# Deriving currents of parallel mutual inductors

This is possibly just a math question, not sure.

In Irwin's textbook "Basic Engineering Circuit Analysis," 7e, there's a quick derivation for current in parallel inductors I don't follow:

The KVL equations are fine and make sense but the solutions for I1 and I2 don't. I generate a half page of algebra (when trying with substitution and subtraction of the linear systems of equations for V) and don't get there and Wolfram Alpha also doesn't come up with that simple of an answer. Is there a simple algebraic answer here?

• I believe that's figure (e) on the bottom right? Look at that circuit instead of (d). It's the same circuit just drawn differently. – KingDuken Oct 9 '18 at 4:06

\begin{align} j\omega L_1I_1 + j\omega MI_2 &= j\omega MI_1 + j\omega L_2I_2 \\ &\Downarrow \\ L_1I_1 + MI_2 &= MI_1 + L_2I_2 \\ &\Downarrow \\ (L_1 - M)I_1 &= (L_2 - M)I_2 \end{align}
So in order to find the relationship between $$\V\$$ and $$\I_1\$$, you use the substitution
\begin{align} V &= j\omega L_1I_1 + j\omega MI_2\\ &\Downarrow\\ V &= j\omega L_1I_1 + j\omega M\cdot \frac{L_1 - M}{L_2 - M}I_1 \\ &= j\omega I_1 \left(L_1 + M\frac{L_1 - M}{L_2 - M} \right) \\ &= j\omega I_1 \left( \frac{L_1(L_2 - M) + M(L_1 - M)}{L_2 - M} \right) \\ &= j\omega I_1 \left(\frac{L_1L_2 - M^2}{L_2 - M}\right) \end{align}
You can do similar steps to find the relationship $$\V\$$ and $$\I_2\$$.