A lot of the conclusions that we might draw depend heavily upon the nature of the pd signal coming from the motor. There are two items discussed below depending on whether we focus on faster or slower rise-time pulses than were assumed in the simulation.
Item 1 - If rise time is dramatically faster (shorter) than I assumed, such that it's shorter than the round trip travel time from pd coupling cap tap to surge cap, then the sensed pulse has an opportunity to reach it's full peak before being affected by the reflection from the surge cap. Thus moving the pd sensing tap closer to the motor may help more than suggested by above analysis if the pulse rise times are shorter.
Item 2 - If rise time is dramatically slower (longer) than I assumed, then the signal may not be affected as much by the coupling cap which acts as a low-pass filter. We can see this because the response to a step hitting the surge cap is 2*(1-exp(-t/tau)) where tau is Z0 * C. That format of the step response can be guessed from the simulations, and can also be proved mathematically. The impulse response would be the derivative of that: 2/tau * exp(-t/tau). The laplace transform of that impulse respose would be 2/(1 + s*tau). Looking in the s-plane, this has a single pole at s = -1/tau. As we vary sinusoidal frequency along the vertical axis, the highest resopnse is at w=0 (low-pass filter) with decreasing response for higher frequencies. Half power bandwidth would occur at w = -1/tau since 45-45-90 triangle tells us the distance from this point to the pole is sqrt(2) higher than at zero frequency and the response is therefore sqrt(2) lower. To convert to hz instead of frequency we divide by 2*pi. So the half-power bandwidth of the low-pass filter associated with our surge capacitor and our associated cabling (as modeled) is 1/(2*pi*tau) = 1/(2*pi*Z0*C) = 1/ (2*pi*9 usec) = 17,700 hz. So for pulses with frequency content on this order or lower, the surge cap may not have dramatic effect (of course the low-pass filter model applies only after the time period required for the round trip time between pd coupling cap and surge cap, but that time would be negligible for the pulses with content below 17,700 hz).
At this point, it seems appropriate to revist the bandwidth of the hi-pass filtering effect associated with the measurement R/C circuit. Input voltage is applied to series R and C and output voltage is measured accrss the resistor. So we have:
Vout/Vin = R/(R+1/(sC)) = s*C*R / (s*C*R+1)
(where these are measurement circuit R and C... R=150K, C=80pF)..
This RC filter is a little more complicated filter than the last one because it includes not only a pole on the negative real axis (at s=-1/(RC) similar to the last pole) but also a zero at the origin. I am not sure how to get a closed form solution of the half-power bandwidth, but numerically I solved it for these specific values using Maple:
> Hfilter:=R/(R+1/(s*C));
> Hfilter1:=subs(s=I*w,Hfilter);
> Hfilter1:=subs([s=I*w,R=150E3, C=80E-12],Hfilter1);
> Hfiltermag:=evalc(abs(Hfilter1));
> whalf:=solve(Hfiltermag=0.707,w);
> Fhalf:=evalf(whalf[2]/(2*Pi));
This results in Fhalf = 13,300 hz.
At first glance (*), it looks like there is not a lot of room between the 13,300hz measurement system hi-pass filter cutoff and the 17,700 surge cap low-pass filter effect.... and therefore it doesn't seem like the surge cap will allow many pulses to be seen (other than those fast ones that reveal themselves before the surge cap low-pass filter kicks in as discussed in item 1). * I do realize that I need to be careful in characterizing waveforms and filters by a single number and will give that some more thought... but interested if anyone else sees a flaw in this logic.
I am left with a lot of questions mostly centering around what kind of pulses and waveforms do we expect to measure. I will certainly contact the PD equipment OEM to discuss this further and gather their insights.
Meanwhile... any other thoughts on the problem or the analysis?
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(2B)+(2B)' ?
