

Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Study Material. Introducing when a circuit has capacitors and inductors other than resistors and sources, the impedance concept will be applied. Circuits with Capacitors and Inductors, Connexions Web site. http://cnx.org/content/m0023/2.12/, Jun 4, 2009. Circuits, Capacitors, Inductors, Don, Johnson, Equation, Elements, Explicit, Voltage, Implicit, Relation, Parallel, Combination, Impedance, Same, Complex, Numbers, Physics, Charles, Steinmetz, Engineering.
Typology: Study notes
1 / 2
This page cannot be seen from the preview
Don't miss anything!
Connexions module: m0023 1
This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License †
Abstract Introducing when a circuit has capacitors and inductors other than resistors and sources, the impedance concept will be applied.
vin
C vout
Figure 1: A simple RC circuit.
Let's consider a circuit having something other than resistors and sources. Because of KVL, we know that vin = vR + vout. The current through the capacitor is given by i = C dv dtout , and this current equals that passing through the resistor. Substituting vR = Ri into the KVL equation and using the v-i relation for the capacitor, we arrive at
RC dvout dt
The input-output relation for circuits involving energy storage elements takes the form of an ordinary dierential equation, which we must solve to determine what the output voltage is for a given input. In contrast to resistive circuits, where we obtain an explicit input-output relation, we now have an implicit relation that requires more work to obtain answers. At this point, we could learn how to solve dierential equations. Note rst that even nding the dierential equation relating an output variable to a source is often very tedious. The parallel and series combination rules that apply to resistors don't directly apply when capacitors and inductors occur. We would have to slog
∗Version 2.12: Jun 4, 2009 11:34 am GMT- †http://creativecommons.org/licenses/by/1.
http://cnx.org/content/m0023/2.12/
Connexions module: m0023 2
our way through the circuit equations, simplifying them until we nally found the equation that related the source(s) to the output. At the turn of the twentieth century, a method was discovered that not only made nding the dierential equation easy, but also simplied the solution process in the most common situation. Although not original with him, Charles Steinmetz^1 presented the key paper describing the impedance approach in 1893. It allows circuits containing capacitors and inductors to be solved with the same methods we have learned to solved resistor circuits. To use impedances, we must master complex numbers. Though the arithmetic of complex numbers is mathematically more complicated than with real numbers, the increased insight into circuit behavior and the ease with which circuits are solved with impedances is well worth the diversion. But more importantly, the impedance concept is central to engineering and physics, having a reach far beyond just circuits.
(^1) http://www.invent.org/hall_of_fame/139.html
http://cnx.org/content/m0023/2.12/