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Application of matrix methods to the solution oftravelling-wave phenomena in polyphase systemsL. M. Wedepohl, Ph.D., B.Sc.(Eng.), Graduate

SynopsisThe interes t in t ravel l ing waves a n d surge phenomena in power sys tems h a s grown considerably becauseof t h e re levance t o power - l ine ca r r i e r commu ni ca t i on a n d protect ion, faul t locat ion, swi tching of unl oadedlines a n d t h e recovery vol tages o n ci rcui t -breakers under shor t - l ine faul t condi t ions .T he paper summar i zes t h e so l u t i on t o t he s ingle-conductor wave equat ion, develops t h e solut ion for the2-conductor case b y class ical methods , a n d then indicates t h e rapidly growing complexi ty of t he p r o b l e mwith increas ing numbers of conductors when solving b y class ical methods . T h e 2-conduc t or case is thensolved b y us i ng t h e m e t h o d of matr ix a lgebra. These resul t s a r e s h o w n to be val id fo r any n u m b e r of con-duc t or s , and t he concep t of surge i mpedance a n d propagat ion coeff ic ient fo r polyphase systems is i n t roduced .The s ingle-phase equat ion is s h o w n to be a par t icular case of the genera l equa t i on a n d t h e similari tybe t ween t h e resul t s of t he two cases is indicated. Examples of the mat r i x me t hod a r e given, including aproof t ha t symmet r i ca l component s a re a par t icular case of the general resul t . T h e paper conc l udes b yindicat ing that t h e m e t h o d is par t icular ly sui table fo r calculat ions carr ied o u t with a digi ta l computer .

Z\fY\

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2 Review of cases of wave propagation insimple systems2.1 Sin gle conductor in presence of an infinite earthplane

The arrangement of a single condu ctor in the p resenceof an infinite earth plane is shown schematically in Fig. 1,(P)ho (P) z n 'A*AAMAA- 1 %

.(P)

Y,,-Ax v, ( pV p )

x = 0 x x= lFig. 1Equivalent circuit of single-conductor systemwhich includes details of one element of the system. Thefundamental equations relating to the element are

0)YtfVr (2)

Henced2Vip)-$- = ZtfYtfvp (3)and ^=YtfZ[l[ (4)Solving for V\p\

V[p) = A exp yxx + B exp yxx . . . . (5)where A and B are arbitrary constants and

yi-VlW (6)V[p) may be seen to consist of two components: onetravelling in the forward direction and the other in thereverse direction.Although only one conductor is being considered, eqn. 5will be rewritten to conform with the notation used for morecomplex cases at a later stage in the paper, namely

y[p) = 5,,[K,(c+> exp -yxx + F, exp Ylx] . (7)This shows the two components of V[p\ The reason for thearbitrary constant S1, j will become apparent for more complexcases. The above equation may be further simplified asfollows:

vy = snv\* (8)whereVU!i = ytc+) e x p _ yiX + yie-) e x p yiX ( 9 )

The current is derived from V[ p) using eqn. 1:

. (10)

7-(0) _ ^ U _ / ^ l l

V) is the difference between the two components ineqn. 9, i.e.V[c) = F < c + > e x p - y x x - V[c~ > e x p y , x . . ( 1 3 )

The equations define completely the general wave equationfor the single-conductor system. It may be noted that eachcomponent voltage produces an associated current, the twobeing related by the so-called characteristic impedance Z[f.It may be also noted that the V^ component travelsfrom right to left in Fig. 1 in accordance with the negativesign of the corresponding current in eqn. 10.The arbitrary constants V{c+) and F / 0 "' may be eliminatedfrom a knowledge of the boundary or terminal conditions ateach end of the system. For example, if Vl0 and /10 are thevoltage and current at the end of the system where x = 0,and V x r and /, r are the corresponding quantities when x = r,^io = ^ . [ ^ ( c + ) + ^, ex p Ylr] (15)S n [ V { c + ) - V } c - > ] . . . . ( 1 6 )

hr = [ ^ J - ' S i i t ^ e x p - Yir - Vf->expyir] (17)By elimination, the usual 4-pole equations of the transmissionline are obtained:

V\r = vio c o s h Yir ~ zuh o sinh yxr . . . . (18)h r = ~ [ZuT1 Pio sinh Ylr + 710 cosh y,r . (19)

The final concept to be considered in this Section is thatof reflection factor. Consider conditions at end r of the systemof Fig. 1. The voltage and current are defined by eqns. 15and 17.The voltage and current are also related by the terminalimpedance Z l l r , Vlr = ZnrIXr. By elimination, it is seen thatwhere

~

exp yxr =Z l lr -Zf?

- yxr (20)

K is known as the reflection factor, since it expresses thereverse travelling wave as a fraction of the forward compo nent.2.2 Two conductors in the presence of an infinite earthplane

Referring to Fig. 2 showing an element of the system,the following equations relate voltage and current in eachphase:dV ip)

dV\p)

dl \p)

PROC. IEE, Vol. 110, No. 12, DECEMBER 1963

(22)

(23)

(24)

(25)2201

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The following equations may be derived by elimination:d2V\p)_ i - = P I l f r + P 1 2 ^ > (26)d2Vkp)-jJ- = P2lV + p2lil (28)d2H>p)- ^ = P l2H + P22l2> (29)

where

(p)

(P) (P)

x = 0

(P) (p)in-

(p ) (p ) (p) (p )

x = l(p)

2,,-Ax-W W A / W V ^

Fig. 2Equivalent circuits of 2-conductor arrangementa Overall relationshipsbl Impedance elementsc Admittance elements

wherePxx =Pn = "P2X =P22 = -

(30)(31)(32)(33)

It is shown in Appendix 11.1 that the solution for thevarious voltages is(34)(35)

yic) = yie+) e x p _ yiX + y(c-) e x p y iXV^ = V2 e x p y2x

+ 4P 12 /> 2 1]}4i >12 P21 ]}

IP 21 ~ P22)2Solving eqns. 22 and 23 fo r I[p) and lg> in terms of V[c\it is shown in Appendix 11.1 that the solution fo r the twocurrents is given by the following ex pressions:

1\P) =. . . (36)

1_ I (37)

x Vl0^ ex p yxxx Vif~^ ex p y2x

2 det Z (p)

V[ c) =

det Z*>The following points regarding the solution should be noted:(a) Two propagation coefficients are present, each beingassociated with a pai r of forward and backward travell ingwaves.(b) Each type of travell ing wave of voltage appears onboth phases in a ratio determined by the system parametersonly. However, the relationship between travell ing waves ofdifferent type is arbitrary.(c) Each travelling wave of voltage is accompanied by acorresponding current. The relationship between voltage andcurrent of a part icular component type in each phase is aconstant and is determined by the system parameters only.These constants have the character of surge impedances, andtheir values to each component in each phase are

de t Z ip)Yi[Zg -

det Z (p)y2[S x2Z%IS22de t Z ip)

det Z w

2202

7(0 _Z21 ~

Z (c) _These relationships are determined from eqns. 36 and 37.The arbitrary constants are eliminated from a knowledge ofthe voltages and Qurrents a t each end of the line, as for thesingle phase case.It may be seen that the line voltages and currents aredetermined in terms of components in a fashion analogousto symmetrical components in symmetrical 3-phase systems.Even for this case the labour in obtaining a solution is fairlygreat and the need for an ordered solution is indicated.The results of this Section could have been expressed instandard form, as was done for the case of a single cpnductorin the previous Section. How ever, the mathematical manipu la-

PROC. IEE, Vol. 110, No. 12, DECEMB ER 1963

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tion involved is tedious, and for this reason the m atrix methodis introduced in the next Section. The equivalence of theresults is demonstrated before proceeding to the general case.3 Ma trix solution to the 2-conductor problem

The main difficulty in solving the 2-phase (and ingeneral multi-phase) problem is due to the fact that second-order rates of change of voltage (and current) in each phaseare a function of the voltages (currents) in all phases (seeeqns. 26-29). Before a classical solution may be obtained,an elimination process must be carried out. Eqns. 22-29 maybe rewritten in matrix form as follows:

(38)dx

dx 2(39)(40)

1dx 2 . . . . (41)and / (/>) are column matrices corresponding to thevoltages and currents of each phase, and Z ( p ) a n d Y^ aresquare matrices, the element at the intersection of row i andcolumn j being the mutual impedance or admittance betweenphases i and ;. It may be noted that both these matrices arebilateral since a passive network is considered, and for thisreason Z\*> = Z an d yO> = Y (tp\ a relationship usedabove in deriving eqn. 41. P is a square matrix previouslyused to simplify the analysis, its elements being defined byeqns. 30-33, i.e. the element at the intersection of row / andcolumn j is

p,j = (42)The matrix solution is based on a linear transformation ofvoltage and subsequent manipulation, so that second-orderdifferential relationships involve diagonal matrices only.Mutual effects are thus eliminated, making a direct solutioncompon ent for component possible. The complete analysis iscarried out in Appendix 11.2, and only the results are givenbelow.Component-voltage matrix K (c) is introduced, related tothe phase-voltage matrix by

(43)Substitution in eqn. 40 and rearrangement yields

dx 2 = y2V (c ) (44)

Matrix S is chosen in such a way that y2 is a diagonalmatrix of the form72 =

As a direct consequence of this relationship, matrix equa-tion 44 separates into two simple second-order equations :

PROC. IEE, Vol. 110, No. 12, DECEMBER 1963

These have the same form as eqn. 3 for the single-conductorcase and have the simple solutionVUA = j/(c+) e x p _ YiX + yu-t e x p y{X # .yu ) = yic+) e x p _ yiX + yic-) e x p yiX

where V[c+ \ V{c+), Vc+ ) and V~ ) are arbitrary constants.The matrix 5 is chosen in such a way that S~ XPS is diagonalsince then a direct solution of the above equations is possible,solving for each component in turn. It is shown in Appendix11.2 that the diagonal relationship S~ [PS y2, where y 2 is adiagonal matrix, is only possible if the following equation issatisfied:

(47)et (P - y2) = 0This relationship is seen to be independent of the S matrix.It is also shown in Appendix 11.2 that the two values of y 2,and hence of y, obtained from eqn. 47 correspond to thetwo values of the propagation coefficient for the 2-phasecase obtained in Section 2.2. (The choice of the symbol y 2for the diagonal matrix anticipated this result.)The elements of column i of the S matrix are then deter-mined by solving the set of homogeneous linear dependentsimultaneous equations given in matrix form by

- yf)s0) = o (48)where S ( /) is a column matrix whose elements correspond tothe elements of the /th column of S. Since the equations aredependent owing to the vanishing of the determinant of thecoefficients, any one value may be chosen arbitrarily and theremaining element solved in terms of this one. It is shown inAppendix 11.2 that the values of the elements 5"i i, S22, Sn a n dS2u determined by making use of eqn. 48, and the phasevoltages, as determined from eqn. 43, are identical with thesolutions previously determined by classical methods. It isfurther shown that the ph ase currents are identical with thosedetermined by classical methods. The phase currents aresolved by inverting the m atrix equation 38, i.e.

1 1dx dx (49)

4 General solution for the polyphase caseThe matrix approach to the solution of the polyphasewave problem was introduced in Section 3. The second-ordermatrix differential equations involving voltage and current,

dx 2 = py(p)and dx 2

(50)(51)

are related to component voltages by the linear trans-formations :y{p) _ (52)/(p) = Q[(c) (53)

where the square matrices S and Q are as yet undefined.Eqns. 52 and 53 express in general terms a relationship whichis well known in work on symmetrical components, in whichcase the phase voltages Vx, V2 and K3 are related to thesequence component voltages V+, V_ and Vo by eqn. 53,where 1 1 1

Also, S = Q in this particular case.2203

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Substituting for V^ and 7 ( p ) in eqns. 50 and 51, the trans-formed equations are obtained:= y2F (54)

= Q-xPtQI{c) = y '2 / ( c ) (55)dx 2

dx 2The matrix method is based on the fact that, by a suitablechoice of S and Q,y2 = S-lPS (56)

and y'2 = Q~ lP tQ (57)are diagonal matrices.Once y and y' have been determined, eqns. 54 and 55are solved as a series of simple wave equations, since mutualeffects have been eliminated. For example,

d2V{*dx 2 = y\Vf (58)(59)

etc., andV\c) = exp - y,JcF /c+ ) + exp . . . (60)

7(c) is determined in a similar fashion.By analogy with the work in Section 3 and Appendix 11.2,it may be seen that, for eqn. 56 to be valid, i.e. y 2 diagonal,= 0 (61)

In eqn. 61, S^ represents column i of the S matrix. Eqn. 61represents in matrix form a system of hom ogenous simul-taneous equations in which the unknowns are the n elementsof column i of the S matrix.For such a system of simultaneous equations to have anon-trivial solution, it is necessary for the determinant of thecoefficients to be zero, i.e.

det (P - y?) = 0 (62)A similar consideration of the solution of the Q matrixrelating component to phase currents given in eqns. 53 and57 will show that, if a diagonal relation in the latter equationis to b e achieved, it is necessary for the following determinentalequation to hold:

det OP, -y'j) = O (63)Since y'2 in eqn. 57 is diagonal, it follows that the matrixP t y '2 differs from matrix P, only by the modification tothe elements in the principal diagonal. Thus

P t - y' 2 = (P - y'2), (64)It is well known that the determinant of a matrix is equal tothe determinant of the transposed matrix, so that

det (P - y'2) = det (P - y'2), (65)By comparing eqns. 62 and 65, it may be seen that y 2 = y '2 .The use of the symbol y 2 for the diagonalization of boththe voltage and current matrices anticipated this importantresult, y 2 does, in fact, define the propagation coefficient foreach component, and for the solution to have a meaning it isnecessary to obtain the same set of values, either from aconsideration of voltages or from a consideration of currents.Once having determined the n values of y 2, the Q and2204

S matrices are solved one column at a time by solving thesystem of homogenous dependent simultaneous equations(P-yf)S a) = o (66)(P , - y2)G (/ ) = 0 (67)

where (/) indicates that the elements of column i of therespective matrices are to be considered. Since these systemsof equations are dependent (owing to the vanishing of thedeterminant of the coefficients), one value in each column isspecified arbitrarily and the remaining elements are evaluatedaccordingly.It may be noted that the n values of y f are known tomathematicians as eigenvalues and the corresponding S^ and

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and propagation coefficient are introduced in relation toboundary conditions in order to complete the analogy betweenthe general solution and the simple solution to the single-conductor wave equation developed in Section 2.5.1 Polyphase surge impedance

Surge impedance in the single-conductor case isgenerally defined as the ratio between voltage and currentin an infinite line, or as the terminal impedance which gives

Zw Z (0 )a reflection factor of zero, i.e. K = " = 0 andZ l l r = Z(eqn.21). Au + ^nPolyphase surge impedance will be defined in the latterway, since the concept of reflection factor is developed at thesame time.

The analysis is carried out in Appendix 11.4, where it isshown that the polyphase surge impedance is

Z = Sy-*S-W> (70)It is also shown that Z (0 ) is a symmetrical matrix, i.e.

The validity of this expression in the single-phase case isevident, since the matrices become elements and

(71)The polyphase surge admittance

y(0) =is also defined. This expression is particularly useful whenanalysing surge impedances under a variety of boundaryconditions since it always yields a physically realizable net-work. Also, at high frequencies, owing to the dominance ofreactive elements in the Z and F matrices, the surge-impedancematrix tends to become almost purely resistive; consequentlyresistive equivalent circuits may be used. Once havingobtained y ( 0 ) by calculation, the effect of differing termina-tions and through-connected lines may be evaluated bymaking use of a network analyser. The equivalent circuithaving been set up, a large number of conditions may beinvestigated very rapidly. This technique is useful both inevaluating surge impedances in relation to circuit-breakerperformance and certain aspects of power-line carrier trans-mission, namely matching the effects due to line traps. Thesetopics will be considered in papers to follow.5.2 Polyphase reflection factor

It is also shown in Appendix 11.4 that the polyphasereflection factor is

(72)This is identical with eqn. 21 for the single-conductor case,since under these circumstances the matrices in eqn. 72

become elements.5.3 Polyphase propagation coefficient

It is shown in Appendix 11.5 that the voltages andcurrents when x = r (right-hand side) may be expressed interms of the values when x = 0 (left-hand side) as follows:

(73)= - Q sinh

^ (74)As before, the equations are valid for the single-conductorcase when the matrices become elements and are therefore

PROC. IEE, Vol. 110, No. 12, DECEMBER 1963

commutative. It is readily seen that eqns. 18 and 19 are aspecial case of eqns. 73 and 74.5.4 Interpretation of polyphase transmission matrices

In this Section it has been shown that the matrixsolution to the polyphase problem includes the single-conductor system as a special case.

Polyphase surge impedance in a multiconductor system isa bilateral set of impedances which define the currents whichwould flow in each conductor of an infinitely long line as aresult of impressing voltages on each of the conductors. Theconverse holds if currents were injected into each of the con-ductors; the surge-impedance matrix would define the voltageof each conductor relative to the earth plane.

Polyphase reflection factor defines the voltage on eachconductor at the terminals of a system travelling in the reversedirection in response to a set of incident voltages arriving inthe forward sense at the terminals. It follows that the totalvoltage expressed in matrix form at the terminals isV=(l +K)V^\ where V is the incident (forward-travelling) phase-voltage matrix. It may be noted that K is,in general, not a diagonal matrix, so that reflected voltagesmay appear on conductors in which there are no incidentvoltages. Exceptions are:(a) all conductors short-circuited when K= 1, so that

the reflected voltage on each conductor is equal andopposite to the incident value, the total voltage being zero(jb) all conductors open-circuited when K= 1, so that thereflected voltage is equal to the incident voltage on eachconductor.

The 4-pole matrix equations 73 and 74 for the multi-conductor system may be useful in solving certain problems.However, in many problems it may be more straightforwardto resolve phase quantities into components from a know-ledge of the S, Q and y matrices since each component hasa definite propagation coefficient, whereas in eqns. 73 and 74propagation is influenced by the relative strengths of com-ponents of each type, which, in turn, are functions of theboundary conditions. No general conclusions may be drawnabout the approach to be usedeach problem has to beexamined before deciding which method is most convenient.

6 Cond uctor systems with planes ofsymmetryIt was shown in Section 4 that the solution to the

general wave equation involved in the first instance thesolution of a polynomial equation of degree equal to thenumber of conductors in the transmission system. Thus, in asingle-circuit 3-phase transmission system, a cubic equationmust be solved, while for a twin-circuit 3-phase system asixth-degree equation arises. The problem is further com-plicated by the fact that in the general case the coefficients ofthe polynomial equation are complex. It is possible to solvea cubic polynomial by trigonometrical methods, but the taskof solving a sixth-degree polynomial with complex coefficientsis almost impossible without the use of a digital computer.

Fortunately it is possible in most practical cases of twin-circuit systems to factorize the polynomial into the productof two cubic equations. The method of approach is quitegeneral and is applicable to all systems which are partiallysymmetrical, irrespective of the number of conductors.

The technique is to factorize det (P y2) into two lower-rank determinants, and since det (P y2) = 0, each of thelower-rank determinants must separately be zero. The methodis detailed in Appendix 11.5, where it is shown that, if thenumber of conductors is even, the determinental equation

2205

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det (P y2) = 0 may be reduced to the solution of twolower-rank equations detCP^ y2) = 0 and det(PB y2) = 0.In the reduced equations the rank of determinants PA andPB is half that of P, so that, for example, in the case of atwin-circuit system, two cubic polynom ials have to be solvedinstead of one of sixth degree. In the case of an odd numberof conductors, it is shown that the reduced determinants haverank (n l)/2 and (n + l)/2, respectively, where n is thenumber of conductors in the system.It may also be seen from A ppendix 11.5 that th e factoriza-tion assumes a particular significance when considering thedistribution of voltages and currents for a given mode ofpropagation, i.e. the values of the elements of one columnof the S and Q matrices corresponding to a particular valueof y. It is shown that all voltages and currents correspondingto the first factor occur in equal and opposite pairs in corre-sponding image conductors, while the voltages and currentscorresponding to the second factor occur in equal and oppo-site pairs in corresponding image conductors. If the numberof conductors is odd , the polynomial of higher degree alwaysyields components of the former ty pe. Also, in the case of anodd nu mber, it is evident that there is no curren t in the centreconductor and the voltage to earth of this conductor is zerofor components of the latter type.It may be noted that, even when digital computers areused for solving problems, a significant saving in time takesplace when factorizing the characteristic polynomial in themanner detailed above. Furthermore, it is readily seen thatthe solution of the equations of a twin-circuit system with aplane of symmetry reduces to the solution of two single-circuit systems. This is valuable in the preparation of digital-computer programmes, which need then only cater for thecase of a single-circuit system with instructions to carry outreductions when dealing with twin-circuit systems.

7 Particular solutions for a 3-phase single-circuit line with a plane of symmetry7.1 Horizontal configuration

In order to complete the paper, the solution to thepartially symmetrical 3-phase system will be developed as anillustration of the matrix method. In such a system, con-ductors 1 and 3 are at an equal height above ground andconductor 2 lies on a line perpendicular to the earth planemidway between 1 and 3.For the system, the impedance and admittance matricesare of the general form7(P)An Zj? nr

(These forms are valid if earth wires are included, providedthat the earth wires are symmetrical with respect to cond uctor 2and the earth plane.)The P matrix will have the general form

p =Pn PnP22 Pn

31 22

where P,y = ZikYkj2206

The determinental equation for y2 is of the form

0 = det (P - y2) = det

= det + Pn - y2)

- y2y2) det ( P n - P 1 3 - y2 )

factorized by the method detailed in Section 6 for a systemwith a plane of symmetry and an odd number of conductors.The three propagation coefficients are= HPn + .22 + ^12 + V[(Pn ~ P22 + Pn

y 2 = { P n + P22 + P1 2 - - ^22 + P12)2+ 8 P 1 3 P 3 I ]}

y\t may be noted that y\ has the simple form y\[Zff - Z$] [yff + Y$] after substituting for P n and P,2in terms of" Z ( p ) and Y (p \ In solving for the S and Q matrices,the first element of each column will be arbitrarily specifiedas being unity. It will be found that these matrices thenhave the simple formri

(These forms are determined by inspection according to therules developed in Section 6.)Only the centre element of the first two columns of eachmatrix is unknown. For the S matrix, these may bedetermined from the first set of three dependent simultaneousequations, i.e.

11S3,

115,,

1- 1

0,Q =

11G31

11G32

1- 1

0

(Pn - P , 2 S 2 f + P12S3r = 0, or,. = S3r =

Similarly it may be shown that

The impedances of each phase to each component may bedetermined from eqn. 122. Thus, for example,ft = ( z n + z12 + z,3

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wherePn= .

The following values will be found for y2:y\ =

These are well known forms in symmetrical-componentanalysis.In solving for the elements of the columns of the S andQ matrices, it will be found that all elements of column 1 ofboth matrices are unity, so that this component is a truezero-sequence wave. However, when considering the remain-ing elements in both matrices, it will be found that the threesimultaneous equations corresponding to each column are allidentical and of the general form Slr + S2r + S2r = 0, andsimilarly for the elements of a column of the Q matrix. Thisis the only restriction on their value, so that any numberof component systems exist. Typical systems which satisfythese requirements are positive- and negative-sequence com-ponents and Clarke a and jS components.In evaluating the impedance of each phase to componentsof each ty pe, the following well know n results will beobtained:

+ / = 2 or 3This illustrative Section has shown the way in which thematrix method is used to solve a typical problem and hasfurther shown that well known symmetrical-componentsystems are a particular case of the general solution.

7.3 Significance in relation to symmetrical-componenttheoryIt has been shown in the previous Section that sym-metrical components in a symmetrical system are in con-formity with the general theory. A problem is then to assessthose circumstances in which it is permissible to use sym-metrical components as an approximation to the actualsolution.In general, it will be necessary to use the exact approachwhen it is required to assess the difference in surge impedancebetween various conductors, mutual effects, and velocity andattenuation factor for various modes of propagation of high-frequency signals on power lines. In most cases it will beobvious when the symmetrical-component approach gives

sufficiently accurate results. These are problem s in applicationand fall outside the scope of the paper.8 Conclusions(a) A completely generalized method of solving travelling-wave phenomena on polyphase lines has been developedby the use of modern methods of matrix algebra.(6) Without the further simplification possible for particularcases, the generalized solutions require digital-computerfacilities for evaluation.(c) The equations are particularly suitable for such digitalcomputation, as the determination of the characteristicand vectors of the matrices form part of many standard

codes. This means that the solution of polynomialequations is an automatic process.PROC. IEE, Vol. 110, No. 12, DECEMBE R 1963

id) Work with preliminary programmes has yielded satis-factory solutions, and this work will be developed andexpanded.(e) With the satisfactory development of program mes, manyinteresting application problems can be examined withoutcomplexity being a restriction.(/ ) The solution s may be used to show the validity ofsymmetrical-component theory in ideal configurationsof symmetry.9 Acknowledgments

The author wishes to thank Messrs. A. Reyrolle andCo. for permission to publish the paper.Thanks are expressed to Mr. F. L. Hamilton (Engineer-in-Charge Research) and Mr. J. B. Patrickson (DeputyEngineer-in-Charge Research) for their helpful discussionand advice.10 References1 FALLOU, J. A.: 'Propagation des courants de haute frequence poly-

phases le long des lignes aeriennes de transport d'energie affecteesde courts-circuits ou de defauts d'isolement', Bull. Soc. Franc. Elect.,1932, Series 5, 2, p. 7872 DE QUERVAIN, A . : 'Carrier links for electricity supply an d distri-but ion ' , Brown Boveri Rev., 1948, 35, p. 1163 GOLDSTEIN, A . : 'Propagation characteristics of power line carrierlinks', ibid., p. 2664 CHEVALLIER, A . : 'Propagation of high-frequency waves along anelectrically long symmetric three-phase line', Rev. gen. Elect., 1945,54, p. 255 ADAMS, G. E. : 'Wave propagation along unbalanced h.v. trans-mission lines', Trans Amer. Inst. Elect. Engrs, 1959, 78, Part HI ,p. 63911 Appendixes11.1 Classical solution for 2-conductor case

Rewriting eqns. 26 and 27 in operational form,(D 2 - PU)V

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where S2l = Sn(y2 - Pn)/P12= Sn{(Pn - P22) -

and (y2 - Pu)Sl2lPl2 = 522or Sl2 = S22P12l(y2-Pu)

u - P22)112 (82)

4 P 1 2 P 2 1 ] } 5 2 2andSl2 _ 2Pl2{P22 - Pn - P 2 2 ) 2 + 4 P 1 2 /> 2 1 ]}

~ P22)2 ~~ P22)2 ++{Pn~ P22 ~

From eqns. 22 and 23,-dV^/dx = Z, ,/

2i (83)

- Z ' f ] (84)

(85)

Hence / | = [ - Z!gdV[p)ldx + Z^If = [_ Zff

From eqns. 80 and 81,dV[p)/dx = - yiSn[V{c+)exp - yxx - F,(c->exp yxx]

- y2^ i2 [^ 2 ( c + ) e x P - 72* - y(2~' ex p y2x]dV(p)Jdx = - yiS2i[V[c+)exp - yxx - V[e^exp yxx]

- Y2S22[Vic+) e x

P - 7ix

~ '/2

c-

)exp y2x]and substituting in eqns. 84 and 85,

. (86)

(87)(p)- Z\p2)2where det Z< p) =

yjc> = ylc+)e x p _ yiX _ yu:-) e x p y^xyg) = ylc+) e x p _ n x _ yu:-i e x p ^

11.2 Transforming matrices to diagonal formFrom eqn. 44

(88)

i.e. (d2/dx2)

If S is so chosen that S~lPS is diagonal, i.e. S~lPS = y2,then d2V^\dx2 = y\ K(c) (89)

y? 0 V\c), . . . . (90)0 y2 V?

This permits a direct solution, sinced2V[c)ldx2 = y\V[c) (91)d2V^\dx2 = yjV 2'c) (92)

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and henceV[ c) = , 4 e x p - y x x + B e x p y ^ x . . . . ( 9 3 )V^ = D exp - y2* + E exp y2x . . . . (94)

where A and B are arbitrary constants and the problem hasbeen reduced to the solution of two elementary waveequations.Rewriting eqn. 88, PS = Sy2 or PS - Sy1 = 0, andexpanding:0 = [(Ai - y\)sn + P12S21 (Pn - rl)sl2 + ^125221lP2lSn + (P 22 - y2)S 2l P2lS l2+(P 22-y 2)S 22 j^For this matrix identity to be true it is necessary for eachelement of the expanded matrix eqn. 95 to be zero.

From this it may be seen that each column of eqn. 95defines a set of homogeneous equations in two of the fourunknowns of the S matrix. For example, taking the left-handcolumn,

(Pn - y2)Sn + P12S2l = 0 (96)^21^.1 + ( ^ 2 2 - 7 ^ 2 1 = 0 (97)

It is well known that, if the determinant of the coefficientsof the unknowns is non-zero, the solution is Sn = S2l = 0 ,which is trivial. Conversely, if the solution is to have ameaning, the determinant of the coefficients must be zero, i.e.

d e t ( P - y 2 ) = 0 (98)Similar reasoning from a consideration of 5"12 and 5"22 wouldyield

de t( P- y2 ) = 0 (99)In fact, both y2 and y\ are contained in the equation

det (P - y2) = 0 (100)since this is a quadratic in y2.

Substituting for the elements of P in eqn. 100 yields(Pn ~ y2)(P22 - 72) - PnP2i = 0

or y\ = [/>n + P22 + V (^ n - J22)2 + 4 ^ 2 1 ]A = H(pn + P22) " V(Pn ~ P22)2 + 4^ 21 ]

It is important to note that this result is not confined tothe 2-phase case. Similar reasoning will show that eqn. 100is valid for any number of phases. The P matrix is modifiedby subtracting y2 from each element along the principaldiagonal; the determinant of the modified matrix is equatedto zero, yielding an nth degree polynomial in y 2. The result iswell known in mathematics, the various values of y 2 beingknown as eigenvalues of the matrix P.

The solution for the elements of S is now straightforward.Owing to the vanishing of the coefficients of S in, for example,eqns. 96 and 97, the values of Sn and iS21 are dependentand it is necessary to specify one element in order to obtainthe second. To conform with the analysis carried out inAppendix 11.1, Sn is specified, so that from eqn. 96,

Similarly,

These values are identical with those obtained by classicalmethods.

Again it is important to note that the results are valid forany number of phases. Once having obtained the n valuesPROC. IEE, Vol. 110, No. 12, DECEMBER 1963

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of y2 from a solution of the nth degree polynomial previouslydiscussed, the elements of the S matrix are solved columnby column using one value of y2 at a time and noting that,because the system of simultaneous equations is dependent,it is necessary to arbitrarily specify one element in eachcolumn.Once having determined the values of elements of the matrix, the phase voltages may be determined from eqn. 43,i.e. y(P) _ gyle)

A exp yxx + B exp yxxD exp y2x + E exp y2x

v\ = s n si2vp s2X s22and expanding,

y[ p) = ^n(^ exp - yxx + e x pSl2(D exp y2x + exp y2x)V (2P) = S2X{A exp y{x + 5 exp y ^ ) +

S22{D exp y 2 x + E ex p y 2 xWith th e chosen values of S, this solution is identical withthat obtained by classical methods (eqns. 34 and 35).11.3 Properties of th e wave equation

Rearranging eqns. 88 and 89, PS = Sy2. Consideringthe elements in column m,

This may be written(101)

(102)Here S^ is column m of the S matrix. Multiplying eqn. 102by S^t, which is a row matrix formed by transposing columnn of the S matrix,S(n)lPS (m) = S(n)S (m)y%, (103)

Similarly, by considering y 2 in place of y2, in eqn. 102, thefollowing equation may be derived:

Transposing,

Taking the difference between eqns. 103 and 104,(104)

(105)Since P ^ P, in the general case, the left-hand side, andconsequently the right-hand side, of eqn. 105 is not zero.This shows that the S matrix does not have the mathe-matical property of orthogonality, i.e. the product of twocolumns taken element by element is not zero. All possibleproducts of this type are contained in the matrix product

(106)and it may therefore be concluded that in the general caseeqn. 106 is not a diagonal matrix. Similar reasoning will showthat, in general, the Q matrix is not orthogonal and further-more that neither the S no r Q matrices are unitary, the term'unitary' meaning that the product of one column of the S(or Q) matrix with the complex conjugate of another columntaken element by element is zero.PROC. IEE, Vol. 110, No. 12, DECEMBER, 1963

Returning to eqn. 102, a matrix product of the followingtype may be formed:(107)

and by considering y 2,

Transposing,Q(n),PSm = Q{n)S{m)yl

Subtracting eqn. 108 from eqn. 107,0 = Q ( / , ) , S ( m ) ( y , 2 - y 2 )

In general, ym ^ yn, and therefore

(108)

(109)

(110)The special case of ym = yn, i.e. of two or more propagationcoefficients being equal, is unlikely to occur in practice, butdoes occur in theory when simplifying assumptions are made,e.g. in 3-phase systems assumed to be completely symmetrical.This special case is dealt with in Section 7.All possible products given by eqn. 110 are contained inthe matrix product Q tS, and it may be concluded that thisproduct is diagonal in form, i.e.

Q,S=D ( I l l )where D is some diagonal matrix.Complex conjugate products may be formed correspondingto eqn. 107, namely

Q(n),PS (m) = Q MlS (m)yl (112)

Taking the complex transpose of both sidesQ(n)tPS (m) = Qw,S(m)y2n (113)

Since ym ^ yn in the general case, it follows thatQ(n)iS(m) ^ 0 and that the Q and S matrices do not form aunitary set.From dV^dx= - S~ lZ^QI^ = - DZI& . . (114)

dl^dx = - Q-XY(P>SV^ = - DyV^ . . (115)where D2 = - S~lZ^Q

D y= -Q~lY^SNow d2V^/dx2 = D2DyV^ (116)and d2l^ldx2 = D yDzlM (117)From eqn. 88,

D zDy = D yDz = y2 (118)For Dz and Dy to form a product which is commutativeand furthermore which is diagonal, y2, it is necessary thatthey should in themselves be diagonal.From eqn. 88, if A is any diagonal matrix,

S-IPS = y2 = A A - y = A y 2 A " ! . . . (119)since diagonal matrices are commutative.Premultiplying eqn. 119 by A " 1 and post-multiplying by A,

A - ^ - ' P S A = y2i .e. (SA^PGSA) = y2

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a n d O S O - ' W ) (120)where S ' = SA (121)

This proves that if S is a solution to eqn. 88, so is S',where S and S' are related by eq n. 121.The surge impedance of each phase to a component of acertain type may be evaluated as follows, remembering thatthe current of a certain type can only be produced by avoltage of the same type as is given by eqns. 114 and 115.The voltage and current in phase r and due to a com-ponent of type s (having propagation coefficient ys) areK(s) = SrsV< c) and Ir(s) = Q r{s)I$ c\ From eqns. 38, 52and 53,

i.e. (d/dx)SrsvP = - Z% QW1= 1Substituting V$e) = V {sc) exp -ypc,

/ to = j(c) e x p _ ysX

Substituting in eqn. 114,

i.e.(125)

where V^ = K ( c + ) exp - yx - V^c~^ exp yxZ =y -lS~ lZW Q (126)Since y~ l is diagonal, and from eqn. 114 Dz is diagonal, itfollows that Z ( c ) is diagonal.11.4 Surge impedance , reflection factor and propagationcoefficient

Consider a set of phase voltages travelling in the positivesense (left to right) in a multiconductor system and encoun-tering a set of terminal impedances, Zn at the right-handterminal when x = r.From eqns. 52, 53, 60 and 125 we can write

Thenand

= I i.Z^Qms )lQ rsys)l= S (122)

The surge impedance in phase r and due to component oftype s is a function of the system impedances Zrm whichare constant, the propagation coefficient ys which is constantand the ratio of elements in column s of matrix Q. Althoughthese elements are arbitrary, their ratio is not, since n 1of these elements were evaluated in terms of the nth, andsince they arise from the solution of a system of linearsimultaneous equations they must all be proportional to thisone element and hence their ratio is determined only by thesystem parameters and the propagation coefficient.Consideration of eqn. 39 would show in a similar fashionthat

so that(123)

(124)

It may be noted that, when m = r, QmJQ rs = SlllsISrs = 1,Z (p) = Zip) and Y(p) Yip)

This shows that the surge impedance in the rth phase isalways a function of the self-impedance and admittance ofthat phase, together with the mutual impedances whosevalues a re modified in accordance with eqn. 124.The matrix Z + expI(rc) = exp ( - yr)I^c+) - exp

Also, the phase voltages at the terminal must be related tothe phase currents byZ,.I^ (127)

Substituting for V[p) and /r(p) ,

/(c) = zrQI(rc)Q exp (yr)/ = - KQ exp ( - y/-) / ( c + ) . . (128)

Where K= [Zr + SZ^Q~^[Zr - SZ^Q~l] . (129)It may be seen that eqn. 128 defines the matrix of reflectedphase currents in terms of the incident phase currents,Polyphase surge impedance is defined in terms of the set ofterminal impedances which gives reflection-free conditions,i.e. K = 0 in eqn. 129. Therefore

Z - -' = 0or (130)From eqn. 126, Z

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and again it may be seen that, for reflection-free termination,the terminal admittance isY =

032)~Kote that Yr=(Zr)

The polyphase wave equation may be defined in terms ofthe terminal conditions Vo, /0 , Vr and lr in the followingmanner:Vo == SZ[/ - lie-)]

Therefore

= S [exp ( - expy - exp

lr=Q [exp ( - yr)I^ + exp (yr)fc^]Substituting for / ( c + ) and /,

Vr = SZ(c>{cosh (yr)[Z^]~ lS~ l VQ - sinh (yr)Q~ % }/,. = {coshCyAOQ- o - sinhCyOtZ^-'S-1^}Vr = SZ cosh (yr)[ZV]-lS-lV0-SZ& sinh (.yr)Q~lI0= Scosh(yr)S-lV0-ZWQsinh(yr)Q-lI0 . (133)Ir = - Q sinh (yr)[ZW]-lS-lV0 + Q cosh (yr)Q-lI0= -Q sinh (yr)G-1[Z]-1Ko + Q cosh (yr)Q~ ll0. . . . (134)It should be noted that in these expressions, exp yr ,exp -= yr, cosh yr and sinh yr are diagonal matrices.

11.5 Sy ste ms with planes of symm etryA partially symmetrical system is defined as one inwhich the conductors are arranged in pairs at the sameheight above ground and at equal horizontal distances oneither side of a reference plane perpendicular to the groundplane. In a system with an odd number of conductors, theodd conductor lies on the reference plane. Most double-circuit transmission systems are partially symmetrical, as aresingle-circuit horizontal systems.Considering first the system with an even number of con-ductors, the Z( p\ y ( P ) and hence P matrices have the

following general form:

P = PaPb

zbZaPbPn

y(P) = Ya Yb

(135)In solving the polyphase wave equation it is necessary toevaluate the n, propagation coefficients by using eqn. 62, i.e.det (P - y2) = 0.It is well known that the determinant of a matrix productis equal to the product of the determinants of the matricestaken in any order, i.e.

det (P - y2) .= det %~ l(P - y2)K ,where K is any regular matrix.PROC. IEE, Vol. 110, No. 12, DECEMBER 1963

Let K = 1 - 11 1 where 1 is the ///2 x /2 unit matrix1 0 00 1 00 0 1

n being the number of conductors.

Substituting for P from eqn. 135,0

0 Pa-P b-y 2\= det {P a +P b- y2).det (P o - Pb - y2) = 0

i.e. det {P a + Pb - y2) = 0 (136)or det (Pa - Pb - y2) = 0 (137)

Eqns. 66 and 67 for solving for the P and Q matrices maybe simplified in the following way:

K-{PKK~lS0) = K-(138)

where K is the same matrix used to simplify det (P y2)

i.e. Pb -y2o Sli)b. . . (139)

= 0 . (140)

(141)where S'^ has been partitioned for comformable matrixmultiplication in eqn. 140.It is evident from eqn. 140 that, if det (P a + Pb - y?) = 0and det (P a Pb y?) = 0, S'0)b = 0 and S[ i)c is obtainedby solving the set of dependent simultaneous equations

(P a + Pb - yj)S'0)a = 0 (142)Similarly, if the converse applies, 5(,)0 = 0 and S[j)b is obtainedby solving the simultaneous equation det {P a Pb y2) = 0Thus 5 ' = 00 (143)and S = KS'

1 - 11 1

s:

sa oo 5;(144)

It is evident from the foregoing treatment that the problemof solving the ^-conductor partially symmetrical system isreduced to the solution of two ^/i-conductor unsymmetricalsystems. All operations and manipulations may be carriedout independently on the reduced systems as if the otherdid not exist, and only when the composite parameters arerequired are the independent values combined by means ofmatrix K.The significance of this result is seen from the structure of

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the S matrix (eqn. 144). For the first n/2 values of y2, propa-gation takes place with voltages (and currents) equal and inphase in each symmetrical pair of conductors, and equal andin antiphase for the remaining n/2 values of y2.In a partially symmetrical system with an odd number ofconductors, the m atrix P will take the general form

where Pa and Pb are square matrices of rank (n l) /2,Pc is a column matrix of (n l)/2 elements, Pd is a rowmatrix of (n l)/2 elements and Pe is a single element. Inorder to simplify the determinental equation det (P y2), amatrix K is introduced as before. In this caseK =

1 1 0- 1 1 0

0 0 11 -10

- 110

002

In K, the first four unit matrices are of rank (n l)/2 andthe final unit matrix is the unit element. K and K~ l aretherefore arranged for conformable multiplication with P :det (P - y2) = det K~\P - y2)K

(P a -P b-y 2) 0 0(P a +Pb- y2) Pc(P e ~ Y2)P.

= det 2P d= det (P a-P6-y2).det

i.e. det ( P a - P 6 -y 2 ) =(p -i_ \

det

2P , (Pe-y2)

or (P a +Pb- y2)

2P dPc

>.-y2) = 0

= 0(145)(146)

In this case the solution of an nth order polynomial hasbeen reduced to the solution of two polynomials of order(n - l)/2 and (n + l)/ 2, respectively.

As before, the elements of the S matrix as indicated ineqn. 138, i.e. K~\P - yj) = 0

or(Pa -yf) o

(P a + Pb- y2)0

(Pe-y2*U)bSU) c

= 0

where S'^ has been portioned for comformable multiplication.It is evident from previous argument that S[^b = S[ ^c 0 if det ( /* , - / * -y ? ) = 0{Pa +Pb~ y2)

(P e - y2)and S'0) a = 0 if detso that Sa 0S'= 0 Sb0 S'c

where there are (n - l)/2 Sa and (n + l)/2 Sb and S' c.Now S = KS'

= 0

110

110

001

Sa00

0siS'c

Sao s: (147)

Two general forms of propagation exist: in the formertype, signals are equal and in antiphase on paired conductorswith no propagation in the odd conductor. This is evidentfrom the fact that the odd conductor lying on the plane ofsymmetry is on an equipotential for magnetic and electricfields and could be removed without affecting the pairedconductors. In the latter type, signals are equal and in phasein paired conductors and the centre conductor is included.An example of odd symmetry is given in Section 7, where itis shown that, in the horizontal single-circuit line, propagationbetween outer phases is independent of the presence of thecentre conductor.

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