Analytical Calculation of the Modes in a Substrate Integrated Waveguide
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Integrated Waveguide
Adan S. Sanchez, Gerardo Romo Systems Research Center-Mexico Intel, Mexico Tlaquepaque, Jal. Mexico adan. sanchez@
Abstract- A first order analytical method is developed to analyze the guided modes in substrate integrated waveguides. The complex propagation constant is derived as a closed expression which is a function of frequency, and geometrical and material parameters. This single equation is able to describe multimode and band gap effects in substrate integrated waveguides. Therefore, the initial design procedure can be greatly simplified by using this analytical model as starting point.
/62= j
16_
h )
(8)
,e-A Etl(p,z) (Fn G(z-IT) + B1"neJA (z-IT))Em,n(p) (1) Ht1(p,z) (F7n e A3(z-IT) -Bmn jeA(z-IT) )Hm,n (p) (2) H1l (p,z) = (F7,n ejA (z-lT) + B_ln7nejA (z-IT) )H7;n (p),
I.
Howard Heck Signaling Technologies Intel Corporation Hillsboro, OR 97124 USA
Bloch's theorem, we obtain an eigenvalue system whose solution provides the propagation constant presented in Sec. III. Finally, Section IV discusses and compares the analytical result with those obtained by 3-D electromagnetic simulations.
Figure 1. Substrate integrated waveguide. Top and bottom ground planes confine the signal in the vertical direction, and two rows of vias confine the field laterally. It can be fabricated using printed circuit board technology
1-4244-0041-4/06/$20.00 ©2006 IEEE.
219
as regions 1 and 2 respectively. We can expand the TEpolarized fields in region (1) of the l-th unit cell in terms of the modes of a rectangular waveguide, E 'n(p), H ,n (p), n H' (p) . Here p is a vector in the xy plane, and the indices t and z denote, respectively, transverse to and along the propagation direction. Also, m and n identify the mode in the rectangular waveguide. Next, we express the TE fields as a sum of forward and backward propagating modes of the rectangular waveguide,
Proceedings of the 6th International Caribbean Conference on Devices, Circuits and Systems, Mexico, Apr. 26-28, 2006
Analytical Calculation of the Modes in a Substrate
In this work, the modes supported by the structure shown in Fig. 2(a) are analytically calculated using a modematching technique. The conductors are assumed to be perfect and the dielectric filling the waveguide is assumed to be lossless and dispersionless. The unit cell can be viewed as composed of two regions, a conventional rectangular waveguide and a parallel plate waveguide, which are denoted
Iபைடு நூலகம்.
MODE MATCHING
INTRODUCTION
Traditionally, chip to chip interconnects have been realized using microstrips and striplines in printed circuit boards. However, at frequencies in the gigahertz range, the signal is degraded by dielectric and radiation losses as well as by crosstalk. To overcome these limitations, waveguiding structures with sidewalls made of periodic arrays of metallic vias have been proposed [1-3]. In this paper, such structures are referred to as substrate integrated waveguides. Structures with a wide bandwidth centered at 50 GHz and fabricated using standard printed circuit board technology have been reported [4]. Due to the high frequency of operation and the complexity of the structure's geometry, high-resolution three-dimensional (3-D) finite-difference time-domain (FDTD) calculations have been required to obtain the transmission characteristics. While the numerical computations can be very accurate, they are time and memory consuming and rarely give physical insight about the effect of all design parameters on the performance of the waveguide. In this work, we present the analytic calculation of the modes that can propagate in a periodic waveguide that has the same fundamental behavior as the substrate integrated waveguide. Namely, the propagation constant is presented as a closed analytical expression that is a function of frequency, and the geometrical and material parameters of the waveguide. The derived equation predicts, straightforwardly, the multimode and band gap behaviors observed in substrate integrated waveguides. In section II, the modes are analyzed applying a mode matching technique. By applying the
where the propagation constant of the rectangular waveguide is given by
It is important to note that the fields in region 2 can be considered independent of the x variable if W >> h. The next step is to apply the boundary conditions to the electric and magnetic fields at the interfaces between regions 1 and 2, namely, at z = (I-1)T and at z=(l-1)T + a. The continuity of the transverse component of the electric and magnetic fields at z = (l-])T leads to
The substrate integrated waveguide is composed of two rows of vias which confine the fields in the lateral direction and top and bottom ground planes which confine the fields in the vertical direction, see Fig. 1. Replacing the metallic cylinders by rectangles, the mathematical analysis is simplified while the fundamental behavior of the waveguide is preserved. Thus the resulting structure can be viewed as a conventional rectangular waveguide with periodic slots on the lateral walls, as shown in Fig. 2(a). The width and the height of the waveguide are denoted by W and h respectively. On the lateral walls, a and b denote the length of the metal and that of the slot in the z direction, respectively.
Adan S. Sanchez, Gerardo Romo Systems Research Center-Mexico Intel, Mexico Tlaquepaque, Jal. Mexico adan. sanchez@
Abstract- A first order analytical method is developed to analyze the guided modes in substrate integrated waveguides. The complex propagation constant is derived as a closed expression which is a function of frequency, and geometrical and material parameters. This single equation is able to describe multimode and band gap effects in substrate integrated waveguides. Therefore, the initial design procedure can be greatly simplified by using this analytical model as starting point.
/62= j
16_
h )
(8)
,e-A Etl(p,z) (Fn G(z-IT) + B1"neJA (z-IT))Em,n(p) (1) Ht1(p,z) (F7n e A3(z-IT) -Bmn jeA(z-IT) )Hm,n (p) (2) H1l (p,z) = (F7,n ejA (z-lT) + B_ln7nejA (z-IT) )H7;n (p),
I.
Howard Heck Signaling Technologies Intel Corporation Hillsboro, OR 97124 USA
Bloch's theorem, we obtain an eigenvalue system whose solution provides the propagation constant presented in Sec. III. Finally, Section IV discusses and compares the analytical result with those obtained by 3-D electromagnetic simulations.
Figure 1. Substrate integrated waveguide. Top and bottom ground planes confine the signal in the vertical direction, and two rows of vias confine the field laterally. It can be fabricated using printed circuit board technology
1-4244-0041-4/06/$20.00 ©2006 IEEE.
219
as regions 1 and 2 respectively. We can expand the TEpolarized fields in region (1) of the l-th unit cell in terms of the modes of a rectangular waveguide, E 'n(p), H ,n (p), n H' (p) . Here p is a vector in the xy plane, and the indices t and z denote, respectively, transverse to and along the propagation direction. Also, m and n identify the mode in the rectangular waveguide. Next, we express the TE fields as a sum of forward and backward propagating modes of the rectangular waveguide,
Proceedings of the 6th International Caribbean Conference on Devices, Circuits and Systems, Mexico, Apr. 26-28, 2006
Analytical Calculation of the Modes in a Substrate
In this work, the modes supported by the structure shown in Fig. 2(a) are analytically calculated using a modematching technique. The conductors are assumed to be perfect and the dielectric filling the waveguide is assumed to be lossless and dispersionless. The unit cell can be viewed as composed of two regions, a conventional rectangular waveguide and a parallel plate waveguide, which are denoted
Iபைடு நூலகம்.
MODE MATCHING
INTRODUCTION
Traditionally, chip to chip interconnects have been realized using microstrips and striplines in printed circuit boards. However, at frequencies in the gigahertz range, the signal is degraded by dielectric and radiation losses as well as by crosstalk. To overcome these limitations, waveguiding structures with sidewalls made of periodic arrays of metallic vias have been proposed [1-3]. In this paper, such structures are referred to as substrate integrated waveguides. Structures with a wide bandwidth centered at 50 GHz and fabricated using standard printed circuit board technology have been reported [4]. Due to the high frequency of operation and the complexity of the structure's geometry, high-resolution three-dimensional (3-D) finite-difference time-domain (FDTD) calculations have been required to obtain the transmission characteristics. While the numerical computations can be very accurate, they are time and memory consuming and rarely give physical insight about the effect of all design parameters on the performance of the waveguide. In this work, we present the analytic calculation of the modes that can propagate in a periodic waveguide that has the same fundamental behavior as the substrate integrated waveguide. Namely, the propagation constant is presented as a closed analytical expression that is a function of frequency, and the geometrical and material parameters of the waveguide. The derived equation predicts, straightforwardly, the multimode and band gap behaviors observed in substrate integrated waveguides. In section II, the modes are analyzed applying a mode matching technique. By applying the
where the propagation constant of the rectangular waveguide is given by
It is important to note that the fields in region 2 can be considered independent of the x variable if W >> h. The next step is to apply the boundary conditions to the electric and magnetic fields at the interfaces between regions 1 and 2, namely, at z = (I-1)T and at z=(l-1)T + a. The continuity of the transverse component of the electric and magnetic fields at z = (l-])T leads to
The substrate integrated waveguide is composed of two rows of vias which confine the fields in the lateral direction and top and bottom ground planes which confine the fields in the vertical direction, see Fig. 1. Replacing the metallic cylinders by rectangles, the mathematical analysis is simplified while the fundamental behavior of the waveguide is preserved. Thus the resulting structure can be viewed as a conventional rectangular waveguide with periodic slots on the lateral walls, as shown in Fig. 2(a). The width and the height of the waveguide are denoted by W and h respectively. On the lateral walls, a and b denote the length of the metal and that of the slot in the z direction, respectively.