The distributed circuit properties based on pores in the etched aluminum oxide film of the electrolytic capacitor have been analysed by the mathematical equation. This paper proposes the selection and the determining method of the parameters appearing in the analysis

Nishida, Hideki

Yamamoto, Tatsuma

Yamamoto, Yoshitake

Yasuhara, Kiyotaka

English

Okayama University Digital Information Repository

Parameters for Analysis on Distributed Circuit Properties of Etched Alunimum Oxide Film

Okayama University Scientific Achievement Repository

Memoirs of the School of Engineering Okayama University Vol. 12, February 1978Parameters for Analysis on Distributed CircuitProperties of Etched Alunimum Oxide FilmTatsuma YAMAMOTO*, Yoshitake YAMAMOTO*,Hideki ~ISHIDA**, and Kiyotaka YASUHARA*(Received November I, 1977)SynopsisThe distributed circuit properties based on pores inthe etched aluminum oxide film of the electrolyticcapacitor have been analysed by the mathematicalequation. This paper proposes the selection and thedetermining method of the parameters appearing in theanalysis.1. IntroductionThe etched aluminum oxide film or the sintered material, used inelectrolytic capacitors, has countless microscopic pores. The circuitproperties due to pores have been analysed by the transmission linemodel and its mathematical equationl)2) • However, the problems to bestudied on the selection and determination of the parameters appearingin analysis are yet left unsolved. The parameters, which cannot bedetermined experimentally, should be excepted from the analysis. Thispaper proposes on the selection and determination of the parameters,which can be simply obtained from the experimental data.2. Results and DiscussionsThe simplified equivalent circuit of the electrolytic capacitor isgenerally given in Fig.l. L is a series inductance which is negligible***Department of Electrical Engineering.Faculty of Education, Tottori University.2930 Tatsuma YAMAMOTO, Yoshitake YAMAMOTO, Hideki NISHIDA, and Kiyotaka YASUHARAEquivalent circuitLZsolFig.lover the frequency range employed.Zsol is an impedance due to theelectrolyte-separator combination.Zsol becomes the pure resistance ROwhen the separator paper is not usedor the frequency is low. R' and C'are the interface impedance due toan aluminum cathode. Since an anodeis not the metal, its interface im-pedance is small enough to neglect.The test capacitors are made by pair of oxide films to make the nonpolartype. Accordingly, in the present study the interface impedance can beneglect. C is a capacitance formed with two foils, in which thegelectrolyte-separator combination is inserted as a dielectric, and Cgis small enough to neglect in -comparison with C Rand C are ans s simpedance based on the et~hed aluminum oxide film, and are determinedby both dielectric properties of the film and distributed circuit pro-perties due to a great many pores on the film.Fig.2 shows the size of a pore and the transmission line model ofcircuit properties due to a pore. The diameter and length of a poreare 2d and i, respectively. The thickness of the film is o. Thesegeometric variables adopt the average values. 2d, ~ and 0 may bedifferent according as an individual pore. However, the averagevalues are usually applied in the analysisl )2). The series impedanceZ = Rs - j (l/wCs ) is expressed by Eq. (1), where the number of poresis N per pair of electrodes.Rs1 Rl Jl./N-=--fwCs12k c(1)(a) (b)Fig.2 Configuration and circuit model of poreParameters for Analysis on Circuit Properties of Alunimum Oxide Filmwherecos! sinh 2A - sin ! sin 2Bf Rcosh 2A cos 2Bsin ~ sinh 2A + </> • 2Bcos '2 s~nf =c conh 2A cos 2BA .f2 k cos </>/2 , B 12k sin </>/2 ,31PI is the resistivity of the electrolyte and E:* = E: l - jE: 2 is thecomplex dielectric constant of the film. IYal is a pseudo-admittancewhich is the quantity relating to a permeability of the film. Sincethe impedance is fortunately the function of only the parametersRl2/N, G22N and C1 2N at a given frequency, these parameters are calledthe fundamental parameters in this paper. The fundamental parametersare contained as a lump in Eq.(l), and are not separated; for example,as Rl and 2/N, etc. and then the fundamental parameters are the macro-scopic quantities defined for pair of electrodes. Next, the ratio ofCsat some standard low frequency f O Hz to that at the other arbitraryfrequency fnHz is given as(C s ) f=f (fc)f=fOnCsr ==(C s ) f=f lXn(fc)f=f0 nwhere lX == (f If )1/2n n 0(-2 )2.1 Films not having the dielectric lossAt first, we presume that the inherent dissipation factor32 Tatsuma YAMAMOTO, Yoshitake YAMAMOTO, Hideki NISHIDA, and Kij'otaka YASUHARA(cot ~ = £2/£1) of the film is zero. £1 becomes a constant value inthis case. We can reduce Eq.(2) to the simple form:esrfc(2bO) (3)a. f (2bOa. )n c nwhere fc(2bO) = lim (f ) f=f f (2bOa. ) = lim (f ) f=fG2+O c 0c n G2+O c n ,2b lim 2k = 16wcsORsOG2+O= ~W(R-2/d)ePl 2bO = (2b) f=f 0e = £/0 RsO RlV3N e sO eliNobtained at high frequency, RsandSince lim f R = lim f = 1 isw+oo w+oo CG2+O G2+Ol/we are proportional to w-l / 2 •sThen, lim R = RsO and lim ew+o s w+o SG2+O G2+Oare obtained at low frequency, where RsO and e sO are constant values.The analysis for the sintered material also is almost the same3). Inthe analyses to date, the adequate considerations relating to theselection and determination of the parameters to describe Eq.(l) havehardlylbeen given. In case of G2 = 0, RsO and e sO 2) or PI ' e, andR-/Id ) were selected by each author as parameters. Although R. H.Broadbentl ) has determined as ~/Id = 0.16, the determination of e wasobscure. The experimental determination of the microscopic parameters(e and R-//,d) is not simple, and is hardly performed in practice.K. Nishitani2) described the experimental determination of RsO and e sO •However, the experimental determination of RsO is very difficultbecause of reasons as follows: (1) since the test sample is capacitiveessentially, the resistive component RsO at low frequency can hardly bemeasured, and (2) although R is obtained by subtracting the electroly-stic resistance RO from the total series resistance, Rs is not simplydetermined because the main part of the total series resistance isalmost RO •To settle these problems, Eq. (3)transcendental function of 2bO ' socan be used effectively. esrthat 2bO can not be expressedis a1L.enUO. 1Parameters for Analysis on Circuit Properties of Alunimum Oxide Film 330.01 0.1 f (kHz) 1 10Fig.3 Theoretical curves of Csrvs. frequencyanalytically using the capacitances at two frequencies. Therefore,the determination of 2bO is performed by means of a graphical solutionas follows: Since Csr is theoretically culculated as a function of2bO and various values of an (n = 1, 2, ---), the graph as shown inFig.3 can be beforehand described. On the other hand, C also can be. srexperimentally determined as the ratio between two capacitance valuesat f O and f n Hz. Therefore, when an experimental curve agrees optimal-ly with a theoretical curve in Fig.3, the unknown quantity 2bO isdetermined as the value of 2bO belonging to its theoretical curve.Next the constant value of C obtained at very low frequency gives thes .value of CsO ' Thus the new fundamental parameters (Rl~/N and Cl~/N)are sim~lY determined from 2bO and CsO ' as Cl~N = CsO and Rl~/N= (2bO) /4TIf OCsO ' It is a remarkable feature that these parameters areobtained from the frequency characteristic of the capacitance alone.The difficulty of the experimental determination of RsO is illustratedby the following example. The break frequency, given as the frequencyat the bend point of the curve of Csr ' decreases with the increase of2bO or PI as shown in Fig.3. Therefore, 4% boric acid solution(H 3B03 = 4%, H20 = 96%, resistivity 277n·m at 17°C) having large valueof PI was used, because the distributed properties appear even at lowerfrequency and hence observations are simply made. Films made atformative voltages 14V and 66V were used as test samples. To make thenonpolar sample, a capacitor was made by pair of films (effective area:lc~2, distance between two films: 0.15mm). The solid lines in Fig.434 Tatsuma YAMAMOTO, Yoshitake YAMAMOTO, Hideki NISHIDA, and Kiyotaka YASUHARA~ experimental value--- theoretical value(2) 66 V-c:-cl100~----;:;::~~~~~---t--~~~--i100.1 f (kHz)10 1...-----l..---L..-I...-L....L..l..Ji...Ll-.-_.l--l-..l-...l....l-LJ..U._~....1_.L..J....l...l...LiLL_____J0.01(1)1ItV(a)~ experimental value- - - theoretical value(1) 14 V10-(2)66VIL.::1.-enU0.10.01 0.1 f(kHz) 10(b)Fig.4 Rs and Cs vs. frequencyshow experimental results on distributed properties, and the dottedlines are theoretical curves calculated from Eq.(l) (G2 = 0 in thiscase), using Rli/N (14V --- 29ln, 66V --- 2720n) and CliN (14V ---8.77~F, 66V --- O.936~F) determined by the above process. In lowParameters for Analysis on Circuit Properties of Alunimum Oxide Film 35frequency, although the adequate agreement in capacitance curves isread, the dotted lines of the resistance agree roughly with solid-1lines. Here 20Hz was used as f O • The phase angles tan l/wCs(RO+Rs )show about 57° (14V) and 79° (66V) at 20Hz, so that the experimentaldetermination of R is considerably difficult even in these specialsexamples. Hence the experimental curves of Rsdo not agree adequatelywith theoretical curves. Since the phase angle of the usual electroly-tic capacitor becomes about 90° in low frequency, the experimentaldetermination of RsO is more difficult in comparison with the above'examples.2.2 Film having the dielectric lossNext, we presume that the inherent dissipation factor of the filmis not zero. In this case, the series capacitance Cs in low frequencyis not constant, but a function of frequency, so that the existence ofthe dissipation can be simply verified, and then the large resistance,due to the dissipation raises the accuracy of the measurement ofresistance. In low frequency, since IYdl is small, Rli/N «l/IYdl,2A «1, and 2B «1 are obtained and moreover we obtain Z * l/Yd fromEq.(l). Namely, the distributed properties due to pores hardly appear,and on the other hand the dielectric properties of the film become adominating factor. The vector locus of G /w and C obtained from thep p .perfect admittance Y = G + jwC = liZ is shown by the mark 0 in Fig.5,p Pwhere G and C can be calculated using the well-known series-parallelp p0.3L;:' 0.280 40::a.-~Q,C> 0.1 200'f\ 400\ 2K --'aK \!'-eoo~0 0·1 0·2 0·3 0·4 0·5 0·6Cp(jJF}Fig.5 Vector locus of G /w and Cp P36 Tatsuma YAMAMOTO, Yoshitake YAMAMOTO, Hideki NISHIDA, ,,;)d Kiyotaka YASUHARAtransformation formula of the impedance. Thi~ locus is obtained onfilm of the test sample made at formative voltage 220V, and deviatesfrom the circular arc in high frequency because of appearance of thedistributed properties. In low frequency, since Y = Yd is obtained,Gp/w and Cp are proportional to E 2 and E l ' respectively. It can beread that this locus is satisfied with the Cole-Cole circular arc law.The fundamental parameters G2~N ~ Gp and Cl~N ~ Cp at some lowfrequencies are determined from Fig.5. To obtain the parameters inhigh frequency, this locus, however, must be extrapolated, using theCole-Cole circular arc law, as shown by the mark. in Fig. 5. ThusG2~N = 0.0175, 0.131, 0.347, 0.303mS, and Cl~N = 0.495, 0.22, 0.165,0.16l~F are obtained at 0.02, 0.2, 2, and 8kHZ, respectively. Sincethe parameters G2~N and Cl Jl.N at f O ;andfnHz can be determined, theright term of Eq. (2) becomes a function of Rl~/N alone. Therefore,if the left term (i.e. experimental value) of Eq.(2) is given, R1Jl./Ncan be calculated using a computer technique. The calculated meanvalue of RIJl./N on the sample of Fig.5 is 450n, as f O = 20Hz, and usingthis value, relative errors (100 x (experimental value of C -srcalculated value of Csr)/calcutated value of Csr) are 13.9, 18.8, 9.8,1.05, and -3.34% at f = 400, 800, 2k, 4k, and 8kHz, respectively.nThe large errors in low frequency arise from the dominant dielectricproperties, in comparison with the distributed properties, asmentioned above.Since the macroscopic fundamental parameters RIJl./N, G2J1.N, and ClJl.N,and their determination method are proposed, the distributed circuitproperties due to pores on the film of the electrolytic capacitorwill be more rationally analysed, or the obscurity appearing in thepast analysis will disappear by the present study.References1) R. H. Broadbent: "Alternating-Current Properties of AluminumFoil Electrolytic Capacitors", Electrochemical Technology, ~(1968),5-6, 163-1662) K. Nishitani: "The AC properties of Electrolytic capacitorsForming Distributed Networks", J. of Electrical Engineering, Japan,89(1969), 970, 1333-13423) J. Vergnolle: "Distributed Network Analysis of Porous ElectrodeCapacitors", J. of Electrochemical Society, !!!(1964), 7, 799-804

Parameters for Analysis on Distributed Circuit Properties of Etched Alunimum Oxide Film

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