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Source and load circuit impedance

Impedance matchingFrom Wikipedia, the free encyclopedia

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

In the case of a complex source impedance ZS and load impedance ZL, maximum power transfer isobtained when

where the asterisk indicates the complex conjugate of the variable. Where ZS represents the characteristicimpedance of a transmission line, minimum reflection is obtained when

The concept of impedance matching found first applications in electrical engineering, but is relevant in other applications in which a form ofenergy, not necessarily electrical, is transferred between a source and a load. An alternative to impedance matching is impedance bridging, inwhich the load impedance is chosen to be much larger than the source impedance and maximizing voltage transfer, rather than power, is thegoal.

Contents

1 Theory1.1 Reflection­less matching1.2 Complex conjugate matching

2 Power transfer3 Impedance­matching devices

3.1 Transformers

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3.2 Resistive network3.3 Stepped transmission line3.4 Filters

4 Power factor correction5 Transmission lines

5.1 Single­source transmission line driving a load6 Electrical examples

6.1 Telephone systems6.2 Loudspeaker amplifiers

7 Non­electrical examples7.1 Acoustics7.2 Optics7.3 Mechanics

8 See also9 Notes10 References11 External links

Theory

Impedance is the opposition by a system to the flow of energy from a source. For constant signals, this impedance can also be constant. Forvarying signals, it usually changes with frequency. The energy involved can be electrical, mechanical, magnetic or thermal. The concept ofelectrical impedance is perhaps the most commonly known. Electrical impedance, like electrical resistance, is measured in ohms. In general,impedance has a complex value; this means that loads generally have a resistance component (symbol: R) which forms the real part of Z and areactance component (symbol: X) which forms the imaginary part of Z.

In simple cases (such as low­frequency or direct­current power transmission) the reactance may be negligible or zero; the impedance can beconsidered a pure resistance, expressed as a real number. In the following summary we will consider the general case when resistance andreactance are both significant, and the special case in which the reactance is negligible.

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Reflection­less matching

Impedance matching to minimize reflections is achieved by making the load impedance equal to the source impedance. If the sourceimpedance, load impedance and transmission line characteristic impedance are purely resistive, then reflection­less matching is the same asmaximum power transfer matching.[1]

Complex conjugate matching

Complex conjugate matching is used when maximum power transfer is required. This is different from reflection­less matching only when thesource or load have a reactive component.

(where * indicates the complex conjugate).

If the source has a reactive component, but the load is purely resistive then matching can be achieved by adding a reactance of the oppositesign to the load. This simple matching network consisting of a single element will usually only achieve a perfect match at a single frequency.This is because the added element will either be a capacitor or an inductor, both of which are frequency dependent and will not, in general,follow the frequency dependence of the source impedance. For wide bandwidth applications a more complex network needs to be designed.

Power transfer

Whenever a source of power with a fixed output impedance such as an electric signal source, a radio transmitter or a mechanical sound (e.g., aloudspeaker) operates into a load, the maximum possible power is delivered to the load when the impedance of the load (load impedance orinput impedance) is equal to the complex conjugate of the impedance of the source (that is, its internal impedance or output impedance). Fortwo impedances to be complex conjugates their resistances must be equal, and their reactances must be equal in magnitude but of oppositesigns. In low­frequency or DC systems (or systems with purely resistive sources and loads) the reactances are zero, or small enough to beignored. In this case, maximum power transfer occurs when the resistance of the load is equal to the resistance of the source (see maximumpower theorem for a mathematical proof).

Impedance matching is not always necessary. For example, if a source with a low impedance is connected to a load with a high impedance thepower that can pass through the connection is limited by the higher impedance. This maximum­voltage connection is a common configurationcalled impedance bridging or voltage bridging, and is widely used in signal processing. In such applications, delivering a high voltage (tominimize signal degradation during transmission or to consume less power by reducing currents) is often more important than maximumpower transfer.

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In older audio systems (reliant on transformers and passive filter networks, and based on the telephone system), the source and load resistanceswere matched at 600 ohms. One reason for this was to maximize power transfer, as there were no amplifiers available that could restore lostsignal. Another reason was to ensure correct operation of the hybrid transformers used at central exchange equipment to separate outgoingfrom incoming speech, so these could be amplified or fed to a four­wire circuit. Most modern audio circuits, on the other hand, use activeamplification and filtering and can use voltage­bridging connections for greatest accuracy. Strictly speaking, impedance matching only applieswhen both source and load devices are linear; however, matching may be obtained between nonlinear devices within certain operating ranges.

Impedance­matching devices

Adjusting the source impedance or the load impedance, in general, is called "impedance matching". There are three ways to improve animpedance mismatch, all of which are called "impedance matching":

Devices intended to present an apparent load to the source of Zload = Zsource* (complex conjugate matching). Given a source with a fixed

voltage and fixed source impedance, the maximum power theorem says this is the only way to extract the maximum power from thesource.Devices intended to present an apparent load of Zload = Zline (complex impedance matching), to avoid echoes. Given a transmission line

source with a fixed source impedance, this "reflectionless impedance matching" at the end of the transmission line is the only way toavoid reflecting echoes back to the transmission line.Devices intended to present an apparent source resistance as close to zero as possible, or presenting an apparent source voltage as highas possible. This is the only way to maximize energy efficiency, and so it is used at the beginning of electrical power lines. Such animpedance bridging connection also minimizes distortion and electromagnetic interference; it is also used in modern audio amplifiersand signal­processing devices.

There are a variety of devices used between a source of energy and a load that perform "impedance matching". To match electricalimpedances, engineers use combinations of transformers, resistors, inductors, capacitors and transmission lines. These passive (and active)impedance­matching devices are optimized for different applications and include baluns, antenna tuners (sometimes called ATUs or roller­coasters, because of their appearance), acoustic horns, matching networks, and terminators.

Transformers

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Transformers are sometimes used to match the impedances of circuits. A transformer converts alternating current at one voltage to the samewaveform at another voltage. The power input to the transformer and output from the transformer is the same (except for conversion losses).The side with the lower voltage is at low impedance (because this has the lower number of turns), and the side with the higher voltage is at ahigher impedance (as it has more turns in its coil).

One example of this method involves a television balun transformer. This transformer converts a balanced signal from the antenna (via 300­ohm twin­lead) into an unbalanced signal (75­ohm coaxial cable such as RG­6). To match the impedances of both devices, both cables must beconnected to a matching transformer with a turns ratio of 2 (such as a 2:1 transformer). In this example, the 75­ohm cable is connected to thetransformer side with fewer turns; the 300­ohm line is connected to the transformer side with more turns. The formula for calculating thetransformer turns ratio for this example is:

Resistive network

Resistive impedance matches are easiest to design and can be achieved with a simple L pad consisting of two resistors. Power loss is anunavoidable consequence of using resistive networks, and they are only (usually) used to transfer line level signals.

Stepped transmission line

Most lumped­element devices can match a specific range of load impedances. For example, in order to match an inductive load into a realimpedance, a capacitor needs to be used. If the load impedance becomes capacitive, the matching element must be replaced by an inductor. Inmany cases, there is a need to use the same circuit to match a broad range of load impedance and thus simplify the circuit design. This issuewas addressed by the stepped transmission line,[2] where multiple, serially placed, quarter­wave dielectric slugs are used to vary a transmissionline's characteristic impedance. By controlling the position of each element, a broad range of load impedances can be matched without havingto reconnect the circuit.

Filters

Filters are frequently used to achieve impedance matching in telecommunications and radio engineering. In general, it is not theoreticallypossible to achieve perfect impedance matching at all frequencies with a network of discrete components. Impedance matching networks aredesigned with a definite bandwidth, take the form of a filter, and use filter theory in their design.

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L networks for narrowband matchinga source or load impedance Z to atransmission line with characteristicimpedance Z0. X and B may each beeither positive (inductor) or negative(capacitor). If Z/Z0 is inside the 1+jxcircle on the Smith chart (i.e. ifRe(Z/Z0)>1), network (a) can beused; otherwise network (b) can beused.[3]

Applications requiring only a narrow bandwidth, such as radio tuners and transmitters, might use a simple tuned filter such as a stub. Thiswould provide a perfect match at one specific frequency only. Wide bandwidth matching requires filters with multiple sections.

L­section

A simple electrical impedance­matching network requires one capacitor and one inductor. Onereactance is in parallel with the source (or load), and the other is in series with the load (or source). If areactance is in parallel with the source, the effective network matches from high to low impedance. TheL­section is inherently a narrowband matching network.

The analysis is as follows. Consider a real source impedance of and real load impedance of . If areactance is in parallel with the source impedance, the combined impedance can be written as:

If the imaginary part of the above impedance is canceled by the series reactance, the real part is

Solving for

If the above equation can be approximated as

The inverse connection (impedance step­up) is simply the reverse—for example, reactance in series with the source. The magnitude of theimpedance ratio is limited by reactance losses such as the Q of the inductor. Multiple L­sections can be wired in cascade to achieve higherimpedance ratios or greater bandwidth. Transmission line matching networks can be modeled as infinitely many L­sections wired in cascade.Optimal matching circuits can be designed for a particular system using Smith charts.

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Coaxial transmission line with one source and one load

Power factor correction

Power factor correction devices are intended to cancel the reactive and nonlinear characteristics of a load at the end of a power line. Thiscauses the load seen by the power line to be purely resistive. For a given true power required by a load this minimizes the true current suppliedthrough the power lines, and minimizes power wasted in the resistance of those power lines. For example, a maximum power point tracker isused to extract the maximum power from a solar panel and efficiently transfer it to batteries, the power grid or other loads. The maximumpower theorem applies to its "upstream" connection to the solar panel, so it emulates a load resistance equal to the solar panel sourceresistance. However, the maximum power theorem does not apply to its "downstream" connection. That connection is an impedance bridgingconnection; it emulates a high­voltage, low­resistance source to maximize efficiency.

On the power grid the overall load is usually inductive. Consequently, power factor correction is most commonly achieved with banks ofcapacitors. It is only necessary for correction to be achieved at one single frequency, the frequency of the supply. Complex networks are onlyrequired when a band of frequencies must be matched and this is the reason why simple capacitors are all that is usually required for powerfactor correction.

Transmission lines

Impedance bridging is unsuitable for RF connections,because it causes power to be reflected back to the sourcefrom the boundary between the high and the lowimpedances. The reflection creates a standing wave if thereis reflection at both ends of the transmission line, whichleads to further power waste and may cause frequency­dependent loss. In these systems, impedance matching isdesirable.

In electrical systems involving transmission lines (such asradio and fiber optics)—where the length of the line is longcompared to the wavelength of the signal (the signalchanges rapidly compared to the time it takes to travel fromsource to load)— the impedances at each end of the linemust be matched to the transmission line's characteristicimpedance ( ) to prevent reflections of the signal at theends of the line. (When the length of the line is short compared to the wavelength, impedance mismatch is the basis of transmission­line

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impedance transformers; see previous section.) In radio­frequency (RF) systems, a common value for source and load impedances is 50 ohms.A typical RF load is a quarter­wave ground plane antenna (37 ohms with an ideal ground plane; it can be matched to 50 ohms by using amodified ground plane or a coaxial matching section, i.e., part or all the feeder of higher impedance).

The general form of the voltage reflection coefficient for a wave moving from medium 1 to medium 2 is given by

while the voltage reflection coefficient for a wave moving from medium 2 to medium 1 is

so the reflection coefficient is the same (except for sign), no matter from which direction the wave approaches the boundary.

There is also a current reflection coefficient; it is the same as the voltage coefficient, except that it has an opposite sign. If the wave encountersan open at the load end, positive voltage and negative current pulses are transmitted back toward the source (negative current means thecurrent is going the opposite direction). Thus, at each boundary there are four reflection coefficients (voltage and current on one side, andvoltage and current on the other side). All four are the same, except that two are positive and two are negative. The voltage reflectioncoefficient and current reflection coefficient on the same side have opposite signs. Voltage reflection coefficients on opposite sides of theboundary have opposite signs.

Because they are all the same except for sign it is traditional to interpret the reflection coefficient as the voltage reflection coefficient (unlessotherwise indicated). Either end (or both ends) of a transmission line can be a source or a load (or both), so there is no inherent preference forwhich side of the boundary is medium 1 and which side is medium 2. With a single transmission line it is customary to define the voltagereflection coefficient for a wave incident on the boundary from the transmission line side, regardless of whether a source or load is connectedon the other side.

Single­source transmission line driving a load

Load­end conditions

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In a transmission line, a wave travels from the source along the line. Suppose the wave hits a boundary (an abrupt change in impedance). Someof the wave is reflected back, while some keeps moving onwards. (Assume there is only one boundary, at the load.)

Let

and be the voltage and current that is incident on the boundary from the source side. and be the voltage and current that is transmitted to the load. and be the voltage and current that is reflected back toward the source.

On the line side of the boundary and and on the load side where , , , , , , and arephasors.

At a boundary, voltage and current must be continuous, therefore

All these conditions are satisfied by

where the reflection coefficient going from the transmission line to the load.

[4][5][6]

The purpose of a transmission line is to get the maximum amount of energy to the other end of the line (or to transmit information withminimal error), so the reflection is as small as possible. This is achieved by matching the impedances and so that they are equal (

).

Source­end conditions

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At the source end of the transmission line, there may be waves incident both from the source and from the line; a reflection coefficient for eachdirection may be computed with

,

where Zs is the source impedance. The source of waves incident from the line are the reflections from the load end. If the source impedancematches the line, reflections from the load end will be absorbed at the source end. If the transmission line is not matched at both endsreflections from the load will be re­reflected at the source and re­re­reflected at the load end ad infinitum, losing energy on each transit of thetransmission line. This can cause a resonance condition and strongly frequency­dependent behavior. In a narrow­band system this can bedesirable for matching, but is generally undesirable in a wide­band system.

Source­end impedance

[7]

where is the one­way transfer function (from either end to the other) when the transmission line is exactly matched at source and load. accounts for everything that happens to the signal in transit (including delay, attenuation and dispersion). If there is a perfect match at the load,

and

Transfer function

where is the open circuit (or unloaded) output voltage from the source.

Note that if there is a perfect match at both ends

and

and then

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Typical push–pull audio tube poweramplifier, matched to loudspeakerwith an impedance­matchingtransformer

.

Electrical examples

Telephone systems

Telephone systems also use matched impedances to minimise echo on long­distance lines. This is related to transmission­line theory. Matchingalso enables the telephone hybrid coil (2­ to 4­wire conversion) to operate correctly. As the signals are sent and received on the same two­wirecircuit to the central office (or exchange), cancellation is necessary at the telephone earpiece so excessive sidetone is not heard. All devicesused in telephone signal paths are generally dependent on matched cable, source and load impedances. In the local loop, the impedance chosenis 600 ohms (nominal). Terminating networks are installed at the exchange to offer the best match to their subscriber lines. Each country hasits own standard for these networks, but they are all designed to approximate about 600 ohms over the voice frequency band.

Loudspeaker amplifiers

Audio amplifiers typically do not match impedances, but provide an output impedance that is lowerthan the load impedance (such as < 0.1 ohm in typical semiconductor amplifiers), for improved speakerdamping. For vacuum tube amplifiers, impedance­changing transformers are often used to get a lowoutput impedance, and to better match the amplifier's performance to the load impedance. Some tubeamplifiers have output transformer taps to adapt the amplifier output to typical loudspeakerimpedances.

The output transformer in vacuum­tube­based amplifiers has two basic functions:

Separation of the AC component (which contains the audio signals) from the DC component(supplied by the power supply) in the anode circuit of a vacuum­tube­based power stage. Aloudspeaker should not be subjected to DC current.Reducing the output impedance of power pentodes (such as the EL34) in a common­cathodeconfiguration.

The impedance of the loudspeaker on the secondary coil of the transformer will be transformed to a higher impedance on the primary coil inthe circuit of the power pentodes by the square of the turns ratio, which forms the impedance scaling factor.

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The output stage in common­drain or common­collector semiconductor­based end stages with MOSFETs or power transistors has a very lowoutput impedance. If they are properly balanced, there is no need for a transformer or a large electrolytic capacitor to separate AC from DCcurrent.

Non­electrical examples

Acoustics

Similar to electrical transmission lines, an impedance matching problem exists when transferring sound energy from one medium to another. Ifthe acoustic impedance of the two media are very different most sound energy will be reflected (or absorbed), rather than transferred across theborder. The gel used in medical ultrasonography helps transfer acoustic energy from the transducer to the body and back again. Without thegel, the impedance mismatch in the transducer­to­air and the air­to­body discontinuity reflects almost all the energy, leaving very little to gointo the body.

The bones in the middle ear provide impedance matching between the eardrum (which is acted upon by vibrations in air) and the fluid­filledinner ear.

Horns are used like transformers, matching the impedance of the transducer to the impedance of the air. This principle is used in both hornloudspeakers and musical instruments. Most loudspeaker systems contain impedance matching mechanisms, especially for low frequencies.Because most driver impedances which are poorly matched to the impedance of free air at low frequencies (and because of out­of­phasecancellations between output from the front and rear of a speaker cone), loudspeaker enclosures both match impedances and preventinterference. Sound, coupling with air, from a loudspeaker is related to the ratio of the diameter of the speaker to the wavelength of the soundbeing reproduced. That is, larger speakers can produce lower frequencies at a higher level than smaller speakers for this reason. Ellipticalspeakers are a complex case, acting like large speakers lengthwise and small speakers crosswise. Acoustic impedance matching (or the lack ofit) affects the operation of a megaphone, an echo and soundproofing.

Optics

A similar effect occurs when light (or any electromagnetic wave) hits the interface between two media with different refractive indices. Fornon­magnetic materials, the refractive index is inversely proportional to the material's characteristic impedance. An optical or wave impedance(that depends on the propagation direction) can be calculated for each medium, and may be used in the transmission­line reflection equation

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to calculate reflection and transmission coefficients for the interface. For non­magnetic dielectrics, this equation is equivalent to the Fresnelequations. Unwanted reflections can be reduced by the use of an anti­reflection optical coating.

Mechanics

If a body of mass m collides elastically with a second body, maximum energy transfer to the second body will occur when the second body hasthe same mass m. In a head­on collision of equal masses, the energy of the first body will be completely transferred to the second body (as inNewton's cradle for example). In this case, the masses act as "mechanical impedances", which must be matched. If and are the massesof the moving and stationary bodies, and P is the momentum of the system (which remains constant throughout the collision), the energy of thesecond body after the collision will be E2:

which is analogous to the power­transfer equation.

These principles are useful in the application of highly energetic materials (explosives). If an explosive charge is placed on a target, the suddenrelease of energy causes compression waves to propagate through the target radially from the point­charge contact. When the compressionwaves reach areas of high acoustic impedance mismatch (such as the opposite side of the target), tension waves reflect back and createspalling. The greater the mismatch, the greater the effect of creasing and spalling will be. A charge initiated against a wall with air behind itwill do more damage to the wall than a charge initiated against a wall with soil behind it.

See also

Power (physics)Reflection coefficientRinging (signal)Standing wave ratioTransmission lineWet Transformer

Notes

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References

Floyd, Thomas (1997), Principles of Electric Circuits (5th ed.), Prentice Hall, ISBN 0­13­232224­2Hayt, William (1989), Engineering Electromagnetics (5th ed.), McGraw­Hill, ISBN 0­07­027406­1Karakash, John J. (1950), Transmission Lines and Filter Networks (1st ed.), MacmillanKraus, John D. (1984), Electromagnetics (3rd ed.), McGraw­Hill, ISBN 0­07­035423­5Sadiku, Matthew N. O. (1989), Elements of Electromagnetics (1st ed.), Saunders College Publishing, ISBN 0030134846Stutzman, Gary; Thiele (2012), Antenna Theory and Design, John Wiley & Sons, ISBN 0470576642Young, E.C., The Penguin Dictionary of Electronics, Penguin, ISBN 0­14­051187­3 (see 'maximum power theorem', 'impedancematching')

External links

Impedance Matching (http://www.antenna­theory.com/tutorial/smith/chart.php) Impedance Matching with the Smith Chart for AntennasUnity Gain and Impedance Matching (http://www.rane.com/note124.html)Impedance matching for microphones: Is it necessary? No. (http://shure.custhelp.com/cgi­bin/shure.cfg/php/enduser/std_adp.php?p_faqid=224)Calculation: Damping of impedance matching ­ connecting Zout and Zin (http://www.sengpielaudio.com/calculator­bridging.htm)

1. Stutzman & Thiele 2012, p. 177, page link (http://books.google.co.uk/books?id=xhZRA1K57wIC&pg=RA1­PA177#v=onepage&q&f=true)2. Chunqui Qian and William W. Brey, "Impedance matching with an adjustable segmented transmission line". Journal of Magnetic Resonance, vol. 199

issue 1 (July 2009), pp. 104­110 (http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WJX­4W2122T­1&_user=5755111&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000000150&_version=1&_urlVersion=0&_userid=5755111&md5=fe79f204b33cf7eb6d03cb89ff250c91) Retrieved 2011­10­29.

3. Pozer, David. Microwave Engineering (3rd ed.). p. 223.4. Kraus (1984, p. 407)5. Sadiku (1989, pp. 505–507)6. Hayt (1989, pp. 398–401)7. Karakash (1950, pp. 52–57)

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Impedance matching: A primer (http://www1.electusdistribution.com.au/images_uploaded/impmatch.pdf)Tutorial on RF impedance matching using the Smith Chart (http://www.maxim­ic.com/appnotes.cfm?appnote_number=742&CMP=WP­6)A description of impedance matching (http://www.maxim­ic.com/appnotes.cfm/appnote_number/1849)Conjugate matching versus reflectionless matching ­ pdf (http://www.ece.rutgers.edu/~orfanidi/ewa/ch12.pdf)Impedance Matching Networks (http://www.advanced­energy.com/upload/file/White_Papers/ENG­WHITE18­270­02.pdf)Java applets demonstrating impedance mismatching (http://www.amanogawa.com/archive/signalintegrityA.html)The impedance transformation along a stepped transmission line (http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WJX­4W2122T­1)

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