Consider an air-filled rectangular waveguide with inside dimensions 5 cm x 2cm. If the wave impedance is 222.24 Ω for TM11 mode of propagation. Then the operating frequency (in GHz)is given byCorrect answer is '10'. Can you explain this answer?

Question

Answer

Calculating the operating frequency for TM11 mode of propagation in the given rectangular waveguide:

Given:
- Inside dimensions of the rectangular waveguide: 5 cm x 2 cm
- Wave impedance for TM11 mode of propagation: 222.24 Ω

Step 1: Determine the cutoff frequency for the TM11 mode
- The cutoff frequency for the TM11 mode can be calculated using the formula:
fc = (c / 2) * sqrt((m/a)^2 + (n/b)^2)
where fc is the cutoff frequency, c is the speed of light in vacuum (3 x 10^8 m/s), m and n are the mode numbers, a and b are the dimensions of the waveguide.
- For TM11 mode, m = 1 and n = 1.
- Substituting the given values into the formula:
fc = (3 x 10^8 / 2) * sqrt((1/0.05)^2 + (1/0.02)^2)
= (1.5 x 10^8) * sqrt(400 + 2500)
= (1.5 x 10^8) * sqrt(2900)
≈ 2.487 x 10^8 Hz

Step 2: Calculate the operating frequency
- The operating frequency can be calculated using the formula:
f = (m/a) * (n/b) * fc
where f is the operating frequency, m and n are the mode numbers, a and b are the dimensions of the waveguide, and fc is the cutoff frequency.
- For TM11 mode, m = 1 and n = 1.
- Substituting the given values into the formula:
f = (1/0.05) * (1/0.02) * 2.487 x 10^8
= 40 * 50 * 2.487 x 10^8
= 4.974 x 10^11 Hz
= 497.4 GHz

Step 3: Convert the operating frequency to GHz
- Dividing the operating frequency by 10^9 to convert it to GHz:
f (GHz) = 497.4 / 10^9
= 0.4974 GHz

Conclusion:
The operating frequency for the TM11 mode of propagation in the given rectangular waveguide is approximately 0.4974 GHz, which can be rounded to 0.5 GHz or 500 MHz. However, the correct answer is '10' GHz, which suggests that there might be an error in the given information or calculation method.

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