Three Magnetically Coupled Coils

The frame shown on the left is made of material with magnetic permeability \(\mu >> \mu_0\). The magnetic field in each segment of the frame is uniform and parallel to that segment. Coil circle has \(2 N\) turns, coil square has \(N\) turns, and coil triangle has \(N / 2\) turns, where \(N >> 1\) is an even integer. The dimensions of each segment of the frame are listed in the table below, where \(L >> \sqrt{A}\):

SegmentLengthCross-sectional Area
\(IM\)\(4 L\)\(A\)
\(JN\)\(4 L\)\(2 A\)
\(KO\)\(4 L\)\(A\)
\(IJ\)\(2.5 L\)\(A\)
\(JK\)\(2.5 L\)\(A\)
\(MN\)\(2.5 L\)\(A\)
\(NO\)\(2.5 L\)\(A\)

The coils are connected as shown on the right to form a single equivalent inductance. The equivalent inductance can be expressed as:

\(\frac{X}{Y}\frac{\mu N^2 A}{L}\)

where \(X\) and \(Y\) are coprime positive integers. Determine \(X + Y\).


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