Generalized network formulation for symmetric double-step waveguide discontinuities

Transcription

1 INT. J. ELECTRONICS, 1989, VOL. 67, NO. 3, Generalized network formulation for symmetric double-step waveguide discontinuities A. K. HAMIDt, I. R. CIRICt and M. HAMIDH The equivalence principle and the moment method, which have previously been employed in the solution of single-step waveguide discontinuities, are extended to find solutions for the field and the scattering matrix coefficients at the junctions of symmetric double-step waveguide discontinuities. Scattering matrix coefficients are computed for representative symmetric double-step waveguide 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 double-step discontinuities in rectangular waveguides has been investigated extensively in the past for a variety of systems such as filters to minimize the reflection coefficient (Cohen 1955), planner dielectric waveguides in electrooptics and at millimeter frequencies (Rozzi 1979), and dielectric resonators in waveguides below and above cut-off (Bladel 1981). The early work was performed using the variational method (Marcuviz 1951, Collin 1960), a quasi-optical theory (Kao 1964), the Schwarz procedure (Goyal 1966), and several numerical methods, some of them based on the moment method solution (Wu and Chow 1972, Sinha 1986). Safavi-Naina and MacPhie (1981) applied the principle of conservation of the complex power to obtain the scattering matrix of a two-port network without matrix inversion. Ray theory has been applied to study the scattering by waveguide discontinuities (Yee and Felsen 1969). The ray theory diagram is complicated and the dimension of the scatterer should be large enough to obtain significant results (Kashyab and Hamid 1971). Rozzi and Mecklenbrauker (1975) proposed a solution based on the variational method and network modelling for interacting inductive irises and steps. A moment method and point matching solution is used by De Smedt and Denturck (1983) to obtain the scattering matrix of a symmetric doublestep discontinuity in terms of that of a single-step discontinuity. In this paper, we present a moment method solution that is straightforward for a symmetric double-step, and can be applied to multiple-step discontinuities (cascades) in rectangular waveguides. The solution is based on the generalized admittance network formulation (Harrington and Mautz 1976) which handles both oversized and ridged discontinuities. Various value of L are considered while the frequency range is from 7 to 19GHz. 2. Formulation of the problem Consider the structure shown in Fig. l. Owing to the symmetry about z = 0, the field analysis can be simplified by considering the two special cases of even and odd Received 31January1989; accepted 24 February t Department of Electrical Engineering, University of Manitoba, Winnipeg, Canada, R3T 1N1. i Presently on leave with the Department of Electrical Engineering, University of Central Florida, Orlando, Florida 32816, U.S.A. OOW-7217/89 $ Taylor & Francis Ltd.

2 428 A. K. Hamid et al. ---_ _ - s_ / I I o_ _s s ;:.z Figure 1. z=- L 2 z=- 2 Geometry of symmetric double-step waveguide discontinuities. 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 (Mittra 1974). Thus, a magnetic wall (open circuit) may be placed at the symmetry plane z = 0. In the case of odd excitation modes, waves opposite in amplitude and phase propagate in regions A and C simultaneously. Since the field in this case is anti-symmetric, this permits an electric wall (short circuit) to be placed at z = 0. 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. 2. The analysis for this type of structure runs parallel to the analysis for single-step waveguide discontinuities, the only difference being that waveguide B is not infinitely long in this case, being bounded by either a magnetic or an electric wall at z = 0, which requires more boundary conditions to be considered in the analysis. 3. Field analysis Consider the structure shown in Fig. 2, 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. Also, due to the presence of the discontinuity, higherorder waveguide modes are excited. The transmitted wave is totally reflected at the z = 0 plane. Thus, the transverse components of the field are given in modal form as follows: E, = l ai exp [ -jra{z + ) ]eai +di exp [jra{z + ) ]eai L bi exp [-jybiz]ebi + L Ebi exp [jybiz]eb;. i i z < /2 /2 < z < 0 (1) aiy,.i exp [ -jya{z + ) ]uz x eai - diy,.i exp [jra{z + ) ]uz x eai H, = z < - L/2 (2) L bilbi exp [-jybiz]uz x ebi - L Ebil/,i exp [jybiz]uz x ebi i i /2 < z < 0

3 Symmetric double-step waveguide discontinuities s z=- 2 "" 1-- v UJ _, UJ z=o (a) J s $ z v I () <{. J::E z=- 2 z =0 (b) Figure 2. (a) Short-circuit bisection. (b) Open-circuit bisection. where ai, bi, and di are the complex coefficients of the transmitted and reflected modes, Yai and Ybi are the modal propagation constants, Y,,i and Y,,i are the modal admittances, and e 0 i and ebi are the modal vectors for the ith 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 (Harrington 1961), the original problem can be divided into two regions as shown in Fig. 3. The fields in the two regions can be modelled in terms of an equivalent magnetic current sheet M placed over the aperture S, with M = Uz x Et at z = - L/2 (3) where uz is the unit normal and Et is the unknown tangential 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.

4 430 A. K. Hamid et al. -II' ::: M 5 5 M z=- 2 > u.. z --7Z " :! u ' "' >- u u.j..'\t_'--_-uj z=- 2 z =O Figure 3. Equivalence for waveguides A and B. 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 Satz = - L/2, requires that aiuz X eai + diuz x eai = M = b; exp [jybi ]uz X ebi I I I + ebi exp [jybi J Uz x ebi (4) Also, the continuity of the tangential magnetic field components across the aperture Satz = - L/2, requires that a; Y,,; Uz x eai - d; Y,,i Uz x eai = b)'i,i exp [jybi ] uz x ebi I I I - ebil'i,i exp [ -jyb; }z x ebi (5) To obtain an approximate solution for (4) and (5), we apply the moment method. 4. Moment method solution To apply the moment method, we define a set of expansion functions {MP} and a set of testing functions {WP} in S. Thus the magnetic current sheet M can be expanded as Q M = L VP MP at z = - L/2 (6) p=l where VP are unknown complex coefficients to be evaluated and 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 V,,haip (i = 1, 2,..., N) (7) p=l 1 Q bi= ( L) ( L) Vphbip exp jybi 2 + exp -jybi 2 p-l (i = 1, 2,..., N) (8)

5 Symmetric double-step waveguide discontinuities 431 where h,;p = l MP Uz x e,; ds (r =a, b) (9) Equation (9) can be written in matrix form as H, = [hrip]nxq Following the procedure outlined by Auda and Harrington (1983) and Hamid et al. (1988), (5) reduces to the generalized admittance form as (10) (11) where I= 2W Y,,a f, = w;y,h, a= [a;]nx 1 (r =a, b) in which Y, is the diagonal matrix of the modal admittances in the corresponding waveguide A or B, W, (r = a, b) is exactly the same as H,, which is given by (10) with MP replaced by, and a and V are the column matrices of the quantities a; and VP, respectively. Let N = Q (since the apertures are identical) and, This leads to, MP=uzxeap ebp=eap P=p=MP (p=l,2,...,q) (16) (12) (13) (14) (15) H, = W, = U (r =a, b) where U is the identity matrix. Equation (11) becomes r exp (iybi )- E exp ( -jybi ) l I=,,+ ( L) ( L)Yi,V exp jybi 2 + E exp -jybi 2 In the case of even excitation (E = 1), (18) reduces to I = [ Y,, + j tan (Yb; L) Yi, Jv while in the case of odd excitation (E = -1), I= [ Y,, - j cot (Yb;L) Yi, Jv (17) (18) (19) (20) Equations (19) and (20) represent (N x N) systems of linear equations. Its solution yields the equivalent magnetic current sheet M in (6). The final step of the analysis

6 432 A. K. Hamid et al. presented is to formulate the scattering matrix for symmetric double-step waveguide discontinuities. 5. Scattering matrix formulation The (2N x 2N) scattering matrix for a symmetric double-step waveguide discontinuity may be written as (De Smedt et al. 1983): S =[Saa Sac]=! [re+ r 0 re - r 0 ] (2l) S,a S,, 2 re-ro re+r 0 where re is the (N x N) reflection coefficient submatrix due to even excitation while r 0 is the reflection coefficient submatrix due to odd excitation. For the even exictation the complex reflection coefficients can be obtained from (7) as where V is obtained from (19) as Substituting (23) into (22), we get d=v-a 1 V = 2 [ Y,, + J. tan (Ybi 2 L) Y,, ]- Y,, a (22) (23) d = 2[ Ya+ j tan (Yb;L) YbJ 1 Y,,a - a (24) Therefore, the submatrix re is obtained in the form re ( ( Yb L) )- 1 = Y,, + j tan --t- Y,, ( Y,, - j tan (Yb L) --t- Y,, ) The submatrix r 0 due to the odd excitation can be obtained in a similar manner, as (25) r 0 = ( Y,, -j cot (Yb;L) Y,, )- 1 ( Y,, + j cot (Yb;L) Yb) (26) 6. Numerical results We consider the structure shown in Fig. 3, where waveguide A has a width wa = mm and a cut-off frequency fc = 6 56 GHz, while waveguide B has a width wb = mm and a cut-off frequency f = 3 28 GHz= 0 5f,. The separation between the junctions is assumed to be mm. Figure 4 shows the magnitude of the reflection coefficient for the TE 10 mode plotted as a function of frequency. Different values of N are considered: N = 5, 7, 9, 11, with N = 11 yielding a satisfactory accuracy. The results are checked against the values calculated by Rozzi and Mecklenbrauker (1975) for the frequency range 7 to 19 GHz. It is seen that the two curves are very close between 7 and 11 4 GHz. However, as the frequency rises, there is considerable deviation between the two curves. The results obtained by the method presented in this paper lead to values of Saa that vanish at specificfrequencies, namely/= 11 2, 14 1, and 17 2GHz. The voltage standing wave ratio due to the impedance mismatch that occurs when two rectangular waveguides with different cross sections are joined together

7 Symmetric double-step waveguide discontinuities oo Variational method " 0.4 ro (l) "Cl ;:! :;::: 0.2 >:: 00 ctl 0.1 Proposed method 0 0 g Freq uency(g Hz) Figure 4. Magnitude of Saa for symmetric double-step with L = mm, wb/w. = may be written as VSWR = l + s 1 - s Figure 5 shows the input VSWR presented as a function of frequency with L as a parameter. It can be seen that by decreasing the value of L from mm to 22 86mm results in increasing the value of VSWR from 1 to 2 8 at the particular frequency 7 GHz. The results presented in Tables 1 and 2 are obtained for wb/w. = 1 2, and propagating frequencies f = 8 and 15GHz. The magnitude of the VSWR is higher at f = 8 GHz, and is approximately constant at higher frequencies. (27) ro > 1.7 = mm - = mm - = mm 1.4 / ',.....\ / \1 "'..: "'---"---"'"---"-'""""..::;...--'-"'='--'="-'-j 7 g F'requency(GHz) Figure 5. Magnitude of VSWR for symmetric double-step with Las a parameter.

8 434 A. K. Hamid et al. Separation, L (cm) s sac VSWR j j j j j j j j l j j j j j j j j Table 1 Separation, L (cm) s s.c VSWR j j j j j j j j j j jo Ol j j jo j j Table 2 7. Discussion and conclusions A moment method solution for the analysis of symmetric double-step discontinuities in waveguides has been presented. The analysis presented could be modified to obtain the scattering of a symmetric double-step in a microstrip. The accuracy of the results obtained is established by comparing with the variational method in Fig. 4. The analysis can be extended to analyse a cascade of multiple-step discontinuities by using the obtained scattering matrix (21) for double-step discontinuities. The results for VSWR have been presented with Las a parameter, and a function of frequency which show in both cases that the magnitude of VSWR is approximately constant at higher frequency. ACKNOWLEDGMENT The authors wish to acknowledge the financial assistance of the Natural Sciences and Engineering Research Council of Canada and the Faculty of Graduate Studies of the University of Manitoba, which made this research possible. REFERENCES AUDA, H., and HARRINGTON, R. F., 1983, A moment method solution for waveguide junction problems. l.e.e.e. Transactions on Microwave Theory and Technology, 31, BLADEL, J. V., 1981, Dielectric resonator in waveguide below cutoff. I.E.E.E. Transactions on Microwave Theory Technology, 29, COirnN, S. B., 1955, Optimum design of stepped transmission-line transformers. l.r.e. Transactions on Microwave Theory Technology, 3, COLLIN, R. E., 1960, Field Theory of Guided Waves (New York: McGraw-Hill).

9 Symmetric double-step waveguide discontinuities 435 DE SMEDT, R., and DENTURCK, B., 1983, Scattering matrix of junction between rectangular waveguides. Proceedings of the Institution of Electrical Engineers, Pt H, 130, GOYAL, K. G., 1966, Analysis of the field structure in corners and tees in rectangular waveguides. M.Sc. thesis, Department of Electrical Engineering, University of Toronto. HAMID, A. K., CIRIC, I. R., and HAMID, M., 1988, Moment method solution of double step discontinuities in waveguides. International Journal of Electronics, 65, HARRINGTON, R. F., 1961, Time-Harmonic Electromagnetic Fields (New York: McGraw-Hill). HARRINGTON, R. F., and MAUTZ, J. R., 1976, A generalized network formulation for aperture problems. I.E.E.E. Transactions on Antennas and Propagation, 24, 87(}-873. KAO, K. C., 1964, Approximate solution of the H-plane right-angled corner in overmoded rectangular waveguide operating in the H 10 mode. Proceedings of the Institution of Electrical Engineers, 3, KASHYAB, S. C., and HAMID, M., 1971, Diffraction by a slit in a thick conducting screen. I.E.E.E. Transactions on Antennas and Propagation, 19, MARCUVITZ, N., 1951, Waveguide Handbook (New York: McGraw-Hill). MITTRA, R., 1974, Computer Techniques in Electromagnetics (New York: Oxford). RozzI, T. E., 1979, Field and network analysis of interacting step discontinuity in planer dielectric waveguides. I.E.E.E. Transactions on Microwave Theory and Technology, 27, Rozzi, T. E., and MECKLENBRAUKER, F. G., 1975, Wide-band network modeling of interacting inductive irises and steps. I.E.E.E. Transactions on Microwave Theory Technology, 23, SAFAVI-NAINA, R., and MACPIDE, R. H., 1981, On solving waveguide junction scattering problems by conservation of complex power technique. I.E.E.E. Transactions on Microwave Theory Technology, 29, SINHA, S., 1986, Analysis of multiple-strip discontinuity in a rectangular waveguide. I.E.E.E. Transactions on Microwave Theory Technology, 34, Wu, S. C., and Cuow, Y. L., 1972, An application of the moment method to the waveguide scattering problem. I.E.E.E. Transactions on Microwave Theory and Technology, 20, YEE, H. Y., and FELSEN, L. B., 1969, Ray optical analysis of electromagnetic scattering in waveguides. I.E.E.E. Transactions on Microwave Theory and Technology, 17,