Impedance matching
Encyclopedia
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 and/or minimize reflections
from the load.
In the case of a complex source impedance ZS and load impedance ZL, maximum power transfer is obtained when
where * indicates the complex conjugate
.
Minimum reflection is obtained when
The concept of impedance matching was originally developed for electrical power
, but can be applied to any other field where a form of energy (not necessarily electrical) is transferred between a source and a load. An alternative to impedance matching is impedance bridging
, where the load impedance is chosen to be much larger than the source impedance and maximizing voltage transfer (rather than power) is the goal.
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
to the source that is in phase
with a sinusoidal source signal and reactance that is out of phase with a sinusoidal source signal. The total impedance (symbol: Z) is the vector sum of the resistance (symbol: R; a real number
) and the reactance (symbol: X; an imaginary number
).
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.
of the transmission line. Although source and load should each be totally resistive for this form of matching to work, the more-general term "impedance" is still used to describe the source and load characteristics. Any reactance present in the source or the load will affect the match.
at a single frequency. In this case,
(where * indicates the complex conjugate
).
If the signals are kept within the narrow frequency range for which the matching network was designed, reflections (in this narrow frequency band only) are also minimized. For the case of purely resistive source and load impedances, all reactance terms are zero and the formula above reduces to
as would be expected.
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 desirable. 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; however, the voltage transfer is higher and less prone to corruption than if the impedances had been matched. 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 and/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.
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 transformer
s, resistor
s, inductor
s, capacitor
s and transmission lines. These passive (and active) impedance-matching devices are optimized for different applications and include balun
s, antenna tuner
s (sometimes called ATUs or roller-coasters, because of their appearance), acoustic horns, matching networks, and terminators
.
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 Turns Ratio = √ [(Load Resistance)/(Source Resistance)].
signals.
Some special situations (such as radio tuners and transmitters) use tuned filters
such as stubs to match impedances at specific frequencies. These can distribute different frequencies to different places in the circuit. In addition, there is the closely related idea of power factor correction devices, 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.
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 (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 chart
s.
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 line
s (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; 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 (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.
Let:
On the line side of the boundary
and 
and on the load side 
where 
, 
, 
, 
, 
, 
, and 
are phasor
s.
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 (
).
, 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.
Note that if there is a perfect match at both ends
and 
and then 
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 amplifier
s 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 MOSFET
s 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.
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.
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 (and because of out-of-phase cancellations between output from the front and rear of a speaker cone), 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 (or the lack of it) affects the operation of a megaphone
, an echo
and soundproofing
.
(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
.
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 in the above mathematical-proof section.
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 spall
ing. 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.
Electronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
, impedance matching is the practice of designing the input impedance
Input impedance
The input impedance of an electrical network is the equivalent impedance "seen" by a power source connected to that network. If the source provides known voltage and current, such impedance can be calculated using Ohm's Law...
of an electrical load (or the output impedance
Output impedance
The output impedance, source impedance, or internal impedance of an electronic device is the opposition exhibited by its output terminals to an alternating current of a particular frequency as a result of resistance, inductance and capacitance...
of its corresponding signal source) to maximize the power transfer and/or minimize reflections
Signal reflection
Signal reflection occurs when a signal is transmitted along a transmission medium, such as a copper cable or an optical fiber, some of the signal power may be reflected back to its origin rather than being carried all the way along the cable to the far end. This happens because imperfections in the...
from the load.
where * indicates the complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
.
Minimum reflection is obtained when
The concept of impedance matching was originally developed for electrical power
Electric power
Electric power is the rate at which electric energy is transferred by an electric circuit. The SI unit of power is the watt.-Circuits:Electric power, like mechanical power, is represented by the letter P in electrical equations...
, but can be applied to any other field where a form of energy (not necessarily electrical) is transferred between a source and a load. An alternative to impedance matching is impedance bridging
Impedance bridging
In electronics, especially audio and sound recording, a high impedance bridging, voltage bridging, or simply bridging connection is one which maximizes transfer of a voltage signal to the load...
, where 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 impedanceElectrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
is perhaps the most commonly known. Electrical impedance, like electrical resistance, is measured in ohms. In general, impedance has a complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
value; this means that loads generally have a resistance
Electrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
to the source that is in phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
with a sinusoidal source signal and reactance that is out of phase with a sinusoidal source signal. The total impedance (symbol: Z) is the vector sum of the resistance (symbol: R; a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
) and the reactance (symbol: X; an imaginary number
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...
).
In simple cases (such as low-frequency or direct-current power transmission) the reactance may be negligible
Negligible
Negligible refers to the quantities so small that they can be ignored when studying the larger effect. Although related to the more mathematical concepts of infinitesimal, the idea of negligibility is particularly useful in practical disciplines like physics, chemistry, mechanical and electronic...
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 (broadband) matching
Impedance matching to minimize reflections and maximize power transfer over a relatively large bandwidth (reflection-less or broadband matching) is the most commonly used. To prevent reflections of the signal back to the source, the load (which must be totally resistive) must be matched to the source impedance (which must also be totally resistive). In this case, if a transmission line is used to connect the source and load together, it must also be the same impedance: Zload = Zline = Zsource, where Zline is the characteristic impedanceCharacteristic impedance
The characteristic impedance or surge impedance of a uniform transmission line, usually written Z_0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm...
of the transmission line. Although source and load should each be totally resistive for this form of matching to work, the more-general term "impedance" is still used to describe the source and load characteristics. Any reactance present in the source or the load will affect the match.
Complex conjugate matching
This is used in cases in which the source and load are both reactive. This form of impedance matching can only maximize the power transfer between a reactive source and a reactive loadExternal electric load
If an electric circuit has a well-defined output terminal, the circuit connected to this terminal is the load....
at a single frequency. In this case,
- Zload = Zsource*
(where * indicates the complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
).
If the signals are kept within the narrow frequency range for which the matching network was designed, reflections (in this narrow frequency band only) are also minimized. For the case of purely resistive source and load impedances, all reactance terms are zero and the formula above reduces to
- Zload = Zsource
as would be expected.
Power transfer
Whenever a source of power with a fixed output impedance such as an electric signal source, a radioRadio
Radio is the transmission of signals through free space by modulation of electromagnetic waves with frequencies below those of visible light. Electromagnetic radiation travels by means of oscillating electromagnetic fields that pass through the air and the vacuum of space...
transmitter
Transmitter
In electronics and telecommunications a transmitter or radio transmitter is an electronic device which, with the aid of an antenna, produces radio waves. The transmitter itself generates a radio frequency alternating current, which is applied to the antenna. When excited by this alternating...
or a mechanical sound (e.g., a loudspeaker
Loudspeaker
A loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...
) operates into a load
External electric load
If an electric circuit has a well-defined output terminal, the circuit connected to this terminal is the load....
, the maximum possible power
Power (physics)
In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...
is delivered to the load when the impedance of the load (load impedance or input impedance
Input impedance
The input impedance of an electrical network is the equivalent impedance "seen" by a power source connected to that network. If the source provides known voltage and current, such impedance can be calculated using Ohm's Law...
) is equal to the complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
of the impedance of the source (that is, its internal impedance or output impedance
Output impedance
The output impedance, source impedance, or internal impedance of an electronic device is the opposition exhibited by its output terminals to an alternating current of a particular frequency as a result of resistance, inductance and capacitance...
). 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
Maximum power theorem
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 be equal to the resistance of the source as viewed from the output terminals...
for a mathematical proof).
Impedance matching is not always desirable. 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; however, the voltage transfer is higher and less prone to corruption than if the impedances had been matched. This maximum-voltage connection is a common configuration called impedance bridging
Impedance bridging
In electronics, especially audio and sound recording, a high impedance bridging, voltage bridging, or simply bridging connection is one which maximizes transfer of a voltage signal to the load...
or voltage bridging, and is widely used in signal processing. In such applications, delivering a high voltage (to minimize signal degradation during transmission and/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
Telephone
The telephone , colloquially referred to as a phone, is a telecommunications device that transmits and receives sounds, usually the human voice. Telephones are a point-to-point communication system whose most basic function is to allow two people separated by large distances to talk to each other...
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
Hybrid coil
A hybrid coil is a transformer that has three windings, and which is designed to be configured as a circuit having four branches, that are conjugate in pairs....
used at central exchange equipment to separate outgoing from incoming speech, so these could be amplified or fed to a four-wire circuit
Four-wire circuit
In telecommunication, a four-wire circuit is a two-way circuit using two paths so arranged that the respective signals are transmitted in one direction only by one path and in the other direction by the other path...
. 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 Rload = Rsource* (complex conjugate matching). Given a source with a fixed voltage and fixed source impedance, the maximum power theoremMaximum power theoremIn 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 be equal to the resistance of the source as viewed from the output terminals...
says this is the only way to extract the maximum power from the source. - Devices intended to present an apparent load of Rload = Rline (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 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 bridgingImpedance bridgingIn electronics, especially audio and sound recording, a high impedance bridging, voltage bridging, or simply bridging connection is one which maximizes transfer of a voltage signal to the load...
connection also minimizes distortionDistortionA distortion is the alteration of the original shape of an object, image, sound, waveform or other form of information or representation. Distortion is usually unwanted, and often many methods are employed to minimize it in practice...
and electromagnetic interferenceElectromagnetic interferenceElectromagnetic interference is disturbance that affects an electrical circuit due to either electromagnetic induction or electromagnetic radiation emitted from an external source. The disturbance may interrupt, obstruct, or otherwise degrade or limit the effective performance of the circuit...
; 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 transformer
Transformer
A transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying current in the first or primary winding creates a varying magnetic flux in the transformer's core and thus a varying magnetic field...
s, resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
s, inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
s, capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
s and transmission lines. These passive (and active) impedance-matching devices are optimized for different applications and include balun
Balun
A balun is a type of electrical transformer that can convert electrical signals that are balanced about ground to signals that are unbalanced , and the reverse. They are also often used to connect lines of differing impedance...
s, antenna tuner
Antenna tuner
An antenna tuner, transmatch or antenna tuning unit is a device connected between a radio transmitter or receiver and its antenna to improve the efficiency of the power transfer between them by matching the impedance of the equipment to the antenna...
s (sometimes called ATUs or roller-coasters, because of their appearance), acoustic horns, matching networks, and terminators
Electrical termination
Electrical termination of a signal involves providing a terminator at the end of a wire or cable to prevent an RF signal from being reflected back from the end, causing interference...
.
Transformers
TransformersTransformer
A transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying current in the first or primary winding creates a varying magnetic flux in the transformer's core and thus a varying magnetic field...
are sometimes used to match the impedances of circuits. A transformer converts alternating current
Alternating current
In alternating current the movement of electric charge periodically reverses direction. In direct current , the flow of electric charge is only in one direction....
at one voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
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
Balun
A balun is a type of electrical transformer that can convert electrical signals that are balanced about ground to signals that are unbalanced , and the reverse. They are also often used to connect lines of differing impedance...
transformer. This transformer converts a balanced signal from the antenna (via 300-ohm twin-lead
Twin-lead
Twin-lead cable is a two-conductor ribbon cable used as a transmission line to carry radio frequency signals.- Characteristics and uses:...
) into an unbalanced signal (75-ohm coaxial cable such as RG-6
RG-6
RG-6/U is a common type of coaxial cable used in a wide variety of residential and commercial applications. The term "RG-6" itself is quite generic and refers to a wide variety of cable designs, which differ from one another in shielding characteristics, center conductor composition, and...
). 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 Turns Ratio = √ [(Load Resistance)/(Source Resistance)].
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 (usually) used to transfer line levelLine level
Line level is a term used to denote the strength of an audio signal used to transmit analog sound between audio components such as CD and DVD players, TVs, audio amplifiers, and mixing consoles, and sometimes MP3 players....
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 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.Some special situations (such as radio tuners and transmitters) use tuned filters
Electronic filter
Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
such as stubs to match impedances at specific frequencies. These can distribute different frequencies to different places in the circuit. In addition, there is the closely related idea of power factor correction devices, 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
Maximum power point tracker
Maximum power point tracking is a technique that grid tie inverters, solar battery chargers and similar devices use to get the maximum possible power from the PV array. Solar cells have a complex relationship between solar irradiation, temperature and total resistance that produces a non-linear...
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
Impedance bridging
In electronics, especially audio and sound recording, a high impedance bridging, voltage bridging, or simply bridging connection is one which maximizes transfer of a voltage signal to the load...
connection; it emulates a high-voltage, low-resistance source to maximize efficiency.
L-section
A simple electrical impedance-matching network requires one capacitor and one inductor. 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 L-section is inherently a narrowbandNarrowband
In radio, narrowband describes a channel in which the bandwidth of the message does not significantly exceed the channel's coherence bandwidth. It is a common misconception that narrowband refers to a channel which occupies only a "small" amount of space on the radio spectrum.The opposite of...
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 asThe 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
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....
of the inductor. Multiple L-sections can be wired in cascade to achieve higher impedance ratios or greater bandwidth. Transmission line
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
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 chart
Smith chart
The Smith chart, invented by Phillip H. Smith , is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency engineering to assist in solving problems with transmission lines and matching circuits...
s.
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 waveStanding wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...
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 line
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
s (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
Characteristic impedance
The characteristic impedance or surge impedance of a uniform transmission line, usually written Z_0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm...
(
) 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 antennaAntenna (radio)
An antenna is an electrical device which converts electric currents into radio waves, and vice versa. It is usually used with a radio transmitter or radio receiver...
(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
Reflection coefficient
The reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A reflection coefficient describes either the amplitude or the intensity of a reflected wave relative to an incident wave...
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 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.
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 (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.)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 phasorPhasor
Phasor is a phase vector representing a sine wave.Phasor may also be:* Phasor , a stereo music, sound and speech synthesizer for the Apple II computer* Phasor measurement unit, a device that measures phasors on an electricity grid...
s.
At a boundary, voltage and current must be continuous, therefore
All these conditions are satisfied by
where :
the reflection coefficientReflection coefficient
The reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A reflection coefficient describes either the amplitude or the intensity of a reflected wave relative to an incident wave...
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 ().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 (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 Telephone systems
TelephoneTelephone
The telephone , colloquially referred to as a phone, is a telecommunications device that transmits and receives sounds, usually the human voice. Telephones are a point-to-point communication system whose most basic function is to allow two people separated by large distances to talk to each other...
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
Hybrid coil
A hybrid coil is a transformer that has three windings, and which is designed to be configured as a circuit having four branches, that are conjugate in pairs....
(2- to 4-wire conversion) to operate correctly. As the signals are sent and received on the same two-wire circuit
Two-wire circuit
In telecommunication, a two-wire circuit is characterized by supporting transmission in two directions simultaneously, as opposed to four-wire circuits, which have separate pairs for transmit and receive. In either case they are twisted pairs. Telephone lines are almost all two wire, while trunks...
to the central office (or exchange), cancellation is necessary at the telephone earpiece so excessive sidetone
Sidetone
Sidetone is audible feedback to someone who is speaking. The term is most used in telecommunication contexts.-Telephony:In telephony, sidetone is the effect of sound that is picked up by the telephone's mouthpiece and in real-time introduced at a low level into the earpiece of the same handset,...
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
Voice frequency
A voice frequency or voice band is one of the frequencies, within part of the audio range, that is used for the transmission of speech.In telephony, the usable voice frequency band ranges from approximately 300 Hz to 3400 Hz...
band.
Loudspeaker amplifiers
Audio amplifier
An audio amplifier is an electronic amplifier that amplifies low-power audio signals to a level suitable for driving loudspeakers and is the final stage in a typical audio playback chain.The preceding stages in such a chain are low power audio amplifiers which perform tasks like pre-amplification,...
s 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
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
amplifiers), for improved speaker damping. For vacuum tube
Vacuum tube
In electronics, a vacuum tube, electron tube , or thermionic valve , reduced to simply "tube" or "valve" in everyday parlance, is a device that relies on the flow of electric current through a vacuum...
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
Transformer
A transformer is a device that transfers electrical energy from one circuit to another through inductively coupled conductors—the transformer's coils. A varying current in the first or primary winding creates a varying magnetic flux in the transformer's core and thus a varying magnetic field...
in vacuum-tube-based amplifiers has two basic functions:
- Separation of the ACAlternating currentIn alternating current the movement of electric charge periodically reverses direction. In direct current , the flow of electric charge is only in one direction....
component (which contains the audio signals) from the DCDirect currentDirect current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...
component (supplied by the power supplyPower supplyA power supply is a device that supplies electrical energy to one or more electric loads. The term is most commonly applied to devices that convert one form of electrical energy to another, though it may also refer to devices that convert another form of energy to electrical energy...
) 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 pentodePentodeA pentode is an electronic device having five active electrodes. The term most commonly applies to a three-grid vacuum tube , which was invented by the Dutchman Bernhard D.H. Tellegen in 1926...
s (such as the EL34EL34The EL34 is a thermionic valve or vacuum tube of the power pentode type. It has an international octal base and is found mainly in the final output stages of audio amplification circuits and was designed to be suitable as a series regulator by virtue of its high permissible voltage between heater...
) 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 MOSFET
MOSFET
The metal–oxide–semiconductor field-effect transistor is a transistor used for amplifying or switching electronic signals. The basic principle of this kind of transistor was first patented by Julius Edgar Lilienfeld in 1925...
s or power transistors
Transistor
A transistor is a semiconductor device used to amplify and switch electronic signals and power. It is composed of a semiconductor material with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals changes the current...
has a very low output impedance. If they are properly balanced, there is no need for a transformer or a large electrolytic capacitor
Electrolytic capacitor
An electrolytic capacitor is a type of capacitor that uses an electrolyte, an ionic conducting liquid, as one of its plates, to achieve a larger capacitance per unit volume than other types. They are often referred to in electronics usage simply as "electrolytics"...
to separate AC from DC current.
Acoustics
Similar to electrical transmission lines, an impedance matching problem exists when transferring sound energy from one medium to another. If the acoustic impedanceAcoustic impedance
The acoustic impedance at a particular frequency indicates how much sound pressure is generated by a given air vibration at that frequency. The acoustic impedance Z is frequency dependent and is very useful, for example, for describing the behaviour of musical wind instruments...
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
Medical ultrasonography
Diagnostic sonography is an ultrasound-based diagnostic imaging technique used for visualizing subcutaneous body structures including tendons, muscles, joints, vessels and internal organs for possible pathology or lesions...
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.
Horns
Horn (acoustic)
A horn is a tapered sound guide designed to provide an acoustic impedance match between a sound source and free air. This has the effect of maximizing the efficiency with which sound waves from the particular source are transferred to the air...
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
Horn (instrument)
The horn is a brass instrument consisting of about of tubing wrapped into a coil with a flared bell. A musician who plays the horn is called a horn player ....
. 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 out-of-phase cancellations between output from the front and rear of a speaker cone), loudspeaker enclosures both match impedances and prevent interference. Sound, coupling with air, from a loudspeaker
Loudspeaker
A loudspeaker is an electroacoustic transducer that produces sound in response to an electrical audio signal input. Non-electrical loudspeakers were developed as accessories to telephone systems, but electronic amplification by vacuum tube made loudspeakers more generally useful...
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 (or the lack of it) affects the operation of a megaphone
Megaphone
A megaphone, speaking-trumpet, bullhorn, blowhorn, or loud hailer is a portable, usually hand-held, cone-shaped horn used to amplify a person’s voice or other sounds towards a targeted direction. This is accomplished by channelling the sound through the megaphone, which also serves to match the...
, an echo
Echo (phenomenon)
In audio signal processing and acoustics, an echo is a reflection of sound, arriving at the listener some time after the direct sound. Typical examples are the echo produced by the bottom of a well, by a building, or by the walls of an enclosed room and an empty room. A true echo is a single...
and soundproofing
Soundproofing
Soundproofing is any means of reducing the sound pressure with respect to a specified sound source and receptor. There are several basic approaches to reducing sound: increasing the distance between source and receiver, using noise barriers to reflect or absorb the energy of the sound waves, using...
.
Optics
A similar effect occurs when lightLight
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
(or any electromagnetic wave) hits the interface between two media with different refractive indices
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
. 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
Fresnel equations
The Fresnel equations , deduced by Augustin-Jean Fresnel , describe the behaviour of light when moving between media of differing refractive indices...
. Unwanted reflections can be reduced by the use of an anti-reflection optical coating
Optical coating
An optical coating is one or more thin layers of material deposited on an optical component such as a lens or mirror, which alters the way in which the optic reflects and transmits light. One type of optical coating is an antireflection coating, which reduces unwanted reflections from surfaces, and...
.
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 (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 in the above mathematical-proof section.
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 spall
Spall
Spall are flakes of a material that are broken off a larger solid body and can be produced by a variety of mechanisms, including as a result of projectile impact, corrosion, weathering, cavitation, or excessive rolling pressure...
ing. 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
- Maximum power theoremMaximum power theoremIn 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 be equal to the resistance of the source as viewed from the output terminals...
- Power (physics)Power (physics)In physics, power is the rate at which energy is transferred, used, or transformed. For example, the rate at which a light bulb transforms electrical energy into heat and light is measured in watts—the more wattage, the more power, or equivalently the more electrical energy is used per unit...
- Reflection coefficientReflection coefficientThe reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A reflection coefficient describes either the amplitude or the intensity of a reflected wave relative to an incident wave...
- Ringing (signal)Ringing (signal)In electronics, signal processing, and video, ringing is unwanted oscillation of a signal, particularly in the step response...
- Standing wave ratioStanding wave ratioIn telecommunications, standing wave ratio is the ratio of the amplitude of a partial standing wave at an antinode to the amplitude at an adjacent node , in an electrical transmission line....
- Transmission lineTransmission lineIn communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
- Wet TransformerWet TransformerA wet transformer is a special type of impedance matching transformer that is used to couple telephone equipment to electronic equipment. It is made with a primary winding that is designed to have a DC voltage impressed upon it...
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