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회로이론-І 2013년2학기 이문석 1

Chapter 7Capacitors 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


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 deviceideal source, op amp

passive: an element that cannot supply an average power that is greater than zero over an infinite time intervalresistor, capacitor, inductor

→

⇒ [unit: C/V or farad(F)] : permittivity


7.1 The Capacitors

Example 7.1 Determine the current i flowing through the capacitor ( C=2 F )

→ 0⇒ 0

2 capacitor: open circuit to DC


7.1 The Capacitors

• Integral Voltage-Current Relationships

⇒ 1 ⇒

1or

1

→ ∞, ∞ 0 ⇒ 1 →

⇒

Example 7.2 Determine the voltage across the capacitor ( C=2 µF )

1 1

15 10 20 10 0

4000 , 0 2 ,

8, 2 ms


7.1 The Capacitors

′ ′ ′12

12

012

• Energy Storage

Power :

Example 7.3 Find maximum energy stored in capacitor.

100 sin 210 10 sin 2

⇒.

10 sin 2 .

2.5

12

12 20 10 100 sin 2

0.1 sin 2

→ maximuat14 ⇒ _ 100


7.1 The Capacitors

1 10 100 2 0.0001 2

20 10 200 cos 2 0.004 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 withtime. 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 throughthe 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 zerotime, for this requires an infinite current through the capacitor. (A capacitor resistsan abrupt change in the voltage across it in a manner analogous to the way aspring resists an abrupt change in its displacement.)

4. A capacitor never dissipates energy, but only stores it. Although this is true for themathematical model, it is not true for a physical capacitor due to finite resistancesassociates with the dielectric as well as packaging.


7.2 The Inductors

• Ideal Inductor Model

Voltage is proportional to the time rate of change of the current producing the magnetic field

[unit] henry (H)

0 → constantcurrent ⇒short circuit to dc

Example 7.4 Determine the inductor voltage. ( L = 3 H )

3


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


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 ⇒ 12 6 cos 5 ′

12

65 sin 5

12

65 sin 5

⇒ 0.6 sin 5 0.6 sin 5 0.6 sin 5 0.6 sin52 1 0.6 sin 5 1.6

12 6 cos 5 0.6 sin 5

⇒ 2 0.6 sin52 1 ⇒ 1 0.6 1.6 ∴ 0.6 sin 5 1.6

Let


7.2 The Inductors

• Energy Storage

Power: ′ ′ ′12

12

0

⇒12

Assume 0

Example 7.7 Find the maximum energy stored in the inductor

12

12 3 12 sin 6 216 sin 6

at 0: 0 0 at 3: 3 216 sin 2 216at 6: 6 0

maximum

0.1 12 sin 6 ⇒ 14.4 sin 6 43.2


7.2 The Inductors

Important Characteristics of an Ideal Inductor

1. There is no voltage across an inductor if the current through it is notchanging 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 voltageacross the inductor is zero, such as when the current through it isconstant.

3. It is impossible to change the current through in inductor by a finiteamount in zero time, for this requires an infinite voltage across theinductor. (An inductor resists an abrupt change in the current across it in amanner analogous to the way a mass resists an abrupt change in itsvelocity.)

4. The inductor never dissipates energy, but only stores it. Although this istrue for the mathematical model, it is not true for a physical inductordue to series resistances.



• Inductors in Series

⋯ ⋯ ⋯

• Inductors in Parallel

1 1 1

1 1

⇒1

1 1 ⋯ 1



• Capacitors in Series

• Capacitors in Parallel

1 1 1

1 1

⇒1

1 1 ⋯ 1

⋯

⋯

⋯



Example 7.8 Simplify the network using series-parallel combinations.

116

13

136

63 2.0

1 2.0 3.0112

13

0.8156

0.8 2

No series or parallelcombinations of either theinductors or the capacitors. can’t be simplified.


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 tothe general RLC circuit, and it follows that the principle of superposition alsoapplies.

3. Initial inductor current and capacitor voltages must be treated as independentsources in applying the superposition principle.

4. Thevénin and Norton’s theorem can be applied to linear RLC circuits, if we wish.


• Integrator

→

⇒1

0

0,

• Differentiator

→

⇒

0,

7.5 Simple Op Amp Circuits with Capacitors


7.6 Duality

• DualityTwo circuits are “dual” if the mesh equations that characterize one of them have thesame mathematical form as the nodal equations that characterize the other.

3 4 4 2 cos 6 ,

4 418 0 5 0

4 418 5 10

3 4 4 2 cos 6 ,

4 418 0 5 0

4 418 5 10

0 10

0 10


7.6 Duality

Voltage ↔ CurrentSerial ↔ ParallelResistor ↔ 1/ResistorCapacitor ↔ Inductor


7.6 Duality

Practice 7.11

Homework : 7장 Exercises 5의 배수 문제 (64번 문제까지)

6108 0.210

10 10 0110 10 0

t

a b

a b b

dedt

i i

i i i dtC

6108 0.1 0.2

10 10 0110 10 0

t

a b

a b b

die idt

i

dtL

ai bi

ba

6

6

6 6

6

10

10

10 10

10

80

80.1

8 0.1 80

16

t

ta a

b b a

t t

t

i e

i ei i

e e

e

회로이론-І 2013년2학기 이문석

이 자료에 나오는 모든 그림은 McGraw·hill 출판사로부터 제공받은 그림을 사용하였음.

All figures at this slide file are provided from The McGraw-hill company.

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