Ch.6 Capacitors & Inductors

Ch.6 Capacitors & Inductors

Capacitor

3

• Capacitor (캐패시터 or 콘덴서)

- Capacitor 내부의 전기장에 전기에너지를 저장할 수 있

도록 고안된 전기소자

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AC

d

메탈필름 세라믹 전해

바리콘

Capacitor

4

• Capacitance : 𝑞 = 𝐶𝑣

- 정전용량의 단위 : Farad = Coulomb/Volt

- Capacitor의 전압-전류 관계

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1 1( ) ( ) ( )

t t

t

dq dv dvq Cv C i C

dt dt dt

v i t dt i t dt v tC C

Capacitor

5

• Instantaneous power to the capacitor

• Energy stored in the capacitor

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dvp vi Cv

dt

( )( )

2

( )( )

22

1( ) ( ) ( )

2

1 1( ( ) 0) or ( )

2 2

v tt t v t

vv

dvw p t dt C v t dt C v t dv Cv

dt

q qw Cv v v

C C

Capacitor

6

• Capacitor의 특징

- A capacitor acts as an open circuit to DC (𝑖 = 𝐶𝑑𝑣

𝑑𝑡)

- The voltage on a capacitor cannot change abruptly

- The ideal capacitor does not dissipate energy

- A non-ideal capacitor can be modeled as the parallel

circuit with ideal capacitor and leakage resistance

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Capacitor

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Example 6.1

Example 6.2

Example 6.3

Capacitor

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Example 6.4

Determine the current through a 200uF capacitor whose voltage is shown in the Fig.

Capacitor

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Example 6.5

Obtain the energy stored in each capacitor in the figure under DC conditions

Series & Parallel Capacitors

10

• Capacitor의 병렬연결

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1 2

1 2

1

1 2

(by KCL)N

N

N k

k

eq N

i i i i

dv dv dv dvC C C C

dt dt dt dt

C C C C

Series & Parallel Capacitors

11

• Capacitor의 직렬연결

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1 2

0 0 01 2

01 2

1 2

(by KVL)

1 1 1( ) ( ) ( )

1 1 1( )

1 1 1 1

N

t t t

t t tN

t

tN

eq N

v v v v

i t dt i t dt i t dtC C C

i t dtC C C

C C C C

Series & Parallel Capacitors

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Example 6.6

Find the equivalent capacitance between terminals a and b of the following circuit :

Series & Parallel Capacitors

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Example 6.7

Find the voltage across each capacitor for the following circuit :

Inductor

14

• Inductor (인덕터 or 코일)

- Inductor 내부의 자기장에 전기에너지를 저장할 수 있

도록 고안된 전기소자

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2N AL

솔레노이드 코일

칩인덕터

초크 코일

Inductor

15

• Inductance : 𝑣 = 𝐿𝑑𝑖

𝑑𝑡

- Inductance의 단위 : [H], Henry

- Inductance의 전압-전류 관계

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1 1( ) ( ) ( )

t t

ti v t dt v t dt i t

L L

Inductor

16

• Instantaneous power to the inductor

• Energy stored in the inductor

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dip vi L i

dt

( )

2

( )

2

1( ) ( ) ( )

2

1( ( ) 0)

2

i tt t t

i

diw p t dt L i t dt L i t di Li

dt

w Li i

Inductor

17

• Inductor의 특징

- A inductor acts as a short circuit to DC (𝑣 = 𝐿𝑑𝑖

𝑑𝑡= 0)

- The current though an inductor can’t change abruptly

- The ideal inductor does not dissipate energy

- A non-ideal inductor can be modeled as the series

circuit with ideal inductor and winding resistance

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Inductor

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Example 6.8

Example 6.9

Inductor

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Example 6.10

Consider the following circuit. Under DC conditions, find: (a) 𝑖, 𝑣𝐶and 𝑖𝐿, (b) the energy stored in the capacitor and inductor

Series & Parallel Inductors

20

• Inductor의 병렬연결

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1 2

1 0 2 0 00 0 0

1 2

1 0 2 0 00

1 2

1 2

(by KCL)

1 1 1( ) ( ) ( )

1 1 1( ) ( ) ( )

1 1 1 1

N

t t t

Nt t t

N

t

Nt

N

eq N

i i i i

vdt i t vdt i t vdt i tL L L

vdt i t i t i tL L L

L L L L

Series & Parallel Inductors

21

• Inductor의 직렬연결

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1 2

1 2

1 2

1 2

(by KVL)N

N

N

eq N

v v v v

di di diL L L

dt dt dt

diL L L

dt

L L L L

Series & Parallel Inductors

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Example 6.11

Find the equivalent inductance of the circuit shown in the following figure

Series & Parallel Inductors

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Example 6.12

For the following circuit, ሻ𝑖(𝑡ሻ = 4(2 − 𝑒−10𝑡 mA. If 𝑖2(0ሻ = −1mA, find (a) ሻ𝑖1(0 ; (b) ሻ𝑣(𝑡ሻ, 𝑣1(𝑡 , and ሻ𝑣2(𝑡 ; (c) ሻ𝑖1(𝑡 and ሻ𝑖2(𝑡

Applications - Integrator

24

• Inverting invertor에 capacitor를 활용하여 적분기 구성

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0

(0 ),

1( )

i o oR C

ti o

o i

v d v dvi i C C

R dt dt

v dvC v v t dt

R dt RC

Applications - Integrator

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Example 6.13

If 𝑣1 = 10cos2𝑡 [mV] and 𝑣2 = 0.5𝑡 [mV], find 𝑣𝑜 in the op-amp circuit in the following figure. Assume that the voltage across the capacitor is initially zero.

Applications - Differentiator

26

• Inverting invertor에 capacitor를 활용하여 미분기 구성

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,i o iC R o

dv v dvi C i v RC

dt R dt

Applications - Integrator

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Example 6.14

Sketch the output voltage for the following circuit. Assume that 𝑣𝑜 = 0 at 𝑡 = 0.

iv

Applications – Analog Computer

28

• Analog computer는 미분방정식을 풀기 위한 목적으로 개

발되었으며, op-amp를 이용하여 구현

•𝑑𝑥

𝑑𝑡는

𝑑2𝑥

𝑑𝑡2를 적분하여, 𝑥는

𝑑𝑥

𝑑𝑡를 적분하여 계산, ሻ𝑓(𝑡 는 외부

의 signal generator를 이용하여 입력

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2

2

2

2

( ), 0

( )

d x dxa b cx f t t

dt dt

d x f t b dx cx

dt a a dt a

Applications – Analog Computer

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Example 6.15

Design an analog computer circuit to solve the differential equation :

Subject to 𝑣𝑜(0ሻ = −4, 𝑑𝑣𝑜

𝑑𝑡|𝑡=0 = 1.

2

22 10sin 4 , 0o o

o

d v dvv t t

dt dt

Applications – Analog Computer

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Example 6.15

Assignment

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Chapter 6

6.13, 6.28, 6.46, 6.57, 6.78

Due date :