Today’s Concepts:A) LC circuits and Oscillation FrequencyB) EnergyC) RLC circuits and Damping

Physics 2112Unit 19

Electricity & Magnetism Lecture 19, Slide 1

C

I

L Q

LC Circuit

Electricity & Magnetism Lecture 19, Slide 2

+

-

dt

dILVL

+

-

C

QVC

Circuit Equation: 0dt

dIL

C

Q

dt

dQI

LC

Q

dt

Qd

2

2

where

Qdt

Qd 22

2

LC

1

)cos()( max tQtQSolution to this DE:

CL

k

x

m

F = -kx

a

Same thing if we notice that and

Electricity & Magnetism Lecture 19, Slide 3

Qdt

Qd 22

2

LC

1

m

kx

dt

xd 22

2

Ck

1

Lm

Lm

Just like in 2111

k

x

m

F = -kx

a

Electricity & Magnetism Lecture 19, Slide 4

Also just like in 2111

22

2

1

2

1kxmv

Total Energy constant (when no friction present)

CL2

2

2

1

2

1LI

C

Q

Total Energy constant (when no resistor present)

CL

I+ +

--

Q and I Time Dependence

Electricity & Magnetism Lecture 19, Slide 5

Charge and Current “90o out of phase”

What is the potential difference across the inductor at t = 0? A) VL = 0 B) VL = Qmax/C C) VL = Qmax/2C

At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. L C

since VL = VC

CheckPoint 1

Electricity & Magnetism Lecture 19, Slide 6

Inductor does not

need current to

have voltage drop!

It’s not lik

e a

resistor!

CheckPoint 2

Electricity & Magnetism Lecture 19, Slide 7

At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. L C

What is the potential difference across the inductor at when the current is maximum? A) VL = 0 B) VL = Qmax/C C) VL = Qmax/2C

Q is max but dQ/dt

is 0!

How much energy is stored in the capacitor when the current is a maximum ? A)  U = Qmax

2/(2C) B) U = Qmax

2/(4C) C) U = 0

CheckPoint 3

Electricity & Magnetism Lecture 19, Slide 8

At time t = 0 the capacitor is fully charged with Qmax and the current through the circuit is 0. L C

Max current means

all of U is in

inductor!

L=0.82H

Example 19.1 (Charged LC circuit)

Unit 19, Slide 9

1 2After being left in position 1 for a long time, at t=0, the switch is flipped to position 2.

All circuit elements are “ideal”.

C=0.01FV=12V

What is the equation for the charge on the upper plate of the capacitor at any given time t?

How long does it take the lower plate of the capacitor to fully discharge and recharge?

R=10W

Example 19.1 (Charged LC circuit)

Unit 19, Slide 10

• How long does it take the lower plate of the capacitor to fully discharge and recharge?

Conceptual AnalysisFill in all the terms in

Strategic AnalysisFind initial currentUse energy conservation to find Qmax

Use L and C to find wUse initial conditions to final f

• What is the equation for the charge on the upper plate of the capacitor at any given time t?

)cos()( max tQtQ

The switch is closed for a long time, resulting in a steady current Vb/R through the inductor.

A. Q(t) = Qmaxcos(ωt) B. Q(t) = Qmaxcos(ωt + π/4) C. Q(t) = Qmaxcos(ωt + π/2) D. Q(t) = Qmaxcos(ωt + π)

Prelecture Question

Electricity & Magnetism Lecture 19, Slide 11

At time t = 0, the switch is opened, leaving a simple LC circuit. Which formula best describes the charge on the capacitor as a function of time?

The capacitor is charged such that the top plate has a charge +Q0 and the bottom plate -Q0. At time t = 0, the switch is closed and the circuit oscillates with frequency w = 500 radians/s.

What is the value of the capacitor C?

A) C = 1 x 10-3 F B)  C = 2 x 10-3 F C)  C = 4 x 10-3 F

L C

L = 4 x 10-3 Hw = 500 rad/s

+ +--

CheckPoint 4

Electricity & Magnetism Lecture 19, Slide 12

Which plot best represents the energy in the inductor as a function of time starting just after the switch is closed?

L C

closed at  t = 0

+Q0

-Q0

CheckPoint 5

Electricity & Magnetism Lecture 19, Slide 13

2

2

1LIU L

When the energy stored in the capacitor reaches its maximum again for the first time after t = 0, how much charge is stored on the top plate of the capacitor?

A)  +Q0

B) +Q0 /2C)  0D) -Q0/2E) -Q0

CheckPoint 6

Electricity & Magnetism Lecture 19, Slide 14

L C

closed at  t = 0

+Q0

-Q0

Current goes to zero twice during one cycle

Q is maximum when current goes to zero

dt

dQI

Add Resistance

Unit 19, Slide 15

1 2

Switch is flipped to position 2.

R

01

2

2

QCdt

dQR

dt

QdL

02 cbtatUse “characteristic equation”

Damped Harmonic Motin

Unit 19, Slide 16

)'cos( teQQ tMax

L

R

2

222' o

Damping factor

Damped oscillation frequency

LCo

1

Natural oscillation frequency

Remember from 2111?

Mechanics Lecture 21, Slide 17

Overdamped

Critically damped

Under damped

Which one do you want for your shocks on your car?

L=0.82H

Example 19.2 (Charged LC circuit)

Unit 19, Slide 18

1 2After being left in position 1 for a long time, at t=0, the switch is flipped to position 2.

The inductor now non-ideal has some internal resistance.

What is the equation for the charge on the capacitor at any given time t?

C=0.01FV=12V

RL=7W

R=10W

How long does it take the lower plate of the capacitor to fully discharge and recharge?

The elements of a circuit are very simple:

This is all we need to know to solve for anything.

But these all depend on each other and vary with time!!

Electricity & Magnetism Lecture 19, Slide 19

dt

dILVL

C

QVC

dt

dQI

IRVR

RCL VVVV

V(t) across each elements

Start with some initial V, I, Q, VL

Now take a tiny time step dt (1 ms)

Repeat…

How would we actually do this?

Electricity & Magnetism Lecture 19, Slide 20

C

QVC

IRVR

dtL

VId L

IdtdQ

CRL VVVV

What would this look like?

In the circuit to the right • R1 = 100 Ω,• L1 = 300 mH, • L2 = 180 mH,• C = 120 μF and • V = 12 V.

Example 19.3

Electricity & Magnetism Lecture 19, Slide 21

The positive terminal of the battery is indicated with a + sign. All elements are considered ideal. Q(t) is defined to be positive if V(a) – V(b) is positive.

What is the charge on the bottom plate of the capacitor at time t = 2.82 ms?

Conceptual AnalysisFind the equation of Q as a function of time

Strategic AnalysisDetermine initial currentDetermine oscillation frequency w0

Find maximum charge on capacitorDetermine phase angle

In the circuit to the right • R1 = 100 Ω,• L1 = 300 mH, • L2 = 180 mH,• C = 120 μF and • V = 12 V.

Example 19.4

Electricity & Magnetism Lecture 19, Slide 22

The positive terminal of the battery is indicated with a + sign. All elements are considered ideal. Q(t) is defined to be positive if V(a) – V(b) is positive.

What is the energy stored in the inductors at time t = 2.82 ms?

Conceptual AnalysisUse conservation of energy

Strategic AnalysisFind energy stored on capacitor using information from Ex 19.3Subtract of the total energy calculated in the clicker question