Message ID: 2465
Entry time: Thu Oct 24 11:42:06 2019
In reply to: 2462
Reply to this: 2485

Author:

Ian MacMillan

Type:

Notes

Category:

FSS

Subject:

FSS Plant Model v2

With discussions with Anchal and some reading

It seems impossible to create a casual circuit with a zero phase shift. (See this for more)

If we have a circuit with an impulse response h(t) and transfer function H(f)=F[h(t)] where H(-f)=H*(f). For the filter to cause no phase shift then ∠H(f)=0 for a complex exponential input for all f. It is also impossible to have a constant phase shift unless that phase shift is zero.

Therefore "filter does not change the phase at all, then H(f) is a real-valued function, and because of the conjugacy constraint, it is also an even function of f. But then its Fourier transform h(t) is an even function of time, and thus the filter cannot be causal (except in trivial cases): if its impulse response is nonzero for any particular t>0, then it is also nonzero for −t (where −t<0)"

Because this can't be done casually, it should be done using a Field Programmable Gate Array. Unfortunately, I don't think we have access to one. I am reading up on the Moku FIR Filter builder to find out if we can program it to do what we want.

Quote:

I have designed a passive circuit that seems to match the ideal transfer functions in shape. Scaling should just be a game of playing with the values of the resistors and capacitors. The phase still seems to be an issue. There is an unwanted phase shift from 0 -> -90.

The next step is trying to finalize the values for the resistors and caps. Possibly model in zero if I have time. Then build and test. Also fix the phase.

Quote:

I have updated the plant model to contain the cavity pole also. Cavity pole is a pair of positive and negative real poles, so it is hard (or maybe impossible) to imitate it exactly with an electronic circuit. Or maybe, my analysis is wrong.

Nevertheless, I have for now made this circuit which has a second-order pole, so it correctly matches the magnitude of the model transfer function up to 1 MHz for both PZT and EOM paths. Note that the elliptical filter is not included in this as we can connect the circuit to Test port 1 which injects just before the filter in LIGO-D0901894. Also, for the gains in EOM path, I had to add some factors to make it the same as the model transfer function. All components are calculated for E12 series resistors and capacitors.

Attached is a pdf of the notebook which contains all the mathematics in latex and a zip file with all files to recreate and further work on this. Ian can use these as support to learn zero further.