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Magnetic field formulation

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Nils227

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Hello,

Assuming that we have a single-phase electrical transmission line (short distance (< 50 km), stranded, non-isolated, made of aluminum conductor steel reinforced), I would like to know the resulting magnetic field (shape, structure, absolute value, and all other possible details) when the line is being injected by different RMS values of an AC current of 50 Hz. It is intended to have a mathematical model formulation of the resulting magnetic field, so that it is possible to know the Tesla value of the magnetic field directly when the RMS value of the injected current is known.

In other terms, I am looking for the Biot-Savart formulation but for an AC current (instead of DC).

Any comments or possible related book chapters will be much appreciated

Thank you
 
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You didn't yet mention an actual geometry. If receiving distance is large compared to strand/massive wire diameter, the difference to the field of a single thin wire is negligible. If not and you are targetting to high accuracy, a 2D FEM solver, e.g. free FEMM tool, would be my solution to model the field of one or more straight conductor(s). Of course you can also use a full featured 3D EM tool like HFSS if you have it at hand.
 
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FvM said:
You didn't yet mention an actual geometry. If receiving distance is large compared to strand/massive wire diameter, the difference to the field of a single thin wire is negligible. If not and you are targetting to high accuracy, a 2D FEM solver, e.g. free FEMM tool, would be my solution to model the field of one or more straight conductor(s). Of course you can also use a full featured 3D EM tool like HFSS if you have it at hand.
Click to expand...
Moin FvM,
Thank you for your reply. I am targeting for high accuracy. Before subjecting such model to a computer based simulator, I would like to calculate the resulting magnetic field by formulas.

The geometry is as follows: a 70 mm2 stranded aluminum transmission line. The distance from it is less than 2 centimeters.

Thank you
 
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If you need hig accuracy you have definitely to go with a simulator.

However I'm still no sure about the actual geometry of the whole system, I mean all the wires and their position in space as well as the measurement point that as far as I've correctly understood it's place from 5 to 15 cm from one of the three (?) conductors.

In any case let's suppose the three wires are placed at the vertex of a triangle and some meters apart one each other. If you go very close to one of them the interaction of the field generated by the other two wires will be almost negligible, but still present (you mention "high accuracy").
If, for instance the wires are 2 meters apart one each other and the measurement point is placed 5 cm from one of them (let me say "A"), then
dA = 0.05 while dB approx=dC approx = 2 m. The phase shift due to the distances will be totally negligible, then we will have phase(A) = 0, phase(B) = 120 deg and phase(C) = 240 deg.

Substituting these values in the formula for B, the square root gives as a result: 19.6, while Biot-Savart applied to the single wire A gives 20.0 corresponding to an error of rouglhy 2%. Since the measurement point is very close to the wire doesn't matter if it is placed insiede or outside the triangle formed by the three wires.
 
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albbg said:
If you need hig accuracy you have definitely to go with a simulator.

However I'm still no sure about the actual geometry of the whole system, I mean all the wires and their position in space as well as the measurement point that as far as I've correctly understood it's place from 5 to 15 cm from one of the three (?) conductors.

In any case let's suppose the three wires are placed at the vertex of a triangle and some meters apart one each other. If you go very close to one of them the interaction of the field generated by the other two wires will be almost negligible, but still present (you mention "high accuracy").
If, for instance the wires are 2 meters apart one each other and the measurement point is placed 5 cm from one of them (let me say "A"), then
dA = 0.05 while dB approx=dC approx = 2 m. The phase shift due to the distances will be totally negligible, then we will have phase(A) = 0, phase(B) = 120 deg and phase(C) = 240 deg.

Substituting these values in the formula for B, the square root gives as a result: 19.6, while Biot-Savart applied to the single wire A gives 20.0 corresponding to an error of rouglhy 2%. Since the measurement point is very close to the wire doesn't matter if it is placed insiede or outside the triangle formed by the three wires.
Click to expand...
Thank you albbg,

Here's a textual geometric sketch:
1. a 3-p overhead electric transmission system
2. the magnetic field is to be measured with regards to only one line of the three (considering that the distance between line-to-line is wide enough, that the "contribution" of each adjacent line is negligible)
3. the considered line (amongst the three, where the remaining two are neglected) is stranded (many filaments are woven together to constitute it)

> I want to calculate the magnetic field, resulting from the passage of an electric current through that stranded line, in an arbitrary point in the space near that transmission line.

> I am searching for a similar mathematical model like the one in this publication: DOI:10.1051/itmconf/20181901010 (Magnetic field of a ribbon busbar of finite length)

You initial (hand-written) mathematical model would be approved for a single solid wire.
The thing is to obtain such a model but for a standed wire:
1692021786097.png


or


1692021809377.png
 
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There is no significant difference between a single solid wire and the stranded wires you proposed.
 
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albbg said:
There is no significant difference between a single solid wire and the stranded wires you proposed.
Click to expand...

Hi aIbbg,
Thank you for your comment.

According to your reply, I can concretely apply the Biot-Savart law to both a single as well as a stranded wire, and get the resulting magnetic field with a good precision: could you please post some studies/publications/papers stating so?

Nevertheless, assuming that B.-S. law can be applied for both wire's geometries: is it possible to derivate the mathematical model (as the one in: DOI:10.1051/itmconf/20181901010) for the magnetic field calculation in a stranded wire (therefore, we can make equation-wise comparisons between it and the original B.-S. thus concluding that minor significance between the two models).

Regards,
 
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