### Double Circuit Inductance and Capacitance

Introduction

In last two articles we developed some useful concepts on transmission line inductance and capacitance. There we discussed the formulas used for calculation of inductance and capacitance of transmission lines for different arrangement of conductors. I advise you to go through those two topics in archive (above links) before proceeding further. In those two articles our discussion was restricted to single phase or one three phase line (single circuit line).  Here we will calculate the inductance and capacitance of double circuit line which you can extend to multi-circuits.
Double circuit GMD and GMR

In high voltage transmission system you will often find transmission towers carrying two or more circuits. In fact numerous multi-circuit lines are already in use. The conductors of circuits are arranged in different configuration depending on the  tower types used. We will calculate inductance for a double circuit line using the formula.

Although the two circuits are in parallel, the inductance of double circuit can not be found by imagining two inductances(for two circuits) in parallel. Actually this is true only  when the magnetic field of one circuit does not link with other circuit conductors.  Hence that is the case when each circuit runs on separate tower and the separation between them is such so that there is negligible or no magnetic field interaction between the circuits.

Due to the closeness of the circuits being on the same tower, the  magnetic and electric field interaction of one circuit on the other requires modified calculation of GMD and GMR.

See the Figure-A below for a double circuit arrangement on a self supporting lattice tower. Here we have chosen the conductors arranged vertically which will simplify our calculation and we will derive the inductance for this arrangement where the phase conductors are also assumed as transposed. As we discuss the topic you will realize that the method can be applied to other arrangements or multi-circuits also.

In the figure one circuit phase conductors are a-b-c and other circuit phase conductors are a'-b'-c'. As shown here  the phase conductors are single rounded conductors.

In the last two articles we have already discussed bundled conductors for inductance and capacitance calculation. Here the double circuit is treated very similar to single circuit or a three phase line with bundled conductors. The conductors a-a' are imagined as bundle conductor for phase A, similarly b-b' and c-c' are imagined as bundle conductors for phases B and C. Of course here the bundle sub-conductors are far away and not bunched unlike the case of twin, triple or quad conductors bundles.

Now we are ready to apply the general formula for calculating Geometric Mean Distance(GMD). For calculating GMD all the distances between the phase conductors are identified. In Fig-A(ii) all the distances between the phases are shown. For example a-b, a-c, a-c', a-b', c-b', c-a' etc.. There are 12 possible distances between conductors of phases as shown. It should be noted that for calculation of GMD the distances a-a', b-b' and c-c' are not taken.

GMD = (Dab . Dab' . Dbc . Dbc' . Dca . Dca' . Da'b . Da'b' . Db'c . Db'c' . Dc'a . Dc'a')1/12

Clearly there are 12 distances and so 12th root of the product of twelve distances are taken.

By rearranging the terms you can also write GMD as below.

GMD = (DAB . DBC . DCA.)1/3

Where DAB = (Dab . Dab' . Da'b . Da'b')1/4   similarly DBC and DCA.

In our example

GMD = (6 . 10 . 6 . 10 . 12 . 8 . 10 . 6 . 10 . 6 . 8 . 12)1/12  meter
= 8.37  meter.

Now let us calculate the Geometric Mean Radius (GMR)

For GMR calculation the method is just similar to bundle conductors. Of course for inductance calculation we require r' and for capacitance r. As already said in previous articles the equivalent radius r' =0.7788r .
Here let us first calculate the GMR of each phase separately for inductance.

GMR= √(Daa' . r')

GMR= √(Dbb' . r')

GMR= √(Dcc' . r')

Daa'  is the distance between the conductors a and a'. Similarly for Dbb'   and  Dcc'

As the phases are transposed so,
GMRL = (GMRa  . GMR. GMRc)1/3

Note: Daa'  Dbb'   and  Dcc'   were not used in GMD calculation but used in GMRL  calculation.
Subscript L is used for GMR of inductance calculation and subscript C for capacitance.

Putting the values of GMD and GMRL   in the equation

L = 2 * 10 -7 ln ( GMD/GMR)

we obtain inductance per meter per phase

In above calculation for GMRL the conductor is circular of radius r. But in actual practice ACSR conductors are mostly used, so for inductance calculation r' is replaced by Ds as supplied by the manufacturer of ACSR conductor.
For capacitance calculation the radius of conductor is used in GMRC formula whether it is one circular conductor or ACSR conductor, for calculation of capacitance,

GMR= √(Daa' . r)

GMR= √(Dbb' . r)

GMR= √(Dcc' . r)

GMRC = (GMRa  . GMR. GMRc)1/3

putting the value of  GMR in the equation,
Cn = 2pk / ln (GMD/ GMRc)

We obtain the value of capacitance ( Cn ) per meter between phase and neutral .
k is the permittivity of air and GMD is already calculated for inductance.

Now you can put the numerical values to get a feel of the Cn so obtained.