Impedance matching

Impedance matching is an electrical method that maximizes the efficiency of power transfer for various reasons.
Contents 
General
Impedance is the resistance of a system to a power source; particularly one that varies with time. The power can be electrical, mechanical, magnetic or even thermal, although electrical impedance is the most common application. In general, impedance is a complex number, which means the sum of a real number and an imaginary number. The real part of an impedance is called resistance, and the imaginary part is called reactance. In simple cases, such as lowfrequency or directcurrent 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.
Impedance matching is the process of adjusting the impedances of the source and load to "match" for one of two effects:
Impedance matching for maximum power transfer (also called conjugate matching) is used to maximize the power dissipated by a load impedance driven from a fixed source impedance. In this case, Z_{load} = Z_{source}* (where * indicates the complex conjugate).
Impedance matching for minimizing reflection (also called reflectionless matching) is used when a source is driving a load that is significantly far away compared to the frequency being sent. To prevent reflections of the signal back to the source, the load impedance must be matched exactly to the line and source impedances. In this case, Z_{load} = Z_{line} = Z_{source}.
For the case of a purely resistive circuit, these two concepts appear the same, which can cause confusion.
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 (i.e. of 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 lowfrequency 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 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 efficient transfer of 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. This was done to maximise power transfer, because there were no amplifiers capable of restoring power once it had been lost. Most modern audio circuits, on the other hand, use active amplification and filtering, and therefore use voltage bridging connections.
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Source_and_load_circuit.png
Image:Source and load circuit.png
To demonstrate this, consider a source whose open circuit voltage is V_{source} and whose internal impedance is R_{source} ohms. Assume this source is connected to a load of R_{load} ohms.
The resulting circuit can be visualised as a perfect voltage source of V_{source} volts driving two series connected resistors (R_{source} and R_{load}) 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 V_{source} and R_{source}, and then try varying R_{load}. 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 P_{load} (the power in the resistor R_{load}) because this is the power that is being transferred from the supply to the load.
 <math>P_\mathrm{load} = I^2 \cdot R_\mathrm{load}
<math>
 <math>I = \frac{V_\mathrm{source}}{R_\mathrm{source} + R_\mathrm{load}}
<math>
 Thus:
 <math>P_\mathrm{load} = \frac{V_\mathrm{source}^2 R_\mathrm{load}}{(R_\mathrm{source} + R_\mathrm{load})^2}
<math>
 Where I is current in the circuit.
We have fixed V_{source} and R_{source}. After some algebra, the power is proportional to
 <math>
{1 \over 1/r + 2 + r} <math>
where r is the impedance ratio
 <math>
r = R_\mathrm{load}/R_\mathrm{source} \,\! <math>
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 <math>1r^{2}<math> which takes the following values:
 Negative for 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 R_{load} = R_{source}. We then get
 <math>P_\mathrm{load} = \frac{1}{4} \cdot \frac{V_\mathrm{source}^2}{R_\mathrm{source}}
<math>
And this is the maximum power that can be transferred into R_{load}, occurring when R_{load} = R_{source}, ie 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,
 <math>
Z_\mathrm{out} = Z_\mathrm{in} \, <math>
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, acoustic horns and terminators used with 10base2 ethernet. (See also: Impedance mismatch.)
Transformers are used to match the impedances of high power circuits. A transformer converts alternating current at one voltage to another voltage, however the power remains the same, except for conversion losses. The side with the lower voltage is attached to the low impedance, because more current can flow through the lower resistance. The side with the higher voltage goes to the higher impedance, because more voltage can get through the higher resistance. The most visible examples are the power transformers used to distribute power from high impedance transmission lines to low impedance retail use.
Resistive impedance matches are the easiest to design. They limit the power deliberately. They are used to transfer lowpower 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 to match impedances for specific frequencies. These can distribute different frequencies to different places in the circuit.
Reflectionless matching
In radiofrequency (RF) systems, a common value for source and load impedances is 50 ohms (the impedance of a quarterwave ground plane antenna). 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 wastage of power. 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 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 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:
 <math>
\Gamma = \left {Z_1  Z_2 \over Z_1 + Z_2} \right <math>
The purpose of a transmission line is to get the maximum amount of energy to the other end of the line, 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.
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:
<math>Z_0 = \frac{V}{I}<math>
Now, at a boundary, e.g. 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, as the new material has a different impedance, <math> V_2 = Z_2 I<math>, which is not the same as the original voltage.
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. This energy can not just 'disappear': it must go somewhere. Due to the impedance difference, it can not 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 magnitude of this reflection wave is given in the formula above.
Telephone systems
Telephone systems use matched impedances to enable a telephone hybrid to transmit and receive at the same volume level. This is not related to transmission lines. The signals are being sent and received down the same wire, and cancellation is necessary for the earpiece so that the signal from the microphone does not override the signal from the other telephone. The devices used are dependent on a 600 ohm source and load impedance.
Loudspeakers
Audio amplifiers never use matched impedances, contrary to myth. The driver amplifier always has an output impedance of < 0.1 ohm and the loudpeaker usually has an input impedance of 4, 8, or 16 ohms and that is really impedance bridging.
Sound
Similar to transmission lines, impedance match problem exists when transferring sound from one medium to another. If the acoustical 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 frequency it is playing, i.e. bigger speakers play louder and deeper (low frequency bass) than small speakers. Oval speakers act like large speakers lengthwise, and like small speakers crosswise.
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 optical coating.
See also
External links
 A good explanation of impedance matching in audio (or the lack thereof) by Elliot Sound Products (http://sound.westhost.com/impedanc.htm#zmatch)
 Clean Signals; TRANSMISSION LINES: why 600 Ω? (http://www.findarticles.com/p/articles/mi_m0JKW/is_2000_Jan/ai_59172236)
 Contains a description of impedance matching (http://www.maximic.com/appnotes.cfm/appnote_number/1849)
 Conjugate matching versus reflectionless matching (http://www.ece.rutgers.edu/~orfanidi/ewa/ch11.pdf) (PDF) taken from Electromagnetic Waves and Antennas (http://www.ece.rutgers.edu/~orfanidi/ewa/)de:Leistungsanpassung