Ch.6 Capacitors & Inductors

Ch.6 Capacitors & Inductors

Capacitor

• Capacitor (캐패시터 or 콘덴서)

- Capacitor 내부의 전기장에 전기에너지를 저장할 수 있 도록 고안된 전기소자

메탈필름 세라믹 전해

Capacitor

• Capacitance : 𝑞 = 𝐶𝑣

- 정전용량의 단위 : Farad = Coulomb/Volt - Capacitor의 전압-전류 관계

0 0

1 1

( ) ( ) ( )

t t

t

dq dv dv

q Cv C i C

dt dt dt

v i t dt i t dt v t

C ^ C

    







Capacitor

• Instantaneous power to the capacitor

• Energy stored in the capacitor

p vi Cv dv

  dt

( ) ( )

2 ( )

( ) 2

2

( ) ( ) ( ) 1

2

1 1

( ( ) 0) or ( )

2 2

t t v t v t

v v

w p t dt C v t dv dt C v t dv Cv dt

q q

w Cv v v

C C

  



   

    

  

Capacitor

• Capacitor의 특징

- A capacitor acts as an open circuit to DC (𝑖 = 𝐶 ^𝑑𝑣

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

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

• 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 dv

C C C C

dt dt dt dt

C C C C



   

 

      

 

    



Series & Parallel Capacitors

• Capacitor의 직렬연결

1 2

0 0 0

1 2

1 2 0

1 2

(by KVL)

1 1 1

( ) ( ) ( )

1 1 1

( )

1 1 1 1

N

t t t

t t t

N t

N t

eq N

v v v v

i t dt i t dt i t dt

C C C

i t dt

C C C

C C C C

   

   

 

     

 

    

  



Series & Parallel Capacitors

Example 6.6

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

Series & Parallel Capacitors

Example 6.7

Find the voltage across each capacitor for the following circuit :

Inductor

• Inductor (인덕터 or 코일)

- Inductor 내부의 자기장에 전기에너지를 저장할 수 있 도록 고안된 전기소자

N2A

솔레노이드 코일 초크 코일

Inductor

• Inductance : 𝑣 = 𝐿 ^𝑑𝑖

𝑑𝑡

- Inductance의 단위 : [H], Henry - Inductance의 전압-전류 관계

0 0

1 1

( ) ( ) ( )

t t

i v t dt t v t dt i t

L ^ L







Inductor

• Instantaneous power to the inductor

• Energy stored in the inductor

p vi L di i dt

 

   

( ) 2

( ) 2

( ) ( ) ( ) 1

2 1 ( ( ) 0)

2

t t t i t

i

w p t dt L di i t dt L i t di Li dt

w Li i

  



   

   

  

Inductor

• Inductor의 특징

- A inductor acts as a short circuit to DC (𝑣 = 𝐿 ^𝑑𝑖

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, 𝑣_𝐶 and 𝑖_𝐿, (b) the energy stored in the capacitor and inductor

Series & Parallel Inductors

• Inductor의 병렬연결

 

1 2

1 0 2 0 0

0 0 0

1 2

1 0 2 0 0

1 2 0

(by KCL)

1 1 1

( ) ( ) ( )

1 1 1

( ) ( ) ( )

1 1 1 1

N

t t t

t t t N

N t

t N N

i i i i

vdt i t vdt i t vdt i t

L L L

vdt i t i t i t

L L L

   

      

 

         

 

    

  



Series & Parallel Inductors

• Inductor의 직렬연결

 

1 2

1 2

1 2

1 2

(by KVL)

N

N

N

eq N

v v v v

di di di

L L L

dt dt dt

L L L di

dt

L L L L

   

   

   

    

Series & Parallel Inductors

Example 6.11

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

Series & Parallel Inductors

Example 6.12

For the following circuit, 𝑖(𝑡ሻ = 4(2 − 𝑒^−10𝑡ሻ mA. If 𝑖₂(0ሻ = −1mA_, find (a) 𝑖₁(0 ; (b) ሻ 𝑣(𝑡ሻ, 𝑣₁(𝑡 , and ሻ 𝑣₂(𝑡 ; (c) ሻ 𝑖₁(𝑡 and ሻ 𝑖₂(𝑡ሻ

Applications - Integrator

• Inverting invertor에 capacitor를 활용하여 적분기 구성

0

(0 ) ,

1 ( )

i o o

R C

i o t

o i

v d v dv

i i C C

R dt dt

v dv

C v v t dt

R dt RC

    

    



Applications - Integrator

Example 6.13

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

Applications - Differentiator

• Inverting invertor에 capacitor를 활용하여 미분기 구성

i , o i

C R o

dv v dv

i C i v RC

dt R dt

     

Applications - Integrator

Example 6.14

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

vi

Applications – Analog Computer

• Analog computer는 미분방정식을 풀기 위한 목적으로 개 발되었으며, op-amp를 이용하여 구현

• ^𝑑𝑥

𝑑𝑡는 ^𝑑2𝑥

𝑑𝑡²를 적분하여, 𝑥는 ^𝑑𝑥

𝑑𝑡를 적분하여 계산, 𝑓(𝑡 는 외부ሻ 의 signal generator를 이용하여 입력

2 2

2 2

( ), 0 ( )

d x dx

a b cx f t t

dt dt

d x f t b dx c dt a a dt a x

   

   

Applications – Analog Computer

Example 6.15

Design an analog computer circuit to solve the differential equation :

Subject to 𝑣_𝑜(0ሻ = −4, ^𝑑𝑣𝑜

𝑑𝑡 |_𝑡=0 = 1.

2

2^o 2 ^o _o 10 sin 4 , 0 d v dv

v t t

dt  dt   

Applications – Analog Computer

Example 6.15

Assignment

Chapter 6

6.13, 6.28, 6.46, 6.57, 6.78 Due date :