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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 1
FachgebietNachrichtentechnische Systeme
N T S
Fundamentals of EE 3
Chapter 3.1Transient processes -
The Laplace transform
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 2
FachgebietNachrichtentechnische Systeme
N T S
3.1.1 Introduction
Use of a complex frequency p for time signals. It applies: p j
pt11 ˆu t Re u e
Thus exponential envelopes are representable for sinusoidal functions in addition to those with constant amplitudes:
Example for RLC resonant circuit:
pt22 ˆu t Re u e ptˆi t Re i e
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 3
FachgebietNachrichtentechnische Systeme
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3.1.1 Introduction
A network analysis shows here that it applies:
t
1
0 t 0
1 20
di t 1u t i t R L i z dzdt C
di t 1 1 1u t i t R L i z dz i z dz u 0 i z dzdt C C C
Solution can be achived by means of: - solution of the integro-differential equations- the Laplace transform
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 4
FachgebietNachrichtentechnische Systeme
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3.1.2 Definition of the Laplace transform
An arbitry function is given: 0 0
( )( ) 0
for tf t
g t for t
f(t) exhibits the following characteristics:
1) f(t) has only finite jumps in the interval 1 20 t t t 2
1
( )t
t
f t dt is limited2) 03) ( ) should drop to zero quickly enough: lim ( ) t
tf t f t e
If these conditions are fulfilled
the Laplace transform of the function f(t) exists:
0
( ) ( ) ( ) ( )
ptf t g t G p g t e dtL L
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 5
FachgebietNachrichtentechnische Systeme
N T S
3.1.2 Definition of the Laplace transform
The relation of original signals and corresponding transform isdescribed by the symbol after DOETSCH:
( )g t ( )G p ( )G p ( )g tRelation with Fourier transform is given for:
- a causal time function and at the same time - a modified time function due to multiplication with
This leads in the integrand of the Fourier Transf. to a term
tet j t pts(t) e e s(t)e
pt0
L s t s t e dt S p
j
1 pt
j
1L S p lim S p e dp s t2 j
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 6
FachgebietNachrichtentechnische Systeme
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3.1.3 Methods for the determination of theoriginal function from the image function
Direct method:
The original function can be determined according to the relationship:
1( ) lim ( )2
j
pt
j
g t G p e dpj
for t > 0
The use of this formula requireshowever some knowledge of the function theory.
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 7
FachgebietNachrichtentechnische Systeme
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3.1.3 Methods for the determination of theoriginal function from the image function
Use of transformation tables:
Example: In a network it applies to the transform of the current:
0 1( )(1 )
Ui tR p p
LLR
with
The original function i(t) of the current is looked for.
Solution: From a transformation table one receives:
1tae
1(1 )p ap
With a it applies then to the original function: 0( ) 1tUi t e
R
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 8
FachgebietNachrichtentechnische Systeme
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3.1.3 Methods for the determination of theoriginal function from the image function
The method of decomposition into partial fractions:
It accepted that itself the image function ( )G p in a brokenrational function:
( )( )( )
Z pG pN p
Order of the nominator polynomial < Order of the denominator polynomial( )N pThe individual partial fractions have thereby the form:
1, 2,3,...( )
kA
kp p
Now for each partial fraction if the respective original function isintended, then results in itself as sum of these the original function g(t).
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 9
FachgebietNachrichtentechnische Systeme
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3.1.3 Methods for the determination of theoriginal function from the image function
The Heaviside expansion theorem:
Conditions:The denominator degree is higher than the nominator degree, The denominator polynomial has only n simple zeros(k = 1)
is a broken rational function, ( )G p
Thus it applies:1
( )( )( )
n AZ pG pN p p p
( ) ( )( ) ( ) for limiting with ( ) v
p p Z pp p G p A p pN p
1 2
1 2
( ) ( )( ) ( ) ( ) ... ...( )
n
n
p p Z p A AA Ap p G p p pN p p p p p p p p p
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 10
FachgebietNachrichtentechnische Systeme
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3.1.3 Methods for the determination of theoriginal function from the image function
• For further solution the L'Hospital rule must be used (if linear factorform is not given):
0'
'
( ) ( )( ) ( ) ( )lim lim
( )( )p p p p
d p p Z pZ p p p Z pdpA d N pN p
dp
Thus it applies:
'
( )lim( )
p p
Z pAN p
Because of the correspondence
1p p
vp te
the original function gives:1
( ) vn
p tvg t A e
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 11
FachgebietNachrichtentechnische Systeme
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3.1.3 Methods for the determination of theoriginal function from the image function
The modified Heaviside expansion theorem: In case of a pole at the origin the following applies:
1
1
( )( )( )
Z pG p
p N p 10pwith
11'
21 1
( )(0)( ) .(0) ( )
v
np tv
v
Z pZg t eN pN p
For the original function arises then:
Example:
1
( ) .1²(1 ²) 2 ²
Z
pUi tpL p p k
L
In a network is the image function of a branch current is as follows:
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 12
FachgebietNachrichtentechnische Systeme
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3.1.3 Methods for the determination of theoriginal function from the image function
• Solution:
11( ) Z p p
'1 1
1 1( ) ²(1 ²) 2 ; ( ) 2 (1 ²)²
pN p p k N p p k
Thus it applies:
322 3
2 2 3
1 1 1
( ) 1 1 12 (1 ²) 2 (1 ²)
p tp tZ
p pUi t e eL p k p k
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 13
FachgebietNachrichtentechnische Systeme
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Fundamentals of EE 3
Chapter 3.2Transient processes -
Application of the Laplacetransform
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 14
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3.2. Application of the Laplace transform
21
2
I p u 01U p I p R L pI p i 0C p p
u 01I p R pLI p I p i 0 LpC p
t
1
0 t 0
1 20
di t 1u t i t R L i z dzdt C
di t 1 1 1u t i t R L i z dz i z dz u 0 i z dzdt C C C
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 15
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3.2. Application of the Laplace transform
• Dissolve for the transform of current and voltage gives forzero-state condition:
1
11
U pI p
R pL 1 pC
U pi t L
R pL 1 pC
1U p I(p) (R pL 1 pC)
11ˆ ˆ ˆû Ri pLi i
pC
For comparison:
NW-Analyses would have given:
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 16
FachgebietNachrichtentechnische Systeme
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3.2. Application of the Laplace transform
Procedure for general solution of the network varaibles:
1. NW-analysis in the time intervall System of integro-differentialequations
2. Laplace transform of the integro-differential equation systemafter specifying the initial values at t = 0 and determination of theLaplace Transforms
3. Dissolution after Laplace transforms of the unknown variables by meansof algebra rewriting
4. Inverse transform to the desired variables.
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 17
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Fundamentals of EE 3Fundamentals of EE 3Chapter 3.3
The classical methodof directly solving
differential equations
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 18
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3.3.1 Introduction
( )Ru t
( )Ri t ( ) ( )R Ru t Ri t
( ) ( ) ( )W p t dt u t i t dt
1) Resisitor :
Repetition to network elements concerning current, voltage, energy and/or instantaneous power p(t) :
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 19
FachgebietNachrichtentechnische Systeme
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L
( )Lu t
( )Li t
( )( )
1( ) ( )
LL
t
L L
di tu t L oderdt
i t u dL
( )
magn0
1W ( ) ( ) ( )²2
Li tt t
Ldip d L i t dt L idi Li tdt
3) Capacitor :
C( )Ci t
( )Cu t
1( ) ( )
t
C Cu t i dC
( )( ) CCdu ti t C
dt
( ) 1( ) ( )²2
Cu tt
el Co
W p d C udu Cu t
2) Coil:
3.3.1 Introduction
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 20
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3.3.1 IntroductionIn the network only finite voltages and currents are possible. Thus: No instantaneous change of the energy of these elements is possible
L( )Li t Li cannot change instantaneously( is constant )Li
0( ) ( )L L for ti t t i t t 0 ( 0) ( 0) L Lt i t i t
C
( )Cu t
0( ) ( )C C for tu t t u t t 0 ( 0) ( 0) C Ct u t u t
Cucannot change instantaneouslyCu
( is constant)
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 21
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3.3.2 The method of the differential equations
- Kirchhoff‘s equations lead to differential equations- These equations have only constant coefficients- Their solutions provide the desired network varaibles- 1. Step: Solution of the system of the homogeneous differential equations
(which represent the natural oscillations of the network)- 2. Step: Add in each case an individual (particular) solution.
Thus the general solution of the system of the inhomogenousdifferential equations is obtained:
( ) ( ) ( )h pi t i t i t ( ) ( ) ( )h pu t u t u t
For a network with DC or constant sinusoidal excitation, the particulardescribe the steady-state conditions at the time t - > ∞
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 22
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3.3.2 The method of the differential equations
Example 1 : Connection of DC voltage to a "RL - circuit".
M
t = 0
i(t)
LR
=0U
At the time t = 0 the network isconnected to a source. The current i(t) iscomputed with the initial condition: ( 0) 0 i t
Solution : For t ³ 0 applies:
0( )( ) di tRi t L U
dt
The differential equation is a usual, linear differential equation of firstorder with constant coefficients, which isinhomogenous.
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 23
FachgebietNachrichtentechnische Systeme
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3.3.2 The method of the differential equations
A) Solution of the homogeneous differential equation:
( )( ) 0di tRi t Ldt
( ) ( ) 0 di t R i t
dt L
On-set: ( )( )t t
hh
di t Ki t Ke edt
1( ) 0
t t tK R Re Ke Ke
L L
with L
R
is the time constant of the circuit
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 24
FachgebietNachrichtentechnische Systeme
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3.3.2 The method of the differential equations
( ) 1 ( ) 0di t i tdt
( )R tL
hi t Ke
B) Individual (particular) solution of the inhomogenous differential equation:
After infinitely long time ( )t the current of the coil L shows its
steady-state value. Then the voltage disappears, i.e. for the current i(t) holds:
0( ) ( )pUi t f t for tR
This corresponds to the case: ( ) 0 , . : 0 L Ldi t d h udt
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 25
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3.3.2 The method of the differential equations
C) Complete solution:
0( ) ( ) ( )RtL
h pUi t i t i t KeR
Initial condition: The coil currentcannot change instantaneously
( 0) ( 0) 0 i t i t
. 0 0
10
RL UKe
R
0UK
R
For the current i(t) thus results: 0( ) (1 )RtLUi t e
R
for t > 0
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 26
FachgebietNachrichtentechnische Systeme
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3.3.2 The method of the differential equations
Remarks on the exponential function: ( )t
f t e
0.1353
0.3679
tt
5
( ) 0.0067t
f t e und e
1
t0
1
0
( )/
i tU R
2 3
After 3 to 5 the turn-on transient is practically terminated
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 27
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3.3.3 Overview
( ) ( )i t or u tL L
Differential equation in i(t)or u(t)
Individualsolution
Initial conditionsFor determining theunknown constants
Solution of thehomogeneousdifferential equation
Solution for i(t) or u(t)
Time domain
Frequencydomain
Inverse LaplaceTransformation
Laplacetransformation
ClassicalMethod
Method of theLaplacetransform
Transform gives rational, algebraic equations for
Solution to ( ) ( )i t or u tL L
Possiblyapplication of partial fractionmethod
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 28
FachgebietNachrichtentechnische Systeme
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Fundamentals of EE 3
Chapter 3.4The method of solving
differential equations bythe Laplace transform
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 29
FachgebietNachrichtentechnische Systeme
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3.4 The method of the Laplace transform• Example 1:
=0U
R L
i(t)
t = 0 Initial condition: ( 0) 0 i t
0 1( ) 1( )Ui tL p p
L
0
0
0
0
( ) 1 ( ) ( )
1 1( ) (0) ( )
1 1( )
Udi t i t tdt L
Up i t i i tL p
Ui t pL p
L L L
L L
L
0( ) 1 ( ) mit Udi t Li tdt L R
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 30
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3.4 The method of the Laplace transform
The inverse transform can be performed by means of the modifiedHeaviside` theorem:
11 1'
21 1 1
( )( ) (0)( ) ( )( ) (0) ( )
np t
v
Z pZ p ZI p i t epN p N p N p
-1L with 1 0 und 2 p n
Thus applies: '0
1 1 2 11 1( ) , ( ) , , ( ) 1 UZ p N p p p N p
L
0 01 1( ) 11 1
t tU Ui t e eL L
withLR
This gives for the current i(t):
0( ) (1 )
tUi t e
R for t > 0
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 31
FachgebietNachrichtentechnische Systeme
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3.4 The method of the Laplace transform
For the inverse transform also the convolution theorem might have been used:
1 11 for 0 1
t
t and ep p
-1 -1L L
0 0 0
0
0
1 ( ) 11
tx
t x tU U Uee dx eL L L
0 1 1( ) 1( )Ui tL p p
-1 -1L LThus it follows:
0 (1 )
tU e
R
2 1 2 1 1 20 0
( ) ( ) ( ) ( ) ( ) ( ) t t
g t g t g x g t x dx g x g t x dx 1 2 1 2( ) ( ) ( ) ( ) G p G p g t g tL
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 32
FachgebietNachrichtentechnische Systeme
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3.4 The method of the Laplace transformExample 2:
At t = 0 the capacitor ischarged to Q0i(t)
C
t = 0
M
S R
0U
0 00
00
0
( )( ) ( ) ( )
( )( )
t
t
Q tRi t U mit Q t Q i x dxC
Q i x dxi t R U
C
0 0
0
1( ) ( ) t U Qi t i x dx
RC R RC
1) 1) General relationsGeneral relations: :
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 33
FachgebietNachrichtentechnische Systeme
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3.4 The method of the Laplace transform2) 2) Solution Solution withwith thethe LaplaceLaplace transformtransform:
0 0
0
1( ) ( ) t U Qi t i x dx
RC R RC
0 0
0 0
0 0 0 0
1( ) ( ) ( )
1( ) 1
1 1( ) 11
Integration theorem
Attenuation theorem
U Qi t i t tRC p R RC
U Qi tRCp R RC p
U Q U QRCpi tR RC RCp p R RC pRC
1L L L
1L
L
0 0( )tRCU Qi t e
R RC
for t > 0 also RC
i(t)
t0
0 0QC
0
00Q UC
00
Q UC
0 0Q
C
Diagram of the results:
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 34
FachgebietNachrichtentechnische Systeme
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3.4 The method of the Laplace transformExample 3 : In this example a series resonant circuit is
connected to a source, with cases a, b and c by an ideal switch S:
a) AC voltage: 0 0ˆ( ) cos( )uu t u t
Desired for all 3 cases: i(t) for t > 0
S '
i(t)M
t = 0 R L C
0 ( )u t b) DC voltage: 0 0( )u t Uc) Mixed case: 0 0 0ˆ( ) cos( )uu t U u t
0) General relations
The loop equation reads:
0( )( ) ( ) ( )C
di tR i t L u t u tdt
and also ( )( ) Cdu ti t Cdt
applies.
0( ) ² ( ) ( ) ( ) C C C
du t d u tRC LC u t u tdt dt
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 35
FachgebietNachrichtentechnische Systeme
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3.4 The method of the Laplace transform• Solution for case 3 (D < 1) and source case b) (DC source)
S
C
LR( )Cu t
( )i t0U
t
0U0 0 0( ) ( )u t U f t
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 36
FachgebietNachrichtentechnische Systeme
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3.4 The method of the Laplace transform
The inhomogenous differential equation is subjected to the Laplace Transform:
0 0 0 0² ( ) ( )2 ² ( ) ²
²C C
Cd u t du tD u t U
dt dt for t > 0
With the initial conditions '(0) 0 (0) 0 C Cu und u
' 0 00 0²² ( ) (0) (0) 2 ( ) (0) ² ( ) C C C C C CUp u t pu u D p u t u u t
p L L L
2
2 0 00 0² ( ) 2 ( ) ( )C C C
Up u t D p u t u tp
L L L
it follows:
• Solution by means of the Laplace Transform for case 3 (D < 1) and sourcecase b) (DC source)
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 37
FachgebietNachrichtentechnische Systeme
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3.4 The method of the Laplace transform
20 0
0 0
( )( ² 2 ²)
C
Uu tp p Dp
L
11
( )( )( )
CZ pu t
pN pL
The original function can be obtained using the modified theorem ofHeaviside:
1 0 0 1 0 0( ) ² , ( ) ² 2 ² Z p U N p p Dp
10 1 2 0
3 0
( ) 2 2 , 0, ( 1 ² )
( 1 ² )
dN p p D p p D j Ddp
p D j D
with
32
3..1 1 0 0 0 0 0 0
'21 1 0 2 2 0 3 3 0
(0) ( ) ² ² ²( )(0) ( ) ² 2( ) 2( )
vp t p tp tvC
v v v
Z Z p U U Uu t e e eN p N p p p D p p D
Thus the original function results to:
( ) :Cu tThus it applies to the Laplace transform of
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 38
FachgebietNachrichtentechnische Systeme
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3.4 The method of the Laplace transform
Here only the case of conjugated complex poles (i.e. D < 1) is delt with. For the inhomogenous differential equation then holds:
t = 0
S R L
C
0 ( )u t ( )i t( )Cu t0 0 0 0
² ( ) ( )2 ² ( ) ² ( )²
C CC
d u t du tD u t u tdt dt
for t > 0, also 0 0ˆ( ) cos( )uu t u t
• Solution for case 3 (D < 1) and source case a) (AC source)
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 39
FachgebietNachrichtentechnische Systeme
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3.4 The method of the Laplace transform
Following is the differential equation for ( ) :Cu t
0 0 0 0² ( ) ( )2 ² ( ) ² ( )
²C C
Cd u t du tD u t u t
dt dt
After Laplace transforming of this equation it results:
' 0 0 0 0² ( ) (0) (0) 2 ( ) (0) ² ( ) ² ( ) C C C C C Cp u t pu u D p u t u u t u t L L L L
0 0 0 0² ( ) 2 ( ) ² ( ) ² ( )C C Cp u t D p u t u t u t L L L L
With the initial conditions: '(0) 0 (0) 0 it follows:C Cu und u
• Solution by means of the Laplace transform for case 3 (D < 1) and sourcecase a) (AC source)
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 40
FachgebietNachrichtentechnische Systeme
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3.4 The method of the Laplace transformThus it applies to the image function of ( ) :Cu t
0 00 0
² ( )( )
² 2 ²
Cu t
u tp Dp
L
L
Concerning the source it holds:
0 0 0cos sinˆ ˆ( ) cos( )
² ²
u uu
pu t u t up
L L
Combining the equations given above leads then to:
0 0 0 0cos sin
ˆ( ) ²² ² ² 2 ²
u uC
pu t u
p p Dp
L
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 41
FachgebietNachrichtentechnische Systeme
N T S
3.4 The method of the Laplace transformThe image function must then be back-transformed then into theoriginal domain e.g. with help of the method of the decompositioninto partial fractions. Here it applies:
( )( )( )
CZ pu tN p
L
and 0 0( ) ² ² ² 2 ² N p p p Dp
with 0 0ˆ( ) ( cos sin ) u uZ p u p
Due to the periodic case (D < 1) it applies:
1 2, p j p j
3,4 1 0 ( 1 ² ) p j D j D
For the first derivative of the denominator it applies: 3
0 0 0( ) 4 6 ² 2( ² ²) 2 ² dN p p Dp p D
dp
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Prof. Dr.-Ing. I. Willms Fundamentals of EE 3 S. 42
FachgebietNachrichtentechnische Systeme
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3.4 The method of the Laplace transformThus finally for the original function holds:
0 0 3 30 0 0 0
cos sin os sinˆ( ) ²4( ) 6 ( )² 2( ² ²) 4( ) 6 ( )² 2( ² ²)( )
j t j tu u u uC
j j cu t u e ej D j j j D j j
1 21 23 2 3 21 0 1 0 1 0 2 0 2 0 2 0
cos sin cos sin4 6 2( ² ²) 2 ² 4 6 2( ² ²) 2 ²
pt p tu u u up pe ep Dp p D p Dp p D
Final problem:
Further simplification to real values is quite complicated !