Impedance matching

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Table of contents

1 General
2 Power transfer
3 Transmission lines
4 Impedance matching devices
5 Sound
6 Light
7 See also

General

Impedance is the resistance of a system to a power source that varies by time. The power can be electrical, mechanical, magnetic or even thermal, although electrical impedance is the most common. If a small impedance is connected to a big impedance, then the power that can pass through the connection is limited by the larger impedance.

Whenever a source of power, such as an electric signal source, a radio transmitter, or even mechanical sound operates into a load, the most power is delivered to the load when the impedance of the load is equal to the internal impedance of the source. Arranging this is called impedance matching.

Power transfer

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. 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.

Thus:

Where I is current in the circuit.

We have fixed V_source and R_source. After some algebra, the power is proportional to 1/(1/r + 2 + r) where r is the impedance ratio R_load/R_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 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

And this is the maximum power that can be transferred into R_load, occurring when R_load = R_source, ie the impedances are matched.

Transmission lines

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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 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 to match impedances for specific frequencies. These can distribute different frequencies to different places in the circuit.

Sound

Sound transfer impedance from a loudspeaker to air is 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.