GATE 2023 IN Q.55: solution and answer

GATE 2023 · INQ.55 · 2 marks · NAT

A sinusoidal current of $i_{1} (t) = 1 sin (200 π t)$ mA is flowing through a 4 H inductor which is mutually coupled to another 5 H inductor carrying

i_{2} (t) = 2 sin (200 π t) mA

as shown in the figure. The coupling coefficient between the inductors is 0.6. The peak energy stored in the circuit is ________ $μ$ J (rounded off to two decimal places).

GATE 2023 IN question 55: Two coils side by side, drawn vertically. — Figure from the GATE 2023 IN paper. It is part of the question.

Answer and solution

ANSWER

The peak energy stored in the coupled inductors is about 17.37 μJ.

Official answer: 17.00 to 18.00Source: Official answer key, GATE 2023 IN (IIT Kanpur)

Dot convention: in the figure, $i_{1}$ enters the dotted (top) end of the 4 H coil and $i_{2}$ enters the dotted (top) end of the 5 H coil. Both currents enter dots, so the mutual fluxes aid each other and the mutual energy term carries a plus sign, $+ M i_{1} i_{2}$ .
Mutual inductance from the coupling coefficient: $M = k L_{1} L_{2} = 0.6 4 \times 5 = 0.6 20 = 2.683 H$
Instantaneous energy stored in a coupled pair (both currents entering dots): $w (t) = \frac{1}{2} L_{1} i_{1}^{2} + \frac{1}{2} L_{2} i_{2}^{2} + M i_{1} i_{2}$
The two currents are in phase: $i_{1} = 1 sin (200 π t) mA$ and $i_{2} = 2 sin (200 π t) mA$ . So every term is proportional to $sin^{2} (200 π t)$ , and the mutual term $M i_{1} i_{2}$ is never negative. The energy is therefore largest when $sin^{2} (200 π t) = 1$ , the instant when $i_{1} = 1 mA$ and $i_{2} = 2 mA$ at the same time.
Put in the peak values: $w_{m a x} = \frac{1}{2} (4) (1 0^{- 3})^{2} + \frac{1}{2} (5) (2 \times 1 0^{- 3})^{2} + (2.683) (1 0^{- 3}) (2 \times 1 0^{- 3}) = 2 μ J + 10 μ J + 5.367 μ J$
Add the three terms: $w_{m a x} = 17.366 μ J \approx 17.37 μ J$ .

KEY IDEA

The energy stored in coupled inductors is $w = \frac{1}{2} L_{1} i_{1}^{2} + \frac{1}{2} L_{2} i_{2}^{2} \pm M i_{1} i_{2}$ , with $M = k L_{1} L_{2}$ . Use the plus sign when both currents enter (or both leave) their dotted ends. Here the two currents are in phase, so all three terms peak together, when both currents are at their peak values.

WATCH OUT FOR

Using the minus sign for the mutual term gives $12 - 5.37 = 6.63 μ J$ . The right sign comes from reading the dots correctly: both currents enter dotted ends here. Other slips: leaving out the mutual term entirely (12 μJ), or taking $M = k L_{1} L_{2}$ or $k (L_{1} + L_{2}) /2$ instead of $k L_{1} L_{2}$ . Some also use rms currents, but the question asks for the peak energy, which needs the peak currents.

From the GATE 2023 IN paper. Every question of this paper · Download the full paper · All 6 Coupled circuits questions, with notes

GATE 2023 IN Q.55: solution and answer

Answer and solution

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