Impedance matching

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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 maximise the power transfer and minimise 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.

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[edit] 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 maximise the voltage transfer, not the power transfer.

[edit] Explanation

The term impedance is used to convey the concept of the opposition of a system to an energy source; particularly one that varies with time. The energy 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 is a complex number, 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 complex 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 significant, and also the special case in which the reactance is negligible.

[edit] Reflectionless or broadband matching

Impedance matching for minimizing reflections and maximising power transfer over a large bandwidth (also called reflectionless matching 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 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, Zload = Zline = Zsource, where Zline 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 reactance in the source or the load will affect the 'match'.

[edit] Complex conjugate matching

This is used in cases where 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,

Zload = Zsource*

(where * indicates the complex conjugate).

If the signals are kept within the narrow frequency range designed for by the matching network, then reflections (in this narrow frequency band only) are also minimised. For the case of purely resistive source and load impedances, all reactance terms are zero and the above formula reduces to

Zload = Zsource

as would be expected.

[edit] 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 operates into a load, the greatest 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 the maximum power theorem article 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 the voltage transfer is higher and less prone to corruption than if the impedances were matched. This maximum voltage connection is a common configuration called impedance bridging or voltage bridging, and is used in signal processing. In such applications, delivering a high voltage (to minimize signal degradation during transmission) and/or consuming less current is often more important than the transfer of maximum power.

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 capable of restoring power once it had been lost. Another reason was to ensure correct operation of the hybrid transformers used at the exchange 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 therefore use voltage bridging connections.

[edit] Mathematical proof

Image:Source and load circuit.png

To demonstrate this, consider a source whose open circuit voltage is Vsource and whose internal impedance is Rsource ohms. Assume this source is connected to a load of Rload ohms.

The resulting circuit can be visualised as a perfect voltage source of Vsource volts driving two series connected resistors (Rsource and Rload) then flowing back to the zero volt terminal on the voltage source.

To see the effects of impedance matching and mismatching, we must fix the values of Vsource and Rsource, and then try varying Rload. Usually, the source impedance cannot be changed, so we are calculating the load impedance for which the greatest amount of available power will be transferred into the load. We will calculate Pload (the power in the resistor Rload) because this is the power that is being transferred from the supply to the load.

P_\mathrm{load} = I^2 \cdot R_\mathrm{load} (from Joule's law)

and

I = \frac{V_\mathrm{source}}{R_\mathrm{source} + R_\mathrm{load}} (from Ohm's law)

where I is current in the circuit. Combining these, we get:

P_\mathrm{load} = \frac{V_\mathrm{source}^2 R_\mathrm{load}}{(R_\mathrm{source} + R_\mathrm{load})^2}

We have fixed Vsource and Rsource. After some algebra, the power is proportional to

{1 \over 1/r + 2 + r}

where r is the impedance ratio

r = R_\mathrm{load}/R_\mathrm{source} \,\!

Note that this function approaches zero as r becomes very small or very large - this indicates that an extreme impedance mismatch results in very little power being transferred to the load.

We are interested in knowing what value of r, and hence of R load, we should use for maximum power transfer. We need to maximise 1/(1/r + 2 + r) which is the same as minimising 1/r + 2 + r. The derivative is 1 − r − 2 which takes the following values:

  • Negative for 0 < r < 1
  • 0 when r = 1
  • Positive for r > 1

This means that as r rises from zero, 1/r + 2 + r falls to some minimum when r = 1 and then increases again. Therefore setting r = 1 minimises 1/r + 2 + r, and maximises 1/(1/r + 2 + r). Setting r = 1 corresponds to setting Rload = Rsource. We then get

P_\mathrm{load} = \frac{1}{4} \cdot \frac{V_\mathrm{source}^2}{R_\mathrm{source}}

And this is the maximum power that can be transferred into Rload, occurring when Rload = Rsource, that is, the impedances are matched.

Strictly speaking, it is not only the real, or resistive, parts of the impedances that are matched, but sometimes the impedances are said to be matched if only the resistance components are matched. If the entire impedances are matched, including reactances,

Z_\mathrm{out} = Z_\mathrm{in} \,

[edit] 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. (See also: Impedance mismatch.)

Transformers are sometimes used to match the impedances of circuits with different impedances. A transformer converts alternating current at one voltage to 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 attached to the low impedance, because this has the lower number of turns, and side with the higher voltage goes to the higher impedance winding with the larger number of turns. Resistive impedance matches are the easiest to design. They limit the power deliberately. They 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 matches, usually built into the structure of the switching element. See resistor.

Some special situations, such as radio tuners and transmitters, use tuned filters, such as stubs, to match impedances for specific frequencies. These can distribute different frequencies to different places in the circuit.

[edit] "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 R1 and real load impedance of R2. If a reactance X1 is in parallel with the source impedance, the combined impedance can be written as:

\frac{j R_1 X_1}{R_1 + j X_1}

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

R_2 = \frac{R_1 X_1^2}{R_1^2 + X_1^2}

Solving for X1

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, 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.

[edit] 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, Z0 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 (the impedance of a quarter-wave ground plane antenna).

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 Z1) will be equal to the waves on the load side (with impedance Z2). 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 Z1 and Z2 so that they are equal (Γ = 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, ZL, than the line, the voltage will be VL = ZLI 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, Γ (from the formula above):

V_0^- = \Gamma V_0^+

[edit] 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.

[edit] Loudspeakers

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 the tube type audio amplifiers, which required impedance matching for proper, reliable operation.

[edit] Acoustical matching

Similar to 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, rather than transferred across the border.

Sound transfer from a loudspeaker to air is related to the ratio of the diameter of the speaker to the wavelength of the sound being reproduced. That is, larger speakers can sound louder and deeper than small speakers. Elliptical speakers act like large speakers lengthwise, and like small speakers crosswise.

[edit] Light

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.

[edit] 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 m1 and m2 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^2m_2}{(m_1+m_2)^2}

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

[edit] See also

[edit] References

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

[edit] External links

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