1P26-

Class 26: Outline

Hour 1:

Driven Harmonic Motion (RLC)

Hour 2:

Experiment 11: Driven RLC Circuit

2P26-

Last Time:Undriven RLC Circuits

3P26-

LC CircuitIt undergoes simple harmonic motion, just like a mass on a spring, with trade-off between charge on capacitor (Spring) and current in inductor (Mass)

4P26-

Damped LC Oscillations

Resistor dissipates energy and system rings down over time

5P26-

Mass on a Spring:Simple Harmonic Motion`

A Second Look

6P26-

Mass on a Spring

2

2

2

20

d xF kx ma m

dt

d xm kxdt

0 0( ) cos( )x t x t

(1) (2)

(3) (4)

We solved this:

Simple Harmonic Motion

What if we now move the wall?Push on the mass?

Moves at natural frequency

7P26-

Demonstration: Driven Mass on a Spring

Off Resonance

8P26-

Driven Mass on a Spring

2

2

2

2

d xF F t kx ma m

dt

d xm kx F tdt

max( ) cos( )x t x t

Now we get:

Simple Harmonic Motion

F(t)

Assume harmonic force:

0( ) cos( )F t F t

Moves at driven frequency

9P26-

Resonance

Now the amplitude, xmax, depends on how close the drive frequency is to the natural frequency

max( ) cos( )x t x t

xmax

Let’s

See…

10P26-

Demonstration: Driven Mass on a Spring

11P26-

Resonance

xmax depends on drive frequency

max( ) cos( )x t x t

xmax Many systems behave like this:

Swings

Some cars

Musical Instruments

…

12P26-

Electronic Analog:RLC Circuits

13P26-

Analog: RLC Circuit

Recall:

Inductors are like masses (have inertia)

Capacitors are like springs (store/release energy)

Batteries supply external force (EMF)

Charge on capacitor is like position,

Current is like velocity – watch them resonate

Now we move to “frequency dependent batteries:”

AC Power Supplies/AC Function Generators

14P26-

Demonstration: RLC with Light Bulb

15P26-

Start at Beginning:AC Circuits

16P26-

Alternating-Current Circuit

• sinusoidal voltage source

0( ) sinV t V t

0

2 : angular frequency

: voltage amplitude

f

V

• direct current (dc) – current flows one way (battery)

• alternating current (ac) – current oscillates

17P26-

AC Circuit: Single Element

0 sin

V V

V t

0( ) sin( )I t I t

Questions:

1. What is I0?

2. What is ?

18P26-

AC Circuit: Resistors

R RV I R

0

0

sin

sin 0

RR

VVI t

R RI t

00

0

VI

R

IR and VR are in phase

19P26-

AC Circuit: Capacitors

C

QV

C

0

20

( )

cos

sin( )

C

dQI t

dtCV t

I t

0 0

2

I CV

0( ) sinCQ t CV CV t

IC leads VC by /2

20P26-

AC Circuit: Inductors

LL

dIV L

dt

0

0

20

( ) sin

cos

sin

LVI t t dtLV tLI t

0 sinL L VdI Vt

dt L L

00

2

VI

L

IL lags VL by /2

21P26-

Element I0

Current vs.

Voltage

Resistance Reactance Impedance

Resistor In Phase

Capacitor Leads

Inductor Lags

AC Circuits: Summary

1CX C

0CCV

LX L

0RV

R

0LV

L

R R

Although derived from single element circuits, these relationships hold generally!

22P26-

PRS Question:Leading or Lagging?

23P26-

Phasor Diagram

Nice way of tracking magnitude & phase:

t

0( ) sinV t V t

0V

Notes: (1) As the phasor (red vector) rotates, the projection (pink vector) oscillates

(2) Do both for the current and the voltage

24P26-

Demonstration: Phasors

25P26-

Phasor Diagram: Resistor

0 0

0

V I R

IR and VR are in phase

26P26-

Phasor Diagram: Capacitor

IC leads VC by /2

0 0

0

1

2

CV I X

IC

27P26-

Phasor Diagram: Inductor

IL lags VL by /2

0 0

0

2

LV I X

I L

28P26-

PRS Questions:Phase

29P26-

Put it all together:Driven RLC Circuits

30P26-

Question of Phase

We had fixed phase of voltage:

It’s the same to write:

0 0sin ( ) sin( )V V t I t I t

0 0sin( ) ( ) sinV V t I t I t

(Just shifting zero of time)

31P26-

Driven RLC Series Circuit

0 sinS SV V t

0( ) sin( )I t I t

0 sinR RV V t

0 2sinL LV V t

0 2sinC CV V t

Must Solve: S R L CV V V V

0 0 0 0 0 0 0What is (and , , )?R L L C CI V I R V I X V I X What is ? Does the current lead or lag ?sV

32P26-

Driven RLC Series Circuit

Now Solve: S R L CV V V V

I(t)

0I 0RV

0LV

0CV

0SV

Now we just need to read the phasor diagram!

VS

33P26-

Driven RLC Series Circuit

0I 0RV

0LV

0CV

0SV

2 2 2 20 0 0 0 0 0( ) ( )S R L C L CV V V V I R X X I Z

1tan L CX X

R

2 2( )L CZ R X X 0

0SVIZ

0 sinS SV V t

0( ) sin( )I t I t

Impedance

34P26-

Plot I, V’s vs. Time

0 1 2 30

+

-/2

+/2

VS

Time (Periods)

0

VC

0

VL

0

VR

0

I

0

0

0 2

0 2

0

( ) sin

( ) sin

( ) sin

( ) sin

( ) sin

R

L L

L C

S S

I t I t

V t I R t

V t I X t

V t I X t

V t V t

35P26-

PRS Question:Who Dominates?

36P26-

RLC Circuits:Resonances

37P26-

Resonance

0 00 2 2

1; ,

( )L C

L C

V VI X L X

Z CR X X

I0 reaches maximum when L CX X

0

1

LC

At very low frequencies, C dominates (XC>>XL): it fills up and keeps the current low

At very high frequencies, L dominates (XL>>XC):the current tries to change but it won’t let it

At intermediate frequencies we have resonance

38P26-

Resonance0 0

0 2 2

1; ,

( )L C

L C

V VI X L X

Z CR X X

C-like: < 0 I leads

1tan L CX X

R

0 1 LC

L-like: > 0 I lags

39P26-

Demonstration: RLC with Light Bulb

40P26-

PRS Questions:Resonance

41P26-

Experiment 11: Driven RLC Circuit

42P26-

Experiment 11: How ToPart I• Use exp11a.ds• Change frequency, look at I & V. Try to find resonance

– place where I is maximum

Part II• Use exp11b.ds• Run the program at each of the listed frequencies to

make a plot of I0 vs.

Part III• Use exp11c.ds • Plot I(t) vs. V(t) for several frequencies