How to calculate iron losses in a steel ring around a conductor?

[OSR]

TL;DR Summary

How to calculate iron losses in steel ring?

Hi

I'm looking for calculation example how to calculate iron losses in steel ring around current carrying conductor.
I have tried to find examples online, but without a suitable result.

 

[Baluncore]

Welcome to PF.

AC current is never passed through a steel ring, or a hole in a steel plate, without an equal and opposite current through the same ring that will cancel the magnetic field. That is required and specified in every wiring code, which is why you cannot find examples in the literature.

As a magnetic transformer core, there is no air gap in the ring to control or limit the inductance of the single conductor primary.

What is your application of such an AC conductor and magnetic topology ?

 

[OSR]

Thanks, my goal is to determine how close a ferromagnetic loop can be to a current-carrying conductor with a high AC current so that there is practically no heating effect. What causes the loop to heat up if it is not due to eddy currents and hysteresis losses?

 

[Baluncore]

Thanks, my goal is to determine how close a ferromagnetic loop can be to a current-carrying conductor with a high AC current so that there is practically no heating effect.

If you have only one side of the AC electric circuit passing through the ring, there will never be zero heating. What path will be taken by the return circuit?

What causes the loop to heat up if it is not due to eddy currents and hysteresis losses?

It is the orientation of those eddy currents that is important. The magnetic field of the AC conductor follows the big circumference of the toroidal ring, a magnetic circuit. The electric eddy currents that heat the ring flow around the small circumference of the toroidal ring, through the hole.

The electric current path is short = 2⋅π⋅r , compared to the longer magnetic path length = 2⋅π⋅R , so the resistance is very low. Skin effect at 50 Hz will limit the depth of that conductive path in the surface of the iron.

 

[OSR]

Return path is outside of the ring. Theoretically yes, there will be always some losses regardless of the size of the ring, but I would assume that in practice, the heating effect on a 50 mm diameter loop and a 500 mm diameter loop would be different.

 

[Baluncore]

... but I would assume that in practice, the heating effect on a 50 mm diameter loop and a 500 mm diameter loop would be different.

Let's call the electric wire a loop, and the iron a ring. Those two circuits, one electric, the other magnetic, are like two interlocked chain links. As you increase the size of one, so the other must become bigger. Scale and separation are important.

The sectional area of the ring, will determine the volume of the loop's B field that will cut the iron ring. That B field, will come from all elements of the electric circuit loop you have not specified.

In effect, you have three circuits, one is the primary electric loop, the second is the magnetic circuit in the ring, the third is the long sheet of short eddy current, flowing in the surface of the iron ring.

 

[OSR]

Ok, here is more information of my test setup.

 

[TonyStewart]

All you have stated so far are some dimensions and low-C Steel.

Steel is a metal alloy much like Curry is a mixed bag of spices. There is a wide range of materials under the same name, not just Fe and C.

"Mild" steel is mostly iron, Fe arranged in random atomic crystals with low carbon, C which can yield high permeability but with a 100:1 wide range on conduction losses from high resistance due to lack of molecular grain orientation of conductor electrons. Alloys Cr, Ni, Mb, Cu, O2, and other trace elements, affect the crystal lattice structures. Also, processing affects electrical loss greatly with how it was annealed, cold rolled or printed as well as the conductor thickness/skin depth ratio.

Each variable affects the mechanical, electrical, magnetic, thermal, and corrosive properties widely.

Although you are only interested in the magnetic, conductive and inductive properties of this 40mm thin ring, keep in mind the wide range of permeabilities, and conductance losses of steel make this impossible to predict without these metallurgical values.

 

[TonyStewart]

How would you apply the formulae to estimate these losses?
Starting with ballpark numbers for ρ = 1e-6 [Ω-m]
R = (ρ • L) / A [Ω] with L=π×D for ring diameter D, cross-sectional area A
A = π • (OD² −ID²) [m]

This calculation doesn't account for eddy current losses, skin effect, permeability, mutual coupling or other factors that could affect the overall thermal losses. For a more detailed and accurate analysis, specialized software or simulations may be required.

Assume relative permeability μr = 1000 and a low mutual coupling factor M due to the PVC gap.

Then temp rise of the iron ring depends on the thermal resistance of the PVC from the ring junction to the ambient or Rja and the power dissipation P= I²R
Mutual coupling is typically calculated using numerical methods like finite element analysis (FEA) or finite difference time domain (FDTD) simulations.

In short ( no pun intended) although the mutual coupling which affects the impedance ratio and current might be 1% ballpark while the conductor resistance of the 125 mm loop might be 5 times the path length compare to the length of conductor = ring width=25mm but perhaps 100 times the electrical resistance and maybe 10~100 times the thermal resistance of PVC compared to the busbar with convection cooling and continuous conductor conduction cooling.

So it could be easily be hotter than the wire.

 

[OSR]

All you have stated so far are some dimensions and low-C Steel.

Steel is a metal alloy much like Curry is a mixed bag of spices. There is a wide range of materials under the same name, not just Fe and C.

"Mild" steel is mostly iron, Fe arranged in random atomic crystals with low carbon, C which can yield high permeability but with a 100:1 wide range on conduction losses from high resistance due to lack of molecular grain orientation of conductor electrons. Alloys Cr, Ni, Mb, Cu, O2, and other trace elements, affect the crystal lattice structures. Also, processing affects electrical loss greatly with how it was annealed, cold rolled or printed as well as the conductor thickness/skin depth ratio.

Each variable affects the mechanical, electrical, magnetic, thermal, and corrosive properties widely.

Although you are only interested in the magnetic, conductive and inductive properties of this 40mm thin ring, keep in mind the wide range of permeabilities, and conductance losses of steel make this impossible to predict without these metallurgical values.

Yes that is true, that steel ring is some unknown piece of water pipe. Is there some good book or source where i can find metallurgical values for some known material?

How would you apply the formulae to estimate these losses?
Starting with ballpark numbers for ρ = 1e-6 [Ω-m]
R = (ρ • L) / A [Ω] with L=π×D for ring diameter D, cross-sectional area A
A = π • (OD² −ID²) [m]

This calculation doesn't account for eddy current losses, skin effect, permeability, mutual coupling or other factors that could affect the overall thermal losses. For a more detailed and accurate analysis, specialized software or simulations may be required.

Assume relative permeability μr = 1000 and a low mutual coupling factor M due to the PVC gap.

Then temp rise of the iron ring depends on the thermal resistance of the PVC from the ring junction to the ambient or Rja and the power dissipation P= I²R

If i understand it right estimated loss is regarding to this is about 0.6 W = 250A^2 * 1.031*10^-5

I tried to estimate the required heating power by measuring the temperature as a function of time.
The temperature change was about 67,8 degrees Celsius over about 20 minutes.

If Q = c·m·ΔT and c=0,46 kJ/(K·kg) then if I'm correct, heating power has to be at least about 1.6 W ?Are these formulas good even estimating eddy current and hysteresis losses in this kind of situation, because i'm getting about 10-20 mW total power loss or is it just that my guess of resistivity and hysteresis co-efficient factor are too off. I'm not also sure did i calculate Bmax correctly, i use Bmax =0.075 T.

I can use Ansys and Maxwell to simulate this, but I want to better understand the theory and get some estimated values before attempting the simulation.

 

[hutchphd]

If you have the instrumentation set up, repeat the experiment up to some max T and then record the cooldown as well. From this you can easilly calculate the final steady state temperature directly I believe.
Does a log plot of that data give a good straight line?

 

[OSR]

If you have the instrumentation set up, repeat the experiment up to some max T and then record the cooldown as well. From this you can easilly calculate the final steady state temperature directly I believe.
Does a log plot of that data give a good straight line?

I created this plot by manually collecting temperature readings every 30 seconds. I need to make some kind of data logger to monitor the cooling process so that I don't have to watch it myself continuously.

 

[hutchphd]

Yes. But are you really only interested in the steady state T rise in the ring? Then I think that is the easiest route to a good number without any physical modeling. Worth a look.

 

[OSR]

I'm not just looking for a answer, i want to know how iron losses can be calculated or estimated through calculations.

 

[OSR]

If you have the instrumentation set up, repeat the experiment up to some max T and then record the cooldown as well. From this you can easilly calculate the final steady state temperature directly I believe.
Does a log plot of that data give a good straight line?

I made this plot manually by collecting temperature points in every 30 s, i need to make some datalogger to record also cooling.

 

[OSR]

I made test with a bit lower 150 A current and recorded also cooling. If we ignore conductor heating effect and look only what plot says, then if i'm right steel ring heating power is about 0.5 W. I would like to verify how the maximum magnetic flux density is calculated in this case, because I get very small values for the total iron losses compared to measured heating power?

 

[Babadag]

If we don't know all the details, for an approximation, we may improvise.
I take the ring as a short pipe of 42 out/37 in 25 mm long.
The average diameter is (42+37)/2=39.2 mm. Since the busbar current is 250 A, I take field intensity H= 250/(PI()*3.95)=20.15 A/cm=2015 A/m and for a mild steel the flux density B=0.55 Wb/m^2.
If the losses for 1.5 T are 3-90 W/kg and the weight of this ring it is Vol[dm^3]*7.85[kg/dm^3]
Vol=pi()*(0.42^2-0.37^2)*0.25/4= 0.007756 dm^3. The weight= 0.007756*7.85= 0.061 kg.
Let’s take the maximum W/kg=90.
Magnetic losses [max.]=90*(0.55/1.5)^2*0.061=0.7381 W
The eddy current is proportional with B^2 and hysteresis with B^1.6.
However, the aluminum busbar evacuates some losses from the part included in the ring.
The resistance of 25mm bar- at 90oC, including skin effect -it is
2.43E-6 ohm. Then the losses are 250^2*2.43/10^6=0.15 W.
Total losses=0.15+0.7381=0.8881 W
The temperature drop through outside pvc tube of 5 mm thick is 0.6 oC and calculating Qconv+Qrad =0.881 W the ring temperature will be 60 oC [if the ambient air temperature is 40oC] or 52oC [if Ta=30oC].

 

[OSR]

Ok i found the problem why i was getting so low values, problem was that i multiplied current with vacuum permeability for no reason when was calculating field intensity. Now I'm getting similar results what I get from previous heat energy calculations, if I adjust steel ring relative permeability between 300-500. I might try to measure hysteresis loop and calculate relative permeability for this steel ring according to this instruction http://info.ee.surrey.ac.uk/Workshop/advice/coils/BHCkt/index.html#top

I did not understand what are you calculating here, can you explain it more:

If the losses for 1.5 T are 3-90 W/kg

Magnetic losses [max.]=90*(0.55/1.5)^2*0.061=0.7381 W