ch.6 capacitors & inductors

Ch.6 Capacitors & Inductors

Capacitor

3

• Capacitor (캐패시터 or 콘덴서)

- Capacitor 내부의 전기장에 전기에너지를 저장할 수 있

도록 고안된 전기소자

Pusan National University

AC

d

메탈필름 세라믹 전해

바리콘

Capacitor

4

• Capacitance : 𝑞 = 𝐶𝑣

- 정전용량의 단위 : Farad = Coulomb/Volt

- Capacitor의 전압-전류 관계


00

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


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


Capacitor

7Pusan National University

Example 6.1

Example 6.2

Example 6.3

Capacitor


Example 6.4

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

Capacitor


Example 6.5

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

Series & Parallel Capacitors

10

• Capacitor의 병렬연결


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


11

• Capacitor의 직렬연결


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



Example 6.6

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



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


2N AL

솔레노이드 코일

칩인덕터

초크 코일

Inductor

15

• Inductance : 𝑣 = 𝐿𝑑𝑖

𝑑𝑡

- Inductance의 단위 : [H], Henry

- Inductance의 전압-전류 관계


00

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


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


Inductor


Example 6.8

Example 6.9

Inductor


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


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


21

• Inductor의 직렬연결


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



Example 6.11

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



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


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



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


,i o iC R o

dv v dvi C i v RC

dt R dt



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


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



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



Example 6.15

Assignment


Chapter 6

6.13, 6.28, 6.46, 6.57, 6.78

Due date :

ch.6 capacitors & inductors

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