Determining inductive reactance from a 3-phase AC circuit

[vwishndaetr]

TL;DR Summary

I am trying to determine the reactance of a circuit if I know the resistance, current, and voltage. I am speculating that current imbalances of parallel runs that are equal in resistance are caused by the differing reactance in the circuit.

I am trying to figure something out, but I am in the first step and just trying to get through my "thought experiment".

Let's say I have a 480V 60hz 3 phase supply feeding a circuit that comprises of nothing but the conductors themselves (effectively a shorted power supply). Consider the circuit below: power supply that has a couple parallel runs between the short and the supply.

I want to figure out what self imposed reactance is being produced by the circuit. More specifically in the parallel runs.

Theoretically, in a perfect world the current in the parallel runs should be half the total current in say phase A (ultimately this will be broken down to 3 single phase circuits, but I wanted to show the whole picture here - so consider everything going forward is about phase A only). If one of the parallel runs draws more current than the other, and the resistance across the two parallel runs is equal, then is it safe to assume that I have some reactance that is creating the current imbalance between the parallel runs? After all, impedance comprises of resistance and reactance. Why else would I have an imbalance? With that being said can I determine what the reactance is if I know the voltage, across the parallel runs, the current in each, and the resistance of each?

I think I have formulas at my disposal to calculate this, but I want to know if my theory makes sense... My electrical knowledge is limited to being in the industry... formal education is mechanical in nature.

TIA for the help, came on here a bunch in my college days when I had other thing explained to me - figured this would be a great place to get some feedback on this.

 

[Baluncore]

What is the "thing" ? If it was a motor, you could do a locked rotor test.
Or the inductance could be measured with an inductance bridge at the frequency of the mains. It does not need to be high voltage.

 

[vwishndaetr]

What is the "thing" ? If it was a motor, you could do a locked rotor test.
Or the inductance could be measured with an inductance bridge at the frequency of the mains. It does not need to be high voltage.

Switchgear. Maybe I made a mistake by trying to outline some arbitrary numbers. Supply is still 3ph 60 Hz, but more like 10V.

Keep in mind, this isn't an installation. I am just trying to wrap my head around some of the theory. So there are no "mains" unless you're referring to the point of supply of the gear, where the supply is literally a power supply.

 

[DaveE]

can I determine what the reactance is if I know the voltage, across the parallel runs, the current in each, and the resistance of each?

Yes. ## \frac{|V|}{|I|} = |R+jX| = \sqrt{R^2 + X^2} ## . You said you know everything but X, right?

I want to know if my theory makes sense

Sorry, I don't think I know what your theory is. Normally, we would assume that your "wires" have no impedance. Then we would explicitly show the salient circuit elements in the schematic. This way everyone is talking about the same problem.

 

[vwishndaetr]

Yes. ## \frac{|V|}{|I|} = |R+jX| = \sqrt{R^2 + X^2} ## . You said you know everything but X, right?Sorry, I don't think I know what your theory is. Normally, we would assume that your "wires" have no impedance. Then we would explicitly show the salient circuit elements in the schematic. This way everyone is talking about the same problem.

Thanks for sharing the equation.

For an AC current carrying wire, a changing magnetic field is produced that ultimately induces an opposing current in the current carrying wire. So the inductive reactance can exist without external influence.

Can the same be said for capacitive reactance? Is there a configuration that can cause capacitive reactance between phases? For example, would two adjacent AC current carrying wires create this affect? Or does capacitive reactance only come into play when capacitors are present?

I don't have a theory. Likely a poor choice of words on my end. I can't explain it any other way... Only going into this with what I know. If I knew enough to ask the right questions then I might be able to figure it out myself. Which I don't, hence why I am asking these.

 

[Baluncore]

Any capacitive or inductive cross coupling between phase conductors will result in a phase shift of the currents in the phases.
I expect you could calculate the coupling, or model it with a FEM simulator.
If your design is symmetrical then it may cancel all the phase shifts due to coupling of the phase shifted sinewaves.

 

[vwishndaetr]

Any capacitive or inductive cross coupling between phase conductors will result in a phase shift of the currents in the phases.
I expect you could calculate the coupling, or model it with a FEM simulator.
If your design is symmetrical then it may cancel all the phase shifts due to coupling of the phase shifted sinewaves.

Yes I am glad you mentioned that!

I was also thinking... if I use an o-scope to measure the phase shift, then that should also let me know whether I have a net capacitive or inductive reactance no? I think I can associate the phase shift angle to a reactance value as well. If I remember right the lead/lag should indicate whether it's a net inductive or capacitive reactance.

 

[DaveE]

I use an o-scope to measure the phase shift, then that should also let me know whether I have a net capacitive or inductive reactance no? I think I can associate the phase shift angle to a reactance value as well. If I remember right the lead/lag should indicate whether it's a net inductive or capacitive reactance.

Yes. You'll need a voltage channel and a current probe. Or, two voltage channels with some know impedances combined with some modelling.

However, the first step in measurement is knowing in advance exactly what you are measuring and why.

 

[vwishndaetr]

Yes. You'll need a voltage channel and a current probe. Or, two voltage channels with some know impedances combined with some modelling.

However, the first step in measurement is knowing in advance exactly what you are measuring and why.

Ok for sure - let me go back to my initial thoughts. I think the o-scope might be a bit more applicable after I answer my initial question.

Right now, this is what I have measured (just looking at A phase):

I_A1 = 2017A
I_A2 = 1998A
I_A was not measured - it's just a sum of the parallel currents.

My end goal is to determine if the imbalance between I_A1 and I_A2 is a result of the difference between resistances R₁ and R₂.

I plan on measuring R₁ and R₂.

If R₁ = R₂, then I can conclude that the imbalance is a result of something other than resistance (ie. inductive reactance from A itself or adjacent phases).
If R₁ not= R₂, then I can use the equation you shared to calculate R and X for each parallel path and for the complete circuit.

Let's table the fact that the example I have above is within 1% error for now. I know am splitting hairs in this example. Other ones the difference is over 5% so a little more substantial.

 

[DaveE]

IA was not measured - it's just a sum of the parallel currents.

Not a simple algebraic sum if this is an AC problem and there is a phase difference between the currents. Otherwise this stipulation implies that the branch impedances can be treated like resistances (although they could be any complex impedance as long as the phase shift is the same for each).

Remember the triangle inequality for complex numbers ##|a| + |b| \geq |a+b| ##.

In general, this is called a current divider problem, and it goes like this:

Let's name the two branch impedances ##Z_{A1}## and ##Z_{A2}##.

Then the parallel combination ##Z_A = Z_{A1} \parallel Z_{A2} = \frac{Z_{A1}Z_{A2}}{Z_{A1}+Z_{A2}}##.

The voltage across ##Z_A## is ##Z_A I_A##. But this is the same as the voltage across each impedance.

So ##V_A = Z_A I_A = Z_{A1} I_{A1} = Z_{A2} I_{A2}##.

##I_A = \frac{Z_{A1}}{Z_A} I_{A1}## or ##I_A = \frac{Z_{A2}}{Z_A} I_{A2} ##

## I_A = \frac{Z_{A1}}{\frac{Z_{A1}Z_{A2}}{Z_{A1}+Z_{A2}}} I_{A1} = \frac{Z_{A1}+Z_{A2}}{Z_{A2}} I_{A1} ##

This can be solved to get ## Z_{A1} = (\frac{I_A}{I_{A1}} - 1) Z_{A2}##

Since we never had the voltage measurement, you can only find the ratio of the two branch impedances.
Also remember that these are complex numbers your data is ##|I_{A1}| ## not ##I_{A1}##.

 

[DaveE]

BTW, I think khan academy has some really good tutorials on AC circuit analysis. You might like those.

 

[Baluncore]

Yes I am glad you mentioned that!

I was also thinking... if I use an o-scope to measure the phase shift, then that should also let me know whether I have a net capacitive or inductive reactance no?

Is the coupling circular? Phase A affects B, B affects C, C affects A. So it might all cancel about the phase circle.

You can calculate the capacitance, inductance and mutual inductance.
Then model the circuit with SPICE and a 3PH supply.
Is this actually a long transmission line model ?
Or is it a short mesh of transformers ?
You can then play with the numbers and see how far you can push the coupling.

 

[vwishndaetr]

BTW, I think khan academy has some really good tutorials on AC circuit analysis. You might like those.

I've started going through some of those. Can you also share what you would consider a good text reference for AC circuit analysis - preferably one that also covers 3ph circuits?

 

[vwishndaetr]

Is the coupling circular? Phase A affects B, B affects C, C affects A. So it might all cancel about the phase circle.

You can calculate the capacitance, inductance and mutual inductance.
Then model the circuit with SPICE and a 3PH supply.
Is this actually a long transmission line model ?
Or is it a short mesh of transformers ?
You can then play with the numbers and see how far you can push the coupling.

I don't have SPICE. Not a long transmission or any transformers involved either. As mentioned previously this is for a switchgear assembly, so predominantly compromised of bus bars as current carrying conductors, with breakers depending on the configuration.

All of my questions essentially revolve around the bus compartment. I also don't necessarily have a problem to solve, yet. I am merely trying to figure out if I have a problem. I want to better understand what effects parallel bars have on each other and the current flow. I figured a good place to start was to break it all down to a simple circuit and work through some of the AC circuit theory to better understand whether construction is affecting current flow... mainly from induced currents from adjacent phases, and also adjacent branches from similar phases.

Hence why my delivery has been so scatter-brained.

Dave's comment above about the current not being an algebraic sum was not obvious to me, so I might've bit off a little more than I can chew. I think I need to go a step back and wrap my head around some of the principles of AC circuits before I start diving into calculations.

With that being said, any other recommendations from you as well? My mathematical background is strong so I would like to have the full picture. I also have a pretty good understanding of DC circuits, so I think I have a "core" in electrical theory as well. 3 phase AC, from an analytical standpoint, is a new space.

 

[DaveE]

I've started going through some of those. Can you also share what you would consider a good text reference for AC circuit analysis - preferably one that also covers 3ph circuits?

Sorry, I don't really know which textbooks are good for this. Maybe someone else does?

You'll want to learn about "phasors" for this sort of problem. Which is jargon for representing AC voltages and impedances at a fixed frequency with complex numbers for their amplitude and phase relationships.

 

[Baluncore]

@vwishndaetr I don't believe you can partition the bar in the way you have done. The bar current will generate a magnetic field that will encircle the cross-section of the bar. That will prevent separation into two different currents.
Skin effect will prevent the magnetic field from penetrating the bar, which is required for separation.
When you analyse the field in the middle of a hypothetically partitioned bar, there will be two counter fields that will cancel and so fuse the conductors back into one bar.

I don't have SPICE. Not a long transmission or any transformers involved either.

Back on the road again; Your conductors are parallel coupled inductors, which in theory makes it a lightly coupled transformer. Also there is capacitance between your inductive conductors, that in theory makes it a short transmission line, so cross-talk can be analysed.

We can help define a model of the situations that you are contemplating. Indeed, you will need a realistic model before doing any numbers. How about drawing a cross section of the conductors and insulation, with some measurements. Also give us an idea of the length of the conductors.

SPICE is available for free as LTspice, now from the AD website. SPICE makes it possible to model capacitance between conductors and coupled inductors. https://en.wikipedia.org/wiki/LTspice

 

[vwishndaetr]

@vwishndaetr I don't believe you can partition the bar in the way you have done. The bar current will generate a magnetic field that will encircle the cross-section of the bar. That will prevent separation into two different currents.
Skin effect will prevent the magnetic field from penetrating the bar, which is required for separation.
When you analyse the field in the middle of a hypothetically partitioned bar, there will be two counter fields that will cancel and so fuse the conductors back into one bar.

When you say partition, do you meant that in the literal sense, meaning in an assembly, or are you talking for the calculation I have presented? Because the way I have it shown is a literal representation of the present construction. If I drew a 3-line to represent the assembly, it would pretty much be what I have shown in my sketch. The paralleling construction is 100% unavoidable in my case.

 

[Baluncore]

When you say partition, do you meant that in the literal sense, meaning in an assembly, or are you talking for the calculation I have presented?

The physical reality must parallel the virtual model, or there is no hope.

At low frequencies, when you cut a narrow longitudinal slot in a flat bar conductor, the ends of the bar remain as circuit nodes, and it makes no difference to the magnetic field. The narrow slots are invisible to the magnetic field model. If you then make different connections along the separate branches, the circuit elements are divided, so you create new circuit nodes and the circuit model must be changed.

You need to present a physical model so that a representative circuit can be extracted.