Investigation on Substrate-Integrated Rectangular Waveguide Short-Circuit Load and Equivalent Rectangular Waveguide Short-Circuit Load
Lei Xu, Wenquan Che, Liang Geng, Dapeng Wang, Kuan
Department of Electrical Engineering Nanjing University of Science & Technology210094 Nanjing, China
Abstract- Substrate integrated rectangular waveguide (SIRW) is an artificial rectangular waveguide (RW) constructed in planar substrate with two rows of periodic metallized posts or slots. In this letter, the theoretical and simulation results are given for the SIRW short load and RW short load, good agreements have been observed
Here R is the cylinder radius and W is the separation between adjacent cylinders. This formula indicates the width equivalence relation of the original SIRW and its equivalent RW. B. The SC (Short-Circuit) Location Equivalence Two SC walls form a cavity; one SC wall then forms a regular short-circuit that we need. The short circuit of our interest here is along the waveguide, the longitudinal direction z. The formula of the equivalent length along z in  is
z0 ' = 1
I. INTRODUCTION The rectangular waveguide structures, as we know, take many advantages over other planar transmission lines for millimeter band applications, but its implementation in planar form seems to be difficult due to its 3D geometry. This difficulty was resolved a few years ago by the proposed SIRW. It is an artificial waveguide structure that is fabricated in planar substrate with periodic metallic cylinders, and preserves much of the properties of a conventional rectangular waveguide (RW)  . A number of passive and active SIRW circuits have then been constructed and demonstrated significant performances . Some empirical formulas have been obtained to give the width equivalence between the original SIRW of cylinder wall and its equivalent RW with solid wall [4-9]. As we know, the simplest circuit component of rectangular waveguide is a short-circuit load, and for SIRW it is formed by a shorting cylinder wall. Following the width equivalence of RW from SIRW , this paper finds the location equivalence of RW short-circuit load (of solid wall) from SIRW short-circuit load (of cylinder wall). II. ANALYTICAL FORMULAS
? βW W ? arcctg ? ln ? 4R ? ? 4
Or the location movement Δz0 between the cylinder shortcircuit wall and the equivalent short-circuit of the solid wall is simply
Δz 0 = z 0 '? z 0 = 1? ? βW W ? π ? ? arcctg ? ln ?? ? 4R ? 2 ? β? ? 4 ? ?
The β in (3) of the traveling wave along z of the waveguide is a function of the frequency ω and the dielectric constant εr of the SIRW substrate, i.e.
β = ω 2 ε r ε 0 μ 0 ? (π / a )2
III. RESULTS AND DISCUSSIONS
A. The Width Equivalence between SIRW and RW In the fundamental TE10 mode, the width equivalence between widths a’ and a, of substrate integrated rectangular waveguide (SIRW) and rectangular waveguide (RW), has been derived in  through the analytical MoM:
a' = 2a
? πW W ? arcctg ? ln ? ? 4a 4 R ?
Here, one SIRW short-circuit load and its equivalent shortcircuit load has been investigated. Fig. 1a is the top view of a SIRW short-circuit load, with width a’ and length L’. The equivalent RW short-circuit load is illustrated in Fig. 1b with equivalent width a and equivalent length L. Noting that, the width a is obtained from (1), while the length L is related to (3), that is: L=L’-Δz0.
implying the accuracy of the width equivalence between the SIRW and RW structures. The input impedances of the SIRW and RW short-circuit loads in two frequency bands have been investigated. The following formulas are used to calculate the input impedances of the SIRW and RW short-circuit loads:
sc ( SIRW ) ' Z in ( L' ) = jZ 0 tan( β ' L' )
Fig.1 (a) the top view of an SIRW short-circuit load
sc ( RW ) Z in ( L ) = jZ 0 tan( β L )
Fig.1 (b) the schematic geometry of an RW short-circuit load
40 Characteristic Impedance(Ohm) 35 30 25 20 15 10 5 8 10 12 14 16 18 20 Rectangular Waveguide (Theory) SIRW (Simulation)
Where Zinsc (SIRW) is the input impedance of the SIRW shortcircuit load, Zinsc (RW) is the input impedance of the RW shortcircuit load. L’ is the length of the SIRW shot-circuit load, L is the length of the RW short-circuit load, as illustrated in Fig. 1. Noting that, the length L of the RW short-circuit load is found ' by L=L’-Δz0. β is the propagation constant of the SIRW, directly obtained from the HFSS simulation, while β is the propagation constant of the equivalent RW, which is calculated using formula (4). As we know, for the terminals of the SIRW and RW are shorted with cylinder wall and solid wall, respectively, the input impedances of these two shortcircuit loads are purely imaginary, only the input reactance is illustrated below. Noting that, in (1a), the characteristic impedance Z0’ is obtained directly from the HFSS simulation of the SIRW, however, in 1(b), the characteristic impedance Z0 is obtained from the waveguide theory . Fig. 3 illustrates the input reactance of the SIRW shortcircuit load and RW short-circuit load in the frequency range 11~14GHz. The length of the SIRW short-circuit load L’=12.569mm, the equivalent RW short-circuit load L=L’Δz0=12.5mm. The dielectric constant of the substrate
70 60 Input Reactance(Ohm) 50 40 30 20 10 0 -10 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 Rectangular Waveguide SIRW
Fig.2 Characteristic impedance of the SIRW and its equivalent RW. The dielectric substrate
ε r = 2.33 ,
SIRW width a’=12.138mm, the cylinder
radius R=0.3mm, cylinder separation W=1mm
An example of SIRW short-circuit load is now taken, the dielectric constant of the substrate ε r = 2.33 , substrate height h=0.254mm, width a’=12.138mm, the cylinder radius R=0.3mm, cylinder separation W=1mm. The characteristic impedance Z0’ of the SIRW can be directly obtained by HFSS simulation, and the characteristic impedance Z0 of the TE10 mode of the rectangular waveguide can be calculated using the well-known formula of wave impedance . The simulated characteristic impedance of the SIRW and the theoretical results of the equivalent RW are illustrated in Fig.2. Obviously, good agreement could be observed,
Fig.3 Input reactance of the SIRW short-circuit load and RW short-circuit load in frequency range 11~14GHz. The dielectric substrate ε r = 2.33 , SIRW width a’=12.138mm, the cylinder radius R=0.3mm, cylinder separation W=1mm, the length of the load L’=12.569mm
ε r = 2.33 , substrate height h=0.254mm, SIRW width a’=12.138mm, the cylinder radius R=0.3mm, cylinder separation W=1mm. Good agreement of the input reactance can be observed, which demonstrates the validity of the equivalence formulas (2) and (3), of the short-circuit load from SIRW to RW. Another example of SIRW short-circuit load is also taken in frequency range 14~20GHz, with length L’=13.569mm, the other parameters are the same with Fig.3. The input impedances of the SIRW short-circuit load and RW shortcircuit load are obtained based on formula (5) and the reactance are illustrated in Fig. 4. Similar agreement is still observed between the short-circuit loads of SIRW and RW structures, these results have further demonstrated the accuracy of the formulas (1) and (2).
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20 10 Input Reactance(Ohm) 0 -10 -20 -30 -40
Rectangular Waveguide SIRW
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Fig.4 Input reactance of the SIRW short-circuit load and RW short-circuit load in frequency range 14~19GHz., the length of the load L’=13.569mm.
IV. CONCLUSIONS Some investigations on the characteristic impedances and input impedances of the SIRW and RW short-circuit loads have been made; the results have further demonstrated the validity and accuracy of the equivalence formulas derived. The equivalence formula is simple and accurate. Such simplicity and accuracy are convenient for the design of small waveguides, of millimeter wave, in a multilayer circuit structure, such as the popular LTCC of the present time. ACKNOWLEDGMENT The authors express their gratitude to the financial support of the National Science Foundation of China under Grant 60471025, the Natural Science Foundation of Jiangsu Province under Grant BK2004135.