12 Transient

7/28/2019 12 Transient

1/39

12 - 1Transient - Basic Relationships


2/39

12 - 2Transient - Switch Open for Negative Time

( ) 0,0


3/39

12 - 3Transient - Switch Closed

.resistancewith

circuitshortaisswitchClosed

0

:KVL( )

( ) 0,357 =+ ttidt

tdi

bygivenandconstantahasequationaldifferentiorderfirstthistosolutionGeneral

( ) ( ) ( )tititi sstr +=

( ) ( ) constantonewithtosolutiongeneral 057 =+ tidt

tditr

tr

inputnoorr.h.s.withbutbeforeasequationsame 0

( ) responsetransientorresponsestatezerogivestitr

( )( ) constantnowithtosolutionparticulara 357 =+ ti

dt

tdiss

ss

( ) responsestatesteadygivestiss


4/39

12 - 4Transient - Steady State Response

( )( ) 0,357 =+ tti

dt

tdi


( ) ( ) constantnowithtosolutionparticulara 357 =+ tidt

tdiss

ss

( )( ) constantonewithtosolutiongeneral 057 =+ ti

dt

tditr

tr

( )

( ) 357 =+ tidttdi

ss

ss

constant

( ) ktiss =try 357 =+ kdtdk

350 =+ k5

3=k

( )5

3=tiss


5/39

12 - 5Transient - Transient Response

( )( ) 0,357 =+ tti

dt

tdi


( ) ( ) ( ) constantnowithtosolutionparticulara, 3575

3=+= ti

dt

tditi ss

ssss

( )( ) constantonewithtosolutiongeneral 057 =+ ti

dt

tditr

tr

( )

( ) 057 =+ tidttdi

tr

tr

( )

( ) dtti

tdi

tr

tr

=57

( )

( ) = dttitdi

tr

tr

5

7

( )[ ] 1ln5

7

httitr +=

( )[ ] 27

5ln h

ttitr +=

( )2

7

5h

t

tr eti+

= 275

h

t

ee

= 75

3

t

eh

=

( ) 0,75

1=

tekti

t

tr


6/39

12 - 6Transient - General Response Derivation

( )( ) 0,357 =+ tti

dt

tdi


( ) ( ) ( ) constantnowithtosolutionparticulara, 3575

3=+= ti

dt

tditi ss

ssss

( )( )

( )1

7

5

1 057 ktidt

tdiekti tr

tr

t

tr constantonewithtosolutiongeneral, =+=

( ) ( ) ( )

( ) ( ) atidt

tdi

tektititi

t

sstr

constantonewithtosolutiongeneraltheis

,

057

05

37

5

1

=+

+=+=

+

( ) ( )[ ]( ) ( )[ ] 357 =++

+titi

dt

titidsstr

sstr( ) ( )[ ] 75

15

3t

sstr ektiti

+=+

+


7/39

12 - 7Transient - Complete Response

( )( ) 0,357 =+ tti

dt

tdi


( ) 0,0


8/39

12 - 8Transient - Current Continuity for Inductor

( ) system,practicalrealanyforSince tv

05

37

5

1 +

tek

t

,

00


9/39

12 - 9Transient - RC Circuit 1

( ).Find tv

0


10/39

12 - 10Transient - RC Circuit 2

:For 0t

KVL( )

( ) 335 =+ tvdt

tdv ( ) ( ) ( )tvtvtv sstr +=

( )( ) 035 =+ tv

dt

tdvtr

trtosolutiongeneral

( )

( ) 35

dt

tv

tdv

tr

tr = ( ) 135

ln ht

tvtr +=

( )1

35h

t

tr etv+

= 135 ht

ee

= 351t

ek

=

( )( ) 335 =+ tv

dt

tdvss

sstointegralparticular

( ) ktvss

=try

[ ][ ] 335 =+ k

dt

kd

[ ] 3=k

( ) 3=tiss0,3351 +=

tek

t

12 11T i t RC Ci it 3 V lt C ti it f C it


11/39

12 - 11Transient - RC Circuit 3 - Voltage Continuity for Capacitor

03351

+

tek

t

,

02


12/39

12 - 12Transient - RL Circuit with Complex Sinusoidal Input 1

( ).Find ti

tje 43

( ) 0,0


13/39

12 - 13

tje 43

Transient - RL Circuit with Complex Sinusoidal Input 2

:For 0t

KVL ( ) ( ) tjetitv 435 =+


( )( ) 057 =+ ti

dt

tditr

trtosolutiongeneral

( )

( ) 7

5dt

ti

tdi

tr

tr = ( ) 17

5ln h

ttvtr +=

( )1

7

5h

t

tr etv+

= 175

h

t

ee

= 75

1

t

ek

=

( )( ) tj

ssss etidt

tdi 4357 =+tointegralparticular

( ) tjss

keti 4=try

[ ] [ ] tjtjtj

ekedt

ked 444

357 =+

( ) [ ] [ ] tjtjtj ekekej 444 3574 =+

( ) 5743

+=

jk( )

0,574

34

7

5

1

++=

t

j

eek

tjt

( )( ) tjeti

dt

tdi 4357 =+

Impedance

( )analysisresp.freq.inAs

12 14Transient RL Circuit with Complex Sinusoidal Input 3


14/39

12 - 14Transient - RL Circuit with Complex Sinusoidal Input 3

tje 43

( )

( )

0,

574

34

7

5

1

+

+=

t

j

eekti

tjt

( ) 0,0


15/39

12 - 15

( )54cos3 +t

( )54cos3 +t

Transient - RL Circuit with Real Sinusoidal Input 1

( ).Find ti

( )54cos3 +t

( )0,0


16/39

12 - 16

tje 43( )54cos3 +t

Transient - RL Circuit with Real Sinusoidal Input 2

:For 0t

KVL ( ) ( ) ( )54cos35 +=+ ttitv ( ) ( ) ( )tititi sstr +=

( )( ) 057 =+ ti

dt

tditr

trtosolutiongeneral

( )

( ) 7

5dt

ti

tdi

tr

tr = ( ) 17

5ln h

ttvtr +=

( )1

7

5h

t

tr etv+

= 175

h

t

ee

= 75

1

t

ek

=

( )( ) ( )54cos357 +=+ tti

dt

tdiss

sstoint.part. ( ) tjtjss eaeati

4

2

4

1

+=try

[ ] [ ]( ) ( )

2357

5454

4

2

4

1

4

2

4

1

++

+=++

+ tjtjtjtjtjtj

eeeaea

dt

eaead

( )( ) ( )54cos357 +=+ tti

dt

tdi

( ) ( )

23

5454 ++ +=

tjtj ee

( )[ ] ( )[ ] tjtj eajeaj 42

4

1 574574

+++=

( )[ ]57423 5

1 +=

j

eaj

( )[ ]57423

5

2 +=

j

ea

j

and22

5

28

tan5

5282

31

+=

jj ee jj aee

5

2

3= *

1a=jj

aee= 5

2

3

( ) ( ) ( )54542

3

2

3 ++ += tjjjtjss eaeaeeti ( )++= 54cos3 ta

Impedance

( )analysisresp.freq.inAs

12 - 17Transient - RL Circuit with Real Sinusoidal Input 3


17/39

12 17Transient RL Circuit with Real Sinusoidal Input 3

tje 43( )54cos3 +t

( ) ( )0,54cos37

5

1 +++=

ttaekti

t

( ) 0,0


18/39

12 18Transient Q1

inductor.theincurrenttheDetermine

12 - 19Transient - Q1


19/39

12 19Transient Q1



20/39

Q

( ).voltagetheDetermine tv



21/39



22/39

( ).voltagetheDetermine tv



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30/39

Write down the differential equation for determining the inductorvoltage.



31/39



32/39

What are the voltages and currents just before and after the switch isclosed?



33/39



34/39

What are the voltages and currents just before and after the switch isclosed?



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12 Transient

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