Definitions

# Characteristic impedance

The characteristic impedance or surge impedance of a uniform transmission line, usually written $Z_0$, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm. The characteristic impedance of a lossless transmission line is purely real, that is, there is no imaginary component ($Z_0=|Z_0|+j0$). Characteristic impedance appears like a resistance in this case, such that power generated by a source on one end of an infinitely long lossless transmission line is dissipated through the line but is not dissipated in the line itself. A transmission line of finite length (lossless or lossy) that is terminated at one end with a resistor equal to the characteristic impedance ($Z_mathrm\left\{L\right\} = Z_0$) appears like an infinitely long transmission line to the source.

## Transmission line model

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

$Z_0=sqrt\left\{frac\left\{R+jomega L\right\}\left\{G+jomega C\right\}\right\}$
where
$R$ is the resistance per unit length,
$L$ is the inductance per unit length,
$G$ is the conductance of the dielectric per unit length,
$C$ is the capacitance per unit length,
$j$ is the imaginary unit, and
$omega$ is the angular frequency.

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

$frac\left\{V^+\right\}\left\{I^+\right\} = Z_0 = -frac\left\{V^-\right\}\left\{I^-\right\}$
where the superscripts $+$ and $-$ represent forward- and backward-traveling waves, respectively.

## Lossless line

For a lossless line, R and G are Zero so the equation for characteristic impedance reduces to

$Z_0 = sqrt\left\{frac\left\{L\right\}\left\{C\right\}\right\}$.

In electric power transmission, the characteristic impedance of a transmission line is expressed in terms of the surge impedance loading (SIL), or natural loading, being the MW loading at which reactive power is neither produced nor absorbed:

$mathit\left\{SIL\right\}=frac\left\{V_mathrm\left\{L-L\right\}\right\}\left\{Z_0\right\}$
in which $V_mathrm\left\{L-L\right\}$ is the line-to-line voltage in volts.

Loaded below its SIL, a line supplies lagging reactive power to the system, tending to raise system voltages. Above it, the line absorbs reactive power, tending to depress the voltage. The Ferranti effect describes the voltage gain towards the remote end of a very lightly loaded (or open ended) transmission line. Underground cables normally have a very low characteristic impedance, resulting in an SIL that is typically in excess of the thermal limit of the cable. Hence a cable is almost always a source of lagging reactive power.

## References

• Guile, A. E. (1977). Electrical Power Systems. ISBN 0-0802-1729-X.
• Pozar, D. M. (2004). Microwave Engineering. 3rd edition, ISBN 0-471-44878-8.
• Ulaby, F. T. (2004). Fundamentals Of Applied Electromagnetics. media edition, ISBN 0-13-185089-X.

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