Mastering Electronics Design

It is customary to consider the output resistance of the non-inverting amplifier as being zero, but why is that? An Op Amp’s own output resistance is in the range of tens of ohms. Still, when we connect the Op Amp in a feedback configuration, the output resistance decreases dramatically. Why?

To answer these questions, let’s calculate the output resistance of the non-inverting amplifier.

It is widely accepted that the output resistance of a device can be calculated using a theoretical test voltage source connected at the device output. The input, or inputs, are connected to ground. Nevertheless, instead of using this method, let’s try a different one: The small signal variation method.

Figure 1 shows the non-inverting amplifier, which drives a load, RL. This circuit has an equivalent Thevenin source as in Figure 2.

Figure 1

Figure 2

From Figure 2, one can see that the output voltage, Vout, can be written as

(1)

If we keep V_TH constant and apply small variations to Vout, by varying RL for example, the Vout variation, noted ΔVout can be written as follows:

(2)

Equation (2) shows that, when the load current increases, the load voltage decreases due to the output resistance. They vary in opposite direction and that is why the negative sign that appears in the Rout calculations is canceled out.

Equation (2) also tells us that we can use a small signal variation method to determine Rout. If, instead of ΔVout and ΔIout we write the small signal notation v_out and i_out, the output resistance becomes

(3)

Let’s apply this method to the non-inverting amplifier.

Figure 3

An ideal Op Amp can be represented as a dependent source as in Figure 3. The output of the source has a resistor in series, Ro, which is the Op Amp’s own output resistance. The dependent source is Ao v_d, where Ao is the Op Amp open-loop gain and v_d is the differential input voltage. The input differential resistance, between the Op Amp inputs, is considered high, so I removed it for simplicity. The same with the common mode input resistances, between the non-inverting input and ground and the inverting input and ground. The non-inverting input is connected to ground, because a fixed value voltage source does not bring any change from a small-signal variation point of view. Thus, we are in line with the general rule that the output resistance of a circuit is calculated with the circuit inputs connected to ground.

Inspecting the loop made by Ao v_d, Ro, and RL, v_out can be expressed as in the following equation.

(4)

where i_out is the small variation load current and i_f is the small variation feedback current.

The differential voltage v_d appears across R1, but with negative sign, so i_f is

(5)

And v_out becomes

(6)

At the same time v_d depends on v_out.

(7)

After replacing v_d in equation (6), the resulting mathematical expression depends on v_out and i_out as in equation (8).

(8)

Based on (3) and (8) Rout is

(9)

Ao is large, about 100000 or 100 dB. Therefore, the second term of the denominator is predominant.

(10)

This proves that the output resistance of the non-inverting amplifier is

(11)

where A_CL=1+R2/R1 and it is the closed-loop gain of the non-inverting amplifier. For a proof of the closed loop gain read this article, How to Derive the Non-Inverting Amplifier Transfer Function.

As equation (11) shows, the output resistance of the non-inverting amplifier is several orders of magnitude smaller than that of the Op Amp, because Ro is divided by the operational amplifier open loop gain. Therefore, the non-inverting amplifier output resistance can be considered zero.

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