ABSTRACT 1. INTRODUCTION 2. FORMULATION OF THE PROBLEM

Transcription

1 ANALYSIS OF MULTIPLE-STEP DISCONTINUITIES IN RECTANGULAR WAVEGUIDES A.K. Hamid, LR. Ciric and M. Hamid Department of Electrical Engineering University of Manitoba Winnipeg, Manitoba, Canada R3T N ABSTRACT The equivalence principle and the moment method which was previously employed in the solution of single-step waveguide discontinuities, is extended to solve for the field and the scattering matrix coefficients at the junctions of multiple-step discontinuities. Scattering matrix coefficients are computed for representative doublestep discontinuities and frequencies. The method is simple and straightforward, and the results obtained are compared with data available in the literature. 1. INTRODUCTION The problem of multiple-step discontinuities in rectangular waveguides has been investigated extensively in the past for a variety of systems such as planner dielectric waveguides in electroptics and at millimeter frequencies [l], filters to minimize the reflection coefficient [], and dielectric resonators in waveguides below and above cutoff [3]. The early work was performed by using the variational method [4,5], Schwarz procedure [6], a quasi-optical theory [7], and several numerical methods, some of them based on the moment method solution [8,9]. Safavi-Naini and MacPhie [ 10] applied the principle of conservation of complex power to obtain the scattering matrix of two-port network without matrix inversion. Ray theory has been applied to study the scattering by waveguides discontinuity [11,1]. The ray theory diagram is complicated and the dimension of the scatterer should be large enough to obtain significant results. Rozzi and Mecklenbrauker proposed a solution based on the variational method and network modeling for interacting inductive irises and steps [13]. A moment method and point matching solution is used by De Smedt and Denturck [14] to obtain the scattering matrix of a double-step discontinuity in terms of that of a single-step discontinuity. In this paper, we present a moment method solution which is straightforward and can be applied to multiple-step discontinuities (cascade) in rectangular waveguides. The solution is based on the generalized admittance network formulation [15] which handles both oversized and ridged discontinuities. Various values of L are considered, however, and the frequency range is 7 and 19 GHz.. FORMULATION OF THE PROBLEM Consider the structure shown in Fig. 1. Due to the symmetry about z=o, the field analysis can be simplified by considering the two special cases of even and odd excitation modes and then superimposing the results. In the case of even excitation modes, waves of equal amplitude but opposite in phase propagate in regions A and C simultaneously [14,16]. Thus, a magnetic wall (open circuit) may be placed at the symmetry plane z=o. In the case of odd excitation modes, waves opposite in amplitude and phase -1-

2 propagate in regions A and C simultaneously. Since the field in this case is antisymmetric, this permits an electric wall (short circuit) to be placed at z=o. Addition of the two excitations results in twice the amplitude of the excitations in waveguide A, and zero excitation in waveguide C. Therefore, the problem is reduced to a single-step discontinuity and terminated alternately by a magnetic or electric wall as shown in Fig.. The analysis for this type of structure runs parallel to the analysis for single-step, the only difference being that waveguide B is not infinitely long in this case, being bounded by an alternate magnetic or electric wall at z=o, which requires more boundary condition to be considered in the analysis. 3. FIELD ANALYSIS Consider the structure shown in Fig., with an incident field propagating in the positive z-direction. A part of it is scattered by the discontinuity and a part is transmitted to waveguide B. The transmitted wave is totally reflected at z=o plane. Thus, the transverse components of the field in modal form are given as follows. z < -L/ -L/ < z < 0 (1) z < -L/ -L/ < z < 0 () ai, bi, and di are complex coefficients of the transmitted and reflected modes, 'Yai and 'Ybi are the modal propagation constants, Y ai and Y bi are the modal admittances, and eai and ebi are the modal vectors for the i th mode, in waveguides A and B, respectively. For the case of an electric wall e = -1 while in the case of a magnetic wall e = 1. Using the equivalence principle [17], the fields in the two regions can be modeled in terms of the equivalent magnetic current sheet M placed over the aperture S, as shown in Fig. 3, with M = UzX E 1 at z = -L/ (3) where Uz is the unit normal and E 1 is the unknown electric field in the aperture S, to be determined. The field in waveguide A is the incident field plus the field produced by the magnetic current sheet M. The field in waveguide B (resonator box) is the total field produced by the magnetic current sheet -M plus the field totally reflected by the magnetic or electric wall at the boundary z = 0. In order to determine the unknown expansion coefficients, we apply the proper boundary conditions. Thus the continuity of the tangential electric field components across the aperture S at z = -L 1, requires that --

3 Also the continuity of the tangential magnetic field components across the aperture S at z = -L /, requires that L aiyaiuzx eai - L diyaiuzx eai = i i To obtain an approximate solution for equations (4) and (5), we apply the moment method. 4. MOMENT METHOD SOLUTION To apply the moment method, we expand the magnetic current sheet M as Q M = L,Vp MP at z=-l/ (6) p=l where VP are unknown complex coefficients to be evaluated, and MP are known vector basis functions. The above summation is limited to a finite number of terms Q. By substituting (6) into (4) and applying the orthogonality condition for the mode functions in each region, we obtain Q ai +di = L, VP haip (i = 1,,..., N) (7) p=l (i = 1,,..., N) (8) where h,ip = f Mp Uz X e,; ds (r = a,b) s Equation (9) can be written in matrix form as H, = [h,ipj Nx Q Following the procedure outlined in [18], equation (5) reduces to the generalized admittance form as where [ l J'Y. -L - J'Ylli. -L - Tt -=+ - e -ee I = Ya +. L. L Yb v e J'Y - - J'Ylli - +ee t=wty a a a - - T Y, - W, Y,H, (r = a,b) (9) (10) (11) (1) (13) -3-

4 lt =fad Nxl V = [Vpl Qx i in which Y, is the diagonal matrix of the modal admittances in the corresponding waveguide A and B, W, (r = a,b) is exactly the same as H,, which is given by (10) with MP replaced by WP, and 71 and V are the column matrices of the quantities a; and VP, respectively. Let N = Q and MP = zxeap, ebp = eap, Wop = Wbp =MP This leads to H, = W, = U (r =a,b) where U is the identity matrix. Equation (11) becomes [ l J"fw-. L. L e -Ee I = Ya +. L. L Yb v -rrwe J"fw- +Ee -rrw- w7 (p=l,,..., Q) In the case of even excitation (E = 1), equation (18) reduces to f [y. ('YbiL)Y] V = a +}tan -- b (14) (15) (16) (17) (18) (19) while in the case of odd excitation (E = -1), [. 'YbiL ] V I= Ya - JCOt (- -) Yb (0) Equations (19) and (0) represent (Nx N) systems of linear equations. Its solution yields the equivalent magnetic current sheet M in equation (6). The final step of the presented analysis is to formulate the scattering matrix for multiple-step discontinuities. 5. SCATTERING MATRIX FORMULATION The ('ln x 'ln) scattering matrix for multiple-step waveguide discontinuity may be written as [14] s = sea [ Saa where re is the reflection coefficient matrix due to even excitation while r 0 is the reflection coefficient matrix due to odd excitation. For the even excitation, the complex reflection coefficients can be obtained from (7) as l=v-ct a r V = [Ya + jt:m ( YbL) Yb Y_-i/ (3) where V is obtained from ( 19) as Substituting (3) into (), we get (1) -4-

5 r [r. it= + jtan ( yl) Yb y_-lf - "ii Therefore, the (N x N) submatrix re is obtained in the form "(/ [ 'Yb L i-l [ l 'Yb L re= a= Ya+ jtan (+)Yb Ya - jtan (+)Yb The submatrix r 0 due to the odd excitation canbe obtained in a similar manner, as 'Yb L 'Yb L ' [ i-l [ l r 0 = Ya - jcot (+)Yb Ya+ jcot (+)Yb (6) (4) (5) 6. RESULTS AND DISCUSSION We consider the structure shown in Fig 3, where waveguide A has a width wa =.86 mm and a cutoff frequency fc = 13.1 GHz, while waveguide B has a width wb = 45.7 mm and a cutoff frequency f 'c = 6.56 GHz = 0.5 f c. The separation between the junctions is assumed to be 7.43 mm. Fig. 4 shows the magnitude of the reflection coefficient is plotted as a function of frequency. Different values of N were considered, N = 5,7,9,11, with N = 11 yielding to a satisfactory accuracy. The results are checked against the values calculated by Rozzi et al for the frequency range 7 to 19 GHz [13]. It is seen that the two curves are very close between 7 and 10.4 GHz. The results obtained by the method presented in this paper lead to values of Saa that vanish at specific frequencies, namely f = 11.,14.1,17. GHz. The voltage standing wave ratio due to the impedance mismatch that occurs when two rectangular waveguides with different cross-sections are joined together may be written as VSWR = 1 +IS I 1 -!Saal The results presented in tables I and II are obtained for wb,wa = 1., and propagating frequencies f = 8 and 15 GHz. The magnitude of the VSWR is higher at f = 8 GHz, and is approximately constant at higher frequencies. REFERENCES [1] T. E. Rozzi,"Field and network analysis of interacting step discontinuity in planner dielectric waveguides", IEEE Trans. Microwave Theory and Tech, vol. MTI- 7, pp April [] S. B. Cohn,"Optimum design of stepped transmission-line transformers", I.R.E. Trans. Microwave Theory Tech., vol. MTT-3, pp. 16-1, April [3] J. V. Bladel,"Dielectric resonator in waveguide below cutoff', IEEE Trans. Microwave Theory Tech., vol. MTT-9, No.4, pp , Apr [4] N. Marcuvitz, Waveguide Handbook, NewYork: McGraw-Hill, [5] R. E. Collin, Fields Theory of Guided Waves, NewYork: McGraw-Hill, [6] K. G. Goyal, Analysis of the field structure in comers and tees in rectangular waveguides, M. Sc. Thesis, Department of Electrical Engineering, University of -5- (7)

6 Toronto, [7) K. C. Kao,"Approximate solution of the H-plane right-angled comer in overmoded rectangular waveguide operating in the H 10 mode", Proc. IEE, vol.ill, No.4, pp , April [8] S. Sinha,"Analysis of multiple-strip discontinuity in a rectangular waveguide", IEEE Trans. Microwave Theory Tech., vol. MTT-34, No.6, pp , June [9] S. C. Wu and Y. L.Chow,"An application of moment method to waveguide scattering problem", IEEE Trans. Microwave Theory Tech., vol. MTT-0, No.11, pp , Nov. 197 [10) R. Safavi-Naina and R. H. MacPhie," On solving waveguide junction scattering problems by conservation of complex power technique", IEEE Trans. Microwave Theory Tech., vol. MTT-9, No.4, pp , Apr [11) H. Y. Yee and L. B. Felsen,"Ray optical analysis of electromagnetic scattering in waveguides", IEEE Trans. Microwave Theory Tech., vol. MTT-17, pp , SepL [1) S. C. Kashyap and M. Hamid,"Diffraction by a slit in a thick conducting screen", IEEE Trans. Antennas Propagat. vol. AP-19, No.4, [13) T. E. Rozzi and W. F. G. Mecklenbrauker,"Wide-band network modeling of interacting inductive irises and steps", IEEE Trans. Microwave Theory Tech., vol. MTT-3, No., pp , Feb [14) R. De Smedt and B. Denturck,"Scattering matrix of junction between rectangular waveguides", IEE Proc., vol.130.pt.h, No., pp , March [15) R. F. Harrington and J. R. Mautz," A generalized network formulation for aperture problems", IEEE Trans. Antennas Propagat. vol.ap-4, pp , Nov [16) R. Miura, Computer techniques in electromagnetics, NewYork: Oxford, [17) R. F. Harrington, Time-Harmonic Electromagnetic Fields, NewYork: McGraw Hill, [18) A. K. Hamid, I. R. Ciric and M.Hamid,"Moment method solution of double step discontinuities in waveguides", Submitted for publication, Int. J. of Electronics. -6-

7 I ' s ---+-I o_@_s_. _s_ ?z -L z - L z - Fie:. I. Geornrlry of mullilep wavquicle s L z - fal = I I= s 1 ---?l '>l v v,_ "" v I.,I -L z -... <.:> Fig.. fa) Shor1-circui1 bistttion. 1b1 Opm<ircuil -l z-- -,--. j:;: z c < ----'>l ':d ' cc,_ v... '----'v'---u; -L z-- Fij!. 3. Equh alence for waveguides A and

8 G G tf.l ::l 0. c: tld (Oj ::s 0.1 I oo Variational method [13) I Proposed method ,-::.t_-.-...:>=:::..._,--..:;:_--T::.c;.-i 7 g Frequency{ GHz) Fig. 4. Mapitude o1 s. for dou witll L = 7.43 mm, wbtw. = 1.. Table I Scancring cofftcicru as a funcuon or scpamion length Table II Scaucting coflicicllls as a function of separation length for wblw = 1.. f = 8 GHz. Separation. L (an) s_ s... VSWR Separation. L (cm) s..., Sac VSWR I -017-j j l j j j ll*j j j0.604 I j S43+j0. I 70 1.()86 s j j j j j j j0.007 O.Ol I l+j0.<l j j II j j II jO.O j O.OO'J-j j0. I j j S4+j j IS j j0.IOl I