APPLIED ELECTROMAGNETISM

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1 Volume 7 June 05 Numbe (ISSN ) Jounal of APPLIED ELECTROMAGNETISM Institute of Communication and Compute Systems Athens - GREECE

2 Volume 7 June 05 Numbe (ISSN ) JOURNAL OF APPLIED ELECTROMAGNETISM Institute of Communication and Compute Systems Athens - GREECE

3 Volume 7 Numbe June 05 TRANS BLACK SEA REGION UNION OF APPLIED ELECTROMAGNETISM (BSUAE) JOURNAL OF APPLIED ELECTROMAGNETISM Institute of Communication and Compute Systems Athens - GREECE

4 Edito: Panayiotis Fangos (Geece), Honoay Edito: Nikolaos K. Uunoglu (Geece), Boad of Associate Editos D. Dimitov (Bulgaia), V. Dumbava (Lithuania), G. Geogiev (Bulgaia), G. Matsopoulos (Geece), Editoial Boad ALBANIA G. Badhyf, C. Pio, ARMENIA H. Bagdasaian, H. Teian, BULGARIA A. Antonov, A. Laaov, S. Savov, GEORGIA R. Zaide, GERMANY M. Geogieva Gosse, GREECE H. Anastassiu, I. Avamopoulos, G. Fikiois, J. Kanellopoulos, G. Kaagiannidis, G. Klios, T. Mathiopoulos, C. Moschovitis, K. Nikita, I. Ouanos, E. Papkelis, J. Sahalos, M. Theologou,

5 N. Tiantafyllou, K. Ksysta, A. Malamou, JORDAN N. Dib, KAZAKSHTAN S. Sautbekov, LITHUANIA L. Svilainis, RUSSIA M. Bakunov, A. Gigoiev, SERBIA B. Reljin, SPAIN E. Gago Ribas, M. Gonale Moales, UNITED KINGDOM G. Goussetis, Publishing Depatment M. Geogieva Gosse, N. Tiantafyllou, K. Ksysta, A. Malamou,

6 Jounal of Applied Electomagnetism Copyight Fom The undesigned I confim that I agee the publication of the aticle in the Jounal of Applied Electomagnetism and the copyight to belong to Tans Black Sea Union of Applied Electomagnetism. I undestand that I have the full ight to euse this manuscipt fo my own puposes. Name: Suname: Addess: Signed: *Please send the pevious fom signed eithe by to pfangos@cental.ntua.g, o by fax to the fax numbe: , attention of Pof. P. Fangos.

7 Addess Institute of Communication and Compute Systems National Technical Univesity of Athens 9, Ioon Polytechniou st Athens - GREECE Tel: (+30) Fax: (+30) 0778, attention of Pof. P. Fangos pfangos@cental.ntua.g Web site:

8 TRANS BLACK SEA REGION UNION OF APPLIED ELECTROMAGNETISM (BSUAE) JOURNAL OF APPLIED ELECTROMAGNETISM (JAE) Volume 7 Numbe June 05 CONTENTS THE METHOD OF AUXILIARY SOURCES FOR THE THIN FILMS SUPERIMPOSED ON THE DIELECTRIC SURFACE I. M. Petoev, V. A. Tabatade and R. S. Zaide The diffaction poblem of the time hamonic electomagnetic wave on the cetain metal-dielectic stuctue was solved by means of the Method of Auxiliay Souces. The stuctue epesents ight paallelepiped dielectic. A lattice of thin pefectly conducting films is supeimposed on the suface of this dielectic. Selecting the paametes of this stuctue, in case of esonance between the lattice elements and the dielectic, some inteesting popeties of the complex mateials in wide ange of the fequency ae evealed. A use fiendly compute pogam has been ceated fo numeical expeiments of the stated poblem; it gives ability to change the stuctues paametes and the incident wave in ode to study the popeties of these kinds of stuctues. SCATTERING BY A NIHILITY ELLIPTIC CYLINDER WITH DIELECTRIC COATING AND UNDER LINE SOURCE ILLUMINATION A. K. Hamid Scatteing by dielectic coated nihility elliptic cylinde is analyed by solving the analogous poblem of scatteing by two dielectic layes elliptic cylinde and unde line souce illumination. The incident, scatteed and tansmitted fields ae expessed in tem of Mathieu functions. Numeical esults ae obtained fo the total fa fields of dielectic coated nihility stip o elliptic cylinde fo diffeent sies and pemittivities. GROUP ANALYSIS OF EQUATIONS OF QUASISTATIONARY ELECTROMAGNETIC FIELD OF HIGH-FREQUENCY INDUCTIVE DISCHARGE OF FINITE LENGTH A. P. Kipichnikov The esults of the goup analysis of equations of quasi-stationay electomagnetic field of high-fequency inductive dischage of finite length, within the fames of which thee have been built Lie algebas of the infinitesimal opeatos, ae given and goups of tansfomations admitted by the equations of an electomagnetic field in dischage ae calculated, and also independent basis scala invaiants (in aggegate foming univesal invaiants) of these tansfomation goups ae found. The equations of an

9 electomagnetic field witten fo new vaiables which scala invaiants of tansfomation goups ae pesented. LOW NOISE MICROWAVE AMPLIFIERS WITH IMPROVED INPUT MATCHING APPLICABLE IN ACTIVE ARRAY ANTENNAS M. P. Tonev 30 The diffeence between the optimal input eflection coefficient fo achieving minimal noise figue and those needed fo pefect matching is usually a challenge fo low noise amplifies design. Nomally amplifies designed to achieve lowest possible noise figue have vey poo input eflection coefficient. This may cause a lot of poblems in case of thei application as a component in moden active phased aay antennas and can be a eason fo significant pefomance degadation. One of the possible solutions is to use balanced amplifie configuation in ode to impove input and output matching. The main disadvantages of this solution ae degadation of the noise figue due to the losses in the input hybid and a bigge occupied space. Anothe appoach fo solving the poblem is pesented in this pape. The non unilateal popeties of the tansistos ae used in ode to obtain minimum noise figue and at the same time conjugate matching. Pactical cicuits and achieved pefomance ae discussed in details. PROVIDING A QUALITY RECEPTION OF DVB-T SIGNALS L. Jodanova, G. Kapov and D. Dobev 37 The pape deals with the influence of channel coding and modulation, tansmitte paametes, distance between tansmitte and eceive and popagation losses on the quality of eceived video infomation in DVB-T systems. Expessions fo calculating of minimum caie-to-noise atio at the input of DVB-T eceive, necessay fo quasi-eo-fee eception, ae given. Mathematical models to detemine the dependence of field stength fom distance to DVB-T tansmitte ae pesented. The dependence of minimum mean field stength at the eceive input fom CNR is deived and the algoithm fo detemination of maximum distance to the tansmitte is descibed.

10 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM The Method of Auxiliay Souces fo the Thin Films Supeimposed on the Dielectic Suface I. M. Petoev, V. A. Tabatade, R. S. Zaide Tbilisi State Univesity, Laboatoy of Applied Electodynamics, 3, Chavchavade Ave. Tbilisi, Geogia Abstact The diffaction poblem of the time hamonic electomagnetic wave on the cetain metal-dielectic stuctue was solved by means of the Method of Auxiliay Souces. The stuctue epesents ight paallelepiped dielectic. A lattice of thin pefectly conducting films is supeimposed on the suface of this dielectic. Selecting the paametes of this stuctue, in case of esonance between the lattice elements and the dielectic, some inteesting popeties of the complex mateials in wide ange of the fequency ae evealed. A use fiendly compute pogam has been ceated fo numeical expeiments of the stated poblem; it gives ability to change the stuctues paametes and the incident wave in ode to study the popeties of these kinds of stuctues.. INTRODUCTION In the moden mico and nano-electonics, the devices with thin films on the dielectic have wide applications. Theefoe, theoetical and numeical study of such stuctues is a vey impotant poblem. In this aticle, the Method of Auxiliay Souces (MAS) is used, in ode to study electomagnetic popeties of such stuctues. It is known, that MAS is efficient fo solution of the diffaction poblem on the voluminous objects. The application of this method deduced to the constuction of two auxiliay sufaces inside and outside of this object. These sufaces epeat the fom of the object and ae shifted fom the sufaces at the cetain distance. The auxiliay souces located on them descibe the field in the

11 THE METHOD OF AUXILIARY SOURCES I. M. PETOEV, V. A. TABATADZE, R. S. ZARIDZE opposite pats of divide space. In case of the thin conducting films at the fist glimpse the ability of the MAS is esticted, because it is not clea how to distibute the auxiliay sufaces. Accoding to the poposed algoithm the given film should be continued in all diection by imaginay suface at a cetain distance. The sie of this imaginay suface is one of the auxiliay paametes, which influence the exactness of the calculation. Fom the both sides of the obtained figue, up and down, thee ae constucted auxiliay sufaces. Field on top of the film is descibed by lowe auxiliay suface and unde the film- by the uppe one. The unknown amplitudes of the auxiliay souces on these sufaces ae detemined by the bounday condition satisfaction, as on the film, also on the imaginay suface. On the both sides of the film tangential component of the total electic film must be eo. The bounday conditions on the imaginay suface ae: the tangential component of the electic and magnetic field must be continuous. Results of successful application of descibed algoithms ae povided in []. This aticle is the continuation of the mentioned wok. Hee, instead of the film in the fee media, thee is consideed a system of such pefectly conducted films, which ae supeimposed on the suface of an odinay dielectic. The motivation in this aticle is also the will to extend the MAS ability to the solution of such kind of poblems efficiently. On the othe hand, using the ight selection of the esonant paametes of the system, it is desiable to get the effects of multiple esonances []. We ae looking such esonance fequencies, when both: the film s sies and the films lattice paametes in total ae esonant togethe with the dielectic s geometic paametes. Exactly in case of such esonances occu the inteesting electomagnetic popeties of the study stuctue, which coespond to the popeties of the composite (complex) mateials [3, 4, 5]. The numeical calculations wee povided using the ceated use fiendly pogam package. Thee wee found esonant paametes of the system, when it has the popeties of the negative efactive index and chiality, in the cetain fequency ange of the incident field. The esults wee obtained by paallel validation of the bounday condition satisfaction.

12 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM. THEORETICAL PART.. Poblem statement. The stuctue unde study epesents dielectic paallelepiped with pemittivity, on the suface of which is supeimposed the lattice of the defined sie ectangle conducting films (figue ). The cones and the vetices of this dielectic ae eplaced by the cylindical and spheical sufaces with small adius of cuvatue, in ode to avoid the occuence of undesiable cuents and fields with the singulaities on the sufaces. This stuctue is illuminated by a hamonic electomagnetic wave with 3 i t e dependence, polaied to the paallel of the films suface. The electomagnetic wave souce can be located outside of the studied stuctue on cetain distance o inside dielectic. Ou aim is to find the diffaction field inside and outside of the stuctue as well as the field in the fa one. The poblem consists in the numeical analysis of this scatteed field and the chaacte of its polaiation, as well as to study the inteesting composite mateials popeties ability of this stuctue on the esonant fequencies. time.. Solution of the poblem. A geneal algoithm fo the poblem solution is povided below. This algoithm can be used fo the moe complicate foms of dielectic and the conducting films. Togethe to the diffaction poblem solution it can be consideed also as an antenna poblem, when the souce of the incident field is inside of this stuctue and we ae inteested in adiated patten in fa field. The poblem is to find the total inne E, H and oute E, H scatteed fields. If the souce of incident field is located outside of the stuctue, then, H H H E E E out Figue. Consideed stuctue and incident field oientation inc in out in, inc out out

13 THE METHOD OF AUXILIARY SOURCES I. M. PETOEV, V. A. TABATADZE, R. S. ZARIDZE in, H H E E If this souce is located inside of the stuctue, then: Hee in out in., H H E E out,, H H H E E E inc E, H and E, H in. ae the fields scatteed by the stuctue outside and inside, espectively. Oute and inne total fields should satisfy the bounday conditions on the dielectic S suface and also on the suface of each conducting plate. On the S suface, total field tangential component continuity is equied. Fom the both side of the conducting films thee is equied the condition, the tangential component of the total field must be eo. As the poblem is 3D, these conditions should be satisfied fo any tangential component on the dielectic and conducting sufaces. If they ae satisfied along two othogonal tangential and vectos, then it will be inc satisfied along any othe tangential (figue )). So: Ein S Eout S, Hin S Hout S, Ein S Eout S, Hin S Hout S. () Along the conducto sufaces total electic field s tangential component must be eo on the both sides of the pieces: out Ein 0 E 0, () Ein 0 Eout 0 Figue. The MAS application 4

14 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM H The application of MAS in ode to find the fields E, H and E, is in moe details descibed in the woks [, 6]. The scheme of auxiliay sufaces constuction and the distibution of the auxiliay souces on them ae pesented on the figue. So, solution of the poblem educed to the solution of the linea algebaic equations to the unknown amplitudes of auxiliay souces. 3. RESULTS OF THE NUMERICAL EXPERIMENTS 3..Evaluation of the solution accuacy. Befoe we get some numeical esults it is necessay to be sue that the solution is accuate and we have to detemine the eo of numeical calculation. Fo this eason, as examples, we estimate the deviation fom the bounday condition satisfaction between the collocation points on the 3 sides of the consideed stuctue, the dimensions of which in the unit of incident wave length is in the given case (figue 3). Anothe 3 sides of the object ae symmetical, with the same values. In this case, inne and oute auxiliay sufaces ae shifted fom the object suface on the distance and in the unit of the wave length. The povided figues coesponds espectively 6 and 36 collocation points on the squaed wave length. It can be clealy seen that the incease of the collocation points numbe deceases the solution eo. Duing numeical expeiments and all calculations wee done when the bounday conditions deviation was not exceed the 3%, which can be the maximum of the numeical eo. a) b) Figue 3. The deviation fom the bounday condition satisfaction fo a) 6 and b) 36 collocation points pe wavelength squaed 5

15 THE METHOD OF AUXILIARY SOURCES I. M. PETOEV, V. A. TABATADZE, R. S. ZARIDZE 3.. The stuctue with the small numbe of the films. An image of the consideed stuctue is pesented on the Fig. 4. This stuctue epesents a dielectic object with the pemittivity 4, on two opposite sides of which thee ae supeimposed 3 squae shape conducted films. The incident field popagates along diection, which makes angle 450 to the X and Y axes, polaied to the paallel of the films sufaces, as it is shown on Fig. 5. Calculations wee made to study multiple esonances and to define optimal paametes fo complex mateials popeties of the stuctue. Figue 4. Consideed stuctue Figue 5. The incident field oientation Figue 6. Scatteed field in fa one Figue 7. The analysis of the polaiation The numeical calculations showed that in case of the dielectic dimension abc and the sie of the film 0.5 (in the units of wavelength), which coesponds to the esonant values, in the patten of the scatteed field appeaed an additional lobe, the diection of which coesponds to the negative efaction (figue 6). 6

16 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM Respectively, the given stuctue with such paametes is simila to the mateial with negative efactive index. The analysis of the polaiation popeties of the scatteed field showed that along two main lobes it has elliptical polaiation with the opposite otations diection (Fig. 7), which coesponds to the chial metamateials. The last one is bette seen on the animation. The nea total field distibution is given on the Fig. 8. Figue 8. The nea total field distibution Figue 9. The dependence of the iadiated powe on the incident field fequency Figue 0. Scatteed field in fa one In ode to get the esonance in wide fequency ange, the iadiated powe dependence on the incident field s fequency in some ange, fo thee diffeent value of the stuctue s width has been studied. As it is shown on the Fig. 9, in case of the width equal to.9 a wide esonance is obseved. At that time the negative efacted lobe inceased, which coesponds to the incease of this effect Fig. 0. Nea field distibution 7

17 THE METHOD OF AUXILIARY SOURCES I. M. PETOEV, V. A. TABATADZE, R. S. ZARIDZE in two mutual pependicula sections is shown on the Fig. a) and b). It can be see the esonant (standing) nea field inside the stuctue (Fig. b). a) b) Figue. Scatteed field in fa one: a) XOY, b) XOZ planes 3.3. The stuctue with the big numbe of the film elements. In ode to incease the effect of negative efaction and chiality, the stuctue with bigge amount of the film elements, than in the pevious case has been consideed. On the uppe side, as on the lowe one we have aleady 9 films with same sies and same distance between them (Fig. ). The dimensions of the dielectic paallelepiped in the units of wavelength ae abc0.35, which yet doesn t coespond to the multiple esonances. Besides this, hee we see small lobe of the negative efaction, also the otation of the polaiation is small. The coesponding distibution of the fa field patten and nea field in two mutual-pependicula sections ae pesented on figues 3, and 4 a), b). Figue. Consideed stuctue Figue 3. The scatteed field in fa one 8

18 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM a) b) Figue 4. Nea field distibution: a) XOY and b) XOZ planes Afte that thee was consideed multiple esonant paametes of the stuctue: abc (in the units of wavelength). In this case, the lobe of negative efaction significantly inceased. The coesponding fa field patten and the nea field distibution ae pesented on the Fig. 5 and 6 Figue 5. The scatteed field in fa one Figue 6. Nea scatteed field distibution The animation of the nea field shows, that next to the ight side of the stuctue thee is the standing wave fomed, which takes place in case of multiple esonance. 4. CONCLUSION The inteaction of the electomagnetic wave with the dielectic stuctues, supeimposed by the conducting thin plane films, has been studied. Selecting the paametes of the given stuctues, in case of the esonance between the lattice elements 9

19 THE METHOD OF AUXILIARY SOURCES I. M. PETOEV, V. A. TABATADZE, R. S. ZARIDZE and the dielectic, thee wee exposed inteesting popeties of the negative efaction and the chiality in compaatively wide fequency ange. The incease of the film object numbe inceases the esonant effects, which has inceased the obseved complex mateials popeties. The numeical calculation was conducted using compute simulation by means of specially ceated use fiendly pogam package. The method of the poblem s numeical solution used the MAS. ACKOWLEDGMENT The wok is done with the financial suppot of the Shota Rustaveli National Science Foundation in the Scope of the gant 5/5. REFERENCES [] I. Petoev, V. Tabatade, D. Kakulia, R. Zaide, application of the Method of Auxiliay Souces fo the thin films and open sufaces Jounal of Communications Technology and Electonics, Moscow, 04, accepted fo publication. [] I. Petoev, V. Tabatade, R. Zaide, The Double and Tiple Resonances Investigation fo the Cassini Lattice into the Dielectic. Jounal of Applied Electomagnetism (JAE), Vol.5, No., 03, pp [3] N. Engheta, R. W. Ziolkowski, Metamateials, Physics and Engineeing Exploations. John Wiley & Sons, Inc., 006. [4] G. V. Eleftheiades, K. G. Balmain, Negative-Refaction Metamateials. Fundamental Pinciples and Applications. Hoboken: John Wiley & Sons, Inc., 005. [5] I.V. Lindell, A.H. Sihvola, S.A. Tetyakov, A.J. Viitanen, Electomagnetic Waves in Chial and Bi-Isotopic Media. Boston: Atech House, 994. [6] I. Petoev, V. Tabatade, R. Zaide, The Method of Auxiliay Souces Applied to Poblems of Electomagnetic Wave Diffaction by Cetain Metal Dielectic Stuctues. Jounal of Communications Technology and Electonics, Vol. 58, No. 5, 03, pp

20 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM Scatteing by a Nihility Elliptic Cylinde with Dielectic Coating and Unde Line Souce Illumination A. K. Hamid Depatment of Electical and Compute Engineeing Univesity of Shajah, Box 77, Shajah, UAE akhamid@shajah.ac.ae Abstact Scatteing by dielectic coated nihility elliptic cylinde is analyed by solving the analogous poblem of scatteing by two dielectic layes elliptic cylinde and unde line souce illumination. The incident, scatteed and tansmitted fields ae expessed in tem of Mathieu functions. Numeical esults ae obtained fo the total fa fields of dielectic coated nihility stip o elliptic cylinde fo diffeent sies and pemittivities.. INTRODUCTION Analytical solution to the poblem of a plane electomagnetic wave scatteing by a dielectic coated PEC elliptic cylinde was investigated by many authos [-], and the solution is late extended to the nonconfocal dielectic case [3]. Axial slot antenna on a dielectic-coated elliptic cylinde was solved by [4]. Sebak obtained a solution to the poblem of scatteing fom dielectic-coated impedance elliptic cylinde [5]. The scatteing by multilayeed dielectic elliptic cylindes was studied by many eseaches [6-]. Recently, scatteing by nihility cicula cylindes and sphees wee studied by many authos by letting the efactive index of the dielectic medium appoaches eo [3-0]. In this pape, the solution of the electomagnetic wave scatteing by a dielectic coated nihility elliptic is obtained by solving the poblem of scatteing by two dielectic layeed elliptic cylinde and letting the efactive index of the inne dielectic laye appoaches eo (elative pemittivity and elative pemeability of the inne medium ae appoximately null-valued, i.e., the electomagntic waves can not popagate in that egion). The analysis and the softwae used fo obtaining the numeical esults have been validated, by calculating the total fa field fo dielectic coated pefectly conducting stip o elliptic cylinde, and showing that these esults ae in full ageement [].

21 SCATTERING BY A NIHILITY ELLIPTIC CYLINDER A. K. HAMID. FORMULATION Conside the case of a line souce illumination in the pesence of two dielectic layeed elliptic cylinde, as shown in Figue. The oute dielectic laye has pemittivity ϵ and pemeability μ while the inne laye is nihility. The semi- majo and semi-mino axes ae a and b of the nihility laye while a and b fo the dielectic coating. Figue. Geomety of the scatteing poblem. It is convenient to define the x and y coodinates of the Catesian coodinate system in tems of u and v coodinates of an elliptical coodinate system also located at the cente of the cylinde in the fom of dependence of x F cosh u cos v and y F sinh usinv. A time jwt e is assumed thoughout the analysis, but suppessed fo convenience. Conside an electic line souce paallel to the -axis and its electic field may be witten whee E H ( k ) i () () () H0 is the second kind Hankel function of eo ode, and k0 = / 0, with 0 being the wavelength. The line souce electic field may be expanded in tems of angula and adial Mathieu functions as i () () em em 0 em 0 om om 0 om 0 m0 m 0 E A R ( c, ) S ( c, ) A R ( c, ) S ( c, ) () whee

22 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM c 0 0 A (4) 4 Rem ( c0, 0) S em ( c0, 0) om om em om N em ( c0) om N em ( c0) [ S em ( c0, )] dv om 0 om / F 4 F F (5) x 0 0 0F (6) 0 0 x 0 0 k F, F is the semifocal length of the elliptical coss section, (, ) o o ae the elliptic coodinates of the line souce. coshu, cosv, angula Mathieu functions of ode m, espectively, adial Mathieu functions of the fist kind, Mathieu functions of the fist kind and functions. (4) R em and N em N and om Sem () R em and and / Som R () om (3) (4) ae the even and odd ae the even and odd (4) Rom ae the even and odd adial ae the even and odd nomalied The scatteed electic field outside the multi dielectic layeed elliptic cylinde ( ) may be expessed in tems of Mathieu functions as follows: s (4) (4) em em 0 em 0 om om 0 om 0 m0 m (7) E B R ( c, ) S ( c, ) B R ( c, ) S ( c, ) B whee em and Bom ae the unknown scatteed field expansion coefficients. The tansmitted electic field into the oute dielectic laye ( ) may be witten as: I () () em em em em em m0 E C R ( c, ) D R ( c, ) S ( c, ) () () ComRom ( c, ) Dom Rom ( c, ) Som( c, ) (8) m 3

23 SCATTERING BY A NIHILITY ELLIPTIC CYLINDER A. K. HAMID whee k F, k c, C em, C om, em D, Dem ae the unknown tansmitted field expansion coefficients, and type. () R em and () R om ae the adial Mathieu functions of the second Similaly, the tansmitted electic field into the inne dielectic laye ( ) may be witten as: whee c k F expansion coefficients. k II () () em em em om om om m0 m (9) E F R ( c, ) S ( c, ) F R ( c, ) S ( c, ), and F em,, Fom ae the unknown tansmitted field The magnetic field components inside and outside the elliptic cylinde can be obtained using Maxwell s equation, i.e.: H H v u j E h v j E h u (0) () whee h F cosh u cos v. The unknown expansion coefficients in equation (7) can be obtained by imposing the bounday conditions at the vaious intefaces. Continuity of the tangential field components at and equie that: i s I E E E () i s I Hv Hv Hv (3) I E II E (4) H I II v Hv (5) Substituting the appopiate equations into equations ()-(5) and applying the othogonality popety will lead to a system of equations which may be solved B numeically fo the unknown scatteed field coefficients em om. 4

24 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM 3. NUMERICAL RESULTS Since the obtained summations constitute the analytic solution, ae infinite in extent, to obtain numeical esults using a digital compute and the associated softwae, these summations have to be tuncated to include only the fist N tems, whee N is an intege popotional to the electical sie and the constitutive paametes of the composite object. The esults given in this pape have been checked fo convegence, and obtained by consideing only the fist 7 tems (i.e. N = 7) of the infinite seies associated with each even and odd function. The obtained numeical esults ae pesented as total fa field fo dielectic coated nihility stip o elliptic cylindes of diffeent sies and pemittivities, fo both TM and TE polaiations [8]. To validate the analysis and the softwae used fo calculating the esults, we have computed the total fa field against the scatteing angle of pefectly conducting dielectic coated elliptic cylinde with.0 b 0.0, and b 0.33 and 0 90 o, 0, a 0.5 a, 0.6, fo TM case [7,]. The numeical esults have been shown in Fig., and they ae in full ageement, veifying the accuacy of the analysis as well as the softwae used fo obtaining the esults. Also in Fig. shows.0, 3 and 4 the total fa field fo a nihility dielectic coated elliptic cylinde with. It can be seen that the total fa field is educed at the scatteing angle 90 degees in the pesence of the nihility egion while it is inceased at the scatteing angle 70 degees. Figue 3 is simila to Figue except fo the TE case. It can be seen that the total fa field is educed significantly at the scatteing angle 90 degees in the pesence of the nihility egion while it is inceased significantly the scatteing angle 70 degees. Figue 4 is simila to except fo line souce location is at 0 field is inceased at the scatteing angle 0 degee and.0 and 3.0 fo 4.0 at the same scatteing angle. 0 o. It can be seen that the total fa while it educed 5

25 E( ) E( ) SCATTERING BY A NIHILITY ELLIPTIC CYLINDER A. K. HAMID =.0, PEC =.0 Nihility = 3.0,Nihility = 4.0, Nihility in degees Figue. Total fa field against the scatteing angle of nihility dielectic coated elliptic cylinde with a 0.5, a 0.6, b 0.0, and b 0.33 and 0 90 o, TM case., 0.5 =.0, PEC =.0 Nihility = 3.0,Nihility = 4.0, Nihility Figue in degees Total fa field against the scatteing angle of nihility dielectic coated elliptic cylinde with a 0.5, a 0.6, b 0.0, and b 0.33, 0 and 0 90 o, TE case. 6

26 E( ) E( ) JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM =.0, PEC =.0 Nihility = 3.0,Nihility = 4.0, Nihility in degees Figue. 4 Total fa field against the scatteing angle of nihility dielectic coated elliptic cylinde with a 0.5 case., a 0.6, b 0.0, and b 0.33, 0 and 0 0 o, TM Fig. 5 Shows total fa field against the scatteing angle of nihility dielectic coated elliptic cylinde with a 0.5 a, 0.58 b 0.5,, and b 0.33, 0 and 0 45 o, TM case..5 =.0, PEC =.0 Nihility = 3.0,Nihility.5 = 4.0, Nihility in degees Figue. 5 Total fa field against the scatteing angle of nihility dielectic coated elliptic cylinde with a 0.5, a 0.58, b 0.5, and b 0.33, 0 and 0 45 o, TM case. 7

27 E( ) SCATTERING BY A NIHILITY ELLIPTIC CYLINDER A. K. HAMID.8 =.0-j = 3.0-j0.5 = 4.0-j0.5 = 5.0-j in degees Figue. 6 Total fa field against the scatteing angle of lossy nihility dielectic coated elliptic cylinde with a 0.5, a 0.58, b 0.5, and b 0.33, 0 and 0 45 o, TM case. 4. CONCLUSIONS Analytical solution of the electomagnetic wave scatteing by nihility dielectic coated elliptic cylindes o stip is obtained fo TM and TE polaiations and unde line souce illumination. The validity and accuacy of the obtained numeical esults wee veified against the case of dielectic coated pefectly conducting elliptic cylinde o stip and the ageement was excellent. Numeical esults wee pesented as a function of the geometical paametes of the cylinde and can be used to design scattees with educed o enhanced backscatteing echo widths. Futhe, the solution is geneal whee the special case of coated nihility cicula cylinde may be obtained by letting the axial atios appoximately equal to.0 while the special case of coated nihility stip may be obtained by letting the thickness of the inne cylinde vanishes (letting the mino axis of the inne cylinde appoaches eo). ACKOWLEDGMENT The autho wishes to acknowledge the suppot povided by the Univesity of Shajah, Shajah, U.A.E. 8

28 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM REFERENCES [] H. A. Ragheb and L. Shafai, "Electomagnetic scatteing fom a dielectic coated elliptic cylinde," Can. J. Phys., vol. 66, pp. 5-, 988. [] J.H. Richmond, "Scatteing by a conducting elliptic cylinde with dielectic coating," Radio Sci, vol.3, pp , 988. [3] H. A. Ragheb, L. Shafai, and M. Hamid, "Plane wave scatteing by a conducting elliptic cylinde coated by a nonconfocal dielectic," IEEE Tans. Antenna and Popag., vol. 39, pp. 8-3, 99. [4] J.H. Richmond, "Axial slot antenna on dielectic coated elliptic cylinde," IEEE Tans. Antenna and Popag., vol. 37, pp. 35-4, 989. [5] A-R. Sebak, "Scatteing fom dielectic coated impedance elliptic cylinde," IEEE Tans. On Antenna and Pogag., vol. 48, pp , 000. [6] R. Holand and V.P. Cable, "Mathieu functions and thei applications to scatteing by a coated stip," IEEE Tans. On Elect. Compt., vol. 34, no., pp. 9-6, 99. [7] S. Caosi, M. Pastoino, and M. Raffetto, "Electomagnetic scatteing by a multilaye elliptic cylinde unde tansvese-magnetic illumination: seies solution in tems of Mathieu function," IEEE Tans. On Antenna and Pogag., vol. 45, no. 6, pp , 997. [8] E. S. Caosi, M. Pastoino, and M. Raffetto, Electomagnetic scatteing by weakly lossy multilaye elliptic cylindes," IEEE Tans. On Antenna and Pogag., vol. 46, no., pp , 998. [9] A. Sebak, L. Shafai, and H. Ragheb, "Electomagnetic scatteing by a two layeed piecewise homogeneous confocal elliptic cylinde," Radio Science, vol. 6, pp. - 9, 99. [0] A-K. Hamid, Multi-Dielectic Loaded Axially Slotted Antenna on Cicula o Elliptic Cylinde, J. of Electomagnetic Waves and Applications, Vol. 0, numbe 9, pp. 59-7, 006. [] A-K. Hamid, Scatteing fom a Semi-Elliptic Channel in a Gound Plane Loaded by a Lossy o Lossless Dielectic Elliptic Shell, Applied Computational Electomagnetics Society Jounal, vol., no. 3, pp , 007. [] A-K. Hamid and M.I. Hussein, Electomagnetic Scatteing by a Lossy Dielectic Coated Elliptic Cylinde, Canadian Jounal of Physics, Vol. 8, no. 5, pp , 003. [3] A. Lakhtakia and J.B. Geddes, Scatteing by a nihility cylinde, AEÜ Intenational Jounal of Electonic and Communications (AEÜ), 6, pp. 6-67, 007. [4] A. Lakhtakia, A., Scatteing by a nihility sphee, Micowave and Optical Technology Lettes, Vol 48, pp , 006. [5] Ahmed, S. and Q. A. Naqvi, Scatteing of Electomagnetic Waves by a Coated Nihility Cylinde, Jounal of Infaed, Millimete and Teahet Waves, Vol. 30, Numbe 0, PP , 009. [6] Ahmed, S. and Q. A. Naqvi, Diective EM Radiation of a Line Souce in the Pesence of a Coated Nihility Cylinde, Jounal of Electomagnetic Waves and Applications, Volume 3, Numbes 5-6, 009, vol., pp ,

29 SCATTERING BY A NIHILITY ELLIPTIC CYLINDER A. K. HAMID [7] A-K. Hamid, EM Scatteing by Nihility Cicula o Elliptical Cylindes with Lossy Metamateials Coating, Jounal of Intenational Review on Modelling and Simulations (IREMOS), vol.4, no., pp , 0. [8] M.H. Hassan, M.J. Mughal, M.M. Ali and Q.A. Naqvi, Electomagnetic fields in a cicula waveguide with DBbounday conditions intenally coated with chial-nihility medium, Intenational Jounal of Applied Electomagnetics and Mechanics 40, 7 35,0 [9] M.H. Hassan, M.J. Mughal and A.R. Faooqi, Electomagnetic fields in a cicula waveguide with DB_-bounday conditions intenally coated with chial-nihility medium, Intenational Jounal of Applied Electomagnetics and Mechanics 40, , 0 [0] Muhammad Khalid, Aqeel A. Syed and Qaisa A. Naqvi, Cicula cylinde with D B, DB and D B bounday conditions placed in chial and chial nihility media, Intenational Jounal of Applied Electomagnetics and Mechanics 44, 59 68, 04 [] F. Cooay and A-K. Hamid, "Scatteing fom a Coated Pefect Electomagnetic Conducting Elliptic Cylinde" Intenational Jounal of Electonics and Communications, AEUE, vol. 66, pp , 0. 0

30 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM GROUP ANALYSIS OF EQUATIONS OF QUASISTATIONARY ELECTROMAGNETIC FIELD OF HIGH-FREQUENCY INDUCTIVE DISCHARGE OF FINITE LENGTH A. P. Kipichnikov Kaan National Reseach Technological Univesity, Kaan, Russia K. Max St. 68, 4005, Kaan, Russia kipichnikov@kstu.u Abstact The esults of the goup analysis of equations of quasi-stationay electomagnetic field of high-fequency inductive dischage of finite length, within the fames of which thee have been built Lie algebas of the infinitesimal opeatos, ae given and goups of tansfomations admitted by the equations of an electomagnetic field in dischage ae calculated, and also independent basis scala invaiants (in aggegate foming univesal invaiants) of these tansfomation goups ae found. The equations of an electomagnetic field witten fo new vaiables which scala invaiants of tansfomation goups ae pesented.. INTRODUCTION Following the system of designations accepted in the wok [], let us ewite the full two-dimensional system of equations of an electomagnetic field in dischage [] E E H c Sin H E ; E H c E Cos H E ; H H H 4 Cos H H H Sin H H E Cos H E c ; H H H Sin H H ()

31 GROUP ANALYSIS OF EQUATIONS A. P. KIRPICHNIKOV H H H Cos 4 E H H Sin H E c H ; H H H Cos H H Sin H H H H ; H H H H Sin Cos H H H H H, as v w Cos u u v Sin y x ; x v Cos u y x ; y w Sin y u Cos y x ; w w u y Sin y Cos y Sin y x; () v v v w w v Cos y v y Sin y ; v v Sin w w y y Cos y. In the equations (), E, E, H, H, H, H ae the amplitude value and cicuit angle of electic cul, longitudinal and coss-magnetied fields within plasmoid espectively; -conductivity in dischage; - oscillation fequency of a HF field; c- speed of light in a vacuum in Gaussian system of units. In the equations () and the following fomulae accoding to [] t ; u E ; v H ; w H ; x E ; y H and H ; с ; 4 с ; index designates adial deivatives of coesponding values, index axial ones.

32 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM. TWO-DIMENSIONAL MODEL OF HFI DISCHARGE The system () witten in this way admits thee-dimensional Lie algeba 3 L of the infinitesimal opeatos the basis of which in a finite-dimensional solutions space of constitutive equations takes the fom X t ; X u v w ; (3) u v w to which the table of geneatos satisfies X x y 3, ~ t u ~ x ~ v ~ y ~ w ~ ~ ~ X X 0 u 0 v 0 w 0 0 X (in the tables the coodinates of opeatos ae designated by the same lettes as vaiables but with a wave on thei top). The thee-paametic Abelian (commutative) goup tansfomations coesponds to the system of basis opeatos (3): (identity tansfomation); 3 G of the following goup); t t (paallel tansfe); (4) a ; ; u u exp a v v exp a w w exp a (dilation and compession x x a 3 ; y y a3 ; a3 (the goup of paallel tansfes) ( a, a, и a 3 paametes of the goup). 3

33 GROUP ANALYSIS OF EQUATIONS A. P. KIRPICHNIKOV 4 It is obvious that in total thee ae 8-3=5 independent basis scala invaiants of the goup (4) which in aggegate foms a univesal invaiant y x y v w v u J,,,, o, in nomal designations (in physical vaiables), H H E H H H H E J,,,,. 3. ONE-DIMENSIONAL MODEL OF HFI DISCHARGE In the same system of designation, the equations of electomagnetic field in a one-dimensional statement of the poblem, obviously, takes the fom E H Sin H c E E ; E H E Cos E H c ; H E H Cos E c 4 ; (5) H ) ( 4 E H Sin H E c o x y Sin v u u ; x y Cos u v x ; x y u Cos v ; x y Sin v u y.

34 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM This system also admits thee-dimensional Lie algeba which 3 L, to the basis vectos of X t ; X u v u v ; X 3 x y the table of opeatos coesponds ~ t u ~ x ~ v ~ y ~ ~ X X 0 u 0 v 0 0 X Specified basis opeatos espond the thee-paametic Abelian goup consisting of tansfomations: 3 G (identity tansfomation); t t (paallel tansfe); (6) a ; u u exp a v v exp a (dilation and compession goup); x x a 3 ; y y a3 (goup of paallel tansfes) ( a, a и a 3 paametes of the goup). Thee ae 6-3=3 independent basis scala invaiants of the goup (6) that fom its univesal invaiant u J,, v y x o 5

35 GROUP ANALYSIS OF EQUATIONS A. P. KIRPICHNIKOV E J,, H E. H 4. MODEL OF HFI DISCHARGE IN NEAR-AXIAL ZONES OF PLASMIC TORCH And, at last, let us conside equations descibing the stuctue of quasistationay electomagnetic field of HFI dischage of finite length in nea-axial ones of plasmoid, i.e. system [3, 4] H H a H 0, 3 ; 4 in which H H 0, a 0, с, having the exact analytical decision [3] H 64a b (,) H (0,0) I0( b) I0( b) Cos( b), 4 ; b 4 H (,) H a b b (0,0) I( b) I( b) Sin( b), 4 b 6 witten by means of modified Bessel functions I 0 and I of eo and fist odes, paamete H b accos L H 0, L, whee L- the length of an estimated aea on the 0,0 coodinate, digitied fom the cental section of a plasmoid. In ou designations it will be the system v w a v, 0 3 ; w w v. Admitted Lie algeba L in this case is two-dimensional, its basis opeatos ae as follows 6

36 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM X t and X v w v w. The table of these opeatos: ~ t v ~ w ~ ~ X X 0 v w 0 Basis opeatos espond to the two-paamete Abelian goup of tansfomations G : (identity tansfomation); t t (paallel tansfe); (7) a ; v v exp a ( a, a paametes of the goup). w w exp a (dilation and compession goup) It is obvious that thee ae 4-= independent basis scala invaiants of the goup (7) which compise the univesal invaiant of this goup w J,, v in common notations (in physical vaiables) H J,. H 7

37 GROUP ANALYSIS OF EQUATIONS A. P. KIRPICHNIKOV 5. DISCUSSION AND CONCLUSION It is inteesting to note that if in systems of equations () and (5) we tansfe to new vaiables epesenting basis scala invaiants of the appopiate goups of tansfomations (these new vaiables may be called goup vaiables), the numbe of equations in these systems can be educed. In paticula, the system () will tansfe to the system of fou equations fo fou unknown values E H H H,, H E, and H H, and system (5) - to the system 4 Sin c c Cos ; 4 Sin c Cos, c whee the numbe of equations (espectively the numbe of unknown values) is educed by half. Like this, the last two equations ae most convenient fo thei subsequent computational solution by means of known methods of the finite diffeence solution of diffeential equations (fo example, by means of implicit diagam with a half weight), of couse, on condition of the coect assignment of these o those bounday conditions. The matte is that in its classical type equations of an electomagnetic field (5) do not admit the computational solution by any standad methods of explicit diagam Eule, Runge-Kutta, Adams and thei vaieties as the fist deivatives of values E, H, and H, and so the ight pats of appopiate equations of the system (5) addess in eo on the axis of plasmoid. REFERENCES [] Ovsyannikov L.V. Goup analysis of diffeential equations (in Russian). Moscow: Nauka, 986. [] Kipichnikov A.P. Limit elations fo HF inductive dischage (in Russian)//Ivestiya VUZov. Fiika No.. P [3] Kipichnikov A.P. The stuctue of quasistationay electomagnetic field of highfequency inductive dischage nea an axis of plasma bunch (in Russian)//Teplofiika Vysokikh Tempeatu (High Tempeatue) V. 33. No.. P

38 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM [4] Kipichnikov A.P. About the stuctue of high-fequency inductive dischage at atmospheic pessue nea an axis of plasma bunch (in Russian)//Teplofiika Vysokikh Tempeatu (High Tempeatue) V. 34. No. 6. P

39 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM Low Noise Micowave amplifies with impoved input matching applicable in active aay antennas M. P. Tonev Technical Univesity of Sofia, Bulgaia Faculty of Tlecommunication, TU-Sofia, Kl.Ohidsky st. 8, 000 Sofia, Bulgaia Abstact The diffeence between the optimal input eflection coefficient fo achieving minimal noise figue and those needed fo pefect matching is usually a challenge fo low noise amplifies design. Nomally amplifies designed to achieve lowest possible noise figue have vey poo input eflection coefficient. This may cause a lot of poblems in case of thei application as a component in moden active phased aay antennas and can be a eason fo significant pefomance degadation. One of the possible solutions is to use balanced amplifie configuation in ode to impove input and output matching. The main disadvantages of this solution ae degadation of the noise figue due to the losses in the input hybid and a bigge occupied space. Anothe appoach fo solving the poblem is pesented in this pape. The non unilateal popeties of the tansistos ae used in ode to obtain minimum noise figue and at the same time conjugate matching. Pactical cicuits and achieved pefomance ae discussed in details.. INTRODUCTION As known in geneal fo evey adio-communications eceive fist stage of input amplifie is matched to achieve minimal noise figue. This often leads to poo input etun loss of the eceive and when it is diectly connected to the eceiving antenna may cause poblems [5]. Usually micowave antennas ae designed to be matched at thei output to 50 ohms. When they see diffeent impedance, fo example as those of pooly matched low-noise amplifie, thei chaacteistics might be changed. This is tue especially concening phased antenna aays. Good example fo such mismatch sensitive system is active antenna aays used in moden mobile satellite communications. Such type of antennas must povide dynamic polaiation contol while on move. In ode to suppot such functionality antennas must compise dual pot antenna elements eceiving signals with two othogonal linea polaiations and two independent summation feed cicuits. In ode to achieve good isolation between these summation cicuits and to avoid phase and amplitude eos in antenna apetue it is need to ensue good matching between all feed line components inside the 30

40 LOW NOISE MICROWAVE AMPLIFIERS M. P. TONEV antenna. So if low noise amplifies at the antenna outputs ae not well matched to thei chaacteistic impedance it may cause significant degadation of the most impotant antenna paametes, such as gain and coss-polaiation isolation. Two possible solutions of the mentioned above poblem ae known in pactice. The fist possible solution is to connect isolato between antenna and the eceive []. This completely solves the poblem with poo input matching of the eceive but degades its nose figue pefomance because of the insetion loss intoduced by the isolato. Fom anothe side using isolato with low insetion loss will significantly incease pice of the device, which is anothe facto that estict using isolatos. Anothe less expensive solution is to use balanced amplifies [6]. Balancing technique is well known method fo impoving etun loss of low noise amplifies, but it also degades noise pefomance due to the input hybid device loss. Othe disadvantages of balanced amplifies ae twice bigge powe consumption and twice bigge occupied space compaed to single ended amplifie. These disadvantages sometimes can make them not applicable. Anothe appoach, using a novel matching technique allowing achieving simultaneously low noise figue and good matching fo such type of amplifies is descibed in the pesented pape.. MICROWAVE TRANSISTOR AS TWO PORT NETWORK At micowave fequencies tansistos ae usually pesented as two pot netwoks, which ae chaacteied with S-paametes []. Such two pot netwok with defined load and souce impedances is pesented schematically on Fig.. Figue. Two-pot netwok with specific souce and load impedances. The thee types of powe gains fo an abitay two-pot netwok connected to souce and load impedances, Z S and Z L may be defined as follows [3]: 3

41 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM Powe Gain = G = P L /P in is the atio of powe dissipated in the load Z L to the powe deliveed to the input of the two-pot netwok. This gain is independent of Z S. Available Gain = G A = P avn /P avs is the atio of the powe available fom the two-pot netwok to the powe available fom the souce. It depends on Z S but not on Z L. Tansduce Powe Gain = G T = P L /P avs is the atio of the powe dissipated in the load to the powe deliveed to the input of two pot netwok. This depends on both Z S and Z L. These definitions diffe pimaily in the way the souce and load ae matched to the two-pot device. If the input and output ae both conjugate matched, then the gain is maximied and G = G A = G T. Equations fo these gains and eflection coefficients in tems of the S paametes of the active device can be defined, efeing to Fig., as follows []: () () (3) whee P G P G G A T L in P P P P S avn avs L in in S S S L S L S S out S L S in S L (4) L Z Z L L Z Z 0, 0 S Z Z S S Z Z 0 0 and (5a) (5b) in out Z Z in in Z Z Z Z out out 0 0 Z Z S 0 0 S SS S SS S L L S S In geneal to obtain good input matching it is necessay input eflection coefficient of two pot netwok in to be close to eo. Fom equation 5a can be seen that in 3

42 LOW NOISE MICROWAVE AMPLIFIERS M. P. TONEV depends on active component paametes (S, S, S ) and on load eflection coefficient L. This means that we can manipulate input eflection coefficient of two pot netwok by changing load impedance. 3. DESIGN OF MICROWAVE LOW NOISE AMPLIFIER WITH IMPROVED INPUT MATCHING The main goal in designing low nose amplifies is obtaining pope eflection coefficient fom input matching netwok. This coefficient must coincide with optimum eflection coefficient fo obtaining minimum noise figue of active component. Usually this coefficient is diffeent fom conjugate machining and this is the eason in of LNA s not to be eo. It is possible to design pope input matching netwok and then to choose load impedance, which will minimie input eflection coefficient of the two pot netwok. In this case amplifie will be simultaneously matched fo obtaining minimum nose figue and conjugate matched with souce. In geneal fo single stage amplifie this will wose output matching. The poblem can be solved by adding second stage in amplifie configuation. Then using inte-stage matching cicuit we can chose pope eflection coefficient to make in 0 and then output matching cicuit can be designed to make out 0. The appoach, pesented above was poven by designing micowave LNA, suitable fo using in Ku band satellite communications woking in fequency band fom 0.7GH to.8gh. It was designed on Alon 5N substate with 0.5mm thickness, dielectic constant 3.38 and loss tangent Used tansistos ae NE30 low noise heteo-junction FET, manufactued by NEC. The S-paametes of tansistos and capacitos, povided by manufactues wee putted in simulato and cicuit simulation was pefomed. The fist step in design is to define input matching cicuit. In Fig. eflection coefficients of input matching cicuit and optimum eflection coefficient of tansisto ae shown. As can be seen, thee is coincidence between them which means that tansisto is matched fo obtaining minimum noise figue. Once input of the amplifie is matched, the inte-stage and output matching cicuit can be designed, using optimiation pocedue in ode to achieve good input and output matching. Simulated S-paametes and noise figue of LNA model ae shown in fig.3 and fig4. 33

43 JAE, VOL. 7, NO., 05 JOURNAL OF APPLIED ELECTROMAGNETISM Figue. Reflection fom input matching cicuit and tansisto opt Figue.3 Simulated S-paametes of LNA Figue.4 Simulated noise Figue of LNA 34