A Capacitively Coupled Field

for 2-Dimensional Distributed

Field Problems

Mapper

Source

Typical va lues of Rs ran ge from 400 to4,000 ohms. The point of maximumvoltage on th e resisti ve surface is thekern el. It is poin t K on the diagram s.Th e maximum surface voltage Ek willbe referred to as the kern el volta g-e.

Measurement Methods

EDWARD O. GILBERTNONMEMBER AlEE

POISSO~ IA ;\ or distributed sourcefield s occur often in prese nt-da y

problems dealin g with such subjects astorsional stress analysis, heat tr an sfer,viscous fluid flow, eddy currents , andelectri c and magnetic fields. Since themathematical solution of Poissonian field shas only been possible in severa l specia lcases, physical ana logues ha ve been de­veloped for 2-dimensional fields. Ahighly success ful physical analogue is th esandbed fluid mapper. I · 3 This paperdescribes an electric analogue mapperwhich was origina ted and developed bythe aut hors . The ana logue ca n givedirectly usable 2-dimensiona l solut ionsof high accuracy.

The Analogue Field Map

The mapper utili zes a resistiv e plotti ngsurface and a coupling ca pacitor. Thecoupling capacitor is formed by th e re­sistive surfa ce and a parallel couplingelectrode . A displacement curre nt act­ing through thi s coupling capacit or sendsa distributed current into the resisti vesurface . As shown in the Appendix,this distributed current flow representsPoisson' s equ ation. Boundary condi­tions are introduced by suita ble conduct­ing terminations of the resist ive surface .The coupling electrode shape and areacorrespond to th ose of th e distributedsource. The displacement curre nt den­sity is readily controlled by th e spaci ngof the couplin g electrode and th e re­sistive surfa ce, Most of thi s paper willdiscuss mappers of const ant displacementcurrent density.

In practice, Teledeltos recordi ng paper(manufactured by the Western UnionTelegraph Company) is used as a re ­sistive surface. A dielectric materialsepara tes it from the coupling electrode,a conducting sheet of aluminum foil orsilver paint. Silver paint is used for theconducting boundary of th e Teledeltospaper. A circular map mad e of the se ma­terials is illustrated in Fig. 1. A uni ­formly distributed source region is repre­sented since the dielectric is unifonn in

ELMER G. GILBERTNONMEMBER AlEE

thickn ess. T his map could , for examp le,represent th e magnetic field within acircular conductor of infinite length .In operati on th e terminal s of the mapper,that is, th e conducting boundar y line ofthe resistive paper and th e conductingsheet, are connected to a source of alter­nating voltage . The resist ance of thepaper is small in comparison with thereactan ce of th e coupling capacitor. Forthi s reason the voltage drop across theresisti ve paper is a small percentage of theap plied voltage and th erefore a subs ta n­tiall y uniform source of current den sityap pears in th e resistive paper over theconducting shee t. This uni fonn curre ntflow represents the pr oblem of the uni ­fonnly di stributed source field. A null ­type measuring circuit is used in con­junction with a probe to plot th e equi ­potential line s on th e resisti ve pa per.These line s correspond to one set of th eor thogo na l sys tem th at is th e solut ion ofth e desired field.

For the capacitively coupled field map­per to be a workable an alogue. the cur­rent density appearing in the resistivepaper must be inver sely proportional todielectric thickness ; this requires thatthe voltage across th e dielectric be thesame at every point. However, themapper work s on the pri nciple of ameasureable voltage dr op across themap surface, The error cau sed by thisimperfect an alogy was mathematicallyinvesti gated for th e circular map andfound to be entirely negligible for anypractical mapper .

Equivalent Circuit

A simplified circui t representation ofthe mapper would be useful in furtherwork. An equival ent circuit is shown inFig. 2. It ha s proved to be highly ac­curate for all maps thus far me asured.The impedance of the coupling cap acitorC is genera lly 100 to 500 times as largeas R s• which represents map surfa ceresistance. The resistance R L is ap ­parent in measurement and is believed todenote loss in the coupling capacitor.

The measureme nt of equipotentiallines presents difficult problems whichmay best be und erst ood by examiningconditions under which a typical mappermight opera te. T he resist ive paper isheld near ground potentia l while thecoupling electrode is excited to a potentialof about 100 volt s. With average condi­tions less than 1 volt will appear acrossthe ~ap surfa ce, In weak field regionsth e voltage gradient is extremely lowand precise measurements must be madeif points are to be located accurately.Min or chan ges in C or R s can producelarge changes in the measured positi onof equi potential line s. For this reasonthe sta bility of mapper components isextremely im portant. Capacitive drift(change in coupling capacity with time)and resist ive drift (cha nge in paper re­sistan ce with time) do exist and increasethe difficulty of accurate measurement.These effects will be discussed more fullylater.

All of the measuring circuits tried wereof the null ty pe. A va cuum- tube volt­meter would be ineffective because of thelow voltag-es pre sent and accuracy re­quired. A null -type detector has provedconvenient because it can give an audiblesigna l not demanding that the operatorbe continuallv shift ing hi s vision fr omth e plotting surface to a visual indicator.The low-volt age level s present indicatethat steps should be taken to minimizestray couplings. This is shown to betrue in actual practice, as without propertechniques deep null s are impossible .The circuit diagram s will show whereshielding is employed . A guard elec­trode placed near the high-potentialcouplin g electrode is in valuable in reduc­ing stray flux. Such a guard electrode issho wn in Fig. 1. A sensitive null indi­ca to r is required in all cases. Head -

P a per 53-193, recommended b y the Al E E Basi~

Sci ence s Co mmi ttee and app roved by th e AI EECo mmi ttee on Techn ical Ope rations for pres enta ­t ion a t t he AlEE So ut hern D istrict M eeting.Louisvill e . K y .. April 22-24 . 1953 . M anuscr iptsub mi tted Octobe r 21. 1952 ; made availabl e forprin ting M arch 6 . 19.';3 .

EDW A RD O. G IL BE RT and ELMER G . G I L BE RT areboth at th e U niversit y of Michi gan , Ann Arbor.M ich .

The autho rs are esp ecially indebted to Prof . A. D .Moore for his hel p and enco urag ement. Th ank sa re a lso due 10 Prof . M . B. St out and the manyothers at the Unive rsi t y of Michigan who havehelped m ake this pa pe r p ossible .

SEPTE:lIBER 1953 Gilbert. Gilbert-r-Capacitiuely Coupled Field Mapper 345

K

R~5

null s and satisfactory accuracy but re­quires bothersome adjustment of theWagner ground circuit for each differentpotential measured . Capacitive and re­sistive drift also cause troublesome shift­ing in the measured positi on of equipoten­tial lines. For these reasons the nextmeasuring method is preferred.

The tran sformer circuit, shown in Fig.4, gives excellent results. Not only doesit use less equipment 'than the previouscircuit but it is abl e to counteract theeffect s of cap acitive drift in the map.

C-coupling ce­pscitence

RL-Ioss resist­dnce

Rs-surfdce re­sistence

K-kerne l

Fig. 2 (right).Equivalent map

circuit

Fig. 1 (left). Acircular map withguard electrode

A - cond uctingbo unda ry

B- resistive ps-per

C-die lectricD - conducting

sheetE- gudrd elec­

trod eK-kernel

A

type s of measuring circuits.The first circuit is shown in Fig. 3.

Th e variable circuit elements R' and Care adjusted so point A has the samepotential as th e kernel. Any fraction ofthe kernel voltage ma y th en be meas­ured by setting th e slider of R at the de­sired percentage of full kernelVolt age.A Wagner ground circuit is used to elimi­nate the effect s of stray detect or capacityto ground. This circuit pr oduces good

phones in conjunction with a battery­operated bridge amplifier make a goodcombination. A selective filter in th edetector amplifier circuit is helpful inreducing th e effects of hum pickup fromnearby a-c lines. A sharp needle is thebest probin g tool since a dull pointmakes positive surface contact difficult .These points are applicable to all of thecircuits used . Map requirements haveled to th e development of three basic

Fig. 4. Trans­former measuring

circuit

c

'---oL-----oL----~A

DETECTOR

----'pII___....J

K

t1-~GI =~

- II- ....JL/

G

DETECTOR

R

Fig. 3 (below). Measuring circuit with Wagner ground

Cs-strdY detector cepecitence to groundG-gudrd elect rodeP- shielded probe

346 Gilbert, Gilbert-Capacitively Coupled Field Mapper SEPTEMBER '1953

r=----",---IIIIII

III

--' '--

'G

Fig. 5 (left). Vacuum-tube measuringcircuit

Fig. 7 (above). The solution of a uniformly distributed source field by twomethods. The closed curves are equipotentials obtained from the capaci­tively coupled mapper. The set of How lines was obtained from a sandbedHuid mapper. The excellent orthogonality of the sets shows close agree-

ment between the two solutions

Fig. 6. A typical experimental setup. The transformer measuring circuit is used

The~transfonner gives a 180-degree phaseshift so that point A may be adjusted tothe same potential as the kernel. Capaci­tor C serves to correct phase shift in thetransformer and its associated circuit.Since the equivalent capacitor of the maphas by far the highest impedance of theseries circuit, it is the main factor in con­trolling the total current through the mapand the transformer primary. Thus, thevoltage across R may be convenientlyadjusted by changing R '. If the re­actance of the map capacitor should in­crease slightly due to drift, then the cur­rent through the circuit would decreaseslightly. This in turn would cause aslight drop in the kernel voltage and inthe voltage across R. Since the poten­tial of the'kernel and point A rise and falltogether, capacitive drift has no effect oncircuit balance. This method does notcompensate for the effects of resistive

drift which may be minimized by properhandling of. the resistive paper. Thecircuit is easy to adjust as there is onlyone control to change in measuring dif­ferent potential lines . The stray ca­pacity Cs can cause trouble if R is toolarge. Operation has been satisfactoryif R is kept below about 500 ohms. Thiscan easily be done by governing the turns­ratio of the transformer. In all mapsmeasured a turns-ratio of nI!n2 equal tofour has been satisfactory. A bridge­type transformer provides excellent isola­tion. It should be noted that the guardelectrode is not returned to ground butis shunted across the oscillator. Thishas been found necessary to make certainthat the transformer carries currentgoverned only by the map capacitor;otherwise, the circuit would not com­pletely counteract capacitive drift.

A vacuum-tube circuit is shown in Fig.

5. The measuring resistance R takesthe fonn of. a load in a cathode-followeramplifier. The step-up transfonnerisnecessary since: the 'voltage gain of thestage is less than unity, and since thecathode-follower input voltage is lessthan the kernel voltage. The gridterminal of the tube is kept in contactwith the map somewhere near the kernel.Map loading is negligible because theinput impedance of the tube is exceed­ingly high. Since the voltage across themeasuring resistance varies as the voltageacross the map, the circuit responds toneither capacitive nor resistive drift.This circuit .is effective in operation onlyif the tube potentials are carefully sta­bilized. The measuring method has allthe advantages of the transformer cir­cuit, but does require more equipment.

The transformer circuit has been foundvery satisfactory and has been used inthe majority of measurements. Withproper control of resistive drift it is ac­curate, simple, and easy to use. Atypical setup is pictured in Fig. 6. Thereare, no doubt, many other methods whichwill give excellent results. The examplesgiven show what can be done with suit­able circuitry. The choice of method willdepend upon the user's requirements.

Causes of Error

Errors in line positions are due to fivecauses: inaccuracy of measuring equip­ment, voltage drop across the map sur­face , nonunifonnity of the dielectric, non­uniformity of the resistive paper, andboundary fringe flux. As mentionedbefore, errors due to surface voltage dropare extremely small in a practical mapper.This type of error is roughly proportional

SEPTEMBER 1953 Gilbert, Gilbert-Capacitively Coupled Field Mapper 347

Fig. 8. Actual mapper representing the magnetic Aux within a rectangular duct of infinitepermeability and length. The duct holds two rectangular, symmetrically placed conductorswith equal, uniform current densities in the same direction. Shown is the left half of the ductwith the centerline at the right. One of the conductors can be seen in the lower left corner.

Shown are Aux lines and the kernel K

to distributed source area, the resistanceper square of the paper, and the admit ­tance per unit area of the capacitor. Fora circular map of moderate area (100square inches), thin paper dielectric(several mills), and usual frequency ofoperation (1,000 cycles per second), themaximum error in line position from thissource is less than 0.01 per cent of the mapdiameter. Since the upward currentdensity is dependent upon the dielectricthickness and the dielectric constant,these factors must be carefully controlledif accuracy is to be achieved. A homo­geneous dielectric of high dimensionalaccuracy is necessary. Care must alsobe taken to insure the uniformity of theadhesive layers which are used to as­semble the map. Another source of erroris resistive paper nonuniformity. Thenonunifonnities of Teledeltos paper willbe discussed in some detail later. At theedge of the conducting sheet the flux tothe conducting paper will not changeabruptly because of fringe flux effects.This fringing can cause a slight error atthe distributed source boundary. It isbelieved that this error is extremely smallin a practical map where the dielectricthickness is small compared to map area.When boundaries of the resistive paperand the coupling electrode coincide, thefringe flux may be eliminated by extend­ing the coupling electrode. This has beendone in Fig. 1.

Conducting Paper

Teledeltos paper has proved to be agood resistive surface medium. It isreadily prepared and maps of greatcomplexity can be built quickly atnegligible cost. Conducting boundariesare simply applied using silver paint.The paper is available in two types,high and low resistance. The low­resistance paper, type L-39, has a re­sistance of approximately 2,000 ohms persquare and is used because it gives lesssurface voltage drop.

Teledeltos paper is not perfectly uni­form in its resistive properties. Themajor variation in resistance seems todepend on direction of current flow. Amaximum difference in resistance occurswhen current flow is first parallel and thenperpendicular to the direction in whichthe paper was rolled. Variations as highas 16 per cent and as low as 2 per centhave been measured in different samples.If selection of the paper were possible,this error could be eliminated or greatlyminimized. The error in line positioncaused by this variation depends uponmap geometry. In most cases the erroris very small. Other relatively minornonuniformities are believed to exist.It is hoped that a resistive paper of highuniformity will be made available formapping purposes. Such a paper wouldhave great value for mapping either

Laplacian or Poissonian fields.An important factor in the use of Tele­

deltos paper is its sensitivity to moisture.Changes in humidity have producedresistance variations in excess of 1 percent. Such variations are the maincause of resistive drift. The drift can beminimized by proper map construction.The face side of Teledeltos has a dullfinish and is quite porous while the re­verse side is shiny and nonporous. If theface side is removed from air contactby cementing it to the dielectric, the driftwill be quite small. Hand moisture cancause noticeable drift and it is advisablenot to touch the map while plotting.If it is desired to eliminate drift com­pletely, it can be done by coating thepaper with a thin layer of acrylic plasticfrom a spray-type bomb. This practiceis not recommended unless necessary sinceit makes probing quite difficult. In mostcases the first precaution is entirely ade­quate.

Dielectric Material

There is a problem in finding dielectricmaterials that are homogeneous and di­mensionally accurate. The problem isfurther complicated if an effort is madeto eliminate capacitive drift. The fonnof mapper construction makes it ex­tremely difficult to construct a stablecoupling capacitor. It is better toeliminate the effects of capacitive driftby suitable measuring circuitry than toeliminate the capacitive drift itself. Ifhigh accuracy is not needed, common di­electrics such as paper and plastics maybe used. A good grade of window glass isan excellent material since it is dimen­sionally accurate (within a few per cent),resistant to attack of adhesives, and easyto obtain. The final choice of a dielec­tric will depend on how severe the user'saccuracy requirements are.

Map Construction

To illustrate possible map-makingtechniques, the construction of a uni­formly distributed source mapper will bedescribed. The dielectric, in this case asheet of glass, is sprayed with acrylicplastic until it is just wet . The Teledeltospaper is then applied and rolled smooth.Considerable pressure on the roller isnecessary to remove excess plastic and toinsure a uniform plastic film. The uni­formity of the film may be checked easilyby viewing the paper through the glass.Since the plastic sets fairly rapidly, somespeed is required in executing these steps.This method of assembly is fast and pro-

Gilbert, Gilbert-Capacitively Coupled Field Mapper SEPTEMBER 1953

Th e volt age drops across the area are

ReFerences

(3)

(2)

(1)

and

I OVI = - lix-

y K oy

st, 1 o' V- = - lixoy K ()y'

Substitute equations 2 and 3 inthe result is

o V liyli V = -liy=I K - -

y oy v lix

1 . FIELDS F RO M FLUID FLOW MAPPERS , A. D.Moore. J ou rnal of Applie d Ph ysi cs , N ew York,N . Y., vo l. 20, Aug . 1949 , pp , 79~804 .

2. SOA P F I L M AND S AND BED M APPER TBCHNJQUE~.A. D . Moore. J our-nal of Appl ied M echallics . Ne wY ork , N . Y., vol. 17, Sept. 1950 , pp. 29 1-98.

3. THE FUR T IfBR D EV E L O PM E N T OF F LU ID l\IAP­PERS, A. D. Moore. A l EE Transactions , vo l. 69 .par t II , 1950, p p. 1615-24.

»t, 1 02Vox = "Kliyox'

02V 02V- + - = - K Jox 2 oy'

which is Poisson's equa tion for two dimen­sions.

If th e capacitively coupled field mapperis to solve 2-dimensional fields of the dis­tributed source type, it must sat isfyPoisson's equat ion. In the following de­velopment the volta ge drop across the re­sistive paper is considered very small incomparison with the applied voltage.Therefore the displacement current densityappearing in the resistive paper is inverselyproportional to the dielectric thickness.Let J equal the displacement current den­sity at any point in the distributed sourcearea. Let K equal the resistance persquare of the resistive pap er. Considerthe sum of the currents leaving a small areain the xy plane as shown in Fig. 9.

Walter E. Rogers (University of Washing­ton , Seattle, Wash. ): This is an import antpap er, and the Gilbert twins are to becommended for their contribution to a fieldin which th ere is such general interest . Ilook forward to th e extension of their de­velopment to nonuniform distributedsources, something which has not beenachieved with the sandbed fluid mapper orthe membran e analogue.

- ---.----

Discussion

Appendix

o V lixli Vx= .--- lix = IxK-

ox liy

1 OVI = -liv-- 'x K J ox

At present the accuracy of the mapperis limited by re sistive paper and dielectricnonunifonnity. Even so, results ob­tain able are excellent. The maximumerror in line position should not exceed1 per cent of the maximum map dimen­sion and average error should be muchless. The time involved in making andusing a mapper such as the one describedis in the order of 3 to 4 hours. The valueof an y potential line can be determinedeasily. In most cases the equipotentiallines are the ones desired in the final solu­tion .

The mapper should easily be applicableto regi ons of nonuniform displacementcurrent density. Though such a map­per has not yet been constructed, littletrouble is anticipated. The method ofcap acit ive coupling might also be appliedto th e electrolytic tank, by-passing theproblem of nonuniform resistive paper.The electrolytic tank mi ght also makepossible the solution of 3-dimensionalfields with axial symmetry.

Conclusion

Results

T he accuracy of the capacitive lycoup led mapper is good . A circular mapas in Fig. 1 was constructed with a glasspla te dielectric with a diam eter of 20.0cent imeters. The maximum erro r inequipotential line positi on was less th an0.1 centimeter. Average error was muchless. Complex field s have been checkedwith sandbed fluid mappers. Such acheck is shown in Fig. 7. The fluid mapis by Prof. A. D . M oore. Excellentor thogonality of equipotential line s andrepresentative fluid flow lines indicatesclose agreement. The fluid flow linesalso seem to originate from the kernelwhich was found by the capacitivelycoupled mapper.

Fig . 8 shows an actual mapper as usedto solve a magnetic field problem. Notehow the symmetry of th e pr oblem enablesthe mapper to be sim plified . The cur­rent enters the resistive paper in thedistributed source area at the lower left.It leaves from the conducting boundariesa t the top, left, and bottom. Here theequipotentials represent magnetic fluxline s. No refraction of flux lines occursat the distributed source boundary .This is as it should be when the per­meabilities are the same on either side ofthe boundary.

gether . When the shield wire is attachedand secured to the mounting cardboard,th e mapper is ready for opera tion .

Fig. 9. An area in the xy plane used in thedevelopment of Poisson's equation

duces results far superior to th ose usingthe more common cements, glues , orpastes. The coupling electrode may beof either silver paint or aluminum foil.The foil is quickly applied, using thesame techni que as is used with th e Tele­deltos paper. If small bubbles appearthey ca n be removed easily , by piercingthem with a pin and th en rolling th emflat. Corr ect foil un iformity will resultin a mirror-like surface. The foil shouldthen be trimmed so that it cover s onlythe area where distributed flow takesplace.

Th e silver paint is useful in the pat chin gof torn foil. Silver paint may replacethe foil but its applicat ion is more time­consuming and costly . The map is thenready for it s associated wires, conductingboundaries, and guard electrode.

Conducting boundaries ar e eas ilyaffixed to the Teledeltos paper with silverpaint. The silver paint is a commerciallyavailabl e product used in printed circuitwork . Wh ere high accuracy is desired,the paint may be applied with a rulingpen. A boundary width of 3/8 inch hasbeen found adequate. Tape is used tohold the wires in contact with th e mapbounda ries. Electric connection is thenmade by daubing the wires to theboundari es with silver paint. T o assurehigh conductivity such connections aremade every few inches. The wires mayhe held more securely if they are kinkedor looped . Conn ecti ons to th e couplingelectrode are made in the same manner.To protect the map and it s connectingwires, it is helpful to mount it on st iffcardboard. A foil guard electrode is cutsligh tly larger than the coupling electrodeand is affixed to the top of the mountingcardboard with cement. A lead is thenattached to the guard electrode and ahole pun ched through the cardboardand gua rd electrode for the capacitorlead . A sheet of paper is used to sepa­rat e th e guard electrode and the bottomof th e coupling electrode ; the comp o­nent s are th en assembled and taped to-

SEPTEMBER 19;)3 Gilbert, Gilbert-s-Capacitioely Coupled Field Mapper 349

I would like to point out that while theelectrica l mea surements discu ssed here arerather elabora te and time-consuming, theyare certainly not more so than are thepresently used methods! of measuring thedisplacement of a soap film or rubbermembrane; nor is the electrolytic tankwithout measurement difficulties. Thecapacitively coupled field mapper is superiort o the sandbed in the matter of fine de taila nd in th e exac t location of the kernels.T his is an opinion which would be sub­stant iated , I think, if the authors andProfessor Moore would submit a photo­graph of th e original sandbed map ofFig. 7 of the paper with th eir answer tot his discussion .

Analo gues are important academicallybecause they are an invaluable aid in thecha llenging problem of t eaching studentsto visualiz e potential fields. We plan toinclude the Gilberts' development alongwith Professor Moore's fluid mappers andsome membrane analogues which are pres­entl y used in our laboratories at the Uni­versit y of Wa shington. To this end , wouldth e authors please state how the loss re­sista nce RL of their equivalent circuit be­comes apparent in the measurements? Isit th at points A and K in Fig. 3 of th epaper are not a t th e same potential unl essR ' is included ?

There is one other detail in the pap erwhich needs clarifying. Are the equi­pot entials, which surround the kernel inFi g. 8, taken at equal increments of poten­t ial with respect to that of the kernel? Int he region of th e kern el, the gradient seemssma ll in compari son with Fig. 7. I observe,however, th at a vert ical line through thekernel and a horizontal line to th e left ofth e kernel are both stream lin es. This indi­ca tes th at th e kernel is correctly located .

R EFERE :-;CE

1. See refer en ce 2 of the paper.

J. H. Fooks (Westinghouse Electric Cor­poration, Sharon, Pa.): The authors areto be commended on their excellent con ­tribution to experimental mapping methods.The thoroughness of their investigationwith regard to errors involved and, par­ti cularl y, their solut ion of the cap acitivedrift problem via the transformer measuringcircuit should clear the way for th e imme­diat e applica t ion of this method to indus­trial field problems.

The possibility of simulat ing "nonuniformd isplacement current densitie s by va ry ingd ielect ric thickness was of particular inter­est. This is readily accomplished in a fluidmapper by using a sandbed of var yingdepth. Offhand , I cannot see how thiscould be done easily in a cap acitivelycoupled map per using a reliably uniformdielectric mat erial such as glass. Con­siderable ca re would have to be given tothe select ion of a more workable material.Perhap s the authors would care to say howthis might be done.

The time involved in constructing andopera t ing a capacitively coupled and afluid mapper is, perhaps, much the same ifthe fluid mapper is carried through to afinished photograph of the field.

Almost all of our application of experi­mental mappers has been confined to 3-di-

mensional fields of axial sy mmetry. Fluidmapping is readily ad aptable to this caseby varying the flow spacing, whereas thecapacitively coupled mapper mu st awaitthe advent of a suitable resist ance paper.It is not inconceivable that such a papermight be forthcoming. The capacitivelycoupled 2-dimensional mapper might beused to approximate the solu t ion of a3-dimensional field or to show field trends.This has frequently been done mathe­matically to simplify methods of solu t ion.

In the past, the tools of experimentalmapping methods and field mapping havenot been used to any great extent in fieldinstruction or problem solving. This hasbeen unfortunate since the visual analysisof fields and field trends provides thequickest and most lucid solut ion of fieldphenomena with reasonable accuracy .

Within the past several years experi­mental mapping has received a tremendousstimulus through the excellent work ofProfessor Moore in the invention and de­velopment of fluid mapper methods. It isgratifying to see the interest in this de­velopment and the trend in new text mate­rial to include sections on pr actical mappingmethods.

It is to be hoped that, with the avail­ability of fluid mappers and similar experi­mental methods such as described in thispaper, the visual simula t ion of field phe­nomena will receive its due recognition .

A. D. Moore (U nivers ity of Michigan, AnnArbor, Mich. ): Learning about sandbedfluid mapper methods for simulat ing thedistributed source, the au t hors soon turnedto the invention and development of thenew analogue they hav e so ably reportedon in their paper.

"T he equipotential lines obtained by theirmethod are orthogonal to the flow linesobtained over the sandbed, in a given case.They have applied their method to severalsandbed cases already available from thewriter's work, and came out with a highdegree of correspondence. This, by theway, constitutes a check on both methods.

Professor Rogers indicates that simula­tion of nonuniform distributed sources hasnot been achieved with sandbed fluidmappers. Rather, it has been achieved butnot yet reported. The writer has built aslab with a rectangular sa nd bed, in which asloping screen makes the sandbed depthvary by a ratio of 3 to 1 from one end tothe other. The flow density appear ing ontop of the bed then is nonuniform, varyingapproximately inversely as the sandbeddepth. In addition to sloping the screen,there are at lea st two other methods forachieving nonuniform distributions withthe sandbed.

The Gilbert contribution is a welcomeaddition to our knowledge dealing withfields . Field problems con st antly ari singare of such range and complexity that weneed all of the analogues we can get.

J. F. Calvert (Northwestern University,Evanston, 111.): The authors are to becongratulated on the development of whatI believe to be an entirely new approachto the solution of 2-dimensional fields withdistributed sources. It would appear thatthey have devised a method which can be

readily used by any engineer and scientistwith normal skill in electric circuit andmeasurement techniques, and it would seemsafe to predict that this method will begenerally accep ted . I hope the authors will(1) show several more examples of the resultsobtained by this technique; (2 ) discuss thepo ssible extension of the equipment to2-dimension al problems where the distrib­uted sources are not of uniform strength;and (3) further di scus s the possible applica­tions to 3-dimensional fields, including theprobable difficulties and limitations as theymay see th em at this time.

W. R. Simmons (Argonne National Labo­ratory, Chicago, Ill .): The analytical solu­tion of problems involving Poissonian ordistributed source fields has always been along and tedious t ask except for those fewcases where known solutions exist. Themethod described in this paper will expe­dite the solution of manv of these 2-dimen­sion al problems. This -method should beof interest to the engineer since most prac­tical engineering problems involving Pois­son ian fields do not have known solutions.Further, it should be noted that the mate­rials required to set up a problem by thismethod are inexpensive and an swers withsufficient accuracy for engineering work canbe rapidly obtained.

The method described in this paper maybe particularly valuable in solving 2-di­mens ional heat transfer problems which in­volve Poi ssoni an fields. This type of prob­lem constantly confronts the nuclear heattran sfer analys t. By suitable constructionof the analogue, heat transfer problemsinvolving any or all of the following condi­tions can be solved: ( 1) Poisson fields;(2) Laplace fields ; (3 ) materials of variousthermal conductivities; and (4 ) convectiveheat tran sfer from a surface to a fluid.

Poisson fields can be built up in a numberof way s. However, the method devised bythe au thors is one of the more ingeniousmethods. The Laplace fields are obvioussince only conduction is involved.

Materials of different thermal conduc­tivities (or resi stivities) can be simulatedby constructing the analogue so that morethan one path of the same shape is availablefor the electric current. These parallelpaths can be electrically connected by care­fully applying silver paint in a discontinuousmanner around the boundaries.

Convective heat transfer from a surfaceto a fluid can be easily included in an ana­logu e constructed from Teledeltos recordingpaper . This is one of the principal advan­tages of an an alogue of this type. This isaccomplished by adding to the surfacewhere convect ive heat transfer takes pl acean addit ional length of Teledeltos recordingpaper. This additional length should beequal to the ratio of thermal conductivityto the heat transfer coefficient (k /h) inorder to properly represent the film resi st­ance. This additional length (k /h) ofconduc t ing paper mu st be cut in strips per­pendicular to the surface from which theconvective heat tran sfer occurs. This in­sures an electric current flow which will beorthogonal to this surface. In this mannerproblems involving 2-dimensional Poisson­ian fields and convective heat transfer fromnonisothermal surfaces can be readilysolved .

350 Gilbert, Gilbert-Capacitively Coupled Field Mapper SEPTEMBER 1953

ELLSWORTH D. COOKASSOCIATE MEMBER AlEE

Reliability in Industrial Electronic

Equipment

The heat transfer section of the ArgonneNational Laboratory has developed a simi­lar type of analogue. However, we ap­proached the problem somewhat differently .We are primarily interested in actual tem­perature differences associated with certainheat flows through Poisson and/or Laplacefields with convective heat transfer from anonisothermal surface. Therefore, we havedeveloped a doc system which gives usdirectly values which are proportional totemperature differences and does not in­volve evaluating a conformal map.

Edward O. Gilbert and Elmer G. Gilbert:The authors appreciate the interest shown inthe capacitively coupled mapper. It wouldbe well to clarify the points mentioned byProfessor Rogers. The potential lines inFig. 8 of the paper have values of 0.8, 0.6,004, and 0.2 Ek with the singular point at0.246 Es, Potential lines in Fig. 7 are at0.9, 0.8, 0.6, 004, and 0.2 e; The presenceof RL is directly known from the settings ofRand R' in Fig. 3. It is apparent that RL

TH E EXPECTATION of the pur­chaser of industrial equipment is no

different from that of the purchaser ofany other form of equipment or, in fact,from our own when we purchase things foruse or pleasure. Reliability is expected.Such reliability, however, includes morethan long life, important as that maybe; it also means long periods of free­dom from readjustment and similar an­noyances. The following observationsrelative to factors which enter into suchreliability were obtained from experiencein the field of industrial electronics, es­pecially as found in the heavy industries.A word of caution is in order a t thispoint : The impression that electronicequipment is hopelessly involved with re­peated failures is not a true one. Whatis reported as in need of correction isactually based upon the known facts thatperformance to date has been such as tocommit industry to the use of electronicmeans, and that the desired improvementsare possible and worthwhile.

It is unfortunate that no magic for­mulae exist by which equipment can beproduced with the desired reliability.The means of accomplishing the objec­tive are known to all of us, but have some-

is not detected by the other circuits. Notethat K and Ek are not known initially, butmust be determined by a series of rapidlyconvergent trials.

The resistive paper modifications dis­cussed by Mr . Simmons should prove veryuseful. Values of resistance per square hadby paralleling resistance sheets are limitedto discrete steps. An array of small, uni­formly spaced conducting dots would pro­vide another approach to the problem .Such an array might be applied to the re­sistive paper using a stencil and silver paint.It is hoped that more versatile resistivematerials will be developed by someone inthe future . The electrolytic tank couldhandle wide changes in resistivity withsome increase in physical complexity.Capacitive coupling to a tilted tank wouldfacilitate the solution of 3-dimensionalproblems with axial symmetry. Here,the dielectric material would form thebottom of the electrolytic tank. Theadditional problems encountered shouldnot be too great as electrolytic tank methodsare well advanced .

thing in common with certain medicinesin that we would prefer not to have touse them. Like medicines, since the idealones are not yet available, we use thoseremedies we know and have.

Since sufficient improvement in com­ponent reliability has not yet occurredwhich automatically would permit amajor improvement in equipment relia­bility, engineers must use such expedientsas their combined experience shows to behelpful. These are the simple, common­sense, but totally unromantic methods ofestablished good engineering practice indesign, installation, maintenance, andservice. What must concern engineersfor the present, therefore, is continuedemphasis on these practices and furtherextension of their use . While all ofthe se functions are important, at thepresent time most emphasis must beplaced on good design; first, of the com­ponent and , second, of the equipment.However, because the methods cited forthe equipment as a whole have a limitedeffectiveness, emphasis must be given tothe encouragement by word and othersupport to the component manufacturersin their efforts to improve the reliability oftheir products.'

The nonuniform distributed source canbe generated in several ways. A multi­layer dielectric could be formed in contourfashion to provide a step approximation to adesired thickness function . Dielectrics suchas celluloid, vinyl plastic, and paper haveproved reasonably successful in constructinguniform source mappers; their extension tomultilayer dielectrics should be convenientand inexpensive. Displacement currenterrors of less than 10 per cent are foreseeable.Even better accuracy would be possiblewith machined dielectrics. A subdividedcoupling electrode and a uniform dielectricwould allow a more versatile approach .Each segment of the coupling electrodecould be assigned a desired current. Onemapper could then solve problems withmany different distributed source configura­tions.

The authors hope that this brief dis­cussion will answer the questions of Pro­fessor Calvert and Mr. Fooks . Develop­ment of the modifications mentioned shouldgreatly increase the usefulness of thecapacitively coupled field mapper.

With so many electronic engineers in­volved in military developments, it some­times comes as a surprise that the field ofindustrial electronics is experiencing asubstantial expansion. With this expan­sion, the need for greater emphasis onreliability is increased. There is a needfor wider understandingof what makes onepiece of equipment reliable and anotherunreliable in each field, and there is aneed for better knowledge of how to keepgood equipment in reliable operation.While the technical subject matter is notmaterially different in the various fieldsof applied electronics, the emphasisplaced On the various elements do differto a marked degree . Thus, in the broad­cast field, it is essential that the programstay on the air ; but in the power field,where production, and even life itself, issometimes jeopardized by failure ofequipment, the need for guaranteed per­fonnance reaches a new order of magni­tude . Perhaps the fact that in Americathe clocks of so much of the nation nowdepend upon the power system will illus­trate this expected freedom from failure.In heavy industry, experience has shownthat failures are most frequently due tofaults of a mechanical nature. Thus,mechanical design is likely to be everybit as important as the purely electronicaspects of the device , and designs which

Paper 53-208, recommended by the AlEE Elec­tronics Committee and approved by the AlEECommittee on Technical Operations for presenta­tion at the AlEE Summer General Meeting,Atlantic City, N. J ., June 15-19, 1953 . Manu­script submitted M arch 6, 1953; made availablefor printing April 8, 1953 .

ELLSWORTH D. COOK is with the General ElectricCompany, Schenectady, N . Y.

SEPTEMBER 1953 Cook-Reliability in Industrial Electronic Equipment 351