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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. A source of electric power such as a generator, amplifier or radio transmitter has a source impedance equivalent to an electrical resistance in series with a frequency-dependent reactance. Likewise, an electrical load such as a light bulb, transmission line or antenna has an impedance equivalent to a resistance in series with a reactance.
The maximum power theorem states that maximum power is transferred from source to load when the load resistance equals the source resistance and the load reactance equals the negative of the source reactance: the reactances cancel each other out with their opposing dependency on frequency. Another way of saying this using complex numbers is the load impedance must equal the complex conjugate of the source impedance. If this condition is met the two parts of the circuit are said to be impedance matched.
In a direct current (DC) circuit, the condition is satisfied if the load resistance equals the source resistance. In an alternating current (AC) circuit the reactance depends on frequency, so circuits which are impedance matched at one frequency may not be impedance matched if the frequency is changed. Impedance matching over a wide band will generally require complex, filter-like structures with many components except in the trivial case of constant source and load resistances when a transformer can be used.
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. Where ZS represents the characteristic impedance 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 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.
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, acoustic, magnetic, optical, 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 (symbol: R) which forms the real part of Z and a reactance component (symbol: X) which forms the imaginary part of Z.
In simple cases (such as low-frequency or direct-current power transmission) 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.
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 is used when maximum power transfer is required, namely
where a superscript * indicates the complex conjugate. A conjugate match is different from a reflection-less match when either the source or load has a reactive component.
If the source has a reactive component, but the load is purely resistive, then matching can be achieved by adding a reactance of the same magnitude but opposite sign to the load. This simple matching network, consisting of a single element, will usually achieve a perfect match at only a single frequency. This is because the added element will either be a capacitor or an inductor, whose impedance in both cases is frequency dependent, and will not, in general, follow the frequency dependence of the source impedance. For wide bandwidth applications, a more complex network must be designed.
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., a loudspeaker) operates into a load, the maximum possible power is delivered to the load when the impedance of the load (load impedance or input impedance) is equal to the complex conjugate of the impedance of the source (that is, its internal impedance or output impedance). 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 (or systems with purely resistive sources and loads) 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 (see maximum power 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 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 (to minimize signal degradation during transmission or to consume less power by reducing currents) is often more important than maximum power transfer.
In older audio systems (reliant on transformers and passive filter networks, and based on the telephone system), 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.
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":
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 (and active) impedance-matching devices are optimized for different applications and include baluns, antenna tuners (sometimes called ATUs or roller-coasters, because of their appearance), acoustic horns, matching networks, and terminators.
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 (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 a higher 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 300-ohm twin-lead) into an unbalanced signal (75-ohm coaxial cable such as RG-6). To match the impedances of both devices, both cables must be connected to a matching transformer with a turns ratio of 2 (such as a 2:1 transformer). 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 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 (usually) used to transfer line level signals.
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 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.
A simple electrical impedance-matching network requires one capacitor and one inductor. In the figure to the right, R1 > R2, however, either R1 or R2 may be the source and the other the load. One of X1 or X2 must be an inductor and the other must be a capacitor. One reactance is in parallel with the source (or load), and the other is in series with the load (or source). If a reactance is in parallel with the source, the effective network matches from high to low impedance.
The analysis is as follows. and real load impedance of . If a reactance is in parallel with the source impedance, the combined impedance can be written as:Consider a real source impedance of
If the imaginary part of the above impedance is canceled by the series reactance, the real part is
Note, , the reactance in parallel, has a negative reactance because it is typically a capacitor. This gives the L-network the additional feature of harmonic suppression since it is a low pass filter too.
The inverse connection (impedance step-up) 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 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.
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 (such as radio and fiber optics)—where the length of the line is long compared to the wavelength of the signal (the signal changes rapidly compared to the time it takes to travel from source to load)— 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. (When the length of the line is short compared to the wavelength, impedance mismatch is the basis of transmission-line impedance transformers; see previous section.) In radio-frequency (RF) systems, a common value for source and load impedances is 50 ohms. A typical RF load is a quarter-wave ground plane antenna (37 ohms with an ideal ground plane); it can be matched to 50 ohms by using a modified 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, which is the negative of the voltage reflection coefficient. If the wave encounters an open at the load end, positive voltage and negative current pulses are transmitted back toward the source (negative current means the current is going the opposite direction). Thus, at each boundary there are four reflection coefficients (voltage and current on one side, and voltage and current on the other side). 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 (unless otherwise 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 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.
In a transmission line, a wave travels from the source along the line. Suppose the wave hits a boundary (an abrupt change in impedance). Some of the wave is reflected back, while some keeps moving onwards. (Assume there is only one boundary, at the load.)
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 (or to transmit information with minimal error), so the reflection is as small as possible. This is achieved by matching the impedances and so that they are equal ().
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.
where is the one-way 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
where is the open circuit (or unloaded) output voltage from the source.
Note that if there is a perfect match at both ends
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 (2- to 4-wire conversion) to operate correctly. As the signals are sent and received on the same two-wire circuit to the central office (or exchange), 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 (nominal). 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.
Audio amplifiers typically do not match impedances, but provide an output impedance that is lower than the load impedance (such as < 0.1 ohm in typical semiconductor amplifiers), 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:
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.
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 (or absorbed), 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 (which is acted upon by vibrations in air) and the fluid-filled inner ear.
Horns in loudspeaker systems are used like transformers in electrical circuits to match the impedance of the transducer to the impedance of the air. This principle is used in both horn loudspeakers and musical instruments. Because most driver impedances are poorly matched to the impedance of free air at low frequencies, loudspeaker enclosures are designed to both match impedance and minimize destructive phase cancellations between output from the front and rear of a speaker cone. The loudness of sound produced in air from a loudspeaker is directly related to the ratio of the diameter of the speaker to the wavelength of the sound being produced: larger speakers can produce lower frequencies at a higher level than smaller speakers. Elliptical speakers are a complex case, acting like large speakers lengthwise and small speakers crosswise. Acoustic impedance matching (or the lack of it) affects the operation of a megaphone, an echo and soundproofing.
A similar effect occurs when light (or any electromagnetic wave) 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 (that depends on the propagation direction) 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.
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 (as in Newton's cradle for example). In this case, the masses act as "mechanical impedances",[ dubious ] which must be matched. If and are the masses of the moving and stationary bodies, and P is the momentum of the system (which remains constant throughout the collision), 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 (explosives). 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 (such as the opposite side of the target), 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.
The characteristic impedance or surge impedance (usually written Z0) 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.
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.
In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected wave to the incident wave, with each expressed as phasors. For example, it is used in optics to calculate the amount of light that is reflected from a surface with a different index of refraction, such as a glass surface, or in an electrical transmission line to calculate how much of the electromagnetic wave is reflected by an impedance. The reflection coefficient is closely related to the transmission coefficient. The reflectance of a system is also sometimes called a "reflection coefficient".
In telecommunications, return loss is a measure in relative terms of the power of the signal reflected by a discontinuity in a transmission line or optical fiber. This discontinuity can be caused by a mismatch between the termination or load connected to the line and the characteristic impedance of the line. It is usually expressed as a ratio in decibels (dB);
In radio engineering and telecommunications, standing wave ratio (SWR) is a measure of impedance matching of loads to the characteristic impedance of a transmission line or waveguide. Impedance mismatches result in standing waves along the transmission line, and SWR is defined as the ratio of the partial standing wave's amplitude at an antinode (maximum) to the amplitude at a node (minimum) along the line.
In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmission must be taken into account. This applies especially to radio-frequency engineering because the short wavelengths mean that wave phenomena arise over very short distances. However, the theory of transmission lines was historically developed to explain phenomena on very long telegraph lines, especially submarine telegraph cables.
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave amplitudes decrease according to the inverse square law as they expand into three dimensional space.
In electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a source with a finite internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals. Moritz von Jacobi published the maximum power (transfer) theorem around 1840; it is also referred to as "Jacobi's law".
In electric and electronic systems, reactance is the opposition of a circuit element to the flow of current due to that element's inductance or capacitance. Greater reactance leads to smaller currents for the same voltage applied. Reactance is similar to electric resistance in this respect, but differs in that reactance does not lead to dissipation of electrical energy as heat. Instead, energy is stored in the reactance, and later returned to the circuit whereas a resistance continuously loses energy.
In electronics, negative resistance (NR) is a property of some electrical circuits and devices in which an increase in voltage across the device's terminals results in a decrease in electric current through it.
In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance, analogous to how conductance & resistance are defined. The SI unit of admittance is the siemens ; the older, synonymous unit is mho, and its symbol is ℧. Oliver Heaviside coined the term admittance in December 1887.
The Smith chart, invented by Phillip H. Smith (1905–1987), and T. Mizuhashi, is a graphical calculator 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. While the use of paper Smith charts for solving the complex mathematics involved in matching problems has been largely replaced by software based methods, the Smith chart is still a very useful method of showing how RF parameters behave at one or more frequencies, an alternative to using tabular information. Thus most RF circuit analysis software includes a Smith chart option for the display of results and all but the simplest impedance measuring instruments can plot measured results on a Smith chart display.
Antenna tuner, matching network, matchbox, transmatch, antenna tuning unit (ATU), antenna coupler, and feedline coupler are all equivalent names for a device connected between a radio transmitter and its antenna, to improve power transfer between them by matching the load impedance of the radio to the combined input impedance of the feedline and the antenna.
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.
The output impedance of an electrical network is the measure of the opposition to current flow (impedance), both static (resistance) and dynamic (reactance), into the load network being connected that is internal to the electrical source. The output impedance is a measure of the source's propensity to drop in voltage when the load draws current, the source network being the portion of the network that transmits and the load network being the portion of the network that consumes.
The standing wave ratio meter, SWR meter, ISWR meter, or VSWR meter measures the standing wave ratio (SWR) in a transmission line. The meter indirectly measures the degree of mismatch between a transmission line and its load. Electronics technicians use it to adjust radio transmitters and their antennas and feedlines to be impedance matched so they work together properly, and evaluate the effectiveness of other impedance matching efforts.
A test probe is a physical device used to connect electronic test equipment to a device under test (DUT). Test probes range from very simple, robust devices to complex probes that are sophisticated, expensive, and fragile. Specific types include test prods, oscilloscope probes and current probes. A test probe is often supplied as a test lead, which includes the probe, cable and terminating connector.
A quarter-wave impedance transformer, often written as λ/4 impedance transformer, is a transmission line or waveguide used in electrical engineering of length one-quarter wavelength (λ), terminated with some known impedance. It presents at its input the dual of the impedance with which it is terminated.
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.
Performance modelling is the abstraction of a real system into a simplified representation to enable the prediction of performance. The creation of a model can provide insight into how a proposed or actual system will or does work. This can, however, point towards different things to people belonging to different fields of work.