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Analysis of an E-plane waveguide T-junction with a quarter-wave transformer using overlapping T-block method and genetic algorithm

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1. 1 Analysis of an E-plane waveguide T-junction with a quarter-wave transformer using overlapping T-block method and genetic algorithm Yong H. Cho Division of…
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  • 1. 1 Analysis of an E-plane waveguide T-junction with a quarter-wave transformer using overlapping T-block method and genetic algorithm Yong H. Cho Division of Information Communication and Radio Engineering Mokwon University 800 Doan-dong, Seo-gu, Daejeon, 302-729, Republic of Korea Phone: +82-42-829-7675 Fax: +82-42-825-5449 E-mail: yhcho@mokwon.ac.kr July 6, 2004 DRAFT
  • 2. 2 Abstract Scattering characteristics of an E-plane waveguide T-junction with a quarter-wave transformer are systematically analyzed with overlapping T-block method. The structure of an E-plane waveguide T- junction with a quarter-wave transformer is divided into two overlapping T-blocks. The geometry of a quarter-wave transformer is optimized with a binary genetic algorithm. Numerical computations are performed to illustrate the rate of convergence and accuracy of a series solution. Keywords T-junction, overlapping T-block, genetic algorithm, Green's function, mode-matching. I. Introduction A waveguide T-junction is a fundamental waveguide structure and has been extensively investigated 1-12]. In 1, 2], electromagnetic performance of a rectangular waveguide T- junction is analyzed using equivalent circuit concepts and admittance matrix in terms of waveguide modes. A closed form of the generalized admittance matrix (GAM) for junc- tions is obtained in 2] without recourse to matching unknown elds. The solutions of a waveguide T-junction associated with the generalized scattering matrix technique are widely utilized for the analysis of waveguide discontinuity problems 3, 4]. The segmen- tation technique in 5, 6] has been utilized to obtain the GAM for junctions. The eld analyses in 5, 6] are performed using the mode-matching method and the dyadic Green's function, respectively. To analyze the E- and H-plane T-junctions, a three plane mode- matching method is proposed in 7]. The Fourier transform 8] and the iteration scheme 9] are also applied to obtain scattering characteristics of an E-plane waveguide T-junction. In 10-12], multimode admittance matrix representation for junctions is obtained in terms of the GAM concept. The theory of cavities, segmentation procedure, and hybrid numer- ical method called BI-RME (Boundary Integral-Resonant Mode Expansion) are utilized for a six-port junction with rounded corners 11, 12]. In the present work, we introduce a novel approach based on overlapping T-block method 13, 14] for waveguide discontinuity problems. The dispersion analyses 13, 14] of overlap- ping T-blocks are extended to the scattering problems of junctions. The structure of an E-plane waveguide T-junction with a quarter-wave transformer is divided into two over- lapping T-blocks. A new formulation of T-blocks for the E-plane waveguide T-junction July 6, 2004 DRAFT
  • 3. 3 with a quarter-wave transformer is introduced, which is di erent from those in 13, 14]. The Hz elds within T-blocks are obtained using the Green's function relation 13] and mode-matching technique. The main advantage of the overlapping T-block method is that scattering relations of junctions are obtained as a closed form without the need of the residue calculus 8] and matrix manipulations 2-5, 10-12]. Our dominant-mode solution is quite accurate and useful for numerical evaluation in a low frequency limit, thus con rming the fast CPU time, increased accuracy, and simple applicability. Next, the geometry of a quarter-wave transformer is optimized with a binary genetic algorithm (GA) 15, 16]. The overlapping T-block method and GA allows us to obtain a simple yet numerically e cient series solution and perform a rigorous computer-aided design. II. Overlapping T-block Analysis Consider an E-plane waveguide T-junction with a quarter-wave transformer shown in Fig. 1. The time-factor e;i!t is suppressed throughout. The quarter-wave transformer (;T=2 < x < T=2 and ;d < y < 0) in Fig. 1 is utilized for impedance matching of an E-plane waveguide T-junction. Based on the overlapping T-block method in 13], we divide the geometry in Fig. 1 into two overlapping T-blocks, that is, subregions (I) and (II) shown in Fig. 2. The superposition of subregions (I) and (II) is the same to the geometry in Fig. 1, when b = b(1), T = 2a(1) = T(2), d = d(1) = b(2), and a = a(2). Note that subregion (I) in Fig. 2(a) is used for Ports 2 and 3 and subregion (II) in Fig. 2(b) does for Port 1. The Hz elds in subregion (I) are represented as HI(1) z (x y) = 1X m=0 q(1) m (1) m (x y) (for ;d(1) < y < 0) (1) HII(1) z (x y) = ; 1X m=0 q(1) m (1) m sin( (1) m d(1)) h H(1) m (x y) + RH(1) m (x y) i (for 0 < y < b(1)) (2) where q(1) m is an unknown modal coe cient, (1) m = q k2 0 ;(a(1) m )2, k0 = !p 0 0, a(1) m = m =(2a(1)), (1) m (x y) = cosa(1) m (x + a(1))cos (1) m (y + d(1)) u(x + a(1)) ;u(x ;a(1))] (3) July 6, 2004 DRAFT
  • 4. 4 H(1) m (x y) = cos (1) m (y ;b(1)) (1) m sin( (1) m b(1)) cosa(1) m (x + a(1)) h u(x + a(1)) ;u(x ;a(1)) i (4) RH(1) m (x y) = 1 0 r A z component = ;r Z J(r0)G(1) H (r r0) dr0 z component = ; Z H(1) m (r0) @ @n h G(1) H (r r0) i dr0 (5) G(1) H (r r0) = 2 b(1) 1X v=0 cos( (1) v y)cos( (1) v y0) v g(1) H (x x0 (1) v ) (6) g(1) H (x x0 ) = ei jx;x0 j ;2i (7) u( ) is a unit step function, A is a magnetic vector potential, J(r0) = H(1) m (r0) ^az ^n, n is the outward normal direction to r0 in H(1) m (r0), 0 = 2, v = 1 (v = 1 2 ), (1) v = v =b(1), (1) v = q k2 0 ;( (1) v )2, G(1) H (r r0) is shown in 17], and g(1) H (x x0 ) is a one-dimensional Green's function. Note that r and r0 in (5) denote an observation point (x y) and a source point (x0 y0) = ( a(1) 0 < y0 < b(1)), respectively. Integrating (5) over 0 < y0 < b(1) at x0 = a(1) yields RH(1) m (x y) = ; 1 b(1) 1X v=0 cos( (1) v y) v h ( (1) v )2 ;(a(1) m )2 i h sgn(x + a(1))ei (1) v jx+a(1) j ;(;1)msgn(x ;a(1))ei (1) v jx;a(1) j i (8) where sgn( ) = 2u( ) ;1. Then, the total Hz eld in subregion (I) is T(1) H (x y) = HI(1) z (x y) + HII(1) z (x y) (9) Similarly, we obtain the Hz elds in subregion (II) as HI(2) z (x y) = 1X m=0 q(2) m (2) m (x y) (for y < 0) (10) HII(2) z (x y) = ; 1X m=0 iq(2) m (2) m h H(2) m (x y) + RH(2) m (x y) i h u(x + T(2)=2) ;u(x ;T(2)=2) i (for 0 < y < b(2)) (11) where q(2) m is an unknown modal coe cient, (2) m = q k2 0 ;(a(2) m )2, a(2) m = m =(2a(2)), (2) m (x y) = cosa(2) m (x + a(2))e;i (2) m y July 6, 2004 DRAFT
  • 5. 5 u(x + a(2)) ;u(x ;a(2))] (12) H(2) m (x y) = cos (2) m (y ;b(2)) (2) m sin( (2) m b(2)) cosa(2) m (x + a(2)) h u(x + a(2)) ;u(x ;a(2)) i (13) RH(2) m (x y) = ; Z H(2) m (r0) @ @n h G(2) H (r r0) i dr0 = ; 1 b(2) 1X v=0 cos( (2) v y) v h ( (2) v )2 ;(a(2) m )2 i h fH(x ;a(2) (2) v ) ;(;1)mfH(x a(2) (2) v ) i (14) G(2) H (r r0) = 2 b(2) 1X v=0 cos( (2) v y)cos( (2) v y0) v g(2) H (x x0 (2) v ) (15) g(2) H (x x0 ) = sin (x< + T(2)=2)sin (T(2)=2 ;x>) sin( T(2)) (16) fH(x x0 ) = sgn(x ;x0) h ei jx;x0 j ;(;1)mei (T(2);jx;x0 j) i 1 ;(;1)mei T(2) (17) x> is the greater of x or x0, and x< is the lesser of x or x0. Then, the total Hz eld in subregion (II) is T(2) H (x y) = HI(2) z (x y) + HII(2) z (x y) (18) Utilizing (9) and (18), we obtain the Hz eld within the E-plane waveguide T-junction in Fig. 1 as TH(x y) = T(1) H (x y) + T(2) H (x y + d) (19) To obtain scattering relations of an E-plane waveguide T-junction, we assume the incident and re ected Hz elds from Ports 1 and 2 as H(1) i (x y) = Is cosa(2) s (x + a) ei (2) s (y+d) + e;i (2) s (y+d)] (20) H(2) i (x y) = Js cos( (1) s y)ei (1) s (x+T=2) (21) where Is and Js are the modal coe cients of the incident Hz elds from Ports 1 and 2, respectively. The superposition shown in (19) and Fig. 2 indicate that the Ex continuities at y = 0 and d are automatically satis ed, due to the presence of PEC at y = 0 and d. As such, we apply the Hz continuities at (;T=2 < x < T=2, y = 0) and (;a < x < a, y = ;d) July 6, 2004 DRAFT
  • 6. 6 to satisfy the boundary conditions. In matching the Hz continuities, it is expedient to introduce a general integration form as I(x y c) = Z x+c x;c h HI z(x0 y) T(1)+T(2) ;HII z (x0 y) T(1)+T(2) i coscl(x0 ;x + c) dx0 = 1X m=0 h q(1) m I(1) H (x y c) ;q(2) m I(2) H (x y + d c) i (22) where cl = l =(2c), I(1) H (x y c) = Z x+c x;c n (1) m (x0 y) + (1) m sin( (1) m d(1)) h H(1) m (x0 y) + RH(1) m (x0 y) io coscl(x0 ;x + c) dx0 (23) I(2) H (x y c) = Z x+c x;c n (2) m (x0 y) + i (2) m h H(2) m (x0 y) + RH(2) m (x0 y) io coscl(x0 ;x + c) dx0 (24) Since the integrands of (23) and (24) are composed of elementary functions, the evaluation of (23) and (24) is trivial. In order to satisfy the Hz eld continuity at (;T=2 < x < T=2, y = 0), we put (x, y) = (0 0) and c = T=2 into (22). We also multiply the incident Hz eld (21) with coscl(x + c) with c = T=2 and integrate over ;T=2 < x < T=2. Then, we obtain the scattering equation as I(0 0 T=2) = Js i (1) s 1 ;(;1)lei (1) s T] ( (1) s )2 ;(a(1) l )2 (25) Similarly, putting (x, y) = (0 d) and c = a into (22), multiplying (20) with coscl(x + c) with c = a, and integrating over ;a < x < a yields I(0 ;d a) = Is2a s sl (26) where sl is the Kronecker delta. It is possible to extend our theory to the analysis of an H-plane waveguide T-junction with the Ez eld representations in 13, 14]. In order to verify our approach, numerical computations of (25) and (26) are performed and compared with results of the iterative solution 9] and Fourier transform method 8]. Fig. 3 shows the convergence characteristics of our series solution, (25) and (26), for an E-plane waveguide T-junction (2a = T in Fig. 1 and s = 0) in terms of a modal index m. Note that Pi2 and Pj in Fig. 3 denote the incident power from Port 2 and the transmitted power to July 6, 2004 DRAFT
  • 7. 7 Port j, respectively. When m 3, our series solution is almost identical with 8, 9]. In a low frequency limit (f < 10 GHz), a dominant-mode solution (m = 0) agrees with 8, 9] within 5 % error. III. Optimization Using Genetic Algorithm Using a binary GA 15, 16], the E-plane waveguide T-junction with a quarter-wave transformer (d and T in Fig. 1) is optimized in terms of re ection coe cient. The tness function of genetic optimization 16] is de ned as tness = 1 1 + perror (27) where error = 1 N NX i=1 e(fi)]2 (28) e(f) = 8 >>< >>: object ;RL(f) ripple when RL(f) < object 0 when RL(f) object (29) RL(f) is a return loss (dB) at frequency f, object in (29) is the goal for a return loss, and ripple in (29) is a permissible error of a return loss in genetic optimization. Note that (28) is de ned in view of the least-square method. The number of tness evaluation (N t) is obtained as N t = Npop h 1 + Ngen(Pc + Pm) i (30) where Npop is the number of population, Ngen is the number of generation, and Pc and Pm denote the probabilities of crossover and mutation, respectively. For initial guess of optimization, the transmission line theory for a quarter-wave transformer is introduced, such as dinit = =(2k0), Tinit = 2 p 2a. When the center frequency of a T-junction is 10 GHz, d = dinit = 7:49 mm and T = Tinit = 9:9 mm for initial guess. Next, we set up the parameters for GA such as object = 20 dB, ripple = 1 dB, bandwidth = 2 GHz, 2a < T < 4 p 2a, =(4k0) < d < =k0, Npop = 30, number of bits = 16, Pc = 0.6, Pm = 0.1, and Ngen = 100. As such, N t = 2130. Elitism and roulette wheel selection are also applied for optimization. It is not until 20 generations (N t = 450) that we obtain the convergent July 6, 2004 DRAFT
  • 8. 8 tness when d = 7:97 mm and T = 11:8 mm. Fig. 4 shows the re ection characteristics of the optimized quarter-wave transformer versus frequency, thus con rming that our optimization scheme is e ective. It is observed in Fig. 4 that the re ection coe cients for an optimal solution are not below 20 dB at any frequency. However, our solution is the best one in terms of the tness function (27). Our computational experiences also indicate that bad starting points from dinit and Tinit a ect little for optimization when d < 5 dinit and T < 5 Tinit. Even though the genetic optimization is applied to a simple structure in Fig. 1, it is possible to extend our theory to other waveguide structures such as coupler, bend, and divider. IV. Conclusions Scattering analysis of an E-plane waveguide T-junction with a quarter-wave transformer is analytically shown using overlapping T-block method. Simple yet rigorous scattering relations for the E-plane waveguide T-junction with a quarter-wave transformer are pre- sented and compared with other results. Optimization is also performed using genetic algorithm. Our approach can be extended to other waveguide structures such as coupler, bend, and divider. July 6, 2004 DRAFT
  • 9. 9 References 1] E. D. Sharp, An exact calculation for a T-junction of rectangular waveguides having arbitrary cross sections," IEEE Trans. Microwave Theory Tech., vol. 15, no. 2, pp. 109-116, Feb. 1967. 2] J. M. Rebollar, J. Esteban, and J. E. Page, Fullwave analysis of three and four-port rectangular waveguide junctions," IEEE Trans. Microwave Theory Tech., vol. 42, no. 2, pp. 256-263, Feb. 1994. 3] F. Arndt, I. Ahrens, U. Papziner, U. Wiechmann, and R. Wilkeit, Optimized E-plane T-junction series power dividers," IEEE Trans. Microwave Theory Tech., vol. 35, no. 11, pp. 1052-1059, Nov. 1987. 4] T. Sieverding, U. Papziner, and F. Arndt, Mode-matching CAD of rectangular or circular multiaperture narrow-wall couplers," IEEE Trans. Microwave Theory Tech., vol. 45, no. 7, pp. 1034-1040, July 1997. 5] F. Alessandri, M. Mongiardo, and R. Sorrentino, Computer-aided design of beam forming networks for modern satellite antennas," IEEE Trans. Microwave Theory Tech., vol. 40, no. 6, pp. 1117-1127, June 1992. 6] F. Alessandri, M. Mongiardo, and R. Sorrentino, A technique for the fullwave automatic synthesis of waveguide components: Application to xed phase shifters," IEEE Trans. Microwave Theory Tech., vol. 40, no. 7, pp. 1484-1495, July 1992. 7] X.-P. Liang, K. A. Zaki, and A. E. Atia, A rigorous three plane mode-matching technique for characterizing waveguide T-junctions, and its application in multiplexer design," IEEE Trans. Microwave Theory Tech., vol. 39, no. 12, pp. 2138-2147, Dec. 1991. 8] K. H. Park, H. J. Eom, and Y. Yamaguchi, An analytic series solution for E-plane T-junction in parallel-plate waveguide," IEEE Trans. Microwave Theory Tech., vol. 42, no. 2, pp. 356-358, Feb. 1994. 9] Y. H. Cho, New iterative equations for an E-plane T-junction in a parallel-plate waveguide using Green's function," Microwave Optical Tech. Lett., vol. 37, no. 6, pp. 447-449, June 2003. 10] V. E. Boria and M. Guglielmi, E cient admittance matrix representation of a cubic junction of rectangular waveguides," IEEE MTT-S Int. Microwave Symp. Digest, vol. 3, pp. 1751-1754, 1998. 11] V. E. Boria, S. Cogollos, H. Esteban, M. Guglielmi, and B. Gimeno, E cient analysis of a cubic junction of rectangular waveguides using the admittance-matrix representation," IEE Proc.-Microw. Antennas Propag., vol. 147, no. 6, pp. 417-422, Dec. 2000. 12] S. Cogollos, V. E. Boria, P. Soto, A. A. San Blas, B. Gimeno, and M. Guglielmi, Direct computation of the admittance parameters of a cubic junction with arbitrarily shaped access ports using the BI-RME method," IEE Proc.-Microw. Antennas Propag., vol. 150, no. 2, pp. 111-119, April 2003. 13] Y. H. Cho and H. J. Eom, Analysis of a ridge waveguide using overlapping T-blocks," IEEE Trans. Microwave Theory Tech., vol. 50, no. 10, pp. 2368-2373, Oct. 2002. 14] Y. H. Cho and H. J. Eom, Overlapping T-block analysis of axially grooved rectangular waveguide," Electron. Lett., vol. 39, no. 24, pp. 1734-1735, Nov. 2003. 15] Z. Michalewicz, Genetic Algorithms+Data Structures=Evolution Programs, 3rd ed., Berlin: Springer-Verlag, 1996. 16] Y. Rahmat-Samii and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms, New York: John Wiley & Sons, 1999. 17] H. J. Eom, Electromagnetic Wave Theory for Boundary-Value Problems: An Advanced Course on Analytical Methods, Berlin: Springer-Verlag, 2004, pp. 218-225. July 6, 2004 DRAFT
  • 10. 10 ;;; ;;; ;;; ;;; ;;;;;;;b e0 ;;; ;;; ;;; ;;; ;;; ;;; ;;; ;;; PEC x y z T d a2 Port 1 Port 2 Port 3 Fig. 1. Geometry of waveguide T-junction with quarter-wave transformer. July 6, 2004 DRAFT
  • 11. 11 ;;; ;;;;;;;b e0 ;;;;;; ;;; ;;;;;; ;;;PEC Port 2 Port 3 d a2T (x',y') (1) (1) x' y' z (1) (1) (a) Subregion (I) ;;; ;;; ;;; ;;; ;;;; ; ;; ; ; ;; PEC T b a2 T (x',y') (2) Port 1 x' y' z (2) (2) (2) (b) Subregion (II) Fig. 2. Subregions of waveguide T-junction with quarter-wave transformer. July 6, 2004 DRAFT
  • 12. 12 5 10 15 20 0 0.2 0.4 0.6 0.8 1 Frequency [GHz] Normalizedpower P2 /Pi2 P1 /Pi2 P3 /Pi2 m = 0 m = 1 m = 2 [9] [8] Fig. 3. Behavior of normalized re ection and transmission powers versus frequency with Is = 0, Js = 1, a = 3:5 mm, and b = 7 mm. 7 8 9 10 11 12 13 −20 −16 −12 −8 −4 0 Frequency [GHz] Reflectioncoefficients[dB] E−plane T−junction original λ/4 transformer optimized λ/4 transformer Fig. 4. Behavior of re ection coe cients versus frequency with Is = 1, Js = 0, a = 3:5 mm, and b = 7 mm. July 6, 2004 DRAFT
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