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12 - 1Transient - Basic Relationships
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12 - 2Transient - Switch Open for Negative Time
( ) 0,0
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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
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12 - 4Transient - Steady State Response
( )( ) 0,357 =+ tti
dt
tdi
( ) ( ) ( )tititi sstr +=
( ) ( ) 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
7/28/2019 12 Transient
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12 - 5Transient - Transient Response
( )( ) 0,357 =+ tti
dt
tdi
( ) ( ) ( )tititi sstr +=
( ) ( ) ( ) 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
7/28/2019 12 Transient
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12 - 6Transient - General Response Derivation
( )( ) 0,357 =+ tti
dt
tdi
( ) ( ) ( )tititi sstr +=
( ) ( ) ( ) 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
+=+
+
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12 - 7Transient - Complete Response
( )( ) 0,357 =+ tti
dt
tdi
( ) ( ) ( )tititi sstr +=
( ) 0,0
7/28/2019 12 Transient
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12 - 8Transient - Current Continuity for Inductor
( ) system,practicalrealanyforSince tv
05
37
5
1 +
tek
t
,
00
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12 - 9Transient - RC Circuit 1
( ).Find tv
0
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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
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12 - 11Transient - RC Circuit 3 - Voltage Continuity for Capacitor
03351
+
tek
t
,
02
7/28/2019 12 Transient
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12 - 12Transient - RL Circuit with Complex Sinusoidal Input 1
( ).Find ti
tje 43
( ) 0,0
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12 - 13
tje 43
Transient - RL Circuit with Complex Sinusoidal Input 2
:For 0t
KVL ( ) ( ) tjetitv 435 =+
( ) ( ) ( )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
=
( )( ) 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
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12 - 14Transient - RL Circuit with Complex Sinusoidal Input 3
tje 43
( )
( )
0,
574
34
7
5
1
+
+=
t
j
eekti
tjt
( ) 0,0
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12 - 15
( )54cos3 +t
( )54cos3 +t
Transient - RL Circuit with Real Sinusoidal Input 1
( ).Find ti
( )54cos3 +t
( )0,0
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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
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12 17Transient RL Circuit with Real Sinusoidal Input 3
tje 43( )54cos3 +t
( ) ( )0,54cos37
5
1 +++=
ttaekti
t
( ) 0,0
7/28/2019 12 Transient
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12 18Transient Q1
inductor.theincurrenttheDetermine
12 - 19Transient - Q1
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12 19Transient Q1
12 - 20Transient - Q2
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Q
( ).voltagetheDetermine tv
12 - 21Transient - Q2
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12 - 22Transient - Q3
7/28/2019 12 Transient
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( ).voltagetheDetermine tv
12 - 23Transient - Q3
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12 - 24Transient - Q4
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12 - 25Transient - Q4
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12 - 26Transient - Q5
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12 - 27Transient - Q5
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12 - 28Transient - Q6
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12 - 29Transient - Q6
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12 - 30Transient - Q7
7/28/2019 12 Transient
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Write down the differential equation for determining the inductorvoltage.
12 - 31Transient - Q7
7/28/2019 12 Transient
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12 - 32Transient - Q8
7/28/2019 12 Transient
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What are the voltages and currents just before and after the switch isclosed?
12 - 33Transient - Q8
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12 - 34Transient - Q9
7/28/2019 12 Transient
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What are the voltages and currents just before and after the switch isclosed?
12 - 35Transient - Q9
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7/28/2019 12 Transient
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12 - 37Transient - Q10
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12 - 38Transient - Q11
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12 - 39Transient - Q11
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