07 first-order · pdf filefirst-order opamp circuits. 2 circuit theory; jieh-tsorng wu the...

1

Jieh-Tsorng Wu

National Chiao-Tung UniversityDepartment of Electronics Engineering

Introduction to Circuit Theory

First-Order Circuits

2012-10-12

Circuit Theory; Jieh-Tsorng Wu27. First-Order Circuits

Outline

1. The Source-Free RC Circuit

2. The Source-Free RL Circuit

3. Singularity Functions

4. Step Response of an RC Circuit

5. Step Response of an RL Circuit

6. First-Order OPAMP Circuits

2

Circuit Theory; Jieh-Tsorng Wu

The Source-Free RC Circuit

37. First-Order Circuits

0 0

10 0 0

( ) Time Constantt

R

RC t

C

dv v dvi i C v

dt R dt RC

v t V e e RCV


Natural Response and Time Constant

The natural response of a circuit refers to the behavior of the circuit itself, with no external sources of excitation.

The time constant of a circuit is the time required for the response to decay by a factor of 1/e or 36.8% of its initial value.


%100674.05

%201832.04

%504979.03

13534.02

36788.0

)( 0

Vtvt

3


Effect of Time Constant



Power and Energy of a RC Circuit


0

0

220

t

tR

tR

v t V e

v t Vi e

R R

R

Vp t vi

R

C

e

2 22 2 2 20 0

0

0 0 0

20

11

2 2

1s , A

2

tt tt t t

R

R

V Vw t pdt e dt e CV e

R R

t w CV

4


The Key for RC Circuit Analysis

Find the initial voltage v(0)=V0 across the capacitor.

Find the time constant =RC.

R is the resistance as seen by the C.



RC Circuit Example 1


0.4

2.5 2.5

5 4

( ) (0)

20 4 (0.1) 0.4 s

129 V A

8 12

15 V

0.7512

eq eq

t tC C

t txx C x

R C

t v e e

vv v

R

v

e ei

‖

5


RC Circuit Example 2


t < 0

t > 0

2

2

0.2

For t<0

(0) 20

1(0) (0)

21

(0) 2.25 J2

For 0

1 9 10

915 V

3 9

(20m) 15

2

10

(0

0m 0.2 s

( ) 15 V)

C

C C

C

eq

eq

t tC

v

Cv

t

R

w

w

R

v t v e e

C


The Source-Free RL Circuit


0 0 T

0 0 0

( ) ime Constant

L

tR L

R

t

di di Rv v L Ri v

dt dt L

i t I e V eL

R

6


Power and Energy of a RL Circuit


0

0

2 20

( )

t

tR

tR

i t I e

v i t I Re

p t v i I Re

L

R

R

2 2 2 2 2 20 0 0

0 0 0

20

11

2 2

1s , A

2

tt tt t t

R

R

w t pdt I Re dt I R e LI e

t w LI


The Key for RL Circuit Analysis

Find the initial voltage i(0)=I0 through the inductor.

Find the time constant =L/R.

R is the resistance as seen by the L.


7


RL Circuit Example 1


1 2 1 2 2 1 1 2 2 1

11

(2/3) (2/3) (2/3)

(2/3)

0.5 3

52( ) 0 3 4 2(

s1 3 2

2 10(0) 10 A 10 V

3 3

5

) 0 2 26

i

1 1

3 3

0.

A3

5

2

o o

o oo eq

o

t t t t

tx

i v i i i v i i i i

v vv i R

i i

di

i i

L

R

i i e e e ev Ldt

ev




t < 0

t > 04

For 0

40(0)

2 (4

16

126 A

12) 12 4

For 0

(12 4) 8

2 1 s

8 4

( ) (0) 6 A

eq

eq

t t

t

R

L

R

i t i e e

t

i

‖

‖

8




t < 0

t > 0

2 A

For 0

22 1 s

2

( ) (0) 2 A

For 0

10(0) (0) 3 (0) 6 V

2 3

3

24

6 3 V A

o

eqeq

t t

t tLoL

t

LR

R

i t i e e

v

t

i

e

v i

vdi

de

iL

t

‖6


Singularity Functions

Singularity functions are functions that either are discontinuous or have discontinuous derivatives.

The three most widely used singularity functions in circuit analysis are

Unit step function, u(t).

Unit impulse function, (t).

Unit ramp function, r(t).


9


Unit Step Functions


0, 0( )

1, 0

tu t

t

00

0

0, ( )

1,

t tu t t

t t

00

0

0, ( )

1,

t tu t t

t t


Unit Step Function and Equivalent Circuits


10


Unit Impulse (Delta) Function


0

0

0, 0

Undefined, 0

0,

( ) 1 ( ') '

0

( )t

td

t u t tdt

t

t dt t dt u t


Unit Impulse (Delta) Functions


( 2) 10 ( ) 4 ( 3( ) )5 t t tf t

11


Unit Impulse Function and Sampling


0 0 0 0

0

0 0 0 0 0 0

( ) ( ) ( ) is sampled at

Let

( )

( ) )

,

(

b b b

a a a

y t f t f t t t

a t b

f t t t dt f t t t dt f t t t dt f

t f t

t

t t t


Unit Ramp Functions


' '

0, 0

, 0

t

r t u t dt tu t

tr t

t t

0

0

0 0

0,

,

r t t

t t

t t t t

0

0

0 0

0,

,

r t t

t t

t t t t

12


Singularity Function Example 1





( ) 10 ( ) 20 ( 2) 10 ( 4)

10 ( ) 20 ( 2) 10 ( 4)

i t u t u t u t

idt r t r t r t

13







( ) 1 ( )u t u t

14


Step Response of an RC Circuit


0

/1 2

/ /11 2

0

1 2 0

2 1 0

At 0, (0)

For 0, (0 ) , and from KCL,

0

0

Let ( )

C R

S

S

t

t tS

S S

t v V

t v V

i i

v VdvC

dt Rdv

RC v Vdt

v t a e a

aRC e a e a V a a V

a V a V VRC

For t > 0


Step Response of an RC Circuit


0

0

0

, 0( )

, 0

If 0, then, for 0

( ) 1

( )

S

tS S

t

t tSs

V tv t

V V V e t

V t

v t V e

Vdv Ci t C V e e

dt R

15


Natural Response and Forced Response


Complete Response = Natural Response + Forced Response

(Stored Energy) (Independent Source)

0

0

0

For 0

( )

1

( ) (

( )

)

( ) 1

tS S

tS

n f

t

tn

ft

S

t

v t V V V e

V e V e

v t v t

v V e

v t V e

t


Transient Response and Steady-State Response


Complete Response = Transient Response + Steady-State Response

(Temporary) (Permanent)

0

0

For 0

( )

( )

( )

tS S

ss

ss

t

S

tt S

t

v t V V V e

v v t

v V

v t V V e

Transient response is the circuit’s temporary response that will die out with time. Steady-state response is the behavior of the circuit a long time after the an

external excitation is applied.

16


Transient Response and Steady-State Response


v(0)

v()

0

0

( ) ( ) (0) ( )

or

( ) ( ) ( ) ( )

(0) is the initial valu

) is the final steady-state value

is the time constant of the circu

e

it

(

t

t t

v t v v v e

v t v v t v e

v

v


RC Circuit Step Response Example 1


/

0.5 0.5

For 0, ( ) (

(0) (0

) (0) ( )

5k15 V

3k 5k( ) 30 V

4k 0.5m 2 sec

( ) 30 [15 30] 30 15 V

) 24

t

t t

v v e

v

t v t v

e

v

v

v

t e

17


RC Circuit Step Response Example 2


t < 0

t > 0

/

0.6 0.6

) (0) ( )

10 V

20( ) 30 20 V

10 201 20 1 5

(10 20)

For 0

sec4 3 4 3

( ) 20 [10 20] 20 10 V

, ( ) (

(0) (0 )

t

t t

v v e

v

v t e e

t v t v

v v

‖


Step Response of an RL Circuit


0

/1 2

/ /11 2 1 2 0

2 1 0

For 0, (0)

For 0, (0 ) / , and from KVL,

Let ( )

S

S

S

S

t

t t S

L R

S S

t i I

t i V R

v v V

diL Ri V

dtVL dv

vR dt R

i t a e a

VaLe a e a a a I

R RV V

a a IL

R R R

For t > 0

18




0( ) 0S tSV Vv t I e t

R R

0If 0, then, for 0

( ) 1

( )

S

SS

t

t t

I t

Vi t e

RVdi L

v t L e V edt R




0

0

( ) ( ) (0) ( )

or

( ) ( ) ( ) ( )

(0) is the initial valu

) is the final steady-state value

is the time constant of the circu

e

it

(

t

t t

i t i i i e

i t i i t i e

i

i

19


RL Circuit Step Response Example 1


/

15 15

For 0,

( ) (

10

) (0) ( )

5 A

10( ) 2 A

51 1 1

sec3 5 15

( ) 2 [5 2] 2 3

(0) 0 )

A

(2

t

t t

t

i t i i e

i

i t e e

i

i i




(I) For 00, ( )t i t

/

2 2

) (0) ( )

40 5 10 A (

(II) For 0< 4, (

) 4 A sec4 6 10 2

( ) 4 [0 4] 4 1

) (

(0 ) (0 ) 10

A

t

ThTh

t t

i i e

Li R

R

i t e

t i t i

i

e

i

20




( 4)/

2 8

1.467( 4)

) (4) ( )

4(1 4(1 A

2 410 20 V

4 2 4 222

2) 63

20 30( ) 2.272 A

22 / 3 11

5 15 sec

22 / 3 22

(III) For 4< ,

( ) (

(4) (4 ) ) ) 4

40

(

( ) 2.727 [4 2.72

4

7]

2

t

t

S

Th

S

Th

Th

t

i i e

e e

V

R

t

i t i

i i

Vi

R

L

R

i t e

‖

1.467( 4).727 1.273 Ate


First-Order Opamp Circuit Example 1


0

1 1

/ /0

/ /00

1

/0 0

1

) 0

) ( ) (0)

(

(0) (

( ) (

( ) ( )

)

Th

t t

t t

f tf

R R

v v e V e

Vdv Ci t C V e e

dt R

v V e

v V v

R C

v t v

Rt R i t

R

21




0

/ /0

/0 0

) 0

) ( )

(0) (

( ) (

(

)

) )

(0

(

fTh f

t t

t

v

R R

v v e V

V v

R C

v t v

t vv V

e

et




0

1

1

1

/

20 20

200 1

) 0

50k 1 0.05

(0) 0

20k3 2 V

10k 20k50

(0) (

(

(

( ) (

( )

k) 1 7 V

20k

( ) ) 5 V

) (0) ( )

5 [0 5] 5

( ) 7 5

5 V

V

o

o

t

t t

t

v V v

v

v t v

t v v

v

v

v v

v v

v v e

e e

etv

22




1 1

3

1 2 3

2 32 3

2 3

/

( ) 2 ( ) V ( ) V

)

) 1

( ) ( ) 2

(0) 0 ( 2

0

( ) (

f fi ab i

fo o

o Th Th

to o

R Rv V t v t

R R

Rv

R R

R RR R C

R R

v t v

t u t u t

Rv

R

R R R

e

‖

07 first-order · pdf filefirst-order opamp circuits. 2 circuit theory; jieh-tsorng wu the...

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