Fig. 1.
Robustness of two electronic operational amplifiers (with and without negative feedback). a A common model of a negative feedback amplifier with typical parameters. y is the output to input u(t), which I take to be unity. Shown also is the unregulated amplifier (circuit inside the rectangle) with input v and output y. This high gain amplifier is manufactured as an integrated transistor circuit. a and c are internal parameters such that c −1 is the time constant and A=a/c is the amplifier gain. Negative feedback is introduced by adding the two resistors R 1 and R 2 in the configuration shown. The circuit is quite complex, but the simple first order model shown is a good representation of its behavior under typical operating conditions. The feedback resistors R 1 and R 2 are supplied by the user and are selected to tune the gain. b The robustness/fragility properties of the two amplifier circuits. For proper comparison, the input to the unregulated amplifier, \(v(t)\equiv \overline {v}\), is chosen so that the corresponding output y ∗ matches that of the negative feedback amplifier. For the feedback amplifier, y ∗ is extremely robust to variations in the parameters a and c, in contrast to the unregulated amplifier. At the same time, y ∗ is quite sensitive to the values of the two resistors, underscoring its robust yet fragile character. c Graphical explanation of the difference in robustness properties of the two amplifiers. For both amplifiers, the abscissa of the point of intersection of the black line and the blue line gives y ∗. In the case of the feedback amplifier, the slope of the blue line is −A β. As A β>>1, one can see that y ∗ will be almost independent of A. Indeed, y ∗ depends almost exclusively on the ratio R 2/R 1, resulting in extreme robustness to A=a/c