**Subject :**Microwave Engineering

## TE Modes in Circular Waveguides

It is commonly assumed that the waves in a circular waveguide are propagating in the positive z direction. Figure shows the coordinates of a circular guide.

**Figure : Coordinates of a circular waveguide
**

The TE_{np} modes in the circular guide are characterized by E_{z} = O. This means that the z component of the magnetic field H_{z} must exist in the guide in order to have electromagnetic energy transmission. A Helmholtz equation for H_{z} in a circular guide is given by :

...(1)

Its solution is given by :

...(2)

which is subject to the given boundary conditions.

For a lossless dielectric, Maxwell's curl equations in frequency domain are given by :

....(3)

In cylindrical coordinates, their components are expressed as :

...(4)

When the differentiation ∂/∂z is replaced by (-jβ_{g}) and the z component of electric field E_{z} by zero, the TE-mode equations in terms of H_{z} in a circular waveguide are expressed as :