Calculating phase currents from line currents in an unbalanced delta

[The Electrician]

I want to be able to optimise the solution to a problem, by transforming forward and backward consistently, without underestimating the circulating current.

It's not possible to calculate (know) what the circulating current is (if there is any) by just knowing the line currents; as DaveE said in post #4, any circulating current has no effect on the line currents.

 

[DaveE]

Y'all are really confusing me with this talk of circulating currents. Help, please explain what you mean explicitly.

The OP's problem is clearly not fully defined, so we say it has a multiplicity of solutions. However, I think it can also be thought of as a multiplicity of problems, each with a unique real world solution. Let's look at some specific examples.

To save me some typing, lets define $V_{XY} \equiv -V_{YX} \equiv V_X - V_Y$

Given that $I_A = -1$, $I_B = 5$, $I_C = -4$

We have the solution in post #19 of $i_{ac}=1$, $i_{ba}=2$, $i_{cb}=-3$
Which is claimed to have no circulating currents because of some "minimum norm" argument that I couldn't really follow.
This solution defines the network (for the applied voltages), with $Z_{AC} = \frac{V_{AC}}{1A}$, etc.

So now, if we add a "circulating current" of $1A$ then for each phase we get $i_{ac}=2$, $i_{ba}=3$, $i_{cb}=-2$.
This defines a different network, $Z_{AC} = \frac{V_{AC}}{2A}$, etc.

Both examples are unique solutions to KVL and KCL for their network. In particular, they each satisfy KVL for the load loop: $i_{ba}Z_{BA}+i_{ac}Z_{AC}+i_{cb}Z_{CB}=0$

So, I claim any passive load network can't have steady state circulating currents, because of KVL, even though I don't really know what they are. Please enlighten me with as few words as possible (equations are preferred to minimize my confusion).

 

[The Electrician]

You are the one who brought up the notion of circulating currents in post #4.

I said in post #25: "As an aside, I've been using a DC circuit in my examples because the arithmetic is easier. I've also used the phrase "circulating currents" but the physics of a DC version of this circuit can't really have circulating currents although the math is consistent as though there were such a thing." The same is true in an AC circuit with loads that are not reactive (pure resistances) so we are in agreement about that.

In post #4 you must have been talking about what could happen in AC circuits with reactive loads. Can you give an example where that would happen?

 

[DaveE]

In post #4 you must have been talking about what could happen in AC circuits with reactive loads. Can you give an example where that would happen?

That post was just wrong. And no, they can't happen with any passive load network in steady state, IMO. Of course that doesn't include transformers or other induced voltages.

But as I said, I'm a bit confused about what others mean when they mention them. It seems like a catch-all phrase like "stray" or "parasitic", but not literally circulating. OTOH, I suppose all currents circulate, what with conservation of charge...

 

[The Electrician]

That post was just wrong. And no, they can't happen with any passive load network in steady state, IMO. Of course that doesn't include transformers or other induced voltages.

But as I said, I'm a bit confused about what others mean when they mention them. It seems like a catch-all phrase like "stray" or "parasitic", but not literally circulating. OTOH, I suppose all currents circulate, what with conservation of charge...

Transformers are passive; do you mean resistive, as post #1 described?

What I mean by "circulating" is this: The delta of branches form a loop. If a positive current of the same value is added to the current in each branch of the delta, the directions of those added currents travel in the same direction around the loop--they "circulate" around the loop.

Post #1 made no mention of impedances or voltages, just line and phase currents. It is possible, given only the phase currents to determine the line currents.

The problem is underdetermined (https://en.wikipedia.org/wiki/Underdetermined_system), so the solution for the line currents is not unique; there an infinite number mathematical solutions, but the physics of the situation may rule out some, or all but one.

For example, if we have a vector of phase currents [1,2,-3] and transform that to line currents as I showed in post #19, we get a line current vector of [-1,5,-4] which transforms back to a phase current vector of [1,2,-3].

We can represent the 3 line or phase currents as a vector, for example phase currents [1,2,-3]. This vector has a Euclidean length (norm) of sqrt(1^2+2^2+3^2) = sqrt(14). If we add 1 to each current (this is adding a circulating current of 1) we have another solution, a current vector [2,3,-2], which has length sqrt(17), larger than the norm of the currents with no circulating current. Of all the infinity of solutions for the phase currents with line currents of [-1,5,-4], the one with no circulating current is [1,2,-3], and it has the smallest norm. AND, it is the only one that is physically possible if circulating phase current is impossible.

The same thing applies in the other direction.
Starting with phase currents of [1,2,-3], perform the transformation to line currents.
We get line currents of [-1,5,-4]; add 100 amps in the direction of the delta to each line current. Then we would have line currents of [99,105,96]. These line currents don't add up to zero, so they are physically impossible. But if we premultiply them by the transformation matrix to convert back to phase currents, we get [1,2,-3]; these added currents are ruled out as solutions by the physics of the situation, but they satisfy the math. Knowing only currents (no impedances, that is), the problem is underdetermined, and it has an infinite number of (mathematical) solutions, but all but one are ruled out by the physics of the circuit if circulating current is impossible.

As an aside , can you answer Baluncore's question in post #23: "Why do you deny the existence of circulating currents in reactive AC systems ?"

 

[DaveE]

Transformers are passive; do you mean resistive, as post #1 described?

No, I meant no induced voltages/currents. No coupling to an energy source. Inductors are OK, coupled inductors are OK as long as they don't couple to an energy source. Honestly "passive" is the best adjective I've got. In my world "passive" means energy is conserved, or dissipated, not created.

If we add 1 to each current (this is adding a circulating current of 1) we have another solution

You have a different unique solution to a different network. The OP needs one more piece of information to fully specify the network. Your "min norm" requirement functions as that piece of information.

So, I remain unsatisfied by your two examples since each defines a different network. You can't just add a constant "circulating current" as a solution to the original network. It is a solution for a member of the set of networks defined in the OP. But, as soon as you add that current you must change the load impedances accordingly to satisfy KVL/KCL. Go back in those examples and finish the problem, calculate each of the phase impedances.

For a single well defined passive load network, I would like a definition of what the circulating currents in that load network are.

As an aside , can you answer Baluncore's question in post #23: "Why do you deny the existence of circulating currents in reactive AC systems ?"

I think either he's wrong or he has a different idea of what a circulating current is than what you will see in this simple example. The real world is much more complicated than this example. In any case, I would ask for his definition as well.

I don't deny that people talk of circulating currents, or that they exist (given a good definition). Most often as "wasted" currents in distribution systems with multiple transformers (like magnetizing currents), multiple sources, and/or induced current from magnetic coupling. For example this, with paralleled inverters.

I don't claim they don't exist anywhere. Just not in this particular problem set. I also am genuinely curious about what y'all think they are. As you can see from my previous posts, I've been a bit confused about this until I thought about it some more and realised the the "solution set" was actually more like a set of separate networks, each with a unique solution. OTOH, I'm getting tired of words and would prefer clear technical explanations.

Based on some brief web searches (like this), I think it is often a pretty ill-defined thing; one of those garbage collection sort of concepts, like "parasitic". In general, current flow that you don't want. I don't seek an answer as much as a definition; the answer will then follow.