1. Lecture 17 Review: RL circuit natural response
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

2. RC circuit natural response – review
Governing equation:
Initial condition:
Response:

3. 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

4. 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

5. Derive governing first order differential equation on previous slide
Determine initial conditions; emphasize that current through inductor cannot change suddenly

6. RL Circuit Natural Response – continued

7. Finish derivation on previous slide
Sketch response on previous slide

8. RL Circuit Natural Response – summary
Inductor current:
Exponential function:
Write i(t) in terms of :

9. 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)

10. First order system natural response – summary
RC circuit:
Solution:
Alternate form of governing equation:
RL circuit:
Solution:
Alternate form of governing equation:

11. General first order system natural response
Governing equation:
Initial condition:
Form of solution:

12. 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

13. 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

14. Example 1
Find v(t), t>0

15. Example 1 – continued
Equivalent circuit, t>0. v(0) = 3V.

16. Example 1 – checking results

17. Example 2
Find iL(t), t>0

18. Example 2 – continued
Equivalent circuit, t>0. iL(0) = 0.33A

19. Example 2 – checking results