# Winding Resistance Challenge

#### GoldDigger

##### Moderator
Staff member
There is no way, elegant or not, to avoid non-linear equations.
I think that using conductances instead of resistances for the delta, as you did, does make the math marginally simpler. I suppose you could call that a way to reframe the physics.

#### xptpcrewx

##### Power System Engineer
I do not think there is any way, elegant or not, to avoid non-linear equations.
I think that using conductances instead of resistances for the delta makes the math marginally simpler.
Agreed. There will still be nasty reciprocals somewhere in need of manipulation/substitution.

#### petersonra

##### Senior Member
Agreed. There will still be nasty reciprocals somewhere in need of manipulation/substitution.
Conductance was one of the things I tried but I never got it to work either. but by then it was like 3 am, so maybe I was not looking at it the right way. I figured there "has" to be a simple answer, like there is for the wye system, but sometimes there is not.

#### xptpcrewx

##### Power System Engineer
Well, as far as the algebra and the denominators, try this:

If x, y, z are the unknown conductances of the coils arranged in a delta, and A, B, and C are the measured conductances from the pairs of corners, then the equations are:

x + 1/(1/y + 1/z) = A

and its two analogues under the 3-cycle (x -> y, y -> z, z -> x ; A -> B ; B -> C ; C -> A).

That becomes

x+ yz/(y+z) = A or
xy + xz + yz = Ay + Az

So you get the three equations:

xy + xz + yz = Ay + Az = Bz + Bx = Cx + Cy

as the lefthand side of the exemplar equation is stable under the 3-cycle.

Which is sufficiently elegant that it makes me wonder if there's an even nicer way to frame the physics to make the math even simpler.

Cheers, Wayne
This is a brilliant observation and result!
It actually resembles the equations for a Y-Δ transform.

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#### wwhitney

##### Senior Member
Where are the NETA/NICET test technicians at???
If this type of calculation is something that technicians actually do in the field, then they can simplify the math for the delta side by taking measurements like this: short, say, terminals 2 and 3 together, then measure the resistance from terminal 1 to the shorting jumper. Repeat for the other 2 symmetric configurations.

That eliminates one of the coils from the measurement, and then the delta math for conductance is the same as the wye math for resistance. I've thought some about whether the value of the above measurements could be easily derived from the measurements in the OP, but I don't see it. If it could, that would be a short mathematical solution to the OP delta question.

Cheers, Wayne

#### xptpcrewx

##### Power System Engineer
If this type of calculation is something that technicians actually do in the field, then they can simplify the math for the delta side by taking measurements like this: short, say, terminals 2 and 3 together, then measure the resistance from terminal 1 to the shorting jumper. Repeat for the other 2 symmetric configurations.

That eliminates one of the coils from the measurement, and then the delta math for conductance is the same as the wye math for resistance. I've thought some about whether the value of the above measurements could be easily derived from the measurements in the OP, but I don't see it. If it could, that would be a short mathematical solution to the OP delta question.

Cheers, Wayne

Technically, shorting out two terminals would still be a delta connection, but just with a lower resistance for the shorted winding. Practically, this would introduce measurement errors as the jumper resistance isn’t zero and the windings most likely already present a lower resistance than a jumper itself. I would imagine measurement repeatability being an issue too - knowing that wire resistance is finicky.

I wouldn’t expect the technician to derive the equations but they should be able to use the general formula.

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#### wwhitney

##### Senior Member
So the units in the OP are milliohms? I'm not familiar with the intricacies of making accurate measurements of low resistances. Would it be feasible to short two terminals 1-2 and then take resistance measurements from terminals 3 to 1 and terminals 3 to 2 to control for the shorting jumper resistance?

Cheers, Wayne

#### xptpcrewx

##### Power System Engineer
So the units in the OP are milliohms? I'm not familiar with the intricacies of making accurate measurements of low resistances. Would it be feasible to short two terminals 1-2 and then take resistance measurements from terminals 3 to 1 and terminals 3 to 2 to control for the shorting jumper resistance?

Cheers, Wayne

I’ve never actually tried it but I’m open to giving it a shot next time I’m in front of a transformer. If anyone has tried it, please share.

Generally, the purpose for conducting such a test is to check for turn-to-turn shorts or poor connections, so you’re basically looking for “small” differences or “discrepancies” between windings against baseline measurements (compared to when the transformer was first placed in service or with respect to measurements taken from other identical units). These measurements aren’t proof by themself (unless the differences are significant enough), but are corroborated together with the results from other tests (i.e. DGA/COQ, TTR, etc.) to infer incipient/existing problems. Given this, introducing jumpers to simplify the analysis is something I’m hesitant about, especially since I already know the formulas.

Note: Knowing the actual individual phase winding resistances is important because the terminal-to-terminal measurements of the delta connection tend to convolute/obscure any differences.

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#### synchro

##### Senior Member
I'm not going to show the numerical solution I had to the second problem, but perhaps I can provide some insights into it.

Let x be the resistance of the H1-H2 winding, y the resistance of the H2-H3 winding, and z the resistance of the H3-H1 winding.
Each of the three resistance measurements establishes its own associated equation in the variables x, y, z.
For each of the three equations I've plotted the surface containing the points x, y, z that are solutions to that particular equation.
The blue surface is defined by the RH1-H2 = 2.428mΩ measurement, purple surface by the RH2-H3 = 2.315mΩ measurement, green surface by the RH3-H1 = 2.445mΩ measurement.
The one point where all three surfaces intersect satisfies all three equations, and therefore it's the correct set of winding resistances x, y, z for the problem.

The following plot is on expanded scale where x, y, and z each range from 2 milliohms to 5 milliohms. This is just to show the curvature of the surfaces due the nonlinear equations, and to give some insight about what the equations represent. A numerical solution would need to be done to get an accurate result.

#### xptpcrewx

##### Power System Engineer
I'm not going to show the numerical solution I had to the second problem, but perhaps I can provide some insights into it.

Let x be the resistance of the H1-H2 winding, y the resistance of the H2-H3 winding, and z the resistance of the H3-H1 winding.
Each of the three resistance measurements establishes its own associated equation in the variables x, y, z.
For each of the three equations I've plotted the surface containing the points x, y, z that are solutions to that particular equation.
The blue surface is defined by the RH1-H2 = 2.428mΩ measurement, purple surface by the RH2-H3 = 2.315mΩ measurement, green surface by the RH3-H1 = 2.445mΩ measurement.
The one point where all three surfaces intersect satisfies all three equations, and therefore it's the correct set of winding resistances x, y, z for the problem.

The following plot is on expanded scale where x, y, and z each range from 2 milliohms to 5 milliohms. This is just to show the curvature of the surfaces due the nonlinear equations, and to give some insight about what the equations represent. A numerical solution would need to be done to get an accurate result.

View attachment 2557517

Awesome visualization of the non-linear nature of this problem. Thanks!

Sent from my iPhone using Tapatalk

#### junkhound

##### Senior Member
non-linear

How is this non-linear? Unless windings are silicon carbide or similar material vs. copper.

#### xptpcrewx

##### Power System Engineer
non-linear

How is this non-linear? Unless windings are silicon carbide or similar material vs. copper.

It’s the system of equations that we are referring to as being nonlinear.

#### junkhound

##### Senior Member
Hmm, nothing except straight lines in Synchro's diagram???

#### xptpcrewx

##### Power System Engineer
Hmm, nothing except straight lines in Synchro's diagram???

I see three curved surfaces.

#### wwhitney

##### Senior Member
The curvature is somewhat subtle, particular if for any given color you only look at the region where the grid lines make squares. But once you then incorporate the region where the grid lines make triangles, the curvature becomes clearer. [BTW, why do the grid lines change like that?]

Cheers, Wayne

#### synchro

##### Senior Member
The curvature is somewhat subtle, particular if for any given color you only look at the region where the grid lines make squares. But once you then incorporate the region where the grid lines make triangles, the curvature becomes clearer. [BTW, why do the grid lines change like that?]

Cheers, Wayne
I believe that the additional lines beyond those forming nearly square grids are due to the intersections of planes with the surface in question, where these planes are perpendicular to and evenly spaced along each axis . For example, the more widely spaced and somewhat horizontal lines on the green surface in the foreground would be intersections with x-y planes that are equally spaced along the z-axis. The lines sloping downward to the right on the blue surface would be intersections with y-z planes that are equally spaced along the x-axis. The lines sloping upward to the right on the purple surface would be intersections with x-z planes that are equally spaced along the y-axis.

If the three surfaces were solutions to linear equations then the surfaces would be planar, and therefore the intersections of the surfaces with the equally spaced planes that are perpendicular to the axes would form equally spaced lines on the surfaces. Clearly that is not the case here.

The increment between each plane that's perpendicular and spaced along the axes was generated by the tool at the link below. I didn't pay attention to what numerical value the increment had or whether it was adjustable.

https://c3d.libretexts.org/CalcPlot3D/index.html

#### wwhitney

##### Senior Member
Sounds like you are saying each surface is ruled in each dimension at the same fixed spacing, so the gaps between ruling lines provides a sense of the rate of change of the different coordinates. I think that matches what I'm seeing.

Thanks,
Wayne

#### synchro

##### Senior Member
Sounds like you are saying each surface is ruled in each dimension at the same fixed spacing, so the gaps between ruling lines provides a sense of the rate of change of the different coordinates. I think that matches what I'm seeing.

Thanks,
Wayne
Yes, I agree. I didn't see any explanation of this within the help file for the tool but I've really just glanced though it. When I first used this tool and saw these lines it looked strange. But I concluded that what I described above is a likely explanation for them.

#### xptpcrewx

##### Power System Engineer
Anyone else want to give this a shot before I post the solution?