Characteristic impedance
Encyclopedia
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
, usually written
, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflectionsReflections of signals on conducting lines
A signal travelling along an electrical transmission line will be partly, or wholly, reflected back in the opposite direction when the travelling signal encounters a discontinuity in the transmission parameters of the line, or at the far end of the line if the line is not correctly terminated in...
. The SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
unit of characteristic impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
is the ohm. The characteristic impedance of a lossless transmission line is purely real, that is, there is no imaginary component (
). 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 transmitted 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 (
) appears to the source like an infinitely long transmission line.Transmission line model
Basic definition:The ratio of voltage applied to the current is called the input impedance; the input impedance of the infinite line is called the characteristic impedance.
where
is the resistanceElectrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
per unit length,
is the inductanceInductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...
per unit length,
is the conductance of the dielectric per unit length,
is the capacitanceCapacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
per unit length,
is the imaginary unitImaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
, and
is the angular frequencyAngular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
.
The voltage and current phasor
Phasor (electronics)
In physics and engineering, a phase vector, or phasor, is a representation of a sine wave whose amplitude and angular frequency are time-invariant. It is a subset of a more general concept called analytic representation. Phasors decompose the behavior of a sinusoid into three independent...
s on the line are related by the characteristic impedance as:
where the superscripts
and 
represent forward- and backward-traveling waves, respectively.Lossless line
For a lossless line, R and G are both zero, so the equation for characteristic impedance reduces to:
The imaginary term j has also canceled out, making Z0 a real expression, and so is purely resistive with a magnitude of
.Surge impedance loading
In electric power transmissionElectric power transmission
Electric-power transmission is the bulk transfer of electrical energy, from generating power plants to Electrical substations located near demand centers...
, the characteristic impedance of a transmission line is expressed in terms of the surge impedance loading (SIL), or natural loading, being the power loading at which reactive power is neither produced nor absorbed:
in which
is the line-to-line voltageVoltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
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
Ferranti effect
The Ferranti effect is a rise in voltage occurring at the receiving end of a long transmission line, relative to the voltage at the sending end, which occurs when the line is energized but there is a very light load or the load is disconnected....
describes the voltage gain towards the remote end of a very lightly loaded (or open ended) transmission line. Underground cable
Cable
A cable is two or more wires running side by side and bonded, twisted or braided together to form a single assembly. In mechanics cables, otherwise known as wire ropes, are used for lifting, hauling and towing or conveying force through tension. In electrical engineering cables are used to carry...
s 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.
See also
- Ampère's circuital law
- Electrical impedanceElectrical impedanceElectrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
- Maxwell's equationsMaxwell's equationsMaxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
- Transmission lineTransmission lineIn communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
- Wave impedanceWave impedanceThe wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields . For a transverse-electric-magnetic plane wave traveling through a homogeneous medium, the wave impedance is everywhere equal to the intrinsic impedance of the medium...