Franclim F. Ferreira, Pedro Guedes de Oliveira, Vítor G. Tavares

Feedback theory

The general structure of a feedback amplifier may be depicted by the signal flow diagram on the right, where x may be a voltage or current signal.

The basic amplifier A (to which the feedback is applied) is supposed to be unilateral and has a forward transmission of gain A, such that xo = A xi.

The feedback block b represents the feedback loop that may vary from a simple direct connection to a complex circuit. The output xo feeds both the load and the feedback block, which we also suppose to be unilateral with transmission gain b (called feedback factor), that delivers a sample of the output signal, xf = b xo.

The way that the output delivers the signal to the feedback block, the signal sampling, may be as a current or as a voltage, i.e., the sampling can be done in series or in shunt.

The feedback signal xf is subtracted to the source signal xs, producing which is the input to the basic amplifier: xi = xs - xf. This operation is called mixing or comparison of the feedback signal and can also be done in series (voltage signals) or in shunt (current signals).

Therefore, the combination of the two types of sampling with the two types of comparison result in four different topologies for the feedback amplifier: series-series, series-shunt, shunt-series and shunt-shunt (meaning comparison-sampling).

Bearing in mind that xo = A xi and xf  = b xo and defining the feedback (or closed loop) gain of the global amplifier as Af = xo / xs, we can easily see that

               that is the fundamental equation of feedback amplifiers.

Take notice that since b = 0 corresponds to opening the feedback loop, the basic amplifier gain A is, after all, the open loop gain of the global amplifier, i.e.:           .

The quantity - b A is appropriately named the loop gain. If it is negative, the feedback is said to be negative or degenerative. We can easily see that if - b A < 0, 1 + b A >1 and therefore Af < A.

It also seems sensible to designate 1 + b A the amount of feedback.

If, in the opposite way, the loop gain is positive, the feedback is also said to be positive or regenerative. There are situations where we can take advantage of this type of configuration but, as far as the study of amplifiers is concerned, the is seldom done.

We should emphasise that with negative feedback, if the loop gain is high, i.e. if b  A » 1, then Af @ 1 / b, which is a very interesting result that shows that the closed loop gain is practically determined by the feedback loop, in general built with passive components the values of which are generally very precise and stable.

Simultaneously Af will be almost independent of A, which is subject to variations of transistor parameters, being only required that b A » 1. That is why integrated OpAmps are built with very high gain.

The analysis of a feedback amplifier can be done using the usual tools for amplifier analysis, i.e. the circuit laws. We can then get the gain as well as the input and output resistances. However, a much more elucidating analysis method uses the feedback model with adequate sampling and comparison types, deriving the gain from the fundamental equation written above.

In what concerns the input and output resistances, it is easy to derive the following expressions (where Rx represents the open loop resistances and Rxf the closed loop ones, being i and o the labels for input and output):

  • with voltage sampling                      Rof = Ro / (1+b A )

  • with current sampling                      Rof = Ro (1+b A )

  • with series comparison                    Rif = Ri (1+b A )

  • with shunt comparison                     Rif = Ri / (1+b A )

These expressions make clear some of the properties of negative feedback that concern input and output resistance manipulation, through the value of b and A. There are still aspects to take into consideration concerning the feedback model, namely the fact that both the amplifier and the feedback loop are not unilateral and also that the load and source resistances have to be dealt with.

To get more insight into these aspects it is worth to see the method for analysing a feedback amplifier.

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