Foundations of Electromagnetic Compatibility with Practical Applications

5Differential Equations

Differential equations are of fundamental importance in electromagnetics because many electromagnetic laws and EMC concepts are mathematically described in the form of differential equations.

We will focus on a selected sample of the differential equations relevant to the subject of EMC. We begin by discussing the first‐order RC and RL circuits and their solutions, and then focus on the second‐order RLC circuits.

We conclude this chapter by presenting several EMC applications described by the differential equations.

5.1 First Order Differential Equations – RC and RL Circuits

5.1.1 RC Circuit

A typical time‐domain RC circuit configuration is shown in Figure 5.1. (In Section 9.3 we will learn how to transform any linear circuit into such a configuration using the Thévenin theorem approach).

Figure 5.1 RC circuit.

At , the switch closes and a dc voltage source, V_T, is connected to a capacitor with an initial voltage of V₀. R_T represents the Thévenin resistance of the circuitry connected to the capacitor.

The differential equation governing the capacitor voltage in this circuit is (Alexander & Sadiku, 2009, p. 274).

(5.1)

Mathematics provides a number of approaches to solving ...