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# Impedance matching

## General

Whenever a source of electicity, such as a signal source or a radio transmitter, operates into a load, optimum 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 Vsource and whose internal impedance is Rsource ohms. Assume this source is connected to a load of Rload ohms.

The resulting circuit can ve 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. 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.

Pload = I^2 * Rload (where I is the current in the circuit and ^2 means "squared")

Since we have fixed Vsource and Rsource, the power is proportional to 1/(1/r + 2 + r) where r is the impedance ratio Rload/Rsource.

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 Rload, 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 - 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 Rload = Rsource. We get

Pload = Vsource^2 / (Rsource * (1 + 2 + 1))

Pload = 1/4 * Vsource^2 / Rsource

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

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