency/Phase discriminator, that's gonna be a little more complicated. Commented Apr 22, 2022 at 21:05
• @robertbristow-johnson understood but I didn't want /need to get into that. Even with the VCO as an integrator, the Loop filter still can have one or more integrators. We can do 3rd order loops Type III loops for example to track ramping frequency etc... etc... I didn't think that was important to the answer as we need to get through a few pages first of other details to get to the points about that :). Similarly I didn't get into the detail of needing a zero for stability when we do introduce one integrator-- - Commented Apr 22, 2022 at 21:40
• It's such an open question and I didn't think I could provide a useful answer concisely as this--- perhaps I didn't! Commented Apr 22, 2022 at 21:42
• it is an open-ended question. it's just that this inherent integrator with the VCO is something that i thought was pretty much a PLL 101 sorta thing. a PLL is sorta a simple standard servo mechanism, but with some sorta nasty twists to it that makes it a little hard to model as a servo. Commented Apr 23, 2022 at 5:34