GATE 2018 EC Q.48: solution and answer

GATE 2018 · ECQ.48 · 2 marks · NAT

The cutoff frequency of $T E_{01}$ mode of an air filled rectangular waveguide having inner dimensions $a$ cm $\times$ $b$ cm $(a > b)$ is twice that of the dominant $T E_{10}$ mode. When the waveguide is operated at a frequency which is 25% higher than the cutoff frequency of the dominant mode, the guide wavelength is found to be 4 cm. The value of $b$ (in cm, correct to two decimal places) is ________.

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Answer and solution

ANSWER

$b = 0.75 cm$ (with $a = 1.5 cm$ ).

Official answer: 0.7 to 0.8Source: Official answer key, GATE 2018 EC (IIT Guwahati)

Cut-off frequencies of an air-filled rectangular guide. For air, $f_{c, mn} = \frac{c}{2} (\frac{m}{a})^{2} + (\frac{n}{b})^{2}$ , where $c = 3 \times 1 0^{8} m/s$ . This gives $f_{c, 10} = \frac{c}{2 a}$ and $f_{c, 01} = \frac{c}{2 b}$ .
Use the given ratio. $f_{c, 01} = 2 f_{c, 10}$ means $\frac{c}{2 b} = 2 \cdot \frac{c}{2 a}$ , so $a = 2 b$ .
Operating frequency. $f = 1.25 f_{c, 10}$ , so $\frac{f _{c, 10}}{f} = \frac{1}{1.25} = 0.8$ . The next modes, $T E_{01}$ and $T E_{20}$ , both cut off at $2 f_{c, 10}$ (because $a = 2 b$ ), which is above $f$ . Only $T E_{10}$ propagates, so the stated guide wavelength belongs to $T E_{10}$ .
Free-space wavelength at $f$ . $λ = \frac{c}{f} = \frac{c}{1.25 ( c /2 a )} = \frac{2 a}{1.25} = 1.6 a$ .
Guide wavelength. $λ_{g} = \frac{λ}{1 - ( f _{c, 10} / f ) ^{2}} = \frac{1.6 a}{1 - 0.64} = \frac{1.6 a}{0.6} = \frac{8 a}{3}$
Solve for $a$ . $\frac{8 a}{3} = 4 cm$ , so $a = 1.5 cm$ .
Find $b$ . $b = \frac{a}{2} = 0.75 cm$ .

KEY IDEA

In an air-filled guide, $f_{c, 10} = c /2 a$ and $f_{c, 01} = c /2 b$ , so the cut-off ratio gives $a = 2 b$ . The guide wavelength is $λ_{g} = λ / 1 - (f_{c} / f)^{2}$ . With $f = 1.25 f_{c}$ we get $f_{c} / f = 0.8$ and a denominator of exactly $0.6$ . Writing $λ = c / f$ in terms of $a$ leaves one unknown.

WATCH OUT FOR

One slip is to take $f_{c} / f = 1.25$ , or $0.75$ ("25% higher" misread), instead of $1/1.25 = 0.8$ . Another is to set the free-space wavelength, rather than the guide wavelength, equal to 4 cm, which gives $a = 2.5$ cm and $b = 1.25$ cm. A third is to stop at $a = 1.5$ cm and report it, when the question asks for $b = a /2$ .

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GATE 2018 EC Q.48: solution and answer

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