Described in this paper are the design considerations of the simple element of the two-dimensional RC distributed constant circuit and its applications for the analysis of the transient heat conduction problems in engineering work. This element is formed by coating with the resistive film the upper surface of the high permittivity ceramics plate whose back side to be silvered. In addition to a resistivity of every elementary area within the resistive film, there exists capacitive coupling between the area and ground. This element can be regarded as a typical two-dimensional RC distributed system and utilized as a simulator for the same dimensional heat conduction system. It has a convenience and high accuracy for the analysis of the transient heat conduction problems in engineering work

Kato, Osamu

Misaki, Takayoshi

English

Okayama University Digital Information Repository

The RC Network Analyzer Using the High Permittivity Ceramics and its Applications

Okayama University Scientific Achievement Repository

MEMOIRS OF THE SCHOOL OF ENGINEERING, OKAYAMA UNIVERSITY; Vol. 1, No.1, MARCH, 1966The RC Network Analyzer Using the High PermittivityCeramics and its ApplicationsTakayoshi MISAKI and Osamu KATODepartment of Electrical Engineering, Okayama University(Received November 20, 1965)Described in this paper are the design considerations of the simple elementof the two-dimensional RC distributed constant circuit and its applications forthe analysis of the transient heat conduction problems in engineering work. Thiselement is formed by coating with the resistive film the upper surface ofthe high permittivity ceramics plate whose back side to be silvered. In addi-tion to a resistivity of e\-ery elementary area within the resistive film, thereexists capacitive coupling between the area and ground_ This element can beregarded as a typical two-dimensional RC distributed system and utilized as asimulator for the same dimensional heat conduction system. It has a conveni-ence and high accuracy for the analysis of the transient heat conductionproblems in engineering work.§ 1. IntroductionThe simulation of the diffusion equation byelectrical means can be accomplished with theresistance and the capacitance. The RC networkelement described in this paper is a simple andtypical two-dimensional RC distributed constantcircuit and has an accuracy good enough as thesimulator of the diffusion equation. In practice.it may be utilized for the analysis of the two-dimensional heat conduction problems in en-gineering work.§ 2. Structure of the elementA cross-sectional view of the element is shownin Fig. 1.Ca.rbon fi.l 1ftSLl ....er electrodeFig. 1. A cross-sectional view of the elementThe base plate of the element is formed bythe high permittivity ceramics, such as. Ti02or BaTiO~. Both surfaces of the plate are suffici-ently polished up.It is coated one side with a carbon film andthe other side with a layer of silver paint.There exists capacitive coupling betweenevery point along the carb: n film having uni-f rm resistivity and the silver electrode. Silverpaint is also preferred as the electrode materialfor boundaries. The shape of the desired beun-dary is painted on the carbon film, and can berepainted in accordance with the problems. Theaccuracy of such a element is determined in themain by thickness variation of the base plate.In general. the magnitude of the capacitanceper unit area is proportional to the permittivityand inverse proportional to the thickness of thedielectric material. Here as the plate of thiselement is formed by the high permittivity cera-mics, it is not necessary to be made the platethin specially. For instance, in the case of theTi02 plate, the element whose thickness is3.4mm. has the capacitance per unit area ofFig. 2. A sample of the element using the titaniumdioxide ceramicsThe RC Network Analyzer 85(2)(1)25 pF/em2• Accordingly, the thickness variationof the element can be decreased enough. A sam-ple is shown in Fig. 2.§ 3. Measurement of the circuit constantsof the elementThe electrical resistance in one specific direc-tion across opposite extremities of area (Ro) andOute,r elec\ode IA) t ~----->I \, Inner electrode IB)~+/! ' :t:::::~Ca.rbo7Lfilm I Base pla.teFig. 3. Thc measuring circuit of the electrical resis-tance per unit square (Raj of the elementthe electrical capacitance per unit area (Co) aremeasured conveniently as follows.As shown in Fig. 3, a pair of electrode is setup and the electrical resistance between bothelectrodes (Ra) is measured by ohm meter.Determing the resistance Ra, the resistance R.is given byR 2rrRa (n)0= In(al b) Jt:where a = inner radius of electrode (A) (em)b = radius of electrode (B) (em)Also the capacitance Co is given byC - eoes (FI em2)o-awhere eo = permittivity of free spaceos = electric permittivityd = dielectric thickness (em)Therefore Co is determined by substituting themeasured value of the dielectric thickness dinto Eq. (2).Measuring results of some samples of theelement are shown in Table 1.Table 1. Measuring results of samplesSampleABI, I Diameter Thickness Co Ro SCeramics I Es__-L_~('!.m~mllJ)'--.-J_~(~mt'mt')'--_~(2!:p"cF.!.-/,::,cm~2j--i_-,,(k=fJ=')_-t_--'(--,cm,---,2)_BaTi03 I 1,500 _I~I~~__ 2.0 660 1 16.6--+----1 ~~---f~~~~--;---TiEh I 100 120 3.4 26 105 1, 140§ 4. The analog approachThe imaginary equivalent circuit of the ele-ment is shown in Fig: 4.These equations can be reformulated as.l~(/;v)+-.L a2v = RoCo§.1!._r or or r2 acp~ at (I;)8cp1/7]R, <pHeat flow in a two-dimensional field is governedbyl.~(R fiB ) +-.L ()2f) = cp_ af} (7)R aR. oR R2 Ccp2 'l arFor evey term in Eq. (6) there is a correspondingterm of the same order of differentiation in Eq.(7)The analogous quantities are summarized below.Electrical ThermalVoltage v TemperatureCurrent i Rate of heat flow uElectrical Thermalresistance R 0 resistanceElectrical Thermalcapacitance Co capacitanceCylindrical Cylindricalcoordinate r, <P coordinateTime t Time- ~~ =Roir (3)-- &v =Roi'l' (4)r~cp~+ oir +-.L &i'l'= _ Co &v (5)r or r cr otFig. 4. The imaginary equivalent circuit of the elementIn cylindrical coordinates, this circuit is gover-ned by86 T. MISAKI and O. KATO (Vol. 1,where r; = thermal conductivity of heat con-ducting mediumc = specific heatp = densityThe circuit shown in Fig. 4 may be regarded asthe analog circuit of the two-dimensional heatconduction system.Eq. (6) has as its solution~ ~vCr, cp, t)=~ ~fm(Amsr)X (Amscosmcpm~O s~l?;\-ms+Bm,sinmq:')e- RoCot +fer, if) (8)where Am', B ms, Ims = constants determined byinitial and boundary con-ditionsm, S = plus integerf (r, cp) = solution in steady statefer, cp) is given by~f(r,if)=~rP(CpcosPq:'+DpsinPif)(9)p-owhere C p , D p = constants determind by boun-dary conditionsp = plus integerIf the results of the theoretical analysis for Eq.(8) coincide with the results obtained by theexperimental analysis shown in Fig. 5 underthe same conditions, the imaginary equivalentcircuit shown in Fig. 4 can be regarded as thetrue equivalent circuit of the element. Cons-equently, the element can be utilized as thesimulator of the two-dimensional heat conduc-tion system.in Table 3 are obtained.Table 2. Numerical values of constants-~--IR~- I Co I Ao~·a A02·a A03·a60mm 1105 k12126pF/c~r 2.40 5.52 8.65Table 3. Calculated values of the solutionobtained by theoretical analysis5 110 I 20 I 30 I 50 i 100I1.00 11.00a 11.00 1.00 1.00 1.00 1.003a/4 0 0.50 0.68 0.83 0.91 0.97 I 1.00a/2 I 0 0.17 0.40 0.67 0.82 0.94 1.00I 0.04 0.22 0.55 0.75 0.92 1.00a/4 I 00 0 0.01 0.16 0.50 0.73 0.91 1.00The experimental results for Fig. 5 is asshown in Fig. 6.1r'=--a2(a)1.00.'0.'0.2(10)(11)Fig. 6. The oscillograms obtained by the experi-mental analysis (a) and those copies (b)40 sot (p.S)(b)10 lOo 5 10Both results are coincident very well. So thatthe circuit shown in Fig. 3 can be representedby the element.Comparing the wave forms in each point onthe same circle, all of these are coincident.Therefore, the uniformity of the element forall direction is as expected.§ 5. Practical setup procedure and applica-tions for the transient heat conductionproblemsAs mentioned above, heat flow in two-dimen-sional fields is governed by Eq. (7). The scalefactor N v, the size ratio N r and the time scaleStep functlcnYolt4j'e 1'tne,a.torIEig. 5. The connection diagram for theexperimental analysisWhen an unit function voltage is applied be-tween the circular electrode set up in r = a andground, the solution by the theoretical analysisis given byvCr, t)= 1-~:E fo(los r) c-:!~ota s~l J.Osfl(JOSa)( ) "'1 2~ JoC'o,r) _~~Stor V r, t = --"'-'------ c ROOOa S~l JOsfl(J.OS a)Since inserting the numerical values of con-stants in Table 2, the calculated results shown1966) The RC Network Analyzer 87(a) (b)HOR 50 SEC/CMFig. 7. The oscillograms obtained by the analoganalysis (Problem 1)where A = radius of iron cylinder(14)factor Nt are given byThese equations are the relations describing thetransformation from the original system vari-ables to the element variables. Substituting Eq.(12) into Eq. (7)l~(r 8v )+l 62v = N y 2~p__av (13)r Br ar r2 6qJ2 Nt 7) atThe follow:equation must be established in orderthat Eq. (7) may coincide Eq. (6).Nr~ H!_ = RoCoNt 7)(b)HOR 100 SEC/CM-----Cro"ruj ~UieVlllt~~'ppJ:";S,de(a)(c)Fig. 9. The oscillograms in r =a/2, (correspondedto R= A/2) obtained by the analog analysis(Problem 2)Fig. 8. The connection diagram for the analoganalysis (Problem 2)(Problem 2)Determine the temperature distribution with.in the endless iron cylinder applied the constanttemperature suddenly on the half surfacecovered from (jJ =0 to (jJ = IT and the ideal atomos·pheric temperature 0 on the other half surface.The magnitude of the physical constants e, p,r; and the values of R o, Co, N r and Nt are thesame as those of problem l.The connection diagram is shown in Fig. 8 andthe analog solutions are shown in Fig. 9. Theresults of the theoretical analysis areAccordingly, the chQice of Ru, Ce, or Ny andNt are made so as to establish Eq. (14). So thatthe heat conduction problems can be analyzedby use of the RC distributed constant circuit,that is, the RC network element.The following are the several problems of thetransient heat conduction.(Problem 1)Determine the temperature distribution with-in the endless iron cylinder on which outer sur-face of R = A = 10 em the constant temeratureis applied suddenly.The magnitude of the physical constants arec = 0.108 (caT/g. deg)p = 'l.8'l (g/cm3 )'1 = 0.18 (eaT/em. s. deg)Using the sample shown in Table 2, the valuesof the circuit constants of the element Ro, Co andthe size ratio N r areRo = 105 (kQ)Co = 26 (PF/ em2)Nr = R/r=A/a=1.67Substituting these values into Eq. (14), the timescale factor Nt is determined as follows.The connection diagram for the analog analysisof this problem is shown in Fig. 5. Imposing thestep function voltage between the circular elect-rode set up in r = a = 6 em and ground, theoscillograms shown in Fig. 7 are obtained as theanalog solutions. Transforming the scales tothe original system, these oscillograms repre-sent the temperature distribution within theiron cylinder.The results of the theoretical analysis are88 T. MISAKI and O. KATO (Va!. 1,Fig. 10. A cross-scetional view of the endlessiron cylinder in the furnaceIron cy .It !1 der:,: . .- ... " .. , .Fu.r na.ce W-o.UHeat sou.rceCnnstant tempera.turere~ullltorof problem 1,For the radiant heat flow part, the analogousquantities are summarized below.Electrical ThermalVoltage VI, V2 Temperature 81, 82Current I, h 12 Radiant heat flow U, UI, U zwhere II = kV1 4 (is constant in this problem)lz = kVZ4I = k<V1 4- V z4)k = constant corresponded to KThe current sources generated the current ([)can be accomplished with the vacuum tube cir-cuit shown in Fig. 11 and the suitable amplifier.~71,e'1 r 1/.0 20010.4.1----I----------f..IJoI-fl-H--+------l0.6.1----I--------tI-!--iY++++----I0.10.2 j-~~_l___/_,i/_ll_Id-~~+_~--I0.3Fig. ll. A nonlinear element using thevacuum tube circuit0.iJl----f-120345610.91.0 I------I------f---/'+I--Hf+-+-----j4.0 £0 6.0 7.0Vi (V)Fig. 12. Nonlinear characteristics of the vacuumtvbe circuit shown in Fig. IIVO(V) ~---~----,--------,-r-r-l----'1.1As shown in Fig. 12, in this vacuum tube cir-cuit, the output voltage through the vacuumtube circuit Vo is proportional to the input vol-(Problem 3)Determine the temperature distribution of theendless iron cylinder in the furnace having thecross section shown in Fig. 10. This problemcontains the heat conduction and radiationphenomena. It follows that the heat energy isradiated from the furnace wall to the iron cylin-der surface and the temperature within the ironcylinder is risen gradually. The heat energy isradiated according to Stefan-Boltzmann's law.And then the radiant heat flow U is given byU=K(81(-tJ24)=Uj-U2 (17)where (kJt = temperature of furnace wall in (OK)82 = temperature of iron cylinder sur-face in (OK)K = constant determined by Stefan-Boltzmann's constant and by geo-metrical shape and relative positionof both surfacesIn this problem, the temperature of the furnacewall is kept constant by the regulator.And then81 = constantso that 814 = constantThe simulation of this heat flow system is asfollows.In the part of the iron cylinder, the magni-tude of the physical constants c, p, r; and thevalues Ro, Co, N r and N 1 are the $ame as those1966) The RC Netivork Analyzer 89tage v, raised to the fourth power.The connection diagram of the analog systemformed by the analogous current sources andthe simulator of the iron cylinder is shown inFig. 13.Table 4. The printed data by the digitalcomputer (~EAC 2203)T i(s) I Ii (deg)I R=A I 3A/4 A/2 I Al4 I 0HOI{ 12. 5 ::>EC/CM25 5397126 5349070 5318186 5262532 523387850 5398225 5366623 5339204 5321833 531491470 5398731 5375688 5353950 5338295 5331315100 5399050 5381644 5364764 5352114 5346292125 5399276 5385967 5372937 5363047 5358448150 5399446 5389226 5379190 5371538 5367968175 5399575 5391717 5383991 5378092 5375337200 5399673 5393628 5387682 5383140 5381018225 5399748 5395098 5390522 5387027 5385393250 5399806 5396228 5392707 5390017 5388760275 5399851 5397098 5394389 5392319 5391351300 5399885 5397767 5395682 5394090 5393345325 5399912 5398281 5396678 5395452 5394879350 5399932 5398678 5397443 5396501 5396060379 5399947 5398982 5398033 5397307 5396968400 I 5399959 5399217 5398486 5397628 5397667425 I 5399969 5399397 5398835 5398406 5398205450 5399976 5399536 5899103 5698773 5398619475 5399981 5399643 5399310 5399056 5398937500 5399985 5399725 5399469 5399273 5399182525 5399989, 5399788 5399591 5399441 5399370550 5399991 5399837 5399685 5399569 5399515575 5399993 5399874 5399758 5399669 5399627600 5399994 5399903 5399813 5399745 5399713625 5399996 5399925 5399856 5399803 5399779650 5399996 5399942 5399889 5399849 5399830675 5399997 5399956 5399915 5899883 5399869700 5399998 5399966 5399934 5399910 5399899725 5399998 5399973 5399949 5399931 5399922750 5399998 5399979 5399961 5399946 5399940775 5399999 5399984 5399970 5399959 5399954800 5399999 5399987 5399976 5399968 5399964825 5399999 5399990 5399982 5399975 5399972850 5399999 5399992 5399986 5399981 5399978875 5399999 5399994 5399989 5399985 5399983900 5399999 5399995 5399991 5399988 5399987925 5399999 5399996 5399993 5399991 5399990950 5399999 5389997 5399995 5399993 5399992975 5399999 5399997 5399996 5399994 53999941000 5399999 5399998 5399996 5399995 539999553~ X 10-2, 52~ X 10-3, Example 5397126~971.26 (deg)Taking a view of the results, such a directanalog method has not necessarily a high accu-racy for these nonlinear problems, but has anallowab~e accuracy in practice (better than 3 percent). If better accuracy is required, it is desired( - l'~The elementS<..-~ n chan1er I/ ..~. I:i:- 12 ... !_._-_ ..l'zRe8u1a.ted constan.tVol t,teFig. 15. The results of the numerical analysisobtained by use of the digital computer (NEAC2203) (Problem 3)The oscillograms are shown in Fig. 14.Fig. 13. The connection diagram for the analoganalysis (Problem 3)E \ ESu.mm.Lnt amp \ Nonli..nea; element\Constant multi.pl~erHOI{ 50 SEC/OilFig. 14. The oscillograms obtained by the analoganalysis (Problem 3)(Accuracy check for problem 3)The results of the numerical analysis obtainedby use of the digital computer (NEAC 2203 madein Japan) are shown in Fig. 15 and in Table 4.90 T. MISAKI and O. KATOto make use of the digital device for the non-linear parts.As it is often required the simplicity of theanalysis rather than the high accuracy in en-gineering work, it may be estimated for theanalysis of these nonlinear problems.§ 6. ConclusionIt is well known that an analog method forthe analysis of the transient phenomena is veryeffective. Here a simple and typical RC distri-buted circuit is represented, and it has anaccuracy good enough for practical use. Also itis very convenient that the circuit constants arecapable to vary by changing the thickness ofthe carbon film and the boundary conditions byrepainting the shape of the silver electrode inaccordance with the problems. The distributedvariabe constants circuit can be represented bychanging the dielectric thickness or the carbonfilm thickness.As above, the design considerations and someapplications of the RC distributed constantscircuit, that is, the RC network elements aresummarized, but for its application it is onlydescribed about few problems. From now on itmust be established the application of the ele-ment.The authors are grateful for the instructionreceived from Dr. Sigenori Hayashi, Prof. ofKyoto University, Dr. Kiyoshi Okazaki, Prof. ofDefense Academy, Mr. Takashi Suzuki, DefenseAgency, and for the useful suggestions givenfrom the references (1), (2). Votes of thanksare also due to Mr. Kaoru Tomono and Miss.Ayako Ogasawara for their assistance in theexperiments and the calculations.References1) W. J. Karplus: Analog Simulation(McGraw-Hill)2) W. J. Ran: Control Eng. (1963 June) 853) T. Misaki, K. Okazaki: J. 1. E. E. of Japan 80 (1960)156.4) T. Misaki, K. Okazaki, T. Suzuki: J. 1. E. E. ofJapan 80 (1960) 14265) K. Okazaki, T. Misaki, T. Yoshioka, Y. Yokoyama:J. 1. E. E. of Japan (1961) 1779

The RC Network Analyzer Using the High Permittivity Ceramics and its Applications

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