fjkaller - Yes you are right that the general pattern flips if I change the phase sequence, as expected.
By physical reasoning, I agree with your comments about effect of stranding. I think it has a pretty dramatic effect on the internal current distribution of large conductors (if these conductors were stranded, you wouldn't see the large current density difference within a single conductor that shows in my powerpoint above), but it has a much smaller effect on sharing among different conductors. So the sharing among conductors predicted using solid conductors is probably not too far away from what we'd see for stranded.
I don't have an easy method for modelling twisted stranding unless I use the approach outlined in "Electromagnetic Modeling By Finite Element Methods" by Bastos and Sadowski. Specifically section 6.4 of that book is "The Skew Effect in Electrical Machines Using 2D Simulation". It provides a means to combine multiple 2-d simulation to create a 3-d simulation of geometry where the position of the conductors gradually changes as we move axially. In the context of the book it is used for modelling skewed squirrel cage rotor bars but I am pretty sure it could be applied to skewed (twisted) stranded wire. But it looks like way more work than I want to try right now. So I am satisfied to think current sharing for stranded is similar by physical reasoning as discussed above.
I have been thinking about whether there would be a way to estimate current sharing analytically without finite element modeling if we made some simplifying assumptions such as uniform current density within a given conductor.
Let's look at a simpler system 2 conductors per phase (although I realize the one in question is 6 conductors per phase). We have a total of 6 conductors c=1..6..call them IA1, IA2, IB1, IB2, IC1, IC2. The previous 6 currents represent 6 complex unknowns. I think we could also get 6 complex equations to solve the unknowns.
The first three equations simply state that the currents within a phase pair add up to the known phase current:
IA1 + IA2 = IAtot
IB1 + IB2 = IBtot
IC1 + IC2 = ICtot
The second three equations simply state that the voltages accross the two conductors of a phase pair is equal.
VA1 =VA2
VB1 = VB2
VC1 = VC2
Now how would we compute the voltage for example VA1:
VA1 = R * IA1 + j * L * IA1 + j * Sum (M1j * Ij) for j=2..6
The Sum represents the voltage induced by mutual inductance to each of the other 5 conductors.
6 complex equations in 6 complex unknowns... should be able to solve it.
L and Mij should be dependent upon the geometry. But finding them is not so straightfroward. L and Mij would be well defined if each of the current variables represented a loop, but in our case the circuits IA1, IA2 etc do not represent loop but instead represent current floiwng in one direction and so the selection of expressions for L and Mij is not so obvious. I think maybe a way around this would be to add a 7th conductor – call it an "imaginary return conductor" which we visualize to carry the return current for each of the conductors... which of course sums to 0 = -IA1 - IA2 - IB1 - IB2 - IC1 - IC2. Now we can view each current as a loop which goes out over one path (for example IA1) and returns over the imaginary return path, and this gives us the advantage of reformulating the problem in terms of loops so we can come up with expressions for L and M from the geometry of the rpoblem. When we add all the loops up of course we have the correct current distribution because the currents all sum to 0 so that current in that imaginary return conductor is 0.
I'll give it a try if I get a chance. Unless someone else has a suggetsion for an easier way to calculate current sharing from geoemtry under simplifying assumptions such as 1 – uniform current density within conductor; (and of course we also have thus far made simplifying assumption that there are no other current paths such as raceway or shields grounded at both ends).
Meanwhile, I am curious how 7anoter4 came up with his numbers for the particular current distributions discussed 27 Feb 09 10:13. Can you explain how you developed those numbers?
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