188bet±¸ÓÃÍøÖ·

Applications of Differential Equations

Electric Circuits

A Theoretical Introduction

As you probably already know, electric circuits can consist of a wide variety of complex components. These may be set up in series, or in parallel, or even as combinations of both. In this laboratory, however, we'll be considering onlyÌýseriesÌýcircuits with especially simple components: resistors, inductors, and capacitors, along with some form of voltage supply.

The mathematics required to deal with such circuits goes a little beyond your high-school physics usage of Ohm's Law. After all, in these circuits the quantities of interest may beÌýchanging, and change implies thatÌýrates of changeÌýmay get involved. It looks like once again we will be seeing equations involvingÌýderivatives—differential equations.

To start with, let's consider the picture of a simple series circuit in which one of each of the components that we mentioned above appears:

Ìý Ìý

In this diagram we see each of the components that we just mentioned. The labels have the following meanings:

• LÌýis aÌýconstantÌýrepresentingÌýinductance, and is measured in henrys
• RÌýis aÌýconstantÌýrepresentingÌýresistance, and is measured in ohms
• CÌýis aÌýconstantÌýrepresentingÌýcapacitance, and is measured in farads
• EÌýrepresents theÌýelectromotive force, and is measured in volts. It is not necessarily a constant, and may be a function of time

Although they don't appear in the diagram, there are a few other quantities that will be involved in our analysis:

• qÌýrepresentsÌýcharge, and is measured in coulombs
• IÌýrepresentsÌýcurrent, and is measured in amperes
• tÌýrepresentsÌýtime, and is measured in seconds

So how do we start to find the relationships over time between these quantities? The key is to useÌýKirchoff's second law, which states:

The sum of the voltage drops across each component in a circuit is equal to the voltage,ÌýE, impressed upon the circuit.

Obviously then, in order to make use of this statement, we need to know what the voltage drop across each component is. Physics has the answer for us! Let's go through the components one at a time:

• the inductor: produces a voltage drop ofÌýL dI/dt, orÌý³¢±õ′.
• the resistor: produces a voltage drop ofÌýRI.
• the capacitor: produces a voltage drop ofÌýq/C.

Restating Kirchoff's second law in abbreviated form, we get the following:

sum of the voltage dropsÌý=ÌýE,

which may be restated as:

inductor voltage dropÌý+Ìýresistor voltage dropÌý+Ìýcapacitor voltage dropÌý=ÌýE,

into which we may substitute the actual voltage drops that we mentioned above, to get:

But, also according to physics,ÌýIÌý=Ìý±ç′, so substituting, we can rewrite the equation purely in terms of the charge,Ìýq, rather than a mixture of charge and current:

or alternatively, if we differentiate equation (1) and use the same substitution, we get an equation purely in terms of current:

We will be mainly concerned with using the last of these three equivalent forms.

Notice that equation (3) isÌýlinearÌýwithÌýconstant coefficients, so in the case when
·¡â€²Ìý=Ìý0, (theÌýhomogeneousÌýcase), it may be solved very easily, even by hand.

The form ofÌý·¡â€²Ìýwill determine the method necessary when solving theÌýnon-homogeneousÌýcase by hand. We would need to use eitherÌýundetermined coefficients, orÌývariation of parameters. (Fortunately, in this laboratory we'll be letting the computer do the work for us.)

We're now going investigate the application of this model to various circuits, paying special interest to the graphs of our solutions, i.e. the graphs of current against time. Let's nowÌýÌýhow we would deal with this type of problem usingÌýMathematica.

If you're lost, impatient, want an overview of this laboratory assignment, or maybe even all three, you can click on the compass button on the left to go to the table of contents for this laboratory assignment.