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GATE 2025 EE – Question 52

Electromagnetic Fields · Inductance, Magnetomotive force, Reluctance, Magnetic circuits, Self and Mutual inductance of simple configurations · 2 marks · Multiple select

Consider two coupled circuits, having self-inductances $L_1$ and $L_2$, that carry non-zero currents $I_1$ and $I_2$, respectively. The mutual inductance between the circuits is $M$ with unity coupling coefficient. The stored magnetic energy of the coupled circuits is minimum at which of the following value(s) of $\dfrac{I_1}{I_2}$ ?

  1. $-\dfrac M{L_1}$
  2. $-\dfrac M{L_2}$
  3. $-\dfrac{L_1}M$
  4. $-\dfrac{L_2}M$

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Correct answer: (A) $-\dfrac M{L_1}$; (D) $-\dfrac{L_2}M$

Explanation

With $x=I_1/I_2$, $W=\tfrac12I_2^2\left(L_1x^2+2Mx+L_2\right)$. Setting $dW/dx=0$ gives $x=-M/L_1$ (a minimum since $L_1>0$). Unity coupling means $M^2=L_1L_2$, so $-\dfrac M{L_1}=-\dfrac{L_2}{M}$. Both options A and D are the same value.