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Study on transmission characteristics in three kinds of deformed finlines based on edge-based finite element method

Publié en ligne: 05 Sep 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 05 Nov 2021
Accepté: 15 May 2022
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Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
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Langues
Anglais
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 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 [8,9,10,11].

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.

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 ε0; the mesh part is the dielectric region, and the dielectric constant is εr; and 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, t1, t2, d1, d2. The deformation amplitude of the model boundary is represented by σ1 − σ8.

According to the Maxwell equation, the electric field and magnetic field in the finlines satisfy the following vector differential equations: {×E=jωμrμ0H×H=jωεrε0E \left\{ {\matrix{ {\nabla \times \vec E = - j\omega {\mu _r}{\mu _0}\vec H} \hfill \cr {\nabla \times \vec H = j\omega {\varepsilon _r}{\varepsilon _0}\vec E} \hfill \cr } } \right. where ε0, μ0, εr and μr represent the permittivity of free space, the permeability of free space, the relative 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): ×(1εr×H)K02μrH=0. \nabla \times \left( {{1 \over {{\varepsilon _r}}}\nabla \times \vec H} \right) - K_0^2{\mu _r}\vec H = 0. n^×(×H)=0onΓ1 \hat n \times (\nabla \times \vec H) = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{on}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\Gamma _1} n^×H=0onΓ2 \hat n \times \vec H = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{on}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\Gamma _2} Γ1 and Γ2 denote conductive wall and magnetic wall, respectively. In the actual calculation process, the boundary condition satisfied by the magnetic field is Eq. (3). After derivation, we get the following variational problem: δF(H)=0. \delta F(\vec H) = 0. where F(H)=12Ω(1εr(t×H).(×Ht)*+1εr(tHZ+jkZHt)(tHZ+jkZHt)*μrk02HH*)dΩ F(\vec H) = {1 \over 2}\int {\int\limits_\Omega {\left( {{1 \over {{\varepsilon _r}}}({\nabla _t} \times \vec H).(\nabla \times {{\vec H}_t}{)^*} + {1 \over {{\varepsilon _r}}}({\nabla _t}{H_Z} + j{k_Z}{{\vec H}_t}) \cdot {{({\nabla _t}{H_Z} + j{k_Z}{{\vec H}_t})}^*} - {\mu _r}k_0^2\vec H \cdot {{\vec H}^*}} \right)d\Omega } }

The following matrix eigenvalue equation is obtained by vector finite element discretisation: F=12e=1M([htehze]T[Se(tt)Se(zt)Se(tz)Se(zz)][htehze]*k02[htehze]T[Te(tt)00Te(zz)][htehze]*) F = {1 \over 2}\sum\limits_{e = 1}^M \left( {{{\left[ {\matrix{ {h_t^e} \cr {h_z^e} \cr } } \right]}^T}\left[ {\matrix{ {{S^e}(tt)} \cr {{S^e}(zt)} \cr } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \matrix{ {{S^e}(tz)} \cr {{S^e}(zz)} \cr } } \right]{{\left[ {\matrix{ {h_t^e} \cr {h_z^e} \cr } } \right]}^*} - k_0^2{{\left[ {\matrix{ {h_t^e} \cr {h_z^e} \cr } } \right]}^T}\left[ {\matrix{ {{T^e}(tt)} \cr 0 \cr } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \matrix{ 0 \cr {{T^e}(zz)} \cr } } \right]{{\left[ {\matrix{ {h_t^e} \cr {h_z^e} \cr } } \right]}^*}} \right) where the elemental matrices are given by Se(tt)=1εreΩe(t×Ne)(t×Ne)TdΩ+β2εreΩe(Ne)(Ne)TdΩ, {S^e}(tt) = {1 \over {\varepsilon _r^e}}\int {\int\limits_{{\Omega ^e}} {({\nabla _t} \times {{\vec N}^e}) \cdot {{({\nabla _t} \times {{\vec N}^e})}^T}d\Omega + {{{\beta ^2}} \over {\varepsilon _r^e}}\int \int_{{\Omega ^e}} ({{\vec N}^e}) \cdot {{({{\vec N}^e})}^T}d\Omega ,} } Se(tz)=β2εreΩe(Ne)(tLe)TdΩ, {S^e}(tz) = {{{\beta ^2}} \over {\varepsilon _r^e}}\int {\int\limits_{{\Omega ^e}} {({{\vec N}^e}) \cdot {{({\nabla _t}{L^e})}^T}d\Omega ,} } Se(zt)=β2εreΩe(tLe)(Ne)TdΩ, {S^e}(zt) = {{{\beta ^2}} \over {\varepsilon _r^e}}\int {\int\limits_{{\Omega ^e}} {({\nabla _t}{L^e}) \cdot {{({{\vec N}^e})}^T}d\Omega ,} } Se(zz)=β2εreΩe(tLe)(tLe)TdΩ, {S^e}(zz) = {{{\beta ^2}} \over {\varepsilon _r^e}}\int {\int\limits_{{\Omega ^e}} {({\nabla _t}{L^e}) \cdot {{({\nabla _t}{L^e})}^T}d\Omega ,} } Te(tt)=μreΩe(Ne)(Ne)TdΩ, {T^e}(tt) = \mu _r^e\int {\int\limits_{{\Omega ^e}} {({{\vec N}^e}) \cdot {{({{\vec N}^e})}^T}d\Omega ,} } Te(zz)=β2μreΩe(Le)(Le)TdΩ. {T^e}(zz) = {\beta ^2}\mu _r^e\int {\int\limits_{{\Omega ^e}} {({L^e}) \cdot {{({L^e})}^T}d\Omega .} }

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.

Using global notation, Eq. (7) can be written as follows: F=12[htehze]T[Se(tt)Se(zt)Se(tz)Se(zz)][htehze]*12k02[htehze]T[Te(tt)00Te(zz)][htehze]* F = {1 \over 2}{\left[ {\matrix{ {h_t^e} \cr {h_z^e} \cr } } \right]^T}\left[ {\matrix{ {{S^e}(tt)} \cr {{S^e}(zt)} \cr } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \matrix{ {{S^e}(tz)} \cr {{S^e}(zz)} \cr } } \right]{\left[ {\matrix{ {h_t^e} \cr {h_z^e} \cr } } \right]^*} - {1 \over 2}k_0^2{\left[ {\matrix{ {h_t^e} \cr {h_z^e} \cr } } \right]^T}\left[ {\matrix{ {{T^e}(tt)} \cr 0 \cr } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \matrix{ 0 \cr {{T^e}(zz)} \cr } } \right]{\left[ {\matrix{ {h_t^e} \cr {h_z^e} \cr } } \right]^*}

Using the Ritz method for variation, we can get the following: [Se(tt)Se(zt)Se(tz)Se(zz)][htehze]=k02[Te(tt)00Te(zz)][htehze]. \left[ {\matrix{ {{S^e}(tt)} \cr {{S^e}(zt)} \cr } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \matrix{ {{S^e}(tz)} \cr {{S^e}(zz)} \cr } } \right]\left[ {\matrix{ {h_t^e} \cr {h_z^e} \cr } } \right] = k_0^2\left[ {\matrix{ {{T^e}(tt)} \cr 0 \cr } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \matrix{ 0 \cr {{T^e}(zz)} \cr } } \right]\left[ {\matrix{ {h_t^e} \cr {h_z^e} \cr } } \right].

After synthesis, we can obtain the following expression: [S][ϕ]=k02[T][ϕ]. [S][\phi ] = k_0^2[T][\phi ]. where [S]=[S(tt)S(zt)S(tz)S(zz)] [S] = \left[ {\matrix{ {S{\kern 1pt} {\kern 1pt} (tt)} \cr {S{\kern 1pt} {\kern 1pt} (zt)} \cr } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \matrix{ {S{\kern 1pt} {\kern 1pt} (tz)} \cr {S{\kern 1pt} {\kern 1pt} (zz)} \cr } } \right] , [T]=[T(tt)00T(zz)] {\kern 1pt} {\kern 1pt} \left[ T \right] = \left[ {\matrix{ {T{\kern 1pt} {\kern 1pt} (tt)} \cr 0 \cr } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \matrix{ 0 \cr {T{\kern 1pt} {\kern 1pt} (zz)} \cr } } \right] , [ϕ]=[hthz] [\phi ] = \left[ {\matrix{ {{h_t}} \cr {{h_z}} \cr } } \right] .

By solving Eq. (16), the transmission characteristics in the three kinds of deformed finlines can be calculated.

Results and discussion
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.

Comparisons of calculation results of cutoff wave numbers of unilateral finline (b/a = 0.5, d/b = 0.25, t/a = 0.01).

εr s/a This Article b/λc Ref. [3] b/λc Error (%) Ref. [12] b/λc Error (%)
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

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 calculation process, for the three kinds of finlines, we assume that εr = 2.55, b/a = 0.5, s/a = 0.3, d1/a = 0.25, d2/a = 0.05, t1/a = 0.01, t2/a = 0.05 and the dielectric substrate is located in the middle of the model. The geometric deformation amplitude of the cross-section in the three kinds of finlines is represented by σ1/a, σ2/a, σ3/a, σ4/a, σ5/a, σ6/a, σ7/a and σ8/a and their variation range is set to 0.01–0.05.

Effect of deformation on the cutoff wavelength characteristics of dominant mode and single-mode bandwidth characteristics of the finlines

When the three kinds of finlines 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 σ1/a, σ2/a, σ3/a, σ4/a, σ5/a, σ6/a, σ7/a and σ8/a may occur when the finlines are used. Limited by space, this paper focuses on the effects of any one and any two deformations of the eight σi/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.

The values of the cutoff wavelength (λc/a) of dominant mode and bandwidth (λc1c2) in three kinds of undeformed finlines.

Unilateral finline Antipodal finline Bilateral finline
λc/a (λc1c2) λc/a (λc1c2) λc/a (λc1c2)
2.9067 2.0202 2.8902 2.0342 3.1089 1.8939

The changes of λc/a and λc1c2 in deformed unilateral finline.

σi/a 0.01 0.02 0.03 0.04 0.05
λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2
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
σi/a, σj/a 0.01 0.02 0.03 0.04 0.05
λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2
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

The changes of λc/a and λc1c2 in deformed antipodal finline.

σi/a 0.01 0.02 0.03 0.04 0.05
λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2
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
σi/a, σj/a 0.01 0.02 0.03 0.04 0.05
λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2
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

The changes of of λc/a and λc1c2 in deformed bilateral finline.

σi/a 0.01 0.02 0.03 0.04 0.05
λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2
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
σi/a, σj/a 0.01 0.02 0.03 0.04 0.05
λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2
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
Summary of the variation trend of transmission characteristics of three kinds of deformed finlines

From Tables 2–5, we can draw the following conclusions.

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.

In terms of deformation amplitude, with the increase of the amplitudes σi/a from 0.01 to 0.05, most of the cutoff wavelength and single-mode bandwidth show a decreasing trend.

There are four situations different from the above Rule 2:

In the first case, the cutoff wavelength and single-mode bandwidth increase as the deformation amplitude increases. For deformed unilateral finline, the details are as follows: σ1/a, σ7/a, σ3/a = σ7/a; for deformed antipodal finline, the details are as follows: σ1/a, σ7/a, σ3/a = σ7/a; for deformed bilateral finline, the details are as follows: σ1/a, σ7/a, σ3/a = σ7/a.

In the second case, the cutoff wavelength changes little and the single-mode bandwidth decreases. For deformed unilateral finline, the details are as follows: σ2/a = σ7/a, σ3/a = σ8/a; for deformed antipodal finline, the details are as follows: σ2/a = σ7/a; for deformed bilateral finline, the details are as follows: σ2/a = σ7/a, σ4/a = σ7/a.

In the third case, the cutoff wavelength changes little and the single-mode bandwidth increases. For deformed unilateral finline, the details are as follows: σ2/a = σ3/a, σ7/a = σ8/a; for deformed antipodal finline, the details are as follows: σ2/a = σ3/a, σ3/a = σ4/a, σ7/a = σ8/a.

In the fourth case, the cutoff wavelength decreases and the single-mode bandwidth changes little. For deformed unilateral finline, the details are as follows: σ3/a = σ4/a; for deformed bilateral finline, the details are as follows: σ3/a = σ4/a, σ7/a = σ8/a.

Careful analysis of the above results shows that the situation in the third point includes the deformation of the dielectric substrate (includes σ3/a or σ7/a). It is concluded that the deformation of the dielectric substrate has a great influence on the cutoff wavelength and single-mode bandwidth.

The effects of deformation on transmission characteristics in the three types of finlines are approximately the same.

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 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.

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.
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.

Comparisons of calculation results of cutoff wave numbers of unilateral finline (b/a = 0.5, d/b = 0.25, t/a = 0.01).

εr s/a This Article b/λc Ref. [3] b/λc Error (%) Ref. [12] b/λc Error (%)
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

The values of the cutoff wavelength (λc/a) of dominant mode and bandwidth (λc1/λc2) in three kinds of undeformed finlines.

Unilateral finline Antipodal finline Bilateral finline
λc/a (λc1c2) λc/a (λc1c2) λc/a (λc1c2)
2.9067 2.0202 2.8902 2.0342 3.1089 1.8939

The changes of λc/a and λc1/λc2 in deformed antipodal finline.

σi/a 0.01 0.02 0.03 0.04 0.05
λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2
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

The changes of of λc/a and λc1/λc2 in deformed bilateral finline.

σi/a 0.01 0.02 0.03 0.04 0.05
λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2
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

The changes of λc/a and λc1/λc2 in deformed unilateral finline.

σi/a 0.01 0.02 0.03 0.04 0.05
λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2 λc/a λc1c2
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

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