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W12D2RC, LR, and

Undriven RLC Circuits;Experiment 4

Today’s Reading Course Notes: Sections 11.7-11.9, 11.10, 11.13.6; Expt. 4: Undriven RLC Circuits

AnnouncementsMath 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 4:RC and RL Circuits

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Mass on a Spring:Simple Harmonic Motion

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DemonstrationMass on a Spring:

Simple Harmonic MotionMass on a Spring (C 2)

http://scripts.mit.edu/~tsg/www/demo.php?letnum=C%202&show=0

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Mass on a Spring(1) (2)

(3) (4)

What is Motion?

Simple Harmonic Motion

x0: Amplitude of Motion

: Phase (time offset)

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Simple Harmonic Motion

Amplitude (x0)

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.

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 (3) Spring (4) Mass

Energy has 2 parts: (Mass) Kinetic and (Spring) Potential

Energy sloshes back

and forth

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LC Circuit

1. Set up the circuit above with capacitor, inductor, resistor, and battery.

2. Let the capacitor become fully charged.

1. Throw the switch from a to b.

1. 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

Charge:

Angular frequency:

Amplitude of charge oscillation:

Phase (time offset):

Simple harmonic oscillator:

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LC Oscillations: Energy

Total energy is conserved !!

Notice relative phases

LC Circuit OscillationSummary

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Adding Damping: RLC Circuits

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Demonstration

Undriven RLC Circuits (Y 190)

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 2

Undriven RLC Circuits

Group 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.

1. 1

2. 2

3. 3

4. 4

Concept Question: Expt. 4In today’s lab the battery turns on and off. Which circuit diagram is most representative of our circuit?

1. 2.

3. 4.

Load lab while waiting…

Concept Question Answer: Expt. 4

Answer: 1.

There is resistance in the circuit (in our non-ideal inductor).

The battery switching off doesn’t break the circuit but allows it to ring down

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? 0 40 80 120

-1.0Q0

-0.5Q0

0.0Q0

0.5Q0

1.0Q0

-1.0I0

-0.5I0

0.0I0

0.5I0

1.0I0

Tlag

Cha

rge

on C

apac

itor

Time (mS)

Charge

Cur

rent

thro

ugh

Cap

acito

r

Current

1. It will increase2. It will decrease3. It will stay the same4. I don’t know

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Concept Question Answer: LC Circuit

Answer 1.TLag will increase.

Putting in a core increases the inductor’s 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.

0 40 80 120-1.0Q

0

-0.5Q0

0.0Q0

0.5Q0

1.0Q0

-1.0I0

-0.5I0

0.0I0

0.5I0

1.0I0

Tlag

Cha

rge

on C

apac

itor

Time (mS)

Charge

Cur

rent

thro

ugh

Cap

acito

r

Current

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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?

0 40 80 120-1.0Q

0

-0.5Q0

0.0Q0

0.5Q0

1.0Q0

-1.0I0

-0.5I0

0.0I0

0.5I0

1.0I0

Tlag

Cha

rge

on C

apac

itor

Time (mS)

Charge

Cur

rent

thro

ugh

Cap

acito

r

Current

1. It will increase (decay more rapidly)2. It will decrease (decay less rapidly)3. It will stay the same4. I don’t 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.

0 40 80 120-1.0Q

0

-0.5Q0

0.0Q0

0.5Q0

1.0Q0

-1.0I0

-0.5I0

0.0I0

0.5I0

1.0I0

Tlag

Cha

rge

on C

apac

itor

Time (mS)

Charge

Cur

rent

thro

ugh

Cap

acito

r

Current