MULTIPLE–LOOP FEEDBACK AMPLIFIERS

In the preceding three chapters, we studied the theory of single-loop feedback amplifiers. The concept of feedback was introduced in terms of return difference. We found that return difference plays an important role in the study of amplifier stability, its sensitivity to the variations of the parameters, and the determination of its transfer and driving-point impedances. The fact that return difference can be measured experimentally for many practical amplifiers indicates that we can include all the parasitic effects in the stability study, and that stability problem can be reduced to a Nyquist plot.

In this chapter, we study multiple-loop feedback amplifiers, which contain a multiplicity of inputs, outputs, and feedback loops. We first review briefly the rules of the matrix signal-flow graph, and then generalize the concept of return difference for a controlled source to the notion of return difference matrix for a multiplicity of controlled sources. For measurement situations, we introduce the null return difference matrix and discuss its physical significance. In particular, we show that the determinant of the overall transfer matrix can be expressed explicitly in terms of the determinants of the return difference and the null return difference matrices, thus generalizing Blackman's formula for the input impedance. This is followed by the derivations of the generalized feedback formulas and the formulation of the multiple-loop feedback theory in terms of the hybrid matrix. The problem of multiparameter sensitivity together with its relation to the return difference matrix is discussed. Finally, we develop formulas for computing multiparameter sensitivity functions.