8 JOURNL OF ELECTRONIC SCIENCE ND TECHNOLOGY OF CHIN, VOL. 7, NO., JUNE 9 On Solving TM n Modal Excitation in a Ka-and Ovemoded Cicula Waveguide by the Consevation of Complex Powe Technique Feng Lan, Xi Gao, and Zong-Jun Shi bstact To measue the adiation popeties of elativistic diffaction geneato (RDG) in Ka-band, a TM n modal excitation model is established, which consists of an ovemoded cicula waveguide and a coaxial line feeding pobe. Using the tansvese E-field mode matching and the consevation of complex powe technique (CCPT), we deduce the scatteing matix at coaxial line to coaxial line and coaxial line to cicula waveguide junctions. Then using the oveall cascaded junction scatteing matix, the numeical esults fo the eflection coefficient of the coaxial line and the powe distibution of TM n multi-modal ae pesented. The numeical esults ae in ageement with HFSS simulation esults and expeimental esults. The analysis shows that by choosing the appopiate position of coaxial line pobe, the powe popotion of the device opeating mode excited in cicula waveguide could be the lagest. Index Tems Coaxial excitation, consevation of complex powe technique, ovemoded waveguide, elativistic diffaction geneato.. Intoduction In high powe millimete wave adiation souce which is diven by elativistic electonic beam, ovemoded cicula waveguide is chosen as output high fequency system usually fo aising device powe capacity [],[]. Fo ascetaining the adiation patten of the conical antenna with ovemoded cicula waveguide, the TM n symmetical multi-modal excitation of a coaxial line pobe-to-cicula waveguide is intoduced. The pobe is placed at the waveguide axes to excite symmetical TM n modes, but the position of pobe tine affects the tansvese efficiency of TEM mode to TM n [] modes. Fo the pupose of discussing the tansvese efficiency of TEM to TM n in Ka-band Manuscipt eceived Januay 6, 8; evised Febuay, 9. This wok was suppoted by the National Natual Science Foundation of China unde Gant No. 657. The authos ae with School of Physical Electonics, Univesity of Electonic Science and Technology of China, Chengdu, 654, China (e-mail: lanf998@tom.com, gao_xi76@6.com, shizongjun@6.com). ovemoded cicula waveguide, a igoous full-wave solution by means of the consevation of complex powe technique (CCPT) and HFSS simulations ae given. The numeical esults obtained by CCPT and HFSS simulation fo a ovemoded cicula waveguide fed by an coaxial pobe and a 5 Ω semiigid coaxial cable show quite good ageement.. Deduction of the Oveall Scatteing Matix by the E-Field Mode Matching and CCPT Fig. illustates the geomety of the foepat of the output window, which includes thee waveguides: the small coaxial line fo z< (Guide ), the lage coaxial line fo <z<l (Guide ), and the ovemoded cicula guide fo l<z<l (Guide ). The stimulant signal is fed-in by TEM mode fom Guide. We assume that the cicula waveguide is long enough, only TM n modes can be excited fo the cicula otay symmety. In paticula, we ae concened with the scatteing of a TEM mode incident fom the small coaxial line and the coaxial pobe tine position whee the dominant mode is TM 4 matching with high fequency system. a b ε ε b ε Guide I. bstact and Index Tems coaxial cicula output line waveguide window Fig.. n ovemoded cicula waveguide with an output conical antenna fed by a coaxial line pobe fo TM n modes excitation. Instead of mode matching the tangential magnetic field in the classical solution [4],[5], the CCPT is invoked and leads to an expession fo the junction s input admittance matix as seen fom the smalle guide to obtain scatteing matices. t each junction ( and ), the scatteing matices S and S fo the modal amplitudes of the TM n modes excited by a TEM mode incident fom Guide ae obtained by the tansvese E-field mode matching and the CCPT. Fist we b pobe L Guide Guide z
LN et al.: On Solving TM n Modal Excitation in a Ka-and Ovemoded Cicula Waveguide by the Consevation of Complex Powe Technique 8 deduced the analytical expessions of the E-filed mode matching matices M and M which ae necessay to deduce S and S. ) M (at the coaxial-coaxial junction ): The nmth element of M fo TM n modes only in two coaxial line is: β nβm M nm π β n ρz( β m ρ) Z( β n ρ) β β n m ( ) ( ) β mρz β nρ Z β mρ whee fo n,,, m,, i,, and q,. β satisfies the tanscendental equations: iq i iq iq i iq b a () J ( β b ) N ( β a) N ( β b ) J ( β a ) () and fo j, : π J j( βiqρ) N( βiqa) N j( βiqρ) J( βiqa) Z j( βiqρ ). () J ( βiqa) J ( βiqbi ) Fo the case of TEM mode incident fom one side at junction, if ( ) N ln i π bi a, then M πn N ln( b a) (4) M ( ) b n πn Z β ρ n a m iq (5) M. (6) ) M (at the coaxial-cicula waveguide junction ): The nmth element of M fo TM n modes only in both sides at junction is whee πβ [ β ρz ( β ρ) J ( β ρ) M m n m n nm bj ( βnb)( βn βm) β ρ J ( β ρ) Z ( β ρ)] b m n m a (7) β b is the nth oot of J () x, fo j, and m. n Z () x If m, n j π J j() x N( βiqa) N j() x J( βiqa). (8) J ( βiqa) J ( βiqbi ) πn M J ( β b ) J ( β a) ]. (9) n n β nbj n ( ) [ β b n ) S and S (scatteing matices at junction and ): The CCPT and the E-filed mode matching matices M, M ae used to obtain the junction s input admittance matices Y and Y and scatteing matices S and S. L L YL ( M ) T Y M () S ( Y + Y ) ( Y Y ) () L L S M ( S + I ) () T ( + L) S Y Y M Y S M S I Y ( M ) T Y M L () (4) (5) S ( Y + Y ) ( Y Y ) (6) L L S M ( S + I ) (7) T ( + L ) S Y Y M Y (8) S M S I t (9) whee YL and YL ae the input admittance matices of and junction netwoks, Y and Y ae the modal admittances of Guide and Guide espectively. I is the unit matix. Scatte matix of the oveall cascaded junction [6] : S S S + S L S G L S () S S G L S () G I LS LS () whee is the backscatteed matix which elates the TEM incident and the TMn modes excited by TEM in t Guide. S is the tansmission matix which elates the TEM incident and the TMn modes excited by TEM fom Guide to Guide. L is the diagonal line matix, whose diagonal element is:,nn γ nl L e () whee γ is the popagation constant of the TM n nth mode in Guide, and l is the length of the pobe tine going though Guide. The TEM eflection coefficient in Guide can be obtained by the backscatteed matix, so: Γ S, (4) The nomalized powe of the backscatteed modes in Guide and the tansmission modes in Guide compaed with incident mode TEM is: Y,m P,mm S (5) Y Y, t,n, nn Y, P S S (6) whee Y i,n is the nth modal chaacteistic admittance in Guide i. ased on CCPT, it comes into existence:
8 JOURNL OF ELECTRONIC SCIENCE ND TECHNOLOGY OF CHIN, VOL. 7, NO., JUNE 9 Re P, m+ P, n m n (7). Numeical and Simulation Results The model paametes we concened ae: ε.ε, ε ε ε, a. mm, b.5 mm, b b 9.8 mm, l+l 4 mm. In ode to avoid the elative convegence poblem, we typically used N /N b /b at junction, N N < N at junction (N denotes mode numbe in Guide, N denotes mode numbe in Guide ). Table : Numeical veification of powe consevation fo the case of f GHz, l / λ. Mode numbe P total N N 5 N N.5 N N 5 N N 5.78 N 5 N N N.9999 N 5 N N N 5.9999 The esults shown in Table indicate that the accuacy is much bette than.% when N 5, N, N N. Typically, we used 5 modes in Guide (N 5) and modes in Guide (N ) to compute the scatteing matix S in (); moeove, in all cases N /N b /b, to avoid elative convegence. To calculate S, one needs to invet a matix of size 5 5 only. Moeove, if only the TM, TM, TM, TM 4 mode can popagate, then most of the N mode fields tansmitted into Guide will have negligible amplitude when they each junction. s a esult, in ou computations involving S, we typically set N N..8.6.4. PT-TM 4 PT-TM PT-TM PT-TM Numeical esults fghz..4.6.8..8.6.4. PT-TM 4 PT-TM PT-TM PT-TM HFSS simulation esults fghz....4.6.8. Fig.. Numeical and HFSS simulation esults of nomalized powe of TM n tansmission modes excited by TEM and TEM eflection in the cicula waveguide as a function pobe length at f GHz: numeical esults and..8.6.4. Numeical esults f4ghz PT-TM 4 PT-TM PT-TM PT-TM..4.6.8..8.6.4. HFSS simulation esults f4ghz PT-TM 4 PT-TM PT-TM PT-TM....4.6.8. Fig.. Numeical and HFSS simulation esults of nomalized powe of TM n tansmission modes excitated by TEM and TEM eflection in the cicula waveguide as a function pobe length at f4 GHz: numeical esults and. Γ..8.6.4.. 4 6 8 4 Fig. 4. Numeical and HFSS simulation esults of TEM eflection coefficient Γ vesus fequency fo l mm. Γ.5.4.... 4 6 8 4 Fig. 5. Numeical and HFSS simulation esults of TEM eflection coefficient Γ vesus fequency l6.4 mm. The numeical esults and simulation esults show quite good coincidence. In Fig. and Fig., we compute the nomalized powe of TM n tansmission excited by TEM
LN et al.: On Solving TM n Modal Excitation in a Ka-and Ovemoded Cicula Waveguide by the Consevation of Complex Powe Technique 8 and TEM eflection in the cicula waveguide as a function pobe length at f GHz and 4 GHz. When f GHz, TEM eflection powe is lowest at. and.7. That coesponds with the highest tansmission powe fom coaxial line to cicula waveguide. When f4 GHz, TEM eflection powe is lowest at.5 and.7. That coesponds with the highest tansmission powe fom coaxial line to cicula waveguide. Fig. 4 and Fig. 5 povide the sweeping fequency cuves of TEM eflection coefficient Γ by numeical calculation and HFSS simulation. The TEM eflection coefficient Γ is lowest at f GHz and 4 GHz when l mm and 6.4 mm accod with the powe distibution cuve. t f GHz and 4 GHz, TM, TM, TM, TM 4 can be tansmitted in the ovemoded cicula waveguide, and the highest tansmission mode takes advantage in the powe distibution mostly. 4. Compaison of Expeimental and Simulation Results fo TEM Reflection Coefficient The RDG output window is connected with an ovemoded cicula waveguide fed by a coaxial pobe, then the additional simulation esults and expeimental esults of coaxial line eflection coefficient S fo the case of f~4 GHz, l5.7 mm ae given in Fig. 6. The lowest eflection fequency is aound 7 GHz. Thei coheence is easonably good except some tiny diffeences, fo cable line attenuation in the gilent87es vecto netwok analyze and impefection of output hon teminal matching. The analysis of the esults demonstates that ou design method is eliable. (d) S S (d) 4 4 4 4 6 8 4 4 4 6 8 4 Fig. 6. Numeical and expeimental esults of TEM eflection coefficient S vesus fequency fo l5.7 mm: simulation esults and expeimental esults. 5. Conclusions The solution of the consevation of complex powe technique (CCPT) fo a TMn modal excitation poblem in Ka-band ovemoded cicula waveguide is given in this pape. Diffeent fom the published papes in this field, elationship between mode numbe and pecision of powe consevation is discussed in this pape. In addition, the compaison among numeical esults, HFSS simulation esults and expeimental esults shows geat ageement. The analysis shows that by choosing the appopiate position of coaxial line pobe, the powe popotion of the highest ode mode excited in cicula waveguide could be the lagest. The CCPT is a eliable method fo modal analysis of scatteing at waveguide junctions [7]-[]. Moeove, CCPT spends much less time than HFSS. t cuent situation, HFSS solution will spend a few hous, howeve, the CCPT solution spends only about min. y means of CCPT and cascading algoithm to obtain the popagation chaacteistic of the peiod cougation waveguide and antenna will be done in the nea futue. Refeences []. N. Vlasov,. G. Shkvaunets, and J. C. Rodges, Ovemoded GW-class suface wave micowave oscillato, IEEE Tans. on Plasma Science, vol. 8, no., pp. 556-56, Jun.. [] F. Lan, X. Gao, Z.-J. Shi, Z.-Q. Yang, and Z. Liang, Measuement of adiation powe fo Ka-band elativistic diffaction geneato, Jounal of Univesity of Electonic Science and Technology of China, vol. 5, no. 5, pp. 777-779, Oct. 6 (in Chinese). [] C. Yu and G. Wen, The TM n multi-modal excitation of a coaxial line pobe-to-cicula waveguide, cta Electonica Sinica, vol. 6, no. 9, pp. 9-9, Sep. 998 (in Chinese). [4] G. G. Gentili, Popeties of TE-TM Mode-Matching Techniques, IEEE Tans. on Micowave Theoy Tech., vol. 9, no. 9, pp. 669-67, Sep. 99. [5] G. V. Eleftheiades,. S. Oma, L. P.. Katehi, and G. M. Rebeiz, Some impotant popeties of waveguide junction genealized scatteing matices in the context of the mode matching technique, IEEE Tans. on Micowave Theoy Tech., vol. 4, no., pp. 896-9, Oct. 994. [6] Y.-H. Liu, H.-F. Li, H. Li, E.-F. Wang, H. Wang, and L. Wang, nalysis of an open cavity with abupt tansition by mode-matching technique, Jounal of Univesity of Electonic Science and Technology of China, vol. 5, no. 4, pp. 494-496, ug. 6 (in Chinese). [7] R. H. Macphie and C. R. Ries, Input impedance of a coaxial line pobe feeding a cicula waveguide in the TM mode,
84 JOURNL OF ELECTRONIC SCIENCE ND TECHNOLOGY OF CHIN, VOL. 7, NO., JUNE 9 IEEE Tans. on Micowave Theoy Tech., vol. 8, no., pp. 4-7, Ma. 99. [8] J. D. Wade and R. H. Macphie, Consevation of complex powe technique fo waveguide junctions with finite wall conductivity, IEEE Tans. on Micowave Theoy Tech., vol. 8, no. 4, pp. 7-78, p. 99. [9] R. R. Mansou and R. H. Macphie, Scatteing at an n-facated paallel-plate waveguide junction, IEEE Tans. on Micowave Theoy Tech., vol., no. 9, pp. 8-85, Sep. 985. [] E. M. Sich and R. H. Macphie, The consevation of complex powe technique and E-plane step-diaphagm junction discontinuities, IEEE Tans. on Micowave Theoy Tech., vol., no., pp. 98-, Feb. 985. [] Y.-W. Zhai and Y.-J. Zhao, nalysis of the inductive iis in ectangula waveguide with the consevation of the complex powe technique, Jounal of Xidian Univesity, vol., no. 4, pp. 6-64, ug. 6 (in Chinese). Feng Lan was bon in Sichuan Povince, China, in 977. He eceived the.s. degee in electonic engineeing and the M.S. degee in optics fom Univesity of Electonic Science and Technology of China (UESTC), Chengdu, China, in and 7, espectively. He is now a lectue with School of Physical Electonics, UESTC. His eseach inteests include high powe micowave and teahetz souces. Xi Gao was bon in Hunan Povince, China, in 977. He eceived the M.S. degee in optics fom UESTC, Chengdu, China, in 6. He is cuently pusuing Ph.D. degee with School of Physical Electonics, UESTC. His eseach inteests include photonic cystal and teahetz souces. Zong-Jun Shi was bon in Sichuan Povince, China, in 974. She eceived the M.S. degee in optics fom UESTC, Chengdu, China, in. She is cuently pusuing Ph.D. degee with School of Physical Electonics, UESTC. He eseach inteests include high powe micowave and teahetz souces.