Characteristic Impedance - Transmission Line Model

Transmission Line Model

The characteristic impedance of a transmission line is the ratio of the voltage and current of a wave travelling along the line. When the wave reaches the end of the line, in general, there will be a reflected wave which travels back along the line in the opposite direction. When this wave reaches the source, it adds to the transmitted wave and the ratio of the voltage and current at the input to the line will no longer be the characteristic impedance. This new ratio is called the input impedance. The input impedance of an infinite line is equal to the characteristic impedance since the transmitted wave is never reflected back from the end. It can be shown that an equivalent definition is: the characteristic impedance of a line is that impedance which when terminating an arbitrary length of line at its output will produce an input impedance equal to the characteristic impedance. This is so because there is no reflection on a line terminated in its own characteristic impedance.

Applying the transmission line model based on the telegrapher's equations, the general expression for the characteristic impedance of a transmission line is:

where

is the resistance per unit length, considering the two conductors to be in series,

is the inductance per unit length,

is the conductance of the dielectric per unit length,

is the capacitance per unit length,

is the imaginary unit, and

is the angular frequency.

Although an infinite line is assumed, since all quantities are per unit length, the characteristic impedance is independent of the length of the transmission line.

The voltage and current phasors on the line are related by the characteristic impedance as:

where the superscripts and represent forward- and backward-traveling waves, respectively. A surge of energy on a finite transmission line will see an impedance of Z₀ prior to any reflections arriving, hence surge impedance is an alternative name for characteristic impedance.