Alenk MLVANVC 1 Brnko KPRVCA 1 Universit of Krgujev Tehnil Fult Ck (1) Clultion of Chrteristi mpene of Eentri Retngulr Coxil Lines Astrt. n this pper the hrteristi impene for oxil lines with retngulr inner n outer onutor onentri eentri n with rotte struture hs een otine using the Equivlent Eletroes Metho (EEM). ur results hve een ompre with those reporte in the literture otine other nltil n numeril methos n those otine using the CMSL Multiphsis softwre pkge n ver goo greement with those results hs een foun. Streszzenie. W rtkule przestwiono olizeni impenji hrkterstznej linii konentrznej z prostokątnm przewonikiem ustuownm entrznie i eksentrznie. lizeni wkonno z wkorzstniem meto ekwiwlentnh elektro EEM. (lizeni impenji hrkterstznej linii konentrznej z eksentrznm przewoem prostokątnm) Kewors: Retngulr oxil line Equivlent Eletroes Metho Chrteristi impene. Słow kluzowe: lini konentrzn impenj hrkterstzn. ntroution As retngulr oxil lines re ommonl use in mirowve tehnolog to trnsmit energ it is ver importnt to know their hrteristi prmeters. The min prmeters re: the pitne per unit length the inutne per unit length n hrteristi impene. Ext nltil solutions for the etermintion of these prmeters re not possile n so numeril methos n pproximte nltil expressions re use to otin them. Sine the erl 1960s severl uthors hve een intereste in nlsing retngulr oxil lines. The first signifint nlsis ws performe Chen [1] using the Shwrz-Christoffel trnsformtion n pproximte nltil expressions for the pitne n inutne per unit length n hrteristi impene of retngulr oxil lines were presente. Vrious other pproximte nltil solutions for the pitne per unit length espeill orner pitne hve een presente in [2]. n 1965 relxtion tehniques for the solution of Lple's eqution in two imensions were pplie [3] to erive the hrteristi impene. The onforml mpping metho n Shwrz-Christoffel trnsformtion were use in 1972 [4] for the lultion of the imensions of line for given hrteristi impene. The step urrent ensit pproximtion hs een use to etermine the hrteristi impene. Costmgn n Fnni in 1992 ompute the hrteristi impenes of vrious oxil lines with retngulr onutors mens of numeril inversion of the Shwrz-Christoffel onforml trnsformtion. A qusi-nltil metho of the multipole theor (MT) metho hs een use for the nlsis of retngulr trnsmission line fmil n the results otine for the hrteristi impene hve een presente in [7]. The Glerkin metho hs een pplie to solve sstem of integrl equtions mens of sis funtions tking into ount the right-ege ehviour of the urrents for lulting the hrteristi impene [8]. Results from 2007 for the pitne per unit length of retngulr oxil lines otine using the Finite Element Metho softwre pkge CMSL hve een presente [9]. n our pper simple numeril proeure lle the Equivlent Eletroes Metho (EEM) hs een propose for the nlsis of retngulr oxil lines. Previousl EEM hs een use to solve stti n qusi-stti eletromgneti fiels n other potentil fiels of theoretil phsis [10 11]. t hs lso eeuessfull use for trnsmission line nlsis [12-15] n ver goo greement etween these results hs een foun omprison with results otine using other nltil n numeril methos. Therefore EEM hs een use in this pper for lulting the hrteristi impene of oxil lines with retngulr inner n outer onutor onentri eentri n with rotte struture. The results otine using EEM hve een ompre with those foun in the literture n those otine using numerous moels of these lines me in the softwre pkge CMSL. EEM Applition EEM hs een pplie to the lultion of the hrteristi impene of oule eentri retngulr oxil line (Fig. 1). This line ontins two retngulr onutors. The inner onutor is isple from the longituinl xis of smmetr in oth horizontl n vertil iretions. Also the inner onutor hs ifferent ngulr position reltive to the outer onutor. The eletri potentils of the onutors re φ 1 n φ 2. Fig.1. Doule eentri retngulr oxil line with rotte inner onutor n ppling EEM eh onutor shoul e reple finite sstem of equivlent eletroes (EE) ple on its surfe. As the sies of the onutors re not the sme size eh hs to e ivie ifferent numer of EE. n our exmple the sies of the onutors with with n hve een ivie N 1 N 2 N 3 n N 4 prts respetivel n the withs of ll of these prts shoul e sme Δx (Fig. 2). The sstems of equivlent eletroes (EE) ple on the surfe of the onutors re: q i i=12 N 1 for one vertil sie of the inner eletroe q j j=12 N 2 for one horizontl sie of the inner eletroe Q n n=12 N 3 for one vertil sie of the outer eletroe n Q m m=12 N 4 for one horizontl sie of p s x 260 PRZEGLĄD ELEKTRTECHNCZNY (Eletril Review) SSN 0033-2097 R. 88 NR 10/2012
the outer eletroe. The totl numer of EE is therefore; N u =2N 1 +2N 2 +2N 3 +2N 4. N 3 x Fig.2. Arrngement of equivlent eletroes x Fig.3. Thin flt strip onutor reple with linril EE N 1 N 4 N 4 x 2r en x Eh prt of the originl onutors in our exmple thin flt strip onutors with with Δx hs een reple linril EE with irulr ross-setion s shown in Fig. 3. The equivlent rius of these EE n e lulte s: r en =Δx/4 [12] where: Δx=/N 1 =/N 2 =/N 3 =/N 4. The equivlent eletroes tht reple these prts hve the sme rius potentil n hrge s the prt of the rel onutor the represent. Axes of EE hve een ple in: x=x n = n for the inner onutor n x=x m = m for the outer onutor where: xn p os sin 2 (1) sin os n 1 1 for n 1 2 N1 nn1 1 xn p os sin (2) 2 1 sin os 2 for n N11 N12 N1N2 xn p os sin 2 (3) sin os nn1n2 1 1 1 for n N1N2 1 N1N 2N1N2 n2n1n2 1 xn p os sin (4) 2 1 sin os 2 for n 2N1N2 12N1N 2N12N2 xm 2 (5) m m 3 1 N 2 x N 1 N 2 N 3 for m 1 2 N3 mn3 1 xm (6) 4 1 m 2 for n N3 1 N3 2 N3 N4 xm 2 (7) m N3 N4 1 m 3 1 for m N3 N4 1 N3 N4 2 2N3 N4 n2n3 N4 1 xm (8) 4 1 m 2 for n 2N3 N4 12N3 N4 2 2N3 2N4. (9) Sstems of EE rete the eletri potentil: N qi 0 ln xx i i i1 4 N Qj ln x xj j j1 4 where: N =2N 1 +2N 2 n N =2N 3 +2N 4 whih shoul stisf the ounr onitions on the surfes of the inner n outer eletroes of the oxil line. B this proeure sstem of liner equtions with the unknown hrges of EE n e otine. N N As φ 1 -φ 2 =U n q i Qj 0 the omplete sstem i1 j1 of liner equtions is: U A11 A1 N B11 B1 1 N q1 U A N 1 A 1 1 N NN B N B NN q (10) 0 C11 C1N D11 D1 N 1 Q1 0 CN 1 C 1 1 QN NN D N D NN 0 1 1 1 1 0 0 where: 1 ln (11) Anm r1 ren mn 4 for x xn n n 12 N m 12 N 1 2 (12) Bnm ln r 2 4 for x xn n n 1 2 N m1 2 N 1 2 (13) Cnm ln r 1 4 for x xm m n1 2 N m1 2 N 1 (14) Dnm ln r 2 r enmn 4 for x xm m n1 2 N m1 2 N PRZEGLĄD ELEKTRTECHNCZNY (Eletril Review) SSN 0033-2097 R. 88 NR 10/20161
2 2 (15) r xx r xx 1 n n 2 m m n δ nm is the Kroneker smol. B solving this sstem the unknown hrges of EE n e etermine. The pitne per unit length n the hrteristi impene n e esil lulte s: (16) N qi i1 r C Z C U C where: is spee of eletromgneti wves in free spe n ε r is the reltive ieletri onstnt of the ieletri meium insie line. Numeril Results EEM hs een pplie to the nlsis of the generl shpe of the oule eentri retngulr oxil line with rotte inner onutor (Fig. 1). The si shpe of retngulr oxil lines hs een nlse t eginning of this setion. Then lines with single or oule eentriit hve een nlse. Lines with rotte inner onutor oth onentri n eentri hve een nlse t the en of this setion. Conentri Lines The line from Fig. 1 hs onentri shpe when p=s=0 n α=0. When ppling numeril metho for the lultion of quntit it is neessr to etermine the onvergene of the otine solution. n the se of EEM the otine solution onverges with n inresing numer of equivlent eletroes. The onvergene of the hrteristi impene for onentri lines hs een presente in Tle 1. Here xil n trnsversl smmetr hs een use so the presente numer of EE is four times greter thn the se without smmetr. Aoring to the results presente in Tle 1 the totl numer of EE in other lultions hs eeet etween 10000 n 16000 whemmetr hs een use or etween 2500 n 4000 without smmetr. This tpe of line hs een nlse mn uthors n the results otine for the hrteristi impene hve een presente in numerous ppers. These results hve een ompre in Tles 2 3 n 4 with the results otine using EEM. Tle 5 presents the omprison of the results for the hrteristi impene of this tpe of line otine using EEM n CMSL. A ver goo greement of ll these results is evient for vrious line imensions. n prtiulr the est greement of the presente results hve een foun etween those otine using EEM n those given in. B nlsing ll these results it n e seen tht the imensions of the line hve gret impt on the hrteristi impene vlue using it to erese s the rtios / n inrese. Tle 2 for ifferent rtios / n when /=1 / [19] 0.05 0.2 121.81 121.63 121.75 121.14 0.1 0.2 109.10 108.53 109.05 108.96 0.2 0.2 91.155 91.20 91.12 92.44 0.1 0.4 79.939 80.32 79.90 80.92 0.2 0.4 67.3788 67.20 67.36 67.62 0.2 0.6 49.816 50.02 49.81 50.54 Tle 3 for ifferent rtios / when -=- /=0.5 / [1] [7] 0.1 79.0583 76.896 79.034 79.124 0.2 66.318 65.717 66.295 66.341 0.3 55.5015 55.367 55.487 55.342 0.4 45.7736 45.759 45.767 45.789 0.5 36.8054 36.815 36.807 36.822 0.6 28.4531 28.468 28.461 28.476 0.7 20.6421 20.661 20.657 20.670 0.8 13.3193 13.342 13.341 13.357 Single Eentri Lines The line from Fig. 1 hs single eentri shpe when p 0 or s 0 n α=0. The results for the hrteristi impene of lines with this shpe hve een presente in Tles 6 n 7. Tle 6 presents the omprison of the results otine using EEM n those given in n ver goo greement with those results hs een otine. The results presente in Tle 7 hve een otine using EEM for vrious eentriities. Tle 4 for ifferent rtios when /=0.5 / =0.1 Z [1] [18] [2] [3 [8] [16] [17] 0.9 27.5145 29.3 27.3 27.7 27.57.57.67.1 27.7 0.75 38.0314 38.7 38.1 38.3 38.03 38.03 38.10 37.8 38.0 0.45 58.4999 58.6 58.8 58.7 58.49 58.49 58.61 59.1 58.6 0.225 86.8448 86.7 87.2 85.9 86.82 86.82 87.27 87.0 86.5 Tle 1 Convergene of the when /=0.2 =0.4 /=0.5 N u 96 39.9148 168 40.4090 336 40.7176 672 40.8972 1344 40.9967 2688 41.0500 5376 41.0779 8064 41.0875 10752 41.0924 13440 41.0953 16128 41.0973 Tle 5 for ifferent rtios when /=0.5 /=0.2 Z CMSL 0.1 75.1266 75.039614 0.2 59.0226 58.953132 0.3 48.5612 48.50593 0.4 41.0920 41.045987 0.5 35.3752 35.336003 0.6 30.6852 30.651742 0.7 26.4790 26.450546 0.8 22.1401 22.117489 0.9 16.3405 16.328868 262 PRZEGLĄD ELEKTRTECHNCZNY (Eletril Review) SSN 0033-2097 R. 88 NR 10/2012
From Tle 7 it n e seen tht inresing the eentriit of the line ereses the vlue of the hrteristi impene. This erese is smller thn the one tht ours when the imensions of the line inrese exept when the eentriit is of the sme orer s the imensions of line. Tle 6 for ifferent rtios / n s/ when /=1 n p=0 / s/ [8] 0.1 0.2 0.35 69.8328 69.54 69.84 69.84 0.2 0.2 0.30 62.2342 62.04 62.24 62.24 0.1 0.4 0.35 47.5623 47.86 47.58 47.58 0.2 0.4 0.30 43.2709 43.27 43.32 43.32 Tle 7 for ifferent rtios n p/ when /=0.5 /=0.2 s=0 p/=0.05 p/=0.1 p/=0.15 p/=0.2 0.1 75.0425 74.7885 74.2494 73.1985 0.2 58.9183 58.6177 57.975 56.7034 0.3 48.4292 48.0362 47.1832 45.4423 0.4 40.9143 40.3566 39.1072 36.3774 0.5 35.1172 34.2537 32.181 26.8044 0.6 30.2714 28.7595 24.4317-0.7 25.7063 22.3359 - - 0.8 20.1924 - - - Doule Eentri Lines The line from Fig. 1 hs oule eentri shpe when p 0 n s 0 n α=0. The results for the hrteristi impene of lines with this shpe hve een presente in Tles 8 9 n 10. Tle 8 presents the omprison of the results otine using EEM n those given in n ver goo greement with those results hs een otine. The results presente in Tles 9 n 10 hve een otine using EEM when eentriit or imensions of the line hve een hnge. A oule eentri line hs smller hrteristi impene thn single eentri line or line without eentriit. Tle 8 for ifferent rtios / p/ n s/ when /=1 / p/ s/ [8] 0.05 0. 2.75 96.6753 97.20 96.64 96.64 0.1 0. 2.5 86.9425 86.78 86.92 86.92 0.1 0.4 1 2.5 64.9165 65.58 64.93 64.93 0.2 0. 2 73.6543 73.54 73.64 73.64 0.2 0.4 1 2 56.6902 56.73 56.68 56.68 0.2 0.6 0 2 43.722 44.04 43.72 43.72 Tle 9 for ifferent rtios p/ n s/ when /=0.5 =0.3 /=0.2 p/ s/=0 s/=0.05 s/=0.1 0 48.5432 44.9082 32.3999 0.05 48.4292 44.8143 32.3566 0.1 48.0362 44.4900 32.2063 0.15 47.1832 43.7830 31.8743 0.2 45.4423 42.3266 31.1728 0.25 41.7792 39.2077 29.5897 0.3 33.1192 31.5779 25.2524 Rotte Strutures The line from Fig. 1 hs rotte shpe when α 0. The results for the hrteristi impene of the rotte strutures with or without eentriit otine using EEM n CMSL hve een presente in Tle 11 n goo greement etween the results is evient. The vrition of the hrteristi impene with rottion ngle otine using EEM hs een presente in Figs. 4 n 5. The hrteristi impene of onentri n eentri lines ereses when inresing the rottion ngle α from 0 to π/2. A smmetril istriution of results hs een otine when the rottion ngle hs vlue etween π/2 n π (Figs. 4 n 5) n similr istriution of results hs een otine for vlues greter thn π. When eentriit of the line is lrge vrition of the hrteristi impene with rottion ngle is not so simple (Figs. 4 n 5). Thus omintion of lrge eentriit n rottion n le to n unesirle inrese of the hrteristi impene. Tle 10 for ifferent rtios n p/ when /=0.5 /=0.2 s/=0.1 p/=0.05 p/=0.1 p/=0.15 p/=0.2 0.1 55.8819 55.7649 55.515 55.0216 0.2 40.9939 40.8706 40.6042 40.0664 0.3 32.3566 32.2063 31.8743 31.1728 0.4 26.6594 26.4541 25.9801 24.8756 0.5 22.5624 22.2476 21.4476 19.0792 0.6 19.3832 18.8165 16.9866-0.7 16.6565 15.2644 - - 0.8 13.7081 - - - 50 48 46 44 42 [] 40 38 36 34 32 p=0 p=1 0 20 40 60 80 100 120 140 160 180 Fig. 4 Vrition of the hrteristi impene with rottion ngle n eentriit when /=0.5 /=0.2 =0.3 s=0. 46 44 42 40 38 [] 36 34 32 30 28 p=3 Fig. 5 Vrition of the hrteristi impene with rottion ngle n eentriit when /=0.5 /=0.2 =0.3 s=0.5. p=2 [Degrees] 0 20 40 60 80 100 120 140 160 180 [Degrees] p=2 p=0 p=1 p=3 PRZEGLĄD ELEKTRTECHNCZNY (Eletril Review) SSN 0033-2097 R. 88 NR 10/20163
Tle 11 for ifferent rottion ngle α n rtio p/ when /=0.5 /=0.2 =0.3 s=0 α CMSL CMSL CMSL CMSL p/=0 p/=0 p/=0.1 p/=0.1 p/=0.2 p/=0.2 p/=0.3 p/=0.3 0 48.5432 48.50593 48.0362 47.998894 45.4423 45.409024 33.1192 33.1197811 π/12 47.9684 47.931376 47.4827 47.446429 44.9868 44.953978 32.5905 32.591744 π/6 46.4629 46.427551 46.0333 45.998132 43.81 43.778111 31.9188 31.916412 π/4 44.6458 44.612034 44.2856 44.254093 42.4255 42.394 32.6391 32.631137 π/3 43.2626 43.233384 42.9617 42.931642 41.4292 41.401904 34.1512 34.137033 5π/12 42.6254 42.599676 42.3611 42.33511 41.0378 41.013704 35.3413 35.327064 π/2 42.4839 42.458483 42.2317 42.207446 40.9782 40.9541723 35.7872 35.773517 Conlusion n this pper EEM is propose for the lultion of the hrteristi impene of retngulr oxil lines oth with eentriit n without it n espeill for lines with rotte inner onutor. The results otine hve een ompre with those foun in the literture. Some of the results hve een ompre with those otine using the CMSL softwre pkge. All the results hve een foun to e in ver goo greement over wie rnge of line imensions eentriit n rottion ngle. The retngulr line is ver importnt in mirowve tehnolog euse mthing impene n trnsitionl struture etween roun oxil line n stripline mirostrip line or other plnr line is ver importnt n therefore this tpe of line must hve well efine hrteristis. ne of these prmeters the hrteristi impene hs een lulte in this pper with ver high ur with respet to other numeril methos previousl use for suh lultions. The results hve een otine for wie rnge of imensions n shpes of the line n thus provies ler piture of how this prmeter hnges with imensions n shpes. t n e onlue tht eentriit n rottion of the inner onutor signifintl impts on the hrteristi impene whih tkes wie rnge of vlues. This llows this tpe of line to hve ontinuousl hrteristi impene whih n e esil lulte using EEM. Compring the results otine using EEM n those foun in the literture it n e seen tht EEM is ver urte metho tht hs no limittions in terms of the imensions n shpes of lines. Aitionll onsiering the entirel new results for lines with rotte inner onutor it n e onlue tht EEM hs signifint vntges over other methos. The pplition of EEM is ver strightforwr n the progrmming is simple n fst without numeril integrtions using onl simple mthemtil opertions. REFERENCES [1] Chen T.S. Determintion of the Cpitne nutne n Chrteristi mpene of Retngulr Lines RE Trnstion on Mirowve Theor n Tehniques 8 (1960) No. 5 pp. 510-519 [2] Cruzn.R. Grver R.V. Chrteristi mpene of Retngulr Coxil Trnsmission Lines EEE Trnstion on Mirowve Theor n Tehniques 12 (1964) No. 5 pp. 488-495 [3] Metlf W.S. Chrteristi mpene of Retngulr Trnsmission Lines Proeeing EE 112 (1965) No. 11 pp. 2033-2039 [4] Rilet H.J. The Ext Dimensions of Fmil of Retngulr Coxil Lines with Given mpene EEE Trnstion on Mirowve Theor n Tehniques 20 (1972) No. 8 pp. 538-541 vnov S.A. Djnkov G.L. Determintion of the Chrteristi mpene Step Current Densit Approximtion EEE Trnstion on Mirowve Tehnolog n Tehnique 32 (1984) No. 4 pp. 450-452 Costmgn E. Fnni A. Anlsis of Retngulr Coxil Strutures numeril nversion of the Shwrz-Christoffel Trnsformtion EEE Trnstion on Mgnetis 28 (1992) No. 2 pp. 1454-1457 [7] Zheng Q. Lin W. Xie F. Li M. Multipole Theor Anlsis of а Retngulr Trnsmission Line Fmil Mirowve n ptil Tehnolog Letters 18 (1998) No. 6 pp. 382-384 [8] Luio M. Pnriello G. Shettino F. Aurte n Effiient Anlsis of Stripline Strutures Mirowve n ptil Tehnolog Letters 43 (2004) No. 1 pp. 14-21 [9] Mus S.M. Siku M.N.. Anlsis of Retngulr Coxil Lines EEE Region 5 Tehnil Conferene Fetteville AR USA (2007) pp. 322-325 [10] Velikovi D.M. Milovnovi A. Eletrostti Fiel of Cue Eletroes Serin Journl of Eletril Engineering 1 (2004) No. 2 pp. 187-198 [11] Milovnovi A. An nfluene of the Petrol Pump for the Atmospheri Eletri Fiel Distriution in the Surrounings of the Vehile Serin Journl of Eletril Engineering 1 (2004) No. 3 pp. 7-14 [12] Velikovi D.M. Equivlent Eletroes Metho Sientifi Review (1996) No. 21-22 pp. 207-248 [13] Velikovi D.M. Milovnovi A. Approximte Clultion of Cpitne Proeeings of V nterntionl GTE Smposium on Numeril Fiel Clultion in Eletril Engineering Grz Austri (1998) pp. 339-344 [14] Rievi N.B. li S.S. Equivlent Eletroe Metho Applition on Anisotropi Miro Strip Lines Clultions nterntionl Conferene on Eletromgnetis in Avne Applitions Torino tli (2007) pp. 998-1001 [15] Milovnovi A.M. Bjeki M.M. Approximte Clultion of Cpitne of Lines with Multiler Meium Journl of Eletril Engineering 62 (2011) No. 5 pp. 249-257 [16] Getsinger W.T. Couple Retngulr Brs etween Prllel Pltes RE Trnstion on Mirowve Theor n Tehniques 10 (1962) No. 1 pp. 65-72 [17] Joines W.T. The hrteristi mpene of Smmetril Strip Trnsmission Lines with unesire Moe Suppresse PhD Dissetttion Duke Universit Durhm NC USA 1964 [18] wkur H. Arkw T. Anlsis of Retngulr Coxil Trnsmission Line (in Jpnese) Reports of the Universit of Eletro-Communitions 28 (1970) pp. 1-8 [19] Chng T.N. Tn C.H. Anlsis of Shiele Mirostrip Line with Finite Metlliztion Thikness the Bounr Element Metho EEE Trnstion on Mirowve Theor n Tehniques 38 (1990) No. 8 pp. 1130-1132 Authors: Dr Alenk Milovnovi E-mil: lenk@tf.kg..rs; Mr Brnko Koprivi E-mil: koprivi@tf.kg..rs. Authors re with Universit of Krgujev Tehnil Fult Ck Deprtment of Eletril n Eletroni Engineering Svetog Sve 65 32000 Ck Seri. 264 PRZEGLĄD ELEKTRTECHNCZNY (Eletril Review) SSN 0033-2097 R. 88 NR 10/2012