Determining output resistance

Determining the output resistance Rout of linear circuits (for example, amplifiers) can often be done by manipulation of the circuit model - often the small-signal equivalent circuit - to highlight those components connected to the output terminals, and by evaluation of the networks that result.

This simple approach falls down in circuits where there is negative feedback - for example in source follower circuits, and in Op-Amp circuits where negative feedback is almost invariably in use.  The inspection method gives incorrect and misleading answers in these cases.

A different method is needed for such circuits.  The method described below is quite general, and can be used perfectly satisfactorily, even in circuits where no negative feedback is applied, and will it lead to the same result as the method of inspection in these simple circuits..

The method for amplifier circuits

We start with a suitable circuit, typically the small-signal equivalent circuit representation.  For Op-Amp circuits this is typically the schematic circuit drawn using the non-ideal Op-Amp symbol.  Then we proceed as follows.  We: 

We must evaluate the value v/i, which is the output resistance Rout.

Why does this work?

Explanation

Whatever the circuit of our amplifier, be it JFET, MOSFET, Op-Amp or some other technology, we can represent it using the generic amplifier form, shown in the pecked rectangle below.  Our aim is to show that if we use the approach presented above, the quantity v/i represents the output resistance of the circuit, which we denote Rout.

If we short-circuit the input to earth, the input signal is by definition 0, and the output from the amplifier stage is also 0. If we represent the circuit by means of our general-purpose amplifier model (see Lecture 15), the voltage generated by its internal voltage generator, Av', is then also zero.

If voltage source Av' is a zero voltage source we can regard it equivalently as a short-circuit. Replacing that generator with a short-circuit, and with the external source v connected, we see that the current that must flow, i, is just v/Rout

Hence Rout = v/i.