Stability of Feedback Circuits

Abstract

Feeding back the output signal to the input does not in all cases establish the equilibrium state expressed by the basic input-output relation

$$ G = \frac{A}{{1 + A\beta }} $$

((3–1))

We can understand this from considering the consequence of reversing the sign of the forward path. Setting its gain to -A results in the following alleged transfer function

$$ G = \frac{{ - A}}{{1 - A\beta }} $$

((3–2))

Although Eq. 3-2 is a valid solution, there is no change for an actual circuit to arrive in this state. The circuit will amplify a small error between the actual and the desired signal, caused for instance by noise or drift, and produce a large output signal. Because of the positive feedback, this in its turn leads to a greater error, resulting in an even larger output signal. The system is said to be instable, since any infinitesimal deviation from the equilibrium condition will cause the output signal to explode. This suggests the following definition for the stability of a linear system: stability is the condition that a bounded input signal results in a bounded response.