Characteristic impedance

 ( Electricity ) 

The characteristic impedance or surge impedance (usually written Z₀) of a uniform transmission line is the ratio of the amplitudes of voltage and Electric current of a wave travelling in one direction along the line in the absence of reflections in the other direction. Equivalently, it can be defined as the input impedance of a transmission line when its length is infinite. Characteristic impedance is determined by the geometry and materials of the transmission line and, for a uniform line, is not dependent on its length. The SI unit of characteristic impedance is the ohm.

The characteristic impedance of a lossless transmission line is purely Real number, with no reactive component (see below). Energy supplied by a source at one end of such a line is transmitted through the line without being Dissipation in the line itself. A transmission line of finite length (lossless or lossy) that is terminated at one end with an impedance equal to the characteristic impedance appears to the source like an infinitely long transmission line and produces no reflections.

The input impedance of an infinite line is equal to the characteristic impedance since the transmitted wave is never reflected back from the end. Equivalently: The characteristic impedance of a line is that impedance which, when terminating an arbitrary length of line at its output, produces an input impedance of equal value. 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 as derived below, the general expression for the characteristic impedance of a transmission line is: $$Z\_0\; =\; \backslash sqrt\{\; \backslash frac\{R\; +\; j\backslash omega\; L\}\{G\; +\; j\backslash omega\; C\}\backslash ,\; \}$$ where

This expression extends to DC by letting $\backslash omega$ tend to 0.

A surge of energy on a finite transmission line will see an impedance of $Z\_0$ prior to any reflections returning; hence surge impedance is an alternative name for characteristic impedance. Although an infinite line is assumed, since all quantities are per unit length, the “per length” parts of all the units cancel, and 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: $$Z\_\backslash text\{0\}\; =\; \backslash frac\{V\_\{(+)\}\}\{I\_\{(+)\}\}\; =-\backslash frac\{V\_\{(-)\}\}\{I\_\{(-)\}\}$$ where the subscripts (+) and (−) mark the separate constants for the waves traveling forward (+) and backward (−). The rightmost expression has a negative sign because the current in the backward wave has the opposite direction to current in the forward wave.

Consider a steady-state problem such that the voltage and current can be written as: Take the positive direction for $V$ and $I$ in the loop to be clockwise. Substitution in the telegraph equations and factoring out the time dependence $\backslash \; e^\{j\; \backslash omega\; t\}$ now gives: with impedance $Z$ and admittance $Y$. Derivation and substitution of these two first-order differential equations results in two uncoupled second-order differential equations: with $k^2\; =\; Z\; Y\; =\; (\; R\; +\; j\; \backslash omega\; L)(G\; +\; j\; \backslash omega\; C\; )=\; (\backslash alpha\; +\; j\; \backslash beta)^2$ and $k\; =\; \backslash alpha\; +\; j\; \backslash beta$ called the propagation constant.

The solution to these types of equations can be written as: with $A$, $A\_1$, $B$ and $B\_1$ the constants of integration. Substituting these constants in the first-order system gives: where $$\backslash frac\{A\}\{A\_1\}\; =\; -\backslash frac\{B\}\{B\_1\}\; =\; \backslash frac\{R\; +\; j\; \backslash omega\; L\}\{k\}\; =\; \backslash sqrt\{\backslash frac\{R+\; j\backslash omega\; L\}\{G\; +\; j\; \backslash omega\; C\}\}\; =\; \backslash sqrt\{\backslash frac\{Z\}\{Y\}\}\; =\; Z\_\{0\}.$$ It can be seen that the constant $Z\_0,$ defined in the above equations has the dimensions of impedance (ratio of voltage to current) and is a function of primary constants of the line and operating frequency. It is called the characteristic impedance of the transmission line.

The general solution of the telegrapher's equations can now be written as: Both the solution for the voltage and the current can be regarded as a superposition of two travelling waves in the $x\_\{+\}$ and $x\_\{-\}$ directions.

For typical transmission lines, that are carefully built from wire with low loss resistance $\backslash \; R\backslash$ and small insulation leakage conductance $\backslash \; G\backslash \; ;$ further, used for high frequencies, the inductive reactance $\backslash \; \backslash omega\; L\backslash$ and the capacitive admittance $\backslash \; \backslash omega\; C\backslash$ will both be large. In those cases, the phase constant and characteristic impedance are typically very close to being real numbers: Manufacturers make commercial cables to approximate this condition very closely over a wide range of frequencies.

Z_0 &= (R + j\ \omega L)\ \operatorname{d}x + \frac{ 1 }{\ (G + j \omega C)\ \operatorname{d}x + \frac{1}{\ Z_0\ }\ } \\[1ex]
Z_0 &= (R + j\ \omega L)\ \operatorname{d}x + \frac{\ Z_0\ }{Z_0\ (G + j \omega C)\ \operatorname{d}x + 1\, } \\[1ex]
Z_0 + Z_0^2\ (G + j\ \omega C)\ \operatorname{d}x &= (R + j\ \omega L)\ \operatorname{d}x + Z_0\ (G + j\ \omega C)\ \operatorname{d}x\ (R + j\ \omega L)\ \operatorname{d}x + Z_0