# Re: [time-nuts] How to properly simulate PLLs?

```Hi,

On 03/07/2018 05:42 PM, Attila Kinali wrote:
> Hi,
>
> I have a small side task, where I need to design a PLL system
> As it is a bit non-conventional, I am not confident that my
> pen and paper analysis is correct and the usuall tool I use
> (Analog's ADPLLsim) doesn't cover it. So my first thought
> was to use spice to simulate the loop. But I am not sure
> how the non-linear effects of the PLL, the divider chains etc
> affect the whole system and whether a spice simulation (which
> would use a linear approximation of a few components) would
> model the system faithfully. Not to mention that this would
> be only valid simulation of the locked state and anything
> that involves the PLL being unlocked (initial lock in process,
> large phase and frequency jumps that cause unlocks) cannot
> be handled at all. Neither would it give me a proper estimate
> of the noise propagation through the system.
>
> So, is there any canonical way how to simulate PLLs?
> If yes, what should I read? (My google foo didn't return
>
```
Now, I called Attila to ask what he was trying to do and crash coarse
him into some stuff. The discussion drifted from there.

I end up writing simulators in C, over and over again, dedicated to each
design. When dealing with phase accumulators, I use a rather simple
acceleration trick which I thought I would share. Rather than simulating
each of the cycles of the phase accumulator, I can fast forward the time
by estimate how many cycles it takes for it to wrap, so if I have the
current phase phi, and the phase accumulator wrap-point phi_w, then the
remaining phase for the cycle is phi_w - phi. Now, the steering word for
the phase accumulator phi_s is what is accumulated each cycle, so we can
figure out how many n cycles it take simply by

n = (phi_w - phi) / phi_s

As this division is assumed to be rounded down, it is actually not
sufficient to wrap the phase around, it would only give the phase just
before wrapping, so it would need one more cycle

n = (phi_w - phi) / phi_s + 1

OK, with this we can update the phase by

phi = phi + n * phi_s

By simulate as if the phase accumulator just wrapped, this method
fast-forwards the simulation and allows to simulate each update of the
PLL. You typically also update the time of the system using

T = T + n * T_s

where T_0 is the period of the phase accumulators clock, to T_0 = 1/fs
somewhere in the initiation code.

Now, typically phase and time needs to be unwrapped over a few variables
not to run into overflow issues that mess with the numerical issues of
simulation, but that is relatively trivial code.

Phase comparison typically is

V_d = phi_in - phi

The PI-loop is trivial

V_i = V_i + I * V_d
V_f = V_i + P * V_d

You want an offset frequency typically

phi_s = V_f + V_f0

That's the basics for building a PI loop simulation.

phi will be the phase-state and V_i the frequency state of the loop. The
phase-state is however best viewed as V_d, the detected phase difference.

Enjoy.

Cheers,
Magnus
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