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Characteristic impedance

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Property of an electrical circuit

This article is about the impedance characterizing transmission lines. For impedance of electromagnetic waves, seeWave impedance. For characteristic acoustic impedance, seeAcoustic impedance.

Atransmission line drawn as two black wires. At a distancex into the line, there is currentphasorI(x) traveling through each wire, and there is avoltage difference phasorV(x) between the wires (bottom voltage minus top voltage). IfZ₀ is thecharacteristic impedance of the line, thenV(x)/I(x) =Z₀ for a wave moving rightward, orV(x)/I(x) = −Z₀ for a wave moving leftward.

Schematic representation of acircuit where a source is coupled to aload with atransmission line having characteristic impedanceZ₀

Thecharacteristic impedance orsurge impedance (usually writtenZ₀) of a uniformtransmission line is the ratio of the amplitudes ofvoltage andcurrent of a wave travelling in one direction along the line in the absence ofreflections in the other direction. Equivalently, it can be defined as theinput 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. TheSI unit of characteristic impedance is theohm.

The characteristic impedance of a lossless transmission line is purelyreal, with noreactive component (seebelow). Energy supplied by a source at one end of such a line is transmitted through the line without beingdissipated in the line itself. A transmission line of finite length (lossless or lossy) that is terminated at one end with animpedance equal to the characteristic impedance appears to the source like an infinitely long transmission line and produces no reflections.

Transmission line model

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The characteristic impedanceZ(ω) of an infinite transmission line at a given angular frequencyω is the ratio of the voltage and current of a pure sinusoidal wave of the same frequency travelling along the line. This relation is also the case for finite transmission lines until the wave reaches the end of the line. Generally, a wave is reflected back along the line in the opposite direction. When the reflected wave reaches the source, it is reflected yet again, adding to the transmitted wave and changing the ratio of the voltage and current at the input, causing the voltage-current ratio to no longer equal the characteristic impedance. This new ratio including the reflected energy is called theinput impedance of that particular transmission line and load.

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.

Schematic ofHeaviside's model of aninfinitesimal segment of transmission line

Applying the transmission line model based on thetelegrapher's equations as derived below,^[1] the general expression for the characteristic impedance of a transmission line is: $Z_{0}={\sqrt {{\frac {R+j\omega L}{G+j\omega C}}\,}}$ where

R is theresistance per unit length, considering the two conductors to bein series,
L is theinductance per unit length,
G is theconductance of the dielectric per unit length,
C is thecapacitance per unit length,
j is theimaginary unitj² = −1, and
ω is theangular frequency.

This expression extends to DC by lettingω tend to 0.

A surge of energy on a finite transmission line will see an impedance ofZ₀ prior to any reflections returning; hencesurge impedance is an alternative name forcharacteristic 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 currentphasors on the line are related by the characteristic impedance as: $Z_{\text{0}}={\frac {V_{(+)}}{I_{(+)}}}=-{\frac {V_{(-)}}{I_{(-)}}}$ 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.

Derivation

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Using the telegrapher's equation

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Main article:telegrapher's equation

Consider one section of the transmission line for the derivation of the characteristic impedance. The voltage on the left would beV and on the right side would beV + dV. This figure is to be used for both the derivation methods.

The differential equations describing the dependence of thevoltage andcurrent on time and space are linear, so that a linear combination of solutions is again a solution. This means that we can consider solutions with a time dependence $e^{j\omega t}$ . Doing so allows to factor out the time dependence, leaving an ordinary differential equation for the coefficients, which will bephasors, dependent on position (space) only. Moreover, the parameters can be generalized to be frequency-dependent.^[2]^[3]

Consider asteady-state problem such that the voltage and current can be written as: ${\begin{aligned}v(x,t)&=V(x)e^{j\omega t}\\[.5ex]i(x,t)&=I(x)e^{j\omega t}\end{aligned}}$ Take the positive direction forV andI in the loop to be clockwise. Substitution in the telegraph equations and factoring out the time dependence $e^{j\omega t}$ now gives: ${\begin{aligned}{\frac {\mathrm {d} V}{\mathrm {d} x}}&=-\left(R+j\omega L\right)I=-ZI,\\[.5ex]{\frac {\mathrm {d} I}{\mathrm {d} x}}&=-\left(G+j\omega C\right)V=-YV,\end{aligned}}$ with impedanceZ andadmittanceY. Derivation and substitution of these twofirst-order differential equations results in two uncoupled second-order differential equations: ${\begin{aligned}{\frac {\mathrm {d} ^{2}V}{\mathrm {d} x^{2}}}&=k^{2}V,\\[.5ex]{\frac {\mathrm {d} ^{2}I}{\mathrm {d} x^{2}}}&=k^{2}I,\end{aligned}}$ withk² =ZY =(R +jωL)(G +jωLC) andk =α +jβ called thepropagation constant.

The solution to these types of equations can be written as: ${\begin{aligned}V(x)&=Ae^{-kx}+Be^{kx}\\[.5ex]I(x)&=A_{1}e^{-kx}+B_{1}e^{kx}\end{aligned}}$ withA,A₁,B andB₁ theconstants of integration. Substituting these constants in the first-order system gives: ${\begin{aligned}A_{1}&={\hphantom {-}}A{\frac {k}{R+j\omega L}}\\[.5ex]B_{1}&=-B{\frac {k}{R+j\omega L}}\end{aligned}}$ where ${\frac {A}{A_{1}}}=-{\frac {B}{B_{1}}}={\frac {R+j\omega L}{k}}={\sqrt {\frac {R+j\omega L}{G+j\omega C}}}={\sqrt {\frac {Z}{Y}}}=Z_{0}.$ It can be seen that the constantZ₀, 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 thecharacteristic impedance of the transmission line.^[1]

The general solution of the telegrapher's equations can now be written as: ${\begin{aligned}v(x,t)&=V(x)e^{j\omega t}=Ae^{-\alpha x}e^{j(\omega t-\beta x)}+Be^{\alpha x}e^{j(\omega t+\beta x)}\\[.5ex]i(x,t)&=I(x)e^{j\omega t}={\frac {A}{Z_{0}}}e^{-\alpha x}e^{j(\omega t-\beta x)}-{\frac {B}{Z_{0}}}e^{\alpha x}e^{j(\omega t+\beta x)}\end{aligned}}$ Both the solution for the voltage and the current can be regarded as a superposition of two travelling waves in thex₍₊₎ andx₍₋₎ directions.

For typical transmission lines, that are carefully built from wire with low loss resistanceR and small insulation leakage conductanceG; further, used for high frequencies, the inductive reactanceωL and the capacitive admittanceωC will both be large. In those cases, thephase constant and characteristic impedance are typically very close to being real numbers: ${\begin{aligned}\beta &\approx \omega {\sqrt {LC}}\\[.5ex]Z_{0}&\approx {\sqrt {\frac {L}{C}}}\end{aligned}}$ Manufacturers make commercial cables to approximate this condition very closely over a wide range of frequencies.

As a limiting case of infinite ladder networks

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Intuition

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Iterative impedance of an infinite ladder of L-circuit sections

Iterative impedance of the equivalent finite L-circuit

Consider an infiniteladder network consisting of a series impedanceZ and a shunt admittanceY. Let its input impedance beZ_IT. If a new pair of impedanceZ and admittanceY is added in front of the network, its input impedanceZ_IT remains unchanged since the network is infinite. Thus, it can be reduced to a finite network with one series impedanceZ and two parallel impedances1/Y andZ_IT. Its input impedance is given by the expression^[4]^[5] $Z_{\mathrm {IT} }=Z+\left({\frac {\ 1\ }{Y}}\parallel Z_{\mathrm {IT} }\right)\$ which is also known as itsiterative impedance. Its solution is: $Z_{\mathrm {IT} }={Z \over 2}\pm {\sqrt {{Z^{2} \over 4}+{Z \over Y}}}\$

For a transmission line, it can be seen as alimiting case of an infinite ladder network withinfinitesimal impedance and admittance at a constant ratio.^[6]^[5] Taking the positive root, this equation simplifies to: $\ Z_{\mathrm {IT} }={\sqrt {{\frac {\ Z\ }{Y}}\ }}\$

Derivation

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Using this insight, many similar derivations exist in several books^[6]^[5] and are applicable to both lossless and lossy lines.^[7]

Here, we follow an approach posted by Tim Healy.^[8] The line is modeled by a series of differential segments with differential series elements (R dx,L dx) and shunt elements (C dx,G dx) (as shown in the figure at the beginning of the article). The characteristic impedance is defined as the ratio of the input voltage to the input current of a semi-infinite length of line. We call this impedanceZ₀. That is, the impedance looking into the line on the left isZ₀. But, of course, if we go down the line one differential lengthdx, the impedance into the line is stillZ₀. Hence we can say that the impedance looking into the line on the far left is equal toZ₀ in parallel withC dx andG dx, all of which is in series withR dx andL dx. Hence: ${\begin{aligned}&Z_{0}=(R+j\omega L)\operatorname {d} \!x+{\frac {1}{(G+j\omega C)\operatorname {d} \!x+{\frac {1}{Z_{0}}}}}\\&Z_{0}=(R+j\omega L)\operatorname {d} \!x+{\frac {Z_{0}}{Z_{0}(G+j\omega C)\operatorname {d} \!x+1}}\\&Z_{0}+Z_{0}^{2}(G+j\omega C)\operatorname {d} \!x=(R+j\omega L)\operatorname {d} \!x+Z_{0}(G+j\omega C)\operatorname {d} \!x\,(R+j\omega L)\operatorname {d} \!x+Z_{0}\end{aligned}}$

The addedZ₀ terms cancel, leaving $Z_{0}^{2}(G+j\omega C)\operatorname {d} \!x=(R+j\omega L)\operatorname {d} \!x+Z_{0}(G+j\omega C)(R+j\omega L)(\mathop {} \!\operatorname {d} \!x)^{2}$

The first-powerdx terms are the highest remaining order. Dividing out the common factor ofdx, and dividing through by the factor(G +jωC), we get $Z_{0}^{2}={\frac {(R+j\omega L)}{(G+j\omega C)}}+Z_{0}(R+j\omega L)\operatorname {d} \!x.$

In comparison to the factors whosedx divided out, the last term, which still carries a remaining factordx, is infinitesimal relative to the other, now finite terms, so we can drop it. That leads to $Z_{0}=\pm {\sqrt {\frac {R+j\omega L}{G+j\omega C}}}\;.$

Reversing the sign± applied to the square root has the effect of reversing the direction of the flow of current.

Lossless line

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The analysis of lossless lines provides an accurate approximation for real transmission lines that simplifies the mathematics considered in modeling transmission lines. A lossless line is defined as a transmission line that has no line resistance and nodielectric loss. This would imply that the conductors act like perfect conductors and the dielectric acts like a perfect dielectric. For a lossless line,R andG are both zero, so the equation for characteristic impedance derived above reduces to: $Z_{0}={\sqrt {{\frac {L}{C}}\,}}\,.$

In particular,Z₀ does not depend any more upon the frequency. The above expression is wholly real, since the imaginary termj has canceled out, implying thatZ₀ is purely resistive. For a lossless line terminated inZ₀, there is no loss of current across the line, and so the voltage remains the same along the line. The lossless line model is a useful approximation for many practical cases, such as low-loss transmission lines and transmission lines with high frequency. For both of these cases,R andG are much smaller thanωL andωC, respectively, and can thus be ignored.

The solutions to the long line transmission equations include incident and reflected portions of the voltage and current: ${\begin{aligned}V&={\frac {V_{r}+I_{r}Z_{c}}{2}}e^{\gamma x}+{\frac {V_{r}-I_{r}Z_{c}}{2}}e^{-\gamma x}\\[1ex]I&={\frac {V_{r}/Z_{c}+I_{r}}{2}}e^{\gamma x}-{\frac {V_{r}/Z_{c}-I_{r}}{2}}e^{-\gamma x}\end{aligned}}$ When the line is terminated with its characteristic impedance, the reflected portions of these equations are reduced to 0 and the solutions to the voltage and current along the transmission line are wholly incident. Without a reflection of the wave, the load that is being supplied by the line effectively blends into the line making it appear to be an infinite line. In a lossless line this implies that the voltage and current remain the same everywhere along the transmission line. Their magnitudes remain constant along the length of the line and are only rotated by a phase angle.

Surge impedance loading

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Inelectric power transmission, the characteristic impedance of a transmission line is expressed in terms of thesurge impedance loading (SIL), or natural loading, being the power loading at whichreactive power is neither produced nor absorbed: $\mathrm {SIL} ={\frac {{V_{\mathrm {LL} }}^{2}}{Z_{0}}}$ in whichV_LL is theroot mean square (RMS) line-to-linevoltage involts.

Loaded below its SIL, the voltage at the load will be greater than the system voltage. Above it, the load voltage is depressed. TheFerranti 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.

Practical examples

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Standard	Impedance (Ω)	Tolerance	Ref.
Category 5	100	±5 Ω	^[9]
USB	90	±15%	^[10]
HDMI	95	±15%	^[11]
IEEE 1394	108	^+3% _−2%	^[12]
VGA	75	±5%	^[13]
DisplayPort	100	±20%	^[11]
DVI	95	±15%	^[11]
PCIe	85	±15%	^[11]
Overhead power line	400	Typical	^[14]
Underground power line	40	Typical	^[14]

The characteristic impedance ofcoaxial cables (coax) is commonly chosen to be50 Ω forRF andmicrowave applications. Coax forvideo applications is usually75 Ω for its lower loss(see alsoNominal impedance § 50 Ω and 75 Ω).

References

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^^a ^b"Derivation of Characteristic Impedance of Transmission line".GATE ECE 2018. 16 April 2016. Archived fromthe original on 9 September 2018. Retrieved9 September 2018.
^Miano, Giovanni; Maffucci, Antonio (2001).Transmission Lines and Lumped Circuits. San Diego: Academic Press. pp. 130–135.ISBN 0-12-189710-9.
^Mooijweer, H. (1971).Microwave Techniques. London: Macmillan Education UK. pp. 74–79.doi:10.1007/978-1-349-01065-3.ISBN 978-1-349-01067-7.
^Feynman, Richard;Leighton, Robert B.;Sands, Matthew."Section 22-6. A ladder network".The Feynman Lectures on Physics. Vol. 2.
^^a ^b ^cLee, Thomas H. (2004). "2.5 Driving-point impedance of iterated structure".Planar Microwave Engineering: A practical guide to theory, measurement, and circuits. Cambridge University Press. p. 44.
^^a ^bFeynman, Richard;Leighton, Robert B.;Sands, Matthew."Section 22-7. Filter".The Feynman Lectures on Physics. Vol. 2.If we imagine the line as broken up into small lengthsΔℓ, each length will look like one section of the L-C ladder with a series inductanceΔL and a shunt capacitanceΔC. We can then use our results for the ladder filter. If we take the limit asΔℓ goes to zero, we have a good description of the transmission line. Notice that asΔℓ is made smaller and smaller, bothΔL andΔC decrease, but in the same proportion, so that the ratioΔL/ΔC remains constant. So if we take the limit of Eq. (22.28) asΔL andΔC go to zero, we find that the characteristic impedancez₀ is a pure resistance whose magnitude is√ΔL/ΔC. We can also write the ratioΔL/ΔC asL₀/C₀, whereL₀ andC₀ are the inductance and capacitance of a unit length of the line; then we have√L₀/C₀.
^Lee, Thomas H. (2004). "2.6.2. Characteristic impedance of a lossy transmission line".Planar Microwave Engineering: A practical guide to theory, measurement, and circuits. Cambridge University Press. p. 47.
^"Characteristic impedance".ee.scu.edu. Archived fromthe original on 2017-05-19. Retrieved2018-09-09.
^"SuperCat OUTDOOR CAT 5e U/UTP"(PDF). Archived fromthe original(PDF) on 2012-03-16.
^"Chapter 2 – Hardware". USB in a NutShell. Beyond Logic.org. Retrieved2007-08-25.
^^a ^b ^c ^d"AN10798 DisplayPort PCB layout guidelines"(PDF).Archived(PDF) from the original on 2022-10-09. Retrieved2019-12-29.
^"Evaluation"(PDF). materias.fi.uba.ar.Archived(PDF) from the original on 2022-10-09. Retrieved2019-12-29.
^"VMM5FL"(PDF). pro video data sheets. Archived fromthe original(PDF) on 2016-04-02. Retrieved2016-03-21.
^^a ^bSingh 2008, p. 212.

Sources

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Guile, A.E. (1977).Electrical Power Systems. Pergamon Press.ISBN 0-08-021729-X.
Pozar, D.M. (February 2004).Microwave Engineering (3rd ed.).ISBN 0-471-44878-8.
Ulaby, F.T. (2004).Fundamentals of Applied Electromagnetics (media ed.). Prentice Hall.ISBN 0-13-185089-X.
Singh, S. N. (23 June 2008).Electric Power Generation: Transmission and Distribution (2 ed.). PHI Learning Pvt. Ltd. p. 212.ISBN 9788120335608.OCLC 1223330325.

External links

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This article incorporatespublic domain material fromFederal Standard 1037C.General Services Administration. Archived fromthe original on 2022-01-22.

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