### Inductance of Composite Conductor Lines

We will consider a single phase 2 wire system. It consists of two conductors say P and Q which are composite conductors. The arrangement of conductors is shown in the Fig. 1. Fig. 1

Conductor P is consisting of x identical, parallel filaments. Each of the filament carries a current of I/x. Conductor Q consists of y filament with each filament carrying a current of - I/y. The conductor Y carries a current of I amps in opposite direction to the current in conductor X as it is forming return path.
The flux linkages of filament say a due to all currents in all the filaments is given by
The inductance of filament a is given by,
The inductance of filament b is given by,
The average inductance of the filaments of conductor P is
The conductor P consists of x number of parallel filaments. If all the filaments are equal inductances then inductance of the conductor would be 1/x times inductance of one filament. All the filaments have different inductances but the inductance of all of them in parallel is 1/x times the average inductance.
Inductance of conductor P is given by,
Substituting the values of La, Lb ..... L in the equation and simplifying the expression we have,
In the above expression the numerator of argument of logarithm is the xy, the root of xy terms. These terms are nothing but products of distance from all the x filaments of conductor P to all the y filaments of the conductor Q.
For each filament in conductor P there are y distances to filaments in conductor Q and there are x filaments in conductor P. The xy terms are formed as a result of product of y distances for each of x filaments. The xyth root of the product of the xy distances is called the geometric mean distance between conductor P and Q. It is termed as Dm or GMD and is called mutual GMD between the conductors.
The denominator of the above expression is the x2 root of x2 terms. There are x filaments and for each filament there are x terms consisting of r' (denoted by Daa, Dbb etc) for that filament times the distance from that filament to every other filament in conductor P.
If we consider the distance Daa then it is the distance of the filament from itself which is also denoted as ra'. This r' of a separate filament is called the self GMD of the filament. It is also called geometric mean radius GMR and identified as Ds.
Thus the above expression now becomes
Comparing this equation with the expression obtained for inductance of a single phase two wire line. The distance between solid conductors of single conductors line is substituted by the GMD between conductors of the composite conductor line. Similarly the GMR (r') of the single conductor is replaced by GMR of composite conductor.
The composite conductors are made up of number of strands which are in parallel. The inductance of composite conductor Q is obtained in a similar manner. Thus the inductance of the line is,
L = Lp + LQ

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