Analysis of Gap Waveguide Structures Using Mode Matching Approach Mladen Vukomanović, University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia [email protected] Abstract – Gap waveguide structures combine the ideas of parallel plate waveguides and electromagnetic band gaps to form novel efficient high frequency waveguiding structures. Since the concept is based on the use of periodic structures with bandgap properties the analysis is very complex. Our proposed analysis method is based on efficient use of Method of Moments and Mode matching techniques to rigorously determine the scattering of each element in the structure. To achieve this, the currents on the elements are first determined by using the Green’s function of a single element in a parallel-plate waveguide. The solution is then expanded to all the elements from which dispersion properties can be determined. Mode matching is the alternative approach that is used to calculate scattering of each element. The results of these approaches are verified using full-wave commercial software. It is important to point out that the results from the developed approach are obtained in a fraction of the time needed by the commercial full-wave software to analyze the same structure. waves, corrugated surface supports or does not support propagating waves along the structure, meaning that it is only a one-dimensional soft surface. Better alternatives are uniform 2D periodic surfaces such as bed of nails surface or mushroom surface (shown in Fig. 2(b)). These structures act as soft in any direction when operating inside a certain frequency bandgap and thus prohibit all surface wave propagation. When such soft surface is paired with a PEC surface in a parallel-plate configuration all lower parallel plate modes will be suppressed if the dimension of the space between the plates is less than a quarter of a wavelength. Essentially, due to the stop band, the lowest order parallel plate modes are in cut-off and the waves are forced to propagate only along the desired directions imposed by ridges or strips as shown Fig. 1. Keywords – Gap Waveguides; Mode Matching; Bed-of-nails I. INTRODUCTION Gap waveguide is a parallel plate waveguiding structure where one wall is replaced by a periodic metamaterial based surface [1] (as shown in Fig. 1). The considered structures are very promising for general microwave applications, but especially for applications above 30 GHz. Applications could include feeding networks for slot arrays, all-metallic transmission lines for use in space, cavity mode suppression for microstrip circuits (for solving packaging problems), etc. [1,2]. The major benefit of this guiding structure compared to the microstrip line is the fact that no dielectric is needed and therefore there are no dielectric related losses. Compared to classical rectangular waveguides this structure can be manufactured more easily and the dimensions can be smaller. This is because instead of using metallic side walls this novel waveguideing mechanism is based on electromagnetic bandgap (EBG) properties of certain periodic surfaces. Essentially EBG surfaces are here used as a practical realization of a soft boundary condition [3]. By definition, soft surface stops waves from propagating, whereas hard surface supports propagating waves. Corrugated surface shown in Fig. 2(a) is a typical example of a surface which can acts as a soft or a hard surface, depending on the direction of propagation. It acts as soft surface when corrugations are orthogonal to direction of propagation, and as hard surface when corrugations are longitudinal (oriented in the same direction as the wave propagation). Depending on the direction of propagating Figure 1. Gap waveguide structure with guiding strip/ridge (a) (b) Figure 2. Other types of EBG surfaces suitable for building gap waveguides. (a) corrugated surface, (b) mushroom surface Since the crucial elements of these structures are periodic surfaces with some inclusions, a general 3D code could be used for the analysis of finite structures. However, this approach can be very time consuming, and therefore it is necessary to introduce an alternative which will keep all the electromagnetic effects, but allow a faster first design. In this paper a most common gap-waveguide problem is analyzed, where the frequency stop band surface is realized 1 using a periodic array of cylindrical pins, so called “bed-ofnails”. The basis of the first presented approach is the Green’s function of a single element from which by using Method of Moments all the currents on all the elements and consequently the scattered field can be determined. Second approach is based on the Mode Matching technique and it gives better results compared to the first approach. Finally, by reducing the number of modes used and by applying acceleration techniques a very efficient and also very accurate solution is obtained. II. In (4a) and (4b), µ0 is vacuum permeability, Jz is current density on the cylinder, J0 is Bessel function of a first kind of order zero, H0 is Hankel function of a first kind of order zero, and ρ is radial distance from the cylinder. Propagation constant kv is defined as: v kv k 2 . h 2 (5) Exact expression for vector potential is shown in (4a) and with it scattered electric field can be calculated as: METHOD OF MOMENT APPROACH A. Cylinder in parallel plate waveguide The starting point of the analysis is metallic (PEC) cylinder inside an infinite parallel plate waveguide shown in Fig. 3. Ez j 2 Az k 2 Az . k 2 z 2 (6) In (4a) current distribution along cylinder is unknown and to find the complete solution PEC boundary condition is applied on the surface of the cylinder: nˆ E 0, Figure 3. Single cylinder inside a parallel plate waveguide First goal was to find current distribution along the pin for a known incident plane wave. The considered incident plane wave is directed along the positive x axis. Assuming that the current distribution along the pin is cylindrically symmetric only the variation along z axis exists: J az a J z z e jt , 0 z L . (7) In (7) E is total electric field on the cylinder and it corresponds to the sum of incident and scattered field. After applying the boundary condition integral equation is obtained where the current distribution Jz is unknown. To compute this current distribution along the cylinder the Method of Moments approach is used. Figure 4 shows the computed current distribution at frequency of 1 GHz. Length of the cylinder L is 8.66 mm, and height of the waveguide h is 12.16 mm. (1) Consequently, vector potential also has only one component: (2) A az Az ( , z ) . To get the exact expression for vector potential Green’s function, a scalar Helmholtz equation has to be solved: 2 Az , z k 2 Az o a J z z . Az , z j a0 2h v Iv cos v 0 L Iv 2 v z J 0 kv a H 0 k v , a , h J 0 kv H 0 2 kv a , a (4a) J z cos ' z z 0 ' v (3) v ' ' z dz , h 1, v 0 , 2, v 0 (4b) Figure 4. Current distribution of a single cylinder in the parallel-plate waveguide The result is expected since at the point where the cylinder is connected with the parallel plate waveguide the current has the maximum value and at the top of the cylinder the current is ideally equal to zero or close to zero. Obtained current distribution allows us to calculate exact values of the scattered electric field. (4c) B. Set of cylinders in parallel plate waveguide One cylinder can be replaced with set of cylinders like in Fig. 5. The function of the cylinders is to act as an artificial 2 magnetic conductor (AMC) in a certain frequency band and the whole structure with the PEC top plate is now the basis of gap waveguide. Figure 5. Gap waveguide based on bed-of-nails AMC It is assumed that the amplitude of all currents on the cylinders is the same, but with a phase difference between them. If only one row of cylinders along the y axis is considered, scattered electric field from that row can be obtained and this result will take into account contributions from all cylinders. Complete formulation for the scattered electric field from one row of cylinders is found to be: Ez a0 2hk 2 e p v 0 jkpDy s in ( ) v v z cos J 0 kv a I v h H 0(2) kv x 2 ( y pDy ) 2 Fig. 7 shows the dispersion diagram of the gap waveguide structure obtained using two methods: Moment Method approach and general EM solver (CST Microwave Studio). Length of the cylinder L is 8.66 mm, height of the waveguide h is 12.16 mm and the period of the pins p is 3.75 mm. From the dispersion diagram it can be seen that the stop band is between 8 GHz and 12 GHz. There is a good agreement between Moment Method and CST results, but only in the first part of the diagram for lower frequencies. For higher frequencies there is a visible deviation between the results. Possible reason for this deviation lies in the fact that the field in the gap between the cylinders and the top PEC plate is perturbed by the current on the cylinder top. For that reason, an alternative approach was developed based on Mode Matching which takes this problem also into account. , (8) where Dy is the distance between cylinders and φ is the angle of each cylinder in cylindrical coordinate system. This procedure can be repeated further for other rows. The total field will be the sum of incident field and scattered field from the structure. To take into account other rows of the periodical structure Bloch theorem is used, as in [5]. It is represented in Fig. 6 where p is the distance between two rows. Figure 7. Dispersion diagram for the gap waveguide calculated using MoM (ABCD) approach and compared to CST simulation results. III. MODE MATCHING APPROACH A. Cylinder in parallel plate waveguide In this method, instead of first calculating the current along the pin and from that the scattered electric field, the electric field will be from the beginning represented as a sum of cylindrical modes. As it is shown in Fig. 8, there is defined electric field in the waveguide and in the gap between the cylinder and the top plate. Figure 6. Bloch theorem scheme Electric and magnetic fields before and after the considered row are observed. Comparing them S-parameters are obtained and consequently ABCD parameters/matrix: E1 A B E 2 ikB p E 2 H1 C D H 2 e H 2 (9) Here, Bloch propagation constant is denoted as kB. From the eigenvalues of ABCD matrix Bloch propagation constant for all frequencies of interest was calculated. Those propagation constants will form the dispersion diagram for this structure. Figure 8. Single pin between parallel plates Electric, oriented in z direction, and magnetic fields between parallel plates, can be expanded in terms of cylindrical modes as: 3 N n z 2 Escat n cos H 0 kn , n 0 hw H scat j N k n z 2 H1 k n . hw 1 k n 0 ncos n (10a) 2 n kn k . hw (11) Electric and magnetic fields in the gap above the cylinder are defined as: N n (12a) Egap n cos z L J 0 kn' , n 0 hg H gap j N k 1 k n 0 ' n n n cos hg z L J1 kn' , (13) Expressions for the incident field that impinges at single pin expanded in terms of cylindrical modes are [6]: N Einc Einc ,n cos( n 0 H inc j k N E inc , n n 0 n z ) J 0 kn , hw A Ei Einc . D2 H i H inc For different values n and m matrix D1 is filled with: hw n z m z , D1 m, n H 0 2 kn a cos cos dz h w hw 0 matrix A with: hw n m z , A m, n J 0 kn' a cos z L cos dz h hw L g (17) (18a) (18b) matrix B with: B m, n hw m n z k 2 H1 kn a cos ( z L) dz , cos kn hg hw L (18c) and matrix D2 with: D2 m, n hw n m k J1 kn' a cos z L cos z L dz . (18d) ' kn h h L g g K matrix contains unknown coefficients α and β. In addition, matrix Ei is filled with: 2 . D1 B (16) (12b) where n is mode number, βn are unknown coefficients for each mode, hg is height of gap between cylinder and a top plate and kn’ is equal to: n kn' k 2 h g Z K I , (10b) Here n is mode number, αn are unknown coefficients for each mode, hw is height of waveguide, k is total propagation constant, η is impedance of the free space, and H0(2) are Hankel functions of second kind of order zero. In addition, N is maximum number of modes and kn is equal to: 2 Mode matching approach implies that we have to match modes in (15a) and (15b) with test functions. The resulting matrix equation is: (14a) w n z m z Ei m, n J 0 kn a cos cos dz , hw hw 0 h and matrix Hi with: hw m n z k H i m, n J1 kn a cos z L dz . cos kn hw L hg (19a) (19b) In (17) Einc and Hinc are amplitudes of incident electric and magnetic field, respectively. B. Row of cylinders in parallel plate waveguide To get closer to a full periodical structure we have to add one row of cylinders. Top view of one row is presented in Fig. 9. 1 n z cos( ) J1 kn . (14b) kn hw where Einc,n are amplitudes of outside incident electric field and J0 and J1 are Bessel functions of order 0 and order 1 respectively. After we have defined all fields, boundary conditions can be applied at the edge of the cylinder (ρ=a), but only in the region above the cylinder: Escat Einc Egap , (15a) Figure 9. One row of cylinders between parallel plates H scat H inc H gap . (15b) When we have more than one cylinder we have to include coupling between them and therefore we have to modify 4 expressions for scattered electric and magnetic fields between parallel plates: N n z 2 jqcos0 p (20a) Escat ncos , H 0 ( q )e h q n 0 w H scat j k N 1 k q n0 n z 2 jqcos 0 p . (20b) H 0 ( q )e hw n cos n Here ρq is radial position of each cylinder and φ0 is angle illustrated in Fig. 9. These series converge very slowly so we used [7] to accelerate them. C. Full periodical structure Main idea to get full periodical structure is to observe each row of cylinders as an individual scatterer, as it is in [8]. Every scatterer is fully described with the scattering matrix. Since all rows are equal the scattering matrix will be equal for all of them. The scattered field of each row can therefore be obtained by multiplying the scattering matrix and the matrix of the incident field on that row. Total incident field that impinges at some row is the sum of outside incident field and scattered fields from all other rows placed before that row: Q Eiin Tik SEkin Ti 0 EI . (21) Figure 10. Dispersion diagram for the gap waveguide calculated using Mode Matching approach and compared to CST simulation results. In comparison with the MoM approach results shown in Fig. 7 it is clear that the Mode matching approach results are in much better agreement with the referent CST MS results. For that reason all further results dealing with the realistic gap waveguide structures will be obtained using the analysis based on Mode Matching approach. k 1 ik Here, Tik is the translation matrix that translates scattered electric field of one row to another row, S is scattering matrix of certain row, Ti0 is translation matrix of incident field, and EI is the outside incident field that impinges on the whole structure. Finally, the same matrix equation is obtained: Z 2 K 2 I 2 , (22) where K2 is a vector of unknown coefficients for the incident field in each row. From these coefficients scattered field of each row of pins at a desired location can be obtained. If we observe total electric field at some points along the structure we can find phase difference between them, from which we can calculate propagation constants for each frequency of interest and present them in the dispersion diagram. Fig. 10 showns the dispersion diagram of the same structure as in Fig. 7. IV. GAP WAVEGUIDE, REAL STRUCTURE In all our examples before, the observed structure was infinite and we performed summations through all elements in a row. When we have a realistic structure we have only a few elements in each row and therefore the summation through all elements will not converge. For this reason we have to observe each element as individual scatterer. We will use expressions from (10a) to (14b). In Fig. 11 is presented dispersion diagram of the real structure where we used twenty rows of cylinders wherein each row contains 9 elements (central element plus four on both sides). Figure 11. Dispersion diagram for the realistic gap waveguide calculated using Mode Matching 5 From Fig. 11 we can see that there is slightly propagation in stop band, i.e. propagation constant is not equal to zero in stop band like it is shown in Fig. 10. This is mostly due to the fact that we have here a finite structure. A. Groove gap waveguide As it is mentioned in the Introduction, gap waveguide structure guides electromagnetic waves if we make some structural modifications like ridges, strips or grooves. This chapter is oriented on groove gap waveguide shown in Fig. 12. Main idea is to guide electromagnetic wave along the groove for frequencies in stop band, thus relaying on the stop band properties to prevent energy leakage from the groove. (a) (b) Figure 12. Groove gap waveguide. (a) top view , (b) side view Propagation constant for rectangular waveguide is equal [10]: 2 f k prop k0 1 c , f (23a) where fc is cut-off frequency of rectangular waveguide and is equal to: fc mn 2 m n , w hw 2 1 2 m, n 0,1, 2 (23b) For TE10 mode, in (23b) m is equal to one, and n is equal to zero. In Fig. 15 is presented dispersion diagram of the groove gap waveguide in stop band and we can see a good agreement between our results and results obtained using CST. However, there is difference between groove waveguide results and normal waveguide, which indicates that the fields in these two structures are not the same, because cylinders are very thin and distance between them is relatively large compared to their dimension, so groove does not act as rectangular waveguide (cylinders does not imitate side walls of r.w.). From Fig. 12 it can be seen that the groove is surrounded by cylinders from both sides and thus only propagation along the groove is allowed in the form of propagating TE/TM modes similar to the modes in a normal rectangular waveguide. 1) First design In this first design we used the same dimensions of the structure as before: length of the cylinder L is 8.66 mm, height of the waveguide h is 12.16 mm and the period of the pins is 3.75 mm. Groove length has to be equal around λ/2 and in order to have propagation above 8 GHz, dimension W from Fig. 12(b) was calculated to be equal 18.66 mm. In Fig. 13 is presented CST design of the structure. Figure 15. Dispersion diagram in stop band of groove gap waveguide Figure 13. First design of groove gap waveguide in CST The results of our approach are also compared with analytical calculation, which are calculated for rectangular waveguide (r.w.) with dimensions as in Fig.14. Further, electric and magnetic fields in the cross section of the groove gap waveguide were calculated in order to compare them with the fields in the normal rectangular waveguide. Field expressions for a rectangular waveguide are as in [11]: Ez E0cos( Hx Hy k x jk y w E0sin( y )e jkx x , w E0cos( y )e jkx x , w y )e jkx x , (24a) (24b) (24c) Figure 14. Front side view of rectangular waveguide 6 k x k 2 k y2 . (24d) Fig. 16 presents a comparison of absolute values of electric field in groove gap waveguide and rectangular waveguide. Electric field in rectangular waveguide is normalized with maximum value of electric field in groove gap waveguide. 2) Second design In the second design we used dimensions of gap waveguide to have stop band between 10 GHz and 20 GHz. Changing the length of the cylinder and height of the waveguide, width and marginal frequencies of the stop band can be approximately calculated. Obtained dimensions are: length of the cylinder L is 5.00 mm, height of the waveguide h is 6.00 mm, radius of the cylinder a is 1.63 mm, period of the cylinders p is 6.50 mm and width of the groove w is 15 mm. In Fig. 18 is presented second design. Figure 18: Second design of groove gap waveguide in CST In Fig. 19 is presented dispersion diagram of that structure in stop band. Figure 16. Electric fields in groove gap waveguide and in rectangular waveguide We can see from Fig. 16 that shapes of electric fields are almost equal. Using electric field two components of magnetic fields were calculated and also compared with analytic solutions. Results are presented in Fig. 17. Figure 19. Dispersion diagram in stop band of groove gap waveguide From Fig. 19 we can see a good agreement between all the results. This indicates that the groove waveguide in this design behaves almost identically as the normal rectangular waveguide. This is further verified with electric and magnetic field plots shown in Figs. 20 and 21. Better agreement in this case is a consequence of the fact that this design with a smaller gap between the pins and the top plate confines the energy much better than the previous design. Figure 17. Magnetic fields in groove gap waveguide and in rectangular waveguide Fig. 17 shows that there is a partial agreement between the results. Significant difference occurs at the edges of the groove because in that area the field is not completely contained due to the fact that there is a small leakage of energy through the pins (since this is not an ideal soft surface). 7 Figure 22. Single element slot in gap waveguide technology [10] VI. Figure 20. Electric fields in groove gap waveguide and in rectangular waveguide Figure 21. Magnetic fields in groove gap waveguide and in rectangular waveguide From Fig. 20 and Fig. 21 can be seen a good agreement between all results. As it is shown in Fig. 17, also here significant difference occurs at the edge of the groove in groove gap waveguide, because in that area the influence of cylinders occurs. V. CONCLUSION Rigorous and efficient method for the analysis of gapwaveguide structures based on the bed-of-nails as the artificial magnetic conductor is presented in this paper. Initial attempt was to use Method of Moments approach to determine the currents and fields inside the structure for single and multiple cylinders. To determine the solution for the periodic array of pins Bloch theory is used to determine the ABCD parameters matrix. Result for this approach was unsatisfactory and therefore another approach based on Mode Matching was pursued. The field in the gap was expanded in to a set of cylindrical modes and so was the field in the waveguide. Using Mode Matching, boundary conditions on the boundary between the gap and the waveguide were applied. From this, dispersion diagram and scattered field could be obtained. Determined dispersion diagrams were compared to full-wave simulation with commercial software and have shown excellent agreement. Also comparison was made between groove gap waveguide and rectangular waveguide. It showed that field confinement inside groove gap waveguide largely depends on the selection of bed of nails parameters. Mode Matching method used to obtain these results was shown to be a very efficient and accurate approach for initial studies of gap waveguide structures. ACKNOWLEDGMENT This work was supported in part by the project ''Passive and Active Metamaterial Structures for Guiding, Scattering and Radiation of Electromagnetic Energy", Unity through Knowledge Fund (UKF), Croatia, 2013. FUTURE WORK: GAP WAVEGUUDE ANTENNAS Gap waveguides are commonly used to feed arrays of antennas. The most common antenna placed on the top plate of gap waveguide is a slot antenna. Antenna is fed by a ridge in ridge gap waveguide as in [10], or with groove in groove gap waveguide. For this reason, the ridge and the slot antenna are two new objects that will be introduced into this analysis approach in future work. Example of a slot antenna placed on a top plate of the gap waveguide is presented in Fig. 22. REFERENCES [1] [2] [3] P.-S. Kildal, E. Alfonso, A. Valero-Nogueira, E. Rajo-Iglesias, “Local metamaterial-based waveguides in gaps between parallel metal plates”, IEEE Antennas and Wireless Propagation letters (AWPL), Volume 8, pp. 84-87, 2009. E. Rajo-Iglesias, A. Uz Zaman, P.-S. Kildal, “Parallel plate cavity mode suppression in microstrip circuit packages using a lid of nails”, IEEE Microwave and Wireless Components Letters, Aug. 2009. E. Pucci, “Gap Waveguide tehnology for Millimeter Wave Applications and Integration with Antennas”, Thesis for the degree of Doctor of Phylosophy, Goteborg, Sweeden, 2013 8 B. Tomasic, “Electric and magnetic sources in the parallel plate waveguide”, IEEE Transaction on Antennas and Propagation, Vol. AP37, No. 11, November 1987. [5] J. Bartolić, Mikrovalna elektronika, Graphis 2011. [6] C.A.Balanis, Advanced Engineering, Electromagnetics, John Wiley & Sons, Inc. 1996 [7] B. Tomasic, “Linear Array of Coaxially Fed Monopole Elements in a Parallel Plate Waveguide-Part I: Theory”, IEEE Transaction on Antennas and Propagation, Vol. AP-36, No. 4, November 1988. [8] H. Esteban. S. Cogollos, C. Bachiller. A. A. San Blas, V. Bona, “A new analytical method for the analysis of multiple scattering problemsusing spectral techniques”, Antennas and Propagation Society International Symposium, 2002. IEEE, vol 2, 82-85, 2002. [9] J.L.Volakis, “Antenna Engineering Hanbook”, chap.9,” Waveguide Slot Antenna Arrays”, The McGraw-Hill Professional, 2007 [10] P.-S. Kildal, A.U. Zaman, “Slot Antenna in Ridge Gap Waveguide Technology”, 6th European Conference on Antennas and Propagation (EUCAP), 3243-3244, 26-30 March 2012 [11] D.G.Dudley, “Mathematical Foundations for Electromagnetic Theory, Wiley-IEEE Press; 1 edition, 1994 [12] M. Bosiljevac, Z. Sipus, P-S. Kildal, "Efficient spectral domain Green's function analysis of novel metamaterial bandgap guiding structures," [4] [13] [14] [15] [16] [17] [18] [19] Proceedings of the Fourth European Conference on Antennas and Propagation (EuCAP), 2010 , pp.1,4, 12-16 April 2010. E.R.Iglesias, P.-S. Kildal, “Groove Gap Waveguide: A Rectangular Waveguide Between Contactless Metal Plates Enabled by Parallel-Plate Cut-Off, Proceedings of the Fourth European Conference on Antennas and Propagation (EuCAP), 2010, pp.1,4,12-16 April 2010 A. Polemi, S. Maci, P.-S. Kildal, “Dispersion Characteristics of a Metamaterial-Based Parallel-Plate Ridge Gap Waveguide Realized by Bed of Nails, IEEE Transaction on Antennas and Propagation, vol. 59, no.3, march 2011. H.Esteban, V.E. Boria, M.Baquero, M.Ferrando, “Generalised iterative method for solving 2D multiscattering problems using spectral techniques, Proceedings of the Microwaves, Antennas and Propagation, pp.73,80, April 1997 P.L.Sullivan, “Analysis of an Aperture Coupled Microstrip Antenna”, IEEE Transaction on Antennas and Propagation, Vol. AP-34, No.8, August 1986 R.W. Lyon, A.J. Sangster, “Efficient moment method analysis of radiating slots in a thick-walled rectangular waveguide”, Microwaves, Optics and Antennas, IEE Proceedings H, pp.197,205, August 1981 CST Microwave Studio 2012, software for simulation of electromagnetic problem Matlab R2012a, tool for tehnical computing 9

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