06 capacitors and inductors - national chiao tung university...2012/09/06 · 7 circuit theory;...

1

Jieh-Tsorng Wu

National Chiao-Tung UniversityDepartment of Electronics Engineering

Introduction to Circuit Theory

Capacitors and Inductors

2012-09-17

Circuit Theory; Jieh-Tsorng Wu26. Capacitors and Inductors

Outline

1. Capacitors

2. Series and Parallel Capacitors

3. Inductors

4. Series and Parallel Inductors

5. Integrators and Differentiators

2

Circuit Theory; Jieh-Tsorng Wu

Capacitor

A capacitor is a passive element designed to store energy in its electric field.

A capacitor consists of two conducting plates separated by an insulator (or dielectric).

Capacitance is the ratio of the charge on one plate of a capacitor to the voltage difference between the two plates, measured in farads (F).

36. Capacitors and Inductors

q C

AC

v

d


Capacitors


Polyester Capacitor Ceramic Capacitor Electrolytic Capacitor

Trimmer Capacitor

3


Nonideal Capacitors


Slope = C

q q

vv

Linear Nonlinear


Capacitor Current-Voltage Characteristic


0

0 0

0

00 Initial Condi

1

1 1 1

tion

t

t t t

t t

dv dvv v idt

dt dt C

v idt idt v t id

dqq C C i C

dt

q tv

tC

C

C

t

C

A capacitor is an open circuit to dc (i=Cdv/dt = 0).

The voltage on a capacitor cannot change abruptly.

4


Power and Energy in Capacitor


( )( )2 2 2

( ) ( )

The energy stored in the capacitor i

The instantaneous

s

1 1 1( ) ( )

2 2 2

Assume the cap

power delivered to the capac

acitor was u

itor is

n

v tv tt t

v v

dvi Cv

dt

dvw pdt C v dt C vdv Cv Cv

p v

t Cvdt

22

charged at , i.e, ( ) 0,

1

2

1

2

t v

w vq

CC

The energy w is stored in the electric field that exists between the plates of the capacitor.

This energy can be retrieved.


Capacitor Example 1


50 0 1

100 50 1 3( )

200 50 3 4

0 otherwise

200

10 mA 0 1

10 mA 1 3( )

10 mA 3 4

0 otherw

F

50

50

50

0 ise

t t

t tv t

t t

dvi C C

dtt

ti t C

t

5


Capacitor Example 2


50 (A) 0 2ms( )

100m (A) 2ms

1( ) 1 mF

t ti t

t

v t idt CC

22

For 0 2ms

1 25( ) (V) ( ) (V) (2m) 100 mV

1mFor 2ms

25

100m 100 (V) 100m 101

( ) 0 (V)

t

tv t v t v

Ct

v t aC

t

t a t t


Capacitor Example 3


1 2

21 1

2 22

21

2 2

34 V 8 V

3 2 41 1

16 mJ2 2

6 2 mA

(2 )

1 1128 mJ

2)

2

4

(4 8

i v v

w C

w C mv

mv

6


Parallel-Connected Capacitors


2

1 2

1

1 2

N

N

eq

N

i i i

dv dv dv dvC C C C

dt dt dt dtdv

C Cd

i

Ct

1 2eq NC C C C


Series-Connected Capacitors


0 0 0 0

0 0

2

1

1 2

0 1 0 0

2

2

0

1

)Let ( ( ( (

1 1

) )

1 1

1 1 1 1

)N

t t t t

t t t t

t t

t

N

eq N

eq tN

v v v

v t t t t

idt idt idt id

v

tC C C C

idt idtC C C C

v v v

1 2

1 2 1 2 1 2

1 1 1 1 1

1 1eqeq N

C CC

C C C C C C C C

7


Combined Capacitor Example 1



Combined Capacitor Example 2


V 5

V 10 V, 15

, Because

C 3.0301010

mF10

m60

1

m30

1

m20

11

33

22

11

332211

3eq

eq

eq

C

qv

C

qv

C

qv

vCvCvCq

vCq

C

C

8


Inductor

An inductor is a passive element designed to store energy in its magnetic field.

An inductor consists of a coil of conducting wire.

Inductance is the property whereby an inductor exhibits opposition to the change of current flowing through it, measured in henrys (H).


2

di

dtv L

N AL

l


Inductors


Solenoidal-Wound

Toroidal

Chip Inductor

Air-Core Iron-Core VariableIron-Core

9


Nonideal Inductors


Slope = L

v v

di/dtdi/dt

Linear Nonlinear


Inductor Current-Voltage Characteristic


0

0 0

0

0 Initial Conditi

1

1

n

1

o

1

t

t t t

t t

divdt

dt L

i vdt vdt i t vdtL

L

L

i

i t

L

v

An inductor is a short circuit to dc (v=Ldi/dt = 0).

The current through an inductor cannot change abruptly.

10


Power and Energy in Inductor


( )( )2 2 2

( ) ( )

The energy stored in the inductor is

1 1 1( ) ( )

2 2 2

Assume no current throug

The instantaneous power delivered to the induc

h the

tor is

i ti tt t

i i

dii Li

dt

diw pdt L i dt C idi Li Li t

d

v

Li

p

t

2

inductor at , i.e, ( ) 0,

1

2

t i

w Li

The energy w is stored in the magnetic field within the inductor. This energy can be retrieved.


Inductor Example


2

2

( ) 10(1 ) V 2 H (0) 2 A

1( ) 5 2.5 A

(0) 2 A ( ) 2 5 2.5 A

At 4, (4) 18 A

t

v t t L i

i t vdt a t tL

i i t t t

t i

11

Circuit Theory; Jieh-Tsorng Wu216. Capacitors and Inductors

2

2 2

2

5

122 A

1 5

12

1 150 J

2 21 1

4 J2 2

10 V1 5

1 10

2 2

C

L

C

C

LL

i i

v

Cv

w Li

w


Series-Connected Inductors


2

1 2

1

1 2

N

N

eq

N

v v v

di di di diL L L

dt dt dt dtdi

L

L

LLt

v

d

1 2eq NL L L L

12


Parallel-Connected Inductors


0 0 0 0

0 0

2

1

1 2

0 1 0 0

2

2

0

1

)Let ( ( ( (

1 1

) )

1 1

1 1 1 1

)N

t t t t

t t t t

t t

t

N

eq N

eq tN

i i i

i t t t t

vdt vdt vdt vd

i

tL L L L

vdt vdtL L L L

i i i

1 2

1 2 1 2 1 2

1 1 1 1 1

1 1eqeq N

L LL

L L L L L L L L


Combined Inductor Example 1


13


Combined Inductor Example 2


2

1 1 2 1

10

2

( ) 4 2 mA

(0) 1 mA

Find ( ( ), ( ),0), ( ), ( ), ( )

ti t e

i

t v ti v t v it i t

1 2 1 2 1 1

10 10

10 101 2 1

10 101 1 2

2 1

00

(0) (0) 4 (0) 1 (0) 5 mA

2 4 5

122 4

(0)

12 5 H 40 200

1(

mV

2

0) 5 3 (mA) 8 3 (mA)4

i

280 mV 0 mV

12 2 3

t teq eq

t t

tttt

div L e e

dt

v v e

i i i i i i i i

L v

v v v v

i v dti e e

i e

e

i

‖

‖

10 (mA)t


Inductor Voltage/Current Division


0 0

1 1 0 2 01 1

2

2 11

1 0 2 0

1 1 2 2

1 21 2

2 1

21 2 1 2

( ) ( ( ) (

Assume (

1 1) )

) ) 0,

( ) (

(

)

t t

t t

s

s

s

i t i t vdt i t i t vdt

i t t

L i

i Li i i

i L

L Lv

L L

i

t L i t

v v vL L L L

2

1 21 2

1 1 2

1 11 2

2

1 2

2

1 2s s

s

di div L v L

dt dtv L

v v vv L

L Lv v v v

L L L L

14


Capacitor Voltage/Current Division


0 0

1 1 0 2 01 2

2

2 11

1 0 2 0

1 1 2 2

1 21 2

2 1

21 2 1 2

( ) ( ( ) (

Assume ( (

( ) (

1 1) )

) ) 0,

)

t t

t t

s

s

s

v t v t idt v t v t idt

v t t

C v t C v t

v Cv v v

v C

C Cv v v v

C C

v

C C C C

1 1 2

1 1

2

1 21 2

1 2 1

12 2

2

2

s s

s s

s

dv dvi C i C

dt dti C

i i ii C

C Ci i i i

C C C C


Important Characteristics of the Basic Elements


15


Integrator


0

( )

( ) (0)

1

1

R C

i a a o

i o

o i

t

o o i

i i

v v d v vC

R dtv dv

CR dt

dv v dtRC

v dtR

tC

v v


Differentiator


( )C

i a a i o

io

oR

d v v v v dv vi i C C

dt R dt Rdv

v RCdt

16


Differentiator Example


1m

0ms 2ms 4ms 6ms

4

2m2

1m 2 V1m

2ms 4ms 6ms 8ms

4

2m2

1m 2 V1m

i io

i

o

i

o

dv dvv RC

dt dtt t

dv

dt

v

t t

dv

dt

v

06 capacitors and inductors - national chiao tung university...2012/09/06 · 7 circuit theory;...

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