2.10: 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. RF and microwave engineers are, of course, electrical engineers who learned how to design circuits using lumped elements. Distributed circuits can be made functionally equivalent to lumped-element circuits, at least over a narrow frequency range. The most important technique for establishing this is equating the \(ABCD\) parameters of the distributed circuit and those of a lumped-element circuit. As will be seen, this technique will be exploited many times in the rest of this book. Distributed circuits enable an RF designer to realize functions that often have superior performance to their lumped-element analogs. At least at microwave frequencies, the loss of lumped elements can be significantly higher than those of distributed elements. Superior performance can be obtained by exploiting the intrinsic behavior of distributed structures. Sometimes functionality can only be conceived of using transmission-line structures, then, often, a low-frequency lumped-element equivalent can be developed. Exploiting the functionality of distributed structures requires a solid understanding of the behavior, modeling, and circuits that can be realized using transmission lines. The technology covered in this chapter provides the theoretical underpinning for all of the other topics covered in this book.

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.

\[\begin{array}{lll}{\text{Reflection coefficient of a load}}&{\text{Load impedance in terms of} }&{\text{Input reflection coefficient of}}\\{ \text{impedance } Z_{L}:}&{\text{reflection coefficient }\Gamma :}&{\text{a lossless line of length }\ell}\\{\Gamma =\Gamma^{V}=\frac{Z_{L}-Z_{\text{REF}}}{Z_{L}+Z_{\text{REF}}}\quad (2.3.6)}&{Z_{L}=Z_{\text{REF}}\frac{1+\Gamma}{1-\Gamma}\quad (2.3.9)}&{\Gamma_{\text{in}}=\Gamma_{L}e^{-\jmath 2\beta\ell}\quad (2.3.16)}\\ {\text{Input reflection coefficient of}}&{\text{Input impedance of a lossless}} &{\text{Input impedance of a lossy}} \\ {\text{a lossy line of length }\ell} &{\text{line}} &{\text{line}} \\ {\begin{aligned}\Gamma_{\text{in}}&=\Gamma_{L}e^{-2\gamma\ell} \\ &=\Gamma_{L}e^{-2\alpha\ell}e^{-2\jmath\beta\ell}\quad (2.5.1) \end{aligned} }&{Z_{\text{in}}=Z_{0}\frac{Z_{L}+\jmath Z_{0}\tan\beta\ell}{Z_{0}+\jmath Z_{L}\tan\beta\ell}\quad (2.3.18)}&{Z_{\text{in}}=Z_{0}\frac{Z_{L}+Z_{0}\tanh\gamma\ell}{Z_{0}+Z_{L}\tanh\gamma\ell}\quad (2.5.17)}\\{\text{Reflection coefficient in terms}} &{\text{VSWR in terms of reflection}}&{} \\ {\text{of VSWR}}&{\text{coefficient}}&{} \\ {|\Gamma|=\frac{\text{VSWR}-1}{\text{VSWR}+1}\quad (2.3.24)}&{\text{VSWR}=\frac{(1+|\Gamma|)}{(1-|\Gamma|)}\quad (2.3.23)}&{}\end{array}\nonumber \]