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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. RC circuit natural response – review. Governing equation: Initial condition: Response:.
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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
RC circuit natural response – review • Governing equation: • Initial condition: • Response:
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
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
Derive governing first order differential equation on previous slide • Determine initial conditions; emphasize that current through inductor cannot change suddenly
Finish derivation on previous slide • Sketch response on previous slide
RL Circuit Natural Response – summary • Inductor current: • Exponential function: • Write i(t) in terms of :
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)
First order system natural response – summary • RC circuit: • Solution: • Alternate form of governing equation: • RL circuit: • Solution: • Alternate form of governing equation:
General first order system natural response • Governing equation: • Initial condition: • Form of solution:
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
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
Example 1 • Find v(t), t>0
Example 1 – continued • Equivalent circuit, t>0. v(0) = 3V.
Example 2 • Find iL(t), t>0
Example 2 – continued • Equivalent circuit, t>0. iL(0) = 0.33A