Coaxial cable

Figure 13.1: coaxial line

Characteristic impedance

The characteristic impedance of a coaxial line can be calculated as follows:

$$Z_L = \dfrac{Z_{F0}}{2\pi\cdot\sqrt{\varepsilon_r}}\cdot\ln{\left(\dfrac{D}{d}\right)}$$ (13.1)

Losses

Overall losses in a coaxial cable consist of dielectric and conductor losses. The dielectric losses compute as follows:

$$\alpha_d = \dfrac{\pi}{c_0}\cdot f\cdot \sqrt{\varepsilon_r} \cdot \tan{\delta}$$ (13.2)

The conductor (i.e. ohmic) losses are specified by

$$\alpha_c = \dfrac{1}{2}\cdot \sqrt{\varepsilon_r} \cdot\left(\dfr... ... + \dfrac{1}{d}}{\ln{\left(\dfrac{D}{d}\right)}}\right)\cdot\dfrac{R_S}{Z_{F0}}$$ (13.3)

with $R_S$ denoting the sheet resistance of the conductor material, i.e. the skin resistance

$$R_S = \sqrt{\pi\cdot f\cdot \mu_r \cdot \mu_o \cdot \rho}$$ (13.4)

Cutoff frequencies

In normal operation a signal wave passes through the coaxial line as a TEM wave with no electrical or magnetic field component in the direction of propagation. Beyond a certain cutoff frequency additional (unwanted) higher order modes are excited.

$$f_{TE} \approx \dfrac{c_0}{\pi\cdot\left(D + d\right)} \;\;\;\;\rightarrow\;\;\;\; \textrm{TE(1,1) mode}$$ (13.5)

$$f_{TM} \approx \dfrac{c_0}{2\cdot\left(D - d\right)} \;\;\;\;\rightarrow\;\;\;\; \textrm{TM(n,1) mode}$$ (13.6)