RF Electronic Devices and Systems

# Coaxial transmission line

A schematic coaxial transmission line is depicted below and widely used both commercially and industrially, especially in connectors.

To obtain the field solution for a coaxial transmission line we have to resolve the Laplace equation for potential with the boundary conditions $\Psi \left(a\right)={V}_{0},\Psi \left(b\right)=0$. The easiest way to do this is to use polar coordinates similar to the ones used in a circular waveguide.

The potential function for a given type of coaxial waveguide is $\Psi \left(\rho ,\phi \right)=\frac{{V}_{0}\mathrm{ln}\frac{b}{\rho }}{in\frac{b}{a}}$. A coaxial line can perform TEM, TE and TM modes. However, in practice the TEM mode is the most common used mode.

Let’s consider the TE mode with , so we must resolve the Laplace equation for ${H}_{Z}$. The solution for ${H}_{Z}$ will be the following: $\mathrm{Hz}\left(\rho ,\phi \right)=\left(\mathrm{Asinn\phi }+\mathrm{Bcosn\phi }\right)\left({\mathrm{CJ}}_{n}‘ \left({k}_{c}\rho \right)+{\mathrm{DY}}_{n}‘ \left({k}_{c}\rho \right)\right){e}^{–\mathrm{j\beta z}}$, where $J$ and $Y$ are Bessel functions. The type of electric field function can be obtained using the boundary conditions. The cut-off number here will be .