A parallel plate waveguide having plate separation b = 14.1mm is partially filled with two lossless dielectrics with permittivities ɛr1 = 2 and ɛr2 = 1.05.How many TM modes can propagate through the guide at the frequency f0?a)Oneb)Twoc)Threed)FourCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

As we have determined in the previous question, the value of f₀ is

f₀ = 12.8 GHz

and the cut-off frequency for TM₁ mode is

(f_c)₁ = 7.52 GHz

So, the cut-off frequency for TM₂ mode will be

(f_c)₂ = 2 (f_c)₁ = 15.04 GHz

Since, the operating frequency f₀ is below the cut-off frequency for TM₂ mode, so, TM₂ mode or the higher modes cannot propagate at the frequency f₀. Therefore, only one mode (TM₁) can propagate at the frequency f₀ through the waveguide.

Answer

Given data:
- Plate separation, b = 14.1 mm
- Permittivity of dielectric 1, εr1 = 2
- Permittivity of dielectric 2, εr2 = 1.05

To find:
- Number of TM modes that can propagate through the guide at frequency f0

Solution:
Step 1: Calculate the cutoff frequency for each dielectric
The cutoff frequency for a waveguide is given by the formula:

fc = c / (2 * b * sqrt(εr))

Where:
- fc is the cutoff frequency
- c is the speed of light in vacuum (3 * 10^8 m/s)
- b is the plate separation
- εr is the relative permittivity of the dielectric

For dielectric 1:
fc1 = c / (2 * b * sqrt(εr1))
= (3 * 10^8 m/s) / (2 * 14.1 * 10^-3 m * sqrt(2))
= 2.007 GHz

For dielectric 2:
fc2 = c / (2 * b * sqrt(εr2))
= (3 * 10^8 m/s) / (2 * 14.1 * 10^-3 m * sqrt(1.05))
= 3.028 GHz

Step 2: Calculate the guided wavelength for each dielectric
The guided wavelength for a waveguide is given by the formula:

λg = λ0 / √(1 - (fc / f0)^2)

Where:
- λg is the guided wavelength
- λ0 is the free-space wavelength (c / f0)
- fc is the cutoff frequency
- f0 is the operating frequency

For dielectric 1:
λg1 = λ0 / √(1 - (fc1 / f0)^2)

For dielectric 2:
λg2 = λ0 / √(1 - (fc2 / f0)^2)

Step 3: Determine the number of TM modes
The number of TM modes that can propagate through the guide is given by the formula:

M = (2 * m + 1) * (2 * n + 1)

Where:
- M is the number of TM modes
- m and n are positive integers

For a TM mode, the guided wavelength must be less than twice the plate separation:

λg < 2='' *='' />

We can substitute the expressions for λg1 and λg2 in the inequality and solve for the number of TM modes.

Conclusion:
In this case, only one TM mode can propagate through the guide at the frequency f0.

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