Dikdörtgen kesitli horn antenler

Abstract

Elektromanyetik alan uygulamalarında oldukça önemli bir yeri olan horn antenler için "Skaler Helmholtz Denklemi"ni değişkenlere ayırma yöntemiyle çözüm imkanı yoktur. Bu nedenle benzer başka problemlerde de yapıldığı gibi bu çalışmada ele alınan horn antenler için, yaklaşık bir yöntem olan "özmod" yöntemi ele alınmış ve dikdörtgen kesitli horn antenler İçin elektromanyetik alan ifadeleri ve radyasyon diyagramları elde edilmeye çalışılmıştır. Bu amaçla, bölüm 2'de; horn antene temel oluşturan daha basit bir yapı, "kama şeklindeki dalga kılavuzları" ele alınmıştır. Bölüm 2. l'de; bu tip yapılar için kartezyen koordinatlarda tam çözüm bulunamadığından, "özmod" yöntemi uygulanarak "Skaler Helmholtz Denklemi" için yaklaşık çözüm bulunmuştur. Bölüm 2.2'de; kama geometrisi için silindirik koordinatlara geçiş yapılmış ve özmod çözümü yeniden elde edilmiştir. Daha sonra silindirik koordinatlarda kama geometrisi için değişkenlere ayırma yöntemiyle bulunabilen tam çözüm, silindirik koordinatlardaki özmod çözümünde değişken dönüşümü yapılarak elde edilmeye çalışılmıştır. Bölüm 2.3'te; kartezyen koordinatlarda elde edilmiş olan yaklaşık özmod çözümünde, bölüm 2.2'dekine benzer bir değişken dönüşümü yapılmış ve silindirik koordinatlarda bulunan çözüme oldukça yakın bir ifade elde edilmiştir. Bu iki ifadenin karşılaştırılmasıyla, kama açısı küçüldükçe, kartezyen koordinatlardaki yaklaşık çözümün, silindirik koordinatlardaki tam çözüme yaklaştığı görülür. Bölüm 3. l'de; kamadan horn antenin daha basit bir şekli olan H-düzlemi hom antenlerine geçiş yapılmış ve bu yapı için "Moment Metodu"ndan yola çıkarak elektromanyetik alan ifadeleri belirlenmiştir. Bölüm 3.2'de; bölüm 3. l'de bulunan sonuçlardan yola çıkarak H-tipi horn antendeki güç dağılımı incelenmiş, güç sakınımının sağlanıp sağlanmadığına bakılarak elde edilen ifadelerin doğruluğu kontrol edilmiştir. Bölüm 4'te; genel tipten horn antenler ele alınmış ve H-düzlemi horn anten için yapılana benzer yolla "öz mod" çözümü elde edilmiştir. Ancak piramidal hornun x ve y doğrultularının her ikisinde de değişim göstermesinden dolayı çözümün elde edilmesi sırasında karşılaşılan özfonksiyonlann iç çarpımları iki katlı integrallerin hesaplanmasını gerektirmektedir. Ayrıca H-plane horn için m gibi tek indisli bir mod parametresi yeterli olmakla birlikte piramidal horn da tanımlanmış olan q mod indisi bir (m.n) çiftine karşı gelmektedir.

In this thesis we consider the non-separable boundary value problem presented by the pyramidal horn antennas of rectangular cross-section. Since coordinate separability is impaired by the imposed boundary conditions this problem can not be solved exactly and one is forced to resort to approximate metods. Here we address this problem utilizing the recently developped "Intrinsic mode (IM) metod" which provides a remarkably good approximation to weakly non-separable problems. For the boundary value problem considered in this thesis the non-separability parameter can be identified as the apex angle(s) of the horn antenna. It therefore becomes crucial to determine the limits of applicability of IM techniques in analyzing horn antennas. To achieve this we begin by considering the H plane sectoral horn antenna. H plane sectoral horn can be solved exactly in case of dominant TE0i mode exitation. This is due to the fact that H-plane horn defines separable problem In the cylindrical polar coordinates, the natural coordinates of the problem. We constract this solution utilizing Galerkin's method and subsequently use the computed results to determine the validity range of IM formulations. The insight gained therefrom is then utilized in applying IM to pyramidal horn antennas for which exact solutions are not available. The construction of two dimensional IM fields for the H plane horn Is equivalent to the IM formulation for the wedge shaped region depicted In figure 1. Scalar Helmholtz equation in cartesian coordinates reads. 2 2 a a, dx* 3z2 W(x,z)=0 + Boundary conditions (1) If one seeks a solution to (1) for the non-penetrable wedge problems, it Is not possible to use the method of "separation of variables" since the boundary conditions render (1) non-separable in the cartesian (x,z) coordinate system. We assume that an approximate solution (IM) to (1) can be expresed as; W(x,z)= iPfef(P)). C0(f(P))X(x,f(P))a(f(P))e,r^HP%dp (2) a(P) and P(z,flp)) are defined as; a(f(P))= 1 9f(P) p ap ^1/2 (3) P(z,f(p))=k0 [z-f(Ç)]dÇ W and C(f(p)), x(x,flp)) and fflJ) are yet to be determined. Substituting from ( 2) into ( 1) gives, (V) 3,2 -ı+kja-p ) -OX X(x,f(P)) C0(f(P))a(f(p))eiP(z'f(P))k0d|3=0 (5) We now Impose the restriction that at the stationary points of the phase function p and % should satisfy the one dimensional wave equation and the local boundary condition, 3,2 - 2+kS<1-Ps).3x X(x,f(Ps)) =0 (6) on z=z, cross section : Y(x,f(B)) I n =0 i a.\ \r// x=0,zitana (7) z=zi Figure 1. Wedge Geometry The solution can be obtained as. X(x,f(ps))=sin(kxx)=sin(\) - ) iz, qrc h Zjtana, q=l,2". Using the relation between the wave numbers, one can write. k2=k2-k2=k2- (8) (9) (10) For convenience we will define a normalized propogation constant §(fkz./'kQ W"1- Vkozı J (11) (VI) Un=XbmGn>1] (72) m=l ( 69 ) and ( 72) can be shown In matrix form as follows; ai u2 u, 1.1 ^2,1 '1,2 G G ı.ı J2,l 3,1 ^1.3 ^.2,2 ^2,3 G -3,2 G 1.2 J2,2 '3,2 G G 1.3 2.3 3.3J '3 b2 Lb3. â=£*b u=G*b => fe=0_1*Ji (73) (74) Thus the formal solution is completed since the field components in both regions are completely determined once a and b are obtained from above and substituted into (38-39) and (44-45). The analysis follows a similiar path for the case of pyramidal horn antenna. Moreover, once again the IM spectral integrals can be evaluated in explicit form. For TE type modes the locally separated scalar wave equation for Hz reads (See fig.5) (75) Hz= ¥(x,y,z) Z(z) 2 2 d d, 2 - 5+-= +k2-p(z) 2 -2+P Z=0 3z2 Subject to boundary conditions Lax-lx=a(Z) Lay and a radiation condition in z. ¥(x,y,z)=0 (76) (77) =0 (78) -V=b(z) Figure.5 Pyramidal Horn Antenna (xm) and solving %\ from (11) yields. -1/2 (12) q' k0<- ' q- =0 Pq In ( 12) Is chosen such that, >(P)' - ^ -I.. Thus pq In (13) Is equal to the "saddle point" that gives the dominant contribution to the integral in ( 5). And flP) will be same with z(P) given by ( 12). Co(flP)) is the normalization function which can be determined as, (13) CjxXdx=l => C0(z(P))=C0(f(p))= f 2 \ ^f(p)tana, 1/2 (14) Then, substituting f(p)=z(p) in ( 3) and ( 4) yields. -3/4 1/2 nn 2 -l 3/2 a(p)=u k03/2(l-P ) =0) [z(p)] (15) P(z,f(P))=k0pz(p)-Dqsin-1p Subsütuting from ( 8), (14), (15) and (16 ) into ( 2) and making some cancellations one obtains. W (x,z)=T- 1 sin^x/ 1-P2j-Ti=« H LtanccJ v J I 2 c J 1-P 1 i[kopz-u sin"1? JdP (16) (17) On the other hand. If one considers the cylindrical polar coordinates, the wave equation is written as. i a a i a, p- + +ko p9pap pW [W(P,<»Wo w(p,4» = o (18) (19) Following same procedure and taking p and instead of z^ respectively in (16) and (15), the solution is given as; W(p,4>)= - si )aj._-!«, -K ^v,312 ilkoPp^-v5"1 PI sin(a) 6)a) [p(p)] e p(P)~ i ' «T' « MP (20) But now \)q is defined as; (VH) Un=XbmGn>1] (72) m=l ( 69 ) and ( 72) can be shown In matrix form as follows; ai u2 u, 1.1 ^2,1 '1,2 G G ı.ı J2,l 3,1 ^1.3 ^.2,2 ^2,3 G -3,2 G 1.2 J2,2 '3,2 G G 1.3 2.3 3.3J '3 b2 Lb3. â=£*b u=G*b => fe=0_1*Ji (73) (74) Thus the formal solution is completed since the field components in both regions are completely determined once a and b are obtained from above and substituted into (38-39) and (44-45). The analysis follows a similiar path for the case of pyramidal horn antenna. Moreover, once again the IM spectral integrals can be evaluated in explicit form. For TE type modes the locally separated scalar wave equation for Hz reads (See fig.5) (75) Hz= ¥(x,y,z) Z(z) 2 2 d d, 2 - 5+-= +k2-p(z) 2 -2+P Z=0 3z2 Subject to boundary conditions Lax-lx=a(Z) Lay and a radiation condition in z. ¥(x,y,z)=0 (76) (77) =0 (78) -V=b(z) Figure.5 Pyramidal Horn Antenna (xm) and solving %\ from (11) yields. -1/2 (12) q' k0<- ' q- =0 Pq In ( 12) Is chosen such that, >(P)' - ^ -I.. Thus pq In (13) Is equal to the "saddle point" that gives the dominant contribution to the integral in ( 5). And flP) will be same with z(P) given by ( 12). Co(flP)) is the normalization function which can be determined as, (13) CjxXdx=l => C0(z(P))=C0(f(p))= f 2 \ ^f(p)tana, 1/2 (14) Then, substituting f(p)=z(p) in ( 3) and ( 4) yields. -3/4 1/2 nn 2 -l 3/2 a(p)=u k03/2(l-P ) =0) [z(p)] (15) P(z,f(P))=k0pz(p)-Dqsin-1p Subsütuting from ( 8), (14), (15) and (16 ) into ( 2) and making some cancellations one obtains. W (x,z)=T- 1 sin^x/ 1-P2j-Ti=« H LtanccJ v J I 2 c J 1-P 1 i[kopz-u sin"1? JdP (16) (17) On the other hand. If one considers the cylindrical polar coordinates, the wave equation is written as. i a a i a, p- + +ko p9pap pW [W(P,<»Wo w(p,4» = o (18) (19) Following same procedure and taking p and instead of z^ respectively in (16) and (15), the solution is given as; W(p,4>)= - si )aj._-!«, -K ^v,312 ilkoPp^-v5"1 PI sin(a) 6)a) [p(p)] e p(P)~ i ' «T' « MP (20) But now \)q is defined as; (VH) In the above equation Sn(fj), Cn((p),Y0 and Hn are defined as follows; Sn((p)=- L=cos(vncp) (46) y0=- =/-=- m yipaa. con / (i 120îc dSJcp) vn i d m Cn(q»=-^-=-7^=sin(vncp) (48) Hn=- -[H^(kp)] (49) and un is, (2n-l)7t v= n=l,2,... n 2a (50) Then using (figure 4); At the cross section x=xq, field components of EMA must be contiuous; E^(x=x0)=Ezu(x=x0) »« H;(x=x0)=H,;(x=x0) b® Substituting from ( 38) and (44) into (51), and ( 39) and (45) into ( 52) yields; oo oo I f!(y)n+R]+S anfn(y)=Z bnSn(9") \\J p" (53) n=2 n=l oo / oo Y^tl-RJ-E Ynanfn=iYov/ p"£ bn[- Cn((p")Hn sincp-+Sn((p-) H>sc|f ] (54) n=2 n=l, _ kp p" ve xo defining aj as; at=l+R (57) ( 53) and (54) can be written as; oo oo I^WH^p- m n=l n=l oo I oo 2Y1f1-E Ynanfn=iYoy P"S bn[- Cn(q>-)Hn rinqf+S^c) H>scp_] (59) n=l n=l kp fn(y) in ( 40) and Sn((p) in ( 46) satisfy orthonormality relations, (XI) f"(y)fm(y)dydz=8",m= z=-a y=-b -a /.a fl n=m.0 n*m Sn(9)Sm(9)pd(pdz=5n(m z=-a «»=. hi=l un" Vn=J'YoV P"2 bms(p~> m=l _ kp In the above equations <. > and Un are defined as follows;.b ra rb = */ J X(y)Y(y)dydz=2a X(y)Y(y)dy z=-a y=-b y=-b '2Y1, n=l un=.0, n>l It will be convenient to Introduce some auxiliary definitions ; c",m= Am=iYov/ P"f - cm(Am> Then,from ( 61) and (63), oo K=1j bmCn m m=l U -Ya=XbmDnm n n n *?? m n,m And writing; ( 70) reduces to. m=l G" "=Y"CL m+D" ". n,m n n,m n,m (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (xn) Un=XbmGn>1] (72) m=l ( 69 ) and ( 72) can be shown In matrix form as follows; ai u2 u, 1.1 ^2,1 '1,2 G G ı.ı J2,l 3,1 ^1.3 ^.2,2 ^2,3 G -3,2 G 1.2 J2,2 '3,2 G G 1.3 2.3 3.3J '3 b2 Lb3. â=£*b u=G*b => fe=0_1*Ji (73) (74) Thus the formal solution is completed since the field components in both regions are completely determined once a and b are obtained from above and substituted into (38-39) and (44-45). The analysis follows a similiar path for the case of pyramidal horn antenna. Moreover, once again the IM spectral integrals can be evaluated in explicit form. For TE type modes the locally separated scalar wave equation for Hz reads (See fig.5) (75) Hz= ¥(x,y,z) Z(z) 2 2 d d, 2 - 5+-= +k2-p(z) 2 -2+P Z=0 3z2 Subject to boundary conditions Lax-lx=a(Z) Lay and a radiation condition in z. ¥(x,y,z)=0 (76) (77) =0 (78) -V=b(z) Figure.5 Pyramidal Horn Antenna (xm) f"(y)fm(y)dydz=8",m= z=-a y=-b -a /.a fl n=m.0 n*m Sn(9)Sm(9)pd(pdz=5n(m z=-a «»=. hi=l un" Vn=J'YoV P"2 bms(p~> m=l _ kp In the above equations <. > and Un are defined as follows;.b ra rb = */ J X(y)Y(y)dydz=2a X(y)Y(y)dy z=-a y=-b y=-b '2Y1, n=l un=.0, n>l It will be convenient to Introduce some auxiliary definitions ; c",m= Am=iYov/ P"f - cm(Am> Then,from ( 61) and (63), oo K=1j bmCn m m=l U -Ya=XbmDnm n n n *?? m n,m And writing; ( 70) reduces to. m=l G" "=Y"CL m+D" ".