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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