# CHAPTER 4

# Solution of Nonlinear Algebraic Circuit Equations

In the presence of nonlinear elements, the network equations can be formulated as a system of nonlinear equations. Solving such systems is not trivial and, in fact, is much harder than solving systems of linear equations. As we will see, the practical approach for solving nonlinear equations is to repeatedly *linearize* them and solve the resulting *linear* systems. In general, nonlinear systems of equations can have a unique solution, no solution, multiple solutions, or an infinity of solutions. Practical methods for solving nonlinear systems can only hope to provide the *approximate* value of “a solution,” if at all; they never provide closed-form solutions, only numerical and approximate ones. We will study the formulation of nonlinear network equations, the general solution methods, and their application to circuit simulation.

## 4.1 NONLINEAR NETWORK EQUATIONS

The need to solve a system of nonlinear equations arises in several ways as part of circuit simulation. For one thing, it comes up under DC Analysis, for finding either the quiescent steady state (*t* = ∞) solution under DC inputs, the initial (*t* = 0) solution required to initiate Transient Analysis, or for finding the DC transfer characteristic, by means of a DC-sweep. As well, the need to solve nonlinear equations arises throughout Transient Analysis, as the circuit response ...