Impedance matching


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 which is equivalent to an electrical resistance in series with a reactance. An electrical load, such as a light bulb, transmission line or antenna similarly has an impedance which is equivalent to a resistance in series with a reactance. The maximum power theorem says 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. Another way of saying this is that 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 circuit, the condition is satisfied if the load resistance equals the source resistance. In an alternating current 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.

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, 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 which forms the real part of Z and a reactance component which forms the imaginary part of Z.
In simple cases the reactance may be negligible or zero; the impedance can be considered a pure resistance, expressed as a real number. In the following summary we will consider the general case when resistance and reactance are both significant, and the special case in which the reactance is negligible.

Reflection-less matching

Impedance matching to minimize reflections is achieved by making the load impedance equal to the source impedance. If the source impedance, load impedance and transmission line characteristic impedance are purely resistive, then reflection-less matching is the same as maximum power transfer matching.

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.

Power transfer

Whenever a source of power with a fixed output impedance such as an electric signal source, a radio transmitter or a mechanical sound operates into a load, the maximum possible power is delivered to the load when the impedance of the load is equal to the complex conjugate of the impedance of the source. For two impedances to be complex conjugates their resistances must be equal, and their reactances must be equal in magnitude but of opposite signs. In low-frequency or DC systems the reactances are zero, or small enough to be ignored. In this case, maximum power transfer occurs when the resistance of the load is equal to the resistance of the source.
Impedance matching is not always necessary. For example, if a source with a low impedance is connected to a load with a high impedance the power that can pass through the connection is limited by the higher impedance. This maximum-voltage connection is a common configuration called impedance bridging or voltage bridging, and is widely used in signal processing. In such applications, delivering a high voltage is often more important than maximum power transfer.
In older audio systems, the source and load resistances were matched at 600 ohms. One reason for this was to maximize power transfer, as there were no amplifiers available that could restore lost signal. Another reason was to ensure correct operation of the hybrid transformers used at central exchange equipment to separate outgoing from incoming speech, so these could be amplified or fed to a four-wire circuit. Most modern audio circuits, on the other hand, use active amplification and filtering and can use voltage-bridging connections for greatest accuracy. Strictly speaking, impedance matching only applies when both source and load devices are linear; however, matching may be obtained between nonlinear devices within certain operating ranges.

Impedance-matching devices

Adjusting the source impedance or the load impedance, in general, is called "impedance matching". There are three ways to improve an impedance mismatch, all of which are called "impedance matching":
There are a variety of devices used between a source of energy and a load that perform "impedance matching". To match electrical impedances, engineers use combinations of transformers, resistors, inductors, capacitors and transmission lines. These passive impedance-matching devices are optimized for different applications and include baluns, antenna tuners, acoustic horns, matching networks, and terminators.

Transformers

s are sometimes used to match the impedances of circuits. A transformer converts alternating current at one voltage to the same waveform at another voltage. The power input to the transformer and output from the transformer is the same. The side with the lower voltage is at low impedance, and the side with the higher voltage is at a higher impedance.
One example of this method involves a television balun transformer. This transformer converts a balanced signal from the antenna into an unbalanced signal. To match the impedances of both devices, both cables must be connected to a matching transformer with a turns ratio of 2. In this example, the 75-ohm cable is connected to the transformer side with fewer turns; the 300-ohm line is connected to the transformer side with more turns. The formula for calculating the transformer turns ratio for this example is:

Resistive network

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

Stepped transmission line

Most lumped-element devices can match a specific range of load impedances. For example, in order to match an inductive load into a real impedance, a capacitor needs to be used. If the load impedance becomes capacitive, the matching element must be replaced by an inductor. In many cases, there is a need to use the same circuit to match a broad range of load impedance and thus simplify the circuit design. This issue was addressed by the stepped transmission line, where multiple, serially placed, quarter-wave dielectric slugs are used to vary a transmission line's characteristic impedance. By controlling the position of each element, a broad range of load impedances can be matched without having to reconnect the circuit.

Filters

are frequently used to achieve impedance matching in telecommunications and radio engineering. In general, it is not theoretically possible to achieve perfect impedance matching at all frequencies with a network of discrete components. Impedance matching networks are designed with a definite bandwidth, take the form of a filter, and use filter theory in their design.
Applications requiring only a narrow bandwidth, such as radio tuners and transmitters, might use a simple tuned filter such as a stub. This would provide a perfect match at one specific frequency only. Wide bandwidth matching requires filters with multiple sections.

L-section

A simple electrical impedance-matching network requires one capacitor and one inductor. 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, and the other is in series with the load. If a reactance is in parallel with the source, the effective network matches from high to low impedance.
The 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
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 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.

Transmission lines

Impedance bridging is unsuitable for RF connections, because it causes power to be reflected back to the source from the boundary between the high and the low impedances. The reflection creates a standing wave if there is reflection at both ends of the transmission line, which leads to further power waste and may cause frequency-dependent loss. In these systems, impedance matching is desirable.
In electrical systems involving transmission lines —where the length of the line is long compared to the wavelength of the signal — the impedances at each end of the line must be matched to the transmission line's characteristic impedance to prevent reflections of the signal at the ends of the line. In radio-frequency systems, a common value for source and load impedances is 50 ohms. A typical RF load is a quarter-wave ground plane antenna ; 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, 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. Thus, at each boundary there are four reflection coefficients. All four are the same, except that two are positive and two are negative. The voltage reflection coefficient and current reflection coefficient on the same side have opposite signs. Voltage reflection coefficients on opposite sides of the boundary have opposite signs.
Because they are all the same except for sign it is traditional to interpret the reflection coefficient as the voltage reflection coefficient. Either end of a transmission line can be a source or a load, so there is no inherent preference for which side of the boundary is medium 1 and which side is medium 2. With a single transmission line it is customary to define the voltage reflection coefficient for a wave incident on the boundary from the transmission line side, regardless of whether a source or load is connected on the other side.

Single-source transmission line driving a load

Load-end conditions

In a transmission line, a wave travels from the source along the line. Suppose the wave hits a boundary. Some of the wave is reflected back, while some keeps moving onwards.
Let
On the line side of the boundary and and on the load side where,,, ,,, and are phasors.
At a boundary, voltage and current must be continuous, therefore
All these conditions are satisfied by
where the reflection coefficient going from the transmission line to the load.
The purpose of a transmission line is to get the maximum amount of energy to the other end of the line, so the reflection is as small as possible. This is achieved by matching the impedances and so that they are equal.
Transfer function
where is the open circuit output voltage from the source.
Note that if there is a perfect match at both ends
and then

Electrical examples

Telephone systems

systems also use matched impedances to minimise echo on long-distance lines. This is related to transmission-line theory. Matching also enables the telephone hybrid coil to operate correctly. As the signals are sent and received on the same two-wire circuit to the central office, cancellation is necessary at the telephone earpiece so excessive sidetone is not heard. All devices used in telephone signal paths are generally dependent on matched cable, source and load impedances. In the local loop, the impedance chosen is 600 ohms. Terminating networks are installed at the exchange to offer the best match to their subscriber lines. Each country has its own standard for these networks, but they are all designed to approximate about 600 ohms over the voice frequency band.

Loudspeaker amplifiers

s typically do not match impedances, but provide an output impedance that is lower than the load impedance, for improved speaker damping. For vacuum tube amplifiers, impedance-changing transformers are often used to get a low output impedance, and to better match the amplifier's performance to the load impedance. Some tube amplifiers have output transformer taps to adapt the amplifier output to typical loudspeaker impedances.
The output transformer in vacuum-tube-based amplifiers has two basic functions:
The impedance of the loudspeaker on the secondary coil of the transformer will be transformed to a higher impedance on the primary coil in the circuit of the power pentodes by the square of the turns ratio, which forms the impedance scaling factor.
The output stage in common-drain or common-collector semiconductor-based end stages with MOSFETs or power transistors has a very low output impedance. If they are properly balanced, there is no need for a transformer or a large electrolytic capacitor to separate AC from DC current.

Non-electrical examples

Acoustics

Similar to electrical transmission lines, an impedance matching problem exists when transferring sound energy from one medium to another. If the acoustic impedance of the two media are very different most sound energy will be reflected, rather than transferred across the border. The gel used in medical ultrasonography helps transfer acoustic energy from the transducer to the body and back again. Without the gel, the impedance mismatch in the transducer-to-air and the air-to-body discontinuity reflects almost all the energy, leaving very little to go into the body.
The bones in the middle ear provide impedance matching between the eardrum and the fluid-filled inner ear.
Horns are used like transformers, matching the impedance of the transducer to the impedance of the air. This principle is used in both horn loudspeakers and musical instruments. Most loudspeaker systems contain impedance matching mechanisms, especially for low frequencies. Because most driver impedances which are poorly matched to the impedance of free air at low frequencies, loudspeaker enclosures both match impedances and prevent interference. Sound, coupling with air, from a loudspeaker is related to the ratio of the diameter of the speaker to the wavelength of the sound being reproduced. That is, larger speakers can produce lower frequencies at a higher level than smaller speakers for this reason. Elliptical speakers are a complex case, acting like large speakers lengthwise and small speakers crosswise. Acoustic impedance matching affects the operation of a megaphone, an echo and soundproofing.

Optics

A similar effect occurs when light hits the interface between two media with different refractive indices. For non-magnetic materials, the refractive index is inversely proportional to the material's characteristic impedance. An optical or wave impedance can be calculated for each medium, and may be used in the transmission-line reflection equation
to calculate reflection and transmission coefficients for the interface. For non-magnetic dielectrics, this equation is equivalent to the Fresnel equations. Unwanted reflections can be reduced by the use of an anti-reflection optical coating.

Mechanics

If a body of mass m collides elastically with a second body, maximum energy transfer to the second body will occur when the second body has the same mass m. In a head-on collision of equal masses, the energy of the first body will be completely transferred to the second body. In this case, the masses act as "mechanical impedances", which must be matched. If and are the masses of the moving and stationary bodies, and P is the momentum of the system, the energy of the second body after the collision will be E2:
which is analogous to the power-transfer equation.
These principles are useful in the application of highly energetic materials. If an explosive charge is placed on a target, the sudden release of energy causes compression waves to propagate through the target radially from the point-charge contact. When the compression waves reach areas of high acoustic impedance mismatch, tension waves reflect back and create spalling. The greater the mismatch, the greater the effect of creasing and spalling will be. A charge initiated against a wall with air behind it will do more damage to the wall than a charge initiated against a wall with soil behind it.