**Subject :**Microwave Engineering

## TM modes in Circular Waveguides

The TM_{np} modes in a circular guide are characterized by H_{z} =0, However, the z component of the electric field E_{z }must exist in order to have energy transmission in the guide. Consequently, the Helmholtz equation for E_{z} in a circular waveguide is given by :

.....(1)

Its solution is given by :

....(2)

which is subject to the given boundary conditions.

The boundary condition requires that the tangential component of electric field E_{z }at r = a vanishes. Consequently :

...(3)

SinceJ_{n}(k_{c}r) are oscillatory functions, there are infinite numbers of roots of J_{n}(k_{c}r).

For H_{z} = 0 and djaz = - jβ_{g}, the field equations in the circular guide, after expanding

are given by

...(4)

Differentiation of Eq. (2) with respect to z and substitution of the result in Eqs. (4) and solving the equation (4) yield the field equations of TM_{np} modes in a circular waveguide:

...(5)

where Zg = Er/ Hφ, = - Eφ/Hr = β_{g}/(ωε) and kc = X_{np}/a have been replace for the wave impedance in the guide and where n = 0,1,2,3, ... and p = 1,2,3,4,.