Circuit Analysis II with MATLAB Applications
Designed for use in a second course in circuit analysis, this text engages a full spectrum of circuit analysis related subjects ranging from the most abstract to the most practical.
TABLE OF CONTENTS Circuit Analysis II with MATLAB Applications
Appendix B: Differential Equations
This appendix is a review of ordinary differential equations. Some definitions, topics, and examples are not applicable to introductory circuit analysis but are included for continuity of the subject, and for reference to more advance topics in electrical engineering such as state variables. These are denoted with an asterisk and may be skipped.
B.1 Simple Differential Equations
In this section we present two simple examples to show the importance of differential equations in engineering applications.
Example B.1
A 1 F capacitor is being charged by a constant current I. Find the voltage v C across this capacitor as a function of time given that the voltage at some reference time t = 0 is V 0.
Solution:
It is given that the current, as a function of time, is constant, that is,
We know that the current and voltage in a capacitor are related by
and for our example, C = 1. Then, by substitution of (B.2) into (B.1) we get
By separation of the variables,
and by integrating both sides of (B.3) we get
where k represents the constants of integration of both sides.
We can find the value of the constant k by making use of the initial condition, i.e., at t = 0, v C = V 0 and (B.4) then becomes
or k = V 0, and by substitution into (B.4),
This example shows that when a capacitor is charged...
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