B.tsoniff Analytical Approach to Calculation of Eigenvectors and Eigenvalues for 5x5 Impedance Matrix

8/18/2019 B.tsoniff Analytical Approach to Calculation of Eigenvectors and Eigenvalues for 5x5 Impedance Matrix

1/23

111Equation Chapter 1 Section 1

Dynamic evaluation of eigenvalues depending on the

parameter

Analytical Approach to Calculation of Eigenvectors

and

Eigenvalues for 5x5 Impedance Matrix

Mathematical background.

(y !"#soni$%

Introduction

#he three phase line &ith t&o grounded shield &ires algeraically descried as

1 1 2 2

1 1 2 2

1 1 2 2

1 1 1 1 1 1 1 2 2

2 2 2 2 1 1 2 2 2

0

0

A AA A AB B AC C AN N AN N

B BA A BB B BC C BN N BN N

C CA A CB B CC C CN N CN N

N A A N B B N C C N N N N N N

N A A N B B N C C N N N N N N

V Z I Z I Z I Z I Z I



Z I Z I Z I Z I Z I

Z I Z I Z I Z I Z I

= + + + +

= + + + +

= + + + +

= + + + +

= + + + +

'1')

ME*+E,-*MA# ("%

or. in matrix form


2/23

1 2

1 2

1 2

11 1 1 1 1 1 2

22 2 2 2 1 2 2

0

0

AA AB AC AN AN A A

BA BB BC BN BN B B

C CA CB CC CN CN C

N N A N B N C N N N N


Z Z Z Z Z V I

Z Z Z Z Z V I

V Z Z Z Z Z I

I Z Z Z Z Z

I Z Z Z Z Z

÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ = × ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷

/1/)

ME*+E,-*MA# ("%

In the impedance matrix

1 2

1 2

1 2

1 1 1 1 1 1 2

2 2 2 2 1 2 2

AA AB AC AN AN

BA BB BC BN BN

CA CB CC CN CN

N A N B N C N N N N

N A N B N C N N N N

Z Z Z Z Z

Z Z Z Z Z

Z Z Z Z Z

Z Z Z Z Z

Z Z Z Z Z

÷ ÷ ÷= ÷ ÷

÷ ÷

0

1)

ME*+E,-*MA# ("%

the diagonal elements ( self2impedance terms% in according to Carson

correction terms formulae. depend on the 3nite Earth resistance R" In

general. it may e said that matrix 0and hence its eigenvaluesparametrically depend on the Earth resistance" 4ynamic evaluation of

eigenvalues depending on the parameter variation is a fundamental

prolem" Mean&hile. dependence of the eigenvalues on the parameters

variation cannot e otained y numerical methods. &hich are generally

not e$ective for prolems &ith the parameters" In such circumstances.

the possiility of an analytical representation of the eigenvalues may e a

decisive factor in the successful solution of the prolem"

Reduction of matrix dimension (theory)


3/23

In the set of equations (1"1% the last t&o equations are homogeneous

(free terms equal 6ero%" In this particular case the numer of equations

may e reduced y the numer of homogeneous equations" #he matrix equation (1"'% may e represented in the loc7 form8

1 2

1 2

1 2

11 1 1 1 1 1 2

22 2 2 2 1 2 2

0

0

AA AB AC AN AN A A

BA BB BC BN BN B B

C CA CB CC CN CN C



Z Z Z Z Z I V

Z Z Z Z Z I V

I Z Z Z Z Z V

I Z Z Z Z Z

I Z Z Z Z Z

÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷= × ÷ ÷ ÷ ÷ ÷ ÷ ÷

÷ ÷ ÷ ÷ ÷

M

M

M

9 : : : : : : M: : :

M

M

515)

ME*+E,-*MA# ("%

Introducing

A

abc B

C

V

V V V

÷

= ÷ ÷

0

00 = ÷

A

abc B

C

I

I I I

÷

= ÷ ÷

1

2

N g

N

I I

I = ÷ ÷

AA AB AC

BA BB BC

CA CB CC

Z Z Z

Z Z Z

Z Z Z

A

÷= ÷ ÷

1 2

1 2

1 2

AN AN

BN BN

CN CN

Z Z

B Z Z

Z Z

÷

= ÷ ÷

1 1 1

2 2 2

N A N B N C

N A N B N C

Z Z Z C

Z Z Z

= ÷ ÷

1 1 1 2

2 1 2 2

N N N N

N N N N

Z Z D

Z Z

= ÷ ÷

&rite do&n loc7 matrix in the form

0

abcabc

g

I V A B I C D

= × ÷ ÷ ÷

;1;) ME*+E,-*MA# ("%


4/23

After matrix multiplication

abc abc g V A I B I = × + ×


5/23

&here A is an( )n n×

complex valued matrix. x and b are vectors in the

complex vector space

nC " Assume that the b vector has t&o 6ero

elements in the last t&o ro&s" #hen (1"% may e &ritten as

........

.......

..

111 12 1n 1

21 22 2n 2 2

n-1,1 n-1,2 n-1,n n-1

nnnn1 n2

ba a ...... a xa a ...... a x b

a a .......... a x 0

x 0a a .............a

÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷

× =M M - M M

1/11/)

ME*+E,-*MA# ("%

#he last t&o equations are constraints y de3nition li7e any homogeneous

equations (free terms equal 6ero%" In this case the order of matrix A may

e reduced to( ) ( )2 2n n− × −

y eliminatingn-1

x and

n x using the

constraint equations

nn-1,1 1 n-1,2 2 n-1,na x + a x .......a x =0 111)ME*+E,-*MA# ("%

and

n,n nn,1 1 n,2 2a x + a x .......a x =0

15115)

ME*+E,-*MA# ("%

*educing the order of the matrix &ill e given in t&o steps8 in the 3rst

step &e eliminaten x . then

n-1 x

" +enerally spea7ing. the elements of

the vector x can e eliminated in any order. since the exchange of t&o

ro&s does not change a matrix"


6/23

,rom (1";%

,

1

,1

1

n n

n

n n j j j

a x a x= −

−

=∑

1;11;) ME*+E,-*MA# ("%

,or the

th

i 2ro& ( i n≠ %

1

, ,1

n

ni j j i n i j

a x a x b+ =−

=∑

111>) ME*+E,-*MA#

("%

4enote

, ,,

,,

i n n ji j

n ni j

a aa

ac −=

'?1'?) ME*+E,-*MA# ("%

,rom (1"11% it follo&s that coe@cients

,i jc

turns 6ero for

i n= or for

j n=

" #hus. the last ro& and the last column removed from the matrix A and

equation (1"% may e &ritten in the form

Cx b='11'1) ME*+E,-*MA# ("%

&here

,

1

1i j j

n

i j

c x b=−

=∑''1'') ME*+E,-*MA# ("%


7/23

#he expression (1"1'% may e &ritten in the structured form as a

determinant of the second order normali6ed y the self2impedance of the

eliminated ground line"

( ) , ,

,,,, , , ,

, ,1 1 i j i n

n nn jn ni j i j i n n j

n n n n

a a

a ac a a a aa a= =−'/1'/)

ME*+E,-*MA# ("%

Expression (1"15% provides &ith the algorithm of the matrix reduction" An

element of the reduced matrix is expressed through the four elements of

the original matrix. rought together into a normali6ed determinant of the

second order" Each of these four elements is. in turn. the normali6eddeterminant of the second order (except the 3rst reduction% as a result of

a previous reduction"

#o eliminaten-1

x the aove procedure should e repeated using

constraint equation (1"%" As a result &e &ill get the matrix

Dx b='1') ME*+E,-*MA# ("%

&here

2

1,i j j

n

i j

d xb−

== ∑

'51'5) ME*+E,-*MA# ("%

1

1 11

, ,

,,1 1, ,

1 i j i n

n nn jd i j n n

c c

c cc

−

− −−− −=';1';)

ME*+E,-*MA# ("%

o& &e have to express the elements of matrix D through the elements

of matrix A"

, ,

,,, ,

1 i j i n

n nn ji j n n

a a

a a

ca

=

'


8/23

1

1

, ,

, ,1,

,

1 i n i n

n n n ni n

n n

a a

a ac

a−

−− =

'=1'=) ME*+E,-*MA# ("%

1, 1,

,,1,

,

1 jn n n

n nn j jn

n n

a a

a ac

a− −

− =

'>1'>) ME*+E,-*MA#

("%

1, 1 1

1

,

, ,1, 1

,

1 nn n n

n n n nnn

n n

a a

a ac

a−− −

−−− =

/?1/?) ME*+E,-*MA#

("%

Sustituting (1"1>% B (1"''% to (1"1=% &e otain

,

1, 1 1,

, 1 ,

,, , , 1

, ,, ,, , 1

1, 1, 1, 1 1,

,, ,,, , 1

,

1 1

1 1

n n

n n n n

n n n n

i ni j i n i n

n n n nn n n nn j n n

n j n n n n n n

n nn n n nn nn j n n

ad

a a

a a

i j

a aa a

a a a aa a

a a a a

a aa aa a

− − −

−

−

−

− − − − −

−

=

/11/1) ME*+E,-*MA# ("%

After simpli3cation

1, 1 1,

,

, 1 ,

,, , , 1

, ,, , 1

1, 1, 1, 1 1,

,,, , 1

1

, n n n nn n

n n n n

i ni j i n i n

n n n nn j n n

n j n n n n n n

n nn nn j n n

d

a aaa a

i j

a aa a

a a a a

a a a a

a aa a

− − −

−

−

−

− − − − −

−

=

/'1/') ME*+E,-*MA# ("%

#he last expression is an embedded determinant" #he factor in front of

the determinant is a constant. &hich depends only on the self and mutual

impedances of the eliminated conductors" #he 3rst term of thedeterminant depends on the ro& and column numers. the second one B

only on the ro& numer. the third one B only on the column numer. and

the fourth one is a constant"


9/23

Double Kron Reduction of impedance matrix

Apply the otained formula for the equation (1"'%"

Introduce the normali6ed coe@cient

1 1 1 2

2 2

2 1 2 2

, ,,

, ,

N N N N N N N

N N N N

Z Z Z

Z Z = ×0

//1//)

ME*+E,-*MA# ("%

o& the impedance matrix0

may e &ritten in the 9ron reduced form8

( )

2 1 2

2 2 2 2 1 2 2

1 1 2

2 2 2 2 2

, , , ,

, , , ,

,

, ,

, , ,

1

i j i N i N i N

N j N N N N N N R

i j

N N j N N N

N j N N N N

Z Z Z Z

Z Z Z Z Z

Z Z

Z Z Z

= ×0 0

/1/)

ME*+E,-*MA# ("%

&here the superscript (R) denotes the doule reduced matrix.

{ }, , ,i j A B C ="

After doule 9ron reduction the original set of 3ve linear equations (1"1%

has een reduced to the set of three linear equations

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

R R R

A AA A AB B AC C

R R R

B BA A BB B BC C

R R RC CA A CB B CC C

V Z I Z I Z I

V Z I Z I Z I

V Z I Z I Z I

= + +

= + +

= + +/51/5)

ME*+E,-*MA# ("%


10/23

&ith corresponding impedance matrix

( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

R R R

AA AB AC

R R R R

BA BB BC

R R R

CA CB CC

Z Z Z

Z Z Z

Z Z Z

÷

= ÷ ÷

÷

0

/;1/;) ME*+E,-*MA#

("%

,or convenience &e drop the superscript (R) in (1"'=%

AA AB AC

BA BB BC

CA CB CC

Z Z Z

Z Z Z

Z Z Z

÷= ÷ ÷

0

/


11/23

directions remain unchanged%" Correspondingly. the entries of matrix A

also depend on the selected asis"If &e consider the vectors x and y &ithout reference to a speci3c asis.

the connection et&een them is the transformation (mapping% descried

y a linear operator

ˆ A

ˆ y A x=/=1/=) ME*+E,-*MA# ("%

&hich is represented y di$erent matrices of the same dimension in

di$erent ases" #hese matrices are called similar and have a numer of

speci3c properties"

:et mapping

ˆ y A x=in an aritrary chosen asis in

nC

is represented ymatrix A so that y Ax=

. and represented y matrix A’ in any other aritrary chosen asis

in the same space. so that

'' ' y x A= &here y’ and x’ are vectors in the

ne& asis" Connection et&een the vectors x’ and x . y’ and y is de3ned

y the transition matrix T from one asis to another

' x T x=/>1/>) ME*+E,-*MA# ("%

' y T y=?1?) ME*+E,-*MA# ("%

&here T is any non2singular matrix"As already mentioned. the matrices A and A’ are similar" #he relation

et&een them can e easily determined" ,rom

'' ' y x A=

'T y A T x=

or

1

' y T A T x−=


12/23

#herefore similar matrices A and A related to each other through the

transformation

1 ' A T A T −=

111) ME*+E,-*MA# ("%

#he similarity transformation guides matrix A from the asis to the asisin the same vector space.

!igenvalue decomposition

As mentioned aove. the impact of operator

ˆ A on the vector x in the

general case is made of t&o independent actions8 the changing the

vectors length and rotation of the vector y a certain angle" If angle of

rotation is 6ero. then operatorˆ A

only change the vectors length. other

&ords

ˆ A x xλ ='1') ME*+E,-*MA# ("%

In this case the vector x is called an eigenvector of the operator

ˆ A

and λ

is called an eigenvalue of the operator corresponding to the given

eigenvector",rom geometric point of vie&. an eigenvector x prescries the direction

&here the operators action is reduced to vectors expansion" #hen the

coe@cient of expansion is an eigenvalue λ"If λ is complex then the geometric interpretation ecomes pointless ut

de3nitions remain in force" In this case only more general algeraic

interpretation exists"

Assume that for the certain asis equation (1"1% may e &ritten in the

matrix form

A x xλ =/1/) ME*+E,-*MA# ("%

#hen the vector x is called an eigenvector of the matrix A and λ is called

an eigenvalue of the matrix corresponding to the given eigenvector"In explicit form


13/23

2 2

2

2 22

2

2

.......................................................................

.......................................................................

1 111 1

1

1

1

n ,n

n ,n

n

a x + a x .......a x = xa x + a x .......a x = x

a

λ

λ

2 21 1 nn,n nn x + a x .......a x = xλ

1)ME*+E,-*MA# ("%

-ne can see no& that there is a product of t&o un7no&n in the right hand

side of each equation8 the eigenvalue λ and the componenti x of the

eigenvector x . that is the prolem of 3nding eigenvalues and

eigenvectors of the matrix A is nonlinear" In addition. the numer of

un7no&ns exceeds the numer of equations y one"

#here is a method of separating λ andi x. ased on the Cramer rule for

solving the set of simultaneous linear equations"

Collecting terms in the right hand side rings (1"/% to the set of linear

homogeneous equations

( )

( ) 2

2

2 22

2

2

.......................................................................

.......................................................................

0

0

111 1

1

1

1

n ,n

n ,n

n

a x + a x .......a x =

a x + a x .......a x =

a

λ

λ

−

−

( )2 2 01 1 n,n nn x + a x ....... a x =λ −515)

ME*+E,-*MA# ("%

In matrix form


14/23

( )

( )

( )

22

..... 0

. 0

1 2

11

n,n

12 1n 1

21 2n 2

nn n

a

a

a

a ...... a x

a ...... a x

x 0a a .......

λ

λ

λ

÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷

−

−

−

× =MM M - M M

;1;)ME*+E,-*MA# ("%

or

( ) 0 A I xλ − =


15/23

&hich al&ays has n complex di$erent or multiple roots"

If all λ are di$erent. then each jλ

corresponds (&ith accuracy up to a

constant factor% to the eigenvector j .

( )1, ... j n=" #hose n vectors are

linearly independent and form the asis in &hich the matrix

1 ' A T A T −=

ta7es the diagonal form" It is easy to prove that

1

2

0 .....0

0 . 0

0 0 n

......

......

.......

A

λ

λ

λ

÷ ÷ ÷ ÷ ÷ ÷

= Λ =M M - M

5?15?) ME*+E,-*MA#

("%

if the transition matrix T made of eigenvector columns"

"ectors #ith the physical dimension and dimension

transformation

In equation

abc abcV I = ×0

51151) ME*+E,-*MA# ("%

vectorsabcV

andabc I

elong to the vector spaces of di$erent physical

dimensions. &hich means that matrix

0

transforms vector from the spaceof currents to the space of voltages"

o&ever. the similarity transformation

1 ' A T A T −=

is de3ned only &ithin

the same linear space and cannot transfer a vector from one space to

another" Since astract algera deals &ith vectors &hich do not have

physical dimensions it is not clear ho& the di$erent physical dimensions

in the left and right hand sides of equation (1"'% may inDuence the

solution of eigenvalue equation" o&ever. the general experience hassho&n that neglecting some sutleties in mathematical reasoning &hile

solving a physical prolem. may lead to fatal errors in the 3nal result"


16/23

#o comply &ith condition that vectorsabc

V and

abc I elong to the space of

the same physical dimension. matrix must e made dimensionless"

4imensionless of matrix0

may e easily achieved y normali6ing all the

elements to a certain numer (numers%. &hich has the dimension ofimpedance. the choice of &hich is more or less aritrary" It is desirale.

ho&ever. to choose the parameters that have a physical meaning" :et us

&rite the equation (1"'% in explicit form

A AA A AB B AC C

B BA A BB B BC C

C CA A CB B CC C

V Z I Z I Z I

V Z I Z I Z I

V Z I Z I Z I

= + +

= + +

= + + 5'15')ME*+E,-*MA# ("%

It ma7es sense to divide each equation y the corresponding diagonal

entry of matrix0

. &hich has clear physical interpretation ( self2

impedance% of the line"

AC A AB A B C

AA AA AA

BC B BA A B C

BB BB BB

C CA CB

A B C CC CC CC

Z V Z I I I

Z Z Z

Z V Z I I I

Z Z Z

V Z Z

I I I Z Z Z

= + +

= + +

= + +5/15/)

ME*+E,-*MA# ("%

In matrix form


17/23

1

1

1

AC AB A

AA AA AA A

BC B BA B

BB BB BB

C C CA CB

CC CC CC

Z Z V

Z Z Z I

Z V Z I

Z Z Z

I V Z Z

Z Z Z

= ×

515)

ME*+E,-*MA# ("%

or

' ''

' ' '

' ' '

1

1

1

AB AC A A

B BA BC B

C C CA CB

Z Z V I

V Z Z I

I V Z Z

= ×

55155)

ME*+E,-*MA# ("%

&here the normali6ed elements are mar7ed y prime"

o& the matrix

' '

' ' '

' '

1

1

1

AB AC

BA BC

CA CB

Z Z

Z Z

Z Z

=

0

5;15;)

ME*+E,-*MA# ("%

is dimensionless and vectors

'

abcV and

abc I elong to the same vector

space"


18/23

#he eigenvalue equation may e &ritten as

' '

' '

' '

1

1 0

1

AB AC

BA BC

CA CB

Z Z

Z Z

Z Z

λ

λ

λ

−

− =

− 5


19/23

o& equation (1"


20/23

3

3 3

3

2

1 2

3

1

1 2 1 2

3

01 2

3

a x x x

a

a x x x x x x

a

a x x x

a

+ + = −

+ + =

= −;/1;/) ME*+E,-*MA# ("%

+. Depressed cubic polynomial

Analytical solution of cuic equation

3 2

3 2 1 0... 0a x a x a x a+ + + + =

;1;)

ME*+E,-*MA# ("%

al&ays starts &ith reducing cuic polynomial to the monic form and the

#schirnhaus transformation &hich removes quadratic term from the

polynomial. ringing it to the depressed one

30 ! "! #+ + =

;51;5) ME*+E,-*MA# ("%

In this case the ietaFs formulae are

3

3 3

3

1 2

1 2 1 2

1 2

0 x x x

x x x x x x "

x x x #

+ + = + + =

= − ;;1;;) ME*+E,-*MA#("%

,. -ubic roots of unity

,ollo&ing the ,undamental theorem of algera the equation

3

1 0ω − = ;


21/23

In trigonometric notation

21 cos 2 sin 2

i $ $ i $ % π π π = + =;=1;=)

ME*+E,-*MA# ("%

2

3 31 0; 1; 2$

i% $ π

= =;>1;>) ME*+E,-*MA#

("%

Correspondingly

0

0

2

31

4

32

1

2 2 1 3cos sin

3 3 2

4 4 1 3cos sin

3 3 2

i

i

%

i% i

i% i

π

π

ω

π π ω

π π ω

= =

− += = + =

+= = + = −


22/23

If the follo&ing relations hold

3bc "ω − =


23/23

1

32 3

1

32 3

4 / 27

2

4 / 272

# # "b

# # "c

+ += ÷

÷

− += ÷ ÷

=11=1)

ME*+E,-*MA# ("%

#here are three complex cue roots among &hich to choose b and c. and

not all sets satisfy the original equation (1"

Documents

B.tsoniff Analytical Approach to Calculation of Eigenvectors and Eigenvalues for 5x5 Impedance Matrix