Method of Orthogonal Potentials Developed for the Analysis of TEM Mode Electromagnetic Resonators

Transcription

1 _00 R.F. Ne #15 NSCL June 1, 005 Jhn incen Mehd f Orhgnal Penials Develed fr he Analysis f TEM Mde Elecrmagneic Resnars INTRODUCTION... DEELOPMENT... 3 E, H FIELD, ω... 4 SUMMARY EQUATIONS... 5 TEST CASE... 6 APPENDIX I (THE.5D FLEXPDE INPUT FILE)... 8 APPENDIX II (THE 3D FLEXPDE INPUT FILE)

2 _00 Inrducin This ne devels a echnique ha simlifies secifying he bundary cndiins fr 3D rblems in he FEA analysis rgram FlexPDE frm PDE Sluins ( I have been using his rgram alms frm he ime i was released under a differen name by a differen cmany. The rgram began as a D scalar slver and has evlved a l in he years wih a 3D versin nw available. A user cmleely describes he rblem equains, bundaries, and bundary cndiins in he inu ex file. As a resul, he rgram is n limied any re-rgrammed rblem se as is rue wih many her rgrams. Addiinally, i cmleely aumaes he finie elemen mesh generain and refinemen rcess. The rgram alies scalar bundary cndiins cnsising f bh alue bundary cndiins and s called Naural bundary cndiins. The naural bundary cndiins are referred as insulaing cndiins and heir effec is based n he aricular frm f he user equains. Since his is a scalar slver, vecr based bundary equains are n sraighfrward unless hey fall n cnsan crdinae surfaces. A cmbinain f hese effecs has led me devel a mehd f slving fr he resnan frequency and fields wihin a 3D TEM mde (ransmissin line mde) resnar ha simlifies and sandardizes he rblem f secifying he bundary cndiins. FlexPDE alies inegrain-by-ars all erms f he user secified arial differenial equain be slved ha cnain secnd-rder derivaives f he sysem variables. As far as elecrmagneic fields are cncerned, his basically means alying eiher Skes s herem r Gauss s law secify he fields n he bundaries. The Naural bundary cndiin (BC) secifies he resuling inegrand. In he fllwing examles, A is a vecr field, u is a scalar field and n is a uni nrmal he enclsing surface r surrunding bundary. ( A ) d ( n A) ds Naural BC he value f ( n A) n he surface S ( A ) ds ( n A) dl Naural BC he value f ( A) S S l ( A ) d ( n A) ds Naural BC he value f ( A) l ( A ) ds ( n A) dl Naural BC he value f ( A) l n n he bundary n n he surface n n he bundary ( A ) d ( n A) ds Naural BC he value f ( n A) S ( u ) d ( n u) ds Naural BC he value f ( n u) l n he surface n he surface

3 _00 Develmen In a TEM mde resnar, n field cmnens are direced alng he wave ah. The TEM cndiin is defined by fields ha bey: E 0 and H 0 In he abve relainshis, he subscri secifies he cmnens and erars exising r acing ransverse he wave direcin. Fr simle rblems, such as hse fund in ms exbks, ne can ake he wave direcin be unifrmly alng ne axis nrmally aken be he z axis. The echnique be develed here is n limied wave ragain alng ne r mre crdinae axis s. Anher cnsequence f he abve relainshis is ha he ransverse fields may be described by he ransverse gradien f a scalar field. E and H m A his in, i is useful review he frm f he sanding wave aern fr he enial field alng a ¼ wave resnan caxial ransmissin line wih uer radius b and inner radius a. b ln r ( r, z) cs b ln a ( Kz) The basic frm f he abve equain is fund in many resnan sysems and may be cas in he mre general frm where in his case, b ln r b ln a and cs( Kz). The subscri is aken mean arallel he wave direcin and as befre means ransverse he wave direcin. Wih hese ideas in mind, he echnique can be develed. 3

4 _00 4 Technique: Slve fr a scalar field () ha can be reslved in a scalar field ( ) reresening he ransverse field cmnens ha bey he elecrsaic field sluin mulilying a scalar field ( ) ha reresens he field behavir alng he wave direcin. The scalar field mus bey he Helmhlz equain; 0 + K 0 + K ( ) K Since nly varies alng he wave direcin and nly varies ransverse he wave direcin (i.e. 0 ) hen he erm ( ) 0. Addiinally, describes a field idenical a charge free elecrsaic field ha is knwn bey Lalace s equain 0. Using his infrmain he equain may be searaed in 0, and 0 + K wih με ω K E, H Field, ω Fr he Elecric Field: E since 0. Ne: E +

5 _00 Fr he Magneic Field: E jωμ H H E jωμ jωμ H + jωμ jωμ since 0 Fr ω: ε μ Slve + K 0 varying K such ha E d H d lume lume Summary Equains 0 ε + K 0 wih K ω μ με such ha E d H d lume lume where E and H jωμ 5

6 0. Z _00 Tes Case The fllwing caxial disk resnar is analyzed in.5d and 3D. The.5D analysis exlis he circularly symmeric cndiin using a sandard echnique, whereas he 3D analysis is dne using he echnique develed here. The resnar is assumed symmeric abu he large circular end surface and shred a he small circular end surface. Y - - X The.5D analysis slves he equain H φ K H φ 0 wih he bundary cndiins ALUE( H φ )0 n he large end and NATURAL( H φ )0 n all her surfaces. The naural bundary cndiin in his case secifies he value f n H ha by Maxwell s equains is equivalen seing he angenial cmnens f E 0. The 3D case bundary cndiins fr are ALUE( )0 n he uer bundaries and ALUE( )1 n he inner bundaries sanning frm he large small caxial ends wih bh he large and small caxial end surfaces se NATURAL( )0. The bundary cndiins fr are ALUE( )1 n he large caxial end surface ALUE( )0 n he small caxial end surface wih NATURAL( )0 n all her surfaces. Aendix I and II rin he inu files fr hese cases fr mre deailed infrmain. The fllwing able summarizes he resuls..5d 3D F(MHz) Q % errr φ The fllwing ls cmare he resuls frm he.5d he 3D analysis. The.5D resuls d n include a enial l because ne is n ssible. 6

7 0. Z 0.75 Y _00.5D Blank, Grid, Hmag, Emag 3D, Grid, Hmag, Emag Quarer Wave Resnar 0.9 ON x0 15:9:5 4/6/05 FlexPDE 4..9 Z x Y XYZ_3D_Disk_Mde: Grid#3 Ndes83 Cells4857 RMS Err 1.3e-4 Sage 1 Quarer Wave Resnar r,z 09:00:07 6/3/05 FlexPDE Z R - X RZ_D_Resnar_Disk: Grid#4 Ndes874 Cells385 RMS Err 1.7e-4 Mde 1 Lambda Quarer Wave Resnar 09:00:07 6/3/05 FlexPDE 4..9 Quarer Wave Resnar 15:9:5 4/6/05 FlexPDE 4..9 Z x Hmag Z x Hmag ON x Scale E R RZ_D_Resnar_Disk: Grid#4 Ndes874 Cells385 RMS Err 1.7e-4 Mde 1 Lambda Y XYZ_3D_Disk_Mde: Grid#3 Ndes83 Cells4857 RMS Err 1.3e-4 Sage 1 Quarer Wave Resnar 09:00:07 6/3/05 FlexPDE 4..9 Quarer Wave Resnar 15:9:5 4/6/05 FlexPDE 4..9 Emag Emag ON x0 Z x Z x R Y RZ_D_Resnar_Disk: Grid#4 Ndes874 Cells385 RMS Err 1.7e-4 Mde 1 Lambda XYZ_3D_Disk_Mde: Grid#3 Ndes83 Cells4857 RMS Err 1.3e-4 Sage 1 7

8 _00 Aendix I (The.5D FlexPDE inu file) TITLE '.5D Quarer Wave Resnar' Crdinaes ycylinder("r","z") SELECT errlim1e-3 mdes hermal_clrs n linegrae ff ARIABLES Hhi DEFINITIONS { Resnar Exens in Meers } r1 0.1 r0. r30.4 r4 L1 0.1 L0. L30.75 { Maerial Cnsans } es e-1 { Farads/m } { Permiiviy f Free Sace} er1.0 { Relaive Permiiviy } mus04*i*1e-7 { Henries/m } { Permeabiliy f Free Sace } mur1.0 { Relaive Permeabiliy } es er*es0 { Resulan Permiiviy } mus mur*mus0 { Resulan Permeabiliy } sigma5.8e+7 { mhs/m } { cnduciviy f cer a 0 degrees C } user1.05 { vls eak } { user desired eak vlage n "user" ah } { Cmued Resuls } megasqr(lambda/(mus*es)) { Angular Frequency } freqmega/(*i) { Frequency } Hvecr(0,0,Hhi) HmagMagniude(H) Er-(1/(mega*es))*dz(Hhi) Ez(1/(mega*es))*(1/r)*dr(r*Hhi) Evecr(Er,Ez) EmagMagniude(E) RRsqr((mus*mega)/(*sigma)) { Surface Resisance } PC(1/)*RR*Sinegral(abs(Hmag)^,'erimeer') { Cnducin Lsses } 8

9 _00 UE(es/)*inegral(Emag^,'caviy') { Sred Elecric Energy } UH(mus/)*inegral(Hmag^,'caviy') { Sred Magneic Energy } Q(mega*UE)/PC { Resnar Qualiy Facr } { User Scaling } -binegral(angenial(e),'user') { lage alng user ah } Kfacrabs(user/) { user scaling facr based n user } PCSPC*Kfacr^ { Scaled Cnducin Lsses } UESUE*Kfacr^ { Scaled Sred Energy } UHSUH*Kfacr^ CS(*UE)/abs()^ { User Pah Shun Caaciance } LS1/(mega^*CS) { User Pah Shun Inducance } RSQ/(mega*CS) { User Pah Shun Resisance } { Equains be Slved } EQUATIONS Hhi: Curl(Curl(Hhi))-lambda*Hhi0! Hhi: Div(Grad(Hhi))+lambda*Hhi0 { Resnar Bundaries and Bundary Cndiins } BOUNDARIES regin 1 'caviy' sar 'erimeer' (r1, L1) Naural(Hhi)0 line (r3, L1) line (r3, 0) alue(hhi)0 line (r4, 0) Naural(Hhi)0 line (r4, L) line (r, L) line (r, L3) line (r1, L3) line finish { Define he User Pah } feaure 1 sar 'User' (r3, 0) line (r4, 0) { Requesed Ouus fr each Mde } PLOTS grid(r,z) cnur(er) ained 9

10 _00 cnur(ez) ained cnur(hmag) ained cnur(emag) ained elevain(emag) ON 'erimeer' elevain(er) frm (r1,0) (r1,l3) elevain(er) n 'user' vecr(e) nrm nis SUMMARY rer(freq) as "frequency" rer(lambda) rer(ues) rer(q) rer(uhs) rer(cs) rer(ls) rer(pcs) rer(ues) rer(user) END 10

11 _00 Aendix II (The 3D FlexPDE inu file) TITLE '3D analysis f a quarer wave case' { Cmmen u crdinaes secin fr recangular case } Crdinaes Caresian3 SELECT errlim5e-4 sages1 hermal_clrs n linegrae ff ARIABLES DEFINITIONS { Resnar Exens in Meers } r1 0.1 r0. r30.4 r4 L1 0.1 L0. L30.75 { Resnar Maxumum Exens } { Maerial Cnsans } es e-1 { Farads/m } { Permiiviy f Free Sace} er1.0 { Relaive Permiiviy } mus04*i*1e-7 { Henries/m } { Permeabiliy f Free Sace } mur1.0 { Relaive Permeabiliy } es er*es0 { Resulan Permiiviy } mus mur*mus0 { Resulan Permeabiliy } ea sqr(mus/es) { imedance f sace } sigma5.8e+7 { mhs/m } { cnduciviy f cer a 0 degrees C } user100 { vls eak } { user desired eak vlage n "user" ah } { Cmued Resuls } lambda sage*0. { Mde 1 eigenvalue }!lambda sage*0. { Mde eigenvalue } megasqr(lambda1/(mus*es)) { Angular Frequency } freqmega/(*i) { Frequency } 11

12 _00 * { lage } E-*Grad()-*Grad() { Elecric Field } H(1/(mega*mus))*crss(grad(),grad()) { Magneic Field } EmagMagniude(E) HmagMagniude(H) UE(es/)*inegral(Emag^) { Sred Elecric Energy } UH(mus/)*inegral(Hmag^) { Sred Magneic Energy } RRsqr((mus*mega)/(*sigma)) { Surface Resisance } PC(1/)*RR*Sinegral(abs(Hmag)^) { Cnducin Lsses } Q(mega*UE)/PC { Resnar Qualiy Facr } errr abs(1-sqr(ue/uh))*100 { +/- errr in resnan frequency } { Equains be Slved } EQUATIONS : Div(Grad()) + Lambda1*0 : Div(Grad())0 { Gemery } EXTRUSION Surface 'Oen' z0 Layer 'Lw_Z' Surface 'S1' zl1 Layer 'High_Z' Surface 'S' zl Layer 'Sem' Surface 'Shr' zl3 { Resnar Bundaries and Bundary Cndiins } BOUNDARIES Surface 'Oen' Naural()0 alue()1 Surface 'Shr' Naural()0 alue()0 regin 1 'Exens' sar 'uer' (r4, 0) alue()0 Naural()0 ARC (CENTER 0,0) ANGLE360 finish Limied regin '1' Layer 'Lw_Z' OID sar 'uer' (r3, 0) alue()1 Naural()0 ARC (CENTER 0,0) ANGLE360 finish Surface 'S1' alue()1 Naural()0 1

13 _00 Limied regin '' Layer 'High_Z' OID sar 'uer' (r1, 0) alue()1 Naural()0 ARC (CENTER 0,0) ANGLE360 finish Limied regin '3' Layer 'Sem' OID sar 'uer' (r1, 0) alue()1 Naural()0 ARC (CENTER 0,0) ANGLE360 finish Limied regin '4' Layer 'Sem' OID sar 'uer' (r4, 0) ARC (CENTER 0,0) ANGLE360 finish sar 'inner' (r, 0) alue()0 Naural()0 ARC (CENTER 0,0) ANGLE360 finish Surface 'S' alue()0 Naural()0 { Requesed Ouus fr each Mde } PLOTS cnur() ON x0 ained cnur() ON x0 ained cnur() ON x0 ained cnur(hmag) ON x0 ained cnur(emag) ON x0 ained SUMMARY rer(freq) as "frequency" rer(pc) rer(q) rer(errr) rer(ue) rer(uh) END 13