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The **characteristic impedance** or **surge impedance** (usually written Z_{0}) of a uniform transmission line 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. Alternatively and 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.

In radio-frequency engineering, a **transmission line** is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

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 characteristic impedance of the line, or if the far end of the line is not terminated in its characteristic impedance. This can happen, for instance, if two lengths of dissimilar transmission lines are joined together.

The **input impedance** of an electrical network is the measure of the opposition to current (impedance), both static (resistance) and dynamic (reactance), into the load network that is *external* to the electrical source. The input admittance (1/impedance) is a measure of the load's propensity to draw current. The source network is the portion of the network that transmits power, and the load network is the portion of the network that consumes power.

- Transmission line model
- Derivation
- Using telegrapher's equation
- Alternative approach
- Lossless line
- Surge impedance loading
- Practical examples
- Coaxial cable
- See also
- References
- External links

The characteristic impedance of a lossless transmission line is purely real, 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 an impedance equal to the characteristic impedance appears to the source like an infinitely long transmission line and produces no reflections.

In electrical and electronic systems, **reactance** is the opposition of a circuit element to a *change* in current or voltage, due to that element's inductance or capacitance. The notion of reactance is similar to electrical resistance, but it differs in several respects.

The characteristic impedance of an infinite transmission line at a given angular frequency is the ratio of the voltage and current of a pure sinusoidal wave of the same frequency travelling along the line. This definition extends to DC by letting tend to 0, and subsists for finite transmission lines until the wave reaches the end of the line. In this case, there will be in general 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, 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^{ [1] }^{ [2] }, the general expression for the characteristic impedance of a transmission line is:

The **telegrapher's equations** are a pair of coupled, linear differential equations that describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver Heaviside who in the 1880s developed the *transmission line model*. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can appear along the line. The theory applies to transmission lines of all frequencies including high-frequency transmission lines, audio frequency, low frequency and direct current.

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 *Z*_{0} prior to any reflections arriving, hence *surge impedance* is an alternative name for characteristic impedance.

The differential equations describing the dependence of the voltage and current 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^{jωt}, and the time dependence will factor out, leaving an ordinary differential equation for the coefficients, which will be phasors depending on space only. Moreover, the parameters can be generalized to be frequency-dependent.^{ [1] }

**Voltage**, **electric potential difference**, **electric pressure **or **electric tension** is the difference in electric potential between two points. The difference in electric potential between two points in a static electric field is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage is named *volt*. In SI units, work per unit charge is expressed as joules per coulomb, where 1 volt = 1 joule per 1 coulomb. The official SI definition for *volt* uses power and current, where 1 volt = 1 watt per 1 ampere. This definition is equivalent to the more commonly used 'joules per coulomb'. Voltage or electric potential difference is denoted symbolically by ∆*V*, but more often simply as *V*, for instance in the context of Ohm's or Kirchhoff's circuit laws.

An **electric current** is a flow of electric charge. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma).

In physics and engineering, a **phasor**, is a complex number representing a sinusoidal function whose amplitude (*A*), angular frequency (*ω*), and initial phase (*θ*) are time-invariant. It is related to a more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor that encapsulates the frequency and time dependence. The complex constant, which encapsulates amplitude and phase dependence, is known as **phasor**, **complex amplitude**, and **sinor** or even **complexor**.

Let

and

The positive directions of V and I are in a loop of clockwise direction.

We find that

and

or

and

These first-order equations are easily uncoupled by a second differentiation, with the results:

and

Both V and I satisfy the same equation. Since ZY is independent of z and t, it can be represented by a constant -k2. The minus sign is included so that k will appear as ±jkz in the exponential solutions of the equation. In fact,

where V+ and V- are the constant of integration, The above equation will be the wave solution for V, and

from the first-order equation.^{ [1] }

If lumped circuit analysis has to be valid at all frequencies, the length of the sub section must tend to Zero.^{ [2] }

Substituting the value of V in the above equation, we get.

Co-efficient of :

Co-efficient of :

Since

It can be seen that, 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 therefore called the “Characteristic Impedance” of the transmission line , often denoted by .^{ [2] }

We follow an approach posted by Tim Healy^{ [3] }. The line is modeled by a series of differential segments with differential series and shunt elements (as shown in the figure above). 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 impedance . That is, the impedance looking into the line on the left is . But, of course, if we go down the line one differential length dx, the impedance into the line is still . Hence we can say that the impedance looking into the line on the far left is equal to in parallel with and , all of which is in series with and . Hence:

The term above containing two factors of may be discarded, since it is infinitesimal in comparison to the other terms, leading to:

and hence to

Reversing the sign of the square root may be regarded as changing the direction of the current.

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 no dielectric loss. This would imply that the conductors act like perfect conductors and the dielectric acts like a perfect dielectric. For a lossless line, *R* and *G* are both zero, so the equation for characteristic impedance derived above reduces to:

In particular, does not depend any more upon the frequency. The above expression is wholly real, since the imaginary term *j* has canceled out, implying that *Z _{0}* is purely resistive. For a lossless line terminated in

The solutions to the long line transmission equations include incident and reflected portions of the voltage and current:

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.

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.

Standard | Impedance (Ω) | Tolerance |
---|---|---|

Ethernet Cat.5 | 100 | ±5 Ω^{ [4] } |

USB | 90 | ±15%^{ [5] } |

HDMI | 95 | ±15%^{ [6] } |

IEEE 1394 | 108 | ^{+3}_{−2}%^{ [7] } |

VGA | 75 | ±5%^{ [8] } |

DisplayPort | 100 | ±20%^{ [6] } |

DVI | 95 | ±15%^{ [6] } |

PCIe | 85 | ±15%^{ [6] } |

The characteristic impedance of coaxial cables (coax) is commonly chosen to be 50 Ω for RF and microwave applications. Coax for video applications is usually 75 Ω for its lower loss.

The **propagation constant** of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, **propagation constant **measures the change undergone by the source quantity as it propagates from one port to the next.

**Electrical impedance** is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term *complex impedance* may be used interchangeably.

In mathematics, **Green's theorem** gives the relationship between a line integral around a simple closed curve *C* and a double integral over the plane region *D* bounded by *C*. It is named after George Green, though its first proof is due to Bernhard Riemann and is the two-dimensional special case of the more general Kelvin–Stokes theorem.

In electronics, **impedance matching** is the practice of designing the input impedance of an electrical load or the output impedance of its corresponding signal source to maximize the power transfer or minimize signal reflection from the load.

The **Smith chart**, invented by Phillip H. Smith (1905–1987), is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. The Smith chart can be used to simultaneously display multiple parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis. The Smith chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis.

**Acoustic impedance** and **specific acoustic impedance** are measures of the opposition that a system presents to the acoustic flow resulting from an acoustic pressure applied to the system. The SI unit of acoustic impedance is the pascal second per cubic metre or the rayl per square metre, while that of specific acoustic impedance is the pascal second per metre or the rayl. In this article the symbol rayl denotes the MKS rayl. There is a close analogy with electrical impedance, which measures the opposition that a system presents to the electrical flow resulting from an electrical voltage applied to the system.

In mathematics, the **Hamilton–Jacobi equation** (**HJE**) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

The **Heaviside condition**, named for Oliver Heaviside (1850–1925), is the condition an electrical transmission line must meet in order for there to be no distortion of a transmitted signal. Also known as the **distortionless condition**, it can be used to improve the performance of a transmission line by adding loading to the cable.

In mathematics, a **metric connection** is a connection in a vector bundle *E* equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. Other common equivalent formulations of a metric connection include:

The **Π pad** is a specific type of attenuator circuit in electronics whereby the topology of the circuit is formed in the shape of the Greek letter "Π".

**Image impedance** is a concept used in electronic network design and analysis and most especially in filter design. The term *image impedance* applies to the impedance seen looking into a port of a network. Usually a two-port network is implied but the concept can be extended to networks with more than two ports. The definition of image impedance for a two-port network is the impedance, *Z*_{i 1}, seen looking into port 1 when port 2 is terminated with the image impedance, *Z*_{i 2}, for port 2. In general, the image impedances of ports 1 and 2 will not be equal unless the network is symmetrical with respect to the ports.

Filters designed using the image impedance methodology suffer from a peculiar flaw in the theory. The predicted characteristics of the filter are calculated assuming that the filter is terminated with its own image impedances at each end. This will not usually be the case; the filter will be terminated with fixed resistances. This causes the filter response to deviate from the theoretical. This article explains how the effects of **image filter end terminations** can be taken into account.

The **primary line constants** are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

**Metal-mesh optical filters** are optical filters made from stacks of metal meshes and dielectric. They are used as part of an optical path to filter the incoming light to allow frequencies of interest to pass while reflecting other frequencies of light.

**Characteristic admittance** is the mathematical inverse of the characteristic impedance. The general expression for the characteristic admittance of a transmission line is:

Laser theory of Fabry-Perot (FP) semiconductor lasers proves to be nonlinear, since the gain, the refractive index and the loss coefficient are the functions of energy flux. The nonlinear theory made it possible to explain a number of experiments some of which could not even be explained, much less modeled, on the basis of other theoretical models; this suggests that the nonlinear theory developed is a new paradigm of the laser theory.

In mathematics, the **exponential response formula** (ERF), also known as **exponential response and complex replacement**, is a method used to find a particular solution of a non-homogeneous linear ordinary differential equation of any order. The exponential response formula is applicable to non-homogeneous linear ordinary differential equations with constant coefficients if the function is polynomial, sinusoidal, exponential or the combination of the three. The general solution of a non-homogeneous linear ordinary differential equation is a superposition of the general solution of the associated homogeneous ODE and a particular solution to the non-homogeneous ODE. Alternative methods for solving ordinary differential equations of higher order are method of undetermined coefficients and method of variation of parameters.

- 1 2 3 "The Telegrapher's Equation".
*mysite.du.edu*. Retrieved 2018-09-09. - 1 2 3 "Derivation of Characteristic Impedance of Transmission line".
*GATE ECE 2018*. 2016-04-16. Retrieved 2018-09-09. - ↑ "Characteristic Impedance".
*www.ee.scu.edu*. Retrieved 2018-09-09. - ↑ "SuperCat OUTDOOR CAT 5e U/UTP" (PDF). Archived from the original (PDF) on 2012-03-16.
- ↑ "USB in a NutShell—Chapter 2—Hardware". Beyond Logic.org. Retrieved 2007-08-25.
- 1 2 3 4 https://www.nxp.com/documents/application_note/AN10798.pdf (PDF) modified 2011-07-04
- ↑ http://materias.fi.uba.ar/6644/info/reflectometria/avanzado/ieee1394-evalwith-tdr.pdf (PDF), modified 2005-06-11
- ↑ http://www.promusic.cz/soubory/File/Downloads/Data%20sheet/Klotz/Kabely%20pro%20video/VMM5FL__e.pdf (PDF) modified 2009-12-07

- Guile, A. E. (1977).
*Electrical Power Systems*. 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.

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