10 PHILIPS TECHNICAL' REVIEW VOLUME 21

THE RESISTANCE NETWORK,A SIMPLE AND ACCURATE AID TO THE SOLUTION OF POTENTIAL PROBLEMS

by J. C. FRANCKEN. 518.5:53.072.13:621.317.729

Among the methods that can be employed to solve the Laplace equation for given boundaryconditions, that involving the use of a resistance network is in many cases highly attractive.It is a method applicable to two-dimensional problems and to three-dimensional problems wherethere is rotational symmetry. Setting-up the boundary conditions is particularly easy, themeasurements do not take up much time, and remarkably accurate results are attainable.

In many branches of physics one is frequentlyconfronted with potential problems, the solution ofwhich involves finding a function cp which satisfiesthe Laplace equation, viz., in rectangular coordi-nates,

Examples of quantities satisfying the equation arethe electrical potential in a space-charge free regionand the gravitational potential in the space betweenthe gravitating masses. The temperature understeady-state conditions of heat flow, and velocitypotential in a non-turbulent stream of incompress-ible fluid are further examples. Usually the valueof the potential function cp on certain closed surfaces,e.g. at solid boundaries, is known. In addition,sometimes the space in which cp must satisfy (1)is entirely enclosed by a surface at which cp isknown. In other cases the space extends to infinity,at which cp approaches a known constant value. Ifcp represents temperature, the constant value willbe the ambient temperature; if it represents electricpotential, the constant value will be earth potential.The example we shall be using to illustrate theemployment of the resistance network is drawnfrom electron optics, and concerns the potentialdistribution in electron guns for television picture-tubes. The surfaces where cp has known values willbe those of the electrodes of the tube.

Only in a few simple cases is it possible to expressthe required potential function explicitly in termsof its given boundary values at certain surfaces.Usually one has to proceed by other methods.Apart from numerical methods, which are nowwidely employed in conjunction with digitalcomputers, analogue techniques are particularlysuited to potential problems. One well-knownanalogue technique involves the use of an electrolytictank. A model of the electrode assembly, often an

(1)

enlarged model, is submerged in a conductingliquid. The potential distribution existing whengiven voltages are applied to the electrodes is notaltered when the model is submerged. The potentialdistribution in the electrolyte is measured with aprobe 1)2)3).The electrolytic tank has proved to he a valuable

aid in the solution of electron-optical problems. Ithas, however, its limitations and drawbacks: theconstruction of the electrode models is oftenlaborious and expensive, and measurements intheir vicinity are inaccurate because the liquid nearthe electrodes rises in consequence of capillarity.For three-dimensional problems with rotationalsymmetry tanks of "wedge" section are often used.A vertical section through this tank has the shapeof a wedge. Electrodes can often be constructedfrom strips of metal, the disadvantages of compli-cated models thus being avoided. On the other handa new disadvantage arises in that measurement nearthe axis of symmetry, precisely the most importantregion, is rendered very inaccurate by the markedcapillary rise at the sloping bottom of the tank. Itis not therefore surprising that, apart from theelectrolytic tank, other analogue techniques havebeen developed for the purpose of determiningpotential distributions. One of these is the resistancenetwork, which is the subject ofthe present articles] ..

1) G. Hepp, Measurements of potentinl by means of the elec-trolytic tank, Philips tech. Rev. 4, 223-230, 1939.

2) N. Warmoltz, Potential distribution at the igniter of arelay valve with mercury cathode, Philips tech. Rev. 8,346-352, 1946.

3) An example of the use of the electrolytic tank for determin-ing temperature distribution is described in F. Reiniger,The study of thermnl conductivity problems by means ofthe electrolytic tank, Philips tech. Rev. 18, 52-60, 1956/57.

4) Another method involves the use of conducting (graphite-surfaced) paper. This method can be employed for solvingtwo-dimensional problems. Electrodes are simulated bycutting them out in copper foil. Conducting paper cannot beused for solving rotationally-symmetric three-dimensionalproblems, whereas the resistance network can he used insuch cases.

1959/60, No. 1 RESISTANCE NETWORK 11

In theory, the solutions it provides are only approx-imate; nevertheless in many cases these solutionsare much more exact than those obtained with theaid of the tank. The resistance network was firstproposed by Hogan 5) and first used by De Packh 6).However, the credit for pointing out the high degreeof accuracy thereby attainable belongs to Lieb-mann 7)8).

Explanation of the principle of the networkbecomes more straightforward if the terms "differ-ential operator" and "finite-difference operator" canhe employed. We therefore start by giving a briefaccount of the two operators. Resistance networksfor two-dimensional and for rotationally-symmetricthree-dimensional problems are then discussed,followed by a description of an actual network ofthe latter kind, and of the manner in which it isused. This will be illustrated with the aid of apractical application. Finally, the implications ofthe fact that the network has finite meshes will bediscussed with reference to two examples,

Differential and difference operators

The sum of the second differerrtial eooefficierrtswith respect to x, y and z of any function w(x,y,z)is often denoted by the symbol \72:

()2W ()2w ()2W()x2 + ()y2 + ()Z2= \72W.

\72 is called the Laplacian operator. The effect ofthe operator on the function w will naturally dependon the point in space at which it is applied (i.e. \72wis again a function of x, y and z). We are now ableto express the Laplace equation (i.e, (1) above) inwords as follows: a function satisfying the equationhas the property that when operated on by the Laplacian\72, the result is zero at every point.

From the differential operator \72 we shall nowderive a finite-difference operator L that, operatingon the same function at the same point, producesalmost the same effect as \72 itself, i.e. Lw ::>.:::I \72W.For this purpose we consider an arbitrary point 0and three pairs of points, Pand Q, Rand S, andT and U, which lie respectively in the 'x~y and zdirections at a distance a on either side of 0 (fig.1).

ó) T. K. Hognn, A general experimental solntion of Poisson'sequation for two independent variables, J. Instn. Engrs.Austr. 15, 89-92, 1943.

6) D. C. de Packh, A resistor network for the approximatesolution of the Laplace equation, Rev. sci. Instr. 18,798-799,1947.

7) G. Liebmann, Solution of partial differential equationswith a resistance network analogue, Brit. J. appl. Phys. 1,92-103, 1950.

S) G. Liebmann, Field plotting and ray tracing in electronoptics, Advances in electronics 2, 101-149, 1950.

We can express the difference between the valueswQ and Wo of w at points Q and 0 by using Taylor'stheorem. This gives us the difference in terms of aand the derivatives of w with respect to x at point 0:

u

a

R a

97",03

Fig. 1. Illustrating the derivation of the difference operator L.

(2)

By replacing a in the above by -a, we obtain asimilar series for Wp - Wo:

(3b)

Adding these two series and solving for ()2W/()X2)o,we obtain:

The differential coefficient ()2wi"ox2)o is thus ex-pressed in terms of the differences (wQ-wo) and(wp - wo) plus a number of correction terms, whosetotal value can be made as small as desired simplyby making a small enough.. ()2W/()y2)O and ("02W/()Z2)O'the other differentialcoefficients occurring in (2) above, can be expressedas differences in an analogous manner. Inserting in(2) the expressions thus obtained, we have:

12 PHILIPS TECHNICAL REVIEW VOLUME 21

Introducing the operator L, we now put:

We see from (5) that (Lw)o is an approximation to(\72W)0' approaching it all the more closely accordingas a is made smaller. L is the finite-difference operatorreferred to above; it is so caIied because L denotesan operation whereby the finite differences Wp - wo,etc. are used.

This procedure for deriving a difference operatorfrom a differential operator can also be followed inthe more general case where the differential equationinvolves the first derivatives tow/ox, etc.) as well asthe second derivatives. By subtracting equations(3a) and (3b) from each other, one can arrive at anexpression for (ow/ox)o involving (wp - wo) and(wQ - wo) and a series of correction terms whosesum approaches zero as a goes to zero. We shallmake use ofthis when dealingwith three-dimensionalproblems with rotational symmetry.

In the special case where w is independent of z,we have WT = Wu = wo, and (6) simplifies to

1(Lw)o =- (wp + wQ+WR+ Ws - 4wo). (7)

a2

We can now go on to discuss the principle of theresistance network.

A resistance network for two-dimensional problems

To start with we shall confine ourselves to two-dimensional cases. There is then one direction -the z-direction say - in which the function beingsought does not vary. In these cases 02rp/OZ2 = 0,and Laplace's equation assumes the form:

To take a definite case, let us consider the exampleshown infig. 2. The three closed outlines SI' S2 and S3

represent sections taken at right angles throughthree infinitely long prisms. On the periphery ofeach prism rp has a known constant value. Theproblem is to find a function rpwhich satisfies (8)

in the area within S3 but outside SI and S2' and whichassumes the prescribed values along sI' S2 and SS.

Over SI' S2 and S3 we place a square grid, the linesof which are parallel to the x and y axes and spacedat intervals of a (see fig. 2). The "grid lines" inter-sect at "grid points". We shall refer to two gridpoints as "adjacent" if their distance apart is themesh width a. In fig. 2 the outlines SI' S2 and S3

have been drawn in such a way that they interseetthe grid only in grid points. This is not a necessarylimitation, but it will simplify discussion, and theexamples we shall be dealing with will subject to it.

y

t

I 53

1/'\1/ 1'\

1/

,'\. SI

S:J - ---

-,

~ _ ..-rl+ ...p!. -4-

a

99290

Fig. 2. The unknown function cp has known values along theoutlines 51' 52 and Sa. Throughout the area inside sa but outside51 and 52 it must satisfy the Laplace equation. It is assumedthat 51' 52 and Sa are such that they interseet a superposedsquare grid (mesh width a) only in grid points.

(8)

We shall refer to grid points located on the outlinesSI' S2 and S3 as "boundary grid points", and to theremaining ones in the area wherein rp has to bedetermined as "internal grid points".The following proposition underlies the principle

of the resistance network employed.If each internal grid point is allotted a value tp"

such that between the value rp* at any such point andthe values at adjacent grid points the relationshipLrp*= ° exists and if at the boundary grid points tp"has the boundary values specified for the requiredfunction rp at those places, then the difference betweenrp*and rp at the internal grid points will approachzero as the mesh width (a) approaches zero.As we have already pointed out, there is a close

connection between the difference operator Landthe differential operator \72; the proposition juststated is accordingly a plausible one. Its truth canbe proved rigorously by demonstrating that at allgrid points rp- tp" is smaller than a certain finite

1959/60, No. 1 RESISTANCE NETWORK 13

quantity that approaches zero as the mesh, width adoes so 9). .For the purpose of finding cp* values, a network

of resistors is built up. We shall see later whatrequirements the component resistors have to satisfy.The junctions of the resistance network will corre-spond to the grid points in fig. 2; accordingly, fourresistors will meet at each junction (jig. 3). We shall

97:123

Fig. 3. Resistors Rl' R2' R3 and R4 meet at 0, a junctionin the resistance network. In the two-dimensional case, PQrepresents a line parallel to the x-axis, and RS a line parallelto the y-axis; in the rotationally-symmetric three-dimensionalcase PQ represents a line parallel to the s-nxis, RS one parallelto the r-axis,

refer to junctions corresponding to boundary gridpoints as "boundary junctions". Between theboundary junctions we may apply voltages that areproportional to the differences between cp values inthe corresponding boundary grid points. If thelowest value of cp at any of the boundary grid pointsis CPminand if we take the potentialof the corre-sponding boundary junction as a datum for meas-uring the potentials Vb of other boundary junctions,then any of these latter potentials is given by

The suffix b to Vand cpindicates that they relate toboundary junctions in the resistance network and

9) See S~ Gerschgorin, Z. angew. Math. Mech. 10, 373-382,1930, orL, Collatz, Numerische Behandlung von Differen-tialgleichungen, Springer, Berlin, 2nd edition, 1955, p. 320et seq. Obtaining numerical solutions of the Laplaceequation likewise often involves the solution (preferablywith the aid of a computer) of the finite-difference equationLq:>·= 0, and the results obtained are accordingly onlyapproximate. However, these methods can be based ondifference operators that are better approximations to theLaplacian than L is, the series of correction terms in (5)beginning with a higher power of a than a2• If terms invol-ving a2 etc. are to be eliminated, additional equations mustof course he available; this means considering not oulyadjacent grid points, but also additional grid points in thevicinity of a given grid point. See for example Collatz,loc. cit., p. 352. .

to the corresponding boundary grid points. 1/ fJ is aconstant of proportionality. We now allot to eachinternal grid point a value

tp" = fJV + CPmin' (10)

where V is the potential measured at the correspond-ing junction in the resistance network. Through aresistor connecting two adjacent junctions in thenetwork, e.g. that between Pand 0 in fig. 3, flowsa current having the value (Vp - Vo)/RI. ApplyingKirchhoff's law to junction 0,we find that

Vp- Vo VQ- Vo VR- Vo Vs- Vo-=--+ +---+ =0.n, R2 Ra R4

From this and from (10) it follows that for any gridpoint 0,

(ll)

Comparison of (ll) with (7) makes it clear that, ifall four resistors have the same value,

(Lcp*)o = 0

at any grid point O. Since, in addition, cp* on theboundary curves has the boundary values laid downfor cp, cp* constitutes an approximation to the re-quired function cp, provided all resistors composingthe network are of the same value. The question asto how close the approximation is will be dealt withat the end of the article.

(9)

Three-dimensional problems with rotational sym-metry

More common than two-dimensional problemsare three-dimensional problems with rotationalsymmetry. If the rectangular coordinates are con-verted to cylindrical coordinates (r, zand D injig. 4), the s-axis being made to coincide with theaxis of symmetry, the Laplace equation assumesthe form:

(12)

z

_Z

97<06

Fig. 4. r, zand 11,cylindrical polar coordinates of a point P.

14 PIHLIPS TECHNICAL REVIEW VOLUME 21

Owing to the rotational symmetry {},does not appearin the equation. As in the two-dimensional case,therefore, we have a differential equation involvingtwo independent variables, these being zand rinstead 0.£ x and y. The Laplacian operator is now

(02 1 0 02

)-+--+-or2 r ör OZ2·

In order to derive a finite-difference operator from(13), we let (13) operate on an arbitrary functionu(z,r), and consider a point 0 and the two pairsof points P,Q and R,S which lie in the zand rdirections respectively, at a distance u on oppositesides of O. Expressing the differential coefficientsas differences, in the same manner as on pp. 11and 12, we arrive at the following:

where

(Mu)o = ~ ~(Up - ua) + (uQ - uo) +a2 ~

+ (1- 2uJ (un- uo) + (1+ 2

uJ(us - uo)}. (15)

The required finite-difference operator M is thusdefined by (15).We now superpose on the z,r-plane a square grid

with a mesh width of u. The mathematical pro-position stated for L on p. 12 is also valid for MlO).The values of rp*that are allotted to the internalgrid points must now satisfy the relationship

Mtp* = 0,

while tp" at the boundary grid points must have theboundary values laid down for rp. tp" will then bean approximation to tp; in other words, tp* willapproach tp as the mesh width u approaches zero.As before, it is possible to build up a resistance

network whose junctions have potentials correspond-ing to tp* values. In order to deduce the requirementsthe resistors will have to satisfy, we shall proceedas before, but this time (ll) must be comparedwith (15). Having done this, we find that rp*

10) For the proof, see Gerschgorin, loc. cito The proof giventhere is a general one valid for all elliptical differentialequations. (8) and (12) are equations of this type.

satisfies the relationship (Mrp*)o= 0 provided that

(13)

2.:2.:~:~=l:I:(I-~):(l+~) (17)n, R2 Ra R4 2r 2,.

{see fig. 3).If the grid is so positioned that the z-axis coincides

with one of the grid lines, then at each of the gridpoints

r =ja,

j being an integer. (17) now becomes:

1 1 1 1-:-:-:-= 2j:2j:(2j-I):(2j+I). (18)Rl R2 Ra R4

In the present case, then, the resistance values mustdecrease with increasing distance from the z-axis.In the network shown in jig. 5, (18) is satisfied forall values of j except j= O. This exception is a pointthat we must look into.

(14)

(16)

97""07

Fig. 5. Part of a network for rotationally-symmetric three-dimensional problems. Resistors meeting at junctions at whichj ¥= 0 have values satisfying the relation (18). For j = 0(junctions on the axis) the resistors satisfy (20).

The resistors lying along the axis

The zero value of j, appropriate to points on theaxis of symmetry (z-axis), gives rise to complica-tions. This is clear for example from the conclusion

1959/60, No. 1 RESISTANCE NETWORK 15

that can he drawn from (18), namely that forresistors meeting at a junction on the axis, Ra musthe negative if RI' R2 and R4 are positive. In addi-tion, (14) and (15) involve indeterminate % termswhen r = 011). These complications can be avoidedby reverting to the finite-difference operator L (see(6)) as an approximation to the Laplace operator forpoints along the z-axis. The reason we abandoned Lin favour of operator M when we took up the three-dimensional case with rotational symmetry was thatthe former would have led to a three-dimensionalnetwork. This objection does not, however, apply topoints on the axis of symmetry. Let us consider apoint 0 on that axis (see fig. 1; we shall assumethat the x-axis in this figure represents the axis ofsymmetry). At point 0, then,

For such a point, therefore, we can rewrite (6) inthe form

Comparison of (ll) with the above expression makesit clear that, in the grid points on the z-axis, tp" willsatisfy Lcp*= 0 provided that

1 1 1 1- . - . - . - - 1 . 1 . 2 . 2 (20)Rl .R

2•Ra .R4 - . . . .

These conditions have in fact been satisfied in thenetwork of fig. 5, where the resistors forming theaxis have the value 2R.The network of fig. 5 is not used in practice;

practical versions extend to one side of the z-axisonly. Such networks are perfectly satisfactory ifthe axial resistors are given a value of 4R insteadof 2R. The validity of this can be confirmed byreasoning as follows. Imagine the 2R resistancealong the z-axis in fig. 5 to have been replaced hytwo 4R resist anc es in parallel, as in fig. 6a. Onaccount of the rotational symmetry of the system,no current flows from the upper portion to the por-tion under the z-axis. The lower portion can th ere-

11) Along the axis, brp/br and the other odd-order derivativeswith respect to r are all zero in consequenceof the symme-try of the system. Hence the middle term of differentialoperator (13) becomes 0/0; similar indeterminacies occurin the correction terms of (14). If the terms of the finite-difference operator (15) (with u replaced by rp) are ar-ranged in a slightly different manner, the expression(a/2r)(rpQ - rpp) appears which likewise has the value 0/0.

_2

gFig. 6. Network (a) is equivalent to that in fig. 5, the 2R resis-tors along the a-axis having been replaced by two parallel 4Rresistors. On account of symmetry, no current flows from thepart of the network below the e-axis to the part above it. Thelower half can therefore he removed without affecting theupper half, as in (b).

fore be omitted (fig. 6b) without making any differ-ence to the upper portion 12).

(19)

Design and use of the resistance network

Resistance networks suitable both for two-dimensional problems and for three-dimensionalproblems with rotational symmetry have beenconstructed in several Philips Laboratories. Thetwo designs are identical apart from the values ofthe resistors and we shall therefore confine ourselvesto describing the network for solving rotationally-symmetric three-dimensional problems. This net-work is constructed according to the arrangementshown in fig. 6b. It extends over 50 meshes in thez-direction and over 25 in the r-direction, and, isaccordingly composed of (51 X 25) + (26 X 50) =2575 resistors in all, which are mounted on the backof a sheet of insulating material (fig. 7). The junc-tions have silver-plated contact pins that passthrough to the front of the panel.

In order to determine the potential distributionin some electrode assembly (for example), thesystem is simulated on the resistance network bylinking the junctions corresponding to the electrodeoutlines with copper wire. In principle it wouldbe possible to apply voltages across the simulatedelectrodes in the manner described above (p. 13, firstcolumn). This is not necessary, however: by the

12) The network of fig. 6b can be arrived at directly by writing(19) in the form:

(Lw)o= ~~(IVP-1VO) + (1VQ-1VO) + :(1Vs-tVo)f.

On comparing (ll) with the above, we obtain the condi-tion

1 1 1 1R-:R :R :R =1:1:0:4-,1 2 3 4

which is satisfied if RI = R2 = 4R, R3 = 00 and R4 =R.

16 PHILlPS TECHNICAL REVIEW VOLUME 21

Fig. 7. Some of the 2575 resistors composing the networkdesigned for three-dimensional problems with rotationalsymmetry.

following simple procedure the required potentialdistribution can be found more conveniently. Oneof the electrodes, G for example (see fig. 8), is con-nected to terminal B of potentiometer AB, and allthe othcr electrodes (K and Al in fig. 8) are con-nected to A, the other potentiometer terminal.A and B are connected up to an accumulator. Tomeasure the potential at, say, the junction P, thelatter is connected to the slide contact of thepotentiometer via a null indicating instrument.Once the slide has been brought to a position where

K A, Cp....._...........__ ........_~D........................... ..................................................... .

~PW.;.(:::::::::::::::::::::::............................................................................

_Z

Fig. 8. Bridge circuit for measuring the potentials of junctionsin the resistance network. K, G and Al are electrode models.The configuration is the same as in fig. 9. AB is a 1000 npotentiometer with a setting error of about 0.01 IJ. The nullindicator is an electronic D.C. millivoltmeter with an internalresistance of 0.6 MIJ.

the null indicator shows a null reading, the poten-tiometer setting indicates the potential differencebetween P and A (or B) as a proportion of thepotential difference between Band A. The valuethus found is therefore the potentialof P whenelectrode G has unit potential and all the otherelectrodes have zero potential. By repeating themeasurements for the other junctions one obtainsthe potential distribution under the above-mentionedcircumstances. One of the other electrodes, Alfor example, is now given an effective potentialofunity and the others are held at zero; the potentialdistribution is then measured again. The potentialdistribution for any given combination of electrodepotentials is then found simply by combining theseresults linearly (superposition). Generalizing, ifthere are n electrodes instead of three, the meas-urements have to be repeated n - 1 times inorder to find, by linear combination of the results,a solution for any given set of electrode poten-tials. The convenience of this method lies in thefact that no adjustment or measurement of voltageis necessary.

In view of the accuracy required, all the resistorswere wound from manganin wire, to tolerances of±0.2%. Liebmann 7) has pointed out that theaverage error arising in the measurement of poten-t.ial, due to inexact resistance values, is much smallerthan the errors in the resistances themselves, beingfrom a tenth to a hundredth thereof. This is aconsequence of the statistical properties of the net-work, whereby errors are levelled out. The tempera-ture coefficient of manganin is so small, the voltageemployed (usually about 2 V) is so low and thephysical dimensions of the resistors are so largethat there is no fear of errors due to heating-up ofthe resistors .

The upper limit to the (in principle, arbitrary)value of R (see fig. 6b) is fixed by the requirementthat the highest value in the network, which is 4R,shall not be an unreasonably high one for wire-wound resistors. On the other hand the smallestresistors must not have too low a value, otherwisethe current through them would be large enoughto set up appreciable potential differences in thecopper wires representing the electrode outlines.In the present networks, R has the value 3600 Q.The extreme resistance values are therefore 4R =14400 Q and Rj50 = 72 Q.A valve voltmeter type GM 601013) serves as the

null indicator. It is a D.e. millivoltmeter combining

97427

13) A. L. Biermasz and A. J. Michels, An electronic D.C.millivoltmeter, Philips tech. Rev. 16, 117-122, 1954/55.

1959/60, No. 1 RESISTANCE NETWORK 17

Fig. 9. The resistance network is mounted on the back of a large board of insulating material.The silver-plated contacts that are visible on the front of the panel are the junctions of thenetwork. The electrode configuration seen on the board corresponds to a problem dealtwith in this article by way of example. Lengths of 2 mm copper wire, representing electrodeoutlines, are attached to the appropriate contacts with metal clips. Electrical potentialsare applied to these electrode models via heavy copper strips lying along the edges of thenetwork. In order to prevent differences of potential arising along the electrode outlines,each is linked to its copper strip by more tban one wire. In the photograph the user hashis right hand on one of the knobs of the decade potentiometer; with his left hand he ispressing a key that enables him to check that the needle of the instrument he is watchingis giving a null reading.

great SenSItIvIty (readings down to 2 [J.V can beobtained) with a high internal resistance (0.6 MO).The "null current" is therefore less than about3 X 10-12 A, which is so small that it makes noperceptible difference to the potential distribution.If it was other than very small it could give rise toappreciable errors, particularly in measurements onthe axis of symmetry, where the highest-valuedresistors lie.

The potentiometer must be very accurate, as itserrors show up unchanged in the results. Thepotentiometer employed had an average accuracyof 1 in 105•

A photograph of the resistance network appearsin fig. 9.

Fields of infinite extent

The region throughout which the required function is presentis by no means always enclosed by a boundary such as S3 infig. 2. Often it extends to infinity, 'P approaching a constantvalue 'P"". One can proceed as follows in such cases. First ofall, the electrodes are plotted on the network on a scale sosmall that the junctions around the edge correspond to gridpoints where 'P is already fairly close to 'P 00' All the junctionsaround the edge are joined up and given a potential corre-sponding to 'Poo' The simulated electrodes are given potentialsin proportion to those of the actual system and an equipeten-tial curve, just enclosing the area within which 'P has to befound, is then determined by measurement. Subsequentlythe scale is increased in such a way that it is still just possibleto accommodate the equipotential curve thus found - whosepotential is now known - on the resistance network; from thenon this curve is treated as an electrode. A similar method can

18 PHILIPS TECHNICAL REVIEW VOLUME 21

bc employed when it.Is desired to investigate only part of theregion where ip is present, and to plot it on a very large scale,so that one or more electrodes or parts thereof fall outside theresistance network.

The same measurement procedure can be followed as before,one of the electredes being given a potential differing fromthat common to other electrodes, and the nett potential dis-tribution being found by linear combination of the results ofsuccessive measurements under these conditions.It is often possible in practice to employ less laborious

methods for circumventing the limitations of the resistancenetwork due to its finite dimensions. We shall return to thispoint when discussing one of its applications.

Example of an application: the determination ofthe cut-off voltage and maximum cathode loadingof electron guns

Often potential distributions are determined toserve as basis for the calculation of electron paths.Here we shall deal with a different example, namelythe measurement of potential distributions withintetrode electron guns for cathode-ray tubes as a steptowards determining the cut-off voltage and themaximum cathode loading. These two quantities areof importance in the design of electron guns.

For our present purposes an electron gun can bestylized as a set of parallel flat electrodes of infiniteextent. The simplest form of gun (the triode gun in.fig. 10) comprises three electrodes, the cathode K,the grid G and the anode ~. Grid and anode havea circular aperture. The centres of the apertureslie on an' axis perpendicular to the cathode, andconsequently the potential field exhibits rotationalsymmetry about this axis. The anode has a positivevoltage VA' of 15 kV, say, with respect to the

/

~AK 97",09

Fig. 10: Diagram to sho'w thc arrangement of electredes in atriode gun. K cathode. G grid. A anode.

cathode. The grid voltage VG (which is also measuredwith respect to the cathode) serves to modulate thebeam current I issuing from the gun. For a givenanode voltage VA' VG can be adjusted to a valuesuch that the beam current is reduced to zero. ThisVG value is always negative, of course; it is termedthe cut-off voltage Vc. It is an important quantity,determining the maximum signal voltage that may

be applied to the grid without its becoming positive(see fig. 11). Moreover, there is a simple relationshipbetween Vc and the maximum current Imax thatthe gun can deliver under these conditions, viz. 14):

~Imax ~ 3X 10-6 Vc"2" ampere (21)

(Vc in volts). It is therefore desirable that somemethod should be available for determining Vc from

Imax

I

I

Fig. ll. The above curve showing the beam current I of atriode gun as a function of the grid voltage VG, the anodevoltage being constant, is a normal IA- VG characteristic for atriode. If excessive spot "blooming" is to be avoided, thesignal voltage must not he allowed to push VG above zero(otherwise grid current would start to flow). The "blacks" inthe signal current corrcspond to - Vc, the cut-off voltage,Hence Vc also represents the maximum signal voltage.

the dimensions of the gun and from the voltageapplied to the anode.

For this purpose we employ a graphical method,the graphs being derived from measurementsperformed with the resistance network. Use can bemade of these graphs in the design of tetrode guns.A tetrode gun has a fourth electrode which ismounted close to the grid in the space between gridand anode (fig. 12). The fourth electrode is given apositive potentialof about 300 V and is known asthe "first anode", the original anode being called"final anode" in order to distinguish between thctwo. In practical cases the voltage on the :finalanode has but little influence on the cut-off voltageand the beam current, and consequently we only

14) See for example M. Ploke, Elementare Theorie der Elek-tronenstraWerzeugung mit Triodensystemen, Z. angew.Phys. 3, 44.1-449, 1951 and 4, 1-12, 1952.

1959/60, No. 1 RESISTANCE NETWORK

need consider the triode portion K-G-Al• Even so,the system possesses properties different to thoseof a normal triode gun. In the latter, grid and anodeare comparatively far apart. Consequently the fieldbetween these electrodes is more or less uniform and

K

Fig. 12. In a tetrode gun the first anode AI' which is given apotentialof about 300 V, is placed close to the grid G. Thefinal anode A2 carries a high tension of about 16 kV.

its strength is given by the potential differencebetween grid and anode divided by the distanceseparating those electrodes. Hence this ratio deter-mines the cut-off voltage, the diameter of,the anodeaperture playing no part at all. In a tetrode gun,on the other hand, the clearance between grid andfirst, anode is small and the field between them isanything but uniform. Consequently the potentialdifference and distance between these electrodeseach exercise a separate effect, and the size of theaperture in the first anode also has' an influence onthe cut-off voltage.

Let us take the following for the potentialdistribution along the axis (assuming tp(O) - 0):

tp(z) = V_;flf(z) + Vc h(z) .

fez) represents the variations in potential along theaxis when VAl = 1 and Vc = 0, and h(z) representsthe corresponding distribution when Vc = 1 andVAl = 0 (jig. 13). The beam current will be cut offwhen the current density in the centre of thecathode (where current density, as a function ofposition on the cathode, always exhibits a maxi-mum) has become zero. Let us assume that theelectrons have no initial velocity on quitting thecathode; if that is so, the beam current will be cutoff the moment that the potential gradient at thecentre of the cathode becomes zero, for the fieldstrength E(O) will then be zero' at that point.Since E along the axis is -"ötpj"öz, we can find thefield strength at the cathode centre by differentiat-ing (22) with respect to z and putting z = 0:

E(O) = -HVAd'(O) + Vc h'(O)~. '. (23)

E(O) becomes zero when Vc • -~j'(O)jh'(O)~ V~l;hence the cut-off voltage is

r (0)Vc=-- VAl = DVAl·h'(O)

97<11

The quantity D = j'(O)jh'(O) is known as the"penetration coefficient" or "Durchgriff".With the aid of the resistance network, curves

f(z) and h(z):»were determined for many differentcombinations of electrode dimensions and separa-tions, D being derived from the slopes of thesecurves at the cathode. Itwas found that their slopesat this point hardly alter in consequence of a changein the axial thickness of the first anode; hence Dis virtually independent of that dimension. Thediscovery was welcome, because it meant oneparameter less to be considered. Our 'investigationswere limited to the case where the openings in gridand first anode have the, same diameter. Since it isonly the.ratios between electrode dimensions thatmatter, the parameters are finally reduced to three.The overall results of the measurements are dis-played in the set of graphs appearing in jig. 14.From these graphs one can determine the pene-tration coefficient of a gun whose dimensions areknown; Vc, the cut-off voltage, can then be calcul-ated with the aid of (24).

'{('?r=0)

15

P(Vr;=O) A,(I1I,=I)/

la

/ / I V /

:- 7I

/

/

rGv

AI=rG, ·r(Z)j.o'

•

(22)

15g

19

(24)

97~2~

20 25 30_z

~ - / -/ I I I/

/ k d I

,/

/ htz) rAI-rG

rGto-.-_

15

10r

i 5

00 5 la 15b

20 25 30____z

Fig. 13. a) Potential distribution f(z) along the central axis inthe triode portion of a tetrode gun when Vc = 0 and VAI = 1.b) Potential distribution' h(z) along the same 'axis whenVc = 1 and VAl = O.

20 PHILIPS TECHNICAL REVIEW VOLUME 21

.Fig. 14. Graphs allowingD, the "penetration coefficient", and 5, the "equivalent anodedistance", to he deterrnined for an electron gun of given dimensions: For the meaning of

. the letters see fig. ,13.

l\. --.. _I~ I 1/·." f\. --..~\'1>~I.'KO~~ r-, .....V'~' ~~'\lj r--.O"",O fj~ I"

" I><'Tl' l-

r"'" .... " ~ l'-q~l--~I- ~...... r-, ...... r-, " 0 i'

l'\r-, ~ ~O)ife Vr--... .~ t- l"'"'" <, ~f>< .~

~~ ~~ c- " 1'..0 <~ r-s'\ r"~ .......... ~ ~v .........'"f\. 1'\ <, ~ J.ó:"" r-, I'-..

1'\ " ...... r- ~~ .........

"0 r-,r-, "r-, ~O~O ' r-,r-, "C?ól~ r-, .........

"ILhrG=0,31

....... "o~ r-, r--... r-.....1''1-... ~ ......."

0,2

.0,1

. 0,1 0,2 0,3

0.4. "- 1'-.. I1 /i"- I .D( J I .,

r- I" SI'Pf6 ~ / Ij

1'\ " OP-"''Y " I I" 0,~ r-,~ .......K 1<J~~

11"- I". I"-.

1\), ."o~-i"~ I'.. ....... :> o{S'LJ~ "~ ~ I'.. k" K ~<?,""~ "- ~ " ~0 ~ ~ <,

", 'k'~'~~o è.,.-

-,~ <,

" ~~

"I'" :::... I'.. ......f. '() ;S ;>r-,<, <, ~ f-.. ~ ~ -, <,

" " -, <, 0<,

~<, <,

1~'ë;=0,51~Keo~ ~~ <,'I.. -, <, ~

0,2

0,1

0,1 0,2 0,3

0,4. " 1--. I1 11 11

'" I'.. .1 ,,<J".. I1

r- r-.....~$_~" ':if Not-, I'.. j I"cr, ~/~o?r-, '" K I t-j,qot-f!_r-, I"-".... r-r-, r-, " v ......1/ ~

J"..~ .......K )1'-...0"0 I .......

r- ...... " .......... 11, r-, ..; q~ }l"-r-, <,

0';0 "I' V r-,r--. i-~ <,

1~1l;=0,81 ....... t-,I""'--

0,2

"

0,1

0,1 0,2 0,3

0,6

r-,JI~ J1/~ ~.!...I-r-; r-, W "'" 17 I

r-, !loo... ~'

~~ ~V

" ....... , f;:: °"0 J~ " ......

" l><l J I.: ~r-, I ~--:$t ~ro;;: ....... o ~I"'" ~ " I"-. K.,.lo. I)<........ ~ ~ " , "'bt' r-....I":~ , r<::

~~~ ~~~

....... I"""~ ~ ,-' I-l\. r-, '()

l\. ...... i""::: t-.... i'. ~ r-; r-r-, l\. l\." ~~1'--......

r-," I"-. ....... '"D~~

~~ , J"..l0!r!! =0.4.1 '1--... l\. r-,

0,1

D,1 0,2 0,3

0,4.

" 1 1l\.

~~

7"\ 1"" ~'~ I- IT~'. I'.. ..... ,_. r-, ~ /I"~ r-, 17 "'" t-... ~_I-- c.,V-<, r-, ~~o ~~ ...... ........ '--" ~~

-

",

I,.;' r--,9"s ~~ <, " ,,~ <, V ........

~ ........... " ~s~ 7'r-,""" ~ ~ r- IS: ~ <, ~ ~

~ 1"-.0,;:;1 .......~o ........ ~~~E'&- -, t>c J".."o,j) ..... ....... r-....lY2rG =0,61 ~ <, r-, "'"

0,2

0,1

0,1 0,2 0,3

" r-, I

" r-, Ir-, ...... ~

I" r-, t......I

" I<, ) /1'- j ~I-

~ ~ r-, 11 I~~qo;...-,r-, ~ ~¥_~ ~ 1'Sj,

~ t-r- ~'V ~~ J-FI-~ "'IC I?r-.... ~ ~I

'" ~V ~ .~~ " i'. ~~-f-~ ~

/ I::)~t-

r-,~ ~ "" r-.... l>"-.o"s Nr-, I"--~;,j ~ NV'~ ~ r'..o.,.lo K:' r-,""'lI~ "" ~ ~ f't-...lY2rG' 1,0 I l'" N I' r-,

0,2

0,1

° 0,1 0,397-413·

1959/60, No. 1 RESISTANCE NETWORK 21

A second .important quantity provided by thesame measurements is the current density in thecentre of the cathode (the maximum, cathode load-ing). It is impermissible for the current density toexceed a certain value at this point, and this natu-rally constitutes a lim:it in design. Using a simplifiedtheory, Ploke 14) has derived the following for thecurrent density at the cathode centre:

-3 aj(O) R::1 4.8x 10 (I Vc)1' X S-2 Afmm2, (25)

in which I is measured in amperes and Vc in volts,and where

1s=--mm.

h'(O)

The parameter s is sometimes called the "equiv-alent anode distance". The reason for the name isthat if a potential difference of Vc+ VG existsbetween cathode and anode in a plane parallel diode(VG < 0), the field strength in the intervening spacebecomes equal to E(O) when the anode-to-cathodedistance is s. This can be deduced from (23) and(24).In order to calculate D we had to determine the

slope h'(O) at the cathode;·we can now use it againto calculate s. Curves from which s can be read offfor a gun ofinov.,;n di~e'nsions also appear in fig. 14.Knowing s, we can work out from (25) the maximumcathode loading for any value of beam current.

Numerical example

Suppose that we want to determine the cut-off voltage andthe maximum cathode loading for a tetrode gun with thedimensions k = 0.15 mm, d = 0.15 mm, 2rG = 2rAI = 0.75mm and 1= 0.35 mm (see fig. 13), and with an acceleratoranode potentialof VAl = 300,V.By calculation, k/2rG = 0.20, d/2r.G = 0.20 and 1/2rG =

0.466. The values of D and s/2rG appropriate to these valuesof k/2rG and d/2rG are now determined from the two graphsin fig. 14 for 1/2rG = 0.4 and·0.5. We then find by linearinterpolation that, for 1/2rG = 0,4.66, D is 0.23 and s/2rG is0.51, from which it follows that s is 0.39 mm. Formula (24)gives V« = 0:23X300 = 69 J{ for the cut-off voltage andformula (25) givesj(O) = 0.4013/5 for the maximum cathodeloading -,If for example the gun provides a beam current of100 fJ.A, so that 1 = 10-4 A, the maximum cathode loadingwill he j(O) = 1.6 X 10-3 A/mm2• •

The U"';'ited dimensions of the resistance network

Figs. 8 and,9 are related to the' example just worked out;they represent, however, the more general case in,which thegrid and first anode apertures are unequal. We may now makesome observations concerning the measures taken to allowfor the fact that the resistance network is not infinitely largein relation to the electrode models plotted upon it. It will heseen from thes~ two figures that the outline of the first. anodehas been extended along two edges of the network, thejunc-

tions on Cl! and DE havirig been wired together. As alreadystated, the results of the measurements' are much the sarnewhether the final anode is present or not, and we can according- _ly leave it off the network. In these circumstances any pointbeyond the first anode and at some distance from its aperturewill have a potential equal to that of the first anode itself. Thiswill certainly he more or less true of points in space corre-sponding to the junctions-along CDand DE. There is thereforeno objection to giving these junctions the said potenrial. Ifthey were not joined up, one would be completely in the darkabout possihle errors arising becau'se the portion of the net-work to the right of the first anode is, as it were, left floating.In the event, the junctions along CD and DE have been short-circuited; and while it is true" that the outline thus traced nolonger conforms.to that of the first anode under investigation,one does at leas~know that the discrepancy will not 'give riseto any serious error.

Our second observation concerns the interelectrode spacesbetween K and G and between G and Al' The network onlyextends over a comparatively short distance in the radialdirection; what sort of error does this give rise to? The resultsof measurements of' potential in the inter-electrode spacesalong the edge of the network (i.e. for maximum r) are re-assuring, for the variation in the s-direction proves to he linèarwithin the required limits of accuraey. This means that theheight ofthe network (its extent in the r-dircction) is adequate.

Implications of the finite mesh widthv ,

Obviously, themesh with a of the theoretical gridcannot be reduced indefinitely, as this would leadto ever larger models on the resistance network.Hence the network will only provide approximatesolutions to the problems worked. out on it. Ingeneral, it is impossible on the basis of purelY.'theoretical reasoning to estimate this fundamentalerror with any degree of accuracy. To get someidea of this accuracy, the network' can be used forworking out a problem whose exact solution isknown, the potential distribution found with thenetwork being compared with the known distribu-tion. Alternatively, measurements can be performedfor ever smaller mesh widths, a better approximationto the correct solution then being found by extra-polation 15) 16). :

Cylin:dri~al capacitor

Here w~""Shallgive some results of'investigationsrelating to a cylindrical capacitor 17) .. For such acapacitor the potentiäl" distribution 'can be workedout exactly. Furphér, it is possible by calculationto find the' solution -that would be obtained with an

15) L. F. Richardson, How to solve diffcrential' equationsapproximately by arithmetic, ·Math. Gazette 12, 415-421,

. 1924/25.16) R. Culver, The use of extrapolation techniques with

electrical network analogue solutions, Brit. J. appl. Phys.3, 376-378, 1952. ,

,17) J. C. Francken, Electron optics of the image iconoscope,thesis Delft, 1953, p. 36 et seq~

22 PlIILlPS TECHNICAL REVIEW VOLUME 21

"ideal" resistance network. The error due to thefinite mesh width can then he determined by com-paring the latter solution with theexact one. Othererrors, like those due to inaccuracies in resistorvalues or in the adjustment of the bridge, are thus.excluded in this comparison.If the ratio of the radii of the inner and outer

electredes is 1: 10 and if the mesh width is madeequal to the inner radius, the error is found to be, atworst, -0.7% of the voltage across the electredes(the worst error arises at the grid points closest tothe inner electrode). Near' the outer electrode the. error is only about -0.07%. If the mesh width ishalved, the errors become about -0.2% and about:__0.02%respectively. By correction of results byextrapolation the errors can be brought down toabout -0.03% and about -0.001% respectively.The problem of the cylindrical capacitor was also

worked out on the actual resistance network, withthe two mesh widths mentioned above. Aftercorrection by extrapolation, the results for junctionsnear the inner and outer electredes differed fromthe calculated true values by -0.035% and-0.008% respectively. Differences between theexperimental values and those calculated from the"ideal" network are due to the other errors referredto above, and have nothing to do with finite mesh. width.

An electrostatic lens

Extensive investigations were also made into thepart played by the finite mesh width when theproblem is to dctermine the potential distributionalong the axis of an electrostatic lens as in fig. 15a.

Fig. IS. a) Lens of a type used in electrostatically focussedpicture-tubes. The electrodes to right and left have the highestpotential in the tube; the middle one is at cathode potential.b) Shape of the same lens when stylized for the purpose ofinvestigating the effect of the finite mesh width on resistance-network measurements of the potential distribution along the,lens axis. The electrode dimension I has been selected as thecharacteristic length to fix the mesh number, i.e. the scale onwhich the lens is to be modelled on the network (mesh numbern = I/a, a being the mesh width of the grid; see fig. 2).

This type of lens is used in television picture-tubes.The two narrower electredes have the highestpotential present in the tube; the middle electrode

is at cathode potential. For the purpose of theinvestigation the lens was stylized, being given thesymmetrical shape indicated in fig. 15b. In thestylized lens the ends of the outside electredes "~reclosed by conducting plates.

The electrodes are "capped" in this way for the same reasonthat led us to introduce further connecting wires into the modelof the first anode dealt with above. Here as before, we have toconsider the inter-electrode gaps. It is possible, amongst otherthings, to short-circuit the entire upper edge of the networkand to give it zero or unit potential. It proves that this has'no appreciable effects on the results. It may be concluded thatthe limited size of the network in ~elation to the spaces betweenthe electrodes, does not prejudice the reliability of the measure-ments.

III

----]!_--- III

I,- --------lI 1I II ----------1,.---t-l._._.,

i r·-=-~_L-lL..J •

Ii

Fig. 16. The lens of fig. 15b plotted on the resistance network.Symmetry makes it unnecessary to model more than a quarterof the lens assembly. Measurements were carried out on threeelectrode models (one at a time, of course) plotted with meshnumbers of 2, 3 and 4, and marked I, 11and III respectivelyin the diagram.

The left-hand edge of the resistance network ismade a mirror plane of symmetry by doubling thevalues of the resistors forming this edge (cf. the

Fig. 17. Variation in potential along the axis of the lens infig. 15b, as determined by means of a resistance network.The scale of this graph is too small to allow any distinctionto bc made between thc slightly different results obtained frommeasurements on the three different models.

1959/60, No. 1 RESISTANCE NETWORK 23

mirror plane along the z-axis in fig. 6b). In eonse-quenee, only a quartet of the lens had to be simulatedon the network. The ~roportions of the stylized elec-trodes were so chosen that the lens could be plottedwith "mesh numbers" that were in the proportionsof 2: 3 : 4. The "mesh number" is the ratio nbetween l, one of the dimensions of the lens, and a,the mesh width of the grid. Any dimension of thelens can be chosen for this purpose provided that itis consistently adhered to. Our choice is indicatedin fig. I5b. The greater the mesh number, the finer

I(n=2)

],0/4.0 + z={Pm °7j 1,0]20

71

t21

41],0080

51

Fig. 18. The relative error (the measured value lpm divided bythe "correct" value lp) in the potential distribution curve offig. 17, plotted as a function of the mesh width a. The quantityall along the abscissa is thereciprocal (l/n) ofthe mesh number.The seven curves are appropriate to seven points along thez-axis. Each has been drawn through three points determinedby resistance-network measurements on three models (I, 11and III in fig. 16). Even when the coarsest grid is used theerrors remain small.

is the grid and the bigger is the electrode model onthe network (fig. 16). The potential values measuredalong the lens axis are plotted in fig. 17. The curveas drawn here is not thin enough to reveal divergen-cies arising from the use of the three mesh numbers.The three values found for each point along the

axis were used to work out, by Culver's method ofextrapolation 16), a value regarded as correct. Thethree measured values of each set were divided bythe "correct" value and the results plotted infig. 18; the three points of each set are joined by asmooth curve. Sets of values appropriate to severalpoints on the z-axis are given. The figure clearlyreveals how slight an effect is exercised by the finitemesh width on the results of measurements with theresistance network.

Summary. The resistance network has won a place beside theelectrolytic tank as an aid to the solution of Laplace's equationfor given boundary conditions. Two kinds of network havepractical importance. The first is useful for solving two-dimensional problems, the second for solvingthree-dimensionalproblems where rotational symmetry exists. In either caseLaplace's equation reduces to a differential equation involvingtwo independent variables only. The network is built up ofresistors, four of which meet at each junction. The junctionscorrespond to the grid points of a hypothetical square gridwhich is imagined to have been set up in the field space.Junctions corresponding to boundaries are given potentia1sproportional to those values which the boundaries are knownto have. Provided the resistors composing the network havethe right values, the remaining junctions will then acquirepotentials that are approximately proportional to the re-quired potential function. Discrepancies from the actualvalues decrease as the mesh width of the grid is reduced tozero. Conditions which must be satisfied by the resistancescomposing the network, in both the two-dimensional and inthe rotationally-symmetric three-dimensional case, are workedout in the course ofthe article. An example ofthe employmentof a network for rotationally-symmetric three-dimensionalproblems is given in which curves are derived which allowthe cut-off voltage and maximum cathode loading oftetrodeelectron guns to be determined. Finally, examples are givento sbow that the errors due to the finite mesh width are verysmall.

,