1. W12D1: RC and LR Circuits
Reading Course Notes: Sections , , , ,
Class 18

2. Announcements Math Review Week 12 Tuesday 9pm-11 pm in 26-152
PS 9 due Week 13 Tuesday April 30 at 9 pm in boxes outside or

3. Outline DC Circuits with Capacitors
First Order Linear Differential Equations
RC Circuits
LR Circuits
Class 15

4. DC Circuits with Capacitors
Week 04, Day 2
DC Circuits with Capacitors 4
Class 09 4

5. Sign Conventions - Capacitor
Moving across a capacitor from the negatively to positively charged plate increases the electric potential
Class 14

6. Power - Capacitor
Moving across a capacitor from the positive to negative plate decreases your potential. If current flows in that direction the capacitor absorbs power (stores charge)
Class 14

7. RC Circuits
Class 15

8. (Dis)Charging a Capacitor
When the direction of current flow is toward the positive plate of a capacitor, then
2. When the direction of current flow is away from the positive plate of a capacitor, then
Class 15

9. Charging a Capacitor What happens when we close switch S at t = 0?
Class 15

10. Charging a Capacitor Circulate clockwise
First order linear inhomogeneous differential equation
Class 15

11. Energy Balance: Circuit Equation
Multiplying by
(power delivered by battery) = (power dissipated through resistor)
+ (power absorbed by the capacitor)
Class 14

12. RC Circuit Charging: Solution
Solution to this equation when switch is closed at t = 0:
(units: seconds)
Class 15

13. Demonstration RC Time Constant Displayed with a Lightbulb (E10)
Class 27

14. Review Some Math: Exponential Decay
Class 23

15. Math Review: Exponential Decay
Consider function A where:
A decays exponentially:
Class 15

16. Exponential Behavior Slightly modify diff. eq.: A “grows” to Af:
Class 23

17. Homework: Solve Differential Equation for Charging and Discharging RC Circuits
Class 15

18. Concept Question: Current in RC Circuit
Class 15

19. Concept Question: RC Circuit
An uncharged capacitor is connected to a battery, resistor and switch. The switch is initially open but at t = 0 it is closed. A very long time after the switch is closed, the current in the circuit is
Nearly zero
At a maximum and decreasing
Nearly constant but non-zero
Class 15

20. Concept Q. Answer: RC Circuit
Answer: 1. After a long time the current is 0
Eventually the capacitor gets “completely charged” – the voltage increase provided by the battery is equal to the voltage drop across the capacitor. The voltage drop across the resistor at this point is 0 – no current is flowing.
Class 15

21. Discharging A Capacitor
At t = 0 charge on capacitor is Q0. What happens when we close switch S at t = 0?
Class 15

22. Discharging a Capacitor
Circulate clockwise
First order linear differential equation
Class 15

23. RC Circuit: Discharging
Solution to this equation when switch is closed at t = 0 with time constant
Class 15

24. Concept Questions: RC Circuit
Class 15

25. Concept Question: RC Circuit
Consider the circuit at right, with an initially uncharged capacitor and two identical resistors. At the instant the switch is closed:
Class 15

26. Concept Question Answer: RC Circuit
Initially there is no charge on the capacitor and hence no voltage drop across it – it looks like a short. Thus all current will flow through it rather than through the bottom resistor. So the circuit looks like:
Class 15

27. Concept Q.: Current Thru Capacitor
In the circuit at right the switch is closed at t = 0. At t = ∞ (long after) the current through the capacitor will be: . 20

28. Con. Q. Ans.: Current Thru Capacitor
Week 10, Day 1
Con. Q. Ans.: Current Thru Capacitor
Answer 1.
After a long time the capacitor becomes “fully charged.” No more current flows into it.
Class 23 28

29. Concept Q.: Current Thru Resistor
In the circuit at right the switch is closed at t = 0. At t = ∞ (long after) the current through the lower resistor will be: . 20

30. Concept Q. Ans.: Current Thru Resistor
Week 10, Day 1
Concept Q. Ans.: Current Thru Resistor
Answer 3.
Since the capacitor is “fullly charged” we can remove it from the circuit, and all that is left is the battery and two resistors. So the current is
Class 23 30

31. Group Problem: RC Circuit
For the circuit shown in the figure the currents through the two bottom branches as a function of time (switch closes at t = 0, opens at t = T>>RC). State the values of current
(i) just after switch is closed at t = 0+
(ii) Just before switch is opened at t = T-,
(iii) Just after switch is opened at t = T+
Class 15

32. Concept Q.: Open Switch in RC Circuit
Week 11, Day 2
Concept Q.: Open Switch in RC Circuit
Now, after the switch has been closed for a very long time, it is opened. What happens to the current through the lower resistor?
It stays the same
Same magnitude, flips direction
It is cut in half, same direction
It is cut in half, flips direction
It doubles, same direction
It doubles, flips direction 20
Class 25 32

33. Con. Q. Ans.: Open Switch in RC Circuit
Answer: 1. It stays the same
The capacitor has been charged to a potential of
so when it is responsible for pushing current through the lower resistor it pushes a current of , in the same direction as before (its positive terminal is also on the left)
Class 15

34. LR Circuits
Class 23

35. Inductors in Circuits Inductor: Circuit element with self-inductance
Ideally it has zero resistance
Symbol:
Class 23

36. E is no longer a static field
Non-Static Fields
E is no longer a static field
Class 23

37. Kirchhoff’s Modified 2nd Rule
If all inductance is ‘localized’ in inductors then our problems go away – we just have:
Class 23

38. Ideal Inductor
BUT, EMF generated by an inductor is not a voltage drop across the inductor!
Because resistance is 0, E must be 0!
Class 23

39. Non-Ideal Inductors
Non-Ideal (Real) Inductor: Not only L but also some R =
In direction of current:
Class 25

40. Circuits: Applying Modified Kirchhoff’s (Really Just Faraday’s Law)
Class 23

41. Sign Conventions - Inductor
Moving across an inductor in the direction of current contributes
Moving across an inductor opposite the direction of current contributes
Class 14

42. LR Circuit Circulate clockwise
First order linear inhomogeneous differential equation
Class 23

43. RL Circuit (units: seconds)
Solution to this equation when switch is closed at t = 0:
(units: seconds)
Class 23

44. RL Circuit
t=0+: Current is trying to change. Inductor works as hard as it needs to to stop it
t=∞: Current is steady. Inductor does nothing.
Class 23

45. Group Problem: LR Circuit
For the circuit shown in the figure the currents through the two bottom branches as a function of time (switch closes at t = 0, opens at t = T>>L/R). State the values of current
(i) just after switch is closed at t = 0+
(ii) Just before switch is opened at t = T-,
(iii) Just after switch is opened at t = T+
Class 15