1 w12d1: rc and lr circuits reading course notes: sections 7.7-7.8, 7.11.3, 11.4-11.6, 11.12.2,...
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W12D1:RC and LR Circuits
Reading Course Notes: Sections 7.7-7.8, 7.11.3, 11.4-11.6, 11.12.2, 11.13.4-11.13.5
AnnouncementsMath Review Week 12 Tuesday 9pm-11 pm in 26-152
PS 9 due Week 13 Tuesday April 30 at 9 pm in boxes outside 32-082 or 26-152
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Outline
DC Circuits with Capacitors
First Order Linear Differential Equations
RC Circuits
LR Circuits
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DC Circuits with Capacitors
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Sign Conventions - Capacitor
Moving across a capacitor from the negatively to positively charged plate increases the electric potential
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Power - CapacitorMoving across a capacitor from the positive to negative plate decreases your potential. If current flows in that direction the capacitor absorbs power (stores charge)
dt
dU
C
Q
dt
d
C
Q
dt
dQVIP
2
2
absorbed
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RC Circuits
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(Dis)Charging a Capacitor1. 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
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Charging a Capacitor
What happens when we close switch S at t = 0?
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Charging a Capacitor
Circulate clockwise
First order linear inhomogeneous differential equation
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Energy Balance: Circuit Equation
Multiplying by
(power delivered by battery) = (power dissipated through resistor)
+ (power absorbed by the capacitor)
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RC Circuit Charging: Solution
Solution to this equation when switch is closed at t = 0:
(units: seconds)
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DemonstrationRC Time ConstantDisplayed with a
Lightbulb (E10)
http://tsgphysics.mit.edu/front/?page=demo.php&letnum=E%2010&show=0
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Review Some Math:Exponential Decay
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Math Review: Exponential Decay
Consider function A where:
A decays exponentially:
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Exponential Behavior
Slightly modify diff. eq.:
A “grows” to Af:
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Homework: Solve Differential Equation for Charging and Discharging RC Circuits
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Concept Question:Current in RC Circuit
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Concept Question: RC CircuitAn 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
1. Nearly zero2. At a maximum and
decreasing3. Nearly constant but non-zero
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Concept Q. Answer: RC Circuit
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.
Answer: 1. After a long time the current is 0
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Discharging A Capacitor
At t = 0 charge on capacitor is Q0. What happens when we close switch S at t = 0?
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Discharging a Capacitor
Circulate clockwise
First order linear differential equation
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RC Circuit: Discharging
Solution to this equation when switch is closed at t = 0 with time constant
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Concept Questions:RC Circuit
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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:
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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:
Answer: 3.
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1. .
2.
3.
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:
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Con. Q. Ans.: Current Thru Capacitor
After a long time the capacitor becomes “fully charged.” No more current flows into it.
Answer 1.
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1. .
2.
3.
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:
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Concept Q. Ans.: Current Thru Resistor
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
.
Answer 3.
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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+
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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?
1. It stays the same2. Same magnitude, flips direction3. It is cut in half, same direction4. It is cut in half, flips direction5. It doubles, same direction6. It doubles, flips direction
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Con. Q. Ans.: Open Switch in RC Circuit
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)
Answer: 1. It stays the same
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LR Circuits
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Inductors in Circuits
Inductor: Circuit element with self-inductance Ideally it has zero resistance
Symbol:
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Non-Static Fields
E is no longer a static field
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Kirchhoff’s Modified 2nd Rule
If all inductance is ‘localized’ in inductors then our problems go away – we just have:
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• BUT, EMF generated by an inductor is not a voltage drop across the inductor!
Ideal Inductor
Because resistance is 0, E must be 0!
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Non-Ideal Inductors
Non-Ideal (Real) Inductor: Not only L but also some R
In direction of current:
=
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Circuits:Applying Modified Kirchhoff’s(Really Just Faraday’s Law)
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Sign Conventions - Inductor
Moving across an inductor in the direction of current contributes
dILdt
Moving across an inductor opposite the direction of current contributes
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LR Circuit
Circulate clockwise
First order linear inhomogeneous differential equation
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RL Circuit
Solution to this equation when switch is closed at t = 0:
(units: seconds)
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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.
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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+