Basu, J

Ghoshal, A K

Sen, S K

English

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S5DETERMINATION OF THE DIELECTRIC CONSTANT OF A  TU BU LAR M ATERIAL A T  3KMc/s8. K. SEN, J. BASU* 4»d A. K. GHOSHALlN8'nTOTJS or Radio Physics & Elec^ onics, Univeksity o f  Calcutta (Received Reptem^ 22, 1964)ABSTRACT. The paper describes two metiiodB for determining the resonant behaviour of cylindrical cavity with a tubular dielectric irujlterial introduced coaxially in it Tlie first method tends to give the exaiit value of the dielectric constant of the material The second one, based on a perturbation theory, yields somewhat approximito resultsExperiments carried out at a microwave frequenijy of 3 KMc/s on two tubes of Pjrrox glass, show that the inaccuracy in determining the dielectric constant arising out of the approximation inherent in the perturbation theory, is, in these cases, so small as to bo com­patible with the inaccuracy duo to experimental limitationsI N T R O D U C T I O NThe treatment on the resonant beliaviour of a cylindriiial cavity when partially filled with a solid dielectric rod, has been given by Horner et a l  (1946). With the htdp of this, an exact evahiation of the dielectric constant of the rod specimen can be made by solving the transcendental equation relating the dielectric constant and the resonant frequency of the cavity with the rod placed coaxially inside. An approximate analysis on the basis of a perturbation theory was also presented by Slater (1946), with which the flielectric con.stant can be measured from the change in resonant frequency of tiu* cavity with and without the specimen. This anal3'^ sis can be utilised to measure the dielectric constant of a fluid (liquid, gas or plasma) in a tubular container (Biondi and Brown, 1949); the evaluation of the dielectric constant of the contaner is not required. For an exact evaluation, however, it is necessary to determine the dielectric constant of the container.In the present work the treatment developed by Horner et d  (1946), as well as the approximate analysis given by Slater (1946), valid for a solid cylindricaldielectric, are extended for a lossless dielectric in the form of a tube. These methods have been used at a microwave frequency of 3 KMc/s to measure the dielectric constant of two pyrex glass tubes, subsequently to be used as plasma containers.T H E D R E T I C A L  C O N S I D E R A T I O N SThe cavity is operated in the lowest frequency mode i.e. TMo,o mode. The boundaries of the cavity are assumed to be perfectly conducting.• Proaont Address : Saha Institute of Nuclear Physics, Calcutta ,601602 8 . K . Sen, J. Baau and A . K . Ohoshal(a) Exact solntionMaxwell’s equations valid for the interior of the cavity are, in cylindrical coordinates (z, r , 0 ). :j ( » / lH g  =(<r+jbik)Ei =dE,d r1 dr  d r (rHg)( 1)where / i ,  k  and <r are the permeability, permittivity and conductivity of the homo­geneous dielectric medium filling the cavity and «  is the angular frequency. All the parameters are expressed in rationalised Af.K.S. units.Solutions of the above equations for E,i and H g  are :K A Jo(Ar)e^ "* volts/metreH g  =  A J-^(K r)ei'"*  amp/metre(2)in which J q< are the Bessel functions of the first kind, A  is the constant of integration governed by the strength of excitation and the propagation constant K  is given byA* =  —j< afi{a-\-jv> k) (3)Let the cavity contain three lossless media (or =  0) 1, 2, and 3, having permit­tivities ij, k^ , kg respectively, as shown in Fig. 1. and permeabilities equal toFig 1 Cavity with glass tubethat of free space, The propagation constants for the media are. from eqn. (3).Determinatwn o f the Dielectric Constant^ etc. 603K.^  =  =  2^ ... (4)Solutions of Maxwell’s equations for the elec^ic and magnetic fields, can be written as follows : >For medium I.. (6 )For medium 2. . .  (6)For medium 3E z, = jcok^m ,  =  [B ^ JtiM +c^Y iiM ]^^“‘... (7)Here B 's  and C 'a  are constants of integration depending upon the strength of excitation and ^0,^1 are the Bessel functions of the second kind. Equation (5) does not contain the second kind Bessel functions as they become infinite at the axis of the cavity.The boundary conditions for the cavity system are :(1) The tangential component of the electric field at the cavity waU vanishes.(2) There is continuity of electric and magnetic fields at the boundaries of the media 3, 2 and 2, 1. Applying these conditions we get from equations (6), (6) and (7)o^ As^ s) ~  ®Now, from equation (4)A = a V ?604 S. K . Sen, J , Basu and A . K . Qhoshaland A2 “  As "x/(8)(9 )Eliminating the constants in equation (8) and using eqn. (9), we get the following transcendental equation relating the dioloctric constants of the different media,[ J o i M -  V i* m M  ] V i 'where F rrr o(^ 3^ 2) ^ ^ o(^ 3^ 2) *^ i(A3^ 2) ^ o(/^ 3^ 3) *^ o(/^ 3^ 3)Assuming that the medium 3 is air, we gcjt a transcendental equation from (10) which gives the dielectric constant of medium 1 in terms of that of 2 and \ico versa, provided tOj the resonant frequency of the composite system is known.Further, if we assume that the dielectric medium 1, enclosed by the tube represented by the medium 2, is air, we can find h^ lk^  =  the dielectric constant of the tubular material from the following equation, the permittivity of air being taken equal to that of free space, ip.[ U P f z ) -  y lly z iP z r d  y i(M )]= [ Vj  ^ rj(/?2^ 2)~ ro(^ S»'2) j Vjp •^ o(Ao*‘i)*^i(A2^ i)~“‘^ o(^ 2^ i)*^i(Ao*'i) j •” (1^ )Determination of the Dielectric Constant^ etc. mwhore Jp' *^ o(^ p^ 2) *^ o(Ao^ 3) ^ o(^0 ‘^z)_and /?u — wy'//J;„ = -  , c being the velocity o f light in free space. c(b) Solution based on a Perturbation Theory -When the olectri<‘. or magnetic field within a cavity is perturbed by insertion of a material within it, a cliango occurs in th^ distribution of the electromagnetic field. Consequently the resonant frequency jef the c-avity changes. This change of resonant frequency, A/, is related to th^ dielectric constant of the materialinserted. \sLet a dielectric tube be placed coaxia^ within the cavity. For TM^ ^^  mode being used, the electric field only is disturbed. Following Slater (1946), the perturbation equation is/«J, (t-l)E M v  "f E-dvJVc( 1 2 )whore Jq if* ilie resonant freciuom^y and E is the elc(;trie field intensity of the i*n- porturbod cavity, A /is tho c^ hangc in fre(picnc v due to the introduction of the tube. Vg refers to intt^^ration over the volume t)f the tube and Vc that over the volume of the (uivity. It sliould, however, be noted that this equation is valid only if th(‘ j)erturbation is small i.e. the dielectric constant of tho material under study is not very high and tho radius of tho tube is small compared to that of tho cavity.Now, using cylindrical coordinates (2, r, 0), the solution of Maxwell’s equation for tho electric field in a cylindrical cavit}*^  operating in mode is, in absenceof any perturbation,E -  J9Jo(A» ... (13)where K^ y is the propagation constant and Z) is a constant of integration depending upon the strength of excitation.From equations (12) and (13)^  M  {e -\ )T)^J^{K^r)dr r^dO-dz h  ■ 2 UQ-D^J,\K,r)dr-rdO-dz• » *>(€ -1  )D^2n-L [ “ rJc^Kor)dr» ft*D^-2n-L{ %JoHKor)dr i 0... (14)iioa S. K . Sen, J . Basu and A . tL. OhoahatL  roprosents tho length of the cavity . r^ , are the internal and external radiio f the tube and is the radius of the cavity (Fig. 1).From epuation (14)A / 1 / 1 '=  -  (e -1 )  JJn ^f rJo^{K^r)dr»-i______r''»I rJ\{K^r)dr( \j^\K„r)dr9Af Jor, , 2A/ ^~L .r.j  rJo^Kor)dr_ i _ .2A/ , “1“ 0^ 3) ] ... (15)As the cavity wall is assumed to be perfectly conducting, the electric field at r =: rg is zero. Therefore, from eauation (13), «/o(/fo 3^) ^^ *nce the cavityis operated in the TM*,io niode, — 2.405, the value at which the first zeroof Jq occurs.Equation (15) becomesJo_________ri [Jo“(2.405-^ 2 j +Ji®( 2.405^ 2. j ] 2.405 j + Ji2^ 2.405.^  j j... (16)E X P E R I M E N TThe block diagram of the experimental arrangement is shown in Fig. 2.iMmCATORFJXED ATjm M TaATTSAmroA Co-40^ Oodb)?}•JmCATOAAHAPTM CAYSTALWWW  ^ ^Di£L£CrJiK rUB£KtySTAOtt OMMMATWf AOAPTSA Qs-ban^ DlJi£CT/OMU. COOPIMAFig 9. Fxperimontal arrangement.AMPTAM ' CAY/ryAAS0M7WTwo pyrox glass tubes wore chosen as dielectric samples and each of them, in turn, was placed coaxially inside the cavity. The resonant frequencies of the cavity with and without each of the samples were measured. The resonant frequency, in each case, is given by the fijequoncy at which the transmission through the cavity is maximum. The frequ^cy meter gives an accurate readingwithin ±0 .3  Mc/sec. f;iU E S U LThe resonant frequency of the empty cajs^ ity in mode as measured ex­perimentally (Jq) is found to be slightly dif^rent from that calculated from its dimensions (/o). The discrepam^y is assume^ to be due to the presence of holes in the cavity wall provided for insiTtion of Ihe samples and the coupling loops. It is tlierefore evident that tlie measured refonant frecpiency of the cavity with a sample (/ ')  is also affected by the holes and needs correction, the corrected value being taken as/ = / ' !  ( / o - / o ' )The f r e q u e n c i e s /  and the difference frequency A / are givenin Table I.Radius of the cavity rg ~  3.837 cm.f?ample 1. Internal radius, — 0.145 cm Externa] radius, rg — 0.233 cm.Sample IT. Internal radius, — 0.380 cm.External radius, — 0.501 cm.The internal radii were cahmlated from the volume of Mer(mry filling the tubes as described by Worsnop and Flint (1961).TABLE IDetermination of the Dielectric Constant, etc. 607Sample/oMc/sec/o’Mc/secS'Mr/seo/Mc/sec Mc/sec1 2992.7 2999.0 2950.7 2944.4 48.3II 2992.7 3004.5 2874.7 2862.9 129.8Equation (11) is used for determining the exact value of the dielectric cons­tant of the samples, puttng to =  whore /  is obtained from Table I. With the help of equation (16) based on the perturbation theory, the dielectric constant is again calculated; /o and A / are given by Table I. Dielectric constant for two pyrex glass tubes as determined by the above methods, are recorded in Table II. Its percentage deviation for the solution based on the perturbation theory, with respect to the value obtained from the exact solution, is also recorded.608 S. K . 8en^ J. Basu arid A . K . Ohoshal, etc.TABLE IISampleIIIDielectric constantfromexactsolutiona4.804.68fromperturbationtlieoryhperct^ntago deviation X  iOOo/^4.864.591.260.22C O N C L U D I N G  R E M A R K SThe value of the dielectric constant of pyrex glass is found to be in conformity with that reported earlier (Von Hippel, 1954; Forsythe, 1956; Knoll, 1959). It appears from Table II that the composition of the two tube samples is slightly different.The result for the tubes, as obtained from the solution based on the pertur­bation theory, differs, by less than 2%, from that given by the exact solution. It may be concluded that the inaccuracy due to the approximation inherent in the perturbation method is, in these cases, so small as to be compatible with the inaccuracy due to experimental limitations. However, if the perturbation is largo, i.e., if the material under study is of high dielectric constant or if the thick­ness of the tubewall is appreciable, compared to the radius of the cavity resonator, the perturbation theory fails to hold. The exact method described in section (a) of theoretical consideration would, then, have to be followed.The dielectric material discussed in this paper is assumed to be completely lossless. For a material with a low loss tangent, the methods described for deter­mining the dielectric constant may still be applicable with a fair degree of accuracy; the loss tangent can be determined by measuring the of the cavity with and without the specimen and applying, in an extended form, the perturbation method (Slater, 1946) or, for a more accurate evaluation, the method descril)ed by Horner et al (1946).A C K N O W L E D G M E N TThe authors are indebted to Prof. J. N. Bhar, D.Sc., F.N.I., for his keen interest in the work and for valuable discussions.Determination o f the Dielectric Constant, etc. 609R E F E R E N C E SBiondi, M. A., and Brown, S. C., 1949, Phys. Rev. 75, 1700Forsythe, Williatn Elmer, 1956, Smithsonian Physiepl Tables, Smithsonian Institution, Washington, 034Homer, F., Taylor, T. A , Dimsrouir, R., |iamb, T., and Jackson, Willis, 1946, Jour Inst Eke Eng, 98, Part Til, 53 !Knoll, Max,, 1959 Materials and Proeelsos of Electron Devices, Springer-Verlag, Borlin/Gottingen/Hoidelberg, 219 »Slater, J. C., 1946, Review of Modern Physij ,^ 18, 441Von Hippel., Arthur H., 1954, Dielectric Mjliterials and Applications, Technology Press of M IT  and John Wiley and Sons,,iTnc., New York, 310Worsnop, B. L., and Flint, H. '1'., 1961, Advaiflicod Pradi( al Physics for Studenls, Methuen and Co. Ltd., London. 26. I

Determination of the Dielectric Constant of a Tubular Material at 3KMc/s

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Determination of the Dielectric Constant of a Tubular Material at 3KMc/s

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