TM modes in Circular Waveguides
The TMnp modes in a circular guide are characterized by Hz =0, However, the z component of the electric field Ez must exist in order to have energy transmission in the guide. Consequently, the Helmholtz equation for Ez in a circular waveguide is given by :
Its solution is given by :
which is subject to the given boundary conditions.
The boundary condition requires that the tangential component of electric field Ez at r = a vanishes. Consequently :
SinceJn(kcr) are oscillatory functions, there are infinite numbers of roots of Jn(kcr).
For Hz = 0 and djaz = - jβg, the field equations in the circular guide, after expanding
are given by
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 TMnp modes in a circular waveguide:
where Zg = Er/ Hφ, = - Eφ/Hr = βg/(ωε) and kc = Xnp/a have been replace for the wave impedance in the guide and where n = 0,1,2,3, ... and p = 1,2,3,4,.