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8/9/2019 Presentation W12D2 Answers
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W12D2RC, LR, and
Undriven RLC Circuits;Experiment 4
Today’s Reading Course Notes: Sections 11.7-11.9, 11.10,
11.13.6; !"t. #: $ndri%en R&C Circuits
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Announcements
'at( Re%ie) *ee+ 13 Tuesday 9"-11 " in 6-1
/S 9 due *ee+ 13 Tuesday at 9 " in o!es outside 3-0 or 6-1
Ne!t Reading 2ssignent *13 Course Notes: Sections 11.-9, 11.1-11.13
2
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Outline
!"erient #: /art 1 RC and &R Circuits
Si"4e 5aronic sci44ator
$ndri%en R&C Circuits
!"erient #: /art $ndri%en R&C Circuits
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RC Circuit Car!in!
So4ution to t(is euation )(en
s)itc( is c4osed at t 8 0: dQ
dt
= −1
RC
Q− Cε ( )
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RC Circuit" Discar!in!
So4ution to t(is euation )(en
s)itc( is c4osed at t 8 0
tie constant:
dQ
dt
= −1
RC
Q
Q(t ) =Qoe−t /RC
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RL Circuit" #ncreasin! Current
So4ution to t(is euation )(ens)itc( is c4osed at t 8 0:
=
t /(
units: seconds
I (t ) = ε
R
(1− e−t /τ )
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RL Circuit" Decreasin! Current
So4ution to t(is euation )(ens)itc( is o"ened at t 8 0:
0−
units: seconds
I (t ) = ε
R
(1− e−t /τ )
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$easurin! %ime Constant
y
2(t
2) = y
2(t
1 + τ ) = y
0e
−(t 1 +τ ) /τ = y
0e
−(t 1
/τ )e−1 = y
1(t
1)e−1
/ic+ a "oint 1 )it(
ind "oint suc( t(at
n t(e 4a you )i44 "4ot sei-4og and =it cur%e a+e sureyou e!c4ude data at ot( ends
y 1(t 1) = y 0e− (t
1/τ )
τ
≡t
2 −t 1
y
2(t
2) = y
1(t
1)e−1
ln( y (t ) / y 0) = lne− (t /τ ) = −(t / τ ) ⇒
ln y (t ) = ln y 0 − (t / τ ) ⇒ τ = −1/slope y (t ) = y
0e− (t /τ ) ⇒
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Experiment 4"
RC and RL Circuits
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$ass on a &prin!"
&imple 'armonic $otion
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Demonstration
$ass on a &prin!"
&imple 'armonic $otion$ass on a &prin! (C 2)
(tt":??scri"ts.it.edu?@tsg?)))?deo."("A4etnu8CB0s(o)80
http://scripts.mit.edu/~tsg/www/demo.php?letnum=C%202&show=0http://scripts.mit.edu/~tsg/www/demo.php?letnum=C%202&show=0
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$ass on a &prin!
F = −kx = ma= md2 x
dt 2
md2 x
dt 2 + kx = 0
x (t ) = x
0
cos(ω 0
t + φ )
1
3 #
*(at is 'otionA
ω 0 =
k
m= Angular freuenc!
Si"4e 5aronic 'otion
x 0: 2"4itude o= 'otion
φ : /(ase tie o==set
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&imple 'armonic $otion
2"4itude !0
x (t ) = x
0cos(ω
0t + φ )
"erio# =
1
freuenc! → T =
1
f
"erio# =2π
angular freuenc!→ T =
2π
ω
"$ase %$ift (φ ) = −π
2
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Concept *uestion" &imple
'armonic Oscillator
*(ic( o= t(e =o44o)ing =unctions x t (as a second
deri%ati%e )(ic( is "ro"ortiona4 to t(e negati%e o= t(e
=unction
x (t ) = Acos2π
T t
÷
x (t ) = Ae−t /T
x (t ) = Aet /T
x (t ) =1
2at 2
2
2&d x x
dt µ −
1.
.
3.
#.
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Concept *uestion Ans+er" &imple
'armonic Oscillator
Ans+er 4
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$ass on a &prin!" Ener!-
x (t ) = x 0 cos(ω 0t + φ )
1 S"ring 'ass 3 S"ring # 'ass
nergy (as "arts: 'ass Dinetic and S"ring /otentia4
K = 12m dx
dt
÷
2
= 12kx
0
2 sin2 (ω 0t + φ )
Us =
1
2
kx 2 =1
2
kx 0
2cos2 (ω 0t + φ )
dx
dt = v x (t ) = −ω 0 x 0 sin(ω 0t + φ )
nergy
s4os(es ac+
and =ort(
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LC Circuit
1. Set u" t(e circuit ao%e)it( ca"acitor, inductor,
resistor, and attery.
. &et t(e ca"acitor ecoe
=u44y c(arged.
1. T(ro) t(e s)itc( =ro a to b.
1. *(at (a""ensA
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LC Circuit
>t undergoes si"4e (aronic otion, Eust 4i+e aass on a s"ring, )it( trade-o== et)een c(arge on
ca"acitor S"ring and current in inductor 'ass.
ui%a4ent4y: trade-o== et)een energy stored in
e4ectric =ie4d and energy stored in agnetic =ie4d.
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Concept * Ans+er" LC Circuit
2ns)er: #. T(e current is
a!iu )(en t(e c(arge on
t(e ca"acitor is Fero
Current and c(arge are e!act4y 90 degrees out o=
"(ase in an idea4 &C circuit no resistance, so )(en
t(e current is a!iu t(e c(arge ust e
identica44y Fero.
LC Ci it &i l ' i
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LC Circuit" &imple 'armonic
Oscillator
d2Q
dt 2 +
1
LCQ = 0 ⇒
Q(t ) = Q0cos(ω
0t + φ )
ω
0 = 1 / LC
C(arge:
2ngu4ar =reuency:
2"4itude o= c(arge osci44ation:
/(ase tie o==set:
Q
C − LdI
dt = 0 ' I = −dQ
dt
φ
Si"4e (aronic osci44ator:
Q
0
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LC Oscillations" Ener!-
U =U
E +U
B =
Q2
2C +
1
2LI 2 =
Q0
2
2C
UE =
Q2
2C
=Q
0
2
2C
÷cos2ω
0t
UB =
1
2
LI 2 =1
2
LI0
2 sin2ω 0t =
Q0
2
2C
÷sin2ω
0t
Tota4 energy is conser%ed GG
Notice re4ati%e "(ases
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LC CircuitOscillation
&ummar-
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Addin! Dampin!"
RLC Circuits
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Demonstration
Undriven RLC Circuits (. 1/0)
http://scripts.mit.edu/~tsg/www/demo.php?letnum=Y%20190&show=0http://scripts.mit.edu/~tsg/www/demo.php?letnum=Y%20190&show=0
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RLC Circuit" Ener!- Can!es
>nc4ude =inite resistance:
'u4ti"4y y
I =
dQ
dt ⇒
ecrease in stored
energy is eua4 to Hou4e
(eating in resistor
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Damped LC Oscillations
Resistor dissi"ates energy
and syste rings do)n o%er
tie. 24so, =reuencydecreases:
Q(t ) =Q
0e−(R/ 2L)t cos(ω t )
ω = ω 02
− (R / 2L)2
ω
0 = 1 / LC > R / 2L
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Experiment 4" art 2
Undriven RLC Circuits
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Appendix" Experiment 4"
art 2Undriven RLC Circuits
roup ro3lem
and Concept *uestions
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ro3lem" LC Circuit
Consider t(e circuit s(o)n in t(e =igure. Su""ose t(e s)itc(
t(at (as een connected to "oint a =or a 4ong tie is sudden4y
t(ro)n to b at t 8 0. ind t(e =o44o)ing uantities:
a t(e =reuency o= osci44ation o= t(e circuit.
t(e a!iu c(arge t(at a""ears on t(e ca"acitor.
c t(e a!iu current in t(e inductor.
d t(e tota4 energy t(e circuit "ossesses as a =unction o= tie t .
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1. 1
.
3. 3
#. #
Concept *uestion" Expt 4>n today’s 4a t(e attery turns on
and o==. *(ic( circuit diagra isost re"resentati%e o= our circuitA
1. .
3. #.
&oad 4a )(i4e )aitingI
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Concept *uestion Ans+er" Expt 4
2ns)er: 1.
T(ere is resistance in t(e circuitin our non-idea4 inductor.
T(e attery s)itc(ing o== doesn’trea+ t(e circuit ut a44o)s it to
ring do)n
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Concept *uestion" LC Circuit
T(e "4ot s(o)s t(e c(argeon a ca"acitor 4ac+ cur%e
and t(e current t(roug( it
red cur%e a=ter you turn
o== t(e "o)er su""4y. >= you"ut a core into t(e inductor
)(at )i44 (a""en to t(e
tie T &ag
A 0 #0 0 10-1.0J
0
-0.J0
0.0J0
0.J0
1.0J0
-1.0>0
-0.>0
0.0>0
0.>0
1.0>0
T4ag
C ( a r g e o n C a " a c i t o r
Tie S
C(arge
C u r r e n t t ( r o u g ( C a " a c i t o r
Current
1. >t )i44 increase. >t )i44 decrease3. >t )i44 stay t(e sae
#. > don’t +no) 34
Concept *uestion Ans+er" LC
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Concept *uestion Ans+er" LC
Circuit
2ns)er 1.
T &ag )i44 increase.
utting in a core increases
(e inductor’s inductance and
ence decreases t(e natura4reuency o= t(e circuit.
o)er =reuency eans
onger "eriod. T(e "(ase )i44
eain at 90K a uartereriod so T
&ag )i44 increase.
0 #0 0 10
-1.0J0
-0.J0
0.0J0
0.J0
1.0J0
-1.0>0
-0.>0
0.0>0
0.>0
1.0>0
T4ag
C ( a r g e o n C a " a c i t o r
Tie S
C(arge
C u r r e n t t ( r o u g ( C
a " a c i t o r
Current
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Concept *uestion" LC Circuit
>= you increase t(eresistance in t(e circuit
)(at )i44 (a""en to rate
o= decay o= t(e "ictured
a"4itudesA0 #0 0 10
-1.0J0
-0.J0
0.0J0
0.J0
1.0J0
-1.0>0
-0.>0
0.0>0
0.>0
1.0>0
T4ag
C
( a r g e o n C a " a c i t o r
Tie S
C(arge
C u r
r e n t t ( r o u g ( C a " a c i t o
r
Current
1. >t )i44 increase decay ore ra"id4y
. >t )i44 decrease decay 4ess ra"id4y3. >t )i44 stay t(e sae#. > don’t +no)
Concept *uestion Ans+er" LC
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Concept *uestion Ans+er" LC
Circuit2ns)er: 1. >t )i44 increase
decay ore ra"id4y
Resistance is )(at dissi"ates "o)er in t(e circuit
and causes t(e a"4itude o= osci44ations todecrease. >ncreasing t(e resistance a+es t(e
energy and (ence a"4itude decay ore ra"id4y.
0 #0 0 10
-1.0J0
-0.J0
0.0J0
0.J0
1.0J0
-1.0>0
-0.>0
0.0>0
0.>0
1.0>0
T4ag
C ( a
r g e o n C a " a c i t o r
Tie S
C(arge
C u r r e n
t t ( r o u g ( C a " a c i t o r
Current