# CHAPTER 5

# Solution of Differential Circuit Equations

In the presence of dynamic (*L* and *C*) elements, the network equations can be formulated as a system of differential equations. Solving such systems is not easy, and gets harder when the dynamic elements are nonlinear. As we will see, the practical approach for solving differential equations is to repeatedly *discretize* them and solve the resulting *algebraic* equations. The need to solve a dynamic network arises as part of the Transient Analysis mode of standard circuit simulators. We will study dynamic elements and the resulting dynamic MNA equations, general solution methods, and their application to circuit simulation.

## 5.1 DIFFERENTIAL NETWORK EQUATIONS

Considering the formulation of the network equations, it should be clear that KCL and KVL remain as linear *algebraic* relationships. The network equations become differential due only to the *dynamic element equations*.

### 5.1.1 Dynamic Elements

The most basic dynamic elements are the familiar two-terminal capacitors and inductors, be they linear or nonlinear, which we will now review. As well, and in order to incorporate internal dynamic elements of multiterminal elements (MTE), we will study some generalizations of the basic *L* and *C* elements.

*Capacitors* If *q(t)* is the charge on a capacitor, then recall that capacitor current is the rate of change of charge, so that:

As well, recall ...