Mastering Electronics Design

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.

Figure 1

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 R_E, 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 R_E (see Figure 2).

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, v_oc is an independent source. If we short-circuit this source, the circuit output resistance as seen from R_E is R_E in parallel with r_th.  Therefore, r_out can be written as in the following equation.

Judging by the usual resistor values of R_source, r_∏ and R_E, the ratio (R_source+r_∏)/R_E is small as compared to β+1. This shows that R_E influence is small in the calculation of the common collector output resistance or, at most, it decreases r_th, which is exactly what we need for a circuit output. If we neglect (R_source+r_∏)/R_E, r_out becomes

which is the equation shown in most articles or textbooks. The output resistance depends mainly on the source resistance R_source, 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.

Figure 3

According to Ohm’s Law, r_input is calculated by dividing the test source voltage at its current. .

Inspecting the circuit in Figure 3, we can see that i_test = i_b.

Also, we notice that R_E receives two currents, i_test and β i_test. As such, from v_test, R_source, r_∏, R_E loop, v_test 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) R_E. In many texts, R_source+r_∏ is neglected, because it is a lot smaller than (β+1) R_E. How small? R_source can be around 10 kΩ, while r_∏, is around 1 kΩ. If R_E is 1 kΩ, and β+1 is 200 Ω, one can easily calculate that R_source+r_∏ can be neglected in the r_input 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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