Impedance matching

Impedance matching is the practice of attempting to make the output impedance of a source equal to the input impedance of the load to which it is ultimately connected, usually in order to maximize the power transfer and minimize reflections from the load. This only applies when both are linear devices. 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 just electrical) is transferred between a source and a load.

Terminology

Sometimes the term "impedance matching" is used loosely to mean "choosing impedances that work well together" instead of "making two impedances complex conjugate". The looser interpretation includes impedance bridging, where the load impedance is much larger than the source impedance. Bridging connections are used to maximize the voltage transfer, not the power transfer.

Explanation

The term impedance is used for the resistance of a system to an energy source. For constant signals, this resistance can also be constant. For varying signals, it usually changes with frequency. The energy involved can be electrical, mechanical, magnetic or even 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, which means that loads generally have a resistance to the source that is in phase with the source signal and a reactance to the source that is in quadrature to the phase of the source. 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 is negligible or zero and the impedance can be considered a pure resistance, expressed as a real number. In the following summary, we will consider the general case when the resistance and reactance are both significant, and also the special case in which the reactance is negligible.

Reflectionless or broadband matching

Impedance matching to minimize reflections and maximise power transfer over a (relatively) large bandwidth (also called reflectionless matching or broadband matching) is the most commonly used. To prevent all reflections of the signal back into the source, the load (which must be totally resistive) must be matched exactly to the source impedance (which again must be totally resistive). In this case, if a transmission line is used to connect the source and load together, Z_load = Z_line = Z_source, where Z_line is the characteristic impedance 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 and all reactance actually 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 reactive. This form of impedance matching can only maximize the power transfer between a reactive source and a reactive load at a single frequency. In this case,

Z_load = Z_source*

(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

Z_load = Z_source

as would be expected.

Power transfer

Main article: Maximum power theorem

Whenever a source of power, such as an electric signal source, a radio transmitter, or even mechanical sound (eg a loudspeaker) operates into a load, the maximum possible power is delivered to the load when the impedance of the load (load impedance) is equal to the complex conjugate of the impedance of the source (that is, its internal 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 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, then the power that can pass through the connection is limited by the higher impedance, but if electrical 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 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 which 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 that 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 best accuracy.

Impedance matching devices

To match electrical impedances, engineers use combinations of transformers, resistors, inductors and capacitors. These impedance matching devices are optimized for different applications, and are called baluns, antenna tuners (sometimes called ATUs or roller coasters because of their appearance), acoustic horns, matching networks, and terminators.

Transformers are sometimes used to match the impedances of circuits with different impedances. 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. Resistive impedance matches are easiest to design. They limit the power deliberately, and are used to transfer low-power signals, such as unamplified audio or radio frequency signals in a radio receiver. Almost all digital circuits use resistive impedance matching which is usually built into the structure of the switching element.

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.

"L" section

One 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 impedance to low impedance. The "L" section is inherently a narrowband matching network.

The analysis is as follows. Consider a real source impedance of $R_{1}$ and real load impedance of $R_{2}$. If a reactance $X_{1}$ is in parallel with the source impedance, the combined impedance can be written as:

${\frac {jR_{1}X_{1}}{R_{1}+jX_{1}}}$

If the imaginary part of the above impedance is completely canceled by the series reactance, the real part is

$R_{2}={\frac {R_{1}X_{1}^{2}}{R_{1}^{2}+X_{1}^{2}}}$

Solving for $X_{1}$

$X_{1}={\sqrt {\frac {R_{2}R_{1}^{2}}{R_{1}-R_{2}}}}$

If $R_{1}\gg R_{2}$ the above equation can be approximated as

$X_{1}\approx {\sqrt {R_{1}R_{2}}}$

The inverse connection, impedance step up, is simply the reverse, e.g. 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 with the use of the Smith chart.

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 impedance and the low impedance. The reflection creates a standing wave, which leads to further power waste. In these systems, impedance matching is essential.

In electrical systems involving transmission lines, such as radio and fiber optics, where the length of the line is large 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, $Z_{0}$ to prevent reflections of the signal at the ends of the line from causing echoes. 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 but can be matched to 50 ohms by using a modified ground plane or a matching network).

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's only one boundary.)

At the boundary, the two waves on the source side of the boundary (with impedance $Z_{1}$) will be equal to the waves on the load side (with impedance $Z_{2}$). The derivatives will also be equal. Using that equality, we solve for all wave functions, getting a reflection coefficient:

$\Gamma ={Z_{2}-Z_{1} \over Z_{1}+Z_{2}}$

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 should be as small as possible. This is achieved by matching the impedances $Z_{1}$ and $Z_{2}$ so that they are equal ($\Gamma =0$).

An electromagnetic wave consists of energy being transmitted down the transmission line. This energy is in two forms, an electric field and a magnetic field, which fluctuate constantly, with a continuing exchange between electrical and magnetic energy. The electric field is due to the voltage over the cross section of the line, perpendicular to the direction the wave is flowing. The magnetic field is due to the current flowing parallel to the direction of the wave.

Assume that voltage and current vary as sine waves. Inside the transmission line, the law of conservation of energy applies: the sum of magnetic and electric energy must always be the same (ignoring the effect of the small amount of energy converted to heat). This means that if the voltage is changing rapidly, the current must also change rapidly.

Now consider two moments: 1). when the current is zero and the voltage is maximum; 2). when the current is maximum and the voltage is zero. The amount of energy stored in the electric field at 1). must be exactly the same as the amount of energy stored in the magnetic field at 2). The ratio between voltage and current at 1). and 2). determines the impedance (Z) of the line:

$Z_{0}={\frac {V}{I}}$

At a boundary, for example, where the line is connected to the receiver, the law of conservation of charge applies. The current just before the boundary must be the same as just after. However, if the circuit at the receiver has a different impedance, $Z_{L}$, than the line, the voltage will be $V_{L}=Z_{L}I$ at the receiver, which is not the same as the original incident voltage $V_{0}^{+}$.

To achieve the voltage difference, an electric field is needed over the boundary. However, energy is needed to form this field, for which a part of the energy of the original wave is used. The remaining energy cannot just 'disappear'; it must go somewhere. Due to the impedance and voltage difference, it cannot go to the other side of the boundary. There remains only one way to go for this energy: back into the transmission line, as a reflection. The voltage of this reflected wave, $V_{0}^{-}$, is calculated from the incident voltage $V_{0}^{+}$ and the reflection coefficient, $\Gamma$ (from the formula above):

$V_{0}^{-}=\Gamma V_{0}^{+}$

Examples

Telephone systems

Telephone systems also use matched impedances to minimise echoes on long distance lines. This is related to transmission lines 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 that excessive sidetone is not heard. All devices used in telephone signal paths are generally dependent on using matched cable, source and load impedances. In the local loop, the impedance chosen is 600 ohm (nominal). Terminating networks are installed at the exchange to try 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 to about 600 ohms over the voice frequency band.

Loudspeaker amplifiers

Modern solid state audio amplifiers do not use matched impedances, contrary to myth. The driver amplifier has a low output impedance such as < 0.1 ohm and the loudspeaker usually has an input impedance of 4, 8, or 16 ohms; many times larger. This type of connection is impedance bridging, and provides better damping of the loudspeaker cone to minimize distortion.

The myth comes from tube audio amplifiers, which required impedance matching for proper, reliable operation. Most of these had output transformer taps to approximately match the amplifier output to typical loudspeaker impedances.

Non-electrical impedance matching examples

Acoustics

Similar to electrical transmission lines, the impedance matching problem exists when transferring sound energy from one medium to another. If the acoustic impedance of the two media are very different, then most of the sound energy will be reflected or absorbed, rather than transferred across the border.

Most loudspeaker systems themselves contain impedance matching mechanisms, especially for low frequencies. Because most driver impedances are poorly matched to the impedance of free air at low frequencies, and because of out-of-phase cancellations between output from the front of a speaker cone and from the rear, loudspeaker enclosures serve both to match impedances and prevent the interference. Sound coupling into 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 higher levels than smaller speakers for this reason. Elliptical speakers are a complex case, acting like large speakers lengthwise, and like small speakers crosswise.

Optics

A similar effect occurs when light (or any electromagnetic wave) transfers between two media with different refractive indices. An optical impedance of each medium can be calculated, and the closer the impedances of the materials match, the more light is refracted rather than reflected from the interface. The amount of reflection can be calculated from 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, the maximum energy transferred to the second body will occur when the second body has the same mass m. For a head-on collision, with 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 $m_{1}$ and $m_{2}$ are the masses of the moving and the stationary body respectively, and P is the momentum of the system, which remains constant throughout the collision, then the energy of the second body after the collision will be E₂:

$E_{2}={\frac {2P^{2}m_{2}}{(m_{1}+m_{2})^{2}}}$

which is analogous to the power transfer equation in the above "mathematical proof" section.

See also

• Maximum power theorem
• Power (physics)
• Electronics
• Impedance bridging

References

• Young, EC, The Penguin Dictionary of Electronics, Penguin, ISBN 0-14-051187-3 (see 'maximum power theorem', 'impedance matching')

External links

• 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