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I am trying to model in LTspice a matrix for symmetrical components. Keeping to the theory, the 120° delays were built with tlines, which means that whatever signal there is at the input, after the delay it becomes x(t-2*π/3), meaning multiplying with a^{2}(=e^{-j*2*π/3}). (to avoid such a long delay for the 240° one, the signal is reversed and then delayed by -60°)

But, after the inverse transformation, the original signals are not restored unless they are sinusoidal (offphase, offset, it doesn't matter). As soon as harmonics come out, the results are off. After further testing, it seems that the 3rd, 9th, 15th, etc harmonics don't seem to be "processed", they show up (if they are present at the input) at the output.

Example:

v_{a}(t)=sin(ωt)+sin(3ωt)/3+sin(5ωt)/5+sin(7ωt)/7+sin(9ωt)/9

after the inverse transform:

v_{+}(t)=sin(ωt)+sin(3ωt)/3+sin(9ωt)/9

I tried a different approach, with all-pass phase shift, but that is only applicable to sine waves (e.g. square wave looks like a too large time-constant differentiation/integration for a ±π/2 delay). Similarly, I tried delay by sin/cos multiplication, so that the 120° phase-shift would be done like this:

v_{a}^{'}(t)=-1/2*v_{a}(t)-√3/2*v_{a}(t-π/2)

where -π/2 comes after a tline delay, but then the square wave from the same example would look like a distorted staircase (true, the π/2 delay applies only to the quadrature fundamental, there would be needed many blocks like these for all the harmonics). However, and this is what puzzles me most: after the inversion with this method, the input signals are back to normal(!), no matter how distorted/unbalanced they are.

So here are my two (four) questions:

- Why doesn't the "classic" way of calculating the symmetrical components completely restore the input signals? Implicit, are the positive/negative/zero components calculated like this good or bad?

- Why does the last method work and, implicit, does that mean the pos/neg/zero components calculated like this are good (or bad)?

Thank you in advance,

Vlad

[edit]This isn't homework/coursework/etc, it's in the right place, I am trying to build a custom block in LTspice[/edit]

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# Three-phase symmetrical components SPICE attempt

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