ELECTRONICS RESEARCH LABORATORY

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1 K. UNIVERSITY DIVISION OF ELE BERKEL CALIFORNIA L ENGINEERING UFORNIA J! ELECTRONICS RESEARC LABORATORY A COAIAL LINE FILLED WIT TWO NON-CONCENTRIC DIELECTRICS h D. J. Angelake 1 : Institte f Engineering Research Series N. 60, Isse N. 102 Nvember 9,1953 W fc»iw aiwska»iiiw#«n»f'',b ' ; ' VI

&j) fi*qt *J»s D. J. Angelaks, Ass't. Prfessr f Electrical Engineering Apprved by: ^^fi^^c^j^u** ***? J. R.

2 .i, DIVISION OF ELECTRICAL ENGINEERING ELECTRONICS RESEARC LABORATORY SerUs N. 60 ISSUO N. 102 ANTENNA. GROUP Reprt N. 21 n Office f Naval Research Cntract N7nr-2952 A COAIAL LINE FILLED WIT TWO NON-CONCENTRIC DIELECTRICS Prepared by? &j) fi*qt *J»s D. J. Angelaks, Ass't. Prfessr f Electrical Engineering Apprved by: ^^fi^^c^j^u** ***? J. R. Whinnery- Vice Chairman, Divisin f Electrical Engineering, in charge f Electrnics Research Labratry 9«M>S»MKMiWMMMIBMI«tt*'.----'«=»^*!W«««^«'*rWM»«W»'"«i«aiW«

31 ABSTRACT An analysis has been made f a caxial transimissicn line cmpsed f tw caxial cylindrical cndctrs* Tw dielectrics fill different anglar prtins f the vlme between the cndctrs* xhe prpagatin

3 31 ABSTRACT An analysis has been made f a caxial transimissicn line cmpsed f tw caxial cylindrical cndctrs* Tw dielectrics fill different anglar prtins f the vlme between the cndctrs* xhe prpagatin cnstants (primarily the gide wavelength) are determined by a resnant cnditin applied t the plane transverse t the directin f prpagatin* Experimental verificatin is given fr near nity vales f the rati f the ter t the inner radis* In additin, an experimental investigatin has been made f the prperties f the gide wavelength as a fnctin f freqency and larger ratis f the radii ;-:~?"jss»''sfc 1 «mm *i»am*.,qjv2tiia';. ;»!''

n I. INTRODUCTION The prpagatin cnstants f caxial transmissin lines with caxial dielectric cylinders have been determined by means f several methds.

4 n I. INTRODUCTION The prpagatin cnstants f caxial transmissin lines with caxial dielectric cylinders have been determined by means f several methds. The prblem has been treated as a bndary vale prblem as well as a 1 2 mdal prblem. ' On the ther hand the cmpsite strctre f caxial cylinders and a wedge-shaped dielectric material des nt lend itself t a simple analysis by means f the sltin f Maxwells' Eqatins siibject t the prper bndary cnditins. A simple apprach which yields gd reslts is made here sing an apprximate mdal methd ;3,yi*^ifr-»*w**^'^,W'"t^ff^^?^^-'-^^.*.---* *«*'» r.*^ *. Mfr^s&;j*j&tt y

5 i..4 ".I. TBORI AND EPERIMENTAL RESULTS The wavegide shwn in Fig, 1(a) cnsists f tw nifrm caxial cylindrical cndctrs the space between the cylinders being partially iiiled with material f dielectric cnstant e r The remaining vlme has a dielectric cnstant e t. The strctre has even symmetry abt the plane AA' and thrgh the se f the bisectin therem can be cnsidered as a "bent" wavegide f width x and length 2vp where pis an average radis. The ends f the "bent" wavegide are pen circited. The reqirement that the ttal admittance lking in bth directins at the pint T yielde the resnant freqencies can be written as fllws?' T *2«Q2 tan [* Jtp]+ T Q1 tan -f (*-#/>"]-0 Y rt and Y,,, are the characteristic admittances f the sectins 2 and v/c UJ. 1 respectively. The wavelengths * - and x -, are the wavelengths f pr- pagatin f the fields int regin 2 and regin 1 respectively f the "bent" wavegide. The angle is shwn in Fig. 1(b). Fr the regin 1, k *01 tap. t 2ir i* f?v )'.1 * and vn,, "- ( t, T Similarly fr regin 2, G? 3 ~z$r [& >- itff and 2n.[ >' i - *g J -3- vpmpapmn

$- =, the abve relatins may be arranged in the fllwing frm: *«/ * T 02/ Y 01 and 2ir, 2n r / *i \ * -T*" 2ir 2ir ft, /*l\*l~*~ ence, the resnant cnditin becmest (1) tanh [s^r-«>']-vfe$- -Ok J If the$

6 Jj Fr bth, x is the free-space wavelength, x is the nknwn resnant wavelength r prpagatin wavelength in the axial directin f Fig. 1(c). Since x. - =, the abve relatins may be arranged in the fllwing frm: *«/ * T 02/ Y 01 and 2ir, 2n r / *i \ * -T*" 2ir 2ir ft, /*l\*l~*~ ence, the resnant cnditin becmest (1) tanh [s^r-«>']-vfe$- -Ok J If the argments are small. 2w I ivft 77 ( --^- V IF syr^r- KW-f^**-,w «* MB^'* r -'»*»w«bmaftku(mar*)w ;»,

7 V. &, Q t-i > n CO s Q CO 2 as (M O W J W Q hi.»»»».wfc1)i«l ;. «&»,-,:!,:.:

The rati f -* then becmes independent f the average radis p- and A l (2) 1 yrrr^ Eqaticn (2) can be btained frm the cnsideratin f the "static" prblem in the diagram f Jig. 2.

-] 2if j * * c i»-tn-b7t are* The eqivalent phase velcity, V, and wavelength f prpagatin, l», V B y^ *A g TUF 71!

8 The rati f -* then becmes independent f the average radis p- and A l (2) 1 yrrr^ Eqaticn (2) can be btained frm the cnsideratin f the "static" prblem in the diagram f Jig. 2. The capacitances per nit length f the regins 1 and 2 are, 2c t O*) ^L " -tn b/a The ttal eqivalent capacitance is. Z **0 tn b/a respectively. C_ - C - C 01 [«*.-] 2if j * * c i»-tn-b7t are* The eqivalent phase velcity, V, and wavelength f prpagatin, l», V B y^ *A g TUF 71! and a Vsct ~ The variatin f the prpagatin wavelength as a fnctin f the amnt f filling by the dielectric as calclated frm the "3tatic" frmla (2) is shwn in Pig. (3)» The experimental pints are fr a cnsiderable nmber f freqencies and fr a rati f the radii: b * These pints fr mst f the freqency range lie within the circles shwn. A detailed investigatin f the deviatin f the rati f ^S,as calclated frm resnant cnditin (1), frm the static apprxi- ' WJWWW miiwf mi* * *«!!«, yk>-', '*

9 ' w i- K U U. z : CC U U Z Q i-» 01 z z a cr Q U J CC J U. 1-4 I. m K O -7- MM «a«uiu*m

$-J sc M \U»-«as "«OS M k. a.$ K a Vi 2 O 1-4 >»- ( > 2 - J J U 2 U.

10 *N II & I-' * CM 00 SB 1- «r> * «fi CM A. II -4 3 U 3 T "N. -J sc M \U»-«as "«OS M k. a. K a Vi 2 O 1-4 >»- ( > 2 - J J U 2 U. W J U > K «r# U I J CO Q» U a O C1 M a : c J «:» -e- 2 I-I 2 c. a.. 2 >- I-) t- 2 U 2 D 3 cr U. r.i «U. I ft O J «>»-*. CM _

natin f (2) as a fnctin f the freqency is smmarized in Fig. U. Fr this set f data, the relative dielectric cnstant f regin 1 is 1.0 (air) and that f regin 2 is 2,8 (l^rsl plastic). The rati is 1.60.

f> indicates this variatin fr several freqencies, As the rati f b t a becmes larger, the "bent" wavegide apprximatin becmes wrse.

11 natin f (2) as a fnctin f the freqency is smmarized in Fig. U. Fr this set f data, the relative dielectric cnstant f regin 1 is 1.0 (air) and that f regin 2 is 2,8 (l^rsl plastic). The rati is (Actal dimensins r b-l/2", a«/l6", freqency range between 1,000 meps and 3,500 meps). The effect f the variatin f the rati - n the wavelength has been investigated fr a large freqency range. Fig. f> indicates this variatin fr several freqencies, As the rati f b t a becmes larger, the "bent" wavegide apprximatin becmes wrse." in the limit f r-*» the physical shape f the dielectric appraches a dielectric wedge. Fig. 6 illstrates that the rati - variatin with freqency is small. The deviatin frm the static apprximatin de t the wavegide prperty is cmpensated in part by a crrectin necessitated by the circlar bend. The mdal strctre f the fields within this partially filled caxial line was examined in detail. The gide wavelength was measred fr the three psitins indicated in Fig. 7. Within experimental accracy, the gide wavelength, the psitin f vltage minima and vltage maxima remained nchanged fr the three psitins. This was tre fr a large nmber f freqencies and fr tw cnditins f lad? e.g. (1) shrt circit and (2) arbitrary resistive lad. By rtating the inner strctre f the caxial cable dielectric assembly, the charge distribtin n the inner srface f the ter cndctr was btained. As is seen in Fig. 8., the charge distribtin des fllw a csinsidal type f variatin frm the plane f symmetry in regin 2 and hyperblic csine variatin in regin lj it is rapidly changing abt the plane f the dielectric discntinity. The rati f charge densities f the tw regins is eqal t the rati f the dielectrics sed. (jrsl plastic,-^- 2.8). The freqency fr this measrement was 2000 meps. The caxial line in which all the measrements were made cnsists f a sltted ter cndctr f 1 inch inner radis. The dielectric sed 1 was a lw-lss ysl plastic f a relative dielectric cnstant f 2.8. The slt thrgh which the search prbe was inserted was cvered by a t sliding shield s that slt mde excitatin was negligible. -9- v-m*7a.rwii*y*>-, *v.r?ia-»&-

' * «*.? i j -53 3 4 2 M U SE M as O SB 1 M O I-* M "~"" O M M at a «US M a. M f> a. M ii M MM OS OS. Mat a, as as 25 as 3 OM O M ea a. W P» M a.

«1= 1= ^ II II it II -«- -e- -e- 1 1 1 1 1 1 1 1 > 11 i i 1 i i i i i i i, i D Lv: i i f - - - - * " - a - M "* a ** I * >, an s ~ w w Z "^ M a

12 ' * «*.? i j M U SE M as O SB 1 M O I-* M "~"" O M M at a «US M a. M f> a. M ii M MM OS OS. Mat a, as as 25 as 3 OM O M ea a. W P» M a.» K O fim t) 03 ** M i t«m I M f 5 / / P V Jl V // / 5 W If // // // // // S // - /' // // // // _ // If h / /' II i // i» - i // - - f r M _j» «1= 1= ^ II II it II -«- -e- -e > 11 i i 1 i i i i i i i, i D Lv: i i f * " - a - M "* a ** I * >, an s ~ w w Z "^ M a - ss -fc JO z + CO O Q CT ^_ ^ O cc h. Id 1- ^ Z In CM O ll Z S M) CL U JC z a.. 5 M4 83 CO U. J (0 w CO (C l-l PS J. z z z»-l In -10- l «MM4t,y

13 fs IS f Q C w O w J Id > W a PS O «U U. U b. [ U U I'M VJ-» -li-

$0.9 *=% O Q STATIC APPRO. C.8 - * =. ""/p 0 0O OO STATIC APPRO. 0.7 \g T *-*\ O O STATIC APPRO.$ 0 ± = 16 a 12-2 8 J J i 1 l I I I I L kb = 2TTb/, 1.2 1.4 1.6 FIG.

14 0.9 *=% O Q STATIC APPRO. C.8 - * =. ""/p 0 0O OO STATIC APPRO. 0.7 \g T *-*\ O O STATIC APPRO O EPERIMENTAL i i L ± = 16 a J J i 1 l I I I I L kb = 2TTb/, FIG. 6 VARIATION OF TE CUIDE-WAVL_i*GT AS A FUNCTION OF FREQUENCY FOR A LARGE RATIO OF TE RADII *»»**»>,,*>..._._..>,.-.

15 U z I I I- w a 2 O W n cc. cc w b. O t. m Pi & a es ^i^ : «->...kji*** * -, ", _!* ";. *

16 ':'.'»M*VM* " FIG. 6 CARGE-DENSITY DISTRIBUTION ON TE INNER SURFACE OF TE OUTER CONDUCTOR FOR A PARTIALLY FILLED COAIAL LINE. -]j^ r«^m--^mmmmrm%imim.»-mnymm&i&&il7;vs:iijz.',. i

CONCLUSION The partially filled caxial cable can be analyzed t a very gcd apprximatin by a simple static methd as lng as the rati f the ter t the inner radis is small.

17 CONCLUSION The partially filled caxial cable can be analyzed t a very gcd apprximatin by a simple static methd as lng as the rati f the ter t the inner radis is small. Even fr freqencies high engh t almst prpagate the next higher mde this apprximate methd is sefl. The methd cld be e xtended t similar prblems which d nt lend themselves t easy sltin by means f exact bndary vale prblem methds* The dielectric wedge btained as a cnseqence f a large rati f the ter t the inner radis prdces a gide wavelength near t that calclated apprximately* i -15- KiiMMiantw...,;-.*.,.

...-.... --...,.,. FJKFBRENCES (1) Bean and Dbsn, Prc. f ga&naj, fl?ctrnlc8 Cnference. 1952 pp. 301-311. (2) Marcvitz, Wavegide andbk (McGx-i^r-ill Bk Craany, Inc.. New Trk), pp.

18 ,.,. FJKFBRENCES (1) Bean and Dbsn, Prc. f ga&naj, fl?ctrnlc8 Cnference pp (2) Marcvitz, Wavegide andbk (McGx-i^r-ill Bk Craany, Inc.. New Trk), pp (3) Marcvitz, Wavegide andbk (McGraw-ill Bk Cmpany, Inc., New Yrk), pp (U) Marcvitz, Wavegide anhnnv (McGraw-ill Bk Cspany, Inc., New Tric), p I ( -16- "' > ft**,'

1 it Distribtin! Chief f Naval Research, Washingtn, D.C. (Cde h27)., 2 Naval Research Labratry, (Cde 2000) 9 Chief f Naval Research, (Cde 1*60) 1 Naval Research Labratry (Cd*?020).

19 1 it Distribtin! Chief f Naval Research, Washingtn, D.C. (Cde h27)., 2 Naval Research Labratry, (Cde 2000) 9 Chief f Naval Research, (Cde 1*60) 1 Naval Research Labratry (Cd*?020). * 2 Naval Research Labratry icde 3U80) 1 BsShips (Cde 836D) 1 BShips (Cde 810)*... 1 Brea f Ordnance (Ref)... 1 Brea f Ordnance (AB3) «1 Brea f Aernatics (EL5>1) 1 Brea f Aernatics (ELI) 1 Chief f Naval Operatins (OD-I4I3) 1 Chief f Naval Operatins (0p-2C) 1 Chief f Naval Operatins (Op-32) 1 Naval Air Develpment Statin, Electrnics Labratry... 1 O.N.R., New Yrk, 1 O.N.R., San Francisc*. 2 O.N.R., Bstn, Massachsetts 1 O.N.R., Chicag, Illinis 1 O.N.R., Pasadena, Califrnia 1 Assistant Naval Attache fr Research. 2 Naval Ordnance Labratry 1 U.S.N.E.L., San Dieg, Califrnia 2 U. S. Naval Academy, Electrical S=g. Sept., Annaplis 1 Cmmandant Cast Gard (SEE)...» 1 Office f the Chief Signal Officer (SIGET) 1 Signal Crps Engineering Labratries, Attar Mr. Wdyard. 1 Electrnics Labratry, Wright Field (MCSSS) 1 AMC Watsn Labratries, Attar Dr. P. Newman... 1 AVC Cambridge Research Labratry, Attar Dr. Spencer. 1 eadqarters, U. S. Air Frce (AEMBR-2) 1 Research and Develpment Bard, Washingtn, D. C. 1 Natinal Brea f Standards 1 Crft Labratry, arvard University, Attar Prf. King 1 Mass. Institte f Technlgy, Attar Prf. L. J. Gn. 1 Stanfrd University, Attnr Dean F. E. Terman. 1 E.S. DepartEent; Illinis University, Attar Prf. Jrdan. 1 Ohi State University Research Fndatin, Dr. Rmsey.. 1 E«E» Department, Crnell University, Attnr Dr.. G. Bker... 1 Staafrd Research Institte, Attnr Dr. J. V. N. Granger 1 Plytechnic Institte f Brklyn, Attar Dr. A. diner 1 Washingtn Sqare Cllege, N.T.U. Math Grp, Attnr Prf. Kline 1 Sqier Signal Labratry, Attnr Vincent J. Kblin. 1 Flight Determinatin Labratry, Attar M. A* Krivanich 1 Department f Cmmerce. 1 Armed Services Technical Infrmatin Agency. $

convection coefficient. The different property values of water at 20 C are given by: u W/m K, h=8062 W/m K

convection coefficient. The different property values of water at 20 C are given by: u W/m K, h=8062 W/m K Practice rblems fr Cnvective Heat Transfer 1. Water at 0 C flws ver a flat late 1m 1m at 10 C with a free stream velcity f 4 m/s. Determine the thickness f bndary layers, lcal and average vale f drag cefficient