**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 $\Psi (a)={V}_{0},\Psi (b)=0$. 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 $\Psi (\rho ,\phi )=\frac{{V}_{0}\mathrm{ln}{\displaystyle \frac{b}{\rho}}}{in\frac{b}{a}}$. 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 ${E}_{Z}=0,{H}_{Z}\ne 0$, so we must resolve the Laplace equation for ${H}_{Z}$. The solution for ${H}_{Z}$ will be the following: $\mathrm{Hz}(\rho ,\phi )=(\mathrm{Asinn\phi}+\mathrm{Bcosn\phi})({\mathrm{CJ}}_{n}\u2018\; ({k}_{c}\rho )+{\mathrm{DY}}_{n}\u2018\; ({k}_{c}\rho )){e}^{\u2013\mathrm{j\beta z}}$, 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 ${k}_{c}=\frac{2}{a+b},{f}_{c}=\frac{{\mathrm{ck}}_{c}}{2\pi \surd \epsilon}$.

The transmission line of parallel planes, stripes and microstripes