Translinear circuits are circuits in which the exponential relationship between the output current and input voltage of a circuit element is exploited to realize various differential or algebraic equations. The precise exponential characteristic of the Bipolar Junction Transistor (BJT) and other devices has been responsible for the increase in popularity of this branch of analog circuits and has stimulated research geared towards implementation of not only linear, but also nonlinear systems in the analog domain. Also, the basic concept of using the exponential characteristic of certain circuit elements has been used in other popular analog components such as bandgap references, Proportional to Absolute Temperature (PTAT) circuits, and constant-gm biasing circuits.

This thesis is concerned with a subclass of translinear circuits, in which the basic translinear element has an output current that is exponentially related to a weighted sum of its input voltages. This multiple-input translinear element (MITE) can be used for the implementation of the same class of functions as traditional translinear circuits. The voltage addition that gives rise to multiplication of currents in traditional translinear circuits is replaced by weighted summation through a capacitive voltage divider. The implementation of algebraic or (algebraic) differential equations using MITEs can be reduced to the implementation of the so-called product-of-power-law (POPL) relationships, in which an output is given by the product of inputs raised to different powers. Hence, the synthesis of these POPL relationships, and their optimization with respect to the relevant cost functions, are very important in the theory of MITE networks.

In this thesis, different constraints on the topology of these POPL networks that result in desirable system behavior are explored and different methods of synthesis, subject to these constraints, are developed. The constraints are usually conditions on certain matrices of the network, which characterize the weights in the relevant MITEs. Some of these constraints are related to the uniqueness of the operating point of the network and the stability of the network. Conditions that satisfy these constraints are developed in this work. The cost functions to be minimized are the number of MITEs and the number of input gates in each MITE. A complete solution to POPL network synthesis is presented here that minimizes the number of MITEs first and then minimizes the number of input gates to each MITE.

A procedure for synthesizing POPL relationships optimally when the number of gates is minimal, i.e., 2, has also been developed here for the single-output case. A MITE structure that produces the maximum number of functions with minimal reconfigurability is developed for use in MITE field-programmable analog arrays. The extension of these constraints to the synthesis of linear filters is also explored, the constraint here being that the filter network should have a unique operating point in the presence of nonidealities. Synthesis examples presented here include nonlinear functions like the inverse tangent function and the Gaussian function which find application in analog implementations of particle filters. Synthesis of dynamical systems is presented here using the examples of a Lorenz system and a sinusoidal oscillator. The procedures developed here provide a structured way to automate the synthesis of nonlinear algebraic functions and differential equations using MITEs.