# W12D1: RC and LR Circuits Reading Course Notes: Sections , , , ,

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

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

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

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

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

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

RC Circuits Class 15

(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

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

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

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

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

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

Review Some Math: Exponential Decay
Class 23

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

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

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

Concept Question: Current in RC Circuit
Class 15

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

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

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

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

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

Concept Questions: RC Circuit
Class 15

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

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

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

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

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

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

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

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

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

LR Circuits Class 23

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

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

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

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

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

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

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

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

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

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

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

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