Download Presentation_26

Class 26: Outline
Hour 1:
Driven Harmonic Motion (RLC)
Hour 2:
Experiment 11: Driven RLC Circuit
P26- 1
Last Time:
Undriven RLC Circuits
P26- 2
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)
P26- 3
Damped LC Oscillations
Resistor dissipates
energy and system
rings down over time
P26- 4
Mass on a Spring:
Simple Harmonic Motion`
A Second Look
P26- 5
Mass on a Spring
(1)
(3)
(2)
(4)
We solved this:
d 2x
F  kx  ma  m 2
dt
2
d x
m 2  kx  0
dt
Simple Harmonic Motion
x(t )  x0 cos(0t   )
Moves at natural frequency
What if we now move the wall?
Push on the mass?
P26- 6
Demonstration:
Driven Mass on a Spring
Off Resonance
P26- 7
Driven Mass on a Spring
Now we get:
d 2x
F  F  t   kx  ma  m 2
dt
2
F(t)
d x
m 2  kx  F  t 
dt
Assume harmonic force:
F (t )  F0 cos(t )
Simple Harmonic Motion
x(t )  xmax cos(t   )
Moves at driven frequency
P26- 8
Resonance
x(t )  xmax cos(t   )
Now the amplitude, xmax, depends on how
close the drive frequency is to the natural
frequency
xmax
Let’s
See…
0

P26- 9
Demonstration:
Driven Mass on a Spring
P26- 10
Resonance
x(t )  xmax cos(t   )
xmax depends on drive frequency
xmax
Many systems behave like this:
Swings
Some cars
Musical Instruments
…
0

P26- 11
Electronic Analog:
RLC Circuits
P26- 12
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
P26- 13
Demonstration:
RLC with Light Bulb
P26- 14
Start at Beginning:
AC Circuits
P26- 15
Alternating-Current Circuit
• direct current (dc) – current flows one way (battery)
• alternating current (ac) – current oscillates
• sinusoidal voltage source
V (t )  V0 sin  t
  2 f : angular frequency
V0 : voltage amplitude
P26- 16
AC Circuit: Single Element
V V
 V0 sin  t
I (t )  I 0 sin(t   )
Questions:
1. What is I0?
2. What is  ?
P26- 17
AC Circuit: Resistors
VR  I R R
VR V0
IR 

sin  t
R R
 I 0 sin   t  0 
V0
I0 
R
 0
IR and VR are in phase
P26- 18
AC Circuit: Capacitors
Q
VC 
C
dQ
I 0   CV0
I C (t ) 
dt




  CV0 cos  t
2
 I 0 sin( t   2 )
IC leads VC by /2
Q(t )  CVC  CV0 sin t
P26- 19
AC Circuit: Inductors
V
I (t ) 
sin  t dt

L
L
dI L
VL  L
dt
0

V0
cos  t
V0
I0 
L
L




 I 0 sin   t  2 
2
IL lags VL by /2
dI L VL V0


sin  t
dt
L
L
P26- 20
AC Circuits: Summary
Element
I0
Current
vs.
Voltage
Resistor
V0 R
R
In Phase
RR
Leads
1
XC 
C
Capacitor
 CV0C
Resistance
Reactance
Impedance
V0 L
Lags
X


L
L
L
Although derived from single element circuits,
these relationships hold generally!
Inductor
P26- 21
PRS Question:
Leading or Lagging?
P26- 22
Phasor Diagram
Nice way of tracking magnitude & phase:
V (t )  V0 sin   t 

V0
t
Notes: (1) As the phasor (red vector) rotates, the
projection (pink vector) oscillates
(2) Do both for the current and the voltage
P26- 23
Demonstration:
Phasors
P26- 24
Phasor Diagram: Resistor
V0  I 0 R
 0
IR and VR are in phase
P26- 25
Phasor Diagram: Capacitor
V0  I 0 X C
1
 I0
C
  2
IC leads VC by /2
P26- 26
Phasor Diagram: Inductor
V0  I 0 X L
 I 0 L


2
IL lags VL by /2
P26- 27
PRS Questions:
Phase
P26- 28
Put it all together:
Driven RLC Circuits
P26- 29
Question of Phase
We had fixed phase of voltage:
V  V0 sin  t I (t )  I 0 sin( t   )
It’s the same to write:
V  V0 sin( t   ) I (t )  I 0 sin  t
(Just shifting zero of time)
P26- 30
Driven RLC Series Circuit
I (t )  I 0 sin(t )
VR  VR 0 sin   t 
VL  VL0 sin   t  2 
VS  V0 S sin   t   
VC  VC 0 sin   t  2

What is I 0 (and VR 0  I 0 R, VL0  I 0 X L , VC 0  I 0 X C )?
What is  ? Does the current lead or lag Vs ?
Must Solve: VS  VR  VL  VC
P26- 31
Driven RLC Series Circuit
V0L
V0S
I(t)
VS
V0C
I0
V0R
Now Solve: VS  VR  VL  VC
Now we just need to read the phasor diagram!
P26- 32
Driven RLC Series Circuit
V0L
V0S

I (t )  I 0 sin(t   )
V0C
VS  V0 S sin   t 
I0
V0R
V0 S  VR 0  (VL 0  VC 0 )  I 0 R  ( X L  X C )  I 0 Z
2
V0 S
I0 
Z
2
Z  R  ( X L  XC )
Impedance
2
2
2
  tan
2
1
 X L  XC 


R


P26- 33
I
I (t )  I 0 sin   t 
0
VR (t )  I 0 R sin   t 
0
VL (t )  I 0 X L sin   t  2 
VL
0
VR
Plot I, V’s vs. Time
VS
VC
+/2
VL (t )  I 0 X C sin   t  2 
-/2
0
VS (t )  VS 0 sin   t   
0
0
+
1
2
3
Time (Periods)
P26- 34
PRS Question:
Who Dominates?
P26- 35
RLC Circuits:
Resonances
P26- 36
Resonance
V0
I0  
Z
V0
R 2  ( X L  X C )2
;
1
X L   L, X C 
C
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
I0 reaches maximum when
0 
X L  XC
1
LC
P26- 37
Resonance
V0
I0  
Z
C-like:
<0
I leads
V0
R 2  ( X L  X C )2
;
1
X L   L, X C 
C
  tan
1
 X L  XC 


R


L-like:
>0
I lags
0  1 LC
P26- 38
Demonstration:
RLC with Light Bulb
P26- 39
PRS Questions:
Resonance
P26- 40
Experiment 11:
Driven RLC Circuit
P26- 41
Experiment 11: How To
Part 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
P26- 42