Show that the input impedance of the line of Problem 7.5;...

Show that the input impedance of the line of Problem $7.5$; that is, the impedance of the line at $x=0$, is given by $$ \begin{array}{r} Z_{i}=Z_{0}\left(\frac{Z_{0} \sinh \gamma l+Z_{L} \cosh \gamma l}{Z_{0} \cosh \gamma l+Z_{L} \sinh \gamma l}\right) \\ \left(\text { Note }: 2 \cosh \gamma l=\mathrm{e}^{\gamma l}+\mathrm{e}^{-\gamma l}\right. \\ \left.2 \sinh \gamma l=\mathrm{e}^{\gamma l}-\mathrm{e}^{-\gamma l}\right) \end{array} $$

Show that the input impedance of the line of Problem $7.5$; that is, the impedance of the line at $x=0$, is given by
$$
\begin{array}{r}
Z_{i}=Z_{0}\left(\frac{Z_{0} \sinh \gamma l+Z_{L} \cosh \gamma l}{Z_{0} \cosh \gamma l+Z_{L} \sinh \gamma l}\right) \\
\left(\text { Note }: 2 \cosh \gamma l=\mathrm{e}^{\gamma l}+\mathrm{e}^{-\gamma l}\right. \\
\left.2 \sinh \gamma l=\mathrm{e}^{\gamma l}-\mathrm{e}^{-\gamma l}\right)
\end{array}
$$

Key Concepts

Transmission Line Theory

This concept covers the analysis of transmission lines, which are specialized cables or structures designed to carry electrical or electromagnetic signals from one point to another with minimal losses. In this theory, key parameters include the characteristic impedance, propagation constant, and the concept of reflection due to impedance mismatches.

Input Impedance

This is the impedance seen by a source at the input of a transmission line. It is influenced by the line's length, its characteristic impedance, and the load impedance at the end of the line. Determining the input impedance is essential when analyzing how signals propagate and reflect along a transmission line.

Characteristic Impedance

This is an intrinsic property of a transmission line that defines the relationship between voltage and current in a traveling wave on the line. It remains constant along a uniform transmission line and is used to determine how energy is transferred and how mismatches bind at interfaces like load terminations.

Propagation Constant

This parameter governs the exponential growth or decay, and phase change, of the wave along the transmission line. It is typically represented by a complex quantity, where its real part corresponds to attenuation and the imaginary part corresponds to phase shift.

Hyperbolic Function Representation

The use of hyperbolic functions, such as sinh and cosh, in the analysis of transmission lines arises from solving the differential equations governing voltage and current along the line. These functions succinctly capture the behavior of signals on lossy or lossless lines over a finite length, thereby simplifying the representation and calculation of input impedance.

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