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 In the case of a complex source impedance ZS and load impedance ZL, maximum power transfer is obtained when where the asterisk indicates the complex conjugate of the variable. 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 of energy, not necessarily electrical, is transferred between a source and a load An alternative to impedance matching is impedance bridging, in which the load impedance is chosen to be much larger than the source impedance and maximizing voltage transfer, rather than power, is the goal 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 For varying signals, it usually changes with frequency. The energy involved can be electrical, mechanical, magnetic or thermal. The concept of electrical 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 which forms the real part of Z and a reactance component which forms the imaginary part of Z In simple cases the reactance may be negligible or zero; the impedance can be considered a pure resistance, expressed as a real number In the following summary we will consider the general case when resistance and reactance are both significant, and the special case in which the reactance is negligible Reflection-less matching Impedance matching to minimize reflections is achieved by making the load impedance equal to the source impedance. If the source impedance, load impedance and transmission line characteristic impedance are purely resistive, then reflection-less matching is the same as maximum power transfer matching Complex conjugate matching Complex conjugate matching is used when maximum power transfer is required. This is different from reflection-less matching only when the source 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 opposite sign 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 operates into a load, the maximum possible power is delivered to the load when the impedance of the load is equal to the complex conjugate of the impedance of the source. For two impedances to be complex conjugates their resistances must be equal, and their reactances must be equal in magnitude but of opposite signs In low-frequency or DC systems the reactances are zero, or small enough to be ignored. In this case, maximum power transfer occurs when the resistance of the load is equal to the resistance of the source Impedance matching is not always necessary For example, if a source with a low impedance is connected to a load with a high impedance the power that can pass through the connection is limited by the higher impedance. This maximum-voltage connection is a common configuration called impedance bridging or voltage bridging, and is widely used in signal processing. In such applications, delivering a high voltage is often more important than maximum power transfer In older audio systems, the source and load resistances were matched at 600 ohms. One reason for this was to maximize power transfer, as there were no amplifiers available that could restore lost signal. Another reason was to ensure correct operation of the hybrid transformers used at central exchange equipment to separate outgoing from incoming speech, so these could be amplified or fed to a four-wire circuit. Most modern audio circuits, on the other hand, use active amplification and filtering and can use voltage-bridging connections for greatest accuracy. Strictly speaking, impedance matching only applies when both source and load devices are linear; however, matching

may be obtained between nonlinear devices within certain operating ranges Impedance-matching devices Adjusting the source impedance or the load impedance, in general, is called “impedance matching”. There are three ways to improve an impedance mismatch, all of which are called “impedance matching”: Devices intended to present an apparent load to the source of Zload = Zsource*. 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 the source Devices intended to present an apparent load of Zload = Zline, 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 to avoid 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 high as possible. This is the only way to maximize energy efficiency, and so it is used at the beginning of electrical power lines. Such an impedance bridging connection also minimizes distortion and electromagnetic interference; it is also used in modern audio amplifiers and signal-processing devices There are a variety of devices used between a source of energy and a load that perform “impedance matching”. To match electrical impedances, engineers use combinations of transformers, resistors, inductors, capacitors and transmission lines. These passive impedance-matching devices are optimized for different applications and include baluns, antenna tuners, acoustic horns, matching networks, and terminators Transformers Transformers are sometimes used to match the impedances of circuits. A transformer converts alternating current at one voltage to the same waveform at another voltage. The power input to the transformer and output from the transformer is the same. The side with the lower voltage is at low impedance, and the side with the higher voltage is at a higher impedance One example of this method involves a television balun transformer. This transformer converts a balanced signal from the antenna into an unbalanced signal. To match the impedances of both devices, both cables must be connected to a matching transformer with a turns ratio of 2. In this example, the 75-ohm cable is connected to the transformer side with fewer turns; the 300-ohm line is connected to the transformer side with more turns. The formula for calculating the transformer 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 an unavoidable consequence of using resistive networks, and they are only used to transfer line level signals Stepped transmission line Most lumped-element devices can match a specific range of load impedances. For example, in order to match an inductive load into a real impedance, a capacitor needs to be used. If the load impedance becomes capacitive, the matching element must be replaced by an inductor In many 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 issue was addressed by the stepped transmission line, where multiple, serially placed, quarter-wave dielectric slugs are used to vary a transmission line’s characteristic impedance. By controlling the position of each element, a broad range of load impedances can be matched without having to reconnect the circuit Filters Filters are frequently used to achieve impedance matching in telecommunications and radio engineering In general, it is not theoretically possible to achieve perfect impedance matching at all frequencies with a network of discrete components Impedance matching networks are designed with a definite bandwidth, take the form of a filter, and use filter theory in their design Applications requiring only a narrow bandwidth, such as radio tuners and transmitters, might use a simple tuned filter such as a stub This would provide a perfect match at one specific frequency only. Wide bandwidth matching requires filters with multiple sections L-section A simple electrical impedance-matching network requires one capacitor and one inductor. One reactance is in parallel with the source, and the other is in series with the load If a reactance is in parallel with the source, the effective network matches from high to low impedance. The L-section is inherently a narrowband matching network The analysis is as follows. Consider a real source impedance of and real load impedance

of . If a reactance 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 is simply the reverse—for example, reactance in series with the source The magnitude of the impedance ratio is limited by reactance losses such as the Q of the inductor Multiple L-sections can be wired in cascade to achieve higher impedance ratios or greater bandwidth. Transmission line matching networks can be modeled as infinitely many L-sections wired in cascade. Optimal matching circuits can be designed for a particular system using Smith charts 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. This causes 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 supplied through the power lines, and minimizes power wasted in the resistance of those power lines. For example, a maximum power point tracker is used to extract the maximum power from a solar panel and efficiently transfer it to batteries, the power grid or other loads. The maximum power theorem applies to its “upstream” connection to the solar panel, so it emulates a load resistance equal to the solar panel source resistance. However, the maximum power theorem does not apply to its “downstream” connection. That connection is an impedance bridging connection; it emulates a high-voltage, low-resistance source to maximize efficiency On the power grid the overall load is usually inductive. Consequently, power factor correction is most commonly achieved with banks of capacitors It is only necessary for correction to be achieved at one single frequency, the frequency of the supply. Complex networks are only required when a band of frequencies must be matched and this is the reason why simple capacitors are all that is usually required for power factor correction Transmission lines Impedance bridging is unsuitable for RF connections, because it causes power to be reflected back to the source from the boundary between the high and the low impedances. The reflection creates a standing wave if there is reflection at both ends of the transmission line, which leads to further power waste and may cause frequency-dependent loss. In these systems, impedance matching is desirable In electrical systems involving transmission lines—where the length of the line is long compared to the wavelength of the signal— the impedances at each end of the line must be matched to the transmission line’s characteristic impedance to prevent reflections of the signal at the ends of the line. In radio-frequency systems, a common value for source and load impedances is 50 ohms. A typical RF load is a quarter-wave ground plane antenna 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, 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 encounters an open at the load end, positive voltage and negative current pulses are transmitted back toward the source. Thus, at each boundary there are four reflection coefficients. All four are the same, except that two are positive and two are negative. The voltage reflection coefficient and current reflection coefficient on the same side have opposite signs. Voltage reflection coefficients on opposite sides of the boundary 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 Either end of a transmission line can be a source or a load, so there is no inherent preference for which side of the boundary is medium 1 and which side is medium 2. With a single transmission line it is customary to define the voltage reflection coefficient for a wave incident on the boundary from the transmission line side, regardless of whether a source or load is connected on the other side Single-source transmission line driving a load Load-end conditions In a transmission line, a wave travels from the source along the line. Suppose the wave hits a boundary. Some of the wave is reflected back, while some keeps moving onwards

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 are phasors 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 The purpose of a transmission line is to get the maximum amount of energy to the other end of the line, so the reflection is as small as possible. This is achieved by matching the impedances and so that they are equal Source-end conditions 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 each direction 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 impedance matches the line, reflections from the load end will be absorbed at the source end. If the transmission line is not matched at both ends reflections from the load will be re-reflected at the source and re-re-reflected at the load end ad infinitum, losing energy on each transit of the transmission line. This can cause a resonance condition and strongly frequency-dependent behavior. In a narrow-band system this can be desirable for matching, but is generally undesirable in a wide-band system Source-end impedance where is the one-way transfer function when the transmission line is exactly matched at source and load. accounts for everything that happens to the signal in transit. If there is a perfect match at the load, and Transfer function where is the open circuit output voltage from the source Note that if there is a perfect match at both ends and and then Electrical examples Telephone systems Telephone systems also use matched impedances to minimise echo on long-distance lines. This is related to transmission-line theory. Matching also enables the telephone hybrid coil to operate correctly. As the signals are sent and received on the same two-wire circuit to the central office, cancellation is necessary at the telephone earpiece so excessive sidetone is not heard. All devices used in telephone signal paths are generally dependent on matched cable, source and load impedances. In the local loop, the impedance chosen is 600 ohms Terminating networks are installed at the exchange to offer the best match to their subscriber lines. Each country has its 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 lower than the load impedance, for improved speaker damping. For vacuum tube amplifiers, impedance-changing transformers are often used to get a low output impedance, and to better match the amplifier’s performance to the load impedance. Some tube amplifiers have output transformer taps to adapt the amplifier output to typical loudspeaker impedances The output transformer in vacuum-tube-based amplifiers has two basic functions: Separation of the AC component from the DC component in the anode circuit of a vacuum-tube-based power stage. A loudspeaker should not be subjected to DC current Reducing the output impedance of power pentodes in a common-cathode configuration The impedance of the loudspeaker on the secondary coil of the transformer will be transformed to a higher impedance on the primary coil in the circuit of the power pentodes by the square of the turns ratio, which forms the impedance scaling factor The output stage in common-drain or common-collector semiconductor-based end stages with MOSFETs or power transistors has a very low output impedance. If they are properly balanced, there is no need for a transformer or a large electrolytic capacitor to separate AC from DC current Non-electrical examples Acoustics Similar to electrical transmission lines, an impedance matching problem exists when transferring sound energy from one medium to another. If the acoustic impedance of the two media are very different most sound energy will be reflected, rather than transferred across the border. The gel used in medical ultrasonography helps transfer acoustic energy from the transducer to the body and back again Without the gel, the impedance mismatch in the transducer-to-air and the air-to-body discontinuity reflects almost all the energy, leaving very little to go into the body

The bones in the middle ear provide impedance matching between the eardrum and the fluid-filled inner ear Horns are used like transformers, matching the impedance of the transducer to the impedance of the air. This principle is used in both horn loudspeakers 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, loudspeaker enclosures both match impedances and prevent interference. Sound, coupling with air, from a loudspeaker is related to the ratio of the diameter of the speaker to the wavelength of the sound being reproduced That is, larger speakers can produce lower frequencies at a higher level than smaller speakers for this reason. Elliptical speakers are a complex case, acting like large speakers lengthwise and small speakers crosswise. Acoustic impedance matching affects the operation of a megaphone, an echo and soundproofing Optics A similar effect occurs when light hits the interface between two media with different refractive indices. For non-magnetic materials, the refractive index is inversely proportional to the material’s characteristic impedance An optical or wave impedance can be calculated for each medium, and may be used in the transmission-line reflection equation to calculate reflection and transmission coefficients for the interface. For non-magnetic dielectrics, this equation is equivalent to the Fresnel equations. Unwanted reflections can be reduced by the use of an anti-reflection 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 has the same mass m. In a head-on collision of equal masses, the energy of the first body will be completely transferred to the second body. In this case, the masses act as “mechanical impedances”, which must be matched. If and are the masses of the moving and stationary bodies, and P is the momentum of the system, the energy of the second body after the collision will be E2: which is analogous to the power-transfer equation These principles are useful in the application of highly energetic materials. If an explosive charge is placed on a target, the sudden release of energy causes compression waves to propagate through the target radially from the point-charge contact. When the compression waves reach areas of high acoustic impedance mismatch, tension waves reflect back and create spalling The greater the mismatch, the greater the effect of creasing and spalling will be. A charge initiated against a wall with air behind it will do more damage to the wall than a charge initiated against a wall with soil behind it See also Power Reflection coefficient Ringing Standing wave ratio Transmission line Wet Transformer Notes References Floyd, Thomas, Principles of Electric Circuits, Prentice Hall, ISBN 0-13-232224-2 Hayt, William, Engineering Electromagnetics, McGraw-Hill, ISBN 0-07-027406-1 Karakash, John J., Transmission Lines and Filter Networks, Macmillan Kraus, John D., Electromagnetics, McGraw-Hill, ISBN 0-07-035423-5 Sadiku, Matthew N. O., Elements of Electromagnetics, Saunders College Publishing, ISBN 0030134846 Stutzman, Gary; Thiele, Antenna Theory and Design, John Wiley & Sons, ISBN 0470576642 Young, E.C., The Penguin Dictionary of Electronics, Penguin, ISBN 0-14-051187-3 External links Impedance Matching Impedance Matching with the Smith Chart for Antennas Unity Gain and Impedance Matching Impedance matching for microphones: Is it necessary? No Calculation: Damping of impedance matching – connecting Zout and Zin Impedance matching: A primer Tutorial on RF impedance matching using the Smith Chart A description of impedance matching Conjugate matching versus reflectionless matching – pdf Impedance Matching Networks Java applets demonstrating impedance mismatching The impedance transformation along a stepped transmission line

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