## Mathematics & Science
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## Applications of Differential Equations## Electric Circuits## A Theoretical IntroductionAs 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 only
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 be 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: In this diagram we see each of the components that we just mentioned. The labels have the following meanings: *L*is a**constant**representing**inductance**, and is measured in henrys*R*is a**constant**representing**resistance**, and is measured in ohms*C*is a**constant**representing**capacitance**, and is measured in farads*E*represents the**electromotive 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: *q*represents**charge**, and is measured in coulombs*I*represents**current**, and is measured in amperes*t*represents**time**, and is measured in seconds
So how do we start to find the relationships over time between these
quantities? The key is to use
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 of*L dI/dt*, or*LI′*.**the resistor**: produces a voltage drop of*RI*.**the capacitor**: produces a voltage drop of*q/C*.
Restating Kirchoff's second law in abbreviated form, we get the following:
which may be restated as:
into which we may substitute the actual voltage drops that we mentioned above, to get:
But, also according to physics,
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) is The form of 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 now go and see how we would deal with this type of problem using |
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ODE Laboratories: A Sabbatical Project by Christopher A. Barker©2017 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA e-mail: cbarker@deltacollege.edu |