W12D2 RC, LR, and Undriven RLC Circuits; Experiment 4

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W12D2RC, LR, andUndriven RLC
CircuitsExperiment 4
Todays Reading Course Notes Sections 11.7-11.9,
11.10, 11.13.6 Expt. 4 Undriven RLC Circuits
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Announcements
Math Review Week 13 Tuesday 9pm-11 pm in
26-152 PS 9 due Week 13 Tuesday at 9 pm in boxes
outside 32-082 or 26-152 Next Reading Assignment
W12D3 Course Notes Sections 11.8-9, 11.12-11.13

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Outline

• Experiment 4 Part 1 RC and LR Circuits
• Simple Harmonic Oscillator
• Undriven RLC Circuits
• Experiment 4 Part 2 Undriven RLC Circuits

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RC Circuit Charging
Solution to this equation when switch is closed
at t 0
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RC Circuit Discharging
Solution to this equation when switch is closed
at t 0 time
constant
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RL Circuit Increasing Current
Solution to this equation when switch is closed
at t 0
(units seconds)
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RL Circuit Decreasing Current
Solution to this equation when switch is opened
at t 0
(units seconds)
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Measuring Time Constant

• Pick a point 1 with
• Find point 2 such that
• By definition then

2) In the lab you will plot semi-log and fit
curve (make sure you exclude data at both ends)
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Experiment 4RC and RL Circuits
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Mass on a SpringSimple Harmonic Motion
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DemonstrationMass on a SpringSimple Harmonic
MotionMass on a Spring (C 2)
http//scripts.mit.edu/tsg/www/demo.php?letnumC
202show0
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Mass on a Spring
(2)
(1)
What is Motion?
(4)
(3)
Simple Harmonic Motion
x0 Amplitude of Motion f Phase (time offset)
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Simple Harmonic Motion
Amplitude (x0)
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Concept Question Simple Harmonic Oscillator
Which of the following functions x(t) has a
second derivative which is proportional to the
negative of the function
1.
2.
3.
4.
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Concept Question Answer Simple Harmonic
Oscillator
Answer 4. By direct calculation, when
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Mass on a Spring Energy
(1) Spring
(2) Mass
(4) Mass
(3) Spring
Energy has 2 parts (Mass) Kinetic and (Spring)
Potential
Energy sloshes back and forth
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LC Circuit

• Set up the circuit above with capacitor,
  inductor, resistor, and battery.
• Let the capacitor become fully charged.
• Throw the switch from a to b.
• What happens?

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LC Circuit
It undergoes simple harmonic motion, just like a
mass on a spring, with trade-off between charge
on capacitor (Spring) and current in inductor
(Mass). Equivalently trade-off between energy
stored in electric field and energy stored in
magnetic field.
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Energy stored in electric field
Energy stored in magnetic field
Energy stored in electric field
Energy stored in magnetic field
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Concept Question LC Circuit
Consider the LC circuit at right. At the time
shown the current has its maximum value. At this
time

1. the charge on the capacitor has its maximum
   value.
2. the magnetic field is zero.
3. the electric field has its maximum value.
4. the charge on the capacitor is zero.

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Concept Q. Answer LC Circuit
Answer 4. The current is maximum when the charge
on the capacitor is zero
Current and charge are exactly 90 degrees out of
phase in an ideal LC circuit (no resistance), so
when the current is maximum the charge must be
identically zero.
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LC Circuit Simple Harmonic Oscillator
Simple harmonic oscillator
Charge Angular frequency Amplitude of charge
oscillation Phase (time offset)
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LC Oscillations Energy
Notice relative phases
Total energy is conserved !!
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LC Circuit OscillationSummary
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Adding Damping RLC Circuits
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DemonstrationUndriven RLC Circuits (Y 190)
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RLC Circuit Energy Changes
Include finite resistance
Multiply by
Decrease in stored energy is equal to Joule
heating in resistor
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Damped LC Oscillations
Resistor dissipates energy and system rings down
over time. Also, frequency decreases
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Experiment 4 Part 2Undriven RLC Circuits
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Appendix Experiment 4 Part 2Undriven RLC
CircuitsGroup Problemand Concept Questions
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Problem LC Circuit

• Consider the circuit shown in the figure.
  Suppose the switch that has been connected to
  point a for a long time is suddenly thrown to b
  at t 0. Find the following quantities

(a) the frequency of oscillation of the
circuit. (b) the maximum charge that appears on
the capacitor. (c) the maximum current in the
inductor. (d) the total energy the circuit
possesses as a function of time t.
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Concept Question Expt. 4
In todays lab the battery turns on and off.
Which circuit diagram is most representative of
our circuit?
1.
2.
3.
4.

1. 1
2. 2
3. 3
4. 4

Load lab while waiting
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Concept Question Answer Expt. 4
1.
Answer
There is resistance in the circuit (in our
non-ideal inductor). The battery switching off
doesnt break the circuit but allows it to ring
down
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Concept Question LC Circuit
The plot shows the charge on a capacitor (black
curve) and the current through it (red curve)
after you turn off the power supply. If you put
a core into the inductor what will happen to the
time TLag?

1. It will increase
2. It will decrease
3. It will stay the same
4. I dont know

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Concept Question Answer LC Circuit

• Answer 1.
• TLag will increase. Putting in a core increases
  the inductors inductance and hence decreases the
  natural frequency of the circuit. Lower
  frequency means longer period. The phase will
  remain at 90º (a quarter period) so TLag will
  increase.

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Concept Question LC Circuit
If you increase the resistance in the circuit
what will happen to rate of decay of the pictured
amplitudes?

1. It will increase (decay more rapidly)
2. It will decrease (decay less rapidly)
3. It will stay the same
4. I dont know

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Concept Question Answer LC Circuit
Answer 1. It will increase (decay more rapidly)
Resistance is what dissipates power in the
circuit and causes the amplitude of oscillations
to decrease. Increasing the resistance makes the
energy (and hence amplitude) decay more rapidly.