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

RL Circuit Natural Response – continued

• 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 1 – checking results

Example 2

• Find iL(t), t>0

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

Example 2 – checking results