Resulting magnetic field due to the passage of an AC current

[Nils227]

TL;DR Summary

1. Value of magnetic field calculation
2. the magnetic field is created by a current flow
3. to get the flux density at an arbitrary point in space near the current carrying conductor

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, the required solution shall be as follows: a mathematical function that has the current's intensity (in Amperes) and frequency (in Hertz), in addition to the cable's length and geometry (thickness, number of stranded wires, etc.) as inputs, then consequently outputs the magnetic flux density (in Tesla) at an arbitrary point in the space near (around) the current carrying conductor (i.e., stranded-wire).

Suppose that the stranded wire lies (coincides) on the x-axis in a Cartesian plane; this wire begins at (0;0;0) and stretches forward the positive x-axis until its length: what would be the flux density at a point P (x;y;z) located in the same plane, that has the following coordinates:

x = 1 meter
y = z = 0.15 meter

Thank you in advance

 

[berkeman]

Welcome to PF.

What have you found in your research so far? What is the reason that you are wanting to calculate this?

Also, a single-phase transmission line has 2 wires, so the B-field that results depends on the distance between those two wires with respect to the measuring distance for the B-field. What are the numbers for the transmission line you are asking about? What is the AC current in this transmission line?

 

[Nils227]

Welcome to PF.

What have you found in your research so far? What is the reason that you are wanting to calculate this?

Also, a single-phase transmission line has 2 wires, so the B-field that results depends on the distance between those two wires with respect to the measuring distance for the B-field. What are the numbers for the transmission line you are asking about? What is the AC current in this transmission line?

Thank you berkeman.
1. I have found in my research nothing more than the Biot-Savart law.
2. Consider that you have a three-phase electrical power transmission system and that it is intended to calculate the flux density at a point P as described resulting from the flow of current in the Phase A conductor, for example (please disregard all statements about a single phase transmission line).
3. AC current of MAX value of 1000 A

 

[berkeman]

If it's 3-phase, there are 3 current-carrying conductors. So you would use the Biot-Savart Law to calculate the vector sum of the 3 B-fields at your points of interest.

https://phys.libretexts.org/Bookshe...3:_Magnetic_Field_due_to_a_Thin_Straight_Wire

 

[Nils227]

If it's 3-phase, there are 3 current-carrying conductors. So you would use the Biot-Savart Law to calculate the vector sum of the 3 B-fields at your points of interest.

https://phys.libretexts.org/Bookshelves/University_Physics/Book:_University_Physics_(OpenStax)/Book:_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)/12:_Sources_of_Magnetic_Fields/12.03:_Magnetic_Field_due_to_a_Thin_Straight_Wire

View attachment 329902

1. Can the Biot-Savart law be applied for alternating currents?
2. If it is a 3-phase system, assumed perfectly balanced, each of the wire carries the same amount of current. Accordingly, I am only concerned about knowing the Tesla value of the current that is passing in the current-carrying conductor A

Thank you

 

[berkeman]

1. Can the Biot-Savart law be applied for alternating currents?

Yes.

2. If it is a 3-phase system, assumed perfectly balanced, each of the wire carries the same amount of current. Accordingly, I am only concerned about knowing the Tesla value of the current that is passing in the current-carrying conductor A

But the vector phases of the 3 conductors are different, so you need to sum those effects to figure out the B-field at some point near the transmission line.

You could do it for just one wire, and you would have the order of magnitude of the B-field value for a 3-phase TL, if that helps.

You still have not said why you are wanting to do this calculation...

 

[berkeman]

You could do it for just one wire, and you would have the order of magnitude of the B-field value for a 3-phase TL, if that helps.

BTW, the real value will be less than this 1-wire value, since the 3 conductors are fairly close together compared to most distances of interest (like to the ground under the TL).

 

[Nils227]

Yes.But the vector phases of the 3 conductors are different, so you need to sum those effects to figure out the B-field at some point near the transmission line.

You could do it for just one wire, and you would have the order of magnitude of the B-field value for a 3-phase TL, if that helps.

You still have not said why you are wanting to do this calculation...

1. Can the Biot-Savart law calculate the resulting magnetic field due to an AC current, flowing in a stranded wire?
2. I want to make this calculation for the following reason: due to the proportionality between the current and its resulting magnetic field, If I know somehow the Tesla value of the latent, I can using this "calculation" know the initial norm of the current that is initially creating it

Thank you

 

[berkeman]

2. I want to make this calculation for the following reason: due to the proportionality between the current and its resulting magnetic field, If I know somehow the Tesla value of the latent, I can using this "calculation" know the initial norm of the current that is initially creating it

Sorry, I'm having trouble parsing this. I'm guessing that you are saying that if you can measure the B-field at some distance from the 3-phase transmission line and you know the dimensions of the wire separation and the distance to the point where you are making your measurement, that you may be able to calculate the power flowing in the AC Mains transmission line. I think that's true, but you will need to do the full calculation for the TL in order to do that.

EDIT/Add -- And you need to know what the nominal AC Mains TL voltage is (in RMS), since the B-field measurement only tells you the AC currents that are flowing, not the voltage or power.

 

[berkeman]

Here is a reference with some numbers for the E-field and B-field values on the ground under big AC Mains power TLs:

https://ec.europa.eu/health/scienti...romagnetic-fields07/l-2/7-power-lines-elf.htm

 

[Nils227]

Sorry, I'm having trouble parsing this. I'm guessing that you are saying that if you can measure the B-field at some distance from the 3-phase transmission line and you know the dimensions of the wire separation and the distance to the point where you are making your measurement, that you may be able to calculate the power flowing in the AC Mains transmission line. I think that's true, but you will need to do the full calculation for the TL in order to do that.

No not really. When I know the B-field resulting at a point P (x;y;z) near a current carrying conductor (due to the passage of a current in that conductor) I can reciprocally know the initial amount of the current, causing hence the creation of that B-field. It is simply the "how" to calculate that B-field having the following inputs:

a. AC current max value
b. geometry of the current carrying conductor (i.e., thickness, stranded configuration, aluminum, etc.)
c. distance from that current carrying conductor

 

[Nils227]

Here is a reference with some numbers for the E-field and B-field values on the ground under big AC Mains power TLs:

https://ec.europa.eu/health/scienti...romagnetic-fields07/l-2/7-power-lines-elf.htm

Thank you but I only need a mathematical model that actually calculates the B-field (I am not interested at this stage to derive a relation between the B-field exposure and human induced diseases)

 

[berkeman]

Thank you but I only need a mathematical model that actually calculates the B-field

I think you have what you need now, no? What is your mathematical background?

 

[berkeman]

(I am not interested at this stage to derive a relation between the B-field exposure and human induced diseases)

BTW, I didn't post that link for that purpose (at least not primarily). I posted it so that once you develop your equations, you can do a "reality check" against the geometry and measurements in that link.

 

[Nils227]

I think you have what you need now, no? What is your mathematical background?

No. The link that you have provided contains 0 equations. Accordingly I cannot say that I have obtained any mathematical-based information. Considering that I have a null mathematical background, I still can be sure that the link that you have provided hasn't got anything to do with maths

 

[berkeman]

No. The link that you have provided contains 0 equations.

That latest link, as I said, was for the purpose of reality checking your final equations. It gives typical B-field values near powerlines.

Considering that I have a null mathematical background, I still can be sure that the link that you have provided hasn't got anything to do with maths

Well, if you have no math background this will be more difficult. My post #4 did contain equations that you can use to do this calculation. So you're saying you want someone else to do this calculation for you? Do you have anybody else working with you on this "project" who has more of a math background?

 

[Nils227]

Thank you for your time and consideration dear berkeman,

"Your" post #4 contains merely a concrete form of Biot-Savart law. If you carefully pay attention to the title "your" Fig. 12.3.1, "your" equation is ONLY valid for calculating the magnetic field emitted around a single straight wire.
I simply want to calculate the magnetic field emitted around a stranded wire composed of multi sub-single wires.
Without any mathematical knowledge, it is clear enough that "your" equations cannot be applied to my concern, which was stated clearly from the beginning of this post.

Thank you again

 

[berkeman]

I simply want to calculate the magnetic field emitted around a stranded wire composed of multi sub-single wires.

Stranded versus solid wire makes no difference for this calculation.

 

[Nils227]

Indeed, it does. each stranded wire results in its unique magnetic field. The eventual (total) magnetic field is the superposition for each of the sub-generated magnetic fields (due to each wire).

 

[berkeman]

I believe one of the assumptions in the link in post #4 is that the wire is thin with respect to the distance away that you are measuring the field. In most situations (especially on the ground under powerlines), this is the case.

 

[Nils227]

Thank you again berkeman,

I did not fully understand the phrase: "is that the wire is thin with respect to the distance away that you are measuring the field". Would be greatly if you could kindly rephrase

 

[berkeman]

It just means that we don't have to integrate over the thickness of the wire. You can see that the problem gets a lot more complicated if you are calculating the field at a distance from the wire that is on the same order as the thickness of the wire. If you are more like 100 wire thicknesses away instead, you can just assume that all the current is flowing in a single thin line.

 

[Nils227]

Thank you for this amazing explanation, berkeman.

What about when I am a centimeter away than the wire which has 70cm of thickness?

 

[berkeman]

What about when I am a centimeter away than the wire which has 70cm of thickness?

A cm away from the surface? Then the situation is much more complicated. Is that one of your scenarios that you want to calculate?

 

[Nils227]

Exactly: I would like to calculate the magnetic field, resulting from the passage of an AC current, in a stranded wire of 70cm, from a distance (with respect to the line) of 1cm.
Thank you

 

[berkeman]

in a stranded wire of 70cm

That's giant! Where in the world do you have such a wire? One of the suspension cables on a bridge or something?

https://www.history.com/topics/landmarks/golden-gate-bridge

 

[Nils227]

That's giant! Where in the world do you have such a wire? One of the suspension cables on a bridge or something?

View attachment 330105
https://www.history.com/topics/landmarks/golden-gate-bridge

Sorry berkeman, the wire thickness is of 70mm2 (7cm2), as bad as can be in math :p

 

[Nils227]

The intended mathematical model is similar to the one in the publication: DOI:10.1051/itmconf/20181901010
But instead of a ribbon busbar (as object geometry) : a stranded wire