IEEE TRANSACTIONS ON EDUCATION, VOL. E-15, NO. 1, FEBRUARY 1972

Analog Mapping of Poissonian Fields

PAUL BIRKE AND REUVEN KITAI, SENIOR MEMBER, IEEE

Abstract-A process is described for analog mapping ofPoissonian fields which is well suited to the teaching labora-tory. A capacitively coupled mapping system is employed inthe form of a sandwich consisting of conducting silver paint, athin tape as dielectric, and conducting paper. The techniqueallows the field from a single-source or from multiple-sourcesto be plotted. Field density information may be obtaineddirectly from the map in the form of differential voltage mea-surements if desired. Some examples are included.

INTRODUCTIONAnalog mapping of two-dimensional Laplacian fields on con-

ducting paper is commonplace in undergraduate laboratories.However, Poissonian field mapping is rare, mainly because ofthe complexity and delicacy of the apparatus required in thepast.[1,2,3] Recently a simple capacitively coupled mappingsystem has been developed for power transformer design [4]which is also particularly suited to the teaching laboratory.The dielectric used is thin adhesive polyester tape (mylar)which is pressed firmly onto the under surface of the conduct-ing paper. The capacitively coupled sources are painted ontothe back surface of the tape, using silver paint. The result is athin flexible robust sandwich. Mylar tape has been found tohave a uniform thickness of about 0.1 mm, so that fringingeffects are negligible. Also the dry pressure-type adhesive doesnot impregnate the conducting paper significantly.At low frequencies where the capacitive reactance is much

greater than the surface resistance of the conducting papercurrent paths, the potential V(x, y) on the paper very nearlysatisfies the Poisson equation

V2V(x,y)=-KJ, (1)

where K is the resistance/square of the paper and J, is thecapacitively coupled current density. [1] Regions on a mapoutside source areas satisfy Laplace's equation.Poissonian maps have found widest application in plotting

magnetic fields due to distributed current sources. At low fre-quencies the magnetic vector potential approaches the Poissoncondition. [5]

V2Az(X, Y ) = -Pjz (2)

Manuscript received August 29, 1970; revised June 1, 1971.This work was supported by the National Research Council ofCanada.P. Birke is with Canadian Westinghouse, Hamilton, Ontario,

Canada.R. Kitai is with McMaster University, Hamilton, Ontario,

Canada.

so that equipotentials on the paper correspond to lines of con-stant A, and K corresponds to ,. A boundary between a paperedge and air corresponds to a boundary between a medium offinite permeability and one of infinite permeability, and J, isanalogous to Jc- Also, since B = curlA a contour of constantmagnitude ofA is a flux streamline so that equipotential lineson the map are analogous to flux lines. Flux densities are ob-tained from

3=AZ a_ (3)

whence fluxes can be obtained by integration. Alternatively Bcan be measured directly on the paper, using a differential typeprobe and a sensitive differential millivoltmeter.Two laboratory mapping systems are now described.

SINGLE-SOURCE ANALOGFig. 1 shows an analog of a circular conductor of radius ro,

carrying uniformly distributed current I, from which the Pois-sonian field within the conductor (r < ro) and the Laplacianfield in air (r > ro) are obtained up to radius 3ro. Using cylin-drical coordinates (r, z, 0)

(4)A= p X dV4r olr

and since A, does not vary with 0 for fixed r,

aAB)0 - -Be

ar

Outside the conductor

Az =- 71[21nr+oa]

and within the conductor

A H -

where i1 = , o- and ,B are constants. [6] For continuity at4irr =ro

1 + -2 In ro.

It is convenient to put ,B = 0 so that A = 0 at r = 0. Hence, forr <ro

(r\2 r(5)

and for r > ro

64

BIRKE AND KITAI: MAPPING OF POISSONIAN FIELDS

mV a.c.

- PAINT

F-

,E

Fig. 1. Analog for determination of the field within and out-side a long circular current-carrying conductor. A Hewlett-Packard oscillator type 200CD and an a.c. millivoltmetertype 3400A were used.

(6)

The mapping system of Fig. 1 was constructed using a 30 de-gree segment of a circle of 46 cm. radius and rO = 15 cms. Thetape under the paper need not be carefully trimmed; only thepainting over the area from F up to radius ro needs to be pre-cise. The second electrode is an arc of silver paint, 1 cm. wide,on the upper paper surface. This map has only one kernelpoint at F so that one terminal of the oscillator may begrounded. The conducting paper should be near ground poten-tial to minimize hand and probe lead capacitance effects. Theprobe used should have a blunt point so as to avoid piercingthe paper and tape. Using a frequency of 10 Hz the oscillatoris adjusted to put F at 100 mV relative to ground G. (The po-tential in the vicinity ofF does not vary markedly so that nospecial care in the positioning of the probe near F is called for.)The measured oscillator voltage is 3.8 V, which is high enoughfor eqn. (1) to apply.A non-metallic scale is laid on the map and potentials mea-

sured along it from F to G. Readings on the millivoltmeter aresubtracted from 100 mV to satisfy eqn. (5). Typical resultsare shown in Fig. 2 where log-log paper is used for the Poissonregion and semi-log paper for the Laplace region. From thesecurves the exponent in eqns. (5) is found to be 2.08, 7 = 30.9from Fig. 2a and 31.2 from Fig. 2b. The measured voltage atrO is 31.5 mV. Also equipotentials may be checked to be arcsof circles. The map can be used many times if handled withreasonable care.

MULTIPLE-SOURCE ANALOGSA two-source analog is shown in Fig. 3 which represents two

long conductors SI, S2 carrying currents in opposite directionsin air, and bounded by high-permeability iron of rectangularcross-section. Maps like these are particularly easily con-structed; source regions S1 and S2 are outlined on both sidesof the rectangular-shaped conducting paper, using a wax pen-cil. The tape is then applied so that each source region iscovered. (Again, no trimming of the tape is necessary.) The

315

20

22

_ /

/

I I

r(cm)

(a)

100 _

0

50 0

30°5 20 30 40 45

r (cm)

(b)Fig. 2. Potential measurements for the map of Figure 1.

Fig. 3. Two-source analog. Currents in sources S1 and S2 arein opposite directions. The oscillator and millivoltmeterused are the same as those in Figure 1.

electrodes are painted on and connections attached.In order to keep the conducting paper near ground potential,

an ungrounded oscillator is used in conjunction with a Wagnerearth system, comprising RI , R2, R3 and C, so that somepoint on the map is at ground potential, depending on thesetting of R2. Typical values ofR 1, R2, R3 are 2,700, 200and 5,000 ohms respectively. C is a decade capacitance box(1 . 1 ,uF max.).Using a substantial oscillator voltage the probe is first held

at any point such as P which is about midway between theelectrodes, and the Wagner earth circuit is adjusted to put Pat ground potential. Then the probe is moved towards the twoelectrodes in turn, and the kernel points k1 , k2 (i.e., the pointsof maximum potential) are located and marked. Points k1 andk2 will always lie somewhere within the electrode boundaries.With the probe held at one of the kernel points (say k1) theWagner earth circuit is now readjusted to put k, at ground po-tential. Then the probe is moved to k2 and the oscillator am-plitude adjusted so that the millivoltmeter reads full-scale onan appropriate range (say 1 OOmV). A check is made to ensurethat the oscillator voltage is substantially greater than thekernel to kernel voltage. Finally, the equipotential field ismapped in say 10% intervals. Typical results are shown by thebroken lines in the figure.The two experiments described are considered to be mean-

ingful to the student and sufficient for the laboratory. If maps

65

IAz = -77 2 In + I , Be = -2i?

-

ro r

IEEE TRANSACTIONS ON EDUCATION, VOL. E-1 5, NO. 1, FEBRUARY 1972

with more than two sources are desired, then the referencesgiven should be consulted.

REFERENCES

[1] Gilbert, E. O. and Gilbert, E. G., "A capacitively-coupledfield mapper for two-dimensional distributed source fieldproblems," A.I.E.E. Trans. (Communication and Elec-tronics), vol. 72, pp. 345-351, 1953.

[2] Lourie, S. I., "Modeling of magnetic fields of trans-formers" (in Russian), Elektrotechnika, pp. 54-57,July 1965.

[3] Lourie, S. I., "Mathematical modeling of magnetic leak-age fields of transformers and reactors on electrocon-ducting paper" (in Russian), Elektritchestvo, pp. 80-86,October 1965.

[4] Birke, P. V. and Palmer, S., "A capacitively-coupledmagnetic flux mapper," I.E.E.E. Trans. Power Apparatusand Systems, vol. PAS-89, pp. 1635-1642, Sept./Oct.1970.

[5] Balmain and Jordan, "Electromagnetic Waves and Ra-diating Systems," Prentice-Hall, 1968, pp. 313-314.

[6] Ibid,p. 93.

Due to budgetary problems, the next paper was not included in the November 1971 Special Issue of this TRANSACTIONS,which was guest edited by Dr. Leon 0. Chua. It is therefore included here.

On the Use ofInteractive Computer Graphics forTeaching Selected Topics fromNonlinear Electronic Circuits

L. 0. CHUA, SENIOR MEMBER, IEEE

Abstract-Four packages of computer-graphics software de-veloped specifically for teaching certain topics from nonlinearelectronic circuits are described; namely, a package calledICAP for studying the signal processing properties of non-linear circuits, a package called MECA for determining thedriving-point- and transfer-characteristic plots of a large classof nonlinear electronic circuits, a package called GAMA forsimulating the circuit models of three-terminal electronic de-vices, and a package called CAPPA for carrying out a phaseplane analysis of second-order autonomous nonlinear systems.Several examples are presented to illustrate the application ofeach of these four computer graphics software packages.

I. INTRODUCTIONNonlinear electronic circuits has long been considered to be

a difficult subject primarily because the solution of most non-linear equations cannot be obtained in closed analytic form.Unlike the case of linear networks, it is extremely difficult to

Manuscript received February 1, 1971; revised May 10,1971. This work was supported in part by NSF Grant GY5300.The author was with the Department of Electrical Engineer-

ing, Purdue University, Lafayette, Ind. 47907. He is now withthe Department of Electrical Engineering and ComputerSciences, University of California, Berkeley, Calif. 94720.

gain "insights" by merely examining the numerical solution ofa nonlinear circuit since its operation may depend critically onboth element parameter values and initial conditions [1], [2].Consequently, the determination of the global qualitative be-havior of nonlinear electronic circuits is usually much moreimportant and meaningful than that of determining a particu-lar solution by numerical methods. While the digital computeris ideal for obtaining quantitative answers, it alone cannot beused efficiently for carrying out a general qualitative analysis.Since the qualitative features of nonlinear circuits are bestportrayed in the form of graphs and curves, it is reasonable toexpect that the understanding of nonlinear electronic circuitscan be greatly enhanced with the help of interactive computergraphics.Our objective in this paper is to describe the computer-

graphics software that has been developed at Purdue Univer-sity, Lafayette, Ind., during the last two years which will allowstudents to carry out a qualitative analysis of a large variety ofnonlinear electronic circuits by communicating and interactingwith a computer-graphics terminal.' The terminal consists of

'These software packages were developed as part of a two-year experiment supported by the National Science Founda-tion to explore the use of computer graphics in electrical en-gineering education. The areas covered by these experimentsare described in [3].

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