1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Equations with Constant Coefficients 279

The electric charge, q, and the current, i = dq/ dt, are functions of time.

The positive quantities of capacitance, C, inductance, L, and resistance, R,

are also functions of time. However, in many instances these quantities are

nearly constant. So we shall assume C, L, and Rare positive constants.

The German physicist Gustav Kirchhoff (1824-~887) stated the following

two laws regarding the behavior of electrical systems:

Kirchhoff's First Law (Current Law) At any junction in a net-

work, the sum of the current flowing into the junction is equal to the

sum of the current flowing out of the junction.

Kirchhoff's Second Law (Voltage Law) The algebraic sum of the

voltage drops around any loop of a network is equal to the algebraic

sum of the impressed electromotive forces around the loop.

And according to the fundamental laws of electricity, the voltage drop across
a resistor is Ri, the voltage drop across a capacitor is q/C, and the voltage
drop across an inductor is Ldi/ dt.

The RLC Circuit Consider the simple RLC circuit consisting of a resistor

with resistance R, an inductor with inductance L, a capacitor with capacitance
C, an electromotive force E(t), and a switch s connected in series as shown
in Figure 6.3. The arrow in the figure provides an orientation for the current
flow in the loop. The current i is positive if it is flowing in the direction of
the arrow and negative if it is flowing in the opposite direction.

~----c

:E:(t) i R

Le>----.

L

Figure 6.3 An RLC Series Circuit

Since there are no junctions in this circuit, we need only apply Kirchhoff's
second law regarding voltage drops to find

(6) Li' +Ri+ ~ =E(t).