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GATE 2018 EE – Question 36

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

A transformer with toroidal core of permeability $\mu$ is shown in the figure. Assuming uniform flux density across the circular core cross-section of radius $r\ll R$, and neglecting any leakage flux, the best estimate for the mean radius $R$ is

Diagram for GATE 2018 EE question 36
  1. $\dfrac{\mu Vr^2N_P^2\omega}{I}$
  2. $\dfrac{\mu Ir^2N_PN_S\omega}{V}$
  3. $\dfrac{\mu Vr^2N_P^2\omega}{2I}$
  4. $\dfrac{\mu Ir^2N_P^2\omega}{2V}$

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Correct answer: (D) $\dfrac{\mu Ir^2N_P^2\omega}{2V}$

Explanation

The primary inductance is $L=\frac{\mu N_P^2A}{2\pi R}=\frac{\mu N_P^2r^2}{2R}$. With $i_p=I\sin\omega t$, $v_p=L\frac{di}{dt}=L\omega I\cos\omega t$, so $V=L\omega I$. So $R=\frac{\mu N_P^2r^2\omega I}{2V}$.