Chapter 7
Capacitors and Inductors
7.1 The Capacitors
7.2 The Inductors
7.3 Inductance and Capacitance Combination
7.4 Consequences of Linearity
7.5 Simple Op Amp Circuits with Capacitors
7.6 Duality
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7.1 The Capacitors
•
Ideal Capacitor Model
active element: an element that is capable of furnishing an average power greater
than zero to some external device
ideal source, op amp
passive: an element that cannot supply an average power that is greater than zero over
an infinite time interval
resistor, capacitor, inductor
→
⇒
[unit: C/V or farad(F)]
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: permittivity
2
7.1 The Capacitors
Example 7.1 Determine the current i flowing through the capacitor ( C=2 F )
→
⇒
0
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0
2
capacitor:
open circuit to DC
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7.1 The Capacitors
•
Integral Voltage-Current Relationships
1
⇒ →
∞,
∞
Example 7.2
1
⇒ 1
⇒
0
→ ⇒
Determine the voltage across the capacitor ( C=2 µF )
1
1
4000 , 0
5
1
or
1
10
20
10
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0 8,
2
,
2 ms
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7.1 The Capacitors
•
Energy Storage
Power :
′
′
0
1
2
′
1
2
1
2
Example 7.3 Find maximum energy stored in capacitor.
.
⇒
1
1
20
2
2
0.1 sin 2 10
→ maximuat
1
⇒ 4
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100 sin 2
10
10
sin 2
.
10
sin 2
2.5
100 sin 2
_
100
5
7.1 The Capacitors
1
10
0.0001
100
2
20
2
10
0.004 cos 2
200 cos 2
0.01257 cos 2
Important Characteristics of an Ideal Capacitor
1. There is no current through a capacitor if the voltage across it is not changing with
time. A capacitor is therefore an open circuit to dc.
2. A finite amount of energy can be stored in a capacitor even if the current through
the capacitor is zero, such as when the voltage across it is constant.
3. It is impossible to change the voltage across a capacitor by a finite amount in zero
time, for this requires an infinite current through the capacitor. (A capacitor resists
an abrupt change in the voltage across it in a manner analogous to the way a
spring resists an abrupt change in its displacement.)
4. A capacitor never dissipates energy, but only stores it. Although this is true for the
mathematical model, it is not true for a physical capacitor due to finite resistances
associates with the dielectric as well as packaging.
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7.2 The Inductors
•
Ideal Inductor Model
[unit] henry (H)
Voltage is proportional to the time rate of change of the current producing the
magnetic field
0 → constantcurrent ⇒short circuit to dc
Example 7.4 Determine the inductor voltage. ( L = 3 H )
3
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7.2 The Inductors
Example 7.5 Determine the inductor voltage. ( L = 3 H )
3
Infinite voltage spikes are required to produce the abrupt changes in the inductor
current
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7.2 The Inductors
•
Integral Voltage-Current Relationships
→ ′
1
1
1
′⇒
1
⇒
1
1
→
′
∞,
∞
0
Example 7.6 Determine the inductor current if i (t=-π/2)=1A. ( L = 2 H )
6 cos 5 ⇒
0.6 sin 5
1
2
⇒
2
1 6
sin 5
2 5
6 cos 5 ′
0.6 sin 5
6 cos 5
0.6 sin
1
2
⇒
0.6 sin 5
0.6 sin
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1 ⇒ 1
0.6 sin 5
1.6
Let
0.6 sin 5
5
2
5
2
1 6
sin 5
2 5
1
0.6
1.6 ∴
0.6 sin 5
1.6
9
7.2 The Inductors
•
Energy Storage
Power:
′
0
′
1
2
1
2
′
1
2
⇒ Assume
0
Example 7.7 Find the maximum energy stored in the inductor
1
2
0.1
12 sin
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6
1
2
3
12 sin
at
0:
0
0 at
3:
3
216 sin
at
6:
6
0
⇒ 216 sin
6
216
2
14.4 sin
6
6
maximum
43.2
10
7.2 The Inductors
Important Characteristics of an Ideal Inductor
1. There is no voltage across an inductor if the current through it is not
changing with time. An inductor is therefore a short circuit to dc.
2. A finite amount of energy can be stored in an inductor even if the voltage
across the inductor is zero, such as when the current through it is
constant.
3. It is impossible to change the current through in inductor by a finite
amount in zero time, for this requires an infinite voltage across the
inductor. (An inductor resists an abrupt change in the current across it in a
manner analogous to the way a mass resists an abrupt change in its
velocity.)
4. The inductor never dissipates energy, but only stores it. Although this is
true for the mathematical model, it is not true for a physical inductor
due to series resistances.
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7.3 Inductance and Capacitance Combination
•
Inductors in Series
⋯
•
⋯
⋯
Inductors in Parallel
1
1
1
⇒
1
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1
1
1
⋯
1
1
12
7.3 Inductance and Capacitance Combination
•
Capacitors in Series
1
1
⇒
1
1
•
1
1
1
⋯
1
1
Capacitors in Parallel
⋯
⋯
⋯
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7.3 Inductance and Capacitance Combination
Example 7.8 Simplify the network using series-parallel combinations.
1
1
6
1
1
2
1
3
0.8
1
5
6
0.8
2
1
2.0
1
3
1
3
6
6
3
2.0
3.0
No
series
or
parallel
combinations of either the
inductors or the capacitors.
 can’t be simplified.
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7.4 Consequences of Linearity
Example 7.9 Write appropriate nodal equations for the circuit.
⇒
1
⇒
′
0
1
′
1
1. Inductors and capacitors are linear elements.
2. The principle of proportionality between source and response can be extended to
the general RLC circuit, and it follows that the principle of superposition also
applies.
3. Initial inductor current and capacitor voltages must be treated as independent
sources in applying the superposition principle.
4. Thevénin and Norton’s theorem can be applied to linear RLC circuits, if we wish.
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7.5 Simple Op Amp Circuits with Capacitors
•
Integrator
0,
→
⇒
•
0,
1
0
Differentiator
→
⇒
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7.6 Duality
•
Duality
Two circuits are “dual” if the mesh equations that characterize one of them have the
same mathematical form as the nodal equations that characterize the other.
3
0
10
4
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2 cos 6 ,
4
4
1
8
4
4
1
8
3
0
4
4
4
0
5
5
0
10
2 cos 6 ,
4
4
1
8
4
4
1
8
0
5
5
0
10
17
7.6 Duality
Voltage
Serial
Resistor
Capacitor
↔
↔
↔
↔
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Current
Parallel
1/Resistor
Inductor
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7.6 Duality
Practice 7.11
6
8e 10 t 

 0.2 
10
  10ia  10ib  0
ia
ib
d
dt
1
10ia  10ib   ib dt  0
C
6
  i  80e 10 t
ia  a  8e
106 t
ib  b  a  0.1 i
a
b
6
8e 10 t  0.1i  0.2 
di
dt
i  10a  10b  0
10a  10b 
6
 8e 10 t  0.1 80e 10
 16e
6
t
106 t
1
b dt  0

L
Homework : 7장 Exercises 5의 배수 문제 (64번 문제까지)
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이 자료에 나오는 모든 그림은 McGraw·hill 출판사로부터 제공받은 그림을 사
용하였음.
All figures at this slide file are provided from The McGraw-hill company.
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