Rectangular waveguide

• A rectangular waveguide is a hollow metallic tube with a rectangular cross section.
• In figure a rectangular waveguide is shown.

• It has cross sectional length a and b and the z-axis is parallel to the axis of the tube.
• Let the conductivity of the metal be infinite and its inner portion be filled with a lossless dielectric medium.

Transverse electric (TE) wave

• For TE mode, E_z (x, y) = 0 everywhere, H_z (x, y) exist.
• So H_z satisfies the wave equation

• Boundary conditions

• The L.H.S. of this equation is only the function of x, and its R.H.S. is only the function of y.
• Since both are equal to each other so they will be separately equal to a constant.

• At x = 0, dX/dx = 0, and at x = a, dX/dx = 0
• At y = 0, dY/dy = 0, and at y = b, dY/dy = 0

• At x = 0, dX/dx = 0

• At x = a, dX/dx = 0

• At y = 0, dY/dy = 0

• At y = b, dY/dy = 0

• Here m and n are two parameters and a set of given values of m, n defines a waveguide mode, known as m, n waveguide mode for TE wave (TE_mn mode)

Cut off frequency and wavelength

• This is cut off wavelength.

• This is cut off frequency.
• The cut off frequency is the frequency at which, and below which no wave propagate through the guide.
• If m = n = 0, then it gives a static field which do not represent a wave propagation, so these values are excluded.
• Thus TE₀₀ mode does not exist.
• If a < b, then the lowest cut off frequency is obtained with m = 0 and n = 1, then

          (ω_c)₀₁= πc / b

• The mode which has lowest cut off frequency is known as principal mode or dominant mode.
• Thus TE₀₁ mode is dominant mode.

Phase velocity and group velocity

• If λ₀ = λ_c , then v_p = ∞
• Thus at cut off frequency the phase velocity become infinite.
• The group velocity is the velocity with which the energy is propagated along the axis of the guide.

• If λ₀ = λ_c , then v_g = 0

Transverse electric and magnetic field intensity

• By using above equations we can determine the values of E_x(x, y), E_y(x, y), H_x(x, y) and H_y(x, y) easily.

Transverse magnetic (TM ) wave