A MXTHOD TO SOLVE VERY LARGE PHYSICAL SYSTUS Ilil USY ...biotek/kron/solvelargephysicalsystems.pdf · A MXTHOD TO SOLVE VERY LARGE PHYSICAL SYSTUS Ilil USY STAGE'S Gabriel Kron Consulting

A MXTHOD TO SOLVE VERY LARGE PHYSICAL SYSTUS Ilil USY STAGE'S

Gabriel Kron Consulting Engineer

General Electric Company Schenectady, X. Y.

Abstract

Physical systems with very large number of variables (saywith tens of thousands of variables) may be solved with already available digital computers by tearing the system apart into a large number of small subdivisions, solving each subdivision separately, afterward interconnecting the partial solutions by a set of transformations to obtain outright the exact solution of the original system, Among the many advantages of the tearing (or tensorial) method is the red 3 ction of the amount of original calculations to a small fraction of about 2/n 9 where n is the number of subdivisions. Another advantage is the reduction of the number of elements in inverse matrices to a fraction smaller than l/fi. The same saving of labor appears also in smaller systems employing slide-rule calculations.

Introduction

The Nethod of Tearing

The purpose of this paper is to call the attention of radio engineers to a novel method of solving physical problems with very large number of variables rapidly and in rather easy stages. The method consists of tearing the original physical system apart into a large number of component systems, so that each part becomes independent of the other part. Next, each partial system is solved independently, as if the other partial systems were non-existent. Afterward, the component solutions themselves are interconnected by a set of transformations _I__- to give outright the solution of the entire system. The remaining work consists of the elimination or solution of the comparatively few constraints that appear at the cuts. The concept of "physical systemll, that can be torn apart, runs the whole gamut of physical sciences and includes both linear and - with certain pre- cautions - non-linear systems,

This paper employs for a simple illustration the solution of the partial differential equations of Maxwell, or rather the solution of their two-dimensional electric circuit model. A more general illustration has already appeared in reference (1) in connection with solving the partial differential equation of Poisson having about 400 variables by tearing the entire system apart into eleven parts. The reader interested in a more detailed philosophy of the method of tearing should consult reference (I). Actual numerical calculations have also been published in references (8) and (9) for studying the losses in the extended transmission systems of several interconnected power companies covering.several States.

This last example cited also illustrates the tearing apart of a physical system whose description requires far more than a set of linear algebraic equations 7 = 32. In transmission system studies the concept of current and impedance do not appear. Their role is replaced by the far mOre difficult

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concepts of real power and loss-coefficient, whose establishment re:?uires a series of assumptions ,and transformations. Loreover, the fj_nal results represent the most economic operating conditions of the entire transmission system. In general it was found in connection with combined steam and hydra-electric system studies that extensive problems of the Calculus of Variations lend themselves readily for solution by the method of tearing.

The Field Equations of Maxwell

Although the solutions of the field ec:uations of ibiaxwell are believed to be more involved than those of Poissonls equation (using either analytical or difference-equation methods), it so happens that in the method of tearing the solution difficulties are interchanged (for two-dimensional problems at least), The steps for solving the steady-state electromgnetic field are much simpler ---..--- than the steps involved in solving a steady heat-flow, even though the electric circuit model configuration also controverts this statement. The simplification is due to the fact that the ground points of all isolated portions of the electromagnetic field are at the same potential. The interconnection of Xaxwellian circuits involves only conventional shunt and series connections, and dispenses with the unfa,miliar Ifinterchange" and l'sneakll paths needed in interconnecting the Poisson circuits. Hence, a repeated development of the method of tearing for a different type of problem in terms of simpler and easier concepts is justified.

The reader will be assumed to have some fleeting acquaintance with the concepts of adding, multiplying and inverting matrices without, however, re- membering the actual rules. The accompanying tensorial concepts of transformations will be stated as needed, without any attempt at justification.

'ir o stant

reference will be made, however, to the lhext "Tensor Analysis of l<ettioorks" L- as TAI\I for more detailed treatment or proof of the more important statements.

Advantapes of Tearing

Several advantages may be claimed for the method of solving explicitly well-nigh any type of extensive physical system by tearing, as.compared to conventional methods of solving explicitly an entire problem as one unit. The following claims refer to exact and explicit solutions and not to iterative or relaxation types of solutions.

1) Existing digital computers may now be used to solve problems with extremily large number of variables that could have not been touched before. If a computer can solve a maximum of say 100 equations within a reasonable time, it can also solve a set of say 10,000 equations by tearing the corresponding physical system (not the equations) into one hundred parts.

2) The accuracy of solutions and the number of necessary significant figures depends now only on the size of the subdivisions (and nature of cuts) and no longer on the size of the original system torn apart.

3) The amount of computation involved in the solution is reduced by a surprizing amount. For instance, by dividing a physical system with an equation I = 3X into n parts

-J2 its inverse calculation % = ZI is

reduced by a factor of about 2/n . If n = 10, the factor is l/SO and with n = 100, the calculations are reduced to a fraction of about 1/5,@Jo* This reduction in the amount of comptation is equally maintained whether slide-rule, desk-calculator, low-speed, or high- speed digital computer is employed.

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4)

5)

6)

If several or all o.f the component subdivisions are identical, the amount of reduction increases by leaps and bounds. (Conventional methods of solution can not profit by the similarity of the component parts of a physical system.)

In solving a set,of N equations, volves ti2 elements.

the conventional inverse usually in- The inethod of tearing.introduces a new type of

inverse, (called "orthogonal" invers 3

) in which the number of elements are reduced to a small fraction of Q , say to ti2/ fi where n is the number of subdivisions. In the example worked out in reference (1) where a system with 400 variables is divided into 11 parts, the 160,000 elements of the conventional i.nverse are reduced to about 30,000; a reduction to less than l/5. In each new set of numerical solution the amount of computation is reduced to this fraction.

If a system already solved is altered in any manner, say is increased or reduced in size, the solution of the altered system need not be started from scratch. Only the solutions of the altered portions are changed, retaining the rest of the solutions unchanged. That is, solutions can participate in the same type of evolutionary growth that the physical systems themselves may undertake.

Solutions of particular problems may be filed away for future use in connection with still larger nroblems. It is possible to store solutions of standardized parts i.n file-cabinets in the same manner as standardized manufactured products are stored in warehouses. Entirely new problems may be solved by simply interconnecting already available partial solutions. Thus, the method of tearing opens up the possibility of mass-production and :nass-consumption of solutions.

All steps in the entire method of analysis are based upon the properties of physical systems and not upon the characteristics of mathematical equations. As a consequence, long drawn-out computations may be checked at almost every step by physical tests (say by Kirchhoff's laws) and any mistake made by the machine, or in the coding, and even in the analysis itself, may easily be detected before the computation proceeds too far.

Nest of the steps in the tearing process consists of a sequence of addition, multiplication, and inversion of matrices. Since all high- speed digital computers have standard routines available for such sequence of steps, the solution of extensive problems by the Lmethod of tearing requires in general no special coding; of the computers.

A Combined Relaxation and Iterative Method

The method of tearing is adaptable not only to exact solutions, but also to iterative and relaxation solutions of physical systems. The smaller subdivisions require far fewer steps of approximations than the original large system does. The solutions of the component parts can afterward be interconnected to .give the solutions of the resultant system.

As a matter of fact the very existence of a physical model suggests a new type of approximation method that combines the judgment accompanying relaxation Mth the repetitive stops inherent in iteration. The physical model used for tearing may be an electrical,mechanical or any other convenient model, including

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the physical system itself. If the impedances (or admittances) of the ultimate smallest components of the model are known numerically, the latter may be +elaxedl by tensorial methods in a series of steps to bring the model into mre easily calculable form. The relaxation is performed on the known impedances z or Y' before any numerical calculations have been made. (In the conventional method the relaxation is performed on the unknown solutions E or I as they are successively approximated.) The relaxed model may then be solved by iteration that converges much faster than the iteration performed on the unrelaxed model (difference ecuations or actual equations). The iterative solutions may include either the inverse matrix z (or Y') or only the final vector E (or f ) as desired.

The matrices associated wi.th the relaxed network (or .:lodel) are either strictly diagonal or contain only very few (usually 2 to 4) non-zero elements in a row so that each step in the iteration involves comparatively few numerical calculations, By combining the method of tearing apart physical systems, the relaxation of networks and the iterative solution with the aid of almost diagonal matrices, the possibility has opened up to solve on already existing digital computers successfully and rapidly immense physical systems (linear or non-linear) involving hundreds of thousands‘and perhaps millions of variables.

New Concepts and New Tools

The possibility of solving very large physical systems quickly and in easy stages with the aid of already existing computers (digital or analobqe) is due to the integration of several apparent&y unrelated scientific concepts into a power- ful universal analytical tool. A detailed description of these concepts is to be found'in reference (I). Some of the salient points of integration may be pointed out as follows:

1)

2)

3)

4)

5)

6)

7)

An electric-circuit (or any other) model of a physical system contains far more information about the system than a set of conventional -- equations do.

The concept of llorthogonalll network supplies all the missing variables and missing equations necessary to define completely a physical system.

The complementary concepts of tearing apart physical systems and interconnecting physical systems, reduce the theory of an infinite variety of physical systems to that of a comparatively few and simple basic elements.

The concepts of tearing and interconnection utilize many of the infor- mations that are missing from the conventional equations (be they difference equations or actual equations, such,as the dynamical equations of Lagrange).

Similarly the process of impedance-re,laxation can not be introduced with conventional equations.

As the entire m&hod of attack is based upon l~physicalls phenomena and not upon properties of ?nathemati.cal" equations, the concept of Ittensor" as the mathematical representation of a physical entity becomes an indispensable analytical tool.

The laws and rules of Tensor Analysis supply a variety of transfor- m&i,ons to pass freely from the 'leouations of state" of a physical

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system (or of its broken up components) to its "equations of solutions" and vice versa at arbitrary 'stages of the analysis.

8) The use of matrices allow easy numerical manipulation of a large number of variables,

9) The existence of high-speed digital computers allows the automatic sequence calculation of a* series of metric operations such as addition, multiplication and division (inversion),

A strong instinct toward a u.nYfied viewpoint of all physical phenomena and,for the unity of all sciences greatly facilitates an understanding of the method of attack shown here and its apnlication to diverse physical phenomena.

Solution of a Subdivision

The Network Equations

Electric circuit models of the electromagnetic field have been developed by the author in reference (3). They wer

77 iscussed fror

PY arious points 0 view

of the radio engineer by Whinnery and Ramo 4 ) McAllister 5 and Shelkunoff Two high-, "ii

6 . frequency models have also been constructed at Stanford University 7 0

Figure 1, containing resistors, inductors, and capacitors represents the field equations of Naxwell for a two-dimensional TE or THwave in any curvilinear orthogonal reference frame.

aB Curl % + dt = 0 Div B = 0' (1)

ai3 Curl H" = dt + T Div D = 0 (2)

The netw.ork of Figure 1 may actually assume any odd shape to represent cavity resonators, wave-guides, radiation fields etc.

The network configuration of Figure 1 may also represent the partial differential equation

Div y grad $8 + a$ + b = 0 (2)

where 7, a and b are known complex functions of two independent real variables u and v.

The components of the matrix y with complex elements represent the horizontal admittances (in the plane of the paper) of Figure 1; the complex scalar 'la" Gresents the vertical adm!ttances (connecting the planar coils to the ground). The complex scalar "bl represents the known currents I impressed at some of the junctions. The unknown fl represents the difference of potential E appearing between each junction point and the ground,

75


The Problem

Two problems will be investigated:

1) Find the value of Z at those junctions only, where the known I are impressed.

2) Find E at all junctions.

At first the simpler first problem is investigated. Afterward the additional steps needed for the second problem are developed.

It should be noted that if the partial differential equation (2) is replaced by a set of difference equations, the latter assumes the form I' = TE where P and Y' are known and E is unknown. The problem is then to establish the inverse eciuations E = zr where z is the inverse of 7. Since I is knolfnz, there- fore E may be calculated by multiplying the matrix ?! with the vector I.

The Subdivision

Let the net:\,ork of Figure 1 be subdivided into four parts, as shown in Figure 2. In order to simplify the presentation, the simpler network of Figure 3 will be studied instead.in detail. One subdivision of Figure 3b is shown in detail in Figure ,!.+a.

The equation r = FE of each subdivision may be solved as 3 = ZI' by x arbitrary method. In order to develop the tensorial concepts in an orderly manner, each subdivision (Figure &a) will be solved here by the tensorial method; that is, by tearing it further apart into its ultimate components, the llcoils't. The collection of these ultimate subdivisions (Figure Lb) is called a ttprimitivelf network (TAN p.86).

The Primitive Netldork '(Frame 1)

Since the problem is to establish the solutions E = ZI of the given network of Figure 1, the sa!ne type of equation will be established outright for the very simple lVpriiriitive It netiqork of Figure &b containing twelve coils, The pri dtive network is so simple that its equations of solutions 3 = may be established b-re inspection as 1 p--

--.-

76


(3)

where each Z is the inverse of the corresponding Y. That is2 Z = l/Y. id0 mutual impedances are assumed to exist between the coils and Zllhas a diagonal matrix, In the general case z,, has complex elements.

The Interconnected I‘:etwork (FrameA

Let the prixitive network of Figure ,!+b be interconnected as shown in Figure 5 (or Figure ha). The interconnected system has still 12 coils but now it has also 9 junction points. (The four ground points shown form one junction point only). Accordingly the network has 9-l = 8 junction-pairs and 12-8 = f+ meshes (TfiN p+ 75).

It should be especially noted that if the boundary points at the cut are temporarily ignored then:

1) The number of junction-pairs is ec,ual to the number of vertical coils (entering the ground).

2) 'The number of meshes is equal to the number of horizontal coils.

The bollndary points at the cuts are assumed to form additional junction- pairs.

Assumed Sleshes and Junction-Pairs

Although the meshes may be assumed quite arbitrarily, for the nest system- atic study it is advantageous to assume vertical meshes only. That is,-each mesh should pass throuj:h two neighboring vertical coils and through one horizontal coil, as shown in Figure 5a.

77


However, only 5 out of the 8 junction-pairs are assumed in Figure 5b because of the following principle (TAN p. 42'7).

"If no impressed currents I exist at some of the junctions, then no equations E = ZI are written for these junctions (as the corresponding Ic are zero) and the corresponding junction-pairs may be ignored."

At those points where the system is cut, an unknown impressed current I appears. Hence, the number of "active" junction-pairs is 5 (.!+ cut'points and 1 impressed current). (Later on in Figure 10 all the 8 junction-pairs are introduced.)

The Connection Tensor ci

In the "primitive " network of Figure 120 appears 12 "old" currents $, while in the interconnected network of Figure jlappeac9 9 "new" currents I . It is possible to establish relations between I and.1 by noticing&at "new" currents pass through each of the coils. Figure ,!+b flows the "old" current i7.

For instance, through coil 7 of Examining Figares 5a and 5b, it is found

that throu h the same coil now flows 2 mesh currents ir and -iq, also 3 junction- currents I E , Ic, and Ie. Hence, the relation between the "old" and "new" currents flowing through coil 7 is

i7 = fr _ iq + Ib + 1' _ Ie

Examining every coil in succession, the following transformation may be established between the'primitive" and "interconnected" networks


The Interconnected System (Frame 2)

The equations of the interconnected system of Figure 5 are The impedance matrix z,, is found from z 11 by the law of

z22 (5)

where the subscript t represents a transposed matrix and the asterisk a gate matrix. conju-

'The multiplication is performed in two steps. The first step is

For the second step the conjugate transpose of c' is


tiultiplying the last two matrices along the directions of the arrows, the resultant Z E, = Z,,I2*22

is found by c8 (zc) as shown. The equations of frame 2 are,

a i5 c a e P q r s

Ia Ib IC Id Ie ip iq .r .S 1. 1 I

66 -95

=? "7 z7 -z7

-%

Ee 0

1

0

0

0

(8)

A check on the correctness of the matrix multiplication is that z,,, must be symmetrical.

In these eruations the junction-pair currents I" to Ie are assumed to be known, while the mesh currents ip to is are unknown. The'voltages impr&sed around the four meshes are zero. The differences of potential Ea to Ee appearing across the five junction-pairs are unknown.

( or The above eciuations E2

t~compoundl~) matrices as = 2;2T2 may also be written in terms of '*partitionedt~

1 3 ii

? J

E = Z,f + E2F

iii fi = ;T3' + Z41 (9)

Elimination of Meshes (Frame 3)

Since the mesh currents i' are unknown and their impressed voltage e is zero, the four z may be eliminated. The elir$.nation of 5 may be done in several ways. In particular:

1) The currents are elirrdnated one at a time. On a digital compter this procedure is the quickest.

2) The currents 1' are eliminated all at once by finding the inverse matrix of Z then performing two matrix xtiplicstions and one addition ac- cording to k&e formula (TAM p. 61).

80


2; = z,-z2(z4)-lz, (10)

used. 3) When g current is eliminated at a time, this same-formula may be

vectors. But then 24 represents.one scalar only. Also Z2 and Z3 are merely

where The elildnation may be performed as a transformation

e2 = 3

q), z E2 22 3

w

This procedure however introduces quantities that cancel afterward.

Solution of One Subdivision

The resultant equations E3 = %,,P' are

(12)

The equivalent circuit of the above solution is given in Figure 6. I 3 consists of five coils extending between the five junction points (at which f

enters) and the ground point. The five coils are coupled by mutual impedances that are not shown on Figure 6a.

Tensor Networks

In order to simplify the analysis and manipulation of a large number of simultaneous equations describing a physical system, the concept of matrix (or rather tensor) has been introduced, represented by a bold-face symbol. Each bold-face symbol such as e represents a large number of conventional numbers expressed along a large number of reference frames. It is possible to perform a long chain of reasoning in terms of tensors only.

Similarly, in order to simplify the analysis of a large variety of extensive electrical networks, it is possible to replace a large number of coils (each performing the same physical role) by one,hypothetical coil, a so-called l%ensorll coil, drawn bold-face in Figure 6b. Each bold-face coil is represented by a bold-face symbol, so that a tensor equation may, be said to represent the performance of a tensor network (TAN p. 480).

81


In Figure 6b all boundary coils (tie-lines) leading to a boundary line are replaced by a %ensor tie-line", Similarly, all generators with known impressed T are replaced by a single bold-face %ensor generator". In the present example the tensor generator happens to have only one component; hence, it is drawn light-face.

Interconnection of the Subdivisions

The llPrimitiveft System Frame 1

Let the four, subdivisions o,r Figure 3b shown again in Figure 7. It is assumed that their solutions E = 21 have already been established. of the four subdivision (the "privitivel'

The equation

written as E system) before interconnection may be

1 = W1

03)

No mutual couplings exists between the subdivisions. such as 4

Each matric equation,

02) * = ZAAT" represents the five scalar equations calculated in equation

Altogether there are 4x5. = 20 scalar equations, corresponding to the 20 coils appearing in the primitive network of Figure ?a. The corresponding tensor network is shown in Figure To.

The Simplicity of Interconnection

The present network of Figure 7 is more complicated than the network representing Poissonls equations, as now additional coils appear that connect every junction of the planar network to the ground. Nevertheless the interconnection of the present type of Maxwell network is far simpler than the interconnection of the networks of Poisson. The simplicity is-due to the fact that now the ground points assumed in each subdivision are at the same potential after inter- connection9 On the other hand in interconnecting the Poisson networks the ground-points (or load-points) of each subdivision assume different absolute potentials.

That is in interconnecting the Poisson networks it is necessary to intro- duce lY.nterchanget' paths to calculate the differences of ground potentials. The interchange paths are absent in the Maxwell network and the five successive transformations are reduced to two only.

The Connection Tensor Cl

As shown in Fil;ure 8,. now not only the neighboring tie-lines are connected in series, but also the several ground points are interconnected by lines with

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E1 2

zero impedance. That is, now every interconnected tie-line_(8 in number) forms a circulating mesh through the ground, as shown at the right of Figure 8a. (In the Poisson network one of the tie-lines connecting two subdivisions always forms an interchange path between two grounds at different potentials.)

The relation between the 20 llold" currents Pi of Figure 7 and the 12 new currents I2 of.Figtqe 8a is (TAN p. 137).

i a i 2 2 4, B a P

The Interconnected System (Frame 2)

where The equations of the interconnected system of Figure 8 are E2 = z2212

The first, step is

83


The second step is

@It =

7 J

("& (Z&) =

m

05)

Each set of matrix multiplication indicated are performed separately and the resulting matrices (such as clt zAA e = z,) are added up.

The above equations are expressed in two groups. One group (4 scalar etluations)along the junction-pairs 3 and the other group (8 scalar equations) along the meshes m.

(17)

The equation is analogous to equation (9). It should be noted that the new voltage vector after interconnection is found as

E2 = (Q, El

84


ECe se %awEAa %aaEAb ‘Cd-Esd E . . . . . . . nC-EDC.

That is, all mesh voltages become zero, since all ground points are short- circuited,

Elimination of lleshes (Frame 3)

The unknown 8 mesh currents i are eliminated with the previous metho s shown, giving the solution of the resultant system of Figure 9 as E = z I 4 . There are only four scalar ec;uations representing the voltages E ac P i#e oss four junctions at which z is impressed.

Solving for the Mesh Currents

Since the component matrices z,, z,, and z of equation (17) are strongly di.agonal with many zero off-diagonal elements, it'is preferable not to perform the elimination, but to leave equation (10) in its given form. That is the solution of T = ?% is represented by the form

E = (“1 - Z2Y4Z3) P (19)

For any given numerical f, the substitutions indicated in equation 19 are to be performed in succession.

The ttorthogonalfl inverse matrix may be represented as

-1 2, z, q 222 = z - -1 3 z4

(20)

differing from (17) by the presence of-the constraint (mesh current) inverse z

k -1 ccc yz. The ~~orthogonal~~ inverse,matrix is strongly diagonal and contains

o ly a small fraction of the elements,that exist in the conventional inverse matr%x.

Thus, another important consequence of the process of tearing is that it requires far less computation to solve simultaneously for both forces and cony straints, than to solve for the forces alone.-

85


Calculation of Junction Potentials

-1 Enlarged Transposed C 2

In the example shown only some of the junctions have a current I impressed upon them. Hitherto it was also assumed that the potentials E of those junctions that have no I, are of no interest.

If the potentials of all junctions have to be known, then Figure 5b is replaced by Figure 10 in which the number of asslImed junction-pairs is 8 instead of 5. Three of the vertical impedances are also assumed as junction-pairs with a potential Ef, E j and E .

Q h However, the corresponding currents If, Ig, and Ih are zero.

Hence, "i, establishing the relations $ = $ T2 reznins unchanged, as given &y equ_ation (4). E2 = 'Cz'i El

But its transpose, ($-) e&ablishilg the relations

be deno e wilf have to be enlarged by three 2 ldditional

by ("i )t to differentiate it from ($=)t. rows. Hence, it will

e

The addition (namely e,) consists only of a unit matrix, rc

The Enlarged Impedance Matrix -I

1 -1 1 1 -1 1

I I I I I I I I I I

1 -1 1

The first product z c" is the same as equation (6). ts s

The second product, however, employs equation z

1 instead of er,uation (7). is the same as 2

Hence, the resultant

tg2the extra voltagez? of equation (8) except it contains more rows, corresponding

These extra three rows are c,, (2 c) where c‘, is the middle matrix of equation (21). L L

86


That is the enlarged zi2 changes from the square form of equation (Cs) to the rectangular form

f J = ii

Elimination of Neshes,

Eliminating the rows and columns of m

The resulting'equations become

E jl =

ii’, 32

(24)

The second row represents the voltages across the additional junction-pairs of one subdivision before interconnection. -.

Interconnection of Subdivisions

The previous steps are simply repeated. That is, the transpose of given in equation (14) is again enlarged by additional rows containing unit

'?i

matrices, EIence, the resultant z again will contain .xore rows than columns. The meshes are eliminated with the2$ormula of eouation (10).

87


If in Z 22 the extra rows are written separately as

b-5)

then equation (24) gives the sought after potentials as

The indicated multiplications should be performed only in terms of nunerical ?.

2)

3)

4)

5)

6)

7)

Bibliography

G, Kron: "A Set of Principles to Interconnect the Solutions of Physical Sys$emstt d, Scheduled to appear in the July 1953 issue of the Journal of Applied Physics.

G. Kron: "Tensor Analysis of Networks", John Wiley and Sons, 1939, (Book).

G. Kron: tlEquivalent Circuit of the Field Equations of Maxwell - I". Proceeding IRE, Vol. 32, Nay 1944, pp. 289-299.

J.R. Whinnery, S. Ramo: ItA New Approach to the Solution of High-Frequency Field ProblemsIt.

J.F. McAllister: "Equivalent Circuits of the Electromagnetic Field"+ GENERAL ELECTRIC REVIEW, Vol. 47, &rch 1941+, pp. o-14,.

S.A. Schelkunoff: "Methods of Electromagnetic Field Analysis". Bell System Technical Journal, Vole 27, 1949, pp. 487-509.

K. Spangenberg, G. Walters, F. Schott: I~Electrical Network Analyzers for the Solution of Electromagnetic Field Problems'to Proceeding IRE, Vol. 37, pp. 724-729, July 1949.

G. Kron: "Tensorial Analysis of Integrated Transmission Systems". Transactions AICEE, Vol. 70, 1951, pp. 1239-46; Vol. 71, 1952, pp* 505-12;. Vol. 71, 1952, ppa 81,!+-21; Vol. 72, 1953,

A.F. Glimm, L.K, Kirchmayer, G.W. Stagg: reanalysis of Losses in Inter- connected Systems". Transactions I=IEE, Vol. 71, 1952, pp. 79640s; Vol. 72, 1953

88


Fig, 1 * Equivalen% circuit of the fie9,d equations of Rxwell -i_n orthogonal curvilinear coordinates,

A) GIVEN NETWORK 6) FOUR SUUOIVISIONS

Fig. 2 - Ftmr subdivisions,

A) “INTERCONNECTED” SYSTEM OI”PRIMSIPIVE” SYSTEM (FRAME :I

Fig, 4 - One stiditision, Fig, 3 - Simple network assumed far illustration,

A) FOUR MESHES 8) FIVE JIJIJCl ION-PAIRS A) SCAL.AR NETWORK 8) TENSOl? NETWORK

89


D c

A B : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

Jsr;z$J. Jsr;z$J. . ..* . . . . . . . . . . . . . . . . . . . . . . . . .._... . ..* . . . . . . . . . . . . . . . . . . . . . . . . .._... JY4d JY4d

\;TENSOR ,TIE-LINE

. . .

:.. . . . . . . . . . . .: . . . . . . . . . . . . . . ..: D G

A) SCALAR NETWORK B) TENSOR NETWORK

Fig. 7 - The ltprimitivelV solution. (FRAME 1)

i

D C

A) SCALAR NETWORK

I

. . . . . .

/

zz\ 2 . . . . . . . .: . 51 i . . . ..-

ID i / ; IC, _ J&\! w

. . . , . . . . . . ..a.... ..‘.......... . . . . . . . . . .

\

6) TENSOR NETWORK

Fig. 8 - The t~interconnectedf~ solution. 0QJQI.E a

Fig, 9 - Final, solution. (ME 3)

b

Fig. 10 - 23troduction of all junction-pairs,

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A MXTHOD TO SOLVE VERY LARGE PHYSICAL SYSTUS Ilil USY ...biotek/kron/solvelargephysicalsystems.pdf · A MXTHOD TO SOLVE VERY LARGE PHYSICAL SYSTUS Ilil USY STAGE'S Gabriel Kron Consulting