Question 2.37: Write a set of coupled equations for the coupled circuit sho...
Write a set of coupled equations for the coupled circuit shown in (a). The iron has a cross-sectional area of A and a relative permeability µr . An equivalent circuit is given in (b).
In this circuit, the mean distances between the intersection points are given by L1, L2 and L3.
From figure (a) , it is possible to draw an equivalent electrical circuit (b) that contains the following elements
R_{1}=\frac{L_{1}}{\mu_{r} \mu_{0} A}, R_{2}=\frac{L_{2}}{\mu_{r} \mu_{0} A}, R_{3}=\frac{L_{3}}{\mu_{r} \mu_{0} A}, F_{1}=N_{1} I_{1} \text {, and} F_{2}=N_{2} I_{2} .
Due to the winding of the wire about the material and the direction of the current in these wires, the magnetic flux from both coils will be in a direction such that the magnetic fluxes will add in the center conductor. Hence, we write from our magnetic circuit using Kirchhoff’s law that
\begin{aligned} &F_{1}=\Psi_{1}\left(R_{1}+R_{2}\right)+\Psi_{2} R_{2} \\&F_{2}=\Psi_{1} R_{2}+\Psi_{2}\left(R_{2}+R_{3}\right) \end{aligned}