lc oscillation
TRANSCRIPT
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Lecture 41 Electromagnetic oscillations andAlternating current/ Chapter # 31
LC oscillations
Damped oscillations in RLC circuits
Alternating current
Simple circuits Resonance and Power in Alternating
current circuits
Transformers
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Review of Voltage DropsAcross Circuit Elements
Idt QVC C= =
Voltage determined by
integralof current andcapacitance
C
I(t)
2
2= =dI d Q
V L Ldt dt
Voltage determined byderivativeof current andinductance
L
I(t)
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Whats Next?
Why and how do oscillations occurin circuits containing capacitors
and inductors? naturally occurring, not driven for now
stored energy capacitive inductive
Where are we going?
Oscillating circuits radio, TV, cell phone, ultrasound, clocks,
computers, GPS
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LC Circuits
Consider the RC and LCseries circuits shown:
Suppose that the circuits areformed att=0 with thecapacitor charged to value Q.
There is a qualitative difference in the time development of thecurrents produced in these two cases. Why??
Consider from point of view of energy!
In the RC circuit, any current developed will cause energy tobe dissipated in the resistor.
In the LC circuit, there is NO mechanism for energydissipation; energy can be stored both in the capacitor andthe inductor!
LCC R++++
- - - -
++++
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LC Circuits - Qualitatively
We discussed RC and RL circuits. In an LC
circuit energy oscillates between the capacitor(E field) and inductor (B field)
22
22 iLU
C
qU BE ==
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LC Oscillations
(qualitative)
LC
+ +
- -
0=I
0QQ +=
LC+ +
- -0=I
0QQ =
LC
0II =
0=Q
LC
0II +=
0=Q
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Oscillations
RC/LC Ci i
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RC/LC Circuits
RC:
current decays exponentially
C R
-It
0
0
I
Q+++
- - -
LC
LC:
current oscillates
I
0
0t
I
Q+++- - -
LC O ill ti
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LC Oscillations(L with finiteR)
IfL has finiteR, then
energy will be dissipated inR.
the oscillations will become damped.
R = 0
Q
0
t t
0
Q
R 0
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10
The Electrical - MechanicalAnalogy
One can make the analogy with mechanicaloscillations (Table 31-1)
qcorresponds to x 1/C corresponds to k icorresponds to v L corresponds to m
)(1)( circuitLCCL
springblockmk ==
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LC Oscillations -
Quantitatively
Block-Spring Oscillator
td
vd
vtd
vd
td
xd
td
xd
td
d
td
vd
td
xdv
2
)( 2
2
2
=
=
=
=
22
2
1
2
1 xkvmUUU sb +=+=
td
dU
=0 td
xd
xktd
xd
td
xd
mtd
xd
xktd
vd
vm +=+= 2
2
)(02
2
nsoscillatiospringblockxk
td
xdm =+
m
kntdisplacemetXx =+= )()cos(
xtXtd
xd 222
2
)cos( =+=
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12
LC Oscillations -
Quantitatively
The LC Oscillator
LCtQq
nsoscillatioLCqCtd
qdL
td
qd
C
q
td
qd
td
qdLtd
qd
C
q
td
idiLtd
Ud
C
qiLUUU EB
1(charge))cos(
)(01
0
22
2
2
2
2
22
=+=
=+
+=+==
+=+=
2
2
td
qd
td
qd
td
d
td
id
tdqdi
=
=
=
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LC Oscillations -
Quantitatively
See also Fig 31-4
)cos( += tQq
1cossin 22 =+
CL
LC
11 2 ==
)sin()sin( +=+== tItQtd
qdi
)(sin2
1
2
)(cos22
2222
222
+==
+==
tQLiL
U
tC
Q
C
q
U
B
E
)(sin2
22
+= tC
QUB
C
QUU EB
2
2
=+
LC O ill ti
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LC OscillationsEnergy Check
The other unknowns ( Q0, ) are found from the initialconditions. E.g., in our original example we assumed initialvalues for the charge (Qi) and current (0). For these values:
Q0 = Qi, = 0.
Question: Does this solution conserve energy?
)(cos21)(
21)( 0
22
0
2 +== tQ
CCtQtUE
)(sin2
1)(2
1)( 0
22
0
2
0
2
+== tQLtLitUB
Oscillation frequency has been found from theloop equation. LC
10 =
E Ch k
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UE
t
0
Energy Check
UB
0t
Energy in Capacitor
)(cos2
1)( 0
22
0 += tQC
tUE
Energy in Inductor
)(sin
2
1)( 0
22
0
2
0 += tQLtUB
LC
10 =
)(sin21)( 0
22
0 += tQC
tUB
CQtUtU BE2
)()(2
0=+Therefore,
P bl
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Problem At t=0 the current flowing through the circuit is
1/2 of its maximum value.
Which of the following plots bestrepresents UB, the energy stored in the
inductor as a function of time?
3A
LC+ +
- -
I
Q
Which of the following is a possible value for the phase , when thecharge on the capacitor is described by: Q(t) = Q0cos(t + )
3B
(a) (b) (c)
00
UB
time
00
UB
time
00
UB
time
(a) = 30 (b) = 45 (c) = 60
P bl ( t)
I
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Problem (cont) At t=0 the current flowing through the circuit is
1/2 of its maximum value.Which of the following plots best represents UB, the
energy stored in the inductor as a function of time?3A
(a) (b) (c)
00
UB
time
00
UB
time
00
UB
time
The key here is to realize that the energy stored in the inductor isproportional to the CURRENT SQUARED.
Therefore, if the current att=0 is 1/2 its maximum value, the energystored in the inductor will be 1/4 of its maximum value!!
LC
+ +
- -
I
Q
P bl ( t)
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Problem (cont) At t=0 the current flowing through the circuit is
1/2 of its maximum value. Which of the following is a possible value for the
phase , when the charge on the capacitor isdescribed by: Q(t) = Q0cos(t + )(a) = 30 (b) = 45 (c) = 60
3B
We are given a form for the charge on the capacitor as a function oftime, but we need to know the current as a function of time.
)sin()( 000 tQdt
dQtI +==
At t = 0, the current is given by: QI sin)0( 00=1 1
max 0 02 2( )I Q= =
Therefore, the phase angle must be given by:21sin = 30=
LC+ +
- -
I
Q
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19
Checkpoint 31-2
A capacitor in an LC oscillator has a maximum potential difference of17 V and a maximum energy of 160 J. When the capacitor has apotential difference of 5 V and energy of 10 J, what are (a) the emf
across the inductor and (b) the energy stored in the magnetic field?
C
qVC =
( )max2
2EEB UC
QUU ==+
td
idLa L =E)(
LE
CV
)sin(
)cos(
+=
+=
tQi
tQq
VVCL 5==E
qLtQLL22 ))cos(( =+=E
CL
12 =
CVCq ==
( ) JJJUUUb EEB 15010160)( max ===
S l P bl )sin( += tQi
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Sample Problem
A 1.5 F capacitor is charged to 57 V. The charging battery is thendisconnected, and a 12 mHcoil is connected in series with the
capacitor so that LC oscillations occur. (a) Assuming that the circuit
contains no resistance, what is the maximum current in the coil?
)sin(
)sin(
+=
+=
tI
tQi
QIa =)(
CVCQ =
CVC= CL1
=CC V
L
CVC
CL== 1
AVHFV
LCI C 637.0)57(
1012105.1 3
6
===
(b) What is the maximum rate (di/dt)max at which the current i changes?
)cos())sin(( +=+= tItd
tIdtdid
sA
FH
A
CL
II
td
id/4750
)105.1)(1012(
637.01
63max
=
===