State variable changes to avoid non computational issues

This paper is about the numerical simulation of nonlinear analog circuits with ”switch ” components, such as diodes. A ”switch ” component is an electrical device that may or may not conduct, depending on the state of the circuit. The problem with ”switch ” components is that the topology of the circuit is variable and so, apparently, it is not possible to describe the system with a single differential equation and solve it using standard numerical methods. This paper shows how to choose an appropriate state variable and over-come the above difficulties. 2 A test example Let’s consider the following circuit. Figure 1: Example circuit If the voltage on the diodes is below their threshold, no current is flowing through the diodes and so no current is flowing into the resistor and the capacitor; the circuit is open. When the voltage gets higher than the diode thresh-old, the circuit becomes a standard RC filter. We have just seen two different topologies that the same circuit can have, depending on its state. The couple of diodes has a memory-less nonlinear trans-fer function, like the one in figure 1. Note that the cur-rent circuit is only an example and so, instead of diodes, we could have other switch components with a different trans-fer function, like the grid of a triode. But the ideas we are going to expose will remain the same. id = f(vd) Figure 2: Nonlinear transfer function Let’s call id the current flowing into the diodes and vd the voltage on the diode. A standard way to proceed into the analysis of the circuit, is to take the current i will be as the state variable of the system. The equations of the system will become: e − iR − vc − vd = 0 vc(t) = 1C ∫ t 0 i(s)ds i(t) = f(vd(t)) Where vc is the voltage on the capacitor and vd is the voltage on the diodes. Remembering that i is the state variable, in order to be integrated, the above system should be written as: e − iR − 1 C ∫