Show that the input impedance of a short-circuited loss-free...

Show that the input impedance of a short-circuited loss-free line of lenght $l$ is given by $$ Z_{i}=i \sqrt{\frac{L_{0}}{C_{0}}} \tan \frac{2 \pi l}{\lambda} $$ and by sketching the variation of the ratio $Z_{i} / \sqrt{L_{0} / C_{0}}$ with $l$, show that for $l$ just greater than $(2 n+1) \lambda / 4, Z_{i}$ is capacitative, and for $l$ just greater than $n \lambda / 2$ it is inductive. (This provides a positive or negative reactance to match another line.)

Show that the input impedance of a short-circuited loss-free line of lenght $l$ is given by
$$
Z_{i}=i \sqrt{\frac{L_{0}}{C_{0}}} \tan \frac{2 \pi l}{\lambda}
$$
and by sketching the variation of the ratio $Z_{i} / \sqrt{L_{0} / C_{0}}$ with $l$, show that for $l$ just greater than $(2 n+1) \lambda / 4, Z_{i}$ is capacitative, and for $l$ just greater than $n \lambda / 2$ it is inductive. (This provides a positive or negative reactance to match another line.)

Key Concepts

Electrical Length and Wavelength

The concept of electrical length, which ties the physical length of the line to the wavelength (?) of the propagating signal, illustrates how periodic variations in impedance occur. By expressing the length in units of ?, one can predict the alternating inductive or capacitive behavior of the line as the length increases.

Reactive Behavior (Inductive and Capacitive Reactance)

Reactive behavior refers to the way the impedance of a transmission line becomes either inductive or capacitive depending on its length relative to the wavelength. If the length is just greater than odd multiples of ?/4, the line exhibits capacitive reactance; conversely, if it is just greater than multiples of ?/2, the line shows inductive reactance. This alternating nature is key for impedance matching in various applications.

Input Impedance of a Short-Circuited Line

For a loss-free transmission line terminated with a short circuit, the input impedance can be derived from the transmission line equations and boundary conditions. This derivation leads to an expression involving the tangent function and reflects how the reactive behavior of the line changes with the line's electrical length.

Transmission Line Theory

Transmission line theory deals with the analysis of electrical signals traveling along a conductor using distributed circuit elements like inductance and capacitance. It provides a framework to understand how signals propagate, reflect, and interact with the load values, with equations that describe voltage and current as functions of position along the line.

Characteristic Impedance

The characteristic impedance of a transmission line, typically given by the square root of the ratio of its distributed inductance (L0) to capacitance (C0), is a fundamental parameter. It represents the impedance seen by a wave traveling along an infinitely long line and is essential for understanding matching conditions and reflections in the line.

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