Transcript Lecture 1 - Digilent Inc.

Lecture 17
•Review:
•RC circuit natural response
•RL circuit natural response
• General first order system natural
response
• First order circuit examples
•Related educational materials:
–Chapter 7.3
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:
• RL circuit:
• Solution:
• Solution:
• Alternate form of governing
equation:
• 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