2.4: Summary

In this chapter a classical treatment of transmission lines was presented. Transmission lines are distributed elements and form the basis of microwave circuits. A distinguishing feature is that they support forward- and backward-traveling waves and they can be used to implement circuit functions.

The most important of the formulas presented in this chapter are listed here. Reflection coefficients are referenced to an impedance \(Z_{0}\), the load impedance is \(Z_{L}\), and a line has a characteristic impedance \(Z_{0}\), physical length \(\ell\), and propagation constant \(\gamma\) (or electrical length in radians of \(\beta\ell\) where \(\ell\) is the physical length of the line.

\[\begin{array}{lll}{\text{Reflection coefficient of a load}}&{\text{Load impedance in terms of} }&{\text{Input reflection coefficient of}}\\{ \text{impedance } Z_{L}}&{\text{reflection coefficient }\Gamma}&{\text{a lossless line of length }\ell}\\{\Gamma =\Gamma^{V}=\frac{Z_{L}-Z_{\text{REF}}}{Z_{L}+Z_{\text{REF}}}\quad (3.3.6)}&{Z_{L}=Z_{\text{REF}}\frac{1+\Gamma}{1-\Gamma}\quad (3.3.8)}&{\Gamma_{\text{in}}=\Gamma_{L}e^{-\jmath 2\beta\ell}\quad (3.3.13)}\\{\text{Reflection coefficient in terms}}&{\text{Input impedance of a lossless}}&{\text{VSWR in terms of reflection}}\\{\text{of VSWR}}&{\text{line}}&{\text{coefficient}}\\{|\Gamma |=\frac{\text{VSWR}-1}{\text{VSWR}+1}\quad(3.3.21)}&{Z_{\text{in}}=Z_{0}\frac{Z_{L}+\jmath Z_{0}\tan\beta\ell}{Z_{0}+\jmath Z_{L}\tan\beta\ell}\quad (3.3.15)}&{\text{VSWR}=\frac{(1+|\Gamma |)}{(1-|\Gamma |)}\quad (3.3.20)}\end{array}\nonumber \]