Circular waveguide

Cylindrical waveguide

• The method of solution of Maxwell’s equation for circular waveguide for obtaining the field configuration is similar to rectangular waveguide.
• Here we use cylindrical coordinate to solve the Maxwell equation.
• In figure a circular waveguide with radius of cross section a is shown.

Transverse electric (TE) wave

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

• Boundary conditions

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

• Here J_n (x) is Bessel function.
• The accepted solution for circular waveguide of Bessel equation is

• For a given value of l, above equation has infinite number of solution for ak’ = 0.
• Let m^th solution of this equation be k’ = s_lm / a.

H_z (r, θ) = J_l (r s_lm / a) [A_l cos l θ + B_l sin l θ]

• Since the transverse electric field intensity is

• By using above equations we can determine the values of E_r (r, θ), E_θ (r, θ), H_r (r, θ) and H_θ (r, θ) easily.

Cut off frequency and wavelength

Guide wavelength

• The wave propagating through the guide have a wavelength, known as guide wavelength.

Transverse magnetic (TM ) wave