RF Design – Why Zo is 50 Ohms? 1 comment

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An air filled coaxial cable designed for maximum power handling has a characteristic impedance of 30\Omega. Whereas, an air-filled coaxial cable designed for lower insertion loss or attenuation(maximum efficiency) has characteristic impedance of 70\Omega. Hence as a compromise between these two factors, around 50\Omega is chosen as a standard for radio communication.

The impedance of a coaxial cable depends on

(1)   \begin{equation*} Z_o = {1 \over 2\pi} {1\over \sqrt{\mu \epsilon}} \ln\left({D \over d} \right) \approx {138 \over \sqrt{\epsilon_r}}\log\left({D \over d} \right) ~~\Omega \end{equation*}

where, D \rightarrow diameter of outer conductor
d \rightarrow diameter of inner conductor
Z_o \rightarrow characteristic impedance of the cable
\epsilon_o \rightarrow permmitivity of free-space
\epsilon_r \rightarrow relative permittivity of dielectric

From Eq.(1), we can conclude that it is the D \over d ratio that defines the characteristic impedance of the cable.

The maximum power handling capacity of air filled coaxial line is set by the dielectric breakdown voltage of air.

Power handling capability of coaxial cable versus it's characteristic impedance

Power handling capability of coaxial cable versus it’s characteristic impedance

For coaxial cable, the electrical field E(r) at a distance r along the radial vector is

(2)   \begin{equation*} E(r) = { V \over r \ln\left({D \over d}\right)} \end{equation*}

The voltage is maximum at the surface of the cable, and is given by

(3)   \begin{equation*} V_p = E_d~d~\ln\left({D \over d}\right) \end{equation*}

where,
E_d \rightarrow dielectric voltage breakdown of air in V/m
V_p \rightarrow Peak voltage at the surface of cable
Therefore, the maximum power handling capacity (P_{max}) of the cable is

(4)   \begin{eqnarray*} P_{max} &=& {V_p^2 \over 2 Z_o} \\ &=&{d^2 E_d^2 [\ln({D \over d})]^2 \over 2 {1 \over 2\pi \sqrt{\mu \epsilon}} \ln({D\over d})} ={\pi D^2 E_d^2 \over {1 \over \sqrt{\mu \epsilon}}} {\ln({D \over d}) \over ({D \over d})^2} \end{eqnarray*}

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