Carson bandwidth rule: Difference between revisions
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{{About|impedance in electrical circuits|impedance of electromagnetic waves|Wave impedance|characteristic acoustic impedance|Acoustic impedance}} | |||
[[File:TransmissionLineDefinitions.svg|thumb|310px|A [[transmission line]] drawn as two black wires. At a distance ''x'' into the line, there is current [[phasor]] ''I(x)'' traveling through each wire, and there is a [[voltage difference]] phasor ''V(x)'' between the wires (bottom voltage minus top voltage). If <math>Z_0</math> is the '''characteristic impedance''' of the line, then <math>V(x) / I(x) = Z_0</math> for a wave moving rightward, or <math>V(x)/I(x) = -Z_0</math> for a wave moving leftward.]] | |||
[[File:Transmission line schematic.svg|thumb|Schematic representation of a [[electrical circuit|circuit]] where a source is coupled to a [[electrical load|load]] with a [[transmission line]] having characteristic impedance <math>Z_0</math>.]] | |||
The '''characteristic impedance''' or '''surge impedance''' of a uniform [[transmission line]], usually written Z<sub>0</sub>, is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in the absence of [[Reflections of signals on conducting lines|reflections]] in the other direction. 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 [[Electrical impedance|impedance]] is the [[Ohm (unit)|ohm]]. | |||
The characteristic impedance of a lossless transmission line is purely [[resistive]], with no [[Electrical reactance|reactive]] component. Energy supplied by a source at one end of such a line is transmitted through the line without being 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. That is to say that, properly terminated, the end of a transmission line produces no reflections. | |||
== Transmission line model == | |||
[[File:Transmission line element.svg|thumb|Schematic representation of the elementary components of a transmission line.]] | |||
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: | |||
:<math>Z_0=\sqrt{\frac{R+j\omega L}{G+j\omega C}}</math> | |||
where | |||
:<math>R</math> is the [[Electrical resistance|resistance]] per unit length, considering the two conductors to be [[in series]], | |||
:<math>L</math> is the [[inductance]] per unit length, | |||
:<math>G</math> is the [[Electrical conductance|conductance]] of the dielectric per unit length, | |||
:<math>C</math> is the [[capacitance]] per unit length, | |||
:<math>j</math> is the [[imaginary unit]], and | |||
:<math>\omega</math> 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 [[Phasor (electronics)|phasor]]s on the line are related by the characteristic impedance as: | |||
:<math>\frac{V^+}{I^+} = Z_0 = -\frac{V^-}{I^-}</math> | |||
where the superscripts <math>+</math> and <math>-</math> represent forward- and backward-traveling waves, respectively. A surge of energy on a finite transmission line will see an impedance of ''Z''<sub>0</sub> prior to any reflections arriving, hence ''surge impedance'' is an alternative name for characteristic impedance. | |||
== Lossless line == | |||
For a lossless line, ''R'' and ''G'' are both zero, so the equation for characteristic impedance reduces to: | |||
:<math>Z_0 = \sqrt{\frac{L}{C}}</math> | |||
The imaginary term ''j'' has also canceled out, making ''Z<sub>0</sub>'' a real expression, and so is purely resistive. | |||
== Surge impedance loading == | |||
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 power loading at which [[reactive power]] is neither produced nor absorbed: | |||
:<math>\mathit{SIL}=\frac{{V_\mathrm{LL}}^2}{Z_0}</math> | |||
in which <math>V_\mathrm{LL}</math> is the line-to-line [[voltage]] in [[volts]]. | |||
Loaded below its SIL, a line supplies 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 [[cable]]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 reactive power. | |||
==See also== | |||
* [[Ampère's circuital law]] | |||
* [[Electrical impedance]] | |||
* [[Maxwell's equations]] | |||
* [[Transmission line]] | |||
* [[Wave impedance]] | |||
== References == | |||
*{{cite book | |||
| last = Guile | |||
| first = A. E. | |||
| title = Electrical Power Systems | |||
| year = 1977 | |||
| isbn = 0-08-021729-X }} | |||
*{{cite book | |||
| last = Pozar | |||
| first = D. M. | |||
| title = Microwave Engineering | |||
| edition = 3rd edition | |||
|date=February 2004 | |||
| isbn = 0-471-44878-8 }} | |||
*{{cite book | |||
| last = Ulaby | |||
| first = F. T. | |||
| title = Fundamentals Of Applied Electromagnetics | |||
| edition = media edition | |||
| year = 2004 | |||
| publisher = Prentice Hall | |||
| isbn = 0-13-185089-X }} | |||
== External links == | |||
{{FS1037C}} | |||
{{DEFAULTSORT:Impedance, Characteristic}} | |||
[[Category:Electricity]] | |||
[[Category:Physical quantities]] | |||
[[Category:Distributed element circuits]] |
Revision as of 22:25, 12 December 2013
29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.
The characteristic impedance or surge impedance of a uniform transmission line, usually written Z0, is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in the absence of reflections in the other direction. 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 resistive, with no reactive component. Energy supplied by a source at one end of such a line is transmitted through the line without being 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. That is to say that, properly terminated, the end of a transmission line produces no reflections.
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.
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.
Surge impedance loading
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 power loading at which reactive power is neither produced nor absorbed:
in which is the line-to-line voltage in volts.
Loaded below its SIL, a line supplies 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 reactive power.
See also
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534