In this article I will show a method to deduce the input and output resistance of the common collector amplifier. The common-collector amplifier is a well known circuit (see Figure 1). It is mostly used as a buffer due to its high input resistance, small output resistance and unity gain. The equations derived in this article are symbolic, as is the derivation of any other formula in this website. Still, even if the resistances’ values are not numeric, the equations are intuitive enough to show the high input, low output resistance property of the amplifier.
The Common-Collector Output Resistance
The output resistance of this amplifier is the resistance seen by the next stage, as looking to the emitter resistor RE, as in Figure 1.
In a previous article, Derive the Transfer Function of the Common Collector Amplifier with Thevenin’s Theorem, I used Thevenin’s Theorem to demonstrate, step by step, how to derive the small-signal transfer function of this amplifier. In the same article, I showed that the common collector amplifier is equivalent to a Thevenin source that feeds RE (see Figure 2).
This source has the open-circuit voltage
and the resistance
The calculation of the common-collector output resistance reduces to the calculation of the output resistance of this circuit, because the two circuits are equivalent.
In Figure 2, voc is an independent source. If we short-circuit this source, the circuit output resistance as seen from RE is RE in parallel with rth. Therefore, rout can be written as in the following equation.
Judging by the usual resistor values of Rsource, r∏ and RE, the ratio (Rsource+r∏)/RE is small as compared to β+1. This shows that RE influence is small in the calculation of the common collector output resistance or, at most, it decreases rth, which is exactly what we need for a circuit output. If we neglect (Rsource+r∏)/RE, rout becomes
which is the equation shown in most articles or textbooks. The output resistance depends mainly on the source resistance Rsource, the transistor input resistance r∏, and it is small, since these two resistor values are divided by a large number, β+1.
The Common-Collector Input Resistance
The input resistance is usually calculated with a test source connected at the amplifier input. Its value can be easily derived if we know the test source and the current it sources into the circuit under test. For this task, let’s replace the transistor with its small-signal equivalent as in Figure 3.
According to Ohm’s Law, rinput is calculated by dividing the test source voltage at its current. .
Inspecting the circuit in Figure 3, we can see that itest = ib.
Also, we notice that RE receives two currents, itest and β itest. As such, from vtest, Rsource, r∏, RE loop, vtest can be written as in the following equation:
So the small signal input resistance of the common-collector amplifier is
This equation shows that the common-collector amplifier has a large input resistance, due to the product (β+1) RE. In many texts, Rsource+r∏ is neglected, because it is a lot smaller than (β+1) RE. How small? Rsource can be around 10 kΩ, while r∏, is around 1 kΩ. If RE is 1 kΩ, and β+1 is 200 Ω, one can easily calculate that Rsource+r∏ can be neglected in the rinput equation.
With a small output resistance and a large input resistance, the common-collector amplifier is mostly used as a buffer. As with saw in a previous article, it has a unity gain, making it an ideal circuit to isolate different stages when designing electronic circuits.
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