Symmetrical components

Question

I am trying to build an abc to positive/negative/zero components transformation block (in LTspice), I did it, but sometimes I get erroneous behaviour: DC levels seem to be "omitted" at the output for an abc > 120 > abc chain.

The schematic I'm using is this:

and the waveforms are these:

You can see that all waveforms are identical, but b/b* and c/c* have a DC difference between them. Here are the differences:

(in1, in2 and in3 are the a, b and c in the previous screenshots)

The signals are these:

• all three phases have unity fundamental + 3rd/3, 5th/5 and 7th/7 all positive harmonics;
• phase a has twice the amplitude;
• phase b has 0.1Vdc;
• phase c has 0.3rad phase lead.

Is this normal? Are DC bias levels "left behind"?

In order to build the schematic for a practical use, I had to maneuver a bit through the equations to avoid the \$\pm \frac{2 \pi}{3} \$ delay, and so I transformed \$e^{\pm j \cdot \frac {2 \pi}{3}} = -\frac{1}{2} \pm j \frac{\sqrt{3}}{2}\$, which left only \$j\$ to take care of, a minor leap for SPICE-kind.

Answer 1

In the meantime, the solution was found: the culprits are the delay lines.

The approach I tried was to have instantaneous values rather than magnitude/angle. For this, I tried to mathematically derive the circuit in order to use only a \$-90^\circ \$ delay instead of the \$\pm 120^\circ \$ ("a"), but here is the catch: symmetrical components are meant to be a phasor analysis, and using the delay line as \$-j\$ , for a \$0.1_{DC}+1\angle 0 \$ input (DC separate for better view) I would get \$-j0.9\$ , which explains the erratic behaviour for different DC unbalance levels.

Lesson learned, hopefully remembered when needed. :)