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![Page 1: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/1.jpg)
Circuits TheoryCircuits TheoryExamplesExamples
Newton-Raphson Method
![Page 2: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/2.jpg)
Formula for one-dimensional case:
Series of successive solutions:
If the iteration process is converged , the limit is the solution of the equationf(x)=0.
)k()k()k()k( xfx'fxx11
,...x,x,x )()()( 210
0)x(f
![Page 3: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/3.jpg)
0xf )(Multidimensional case:
)k()k()k()k( xfxJxx11
where:
n
nnn
n
n
x)(f
x)(f
x)(f
x)(f
x)(f
x)(f
x)(f
x)(f
x)(f
xxx
xxx
xxx
xJ
21
2
2
2
1
2
1
2
1
1
1
)()k(
kxxxJxJ
JACOBIANMATRIX
![Page 4: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/4.jpg)
ALGORALGORITHMITHM
STEP 0 )o(0k x STARTING POINT
STEP 1 )k()k( , xJxfCalculate
STEP 2 Solve the equation:
)()()()1()1()()(
)()1()(
,, kk xfbxxyxJA
byA
kkkkk
kkk
STEP 3 find )()1()1( kkk xyx
check STOP conditions
If the current solution is not acceptable:1kk
GO TO 1
![Page 5: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/5.jpg)
EXAMPLE of STOP PROCEDUREEXAMPLE of STOP PROCEDURE
1
1 )( ky
2
1 )( kxf
NNoo
NoNo
k=k+1GOTO 1
YesYes
YesYes
*)1k( xx STOP
![Page 6: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/6.jpg)
• Stop condition parameter 1
1
212
1
1
1
1 )k(
n
k
n
)k(kk xx...xxy
• Stop condition parameter
2
2121
11 k
nn
k xf...xf1kxf
2
![Page 7: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/7.jpg)
Numerical EXAMPLESNumerical EXAMPLES
Example 1
![Page 8: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/8.jpg)
Solve the following set of nonlinearequation using the Newton’s Method:
02
0143
01023
2
3
3
21
3
2
2
3
1
32
2
1
xxx
xxx
xxx
T321 xxx,)( x0xf
![Page 9: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/9.jpg)
2
143
1023
2
3
3
213
3
2
2
3
12
32
2
11
xxx)(f
xxx)(f
xxx)(f
x
x
x
Starting point (first approximation):
T)0( 111x
T)0( 194)(f xCalculate:
![Page 10: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/10.jpg)
)()(' 0xx
0 xJxf
1x1x1x
322
221
1
3
2
1x2x31
3x2x3
12x6
231
323
126
)0()1()( xfyxJ 0 where: )0()1()1( xxy
![Page 11: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/11.jpg)
1
9
4
231
323
126
)1(3
)1(2
)1(1
y
y
y
123
9323
426
)1(3
)1(2
)1(1
)1(3
)1(2
)1(1
)1(3
)1(2
)1(1
yyy
yyy
yyy (1a)
(1b)
(1c)
![Page 12: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/12.jpg)
3)0(
333)0(
232)0(
131
2)0(
323)0(
222)0(
121
1)0(
313)0(
212)0(
111
byayaya
byayaya
byayaya
(1a)
(1b)
(1c)
Let us assume )0(yy
3333232131
2323222121
1313212111
byayaya
byayaya
byayaya
(1a)
(1b)
(1c)
![Page 13: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/13.jpg)
Gauss elimination computer scheme
STEPSTEP 1 ELIMI 1 ELIMINATE NATE y y11 fromfrom b i c b i c:
1y2y3y
9y3y2y3
4yy2y6
321
321
321
Multiply by
and add to 1b63
aa
11
21
![Page 14: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/14.jpg)
7y25
y
9y3y2y3
2y21
yy3
32
321
321
![Page 15: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/15.jpg)
1y2y3y
9y3y2y3
4yy2y6
321
321
321
Multiply by
and add to 1c61
61
a
a
11
31
31
y6
11y
310
1y2y3y32
y61
y31
y
32
321
321
![Page 16: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/16.jpg)
New set : )()( 22 byA
)2(33
)2(332
)2(32
)2(23
)2(232
)2(22
)2(13
)2(132
)2(121
)2(11
byaya
byaya
byayaya
(2a)
(2b)
(2c)
31
y6
11y
310
7y25
y1
4yy2y6
32
32
321
(2a)
(2b)
(2c)
![Page 17: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/17.jpg)
31
y6
11y
310
7y25
y1
4yy2y6
32
32
321
(2a)
(2b)
(2c)
Elimination scheme repeat for equations 2b i 2c:
Multiply by
add o 2c1
3/10a
a)2(
22
)2(32
370
y325
y310
32
371
y661
3
![Page 18: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/18.jpg)
)3()3( byA
)3(33
)3(33
)3(23
)3(232
)3(22
)3(13
)3(132
)3(121
)2(11
bya
byaya
byayaya
(3a)
(3b)
(3c)
371
y661
7y25
y1
4yy2y6
3
32
321
(3a)
(3b)
(3c)
![Page 19: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/19.jpg)
Back substitution part:
328.261
142ab
y )3(33
)3(3
3
Setting y3 to 3b:
61142
y
7y25
y1
4yy2y6
3
32
321
Multiply by
add to 3b
25
a )3(23
![Page 20: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/20.jpg)
61142
y
6172
y
4yy2y6
3
2
321
1a )3(13
2a )3(12
328.2
180.1
115.0
y
y
y
100
010
001
3
2
1
![Page 21: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/21.jpg)
3283
1802
8150
0
3
1
3
0
2
1
2
0
1
1
1
.
.
.
xy
xy
xy
)()(
)()(
)()(
)( 1x
Because )()()( 011 xxyy
It is the first calculated approximation of the solution.Next iterations form a converged series:
006.3
010.2
002.1)2(x
3
2
1)3(x *)4(
3
2
1
xx
![Page 22: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/22.jpg)
ExampleExample 2 2
Nonlinear circuit having two variables (node voltages)
![Page 23: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/23.jpg)
R 3
VS 3
R2
j1 j
4
5i
6iv
5v
6
1
2
i 3
i2
e1
e2
![Page 24: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/24.jpg)
Data:
1)(
)(
6666
25
35555
kvedvgi
cbvavvgi
VkAd
AcV
Ab
V
Aa
VvAjAj
RR
S
11,1
,1,1,1
,3,4,1
,3,2
23
341
32
![Page 25: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/25.jpg)
R 3vS 3
R2
j1 j
4
5i
6iv
5v
6
1
2
Nodal equations:
013
312215
2
1
jR
veeeeg
R
e S1
2 04
3
31226215
j
R
veeegeeg S
![Page 26: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/26.jpg)
013
312215
2
1
jR
veeeeg
R
e S
043
31226215
j
R
veeegeeg S
Jacobian matrix:
2
)(2)(31
)(2
)(31
)(2)(31
)(2)(3
11
)(
212
213
21
221
3
212
213
212
21
32
kedke
eebeeaR
eeb
eeaR
eebeeaR
eebeea
RR
eJ
![Page 27: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/27.jpg)
We choose starting vector:
0
0)0(e
4
1)( )0(ef
Calculate:
333.1333.0
333.0833.0)( )0(eJ
![Page 28: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/28.jpg)
Applying N-R scheme:
)0()1()( efyeJ 0 where: )0()1()1( eey
4
1
333.1333.0
333.0833.0)1(
2
)1(1
y
y
hence:
6673
66721
2
1
1
.
.
y
y)(
)(
667.3
667.2)0(
2)1(
2
)0(1
)1(1
)1(2
)1(1
ey
ey
e
e
![Page 29: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/29.jpg)
STOP CRITERIA not satisfied:
1211
1 100010 .,.,y )(
455.34
0)( )1(ef
k=k+1:
455.40333.1
333.1833.1)( )1(eJ
![Page 30: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/30.jpg)
Second NR iteration
)1()2()1( efyeJ
where:)1()2()2( eey
46311
10
455403331
333183312
2
2
1
.
.
y
y
..
..)(
)(
hence:
8730
63502
2
2
1
.
.
y
y)(
)(
794.2
032.2)1(
2)2(
2
)1(1
)2(1
)2(2
)2(1
ey
ey
e
e
![Page 31: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/31.jpg)
for k=7: )6()7()6( efyeJ where: )6()7()7( eey
002.0
003.0
105.6885.0
718.0225.1)7(
2
)7(1
y
y
hence:
47
2
7
1
101811
0010
.
.
y
y)(
)(
629.1
807.1)6(
2)6(
2
)6(1
)6(1
)7(2
)7(1
ey
ey
e
e
![Page 32: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/32.jpg)
Because:
6
6)(
10653.2
10689.2)( 7ef
629.1
807.1*2
*1
)7(2
)7(1
e
e
e
e
2
6
272
712
272
711
7
10777.3
,,
eefeefef
![Page 33: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/33.jpg)
Briefly about:Briefly about:
Iterative models of nonlinear elements
![Page 34: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/34.jpg)
Iterative NR model of nonlinear resistor (voltage Iterative NR model of nonlinear resistor (voltage controled)controled)
vfi i i
v v
![Page 35: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/35.jpg)
circuit
11 '' kkkkkk vvfvvfii
ki~ kG kkkk vvfii '~ kk vfG '
From NR method:From NR method:
![Page 36: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/36.jpg)
Model iterowany opornika (Model iterowany opornika (66))
11 ~ kkkk vGii
ki~
kG 1ki 1kv
![Page 37: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/37.jpg)
ExampleExample 3 3
Newton-RaphsonNewton-Raphson
Iterative model method
![Page 38: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/38.jpg)
R 3
VS 3
R2
j1 j
4
5i
6iv
5v
6
1
2
i 3
i2
e1
e2
![Page 39: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/39.jpg)
Data:
1)(
)(
6666
25
35555
kvedvgi
cbvavvgi
VkAd
AcV
Ab
V
Aa
VvAjAj
RR
S
11,1
,1,1,1
,3,4,1
,3,2
23
341
32
![Page 40: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/40.jpg)
Scheme for (k+1) iterationScheme for (k+1) iteration
v S3
R 2
R3
j1 j4
v 5(k+1)
i5(k+1)
G5
(k)
(k)i5~ (k)
i6~ G 6
(k)
v 6(k+1)
i 6(k+1)
1
2
11
ke 12
ke
![Page 41: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/41.jpg)
0~51
3
31
11
2
5
12
11
2
11
k
Skk
k
kkk
ij
R
vee
R
ee
R
e1
v S3
R 2
R3
j1 j4
v 5(k+1)
i5(k+1)
G5
(k)
(k)i5~ (k)
i6~ G 6
(k)
v 6(k+1)
i 6(k+1)
1
2
11
ke 12
ke
![Page 42: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/42.jpg)
2
0~~654
3
31
11
2
6
12
5
12
11
kk
Skk
k
k
k
kk
iij
R
vee
R
e
R
ee
v S3
R 2
R3
j1 j4
v 5(k+1)
i5(k+1)
G5
(k)
(k)i5~ (k)
i6~ G 6
(k)
v 6(k+1)
i 6(k+1)
1
2
11
ke 12
ke
![Page 43: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/43.jpg)
0~51
3
31
11
2
5
12
11
2
11
k
Skk
k
kkk
ij
R
vee
R
ee
R
e1
2
0~~654
3
31
11
2
6
12
5
12
11
kk
Skk
k
k
k
kk
iij
R
vee
R
e
R
ee
![Page 44: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/44.jpg)
3
351
53
12
352
11
~
11111
R
vij
RRe
RRRe
Sk
kk
kk
1
2
3
3
654
356
12
53
11
11
11111
R
v
RRj
RRRe
RRe
Skk
kkk
kk
![Page 45: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/45.jpg)
• For starting vector:
0
00
2
0
10
v
v)(v
0' 02
015
05 eegG
02
01
05 eev
1'~ 05
05
05
05 vvfii
055
05 vgi
• We calculate parameters of the models:
![Page 46: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/46.jpg)
0' 026
06 egG
02
06 ev
0'~ 06
066
06
06 vvgii
066
06 vgi
• For nonlinear element g6:
![Page 47: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/47.jpg)
Linear equations for the first approximationLinear equations for the first approximation::
4
1
333.1333.0
333.0833.01
2
11
e
e
667.3
667.2)1(
2
)1(1
e
e
Solution for k=1=i5
x1y11
![Page 48: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/48.jpg)
Second stepSecond step
371.110
0
468.40333.1
333.1833.12
2
21
e
e
794.2
032.2)2(
2
)2(1
e
e
Solution for k=2
![Page 49: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/49.jpg)
Briefly about:Briefly about:
Forward Euler Method (Explicit)
Backward Euler Method (Implicit)
![Page 50: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/50.jpg)
Forward Euler Method (Explicit)
),( 111 kkkk txfhxx
Backward Euler Method (Explicit)
),(1 kkkk txfhxx
![Page 51: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/51.jpg)
Backward Euler Method (Explicit) is based on the following Taylor series expansion
2
1
hdt
dxhtx
htxx
ktk
kk
),(1 kkkk txfhxx
![Page 52: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/52.jpg)
EX A M PLE .
E
L
CR
i (t)L
t=0
Cu (t)v (t)
Cvs
HL
FC
R
VvVv CS
25.0
1
200
200)0(,100
![Page 53: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/53.jpg)
eudt
diL C
L (1)
R
u
dt
duCi CC
L (2)
State vector:
2
1
L
C
x
x
i
ux (3)
![Page 54: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/54.jpg)
f r o m ( 1 ) a n d ( 2 ) :
SCL
LCc
vL
vLdt
di
iC
vRCdt
du
11
11
( 4 )
o r :
SvLx
x
L
CRC
1
0
01
11
2
1x ( 5 )
![Page 55: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/55.jpg)
,.
L
CRC
04
101050
01
1164
A
( 6 )
VvvL
S 100,4
010
B ( 7 )
I n i t i a l c o n d i t i o n s :
5.0
2001
200
0
0
0
0
2
1
Rvi
u
x
x
sL
Cox . ( 8 )
![Page 56: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/56.jpg)
F E M :
11
11 )(
kk
kkk
h
fh
xx
xxx
![Page 57: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/57.jpg)
Dla n=1, korzystając z wzoru (9) i uwzględniając, że V200u0u )0(CC i A5.0i)0(i )0(LL otrzymamy:
eL
1u
L
1hii
iC
1u
RC
1huu
)0(C)0(L)1(L
)0(L)0(C)0(C)1(C
(11)
![Page 58: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/58.jpg)
F o r t h e s t e p h = 1 0 - 4 :
46.0400800105.0
15010
5.0
10200
20010200
4)1(
664
)1(
L
C
i
v
N e x t s t e p k = 2 :
44.04006001046.0
12110
46.0
10200
15010150
4)2(
664
)2(
L
C
i
v
![Page 59: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/59.jpg)
vC(tk)
![Page 60: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/60.jpg)
iL(tk)
Table for 10 iterations n 0 1 2 3 4 5 6 7 8 9 vC
200 150 121 104.5 95.41 90.685 88.506 87.789 87.891 88.429
iL 0.5 0.46 0.44 0.4316 0.4298 0.4316 0.4354 0.4399 0.4448 0.4497
![Page 61: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/61.jpg)
B E M :
n1n
n1nn
h
)(fh
xx
xxx
SkCkLkL
kLkCkCkC
vL
vL
hii
iC
vRC
huv
11
11
)()1()(
)()()1()(
![Page 62: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/62.jpg)
O r i n m a t r i x f o r m :
)e(h
)(h
n1n
nn1nn
BAxx
BuAxxx
f o r k = 1 , s e t t i n g : Vvv CC 2000 )0(
Aii LL 5.0)0( )0(
4004105.0
101020010200
)1(4
)1(
6)1(
6)1(4
)1(
CL
LCC
vi
ivv
solving in terms of )1(Cv and )1(Li :
935.164)1( Cv 474.0i )1(L
![Page 63: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/63.jpg)
h=0.0001.
vc(tk)
![Page 64: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/64.jpg)
iL(tk)
![Page 65: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/65.jpg)
Example with nonlinear capacitorExample with nonlinear capacitor
• FEM
211
11
RRv
R
v
dt
dqc
S
0)0(,2,1,10 221 qqvRRVv cS
2410 qq
)410( 211
kqhqqkk
![Page 66: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/66.jpg)
FEM stepsFEM steps
110*1.0)410( 2001
qhqq
6.1)410(1.01)410( 2112
qhqq
5811.1)410( 2889
qhqq
![Page 67: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/67.jpg)
BEM step 1BEM step 1
)410( 2101
qhqq
0
)410(
11
11011
2
qfequationnonlinearofsolution
qqhqqf
0
)410(
1
01
2
xfequationnonlinearofsolution
xxhqxf
7655644.01
q
![Page 68: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/68.jpg)
00 x
Using N-R method with starting point Using N-R method with starting point
)1(1)1()1()(11
' kkkk xfxfxx
1101.010 1)1( x
12)1(
011)1()( 4101.018.0 kkkkk xxqxxx
7777.0)161.0(18.01 1)2( x
7656.0)4( x
00 x
![Page 69: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/69.jpg)
BEM step 2BEM step 2
)410( 2212
qhqq
0
)410(
22
22122
2
qfequationnonlinearofsolution
qqhqqf
0
)410(
2
12
2
xfequationnonlinearofsolution
xxhqxf
1947.12
q
after N-R procedure with new starting point
![Page 70: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/70.jpg)
Using N-R method with starting point Using N-R method with starting point
2403.1)1( x
1947.1)4( x
7656.00 x
02)0(1
10)0()1( 410(1.018.0 xxqxxx
![Page 71: Circuits Theory Examples Newton-Raphson Method. Formula for one-dimensional case: Series of successive solutions: If the iteration process is converged,](https://reader035.fdocuments.net/reader035/viewer/2022062305/5697bf931a28abf838c8f579/html5/thumbnails/71.jpg)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6