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Electrical Circuits (2)
Lecture 8
Transient Analysis Part(2)
Dr.Eng. Basem ElHalawany
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Electrical Circuits (2) - Basem ElHalawany 2
Second-Order RLC Transient (Step Response) Second-Order RLC Transient (Step Response)
The Switch “S” is closed at t=0 Applying KVL will produce the following
Integro-Differential equation:
Differentiating, we obtain
This second order, linear differential equation is of the homogeneous type with a particular solution of zero. This second order, linear differential equation is of the homogeneous type with a particular solution of zero.
The complementary function can be one of three different types according to the roots of the auxiliary equation which depends upon the relative magnitudes of R, L and C.
The complementary function can be one of three different types according to the roots of the auxiliary equation which depends upon the relative magnitudes of R, L and C.
012
LC
mL
Rm
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Electrical Circuits (2) - Basem ElHalawany 3
Second-Order RLC Transient (Step Response) Second-Order RLC Transient (Step Response)
012
LC
mL
Rm
02 2002 mm
We can Rewrite the auxiliary equation as:
2
0
0
1
2
LC
L
R
frequency natural undamped :
ratio dampling expontial :
0
The roots of the equation (or natural frequencies): The roots of the equation (or natural frequencies):
1
1
2
002
2
001
m
m
LCL
R
L
Rm
LCL
R
L
Rm
1
422
1
42
2
2
2
2
1
0
2
2
0
2
1
m
m
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4
Case 1: Overdamped, Case 1: Overdamped,
o
LC22
1
2L
R
1
Natural response is the sum of two decaying exponentials:
tmtm
tr eKeKi21
21
1
1
2
002
2
001
m
m
Second-Order RLC Transient (Step Response) Second-Order RLC Transient (Step Response)
unequal and real are , 21 mm
Case 2: Critically damped, Case 2: Critically damped, equal. and real are , 21 mm
o
LC22
1
2L
R
1
021 mm
)()( 211 tBBetxtm
c Use the initial conditions to get the constants
Use the initial conditions
to get the constants
Usually it is reduced to: Usually it is reduced to: tm
c etBtx1..)(
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5 Second-Order RLC Transient (Step Response) Second-Order RLC Transient (Step Response)
Case 3: Underdamped, Case 3: Underdamped,
conjugate. andcomplex are , 21 mm
Natural response is an exponentially damped oscillatory response:
)}sin()cos({ 21 tAtAei ddt
tr
)1()(
)1()(
2
002
2
001
jjm
jjm
d
d
o
LC22
1
2L
R
1
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Electrical Circuits (2) - Basem ElHalawany 6
tri
t t
Critically
damped
Overdamped Underdamped
(ringing)
Envelope
tri
The current in all cases contains the exponential decaying factor (damping factor) assuring that the final value is zero
In other words, assuring that the complementary function decays in a relatively short time.
The current in all cases contains the exponential decaying factor (damping factor) assuring that the final value is zero
In other words, assuring that the complementary function decays in a relatively short time.
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Electrical Circuits (2) - Basem ElHalawany 7
Alternating Current Transients Alternating Current Transients
RL Sinusoidal Transient
1. Complementary (Transient) Solution is the solution of the homogeneous 1st order DE 1. Complementary (Transient) Solution is the solution of the homogeneous 1st order DE
The same as before, The auxiliary equation is : 0L
Rm
2. Particular (Steady-State) Solution 2. Particular (Steady-State) Solution
The steady-state value of the current for ac source is : The steady-state value of the current for ac source is :
))/(( tan1
22
max RLwtSin
L
I
RX
Vss
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Electrical Circuits (2) - Basem ElHalawany 8
Alternating Current Transients Alternating Current Transients
RL Sinusoidal Transient
Use the initial condition to find the value of c Use the initial condition to find the value of c
Substituting by the constant values, we get: Substituting by the constant values, we get:
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Electrical Circuits (2) - Basem ElHalawany 9
Alternating Current Transients Alternating Current Transients
RC Sinusoidal Transient RC Sinusoidal Transient
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Electrical Circuits (2) - Basem ElHalawany 10
Alternating Current Transients Alternating Current Transients
RLC Sinusoidal Transient RLC Sinusoidal Transient
Particular (Steady-State) Solution Particular (Steady-State) Solution
Complementary(Transient) Solution Complementary(Transient) Solution
The complementary function is identical to that of the DC series RLC circuit examined previously where the result was overdamped, critically
damped or oscillatory, depending upon R, L and C.
The complementary function is identical to that of the DC series RLC circuit examined previously where the result was overdamped, critically
damped or oscillatory, depending upon R, L and C.
For the complete analysis Check Chapter 16 Schaum Series (Old version)
For the complete analysis Check Chapter 16 Schaum Series (Old version)
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Electrical Circuits (2) - Basem ElHalawany 11
Transient Analysis using Laplace Transform Transient Analysis using Laplace Transform
Solving differential equations Circuit analysis (Transient and general circuit
analysis) Digital Signal processing in Communications and Digital Control
Laplace transform is considered one of the most important tools in Electrical Engineering
It can be used for:
Laplace transform is considered one of the most important tools in Electrical Engineering
It can be used for:
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Electrical Circuits (2) - Basem ElHalawany 12
Transient Analysis using Laplace Transform Transient Analysis using Laplace Transform
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Electrical Circuits (2) - Basem ElHalawany 13
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Electrical Circuits (2) - Basem ElHalawany 14
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Electrical Circuits (2) - Basem ElHalawany 15
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Electrical Circuits (2) - Basem ElHalawany 16
The switch “S” is closed at t = 0 to allow the step voltage to excite the circuit Apply KVL to the circuit in figure: The switch “S” is closed at t = 0 to allow the step voltage to excite the circuit Apply KVL to the circuit in figure:
First-Order RL Transient (Step-Response) First-Order RL Transient (Step-Response)
s
VisIsLsIR )]0()(.[)(.
Apply Laplace Transform on both sides
i(0) = 0 >> initial value of the current at t = 0
s
VsLRsI ]).[(
][][)(
LRss
LV
sLRs
VsI
Apply the inverse Laplace Transform technique to get the expression of the current i(t)
Apply the inverse Laplace Transform technique to get the expression of the current i(t)
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17
First-Order RL Transient (Step-Response) First-Order RL Transient (Step-Response)
Use the partial fraction technique Use the partial fraction technique
LRs
A
s
A
LRss
LV
sI
21
][)(
Multiply both sides by Multiply both sides by ).( LRss
L
RAsAA
sAL
RsAL
V
.).(......
.).(
11
1
2
2
both sides Compare the coefficients
RVA
1 RVA 2
)11
()(
LRssR
VsI
So, the current in s-domain is given by: So, the current in s-domain is given by:
Apply the inverse Laplace transform : Apply the inverse Laplace transform : 0);1()(
teR
Vti
tL
R
R
VsILRsA
R
VssIA
LRs
s
/2
01
|)()/(
|)(
OR
The same as last lecture The same as last lecture
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18 First-Order RL Transient (Discharge) First-Order RL Transient (Discharge)
The RL circuit shown in Figure contains an initial current of (V/R)
The Switch “S” is moved to position”2” at t=0
The same as before
0)]0()(.[)(. isIsLsIR
Apply Laplace Transform on both sides
i(0) = V/R >> initial value of the current at t = 0
][)(
LRs
RV
sI
0])(.[)(. RVsIsLsIR
Apply the inverse Laplace transform : Apply the inverse Laplace transform :
0;..)(
teIeR
Vti
tL
R
o
tL
R
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Electrical Circuits (2) - Basem ElHalawany 19
First-Order RC Transient (Step-Response) First-Order RC Transient (Step-Response)
o Assume the switch S is closed at t = 0 o Apply KVL to the series RC circuit shown:
s
VsIR
s
v
cs
sI c )(.])0()(
[
Apply Laplace Transform on both sides
Vc(0) = 0 >> initial value of the voltage at t = 0
VtiRvdttic
c )(.)]0().(1
[
s
V
csRsI ]
1).[(
]1[]1[)(
cRs
RV
csR
sV
sI
Apply the inverse Laplace Transform technique to get the expression of the current i(t)
Apply the inverse Laplace Transform technique to get the expression of the current i(t)
0;)(
1
teR
Vti
tRC
The same as last lecture The same as last lecture
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20
Second-Order RLC Transient (Step Response) Second-Order RLC Transient (Step Response)
The Switch “S” is closed at t=0 Applying KVL will produce the following
Integro-Differential equation:
The Switch “S” is closed at t=0 Applying KVL will produce the following
Integro-Differential equation:
s
VsIRisIsL
s
v
cs
sI c )(.)]0()(..[])0()(
[
Apply Laplace Transform on both sides Apply Laplace Transform on both sides
VtiRdt
tdiLvdtti
cc )(.
)()]0().(
1[
Assume: vc(0) = 0 & i(0) = 0 Assume: vc(0) = 0 & i(0) = 0
]1).([]1[)(
2
LcLRss
LV
cssLR
sV
sI
To convert this to time-domain, it will depend on the roots of the denominator which could be expressed as:
To convert this to time-domain, it will depend on the roots of the denominator which could be expressed as:
(s-S1).(s-S2) >>>>> similar to last lecture (m-m1).(m-m2)
012
LC
mL
Rm
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Electrical Circuits (2) - Basem ElHalawany 21
LCL
R
L
RS
1)
2(
2
2
2,1
12
002,1 S
2
0
2
2,1 Sfrequency natural undamped :
ratio dampling expontial :
0
Second-Order RLC Transient (Step Response) Second-Order RLC Transient (Step Response)
212 ]1).([
)(SS
B
SS
A
LcLRss
LV
sI
Apply Partial Fraction: Apply Partial Fraction:
AL
VsISSB
L
VsISSA
o
ss
o
ss
1.2|)().(
1.2|)().(
22
21
2
1
]11
.[1.2
)(21
2 SSSSL
VsI
o
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22
Second-Order RLC Transient (Step Response) Second-Order RLC Transient (Step Response)
Apply Partial According to the values of the roots , we have 3 scenarios: Apply Partial According to the values of the roots , we have 3 scenarios:
].[1.2
)( 212
tStS
o
eeL
Vti
1. Over-damped Case 1. Over-damped Case i.e. Two real distinct roots
ttS oetL
Vet
L
Vti
....)( 1
1. Critically-damped Case 1. Critically-damped Case i.e. Two real equal roots
222
12 )()()(]1).([
)(oS
LV
S
LV
SS
LV
LcLRss
LV
sI
L
RSS
221
o
Convert by inverse L.T: Convert by inverse L.T: at
n
ne
n
t
aS
.
)!1()(
1 1
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23
Second-Order RLC Transient (Step Response) Second-Order RLC Transient (Step Response)
)sin(......
)]sin(.cos)cos(.[sin......
)]sin()cos([......
)]sin()cos([)(
.
.
21.
21
.
tAe
ttAe
tA
At
A
AAe
tAtAeti
d
t
dd
t
dd
t
dd
t
3. Under-damped Case 3. Under-damped Case i.e. Two Complex-conjugate roots
djS 2
0
2
2,1
1A
2A
A
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RL Sinusoidal Transient RL Sinusoidal Transient
R = 5 ohms, L = 0.01 H, Vm = 100 volts, ф = 0, ω = 500 R = 5 ohms, L = 0.01 H, Vm = 100 volts, ф = 0, ω = 500
22100)]0()(.[)(.
sisIsLsIR
Apply Laplace Transform on both sides
i(0) = 0 >> initial value of the current at t = 0
22100)]().[(
sL
RsLsI
)500).((
105
)).(.(
.100)(
22
6
22
ss
x
L
RssL
sI
jS
jS
S
3
2
1 500
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25
RL Sinusoidal Transient RL Sinusoidal Transient Use Partial Fraction: Use Partial Fraction:
500)500).((
105)( 321
22
6
s
A
js
A
js
A
ss
xsI
OR 500
3
22
21
s
A
s
BsB
tettti 500.10)500cos(.10)500sin(.10)(
Compare to find the constants:
Use Inverse L.T. Use Inverse L.T.
500
10
500.10
500
500.10
500
10]500[10)(
222222
ss
s
sss
ssI
tettti 500.10)]500cos()500.[sin(10)(
1
1
2)1(1 22 A
o45)1
1(tan 1
trss
to IIetti .500.10)45500sin(.2.10)(
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Check EXAMPLE 11–12 ,13 Miller Check EXAMPLE 11–12 ,13 Miller 26
Examples Examples
The capacitor of Figure 11–24(a) is uncharged. The switch is moved to position 1 for 10 ms, then to position 2, where it remains.
The capacitor of Figure 11–24(a) is uncharged. The switch is moved to position 1 for 10 ms, then to position 2, where it remains.
Note that discharge is more rapid than charge since Note that discharge is more rapid than charge since