ABSTRACT
Nonlinear synthesis and design can be informally defined as a constructive procedure to interconnect components from a catalog of available primitives, and to assign values to their constitutive parameters to meet a specific nonlinear relationship among electrical variables. This relationship is represented as an implicit integrodifferential operator, although we primarily focus on the synthesis of explicit algebraic functions,
y ¼ f (x) (8:1)
where y is voltage or current f () is a nonlinear real-valued function x is a vector with components that include voltages and currents
This synthesis problem is found in two different circuit-related areas: device modeling [8,76] and analog computation [26]. The former uses ideal circuit elements as primitives to build computer models of real circuits and devices (see Chapter 7). The latter uses real circuit components, available either off the shelf or integrable in a given fabrication technology, to realize hardware for nonlinear signal processing tasks. We focus on this second area, and intend to outline systematic approaches to devise electronic function generators. Synthesis relies upon hierarchical decomposition, conceptually shown in Figure 8.1, which encompasses several subproblems listed from top to bottom:
. Realization of nonlinear operators (multiplication, division, squaring, square rooting, logarithms, exponentials, sign, absolute value, etc.) through the interconnection of primitive components (transistors, diodes, operational amplifiers, etc.)
. Realization of elementary functions (polynomials, truncated polynomials, Gaussian functions, etc.) as the interconnection of the circuit blocks devised to build nonlinear operators
. Approximation of the target as a combination of elementary functions and its realization as the interconnection of the circuit blocks associated with these functions
Figure 8.1 illustrates this hierarchical decomposition of the synthesis problem through an example in which the function is approximated as a linear combination of truncated polynomials [30], where realization involves analog multipliers, built by exploiting the nonlinearities of bipolar junction transistors (BJTs) [63]. Also note that the subproblems cited above are closely interrelated and, depending on the availability of primitives and the nature of the nonlinear function, some of these phases can be bypassed. For instance, a logarithmic function can be realized exactly using BJTs [63], but requires approximation if our catalog includes only field-effect transistors whose nonlinearities are polynomic [44].
