Rectangular Waveguide MCQ Quiz - Objective Question with Answer for Rectangular Waveguide - Download Free PDF

Explanation:

Waveguide and Guide Wavelength

A waveguide is a physical structure that guides electromagnetic waves from one point to another. It is commonly used for high-frequency signal transmission, such as microwave and RF (radio frequency) signals.

The guide wavelength refers to the wavelength of the signal within the waveguide, which differs from the wavelength in free space due to the boundary conditions imposed by the waveguide structure.

Guide Wavelength Formula:

The relationship between the guide wavelength (λ_g), the free space wavelength (λ₀), and the cutoff wavelength (λ_c) is given by the formula:

λ_g = λ₀ / √(1 - (λ₀ / λ_c)²)

Where:

• λ_g: Guide wavelength
• λ₀: Free space wavelength (c/f, where c is the speed of light and f is the frequency)
• λ_c: Cutoff wavelength of the waveguide, determined by its dimensions and the operating mode

From this equation, it is evident that λ_g is always longer than λ₀ when the operating frequency is above the cutoff frequency (ensuring wave propagation).

The correct option is: Option 1: Guide wavelength is longer than free space wavelength.

Concept:

The cut-off frequency of a rectangular waveguide in dominant TE₁₀ mode is given by:

, where:

c = speed of light = , and a = broader dimension of the waveguide

Calculation:

Given:

Dimensions of waveguide = 1 cm × 0.5 cm

So, a = 1 cm = 0.01 m

Using the formula,

Correct Answer: 2) 15 GHz

Concept:

For distortionless transmission line, we have

Characteristic impedance:

   ---(1)

Attenuation constant:

       --- (2)

Calculation:

From equation (1) and (2), we get

Putting the values of R and Z0, we get

Concept:

Cutt off frequency of TE_mn / TMmn mode

Where;

c → Speed of light

a,b → Dimensions of walls

Calculation:

GIven;

a = 2b

Cut off frequency for TE₀₂ mode

f_C02 = 12 Ghz

a = 2 × 2.5 = 5 cm

Cut off frequency for TM₁₁ mode

Concept:

The dominant mode in a particular waveguide is the mode having the lowest cut-off frequency.

The cut-off frequency for a rectangular waveguide with dimension ‘a (length)’ and ‘b (width)’ is given as:

'm' and 'n' represents the possible modes.

Application:

For a standard rectangular waveguide a > b;

The minimum frequency is obtained when m = 1 and n = 0,

• For a rectangular waveguide with b > a, TE₀₁ will be the dominant mode with the lowest cut-off frequency.
• TEM mode cannot exist in Hollow conductor waveguide
• Circular and rectangular are hollow waveguides hence there is no TEM mode in them, they can support only TE or TM but not TEM mode
• The transmission line, parallel plate waveguide, coaxial cable can have TEM mode

Waveguides only allow frequencies above cut-off frequency and do not pass below the cut-off frequencies.

Hence it acts as a high pass filter.

The cut off frequency is given as:

Where a and b are the dimensions of the waveguide (a>b)

m and n are mode numbers TE_mn

In a dispersive medium, the group velocity is less than the phase velocity only.

Derivation:

Phase velocity is defined as:

β is the phase constant defined as:

Using  where c =  speed of light, the above expression becomes:

Also 

Using , we get:

Since -1 ≤ cos θ ≤ 1

∴ Vp > c

Group velocity is given by:

Vg = c cos θ

Vg 

Conclusion: 

The phase velocity is always greater than the speed of light and group velocity is always less than the speed of light. Hence, the group velocity is less than the phase velocity 

Extra Information:

For a Non-dispersive medium:

A one-dimensional wave defined as:

U(x, t) = A0 sin (ωt – kx + ϕ) has a phase angle (θ) of ωt – kx + ϕ

In general, the phase is constant,

i.e. 

Group velocity is defined as:

 .

Dispersion is when the distinct phase velocities of the components of the envelope cause the wave packet to “Spread out” over time.

When there is no dispersion derivative term is 0 and

 Vp = Vg

The wavelength in a waveguide is considered as a wavelength in the direction of wave propagation and its dependence on wave frequency is defined as follows:

Where λ₀ is a wavelength in free space at a given frequency and λ_c stands for the cutoff wavelength for given waveguide dimensions and waveguide mode.

The wavelength of a wave in a waveguide is greater than in free space. The phase velocity is greater than the speed of light.

Concept:

• The dominant mode in a particular waveguide is the mode having the lowest cut-off frequency.
• Since the frequency and wavelength have an inverse relationship given as:

          , the dominant mode is the mode that will have the highest wavelength as well.

The cut-off frequency for a rectangular waveguide with dimension ‘a (length)’ and ‘b (width)’ is given as:

'm' and 'n' represents the possible modes.

• For a rectangular waveguide with b > a, TE01 will be the dominant mode with the lowest cut-off frequency.
• TEM mode cannot exist in Hollow conductor waveguide
• Circular and rectangular are hollow waveguides hence there is no TEM mode in them, they can support only TE or TM but not TEM mode
• The transmission line, parallel plate waveguide, coaxial cable can have TEM mode

Concept:

Degenerate modes

Whenever the two or more modes having the same cut-off frequency, they are said to be degenerate modes.

For a rectangular waveguide, the cutoff frequency is defined as:

m, n: Mode of operation

a, b: Dimensions of the waveguide

In a rectangular waveguide, TEmn and TMmn modes are always degenerate.

The above cutoff frequency is the same for both TE and TM waves.

For TE waves, E_z = 0 and TM waves have H_z = 0.

Concept:

• All electromagnetic waves consist of electric and magnetic fields propagating in the same direction of travel, but perpendicular to each other.
• Along the length of a normal transmission line, both electric and magnetic fields are perpendicular (transverse) to the direction of wave travel. This is known as the principal mode, or TEM (Transverse Electric and Magnetic) mode.
• This mode of wave propagation can exist only where there are two conductors, and it is the dominant mode of wave propagation where the cross-sectional dimensions of the transmission line are small compared to the wavelength of the signal.
• The cutoff frequency of TEM wave is zero. 

A rectangular waveguide is a hollow metal tube with a rectangular cross-section.

The conducting walls of the waveguide confine the electromagnetic fields and thereby guide the electromagnetic wave.

Microwave energies propagate the length of the waveguide by the reflection of its sidewalls.

Concept:

The dominant mode in a particular waveguide is the mode having the lowest cut-off frequency.

The cut-off frequency for a rectangular waveguide with dimension ‘a (length)’ and ‘b (width)’ is given as:

'm' and 'n' represents the possible modes.

c = speed of light = 3 × 1010 cm/s

Calculation:

TE₁₀ mode means m = 1, n = 0

The cut - off frequency of the dominant mode TE10 of the rectangular waveguide is:

Where a is the dimension of the inner broad wall

= 15 mm

Group Velocity:

The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes propagates through space.

Group velocity,

Important Points

Phase Velocity:

The phase velocity of a wave is the rate at which the phase of the wave propagates in space.

Phase velocity