Question 2.8: USING THE WHEELER INCREMENTAL INDUCTANCE RULE TO FIND THE AT...
USING THE WHEELER INCREMENTAL INDUCTANCE RULE TO FIND THE ATTENUATION CONSTANT Calculate the attenuation due to conductor loss of a coaxial line using the Wheeler incremental inductance rule.
From (2.32)\ Z_{0}=\frac{V_{o}}{I_{o}} =\frac{E_{\rho }\ln b/a}{2\pi H_{\phi }} =\frac{\eta \ln b/a}{2\pi } =\sqrt{\frac{\mu }{\epsilon } } \frac{\ln b/a}{2\pi } ,
the characteristic impedance of the coaxial line is
\ Z_{0}=\frac{\eta }{2\pi } \ln \frac{b}{a} .
From the incremental inductance rule of the form given in (2.106), the attenuation due to conductor loss is
\ \alpha _{c}=\frac{R_{s}}{2Z_{0}\eta } \frac{dZ_{0}}{dl} =\frac{R_{s}}{4\pi Z_{0}} \left(\frac{d\ln b/a}{db} -\frac{d\ln b/a}{da}\right) =\frac{R_{s}}{4\pi Z_{0}}\left(\frac{1}{a}+\frac{1}{b} \right) ,which is seen to be in agreement with the result of Example 2.7. The negative sign on the second differentiation in this equation is because the derivative for the inner conductor is in the -ρ direction (receding wall).