Impedance matching

In electrical engineering, impedance matching is the practice of designing or adjusting the input impedance or output impedance of an electrical device for a desired value. Often, the desired value is selected to maximize power transfer or minimize signal reflection. For example, impedance matching typically is used to improve power transfer from a radio transmitter via the interconnecting transmission line to the antenna. Signals on a transmission line will be transmitted without reflections if the transmission line is terminated with a matching impedance.
Techniques of impedance matching include transformers, adjustable networks of lumped resistance, capacitance and inductance, or properly proportioned transmission lines. Practical impedance-matching devices will generally provide best results over a specified frequency band.
The concept of impedance matching is widespread 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, such as in acoustics or optics.

Theory

Impedance characterizes the ratio of energy components in a given domain. 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, electromagnetic, 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 and a reactance component which forms the imaginary part. This impedance characterizes the ratio between the voltage and current components in the medium.
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.

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.

Phase-shifting matching network

If a lossless reciprocal two-port network matches a load impedance to a source impedance with transmission phase shift , then the transmission matrix of the matching network can be expressed as
where and are real and imaginary part of complex number. is the complex conjugate of.
If a lossless reciprocal two-port network matches a resistive load impedance to a resistive source impedance with a transmission phase shift of , then the transmission matrix of the matching network can be expressed as
If both source and load are equal resistive impedance, then the ABCD parameters can be written as
which is a transmission line with characteristic impedance and electrical length

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 delivering a high voltage is more important than maximizing power transfer, then impedance bridging or voltage bridging is often used.
In older audio systems, the source and load resistances were matched at 600 ohms. One reason for this was to maximize power transfer, as there were no amplifiers available that could restore lost signal. Another reason was to ensure correct operation of the hybrid transformers used at central exchange equipment to separate outgoing from incoming speech, so these could be amplified or fed to a four-wire circuit. Most modern audio circuits, on the other hand, use active amplification and filtering and can use voltage-bridging connections for greatest accuracy. Strictly speaking, impedance matching only applies when both source and load devices are linear; however, matching may be obtained between nonlinear devices within certain operating ranges.

Impedance-matching devices

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

Devices intended to present an apparent load to the source of Z_load = Z_source*. Given a source with a fixed voltage and fixed source impedance, the maximum power theorem says this is the only way to extract the maximum power from the source.
Devices intended to present an apparent load of Z_load = Z_line, to avoid echoes. Given a transmission line source with a fixed source impedance, this "reflectionless impedance matching" at the end of the transmission line is the only way to avoid reflecting echoes back to the transmission line.
Devices intended to present an apparent source resistance as close to zero as possible, or presenting an apparent source voltage as high as possible. This is the only way to maximize energy efficiency, and so it is used at the beginning of electrical power lines. Such an impedance bridging connection also minimizes distortion and electromagnetic interference; it is also used in modern audio amplifiers and signal-processing devices.

There are a variety of devices used between a source of energy and a load that perform "impedance matching". To match electrical impedances, engineers use combinations of transformers, resistors, inductors, capacitors and transmission lines. These passive impedance-matching devices are optimized for different applications and include baluns, antenna tuners, acoustic horns, matching networks, and terminators.

Transformers

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 allows interfacing a balanced line and an unbalanced line. To match the impedances, both cables must be connected to a matching transformer with a turns ratio of 2:1. In this example, the 300-ohm line is connected to the transformer side with more turns; the 75-ohm cable is connected to the transformer side with fewer 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.