Question 2.1: TRANSMISSION LINE PARAMETERS OF A COAXIAL LINE The fields of...
TRANSMISSION LINE PARAMETERS OF A COAXIAL LINE The fields of a traveling TEM wave inside the coaxial line of Figure 2.3 can be expressed as
\ \bar{E} =\frac{V_{0}\hat{\rho } }{\rho \ln b/a} e^{-\gamma z}
\ \bar{H} =\frac{I_{0}\hat{\phi } }{2\pi\rho} e^{-\gamma z},
,where γ is the propagation constant of the line. The conductors are assumed to have a surface resistivity R_{s} ,and the material filling the space between the conductors is assumed to have a complex permittivity \ \epsilon =\epsilon ^{\prime } -j\epsilon ^{\prime \prime } and a permeability \ \mu =\mu _{0} \mu _{r}. Determine the transmission line parameters.
From (2.17)\ L=\frac{\mu }{\left|I_{0}\right| ^{2} }\int_{s}^{}{\bar{H}.\bar{H}^{*}ds } H/m.–(2.20)\ G=\frac{\omega \acute{\acute{\epsilon }} }{\left|V_{0}\right| ^{2} }\int_{s}^{}{\bar{E}.\bar{E}^{*}ds }S/m. and the given fields the parameters of the coaxial line can be calculated as
\ L=\frac{\mu }{\left(2\pi \right)^{2} }\int_{\phi =0}^{2\pi }{\int_{\rho =a}^{b}{\frac{1}{\rho ^{2}} } } \rho d\rho d\phi =\frac{\mu }{2\pi } \ln b/a H/m,\ C=\frac{ \epsilon ^{ \prime } }{\left(\ln b/a\right) ^{2} }\int_{\phi =0}^{2\pi }{\int_{\rho =a}^{b}{\frac{1}{\rho ^{2}} } } \rho d\rho d\phi =\frac{2\pi \epsilon ^{\prime } }{\ln b/a } F/m,
\ R=\frac{R_{s} }{\left(2\pi\right) ^{2} }\left\{\int_{\phi =0}^{2\pi }\frac{1}{a^{2} }ad\phi+ {\int_{\phi =0}^{2\pi } \frac{1}{b^{2} } } b d\phi \right\} =\frac{R_{s} }{2\pi } \left(\frac{1}{a}+\frac{1}{b} \right) \Omega /m,
\ G=\frac{\omega \epsilon ^{\prime \prime } }{\left(\ln b/a\right)^{2} }\int_{\phi =0}^{2\pi }{\int_{\rho =a}^{b}{\frac{1}{\rho ^{2}} } } \rho d\rho d\phi=\frac{2\pi \omega\epsilon ^{\prime \prime } }{\ln b/a} S/m.
Table 2.1 summarizes the parameters for coaxial, two-wire, and parallel plate lines. As we will see in the next chapter, the propagation constant, characteristic impedance, and attenuation of most transmission lines are usually derived directly from a field theory solution; the approach here of first finding the equivalent circuit parameters (L, C, R, G) is useful only for relatively simple lines. Nevertheless, it provides a helpful intuitive concept for understanding the properties of a transmission line and relates a transmission line to its equivalent circuit model.
| COAX
|
TWO-WIRE
|
PARALLEL PLATE
|
|
| L | \ \frac{\mu }{2\pi } \ln \frac{b}{a} | \ \frac{\mu }{\pi } \cos h^{-1} \left(\frac{D}{2a} \right) | \ \frac{\mu d}{w } |
| C | \ \frac{2\pi\acute{\epsilon } }{\ln b/a} | \ \frac{\pi\acute{\epsilon } }{ \cos h^{-1}\left(D/2a\right) } | \ \frac{\acute{\epsilon } w }{d} |
| R | \ \frac{R_{s} }{2\pi } \left(\frac{1}{a}+\frac{1}{b} \right) | \ \frac{R_{s} }{\pi a} | \ \frac{2R_{s}}{w } |
| G | \ \frac{2\pi \omega \acute{\acute{\epsilon }} }{\ln b/a} | \ \frac{\pi \omega \acute{\acute{\epsilon } } }{\cos h^{-1}\left(D/2a\right)} | \ \frac{\omega \acute{\acute{\epsilon }}w }{d} |
