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Theory of electric networks: The two-point resistance and impedance
F. Y. Wu
Northeastern UniversityBoston, Massachusetts USA
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Z
Impedance network
?Z
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Ohm’s law
Z
V
I
I
VZ
Combination of impedances
1z 2z
1z
2z
21 zzz
21
111
zzz
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In the phasor notation, impedance for inductance L is
Ljz
Impedance for capacitance C is
Cjz /1
where 1j .
Impedances
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=
-Y transformation: (1899)
z
zzZ Z
Z Zz
ZZ
ZZz
3
12
)2(2
Star-triangle relation: (1944)
1
32
Ising model
J
JJR
R
R
1
2 3
=
)()( 133221321
RJ Fee =
)(cosh2 321 J
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1r
2r3r2R
3R
1R
-Y relation (Star-triangle, Yang-Baxter relation)
A.E. Kenelly, Elec. World & Eng. 34, 413 (1899)
321
321 RRR
RRr
321
132 RRR
RRr
321
213 RRR
RRr
133221
321321
11 111
111)(
1
rrrrrr
rrrrrr
rR
133221
321321
22 111
111)(
1
rrrrrr
rrrrrr
rR
133221
321321
33 111
111)(
1
rrrrrr
rrrrrr
rR
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1
2 3
4
z1 z1
z1
z1
z2
?13 Z
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1
2 3
4
z1 z1
z1
z1
z2
3
1
2 3
1
3
1
113 zZ ?13 Z
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1
2 3
4
r1 r1
r1
r1
r2
3
1
2 3
1
3
1113 rR
?13 R
I
I/2I/2
I/2
I/2I
1
2 3
4
r1 r1
r1
r1
r2
112
112
1
13 rI
IrIrR
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1
2
r
r
r
r
r
r
r
r
r
r
r
r
?12 R
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1
2
r
r
r
r
r
r
r
r
r
r
r
r
rI
VR
IrrI
rI
rI
V
6
56
5
363
1212
12
I
I/3
I/3
I/3
I/3
I
1
2
r
r
r
r
r
r
r
r
r
r
r
r
?12 R
I/3
I/6I/6
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Infinite square network
I/4I/4
I/4I/4
I
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V01=(I/4+I/4)r
I/4I/4
I/4I/4
I
I
I/4
201
01
r
I
VR
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Infinite square network
2
1
38
2
1
2
24
2
17
4
2
4
2 2
14
2
14
0
24
2
17 4
3
46
43
46
823
2
1
3
4
2
1
3
4
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Problems:
• Finite networks• Tedious to use Y- relation
1
2
r
rR
)7078.1(
027,380,1
898,356,212
(a)
(b) Resistance between (0,0,0) & (3,3,3) on a 5×5×4 network is
r
rR
)929693.0(
225,489,567,468,352
872,482,658,687,327
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I0
1
4
3
2 Kirchhoff’s law
z01z04
z02
z03
04
40
03
30
02
20
01
10
040302010
z
VV
z
VV
z
VV
z
VV
IIIII
Generally, in a network of N nodes,
N
ijjji
iji VV
zI
,1
1
Then set )( iii II I
VVZ
Solve for Vi
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2D grid, all r=1, I(0,0)=I0, all I(m,n)=0 otherwise
I0
(0,0)
(0,1) (1,1)
(1,0)
00,0,),(4)1,()1,(),1(),1( InmVnmVnmVnmVnmV nm
Define
)1()(2)1()(
)1()()(2
nfnfnfnf
nfnfnf
n
n
Then 00,0,22 ),()( InmV nmnm
Laplacian
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Harmonic functionsRandom walksLattice Green’s functionFirst passage time
• Related to:
• Solution to Laplace equation is unique
• For infinite square net one finds
2
0
2
02 2coscos
)(exp
)2(2
1),(
nmiddnmV
• For finite networks, the solution is not straightforward.
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General I1 I2
I3
N nodes
,1
,1,1
N
ijjjijii
N
ijjji
iji VYVYVV
zI
ijij
N
ijj iji z
Yz
Y1
,1
,1
NNNNN
N
N
I
I
I
V
V
V
YYY
YYY
YYY
2
1
2
1
21
2221
1121
The sum of each row or column is zero !
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Properties of the Laplacian matrix
All cofactors are equal and equal to the spanning tree generating function G of the lattice (Kirchhoff).
Example1
2 3
y3
y1
y2 G=y1y2+y2y3+y3y1
2112
1313
2332
yyyy
yyyy
yyyy
L
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Spanning Trees:
x
x
x
x
xx
y y
y
y
y
y
y
y
xS.T all
21),( nn yxyxG
G(1,1) = # of spanning trees
Solved by Kirchhoff (1847) Brooks/Smith/Stone/Tutte (1940)
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1
4
2
3x
x
y y G(x,y)= +
x
x
x
xx
x
+ +yyyy y y
=2xy2+2x2y
yxxy
xyxy
yyxx
yxyx
yxL
0
0
0
0
),(
1 2 3 4
1
2
3
4
LN
yxG of seigenvalue nonzero ofproduct 1
),(
N=4
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General I1 I2
I3
N nodes
,1
,1,1
N
ijjjijii
N
ijjji
iji VYVYVV
zI
ijij
N
ijj iji z
Yz
Y1
,1
,1
NNNNN
N
N
I
I
I
V
V
V
YYY
YYY
YYY
2
1
2
1
21
2221
1121
The sum of each row or column is zero !
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I2I1
IN
network
Problem: L is singular so it cannot be inverted.
Day is saved:
Kirchhoff’s law says 01
N
jjI
Hence only N-1 equations are independent → no need to invert L
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NNNNN
N
N
I
I
I
V
V
V
YYY
YYY
YYY
2
1
2
1
21
2221
1121
Solve Vi for a given I
Kirchhoff solutionSince only N-1 equations are independent, we can set VN=0 & consider the first N-1 equations!
1
2
1
1
2
1
12,11,1
1,2221
1,1121
NNNNN
N
N
I
I
I
V
V
V
YYY
YYY
YYY
The reduced (N-1)×(N-1) matrix, the tree matrix, now has an inverse and the equation can be solved.
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0
0
131211
131211
131211
zcycxc
zbybxb
Izayaxa
0
0
333231
232221
131211 I
z
y
x
aaa
aaa
aaa
333231
232221
131211
aaa
aaa
aaa
3332
2322
1312
0
0
aa
aa
aaI
x
3331
2321
1311
0
0
aa
aa
aIa
y
0
0
3231
2221
1211
aa
aa
Iaa
z
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Example1
2 3
y3
y1
y2
2112
1313
2332
yyyy
yyyy
yyyy
L
133221211
1311 yyyyyy
yyy
yyyL
2112 yyL
133221
21
1
1212 yyyyyy
yy
L
Lz
32112
111
zzzz
or
The evaluation of L & L in general is not straightforward!
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)( iii II
I I Kirchhoff result:
Writing
L
LZ
Where L is the determinant of the Laplacian with the -th row & column removed.
L= the determinant of the Laplacian with the -th and -th rows & columns removed.
But the evaluation of Lfor general network is involved.
trees)spanning(
) and rootsh forest wit (spanning
G
G
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NNNNN
N
N
I
I
I
V
V
V
YYY
YYY
YYY
2
1
2
1
21
2221
1121
)(
)(
)(
For resistors, z and y are real so L is Hermitian, we can then consider instead the eigenvalue equation
Solve Vi () for given Ii and set =0 at the end.
This can be done by applying the arsenal of linear algebra and deriving at a very simple result for 2-point resistance.
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Eigenvectors and eigenvalues of L
1
1
1
0
1
1
1
21
2221
1121
NNN
N
N
YYY
YYY
YYY
L
0 is an eigenvalue with eigenvector
1
1
1
1
N
L is HermitianL has real eigenvaluesEigenvectors are orthonormal
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IGV
IVL
)()(
)()(
Consider
where1)]([)( LG
i
i
2i
1 :)( of sEigenvalue
:)( of sEigenvalue
, ,0 :)0( of sEigenvalue
G
L
L
LLet
This gives
N
i i
ii
NG
2
*1
)(
Z and
0 sinceout drops 1
Term i
iIN
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Example
1
2 3
4
r1 r1
r1
r1
r2
21121
111
21211
111
2
20
2
02
ccccc
ccc
ccccc
ccc
L
)1,0,1,0(2
1 ),(2
)0101(2
1 ,2
)1111(2
1 ,4
3214
313
212
cc
,,,c
,,,c
)(4
)32()(
1)(
1)(
1
)(1
)(1
)(1
21
21124441
4
23431
3
22421
214
12
43414
23331
3
22321
213
rr
rrrr
rr
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0 and
,,2,1
1
Ni
L iii
For resistors let
iN
i
i
i
2
1
= orthonormal
Theorem for resistor networks:
2
2
1
ii
N
i i
R
This is the main result of FYW, J. Phys. A37 (2004) 6653-6679 whichmakes use of the fact that L is hermitian and is orthonormal
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Corollary:
)1
)((1 2
22
ii
N
i i
N
iiN
) and rootsh forest wit spanning( G
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Example: complete graphs
111
111
111
1
N
N
N
rL
N=3
N=2
N=4
110
121 ),/2exp(1
121 ,
,00
,N-,,α
,,N-,,nNniN
,N-,,nr
N
n
n
rN
R2
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1 2 3 N-1r rr r N
100
021
011
1
rL
r
Nn
Nn
Nn
N
rR
N
i
1
1
2
cos1
)21
cos()21
cos(
N
n
N
N
N
n
n
n
)2
1cos(
2
1
cos12
0
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If nodes 1 & N are connected with r (periodic boundary condition)
][ /1
2cos1
/2exp/2exp
2
1
1
2Per
Nr
Nn
NniNni
N
rR
N
iαβ
NniN
N
n
n
n
/2exp1
2cos12
201
021
112
1
rL
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New summation identities
1
0 coscos
cos1
)(N
n
Nn
Nn
l
NlI
NlNN
lNlI
l
20 ,2/cosh4
)1(1
sinh
11
sinhsinh
)cosh()(
221
NlN
lNlI
0 ,
2/sinhsinh
)2/cosh()(2
New product identity
2sinh
2coscosh
1
0
2 N
N
nN
n
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M×N network
N=6
M=5
r
s
sNMNMNM TIs
ITr
L 11
1000
0210
0121
0011
NT IN unit matrix
1,,2,1 ,2
1cos
2
0 ,1
2cos1
22cos1
2
)(
)()(),(
),(
NN
n
N
N
N
n
rM
m
s
Nn
Nn
Mmnm
nm
s
r
rr
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M, N →∞
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Resistance between two corners of an N x N
square net with unit resistance on each edge
2ln
2
141ln
4NRNxN
......082069878.0ln4
N
where ...5772156649.0 Euler constant
N=30 (Essam, 1997)
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Finite lattices
Free boundary condition
Cylindrical boundary condition
Moebius strip boundary condition
Klein bottle boundary condition
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Klein bottleMoebius strip
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Free
Cylinder
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Klein bottle
Moebius strip
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Klein bottle
Moebius strip
Free
Cylinder
Torus
)3,3)(0,0(R on a 5×4 network embedded as shown
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Resistance between (0,0,0) and (3,3,3) in a 5×5×4 network with free boundary
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In the phasor notation, impedance for inductance L is
Ljz
Impedance for capacitance C is
Cjz /1
where 1j .
Impedances
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For impedances, Y are generally complex and the matrixL is not hermitian and its eigenvectors are not orthonormal; the resistor result does not apply.
But L^*L is hermitian and has real eigenvelues.We have
N1,2,..., , 0 , ^* α LL
with
.
1
:
:
1
1
N
1 0, 11
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N1,2,..., , 0 , ^* α LL
Theorem
Let L be an N x N symmetric matrix with complexMatrix elements and
Then, there exist N orthnormal vectors u
satisfying the relation
NuuL ,...,2,1 *,
where * denotes complex conjugate and
real. ,
ie
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Remarks:
For nondegenerate one has simply
u
For degenerate
,
, one can construct
as linear combinations of u
ieLv *)(
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0 and
,,2,1
*
1
Ni
uLu iii
For impedances let
iN
i
i
i
u
u
u
u
2
1
= orthonormal
Theorem for impedance networks:
2,0 if ,
2 ,0 if ,)(1 2
2
i
iuuZ
i
iii
N
i i
This is the result of WJT and FYW, J. Phys. A39 (2006) 8579-8591
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The physical interpretation of Z
is the occurrence of a resonance such asin a parallel combination of inductance Land capacitance C the impedance is
Z =
Lj
Cj
Cj
Lj
))((
LC/1at , =
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Generally in an LC circuit there can exist multiple resonances at frequencies where . 0i
In the circuit shown, 15 resonance frequencies at
M=6N=4
.3,..,1;5,..,1 ,1
)2/sin(
)2/sin( nm
LCMm
Nnmn
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Summary
• An elegant formulation of computing two-point impedances in a network, a problem lingering since the Kirchhoff time.
• Prediction of the occurrence of multi-resonances in a network consisting of reactances L and C, a prediction which may have practical relevance.
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FYW, J. Phys. A 37 (2004) 6653-6673
W-J Tzeng and FYW, J. Phys. A 39 (2006), 8579-8591
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