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Study on Transmission Characteristics in Three Kinds of Deformed Finlines Based on Edge-Based Finite Element Method

Study on Transmission Characteristics in Three Kinds of Deformed Finlines Based on Edge-Based... The influence of model deformation on the transmission characteristics in unilateral finline, antipodal finline and bilateral finline is discussed using edge-based finite element method (FEM). The deformation is considered with respect to the eight boundaries of the model, and their amplitude is set to 0.01–0.05; the transmission characteristics include the cutoff wavelength characteristics of the dominant mode and the single-mode bandwidth characteristics. The results show a decreasing trend when the vacuum area is deformed, while they show a variety of possibilities when the loading region is deformed. These numerical results have strong guiding significance for the influence of the deformation of finlines on the overall device. Keywords: Cutoff wavelength characteristics of dominant mode, single-mode bandwidth characteristics, deformed unilateral finline, deformed antipodal finline, deformed bilateral finline AMS 2010 code: 35J60. 1 Introduction As new millimetre-wave transmission lines, finlines are widely used in microwave devices, such as directional couplers, phase shifters and filters, because of their long cutoff wavelength characteristics and the single-mode bandwidth characteristics, small attenuation and loss, weak dispersion and easy connection with solid-state devices. Since Saad and Begemann [1] studied the antipodal finline in 1977, academic research on it has also been in a hot state. For example, in 1981, Beyer [2] studied the grounding finline. In 1983, Sharma and Hoefer [3] discussed the empirical formula of finline design. In 2002, Zheng and Song [4] and, in 2003, Lu and Leonard [5] used multilevel theory and node finite element method (FEM) to calculate the partial transmission Corresponding author. Email address: hend.elzefzafy@asu.edu.bh ISSN 2444-8656 doi:10.2478/amns.2022.1.00021 Open Access. © 2022 Sun et al., published by Sciendo. This work is licensed under the Creative Commons Attribution alone 4.0 License. 36 Sun and Elzefzafy. Applied Mathematics and Nonlinear Sciences 8(2023) 35–44 characteristics of finlines. In 2011, Sun and his team calculated the partial transmission characteristics of uni- lateral finline and antipodal finline [6, 7]. In recent years, many researchers have carried out a huge amount of research on the applications of finline [8–11]. Deformation of the model may occur during the actual use of any device, and the finline is no exception. Deformation of the geometric model of finlines will inevitably lead to a change of transmission characteristics. However, research on non-uniform deformation transmission lines has not been carried out. Therefore, we mainly calculate the cutoff wavelength characteristics of the dominant mode and the single-mode bandwidth characteristics in deformed unilateral finline, deformed antipodal finline and deformed bilateral finline by using the edge-based FEM, and the effects of deformation on these transmission characteristics are analysed in detail. 2 Theoretical analysis The cross-section of finlines before and after deformation is shown in Fig. 1; the white part is the vacuum region, the dielectric constant is ε ; the mesh part is the dielectric region, and the dielectric constant is ε ; and 0 r the black part is the fin. The position and size of the fins and the filling area are represented by the symbols a, b, s, t , t , d , d . The deformation amplitude of the model boundary is represented by σ − σ . 1 2 1 2 1 8 According to the Maxwell equation, the electric field and magnetic field in the finlines satisfy the following vector differential equations: ⃗ ⃗ ∇× E = − jωµ µ H (1) ⃗ ⃗ ∇× H = jωε ε E r 0 where ε , µ , ε and µ represent the permittivity of free space, the permeability of free space, the relative 0 0 r r permittivity and the relative permeability, respectively. In order to calculate the transmission characteristics of the finline, it is necessary to establish a functional vector formula based on the magnetic field as the working variable. Therefore, the vector Helmholtz equation based on magnetic field is obtained from Eq. (1): ⃗ ⃗ ∇× ∇× H − K µ H = 0. (2) n ˆ× (∇× H) = 0 on Γ (3) n ˆ× H = 0 on Γ (4) Γ and Γ denote conductive wall and magnetic wall, respectively. In the actual calculation process, the boundary 1 2 condition satisfied by the magnetic field is Eq. (3). After derivation, we get the following variational problem: δ F(H) = 0. (5) where 1 1 1 ∗ ∗ 2 ∗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ F(H) = ∫ ∫ (∇ × H).(∇× H ) + (∇ H + jk H )· (∇ H + jk H ) − µ k H · H dΩ (6) t t t Z Z t t Z Z t r 2 ε ε r r The following matrix eigenvalue equation is obtained by vector finite element discretisation: T ∗ T ∗ e e e e e e e h S (tt) S (t z) h h T (tt) 0 h t t 2 t t F = − k (7) ∑ e e e e 0 e e e h S (zt) S (zz) h h 0 T (zz) h z z z z e=1 where the elemental matrices are given by 1 β e e e T e e T ⃗ ⃗ ⃗ ⃗ S (tt) = ∫ ∫ (∇ × N )· (∇ × N ) dΩ+ ∫ ∫ (N )· (N ) dΩ, (8) t t e e ε e ε e Ω Ω r r Transmission characteristics in deformed finlines 37 Fig. 1 Cross-sections of undeformed and deformed samples of three kinds of finlines. (A) Unilateral finline, (B) Deformed unilateral finline, (C) Antipodal finline, (D) Deformed antipodal finline, (E) Bilateral finline and (F) Deformed bilateral finline. e e e T S (t z) = ∫ ∫ (N )· (∇ L ) dΩ, (9) ε e e e e T S (zt) = ∫ ∫ (∇ L )· (N ) dΩ, (10) ε e e e e T S (zz) = ∫ ∫ (∇ L )· (∇ L ) dΩ, (11) t t ε e e e e e T ⃗ ⃗ T (tt) = µ ∫ ∫ (N )· (N ) dΩ, (12) e 2 e e e T T (zz) = β µ ∫ ∫ (L )· (L ) dΩ. (13) Ω 38 Sun and Elzefzafy. Applied Mathematics and Nonlinear Sciences 8(2023) 35–44 Using global notation, Eq. (7) can be written as follows: T ∗ T ∗ e e e e e e e 1 1 h S (tt) S (t z) h h T (tt) 0 h t t 2 t t F = − k (14) e e e e 0 e e e h S (zt) S (zz) h h 0 T (zz) h 2 2 z z z z Using the Ritz method for variation, we can get the following: e e e e e S (tt) S (t z) h T (tt) 0 h t t = k . (15) e e e e e S (zt) S (zz) h 0 T (zz) h z z After synthesis, we can obtain the following expression: [S][φ] = k [T][φ]. (16) S (tt) S (t z) T (tt) 0 h where [S] = , [T] = ,[φ] = . S (zt) S (zz) 0 T (zz) h By solving Eq. (16), the transmission characteristics in the three kinds of deformed finlines can be calculated. 3 Results and discussion 3.1 Validation of method In this part, we first verify the correctness of the edge-based FEM with the magnetic field as the working variable. Therefore, we use the method derived in this paper to calculate the main mode cutoff wavelength of the unilateral finline; the computed results and comparisons between the literature data [3, 12] are shown in Table 1. The relative errors are within 2%, which shows that the calculation method in this paper is feasible. Then, we calculate the effects of deformation on the cutoff wavelength characteristics of dominant mode and single-mode bandwidth characteristics in unilateral finline, antipodal finline and bilateral finline. In the calcula- tion process, for the three kinds of finlines, we assume that ε = 2.55, b/a = 0.5, s/a = 0.3, d /a = 0.25, d /a = r 1 2 0.05, t /a = 0.01, t /a = 0.05 and the dielectric substrate is located in the middle of the model. The geometric 1 2 deformation amplitude of the cross-section in the three kinds of finlines is represented by σ /a, σ /a, σ /a, 1 2 3 σ /a, σ /a, σ /a, σ /a and σ /a and their variation range is set to 0.01–0.05. 4 5 6 7 8 Table 1 Comparisons of calculation results of cutoff wave numbers of unilateral finline ( b/a = 0.5, d/b = 0.25, t/a = 0.01). εεε s/a This Article b b b/λλλ Ref. [3] b b b/λλλ Error (%) Ref. [12] b b b/λλλ Error (%) r r r c c c c c c c c c 2.22 0.25 0.1551 0.15457 0.34 0.15597 0.56 0.125 0.1605 0.16140 0.56 0.16218 1.04 0.0625 0.1688 0.16925 0.27 0.16996 0.68 3.0 0.25 0.1395 0.13908 0.30 0.14088 0.98 0.125 0.1479 0.14756 0.23 0.14884 0.63 0.0625 0.1581 0.15799 0.07 0.15992 1.14 Table 2 The values of the cutoff wavelength (λ /a) of dominant mode and bandwidth (λ /λ ) in three kinds of c2 c c1 undeformed finlines. Unilateral finline Antipodal finline Bilateral finline λλλ ///a a a (λλλ ///λλλ ) λλλ ///a a a (λλλ ///λλλ ) λλλ ///a a a (λλλ ///λλλ ) c c c c c c c c2 c c c c c2 c c c c c2 c c c1 c c c1 c c c1 2.9067 2.0202 2.8902 2.0342 3.1089 1.8939 Transmission characteristics in deformed finlines 39 3.2 Effect of deformation on the cutoff wavelength characteristics of dominant mode and single-mode bandwidth characteristics of the finlines When the three kinds of nlines fi are not deformed, the characteristics of the cutoff wavelength of dominant mode and the single-mode bandwidth characteristics are shown in Table 2. Any one or several deformations of σ /a, σ /a, σ /a, σ /a, σ /a, σ /a, σ /a and σ /a may occur when the finlines are used. Limited by space, 1 2 3 4 5 6 7 8 this paper focuses on the effects of any one and any two deformations of the eight σ /a on the transmission characteristics in the three kinds of finlines, and other changes can be calculated by analogy. The calculation results are presented in Tables 3–5. 3.3 Summary of the variation trend of transmission characteristics of three kinds of deformed finlines From Tables 2–5, we can draw the following conclusions. 1. As a whole, for the three kinds of finlines, both the values of the cutoff wavelength and the single-mode bandwidth decrease with the boundary deformation of the vacuum region, while they increase with the boundary deformation of the dielectric substrate. 2. In terms of deformation amplitude, with the increase of the amplitudes σ /a from 0.01 to 0.05, most of the cutoff wavelength and single-mode bandwidth show a decreasing trend. 3. There are four situations different from the above Rule 2: (1) In the first case, the cutoff wavelength and single-mode bandwidth increase as the deformation ampli- tude increases. For deformed unilateral finline, the details are as follows: σ /a, σ /a, σ /a = σ /a; 1 7 3 7 for deformed antipodal finline, the details are as follows: σ /a, σ /a, σ /a = σ /a; for deformed 1 7 3 7 bilateral finline, the details are as follows: σ /a, σ /a, σ /a = σ /a. 1 7 3 7 (2) In the second case, the cutoff wavelength changes little and the single-mode bandwidth decreases. For deformed unilateral finline, the details are as follows: σ /a = σ /a, σ /a = σ /a; for deformed 2 7 3 8 antipodal finline, the details are as follows: σ /a = σ /a; for deformed bilateral finline, the details 2 7 are as follows: σ /a = σ /a, σ /a = σ /a. 2 7 4 7 (3) In the third case, the cutoff wavelength changes little and the single-mode bandwidth increases. For deformed unilateral finline, the details are as follows: σ /a = σ /a, σ /a = σ /a; for deformed 2 3 7 8 antipodal finline, the details are as follows: σ /a = σ /a, σ /a = σ /a, σ /a = σ /a. 2 3 3 4 7 8 (4) In the fourth case, the cutoff wavelength decreases and the single-mode bandwidth changes little. For deformed unilateral finline, the details are as follows: σ /a = σ /a; for deformed bilateral finline, 3 4 the details are as follows: σ /a = σ /a, σ /a = σ /a. 3 4 7 8 4. Careful analysis of the above results shows that the situation in the third point includes the deformation of the dielectric substrate (includes σ /a or σ /a). It is concluded that the deformation of the dielectric 3 7 substrate has a great influence on the cutoff wavelength and single-mode bandwidth. 5. The effects of deformation on transmission characteristics in the three types of finlines are approximately the same. 4 Conclusion In this paper, we mainly study the effects of the deformation of unilateral finline, antipodal finline and bilateral finline on their transmission characteristics by using edge-based FEM. The cutoff wavelength of the dominant mode and single-mode bandwidth are calculated in detail. The results show that the deformation of the three types of finlines greatly changes their transmission characteristics, which are as follows: after the 40 Sun and Elzefzafy. Applied Mathematics and Nonlinear Sciences 8(2023) 35–44 boundary deformation of the vacuum region, both the cutoff wavelength of dominant mode and single-mode bandwidth mostly decrease; while after the boundary deformation of the loading region, the changes of the cutoff wavelength and single-mode bandwidth are more complex. These results will be used as references for the application of finlines in new microwave and millimetre-wave devices. Table 3 The changes of λ /a and λ /λ in deformed unilateral finline. c c1 c2 σ /a σ σ //a a 0.01 0.02 0.03 0.04 0.05 i i i λ /a λ /λ λ /a λ /λ λ /a λ /λ λ /a λ /λ λ /a λ /λ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 i = 1 2.9196 2.0043 2.9071 1.9942 2.8945 1.9838 2.8817 1.9732 2.8688 1.9621 i = 2 2.9269 2.0104 2.9213 2.0061 2.9158 2.0011 2.9106 1.9944 2.8816 1.9809 i = 3 2.9355 2.0350 2.9391 2.0505 2.9434 2.0620 2.9492 2.0712 2.9558 2.0794 i = 4 2.9210 2.0066 2.9100 1.9992 2.8992 1.9918 2.8882 1.9843 2.8775 1.9771 i = 5 2.9127 2.0010 2.8930 1.9873 2.8735 1.9736 2.8536 1.9596 2.8331 1.9453 i = 6 2.9178 2.0070 2.9074 2.0003 2.8960 1.9926 2.8851 1.9851 2.8765 1.9780 i = 7 2.9350 2.0368 2.9387 2.0537 2.9429 2.0659 2.9474 2.0749 2.9543 2.0833 i = 8 2.9059 2.0100 2.8983 2.0049 2.8944 1.9989 2.8888 1.9924 2.8834 1.9846 σ σ σ ///a a a, σ σ σ ///a a a 0.01 0.02 0.03 0.04 0.05 i i i j j j λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 i, j = 1, 2 2.9091 2.0023 2.8907 1.9872 2.8719 1.9717 2.8523 1.9550 2.8312 1.9349 i, j = 1, 3 2.9179 2.0274 2.9085 2.0341 2.8996 2.0386 2.8909 2.0420 2.8850 2.0430 i, j = 1, 4 2.8766 2.0047 2.8395 1.9834 2.8236 1.9719 2.7937 1.9495 2.7755 1.9323 i, j = 1, 5 2.9003 1.9909 2.8683 1.9672 2.8365 1.9435 2.8042 1.9193 2.7715 1.8946 i, j = 1, 6 2.9037 1.9987 2.8872 1.9800 2.8641 1.9625 2.8312 1.9425 2.8191 1.9260 i, j = 1, 7 2.9225 2.0270 2.9132 2.0349 2.9035 2.0388 2.8954 2.0418 2.8894 2.0446 i, j = 1, 8 2.9099 1.9981 2.8934 1.9830 2.8692 1.9699 2.8580 1.9509 2.8371 1.9321 i, j = 2, 3 2.9254 2.0337 2.9235 2.0500 2.9228 2.0630 2.9207 2.0748 2.9222 2.0828 i, j = 2, 4 2.8904 2.0063 2.8755 1.9940 2.8611 1.9727 2.8437 1.9680 2.8297 1.9415 i, j = 2, 5 2.9017 1.9967 2.8747 1.9780 2.8358 1.9577 2.8341 1.9401 2.8096 1.9205 i, j = 2, 6 2.8999 2.0072 2.8816 1.9958 2.8675 1.9778 2.8529 1.9727 2.8401 1.9533 i, j = 2, 7 2.9230 2.0312 2.9220 2.0403 2.9218 2.0376 2.9009 2.0339 2.9008 2.0170 i, j = 2, 8 2.9002 2.0055 2.8812 1.9974 2.8739 1.9833 2.8608 1.9723 2.8477 1.9613 i, j = 3, 4 2.8989 2.0339 2.8939 2.0449 2.8869 2.0517 2.8822 2.0574 2.8757 2.0609 i, j = 3, 5 2.8897 2.0280 2.8750 2.0326 2.8627 2.0315 2.8473 2.0299 2.8351 2.0298 i, j = 3, 6 2.9109 2.0329 2.9045 2.0398 2.8967 2.0424 2.8909 2.0430 2.8888 2.0444 i, j = 3, 7 2.9133 2.0654 2.9197 2.1150 2.9295 2.1689 2.9399 2.2237 2.9535 2.2693 i, j = 3, 8 2.9075 2.0272 2.9027 2.0338 2.9052 2.0335 2.9017 2.0316 2.9054 2.0257 i, j = 4, 5 2.8786 1.9968 2.8487 1.9772 2.8165 1.9534 2.7846 1.9317 2.7508 1.9078 i, j = 4, 6 2.8934 2.0037 2.8713 1.9879 2.8482 1.9722 2.8261 1.9576 2.8059 1.9413 i, j = 4, 7 2.8986 2.0291 2.8902 2.0360 2.8849 2.0400 2.8790 2.0405 2.8762 2.0403 i, j = 4, 8 2.8882 2.0038 2.8748 1.9925 2.8566 1.9797 2.8402 1.9665 2.8289 1.9520 i, j = 5, 6 2.8841 1.9976 2.8526 1.9751 2.8225 1.9545 2.7914 1.9323 2.7566 1.9082 i, j = 5, 7 2.8876 2.0227 2.8742 2.0249 2.8601 2.0254 2.8452 2.0219 2.8335 2.0202 i, j = 5, 8 2.8802 1.9954 2.8602 1.9786 2.8371 1.9597 2.8107 1.9401 2.7881 1.9196 i, j = 6, 7 2.9059 2.0324 2.8958 2.0406 2.8932 2.0487 2.8886 2.0530 2.8820 2.0548 i, j = 6, 8 2.8952 2.0041 2.8795 1.9909 2.8625 1.9780 2.8480 1.9632 2.8258 1.9471 i, j = 7, 8 2.9057 2.0321 2.9010 2.0481 2.9011 2.0622 2.9015 2.0744 2.9014 2.0827 Transmission characteristics in deformed finlines 41 Table 4 The changes of λ /a and λ /λ in deformed antipodal finline. c c1 c2 σ σ σ ///a a a 0.01 0.02 0.03 0.04 0.05 i i i λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 i = 1 2.8774 2.0200 2.8619 2.0086 2.8427 1.9999 2.8263 1.9875 2.8110 1.9757 i = 2 2.8866 2.0199 2.8828 2.0078 2.8767 1.9997 2.8707 1.9896 2.8638 1.9786 i = 3 2.8964 2.0488 2.9006 2.0585 2.9055 2.0649 2.9100 2.0714 2.9176 2.0758 i = 4 2.8860 2.0264 2.8779 2.0212 2.8691 2.0151 2.8587 2.0079 2.8485 2.0030 i = 5 2.8754 2.0232 2.8603 2.0114 2.8453 2.0001 2.8300 1.9883 2.8144 1.9767 i = 6 2.8920 2.0207 2.8876 2.0130 2.8801 2.0062 2.8749 1.9959 2.8671 1.9778 i = 7 2.8969 2.0481 2.9006 2.0571 2.9051 2.0630 2.9101 2.0679 2.9163 2.0733 i = 8 2.8923 2.0266 2.8831 2.0207 2.8737 2.0129 2.8644 2.0077 2.8546 2.0003 σ /a σ /a σ σ //a a, σ σ //a a 0.01 0.02 0.03 0.04 0.05 i i i j j j λ /a λ /λ λ /a λ /λ λ /a λ /λ λ /a λ /λ λ /a λ /λ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 i, j = 1, 2 2.8737 2.0063 2.8509 1.9879 2.8275 1.9684 2.8022 1.9475 2.7771 1.9257 i, j = 1, 3 2.8807 2.0385 2.8676 2.0386 2.8565 2.0334 2.8448 2.0298 2.8335 2.0260 i, j = 1, 4 2.8706 2.0154 2.8463 1.9991 2.8218 1.9817 2.7965 1.9637 2.7735 1.9479 i, j = 1, 5 2.8632 2.0081 2.8321 1.9859 2.8022 1.9631 2.7714 1.9394 2.7367 1.9200 i, j = 1, 6 2.8740 2.0086 2.8543 1.9889 2.8299 1.9711 2.8083 1.9504 2.7848 1.9292 i, j = 1, 7 2.8816 2.0355 2.8713 2.0372 2.8559 2.0362 2.8479 2.0355 2.8389 2.0317 i, j = 1, 8 2.8755 2.0137 2.8485 1.9942 2.8220 1.9768 2.7965 1.9579 2.7676 1.9414 i, j = 2, 3 2.8915 2.0398 2.8895 2.0491 2.8868 2.0531 2.8849 2.0555 2.8843 2.0578 i, j = 2, 4 2.8815 2.0098 2.8639 1.9939 2.8498 1.9796 2.8370 1.9641 2.8208 1.9462 i, j = 2, 5 2.8744 2.0043 2.8537 1.9866 2.8312 1.9669 2.8090 1.9475 2.7905 1.9266 i, j = 2, 6 2.8829 2.0089 2.8715 1.9970 2.8562 1.9842 2.8449 1.9714 2.8326 1.9579 i, j = 2, 7 2.8896 2.0270 2.8892 2.0251 2.8872 2.0194 2.8843 2.0133 2.8855 2.0040 i, j = 2, 8 2.8777 2.0176 2.8689 2.0008 2.8521 1.9875 2.8339 1.9735 2.8177 1.9576 i, j = 3, 4 2.8856 2.0479 2.8852 2.0522 2.8805 2.0586 2.8779 2.0625 2.8756 2.0678 i, j = 3, 5 2.8813 2.0394 2.8711 2.0413 2.8596 2.0390 2.8516 2.0380 2.8416 2.0359 i, j = 3, 6 2.8924 2.0326 2.8871 2.0305 2.8878 2.0255 2.8869 2.0189 2.8866 2.0120 i, j = 3, 7 2.8993 2.0781 2.9057 2.1245 2.9124 2.1699 2.9242 2.2043 2.9395 2.2370 i, j = 3, 8 2.8938 2.0394 2.8867 2.0399 2.8831 2.0384 2.8787 2.0370 2.8757 2.0353 i, j = 4, 5 2.8713 2.0152 2.8464 1.9959 2.8193 1.9797 2.7942 1.9613 2.7696 1.9421 i, j = 4, 6 2.8813 2.0152 2.8637 2.0035 2.8475 1.9903 2.8321 1.9773 2.8139 1.9627 i, j = 4, 7 2.8900 2.0412 2.8843 2.0413 2.8788 2.0409 2.8737 2.0406 2.8720 2.0376 i, j = 4, 8 2.8835 2.0209 2.8655 2.0097 2.8477 1.9996 2.8295 1.9910 2.8110 1.9797 i, j = 5, 6 2.8743 2.0090 2.8515 1.9902 2.8287 1.9715 2.8057 1.9513 2.7803 1.9315 i, j = 5, 7 2.8823 2.0360 2.8712 2.0357 2.8589 2.0325 2.8480 2.0282 2.8408 2.0255 i, j = 5, 8 2.8780 2.0148 2.8536 1.9970 2.8278 1.9808 2.8027 1.9646 2.7787 1.9460 i, j = 6, 7 2.8922 2.0422 2.8900 2.0503 2.8863 2.0537 2.8862 2.0555 2.8876 2.0583 i, j = 6, 8 2.8892 2.0133 2.8727 1.9977 2.8549 1.9818 2.8399 1.9660 2.8250 1.9491 i, j = 7, 8 2.8947 2.0475 2.8887 2.0540 2.8842 2.0613 2.8804 2.0656 2.8768 2.0681 Acknowledgments This work was supported by the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (grant number 2020QYY03) and supported by the Open Research Fund Program of Data Recovery Key Laboratory of Sichuan Province (grant number DRN2108) and 42 Sun and Elzefzafy. Applied Mathematics and Nonlinear Sciences 8(2023) 35–44 Table 5 The changes of of λ /a and λ /λ in deformed bilateral finline. c c1 c2 σ σ σ ///a a a 0.01 0.02 0.03 0.04 0.05 i i i λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 i = 1 3.0963 1.8855 3.0798 1.8801 3.0642 1.8748 3.0455 1.8695 3.0281 1.8641 i = 2 3.1059 1.8866 3.0970 1.8801 3.0856 1.8768 3.0768 1.8709 3.0660 1.8644 i = 3 3.1164 1.8986 3.1170 1.9010 3.1209 1.9041 3.1246 1.9090 3.1263 1.9090 i = 4 3.1083 1.8846 3.0962 1.8809 3.0880 1.8770 3.0772 1.8710 3.0612 1.8702 i = 5 3.1007 1.8846 3.0833 1.880 3.0668 1.8758 3.0506 1.8705 3.0338 1.8653 i = 6 3.1076 1.8856 3.0982 1.8806 3.0870 1.8769 3.0783 1.8712 3.0660 1.8668 i = 7 3.1147 1.9001 3.1180 1.9029 3.1191 1.9055 3.1195 1.9081 3.1261 1.9115 i = 8 3.1064 1.8856 3.0972 1.8802 3.0861 1.8742 3.0755 1.8704 3.0648 1.8655 σ /a σ /a σ σ //a a, σ σ //a a 0.01 0.02 0.03 0.04 0.05 i i i j j j λ /a λ /λ λ /a λ /λ λ /a λ /λ λ /a λ /λ λ /a λ /λ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 i, j = 1, 2 3.0905 1.8782 3.0601 1.8681 3.0028 1.8446 3.0004 1.8474 2.9679 1.8270 i, j = 1, 3 3.1000 1.8927 3.0768 1.8925 3.0666 1.8848 3.0509 1.8784 3.0354 1.8722 i, j = 1, 4 3.0863 1.8807 3.0597 1.8716 3.0335 1.8612 3.0067 1.8486 2.9796 1.8355 i, j = 1, 5 3.0820 1.8803 3.0495 1.8692 3.0126 1.8586 2.9790 1.8432 2.9441 1.8200 i, j = 1, 6 3.0824 1.8836 3.0724 1.8744 3.0353 1.8588 3.0084 1.8471 2.9809 1.8351 i, j = 1, 7 3.0982 1.8919 3.0838 1.8893 3.0648 1.8837 3.0519 1.8782 3.0365 1.8729 i, j = 1, 8 3.0923 1.8786 3.0617 1.8673 3.0306 1.8582 2.9991 1.8449 2.9669 1.8293 i, j = 2, 3 3.1598 1.9349 3.1118 1.8990 3.0948 1.8979 3.0853 1.8965 3.0783 1.8927 i, j = 2, 4 3.0996 1.8813 3.0775 1.8709 3.0590 1.8606 3.0374 1.8506 3.0208 1.8382 i, j = 2, 5 3.0840 1.8837 3.0666 1.8681 3.0374 1.8619 3.0099 1.8489 2.9838 1.8365 i, j = 2, 6 3.1001 1.8824 3.0795 1.8723 3.0595 1.8643 3.0392 1.8557 3.0197 1.8455 i, j = 2, 7 3.1090 1.8920 3.1024 1.8894 3.0898 1.8847 3.0872 1.8815 3.0873 1.8778 i, j = 2, 8 3.1065 1.8791 3.0866 1.8692 3.0590 1.8625 3.0386 1.8525 3.0159 1.8431 i, j = 3, 4 3.1103 1.8961 3.1018 1.8948 3.0921 1.8941 3.0879 1.8918 3.0791 1.8886 i, j = 3, 5 3.1004 1.8941 3.0841 1.8901 3.0725 1.8844 3.0560 1.8804 3.0431 1.8758 i, j = 3, 6 3.1071 1.8925 3.1011 1.8906 3.0916 1.8869 3.0858 1.8825 3.0805 1.8783 i, j = 3, 7 3.1170 1.9212 3.1219 1.9547 3.1264 1.9894 3.1331 2.0243 3.1370 2.0637 i, j = 3, 8 3.1073 1.8907 3.0982 1.8874 3.0938 1.8862 3.0848 1.8812 3.0797 1.8762 i, j = 4, 5 3.0895 1.8810 3.0653 1.8714 3.0365 1.8624 3.0097 1.8510 2.9811 1.8375 i, j = 4, 6 3.0979 1.8806 3.0749 1.8730 3.0583 1.8642 3.0387 1.8544 3.0179 1.8449 i, j = 4, 7 3.1061 1.8946 3.1015 1.8898 3.0893 1.8903 3.0874 1.8850 3.0865 1.8809 i, j = 4, 8 3.0973 1.8809 3.0812 1.8713 3.0602 1.8647 3.0378 1.8561 3.0186 1.8456 i, j = 5, 6 3.0888 1.8811 3.0640 1.8705 3.0374 1.8605 3.0107 1.8502 2.9811 1.8361 i, j = 5, 7 3.1010 1.8901 3.0852 1.8905 3.0730 1.8853 3.0590 1.8813 3.0462 1.8747 i, j = 5, 8 3.0934 1.8796 3.0677 1.8694 3.0382 1.8618 3.0107 1.8483 2.9852 1.8348 i, j = 6, 7 3.1083 1.8963 3.1012 1.8965 3.0943 1.8942 3.0873 1.8933 3.0805 1.8928 i, j = 6, 8 3.1005 1.8793 3.0795 1.8688 3.0539 1.8612 3.0391 1.8496 3.0197 1.8376 i, j = 7, 8 3.1117 1.8987 3.1018 1.8994 3.0935 1.8973 3.0860 1.8964 3.0773 1.8936 supported by the Key projects of Leshan Science and Technology Bureau (grant number 20GZD022). Transmission characteristics in deformed finlines 43 References [1] Saad A.M.K, Begemann G. Electrical performance of finlines of various configurations, Microwave Opt Acoust, 1, (1977), 81–88. [2] A. Beyer. Analysis of the characteristics of an earthed fin line, IEEE Trans Microwave Theory Techn MTT-29, (1981), 676–680. [3] A.K. Sharma and W.J.R. Hoefer. Empirical expressions for fin-line design, IEEE Trans Microwave Theory Techn MTT-31, (1983), 350–356. [4] Q.H. Zheng and M. Song. Application of the multipole theory method to the analysis of finlines, Microwave Optical Technology Lett., vol. 35, pp. 100–102, Oct. 2002. [5] M. Lu and P.J. Leonard. On the field patterns of the dominant mode in unilateral finline by finite-element method, Microwave Opt Technol Lett, 38(2003), 193–195. [6] Hai Sun, Yu-jiang Wu and Zhou-sheng Ruan. A study of transmission characteristics in elliptic-shaped microshield lines, Journal of Electromagnetic Waves and Applications, Vol. 25, No. 17, 2011, pp. 2353–2364. [7] Sun H and Wu Y J. Finite element method analysis of the cutoff wavelength of dominant mode in unilateral finline [J]. Journal of Infrared, Millimeter and Terahertz Waves, 2011, 32(1):34–39. [8] Tan B K. Design of unilateral finline SIS mixer [J], Springer International Publishing, 2016. [9] Rao, V. Madhusudana and Rao, B. Prabhakara. Design of triple band pass filters using three-coupled finline and meta- materials. International Journal of Engineering Research, 2017(6):149–153. [10] Yuta Mizuno, Kunio Sakakibara, Nobuyoshi Kikuma and Kojiro Iwasa. Design of millimeter-wave 4 × 4 butler matrix for feeding circuit of multi-beam antenna using finline in multilayer substrate [C], 12th European Conference on Antennas and Propagation (EuCAP 2018). [11] Khodja, A., Yagoub, M., Touhami, R. and Baudrand, H. Systematic characterization of isotropic unilateral finlines for millimeter-wave applications [C], International Conference on Advanced Electrical Engineering (ICAEE 2019). [12] Mai Lu, Paul J. Leonard, Duo-Wang Fan and Fu-Yong Xu. On the cutoff wavelength of rectangular-shaped microshield line by edge element method, International Journal of Infrared and Millimeter waves, 25(10), 2004, 1469–1479. 44 Sun and Elzefzafy. Applied Mathematics and Nonlinear Sciences 8(2023) 35–44 This page is intentionally left blank http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Nonlinear Sciences de Gruyter

Study on Transmission Characteristics in Three Kinds of Deformed Finlines Based on Edge-Based Finite Element Method

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Abstract

The influence of model deformation on the transmission characteristics in unilateral finline, antipodal finline and bilateral finline is discussed using edge-based finite element method (FEM). The deformation is considered with respect to the eight boundaries of the model, and their amplitude is set to 0.01–0.05; the transmission characteristics include the cutoff wavelength characteristics of the dominant mode and the single-mode bandwidth characteristics. The results show a decreasing trend when the vacuum area is deformed, while they show a variety of possibilities when the loading region is deformed. These numerical results have strong guiding significance for the influence of the deformation of finlines on the overall device. Keywords: Cutoff wavelength characteristics of dominant mode, single-mode bandwidth characteristics, deformed unilateral finline, deformed antipodal finline, deformed bilateral finline AMS 2010 code: 35J60. 1 Introduction As new millimetre-wave transmission lines, finlines are widely used in microwave devices, such as directional couplers, phase shifters and filters, because of their long cutoff wavelength characteristics and the single-mode bandwidth characteristics, small attenuation and loss, weak dispersion and easy connection with solid-state devices. Since Saad and Begemann [1] studied the antipodal finline in 1977, academic research on it has also been in a hot state. For example, in 1981, Beyer [2] studied the grounding finline. In 1983, Sharma and Hoefer [3] discussed the empirical formula of finline design. In 2002, Zheng and Song [4] and, in 2003, Lu and Leonard [5] used multilevel theory and node finite element method (FEM) to calculate the partial transmission Corresponding author. Email address: hend.elzefzafy@asu.edu.bh ISSN 2444-8656 doi:10.2478/amns.2022.1.00021 Open Access. © 2022 Sun et al., published by Sciendo. This work is licensed under the Creative Commons Attribution alone 4.0 License. 36 Sun and Elzefzafy. Applied Mathematics and Nonlinear Sciences 8(2023) 35–44 characteristics of finlines. In 2011, Sun and his team calculated the partial transmission characteristics of uni- lateral finline and antipodal finline [6, 7]. In recent years, many researchers have carried out a huge amount of research on the applications of finline [8–11]. Deformation of the model may occur during the actual use of any device, and the finline is no exception. Deformation of the geometric model of finlines will inevitably lead to a change of transmission characteristics. However, research on non-uniform deformation transmission lines has not been carried out. Therefore, we mainly calculate the cutoff wavelength characteristics of the dominant mode and the single-mode bandwidth characteristics in deformed unilateral finline, deformed antipodal finline and deformed bilateral finline by using the edge-based FEM, and the effects of deformation on these transmission characteristics are analysed in detail. 2 Theoretical analysis The cross-section of finlines before and after deformation is shown in Fig. 1; the white part is the vacuum region, the dielectric constant is ε ; the mesh part is the dielectric region, and the dielectric constant is ε ; and 0 r the black part is the fin. The position and size of the fins and the filling area are represented by the symbols a, b, s, t , t , d , d . The deformation amplitude of the model boundary is represented by σ − σ . 1 2 1 2 1 8 According to the Maxwell equation, the electric field and magnetic field in the finlines satisfy the following vector differential equations: ⃗ ⃗ ∇× E = − jωµ µ H (1) ⃗ ⃗ ∇× H = jωε ε E r 0 where ε , µ , ε and µ represent the permittivity of free space, the permeability of free space, the relative 0 0 r r permittivity and the relative permeability, respectively. In order to calculate the transmission characteristics of the finline, it is necessary to establish a functional vector formula based on the magnetic field as the working variable. Therefore, the vector Helmholtz equation based on magnetic field is obtained from Eq. (1): ⃗ ⃗ ∇× ∇× H − K µ H = 0. (2) n ˆ× (∇× H) = 0 on Γ (3) n ˆ× H = 0 on Γ (4) Γ and Γ denote conductive wall and magnetic wall, respectively. In the actual calculation process, the boundary 1 2 condition satisfied by the magnetic field is Eq. (3). After derivation, we get the following variational problem: δ F(H) = 0. (5) where 1 1 1 ∗ ∗ 2 ∗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ F(H) = ∫ ∫ (∇ × H).(∇× H ) + (∇ H + jk H )· (∇ H + jk H ) − µ k H · H dΩ (6) t t t Z Z t t Z Z t r 2 ε ε r r The following matrix eigenvalue equation is obtained by vector finite element discretisation: T ∗ T ∗ e e e e e e e h S (tt) S (t z) h h T (tt) 0 h t t 2 t t F = − k (7) ∑ e e e e 0 e e e h S (zt) S (zz) h h 0 T (zz) h z z z z e=1 where the elemental matrices are given by 1 β e e e T e e T ⃗ ⃗ ⃗ ⃗ S (tt) = ∫ ∫ (∇ × N )· (∇ × N ) dΩ+ ∫ ∫ (N )· (N ) dΩ, (8) t t e e ε e ε e Ω Ω r r Transmission characteristics in deformed finlines 37 Fig. 1 Cross-sections of undeformed and deformed samples of three kinds of finlines. (A) Unilateral finline, (B) Deformed unilateral finline, (C) Antipodal finline, (D) Deformed antipodal finline, (E) Bilateral finline and (F) Deformed bilateral finline. e e e T S (t z) = ∫ ∫ (N )· (∇ L ) dΩ, (9) ε e e e e T S (zt) = ∫ ∫ (∇ L )· (N ) dΩ, (10) ε e e e e T S (zz) = ∫ ∫ (∇ L )· (∇ L ) dΩ, (11) t t ε e e e e e T ⃗ ⃗ T (tt) = µ ∫ ∫ (N )· (N ) dΩ, (12) e 2 e e e T T (zz) = β µ ∫ ∫ (L )· (L ) dΩ. (13) Ω 38 Sun and Elzefzafy. Applied Mathematics and Nonlinear Sciences 8(2023) 35–44 Using global notation, Eq. (7) can be written as follows: T ∗ T ∗ e e e e e e e 1 1 h S (tt) S (t z) h h T (tt) 0 h t t 2 t t F = − k (14) e e e e 0 e e e h S (zt) S (zz) h h 0 T (zz) h 2 2 z z z z Using the Ritz method for variation, we can get the following: e e e e e S (tt) S (t z) h T (tt) 0 h t t = k . (15) e e e e e S (zt) S (zz) h 0 T (zz) h z z After synthesis, we can obtain the following expression: [S][φ] = k [T][φ]. (16) S (tt) S (t z) T (tt) 0 h where [S] = , [T] = ,[φ] = . S (zt) S (zz) 0 T (zz) h By solving Eq. (16), the transmission characteristics in the three kinds of deformed finlines can be calculated. 3 Results and discussion 3.1 Validation of method In this part, we first verify the correctness of the edge-based FEM with the magnetic field as the working variable. Therefore, we use the method derived in this paper to calculate the main mode cutoff wavelength of the unilateral finline; the computed results and comparisons between the literature data [3, 12] are shown in Table 1. The relative errors are within 2%, which shows that the calculation method in this paper is feasible. Then, we calculate the effects of deformation on the cutoff wavelength characteristics of dominant mode and single-mode bandwidth characteristics in unilateral finline, antipodal finline and bilateral finline. In the calcula- tion process, for the three kinds of finlines, we assume that ε = 2.55, b/a = 0.5, s/a = 0.3, d /a = 0.25, d /a = r 1 2 0.05, t /a = 0.01, t /a = 0.05 and the dielectric substrate is located in the middle of the model. The geometric 1 2 deformation amplitude of the cross-section in the three kinds of finlines is represented by σ /a, σ /a, σ /a, 1 2 3 σ /a, σ /a, σ /a, σ /a and σ /a and their variation range is set to 0.01–0.05. 4 5 6 7 8 Table 1 Comparisons of calculation results of cutoff wave numbers of unilateral finline ( b/a = 0.5, d/b = 0.25, t/a = 0.01). εεε s/a This Article b b b/λλλ Ref. [3] b b b/λλλ Error (%) Ref. [12] b b b/λλλ Error (%) r r r c c c c c c c c c 2.22 0.25 0.1551 0.15457 0.34 0.15597 0.56 0.125 0.1605 0.16140 0.56 0.16218 1.04 0.0625 0.1688 0.16925 0.27 0.16996 0.68 3.0 0.25 0.1395 0.13908 0.30 0.14088 0.98 0.125 0.1479 0.14756 0.23 0.14884 0.63 0.0625 0.1581 0.15799 0.07 0.15992 1.14 Table 2 The values of the cutoff wavelength (λ /a) of dominant mode and bandwidth (λ /λ ) in three kinds of c2 c c1 undeformed finlines. Unilateral finline Antipodal finline Bilateral finline λλλ ///a a a (λλλ ///λλλ ) λλλ ///a a a (λλλ ///λλλ ) λλλ ///a a a (λλλ ///λλλ ) c c c c c c c c2 c c c c c2 c c c c c2 c c c1 c c c1 c c c1 2.9067 2.0202 2.8902 2.0342 3.1089 1.8939 Transmission characteristics in deformed finlines 39 3.2 Effect of deformation on the cutoff wavelength characteristics of dominant mode and single-mode bandwidth characteristics of the finlines When the three kinds of nlines fi are not deformed, the characteristics of the cutoff wavelength of dominant mode and the single-mode bandwidth characteristics are shown in Table 2. Any one or several deformations of σ /a, σ /a, σ /a, σ /a, σ /a, σ /a, σ /a and σ /a may occur when the finlines are used. Limited by space, 1 2 3 4 5 6 7 8 this paper focuses on the effects of any one and any two deformations of the eight σ /a on the transmission characteristics in the three kinds of finlines, and other changes can be calculated by analogy. The calculation results are presented in Tables 3–5. 3.3 Summary of the variation trend of transmission characteristics of three kinds of deformed finlines From Tables 2–5, we can draw the following conclusions. 1. As a whole, for the three kinds of finlines, both the values of the cutoff wavelength and the single-mode bandwidth decrease with the boundary deformation of the vacuum region, while they increase with the boundary deformation of the dielectric substrate. 2. In terms of deformation amplitude, with the increase of the amplitudes σ /a from 0.01 to 0.05, most of the cutoff wavelength and single-mode bandwidth show a decreasing trend. 3. There are four situations different from the above Rule 2: (1) In the first case, the cutoff wavelength and single-mode bandwidth increase as the deformation ampli- tude increases. For deformed unilateral finline, the details are as follows: σ /a, σ /a, σ /a = σ /a; 1 7 3 7 for deformed antipodal finline, the details are as follows: σ /a, σ /a, σ /a = σ /a; for deformed 1 7 3 7 bilateral finline, the details are as follows: σ /a, σ /a, σ /a = σ /a. 1 7 3 7 (2) In the second case, the cutoff wavelength changes little and the single-mode bandwidth decreases. For deformed unilateral finline, the details are as follows: σ /a = σ /a, σ /a = σ /a; for deformed 2 7 3 8 antipodal finline, the details are as follows: σ /a = σ /a; for deformed bilateral finline, the details 2 7 are as follows: σ /a = σ /a, σ /a = σ /a. 2 7 4 7 (3) In the third case, the cutoff wavelength changes little and the single-mode bandwidth increases. For deformed unilateral finline, the details are as follows: σ /a = σ /a, σ /a = σ /a; for deformed 2 3 7 8 antipodal finline, the details are as follows: σ /a = σ /a, σ /a = σ /a, σ /a = σ /a. 2 3 3 4 7 8 (4) In the fourth case, the cutoff wavelength decreases and the single-mode bandwidth changes little. For deformed unilateral finline, the details are as follows: σ /a = σ /a; for deformed bilateral finline, 3 4 the details are as follows: σ /a = σ /a, σ /a = σ /a. 3 4 7 8 4. Careful analysis of the above results shows that the situation in the third point includes the deformation of the dielectric substrate (includes σ /a or σ /a). It is concluded that the deformation of the dielectric 3 7 substrate has a great influence on the cutoff wavelength and single-mode bandwidth. 5. The effects of deformation on transmission characteristics in the three types of finlines are approximately the same. 4 Conclusion In this paper, we mainly study the effects of the deformation of unilateral finline, antipodal finline and bilateral finline on their transmission characteristics by using edge-based FEM. The cutoff wavelength of the dominant mode and single-mode bandwidth are calculated in detail. The results show that the deformation of the three types of finlines greatly changes their transmission characteristics, which are as follows: after the 40 Sun and Elzefzafy. Applied Mathematics and Nonlinear Sciences 8(2023) 35–44 boundary deformation of the vacuum region, both the cutoff wavelength of dominant mode and single-mode bandwidth mostly decrease; while after the boundary deformation of the loading region, the changes of the cutoff wavelength and single-mode bandwidth are more complex. These results will be used as references for the application of finlines in new microwave and millimetre-wave devices. Table 3 The changes of λ /a and λ /λ in deformed unilateral finline. c c1 c2 σ /a σ σ //a a 0.01 0.02 0.03 0.04 0.05 i i i λ /a λ /λ λ /a λ /λ λ /a λ /λ λ /a λ /λ λ /a λ /λ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 i = 1 2.9196 2.0043 2.9071 1.9942 2.8945 1.9838 2.8817 1.9732 2.8688 1.9621 i = 2 2.9269 2.0104 2.9213 2.0061 2.9158 2.0011 2.9106 1.9944 2.8816 1.9809 i = 3 2.9355 2.0350 2.9391 2.0505 2.9434 2.0620 2.9492 2.0712 2.9558 2.0794 i = 4 2.9210 2.0066 2.9100 1.9992 2.8992 1.9918 2.8882 1.9843 2.8775 1.9771 i = 5 2.9127 2.0010 2.8930 1.9873 2.8735 1.9736 2.8536 1.9596 2.8331 1.9453 i = 6 2.9178 2.0070 2.9074 2.0003 2.8960 1.9926 2.8851 1.9851 2.8765 1.9780 i = 7 2.9350 2.0368 2.9387 2.0537 2.9429 2.0659 2.9474 2.0749 2.9543 2.0833 i = 8 2.9059 2.0100 2.8983 2.0049 2.8944 1.9989 2.8888 1.9924 2.8834 1.9846 σ σ σ ///a a a, σ σ σ ///a a a 0.01 0.02 0.03 0.04 0.05 i i i j j j λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 i, j = 1, 2 2.9091 2.0023 2.8907 1.9872 2.8719 1.9717 2.8523 1.9550 2.8312 1.9349 i, j = 1, 3 2.9179 2.0274 2.9085 2.0341 2.8996 2.0386 2.8909 2.0420 2.8850 2.0430 i, j = 1, 4 2.8766 2.0047 2.8395 1.9834 2.8236 1.9719 2.7937 1.9495 2.7755 1.9323 i, j = 1, 5 2.9003 1.9909 2.8683 1.9672 2.8365 1.9435 2.8042 1.9193 2.7715 1.8946 i, j = 1, 6 2.9037 1.9987 2.8872 1.9800 2.8641 1.9625 2.8312 1.9425 2.8191 1.9260 i, j = 1, 7 2.9225 2.0270 2.9132 2.0349 2.9035 2.0388 2.8954 2.0418 2.8894 2.0446 i, j = 1, 8 2.9099 1.9981 2.8934 1.9830 2.8692 1.9699 2.8580 1.9509 2.8371 1.9321 i, j = 2, 3 2.9254 2.0337 2.9235 2.0500 2.9228 2.0630 2.9207 2.0748 2.9222 2.0828 i, j = 2, 4 2.8904 2.0063 2.8755 1.9940 2.8611 1.9727 2.8437 1.9680 2.8297 1.9415 i, j = 2, 5 2.9017 1.9967 2.8747 1.9780 2.8358 1.9577 2.8341 1.9401 2.8096 1.9205 i, j = 2, 6 2.8999 2.0072 2.8816 1.9958 2.8675 1.9778 2.8529 1.9727 2.8401 1.9533 i, j = 2, 7 2.9230 2.0312 2.9220 2.0403 2.9218 2.0376 2.9009 2.0339 2.9008 2.0170 i, j = 2, 8 2.9002 2.0055 2.8812 1.9974 2.8739 1.9833 2.8608 1.9723 2.8477 1.9613 i, j = 3, 4 2.8989 2.0339 2.8939 2.0449 2.8869 2.0517 2.8822 2.0574 2.8757 2.0609 i, j = 3, 5 2.8897 2.0280 2.8750 2.0326 2.8627 2.0315 2.8473 2.0299 2.8351 2.0298 i, j = 3, 6 2.9109 2.0329 2.9045 2.0398 2.8967 2.0424 2.8909 2.0430 2.8888 2.0444 i, j = 3, 7 2.9133 2.0654 2.9197 2.1150 2.9295 2.1689 2.9399 2.2237 2.9535 2.2693 i, j = 3, 8 2.9075 2.0272 2.9027 2.0338 2.9052 2.0335 2.9017 2.0316 2.9054 2.0257 i, j = 4, 5 2.8786 1.9968 2.8487 1.9772 2.8165 1.9534 2.7846 1.9317 2.7508 1.9078 i, j = 4, 6 2.8934 2.0037 2.8713 1.9879 2.8482 1.9722 2.8261 1.9576 2.8059 1.9413 i, j = 4, 7 2.8986 2.0291 2.8902 2.0360 2.8849 2.0400 2.8790 2.0405 2.8762 2.0403 i, j = 4, 8 2.8882 2.0038 2.8748 1.9925 2.8566 1.9797 2.8402 1.9665 2.8289 1.9520 i, j = 5, 6 2.8841 1.9976 2.8526 1.9751 2.8225 1.9545 2.7914 1.9323 2.7566 1.9082 i, j = 5, 7 2.8876 2.0227 2.8742 2.0249 2.8601 2.0254 2.8452 2.0219 2.8335 2.0202 i, j = 5, 8 2.8802 1.9954 2.8602 1.9786 2.8371 1.9597 2.8107 1.9401 2.7881 1.9196 i, j = 6, 7 2.9059 2.0324 2.8958 2.0406 2.8932 2.0487 2.8886 2.0530 2.8820 2.0548 i, j = 6, 8 2.8952 2.0041 2.8795 1.9909 2.8625 1.9780 2.8480 1.9632 2.8258 1.9471 i, j = 7, 8 2.9057 2.0321 2.9010 2.0481 2.9011 2.0622 2.9015 2.0744 2.9014 2.0827 Transmission characteristics in deformed finlines 41 Table 4 The changes of λ /a and λ /λ in deformed antipodal finline. c c1 c2 σ σ σ ///a a a 0.01 0.02 0.03 0.04 0.05 i i i λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 i = 1 2.8774 2.0200 2.8619 2.0086 2.8427 1.9999 2.8263 1.9875 2.8110 1.9757 i = 2 2.8866 2.0199 2.8828 2.0078 2.8767 1.9997 2.8707 1.9896 2.8638 1.9786 i = 3 2.8964 2.0488 2.9006 2.0585 2.9055 2.0649 2.9100 2.0714 2.9176 2.0758 i = 4 2.8860 2.0264 2.8779 2.0212 2.8691 2.0151 2.8587 2.0079 2.8485 2.0030 i = 5 2.8754 2.0232 2.8603 2.0114 2.8453 2.0001 2.8300 1.9883 2.8144 1.9767 i = 6 2.8920 2.0207 2.8876 2.0130 2.8801 2.0062 2.8749 1.9959 2.8671 1.9778 i = 7 2.8969 2.0481 2.9006 2.0571 2.9051 2.0630 2.9101 2.0679 2.9163 2.0733 i = 8 2.8923 2.0266 2.8831 2.0207 2.8737 2.0129 2.8644 2.0077 2.8546 2.0003 σ /a σ /a σ σ //a a, σ σ //a a 0.01 0.02 0.03 0.04 0.05 i i i j j j λ /a λ /λ λ /a λ /λ λ /a λ /λ λ /a λ /λ λ /a λ /λ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 i, j = 1, 2 2.8737 2.0063 2.8509 1.9879 2.8275 1.9684 2.8022 1.9475 2.7771 1.9257 i, j = 1, 3 2.8807 2.0385 2.8676 2.0386 2.8565 2.0334 2.8448 2.0298 2.8335 2.0260 i, j = 1, 4 2.8706 2.0154 2.8463 1.9991 2.8218 1.9817 2.7965 1.9637 2.7735 1.9479 i, j = 1, 5 2.8632 2.0081 2.8321 1.9859 2.8022 1.9631 2.7714 1.9394 2.7367 1.9200 i, j = 1, 6 2.8740 2.0086 2.8543 1.9889 2.8299 1.9711 2.8083 1.9504 2.7848 1.9292 i, j = 1, 7 2.8816 2.0355 2.8713 2.0372 2.8559 2.0362 2.8479 2.0355 2.8389 2.0317 i, j = 1, 8 2.8755 2.0137 2.8485 1.9942 2.8220 1.9768 2.7965 1.9579 2.7676 1.9414 i, j = 2, 3 2.8915 2.0398 2.8895 2.0491 2.8868 2.0531 2.8849 2.0555 2.8843 2.0578 i, j = 2, 4 2.8815 2.0098 2.8639 1.9939 2.8498 1.9796 2.8370 1.9641 2.8208 1.9462 i, j = 2, 5 2.8744 2.0043 2.8537 1.9866 2.8312 1.9669 2.8090 1.9475 2.7905 1.9266 i, j = 2, 6 2.8829 2.0089 2.8715 1.9970 2.8562 1.9842 2.8449 1.9714 2.8326 1.9579 i, j = 2, 7 2.8896 2.0270 2.8892 2.0251 2.8872 2.0194 2.8843 2.0133 2.8855 2.0040 i, j = 2, 8 2.8777 2.0176 2.8689 2.0008 2.8521 1.9875 2.8339 1.9735 2.8177 1.9576 i, j = 3, 4 2.8856 2.0479 2.8852 2.0522 2.8805 2.0586 2.8779 2.0625 2.8756 2.0678 i, j = 3, 5 2.8813 2.0394 2.8711 2.0413 2.8596 2.0390 2.8516 2.0380 2.8416 2.0359 i, j = 3, 6 2.8924 2.0326 2.8871 2.0305 2.8878 2.0255 2.8869 2.0189 2.8866 2.0120 i, j = 3, 7 2.8993 2.0781 2.9057 2.1245 2.9124 2.1699 2.9242 2.2043 2.9395 2.2370 i, j = 3, 8 2.8938 2.0394 2.8867 2.0399 2.8831 2.0384 2.8787 2.0370 2.8757 2.0353 i, j = 4, 5 2.8713 2.0152 2.8464 1.9959 2.8193 1.9797 2.7942 1.9613 2.7696 1.9421 i, j = 4, 6 2.8813 2.0152 2.8637 2.0035 2.8475 1.9903 2.8321 1.9773 2.8139 1.9627 i, j = 4, 7 2.8900 2.0412 2.8843 2.0413 2.8788 2.0409 2.8737 2.0406 2.8720 2.0376 i, j = 4, 8 2.8835 2.0209 2.8655 2.0097 2.8477 1.9996 2.8295 1.9910 2.8110 1.9797 i, j = 5, 6 2.8743 2.0090 2.8515 1.9902 2.8287 1.9715 2.8057 1.9513 2.7803 1.9315 i, j = 5, 7 2.8823 2.0360 2.8712 2.0357 2.8589 2.0325 2.8480 2.0282 2.8408 2.0255 i, j = 5, 8 2.8780 2.0148 2.8536 1.9970 2.8278 1.9808 2.8027 1.9646 2.7787 1.9460 i, j = 6, 7 2.8922 2.0422 2.8900 2.0503 2.8863 2.0537 2.8862 2.0555 2.8876 2.0583 i, j = 6, 8 2.8892 2.0133 2.8727 1.9977 2.8549 1.9818 2.8399 1.9660 2.8250 1.9491 i, j = 7, 8 2.8947 2.0475 2.8887 2.0540 2.8842 2.0613 2.8804 2.0656 2.8768 2.0681 Acknowledgments This work was supported by the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (grant number 2020QYY03) and supported by the Open Research Fund Program of Data Recovery Key Laboratory of Sichuan Province (grant number DRN2108) and 42 Sun and Elzefzafy. Applied Mathematics and Nonlinear Sciences 8(2023) 35–44 Table 5 The changes of of λ /a and λ /λ in deformed bilateral finline. c c1 c2 σ σ σ ///a a a 0.01 0.02 0.03 0.04 0.05 i i i λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ λλλ ///a a a λλλ ///λλλ c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 i = 1 3.0963 1.8855 3.0798 1.8801 3.0642 1.8748 3.0455 1.8695 3.0281 1.8641 i = 2 3.1059 1.8866 3.0970 1.8801 3.0856 1.8768 3.0768 1.8709 3.0660 1.8644 i = 3 3.1164 1.8986 3.1170 1.9010 3.1209 1.9041 3.1246 1.9090 3.1263 1.9090 i = 4 3.1083 1.8846 3.0962 1.8809 3.0880 1.8770 3.0772 1.8710 3.0612 1.8702 i = 5 3.1007 1.8846 3.0833 1.880 3.0668 1.8758 3.0506 1.8705 3.0338 1.8653 i = 6 3.1076 1.8856 3.0982 1.8806 3.0870 1.8769 3.0783 1.8712 3.0660 1.8668 i = 7 3.1147 1.9001 3.1180 1.9029 3.1191 1.9055 3.1195 1.9081 3.1261 1.9115 i = 8 3.1064 1.8856 3.0972 1.8802 3.0861 1.8742 3.0755 1.8704 3.0648 1.8655 σ /a σ /a σ σ //a a, σ σ //a a 0.01 0.02 0.03 0.04 0.05 i i i j j j λ /a λ /λ λ /a λ /λ λ /a λ /λ λ /a λ /λ λ /a λ /λ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ λλ //a a λλ //λλ c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 c c c c c c1 c c c2 i, j = 1, 2 3.0905 1.8782 3.0601 1.8681 3.0028 1.8446 3.0004 1.8474 2.9679 1.8270 i, j = 1, 3 3.1000 1.8927 3.0768 1.8925 3.0666 1.8848 3.0509 1.8784 3.0354 1.8722 i, j = 1, 4 3.0863 1.8807 3.0597 1.8716 3.0335 1.8612 3.0067 1.8486 2.9796 1.8355 i, j = 1, 5 3.0820 1.8803 3.0495 1.8692 3.0126 1.8586 2.9790 1.8432 2.9441 1.8200 i, j = 1, 6 3.0824 1.8836 3.0724 1.8744 3.0353 1.8588 3.0084 1.8471 2.9809 1.8351 i, j = 1, 7 3.0982 1.8919 3.0838 1.8893 3.0648 1.8837 3.0519 1.8782 3.0365 1.8729 i, j = 1, 8 3.0923 1.8786 3.0617 1.8673 3.0306 1.8582 2.9991 1.8449 2.9669 1.8293 i, j = 2, 3 3.1598 1.9349 3.1118 1.8990 3.0948 1.8979 3.0853 1.8965 3.0783 1.8927 i, j = 2, 4 3.0996 1.8813 3.0775 1.8709 3.0590 1.8606 3.0374 1.8506 3.0208 1.8382 i, j = 2, 5 3.0840 1.8837 3.0666 1.8681 3.0374 1.8619 3.0099 1.8489 2.9838 1.8365 i, j = 2, 6 3.1001 1.8824 3.0795 1.8723 3.0595 1.8643 3.0392 1.8557 3.0197 1.8455 i, j = 2, 7 3.1090 1.8920 3.1024 1.8894 3.0898 1.8847 3.0872 1.8815 3.0873 1.8778 i, j = 2, 8 3.1065 1.8791 3.0866 1.8692 3.0590 1.8625 3.0386 1.8525 3.0159 1.8431 i, j = 3, 4 3.1103 1.8961 3.1018 1.8948 3.0921 1.8941 3.0879 1.8918 3.0791 1.8886 i, j = 3, 5 3.1004 1.8941 3.0841 1.8901 3.0725 1.8844 3.0560 1.8804 3.0431 1.8758 i, j = 3, 6 3.1071 1.8925 3.1011 1.8906 3.0916 1.8869 3.0858 1.8825 3.0805 1.8783 i, j = 3, 7 3.1170 1.9212 3.1219 1.9547 3.1264 1.9894 3.1331 2.0243 3.1370 2.0637 i, j = 3, 8 3.1073 1.8907 3.0982 1.8874 3.0938 1.8862 3.0848 1.8812 3.0797 1.8762 i, j = 4, 5 3.0895 1.8810 3.0653 1.8714 3.0365 1.8624 3.0097 1.8510 2.9811 1.8375 i, j = 4, 6 3.0979 1.8806 3.0749 1.8730 3.0583 1.8642 3.0387 1.8544 3.0179 1.8449 i, j = 4, 7 3.1061 1.8946 3.1015 1.8898 3.0893 1.8903 3.0874 1.8850 3.0865 1.8809 i, j = 4, 8 3.0973 1.8809 3.0812 1.8713 3.0602 1.8647 3.0378 1.8561 3.0186 1.8456 i, j = 5, 6 3.0888 1.8811 3.0640 1.8705 3.0374 1.8605 3.0107 1.8502 2.9811 1.8361 i, j = 5, 7 3.1010 1.8901 3.0852 1.8905 3.0730 1.8853 3.0590 1.8813 3.0462 1.8747 i, j = 5, 8 3.0934 1.8796 3.0677 1.8694 3.0382 1.8618 3.0107 1.8483 2.9852 1.8348 i, j = 6, 7 3.1083 1.8963 3.1012 1.8965 3.0943 1.8942 3.0873 1.8933 3.0805 1.8928 i, j = 6, 8 3.1005 1.8793 3.0795 1.8688 3.0539 1.8612 3.0391 1.8496 3.0197 1.8376 i, j = 7, 8 3.1117 1.8987 3.1018 1.8994 3.0935 1.8973 3.0860 1.8964 3.0773 1.8936 supported by the Key projects of Leshan Science and Technology Bureau (grant number 20GZD022). Transmission characteristics in deformed finlines 43 References [1] Saad A.M.K, Begemann G. Electrical performance of finlines of various configurations, Microwave Opt Acoust, 1, (1977), 81–88. [2] A. Beyer. Analysis of the characteristics of an earthed fin line, IEEE Trans Microwave Theory Techn MTT-29, (1981), 676–680. [3] A.K. Sharma and W.J.R. Hoefer. Empirical expressions for fin-line design, IEEE Trans Microwave Theory Techn MTT-31, (1983), 350–356. [4] Q.H. Zheng and M. Song. Application of the multipole theory method to the analysis of finlines, Microwave Optical Technology Lett., vol. 35, pp. 100–102, Oct. 2002. [5] M. Lu and P.J. Leonard. On the field patterns of the dominant mode in unilateral finline by finite-element method, Microwave Opt Technol Lett, 38(2003), 193–195. [6] Hai Sun, Yu-jiang Wu and Zhou-sheng Ruan. A study of transmission characteristics in elliptic-shaped microshield lines, Journal of Electromagnetic Waves and Applications, Vol. 25, No. 17, 2011, pp. 2353–2364. [7] Sun H and Wu Y J. Finite element method analysis of the cutoff wavelength of dominant mode in unilateral finline [J]. Journal of Infrared, Millimeter and Terahertz Waves, 2011, 32(1):34–39. [8] Tan B K. Design of unilateral finline SIS mixer [J], Springer International Publishing, 2016. [9] Rao, V. Madhusudana and Rao, B. Prabhakara. Design of triple band pass filters using three-coupled finline and meta- materials. International Journal of Engineering Research, 2017(6):149–153. [10] Yuta Mizuno, Kunio Sakakibara, Nobuyoshi Kikuma and Kojiro Iwasa. Design of millimeter-wave 4 × 4 butler matrix for feeding circuit of multi-beam antenna using finline in multilayer substrate [C], 12th European Conference on Antennas and Propagation (EuCAP 2018). [11] Khodja, A., Yagoub, M., Touhami, R. and Baudrand, H. Systematic characterization of isotropic unilateral finlines for millimeter-wave applications [C], International Conference on Advanced Electrical Engineering (ICAEE 2019). [12] Mai Lu, Paul J. Leonard, Duo-Wang Fan and Fu-Yong Xu. On the cutoff wavelength of rectangular-shaped microshield line by edge element method, International Journal of Infrared and Millimeter waves, 25(10), 2004, 1469–1479. 44 Sun and Elzefzafy. Applied Mathematics and Nonlinear Sciences 8(2023) 35–44 This page is intentionally left blank

Journal

Applied Mathematics and Nonlinear Sciencesde Gruyter

Published: Jan 1, 2023

Keywords: Cutoff wavelength characteristics of dominant mode; single-mode bandwidth characteristics; deformed unilateral finline; deformed antipodal finline; deformed bilateral finline; 35J60

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