Lecture 17
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Transcript of Lecture 17
![Page 1: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/1.jpg)
Lecture 17•Review:
•RC circuit natural response•RL circuit natural response•General first order system natural response•First order circuit examples•Related educational modules:
–Section 2.4.3
![Page 2: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/2.jpg)
RC circuit natural response – review• Governing equation:
• Initial condition:
• Response:
![Page 3: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/3.jpg)
RL circuit natural response – overview• No power sources
• Circuit response is due to energy initially stored in the inductor– i(t=0) = I0
• Inductor’s initial energy is dissipated through resistor after switch is closed
![Page 4: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/4.jpg)
RL Circuit Natural Response• Find i(t), t>0 if the current through the inductor prior to
motion of the switch is i(t=0-) = I0
![Page 5: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/5.jpg)
• Derive governing first order differential equation on previous slide
• Determine initial conditions; emphasize that current through inductor cannot change suddenly
![Page 6: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/6.jpg)
RL Circuit Natural Response – continued
![Page 7: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/7.jpg)
• Finish derivation on previous slide• Sketch response on previous slide
![Page 8: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/8.jpg)
RL Circuit Natural Response – summary• Inductor current:
• Exponential function:
• Write i(t) in terms of :
![Page 9: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/9.jpg)
• Notes:• L and R set time constant• Increase L => Time constant increases )more
energy to dissipate)• Decreasing R => time constant increases
(energy dissipates more slowly)
![Page 10: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/10.jpg)
First order system natural response – summary • RC circuit:
• Solution:
• Alternate form of governing equation:
• RL circuit:
• Solution:
• Alternate form of governing equation:
![Page 11: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/11.jpg)
General first order system natural response• Governing equation:
• Initial condition:
• Form of solution:
![Page 12: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/12.jpg)
Checking results
• Our analyses are becoming more mathematically complex
• Checking your results against expectations about the circuit’s physical behavior is essential!• For first order circuits, it is often possible to determine
the circuit response directly from the circuit itself• However, I recommend doing the math and using the
circuit physics to double-check the math
![Page 13: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/13.jpg)
1. Checking the time constant
• Governing equation:
• RC circuit time constant:
• RL circuit time constant:
• Note:• In the time constant
expressions, the resistance is the equivalent resistance seen by the energy storage element
• An outcome of Thévenin’s theorem
![Page 14: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/14.jpg)
Example 1
• Find v(t), t>0
![Page 15: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/15.jpg)
Example 1 – continued
• Equivalent circuit, t>0. v(0) = 3V.
![Page 16: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/16.jpg)
Example 1 – checking results
![Page 17: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/17.jpg)
Example 2
• Find iL(t), t>0
![Page 18: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/18.jpg)
Example 2 – continued• Equivalent circuit, t>0. iL(0) = 0.33A
![Page 19: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/19.jpg)
Example 2 – checking results
![Page 20: Lecture 17](https://reader035.fdocuments.net/reader035/viewer/2022081008/56813022550346895d95a647/html5/thumbnails/20.jpg)