Hi,
I've been tring to get MathCAD to allow additional units to be defined, such as PUs (per Unit), and other dimensions.
They are very important for two reasons, both social engineering - everyone likes 'percent of rated' which is what PUs are, and they make the matrix maths much more stable as you can raise matrices to powers without the numbers exploding or vanishing too quickly.
They also work despite transformer galvanic isolation in the ESI (electrical supply industry) - one unit of power being fed down a transmission line at its rated voltage and current, no matter how many transformetr voltage level changes.
I'm reading this as a 50Hz transmission system, at 63.5kV, 131Amp : 8.32MW.
I'm quessing that the 11th power is simply an implied voltage limiter (it only works well if used in PUs - see rasing to powers above - it clips at unity) that is slightly beyond the 10th harmonic in a THD (Total Harmonic Distortion) calculation. It is possible to get mathcad to add a real limiter see some of the old work by Tom Gutma on the old collab.
I doubt you want any mathematicians discussion on what a 'spectrum' is in n-dimensional matrix spectral theorems (and I'm not the one who knows the answers either, I just know they are lurking out there). So I presume we are talking about normal time domain repetative vaveforms which are them converted to being frequencies, and looking at the amplitude spectrum.
I also presume that the initial transient isn't what you want to analyse - that is because the spectrum will vary depending on how much of the transient you include. That means you should reset the intial u(0) to be approx the peak voltage, so that it starts stable (i.e find the sample point near the end where the voltage reate u'=0, so you can match the t=0 case)
Having done / understood all that, you can now do the PSD (Power Spectral Density) of your wave form. You should have no initial transient, and you should be able to see nice repetition of the waveform. If you get it right, you should be able to change the settings so that you have much smaller steps, but only capture one or two cycles (note the transient decay may mean you want a few more cycles so you can check the initial conditions match the end conditions).
You now simply 'copy' the one cycle of voltage samples to the FFT/fft routine.
One of them produces the voltage amplitude at each spot frequency. You can tell by doubling the number of samples (per cycle), and getting the same result.
The other produces root power 'density', that is, it has fancy normalisation factors [most folk can't see the problem] that are based on the number of samples, rather than the frequency range.
The PSD will (essentially) be in Volts per root Hz, that is, if V^2/R is calculated you get Power/unit bandwidth, or power density. Obviously you need to convert to your choice of PU bandwidth (is it in PU (Watts|Volts) per [Hz|PU BW] ?).
Sccond aside : the normalilsation argument partly depends on how wide band noise is defined/simulated/aliased, and assumptions about band limited signals.
