Lecture 17

1. 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

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