GATE 2020 IN Q.54: solution and answer

GATE 2020 · INQ.54 · 2 marks · NAT

In the Maxwell-Wien bridge shown, the detector D reads zero when $C_{1} = 100 nF$ and $R_{1} = 100 k Ω$ . The Q factor of the coil is ____

GATE 2020 IN question 54: A four-arm bridge drawn as a diamond with a top node, a bottom node, a left node and a right node. — Figure from the GATE 2020 IN paper. It is part of the question.

Answer and solution

ANSWER

The Q factor of the coil is 10.

Official answer: 10Source: Official answer key, GATE 2020 IN (IIT Delhi)

Identify the arms. The source $2 cos (1000 t) V$ feeds the top and bottom nodes, and D sits between the left and right nodes. The opposite pairs are therefore ( $R_{1} ∥ C_{1}$ , coil) and ( $R_{2}$ , $R_{4}$ ).
Balance condition (D reads zero): the products of opposite-arm impedances are equal, $Z_{1} Z_{x} = R_{2} R_{4}$ , where $\frac{1}{Z _{1}} = \frac{1}{R _{1}} + j ω C_{1}$ and $Z_{x} = R_{x} + j ω L_{x}$ .
Solve for the coil impedance: $R_{x} + j ω L_{x} = \frac{R _{2} R _{4}}{Z _{1}} = R_{2} R_{4} (\frac{1}{R _{1}} + j ω C_{1})$
Equate real parts: $R_{x} = \frac{R _{2} R _{4}}{R _{1}}$ . Equate imaginary parts: $ω L_{x} = ω R_{2} R_{4} C_{1}$ , so $L_{x} = R_{2} R_{4} C_{1}$ .
Q factor of the coil: $Q = \frac{ω L _{x}}{R _{x}} = \frac{ω R _{2} R _{4} C _{1}}{R _{2} R _{4} / R _{1}} = ω C_{1} R_{1}$ $R_{2}$ and $R_{4}$ cancel, which is why the question does not give their values.
The source $2 cos (1000 t)$ gives $ω = 1000 rad/s$ . This is the angular frequency, not $1000 Hz$ .
$Q = 1000 \times (100 \times 1 0^{- 9}) \times (100 \times 1 0^{3}) = 10$

KEY IDEA

At balance in a Maxwell-Wien (inductance-capacitance) bridge, the coil's $L_{x} / R_{x}$ ratio equals $C_{1} R_{1}$ of the parallel RC arm opposite it. So $Q = ω L_{x} / R_{x} = ω C_{1} R_{1}$ depends only on $ω$ , $C_{1}$ and $R_{1}$ , and the ratio arms $R_{2}$ , $R_{4}$ cancel.

WATCH OUT FOR

Using $ω = 2 π \times 1000$ , which gives $Q \approx 62.8$ . The source is written as $cos (1000 t)$ , so $1000$ is already $ω$ in rad/s. A second mistake is to treat $R_{1}$ and $C_{1}$ as a series combination instead of a parallel one, which gives $Q = 1/ (ω C_{1} R_{1}) = 0.1$ .

From the GATE 2020 IN paper. Every question of this paper · Download the full paper · All 14 Bridges questions, with notes

GATE 2020 IN Q.54: solution and answer

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