chapter6c2
TRANSCRIPT
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Chapter 6. Capacitance andinductance
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Contents
1. Capacitors
2. Inductors
3. Capacitor and inductor combinations
4. RC operational amplifier circuits
5. Application examples
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Power supply board
PC motherboard
inductor capacitor
inductor
capacitor
Cell phone
Usage of inductors and capacitors
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1. Capacitors
Capacitance is defined to be the ratio of charge to voltage difference.
Used to store charges
Used to store electrostatic energy
0V
0Q
q
SV SV
SVSVVE
If the voltage difference between the
terminals of the capacitor is equal to
the supply voltage, net flow of
charges becomes zero.
V
QC
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Electrons(-) are absorbed.(+) charges are generated
Electrons(-) are generated.(+) charges are absorbed.
Generation of charges : battery
e2ZnZn 2
234 HNH222NH
e
Electrons are generated via
electro-chemical reaction.
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Used to store charges
Used to store electrostatic energy
Slow down voltage variation
Usage of capacitors
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Type of capacitors
t
0
)(1
)( dttiC
t
http://www.google.co.kr/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=6ZZX4Racl1o-sM&tbnid=ZOMpPYoONnjwqM:&ved=0CAUQjRw&url=http://variable-capacitors.blogspot.com/&ei=X9HvUtbPCOmSiQem_IGQBQ&psig=AFQjCNF7BjRZ_I3oSPZ6jAia_i8E2D2F_g&ust=1391534331654366 -
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i
t
t0
0
t
diC
)(1
irelation of capacitors
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)(ti
Example 6.2
The voltage across a 5-F capacitor has the waveform shown in Fig. 6.4a. Determine
the current waveform.
dt
dCi
mst
mstt
mstt
t
80
8696102
24
60106
24
)( 3
3
mst
mstmA
mstmA
ti
80
8660
6020
)(
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Capacitor voltage cannot change instantaneouslydue to finite current supply.
Properties of capacitors
SV
i
t
dt
dCi
SR
CC
In steady state, capacitor behaves as if open circuited.
SV
iSR
0
dt
dCi DC
)()( 00 tt
0t
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Example 6.3
Determine the energy stored in the electric field of the capacitor in Example 6.2 at
t=6 ms.
][1440)]([2
1)(
2JtCtW
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The current in an initially uncharged 4F capacitor is shown in Fig. 6.5a. Let us
derive the waveforms for the voltage, power, and energy and compute the energy
stored in the electric field of the capacitor at t=2 ms.
Example 6.4
mst
mst
mstt
ti
40
428
20102
16
)(
3
mst
msttdx
msttxdx
tt
t
40
42824)8(4
1
2010001084
1
)(0
0
23
mst
mstt
mstt
ttitp
40
426416
208
)()()(
3
mst msttt
mstt
dxxptW
t
40421012810648
202
)()(
1292
4
0
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2. Inductors
dt
diLtL )(
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Two important laws on magnetic field
Current generates magnetic field(Biot-Savart Law)
inducedV
Time-varying magnetic field generates
induced electric field that opposes the
variation. (Faradayslaw)
Current
Current
B-field
B-field
V
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Biot-Savart Law Faradays Law
rrRRr
B
,4
2C R
Id
LidS
aB
dt
dV
ind
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Self induced voltage
dt
diL
dt
dV
ind
The induced voltage is generated such that it opposes the applied magnetic flux.
The inductor cannot distinguish where the applied magnetic flux comes from.
If the magnetic flux is due to the coil itself, it is called that the induced voltage
is generated by self-inductance.
=
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Frequently used formulas on inductors
dt
diL
t
t
t
t
tt
dL
ti
dL
dL
dL
i
0
0
0
)(1
)(
)(1
)(1
)(1
0
2
2
1)(
)()()()( Li
dt
dti
dt
tdiLtittp
22)]([
2
1
2
1)()()( tiLdLi
d
dditW
tt
i
NL
Energy :
Power :
Voltage :
Inductance :
Current :
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Properties of inductors
SV
i
t
i
dt
diL
SR
LL
In steady state, inductor behaves as if short circuited.
SV
iSR
0dt
diL DC
Inductor current cannot change instantaneouslydue to finite current supply.
0t
)()( 00 titi
20
6
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Find the total energy stored in the circuit of Fig. 6.8a.
Example 6.5
0815254273
09
36
9
111
11
CCC
CC
VVV
VV
][8.109
62.16],[2.16
21VVVV CC
][8.19
],[2.16
91
2
1
1A
VIA
VI
C
L
C
L
][44.1)2.1)(102(2
1 231 mJWL
][48.6)8.1)(104(2
1 232
mJWL
][62.2)2.16)(1020(2
1 261
mJWC
][92.2)8.10)(1050(2
1 262 mJWC
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E l 6 7
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Example 6.7
The current in a 2-mH inductor is
][)377sin(2)( Atti
Determine the voltage across the inductor and the energy stored in the inductor.
][)377cos(508.1)]377sin(2[
)102()( 3
Vtdt
td
dt
diLtL
][)377(sin004.0)]377sin(2)[102(2
1)]([
2
1)( 2232 JtttiLtWL
23
E l 6 8
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Example 6.8
The voltage across a 200-mH inductor is given by the expression
00
0][)31()(
3
t
tmVett
t
Let us derive the waveforms for the current, energy, and power.
00
0][5)31(200
10 30
33
t
tmAtedxexi
tt
x
00
0][)31(5)()()(
6
t
tWetttittp
t
000][5.2)]([
21)(
62
2
ttJettiLtW
t
24
C it d i d t ifi ti
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Capacitor and inductor specifications
Standard tolerance
values are ; 5%, ;
10%, and ; 20%.
Tolerances are
typically 5% or
10% of the
specified value.
25
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Example 6.10
The capacitor in Fig. 6.11a is a 100-nF capacitor with a tolerance of 20%. If the
voltage waveform is as shown in Fig. 6.11b, let us graph the current waveform for the
minimum and maximum capacitor values.
dt
dCi
26
6 3 C it d I d t C bi ti
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6.3 Capacitor and Inductor Combinations
=
N
N
i iS CCCCC
11111
211
N
N
i
iP CCCCCC
321
1
=
27
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=
=
N
N
i iP LLLLLL
111111
3211
N
N
i
iS LLLLLL
321
1
28
6 4 RC O ti l A lifi Ci it
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6.4RC Operational Amplifier Circuits
Op-amp differentiator
Ci
iRdt
dC o
2
11 )(
0,0 i
dt
tdCRo
)(112
29
O i t t
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Op-amp integrator
i
dt
dC
R o )(2
1
1
0,0 idt
tdC
R
o )(2
1
1
0)0(
)(1
)0()(1
)(1
0 1
210
1
21
1
21
o
t
o
tt
o dxxCR
dxxCR
dxxCR
30
Example 6 17
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Example 6.17
The waveform in Fig. 6.26a is applied at the input of the differentiator circuit shown
in Fig. 6.25a. If R2=1 kand C1=2 F, determine the waveform at the output of the
op-amp.
mstV
mstV
dt
td
dt
tdCRo
105][4
50][4)(10)2(
)( 13112
31
Example 6 18
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Example 6.18
If the integrator shown in Fig. 6.25b has the parameters R1=5 k and C2=0.2F,
determine the waveform at the op-amp output if the input waveform is given as in Fig.
6.27a and the capacitor is initially discharged.
][1.020
][1.002010)20(10)(
1 33
0 1
21 stt
stttdxx
CR
t
o