Single-Phase Double Circuit Transmission Line

[ProtoMan]

Homework Statement

There are 4 wires with diameter d arranged in a way such that they form a square shape, with the wires on the corners. The horizontal distance is kD while the vertical distance is D. Prove that the inductance per meter of each conductor is

1/2 + 2ln { [2kd√(1 + k^2)] / d } x10^-7 H/m

Homework Equations

2x10^-7 ln(GMD / GMR)

The Attempt at a Solution

I managed to get the 1/2 +2ln part but I do not know how to get the GMD and GMR. The wires has no labels like A1,A2,B1,B2. I searched the net and all I can see is three-phase double circuit. I tried getting the GMD this way by using the six distance available:

GMD = {(D)(D)(kD)(kD)[D√(1 + k^2)][D√(1 + k^2)]} ^ 1/6

but I'm just stuck afterwards since I don't know how to get GMR,

 

[Baluncore]

Welcome to PF.

Four wire RF transmission lines are usually driven with diagonal wires cross connected.

References to GMD are hard to find. They were originally used by J C Maxwell.
There is an excellent reference to the use of GMD in; "The Theory and Practice of Absolute Measurements in Electricity and Magnetism". By Andrew Gray. 1893. Volume 2. Section 2. Calculation of coefficients of induction. Start at page 288. Volume 1 is mainly pictures of equipment. Now out of copyright, you should find a .pdf copy via; https://archive.org/search.php?query=

Let us know how you get on.

 

[Babadag]

In my opinion the inductance it is the magnetic flux produced by the currents of opposite sense that means between 1 and 2 and 1 to 3 [if the forward currents are 1 and 3 and return currents are 2 and 4.] divided by current. If we consider solid wire GMR=d/2*.7788=r*e^(-1/4)
L1_2=2*10^(-7)*LN(2*kD/d/0.7788) or L1_2=2*10^(-7)*[LN(2*kD/d)+1/4)
L1_4=2*10^(-7)*[LN(2*D*√(1+k^2)/d)+1/4)
Total L1_2+L1_4= 2*10^(-7)*{ LN[2*kD^2*√(1+k^2)/d]+1/2)}

 

[Babadag]

 

[Baluncore]

This is a good site for info on GMD and inductance calculation.
http://www.g3ynh.info/zdocs/magnetics/part_1.html

See this thesis "Study of the Method of Geometric Mean Distances Used in Inductance Calculations."
http://scholarsmine.mst.edu/masters_theses/6747/

Also; Geometric Mean Distance, Its derivation and application in inductance calculations, Robert Weaver. 2016.
http://electronbunker.ca/eb/CalcMethods4a.htmlEarly Books and Papers on the Electromagnetic Field, some include use and discussion of GMD..

This is where it started.
1864. A Dynamical Theory of the Electromagnetic Field. By J. Clerk Maxwell.
https://ia600500.us.archive.org/0/items/philtrans03147378/03147378.pdf

1873. A Treatise on Electricity and Magnetism, Vol 1. Maxwell J.C. https://ia902302.us.archive.org/25/items/ATreatiseOnElectricityMagnetism-Volume1/Maxwell-ATreatiseOnElectricityMagnetismVolume1_text.pdf

1873. A Treatise on Electricity and Magnetism, Vol 2. Maxwell J.C. https://ia600304.us.archive.org/9/items/ATreatiseOnElectricityMagnetism-Volume2/Maxwell-ATreatiseOnElectricityMagnetismVolume2_text.pdf

This paper explains how electrons travel on the wires, while energy travels externally in the guided EM fields. According to Oliver Lodge, Poynting recognised this from Maxwells equations.
1883. On the Transfer of Energy in the Electromagnetic Field. J.H.Poynting.
https://ia800303.us.archive.org/22/items/philtrans03617950/03617950.pdf

1889. Modern Views of Electricity. Oliver J. Lodge. 3rd Edn 1907.
https://ia600301.us.archive.org/21/items/cu31924031233061/cu31924031233061.pdf

1891. The Electromagnet and Electromagnetic Mechanism. Silvanus Phillips Thompson.

1893. Absolute Measurements in Electricity and Magnetism. Vol 1. Andrew Gray. (Vol 1 is instruments).

Excellent on GMD, Section II, Calculation of Coefficients of Induction. Starts on page 288.
1893. Absolute Measurements in Electricity and Magnetism. Vol 2. Andrew Gray. (Vol 2 is theory).
https://archive.org/download/in.ernet.dli.2015.503668/2015.503668.Absolute-Measurements_text.pdf

GMD calculations are covered here in Chapter XIII.
1921. Absolute Measurements in Electricity and Magnetism. (Revised). Andrew Gray.
https://ia600200.us.archive.org/9/items/absolutemeasurem00grayuoft/absolutemeasurem00grayuoft.pdf

1898. A Treatise on Magnetism and Electricity. Vol 1. Andrew Gray.
https://archive.org/download/treatiseonmagnet030894mbp/treatiseonmagnet030894mbp.pdf
1898. A Treatise on Magnetism and Electricity. Vol 2. Andrew Gray.

 

[Babadag]

I think the following may be useful also:
http://tkne.net/downloads/power/transmissionlines1/transmission%20lines/OVERHEAD.pdf
I forgot to subtract F1_3 and to multiply by I/2.So the actual formulae are[I hope]:
we have to subtract 1_3 flux indeed:
F1_2=2*10^(-7)*I/2[LN(2*kD/d)+1/4]
F1_4=2*10^(-7)*I/2[LN(2*D*√(1+k^2)/d)+1/4]
F1_3=2*10^(-7)*I/2[LN(2*D/d)+1/4]
Total F1=2*10^(-7)*I*LN√[2*kD*√1+k^2)/d']
L1=F1/(I)=2*10^(-7)*LN√[2*kD*√1+k^2)/d+1/8]
If GMD=√ [kD*D√(1+k^2)] and GMR=√d’/2*D d'=d*e^-1/4=d*0.7788 for solid wire then
L1=2/10^7*LN{√ [kD*D√(1+k^2)]/ √d’/2*D]
L1=2*10^(-7)*[LN√[2kD*√(1+k^2)/d]+1/8)

 

[Baluncore]

In my opinion the inductance it is the magnetic flux produced by the currents of opposite sense that means between 1 and 2 and 1 to 3 [if the forward currents are 1 and 3 and return currents are 2 and 4.]

A single-phase double-circuit transmission-line would not be balanced unless the diagonally opposite wires were tied together.