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.
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
According to the Maxwell equation, the electric field and magnetic field in the finlines satisfy the following vector differential equations:
The following matrix eigenvalue equation is obtained by vector finite element discretisation:
Using global notation, Eq. (7) can be written as follows:
Using the Ritz method for variation, we can get the following:
After synthesis, we can obtain the following expression:
By solving Eq. (16), the transmission characteristics in the three kinds of deformed finlines can be calculated.
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 (
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
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
The values of the cutoff wavelength (
( |
( |
( |
|||
---|---|---|---|---|---|
2.9067 | 2.0202 | 2.8902 | 2.0342 | 3.1089 | 1.8939 |
The changes of
2.9196 | 2.0043 | 2.9071 | 1.9942 | 2.8945 | 1.9838 | 2.8817 | 1.9732 | 2.8688 | 1.9621 | |
2.9269 | 2.0104 | 2.9213 | 2.0061 | 2.9158 | 2.0011 | 2.9106 | 1.9944 | 2.8816 | 1.9809 | |
2.9355 | 2.0350 | 2.9391 | 2.0505 | 2.9434 | 2.0620 | 2.9492 | 2.0712 | 2.9558 | 2.0794 | |
2.9210 | 2.0066 | 2.9100 | 1.9992 | 2.8992 | 1.9918 | 2.8882 | 1.9843 | 2.8775 | 1.9771 | |
2.9127 | 2.0010 | 2.8930 | 1.9873 | 2.8735 | 1.9736 | 2.8536 | 1.9596 | 2.8331 | 1.9453 | |
2.9178 | 2.0070 | 2.9074 | 2.0003 | 2.8960 | 1.9926 | 2.8851 | 1.9851 | 2.8765 | 1.9780 | |
2.9350 | 2.0368 | 2.9387 | 2.0537 | 2.9429 | 2.0659 | 2.9474 | 2.0749 | 2.9543 | 2.0833 | |
2.9059 | 2.0100 | 2.8983 | 2.0049 | 2.8944 | 1.9989 | 2.8888 | 1.9924 | 2.8834 | 1.9846 |
2.9091 | 2.0023 | 2.8907 | 1.9872 | 2.8719 | 1.9717 | 2.8523 | 1.9550 | 2.8312 | 1.9349 | |
2.9179 | 2.0274 | 2.9085 | 2.0341 | 2.8996 | 2.0386 | 2.8909 | 2.0420 | 2.8850 | 2.0430 | |
2.8766 | 2.0047 | 2.8395 | 1.9834 | 2.8236 | 1.9719 | 2.7937 | 1.9495 | 2.7755 | 1.9323 | |
2.9003 | 1.9909 | 2.8683 | 1.9672 | 2.8365 | 1.9435 | 2.8042 | 1.9193 | 2.7715 | 1.8946 | |
2.9037 | 1.9987 | 2.8872 | 1.9800 | 2.8641 | 1.9625 | 2.8312 | 1.9425 | 2.8191 | 1.9260 | |
2.9225 | 2.0270 | 2.9132 | 2.0349 | 2.9035 | 2.0388 | 2.8954 | 2.0418 | 2.8894 | 2.0446 | |
2.9099 | 1.9981 | 2.8934 | 1.9830 | 2.8692 | 1.9699 | 2.8580 | 1.9509 | 2.8371 | 1.9321 | |
2.9254 | 2.0337 | 2.9235 | 2.0500 | 2.9228 | 2.0630 | 2.9207 | 2.0748 | 2.9222 | 2.0828 | |
2.8904 | 2.0063 | 2.8755 | 1.9940 | 2.8611 | 1.9727 | 2.8437 | 1.9680 | 2.8297 | 1.9415 | |
2.9017 | 1.9967 | 2.8747 | 1.9780 | 2.8358 | 1.9577 | 2.8341 | 1.9401 | 2.8096 | 1.9205 | |
2.8999 | 2.0072 | 2.8816 | 1.9958 | 2.8675 | 1.9778 | 2.8529 | 1.9727 | 2.8401 | 1.9533 | |
2.9230 | 2.0312 | 2.9220 | 2.0403 | 2.9218 | 2.0376 | 2.9009 | 2.0339 | 2.9008 | 2.0170 | |
2.9002 | 2.0055 | 2.8812 | 1.9974 | 2.8739 | 1.9833 | 2.8608 | 1.9723 | 2.8477 | 1.9613 | |
2.8989 | 2.0339 | 2.8939 | 2.0449 | 2.8869 | 2.0517 | 2.8822 | 2.0574 | 2.8757 | 2.0609 | |
2.8897 | 2.0280 | 2.8750 | 2.0326 | 2.8627 | 2.0315 | 2.8473 | 2.0299 | 2.8351 | 2.0298 | |
2.9109 | 2.0329 | 2.9045 | 2.0398 | 2.8967 | 2.0424 | 2.8909 | 2.0430 | 2.8888 | 2.0444 | |
2.9133 | 2.0654 | 2.9197 | 2.1150 | 2.9295 | 2.1689 | 2.9399 | 2.2237 | 2.9535 | 2.2693 | |
2.9075 | 2.0272 | 2.9027 | 2.0338 | 2.9052 | 2.0335 | 2.9017 | 2.0316 | 2.9054 | 2.0257 | |
2.8786 | 1.9968 | 2.8487 | 1.9772 | 2.8165 | 1.9534 | 2.7846 | 1.9317 | 2.7508 | 1.9078 | |
2.8934 | 2.0037 | 2.8713 | 1.9879 | 2.8482 | 1.9722 | 2.8261 | 1.9576 | 2.8059 | 1.9413 | |
2.8986 | 2.0291 | 2.8902 | 2.0360 | 2.8849 | 2.0400 | 2.8790 | 2.0405 | 2.8762 | 2.0403 | |
2.8882 | 2.0038 | 2.8748 | 1.9925 | 2.8566 | 1.9797 | 2.8402 | 1.9665 | 2.8289 | 1.9520 | |
2.8841 | 1.9976 | 2.8526 | 1.9751 | 2.8225 | 1.9545 | 2.7914 | 1.9323 | 2.7566 | 1.9082 | |
2.8876 | 2.0227 | 2.8742 | 2.0249 | 2.8601 | 2.0254 | 2.8452 | 2.0219 | 2.8335 | 2.0202 | |
2.8802 | 1.9954 | 2.8602 | 1.9786 | 2.8371 | 1.9597 | 2.8107 | 1.9401 | 2.7881 | 1.9196 | |
2.9059 | 2.0324 | 2.8958 | 2.0406 | 2.8932 | 2.0487 | 2.8886 | 2.0530 | 2.8820 | 2.0548 | |
2.8952 | 2.0041 | 2.8795 | 1.9909 | 2.8625 | 1.9780 | 2.8480 | 1.9632 | 2.8258 | 1.9471 | |
2.9057 | 2.0321 | 2.9010 | 2.0481 | 2.9011 | 2.0622 | 2.9015 | 2.0744 | 2.9014 | 2.0827 |
The changes of
2.8774 | 2.0200 | 2.8619 | 2.0086 | 2.8427 | 1.9999 | 2.8263 | 1.9875 | 2.8110 | 1.9757 | |
2.8866 | 2.0199 | 2.8828 | 2.0078 | 2.8767 | 1.9997 | 2.8707 | 1.9896 | 2.8638 | 1.9786 | |
2.8964 | 2.0488 | 2.9006 | 2.0585 | 2.9055 | 2.0649 | 2.9100 | 2.0714 | 2.9176 | 2.0758 | |
2.8860 | 2.0264 | 2.8779 | 2.0212 | 2.8691 | 2.0151 | 2.8587 | 2.0079 | 2.8485 | 2.0030 | |
2.8754 | 2.0232 | 2.8603 | 2.0114 | 2.8453 | 2.0001 | 2.8300 | 1.9883 | 2.8144 | 1.9767 | |
2.8920 | 2.0207 | 2.8876 | 2.0130 | 2.8801 | 2.0062 | 2.8749 | 1.9959 | 2.8671 | 1.9778 | |
2.8969 | 2.0481 | 2.9006 | 2.0571 | 2.9051 | 2.0630 | 2.9101 | 2.0679 | 2.9163 | 2.0733 | |
2.8923 | 2.0266 | 2.8831 | 2.0207 | 2.8737 | 2.0129 | 2.8644 | 2.0077 | 2.8546 | 2.0003 |
2.8737 | 2.0063 | 2.8509 | 1.9879 | 2.8275 | 1.9684 | 2.8022 | 1.9475 | 2.7771 | 1.9257 | |
2.8807 | 2.0385 | 2.8676 | 2.0386 | 2.8565 | 2.0334 | 2.8448 | 2.0298 | 2.8335 | 2.0260 | |
2.8706 | 2.0154 | 2.8463 | 1.9991 | 2.8218 | 1.9817 | 2.7965 | 1.9637 | 2.7735 | 1.9479 | |
2.8632 | 2.0081 | 2.8321 | 1.9859 | 2.8022 | 1.9631 | 2.7714 | 1.9394 | 2.7367 | 1.9200 | |
2.8740 | 2.0086 | 2.8543 | 1.9889 | 2.8299 | 1.9711 | 2.8083 | 1.9504 | 2.7848 | 1.9292 | |
2.8816 | 2.0355 | 2.8713 | 2.0372 | 2.8559 | 2.0362 | 2.8479 | 2.0355 | 2.8389 | 2.0317 | |
2.8755 | 2.0137 | 2.8485 | 1.9942 | 2.8220 | 1.9768 | 2.7965 | 1.9579 | 2.7676 | 1.9414 | |
2.8915 | 2.0398 | 2.8895 | 2.0491 | 2.8868 | 2.0531 | 2.8849 | 2.0555 | 2.8843 | 2.0578 | |
2.8815 | 2.0098 | 2.8639 | 1.9939 | 2.8498 | 1.9796 | 2.8370 | 1.9641 | 2.8208 | 1.9462 | |
2.8744 | 2.0043 | 2.8537 | 1.9866 | 2.8312 | 1.9669 | 2.8090 | 1.9475 | 2.7905 | 1.9266 | |
2.8829 | 2.0089 | 2.8715 | 1.9970 | 2.8562 | 1.9842 | 2.8449 | 1.9714 | 2.8326 | 1.9579 | |
2.8896 | 2.0270 | 2.8892 | 2.0251 | 2.8872 | 2.0194 | 2.8843 | 2.0133 | 2.8855 | 2.0040 | |
2.8777 | 2.0176 | 2.8689 | 2.0008 | 2.8521 | 1.9875 | 2.8339 | 1.9735 | 2.8177 | 1.9576 | |
2.8856 | 2.0479 | 2.8852 | 2.0522 | 2.8805 | 2.0586 | 2.8779 | 2.0625 | 2.8756 | 2.0678 | |
2.8813 | 2.0394 | 2.8711 | 2.0413 | 2.8596 | 2.0390 | 2.8516 | 2.0380 | 2.8416 | 2.0359 | |
2.8924 | 2.0326 | 2.8871 | 2.0305 | 2.8878 | 2.0255 | 2.8869 | 2.0189 | 2.8866 | 2.0120 | |
2.8993 | 2.0781 | 2.9057 | 2.1245 | 2.9124 | 2.1699 | 2.9242 | 2.2043 | 2.9395 | 2.2370 | |
2.8938 | 2.0394 | 2.8867 | 2.0399 | 2.8831 | 2.0384 | 2.8787 | 2.0370 | 2.8757 | 2.0353 | |
2.8713 | 2.0152 | 2.8464 | 1.9959 | 2.8193 | 1.9797 | 2.7942 | 1.9613 | 2.7696 | 1.9421 | |
2.8813 | 2.0152 | 2.8637 | 2.0035 | 2.8475 | 1.9903 | 2.8321 | 1.9773 | 2.8139 | 1.9627 | |
2.8900 | 2.0412 | 2.8843 | 2.0413 | 2.8788 | 2.0409 | 2.8737 | 2.0406 | 2.8720 | 2.0376 | |
2.8835 | 2.0209 | 2.8655 | 2.0097 | 2.8477 | 1.9996 | 2.8295 | 1.9910 | 2.8110 | 1.9797 | |
2.8743 | 2.0090 | 2.8515 | 1.9902 | 2.8287 | 1.9715 | 2.8057 | 1.9513 | 2.7803 | 1.9315 | |
2.8823 | 2.0360 | 2.8712 | 2.0357 | 2.8589 | 2.0325 | 2.8480 | 2.0282 | 2.8408 | 2.0255 | |
2.8780 | 2.0148 | 2.8536 | 1.9970 | 2.8278 | 1.9808 | 2.8027 | 1.9646 | 2.7787 | 1.9460 | |
2.8922 | 2.0422 | 2.8900 | 2.0503 | 2.8863 | 2.0537 | 2.8862 | 2.0555 | 2.8876 | 2.0583 | |
2.8892 | 2.0133 | 2.8727 | 1.9977 | 2.8549 | 1.9818 | 2.8399 | 1.9660 | 2.8250 | 1.9491 | |
2.8947 | 2.0475 | 2.8887 | 2.0540 | 2.8842 | 2.0613 | 2.8804 | 2.0656 | 2.8768 | 2.0681 |
The changes of of
3.0963 | 1.8855 | 3.0798 | 1.8801 | 3.0642 | 1.8748 | 3.0455 | 1.8695 | 3.0281 | 1.8641 | |
3.1059 | 1.8866 | 3.0970 | 1.8801 | 3.0856 | 1.8768 | 3.0768 | 1.8709 | 3.0660 | 1.8644 | |
3.1164 | 1.8986 | 3.1170 | 1.9010 | 3.1209 | 1.9041 | 3.1246 | 1.9090 | 3.1263 | 1.9090 | |
3.1083 | 1.8846 | 3.0962 | 1.8809 | 3.0880 | 1.8770 | 3.0772 | 1.8710 | 3.0612 | 1.8702 | |
3.1007 | 1.8846 | 3.0833 | 1.880 | 3.0668 | 1.8758 | 3.0506 | 1.8705 | 3.0338 | 1.8653 | |
3.1076 | 1.8856 | 3.0982 | 1.8806 | 3.0870 | 1.8769 | 3.0783 | 1.8712 | 3.0660 | 1.8668 | |
3.1147 | 1.9001 | 3.1180 | 1.9029 | 3.1191 | 1.9055 | 3.1195 | 1.9081 | 3.1261 | 1.9115 | |
3.1064 | 1.8856 | 3.0972 | 1.8802 | 3.0861 | 1.8742 | 3.0755 | 1.8704 | 3.0648 | 1.8655 |
3.0905 | 1.8782 | 3.0601 | 1.8681 | 3.0028 | 1.8446 | 3.0004 | 1.8474 | 2.9679 | 1.8270 | |
3.1000 | 1.8927 | 3.0768 | 1.8925 | 3.0666 | 1.8848 | 3.0509 | 1.8784 | 3.0354 | 1.8722 | |
3.0863 | 1.8807 | 3.0597 | 1.8716 | 3.0335 | 1.8612 | 3.0067 | 1.8486 | 2.9796 | 1.8355 | |
3.0820 | 1.8803 | 3.0495 | 1.8692 | 3.0126 | 1.8586 | 2.9790 | 1.8432 | 2.9441 | 1.8200 | |
3.0824 | 1.8836 | 3.0724 | 1.8744 | 3.0353 | 1.8588 | 3.0084 | 1.8471 | 2.9809 | 1.8351 | |
3.0982 | 1.8919 | 3.0838 | 1.8893 | 3.0648 | 1.8837 | 3.0519 | 1.8782 | 3.0365 | 1.8729 | |
3.0923 | 1.8786 | 3.0617 | 1.8673 | 3.0306 | 1.8582 | 2.9991 | 1.8449 | 2.9669 | 1.8293 | |
3.1598 | 1.9349 | 3.1118 | 1.8990 | 3.0948 | 1.8979 | 3.0853 | 1.8965 | 3.0783 | 1.8927 | |
3.0996 | 1.8813 | 3.0775 | 1.8709 | 3.0590 | 1.8606 | 3.0374 | 1.8506 | 3.0208 | 1.8382 | |
3.0840 | 1.8837 | 3.0666 | 1.8681 | 3.0374 | 1.8619 | 3.0099 | 1.8489 | 2.9838 | 1.8365 | |
3.1001 | 1.8824 | 3.0795 | 1.8723 | 3.0595 | 1.8643 | 3.0392 | 1.8557 | 3.0197 | 1.8455 | |
3.1090 | 1.8920 | 3.1024 | 1.8894 | 3.0898 | 1.8847 | 3.0872 | 1.8815 | 3.0873 | 1.8778 | |
3.1065 | 1.8791 | 3.0866 | 1.8692 | 3.0590 | 1.8625 | 3.0386 | 1.8525 | 3.0159 | 1.8431 | |
3.1103 | 1.8961 | 3.1018 | 1.8948 | 3.0921 | 1.8941 | 3.0879 | 1.8918 | 3.0791 | 1.8886 | |
3.1004 | 1.8941 | 3.0841 | 1.8901 | 3.0725 | 1.8844 | 3.0560 | 1.8804 | 3.0431 | 1.8758 | |
3.1071 | 1.8925 | 3.1011 | 1.8906 | 3.0916 | 1.8869 | 3.0858 | 1.8825 | 3.0805 | 1.8783 | |
3.1170 | 1.9212 | 3.1219 | 1.9547 | 3.1264 | 1.9894 | 3.1331 | 2.0243 | 3.1370 | 2.0637 | |
3.1073 | 1.8907 | 3.0982 | 1.8874 | 3.0938 | 1.8862 | 3.0848 | 1.8812 | 3.0797 | 1.8762 | |
3.0895 | 1.8810 | 3.0653 | 1.8714 | 3.0365 | 1.8624 | 3.0097 | 1.8510 | 2.9811 | 1.8375 | |
3.0979 | 1.8806 | 3.0749 | 1.8730 | 3.0583 | 1.8642 | 3.0387 | 1.8544 | 3.0179 | 1.8449 | |
3.1061 | 1.8946 | 3.1015 | 1.8898 | 3.0893 | 1.8903 | 3.0874 | 1.8850 | 3.0865 | 1.8809 | |
3.0973 | 1.8809 | 3.0812 | 1.8713 | 3.0602 | 1.8647 | 3.0378 | 1.8561 | 3.0186 | 1.8456 | |
3.0888 | 1.8811 | 3.0640 | 1.8705 | 3.0374 | 1.8605 | 3.0107 | 1.8502 | 2.9811 | 1.8361 | |
3.1010 | 1.8901 | 3.0852 | 1.8905 | 3.0730 | 1.8853 | 3.0590 | 1.8813 | 3.0462 | 1.8747 | |
3.0934 | 1.8796 | 3.0677 | 1.8694 | 3.0382 | 1.8618 | 3.0107 | 1.8483 | 2.9852 | 1.8348 | |
3.1083 | 1.8963 | 3.1012 | 1.8965 | 3.0943 | 1.8942 | 3.0873 | 1.8933 | 3.0805 | 1.8928 | |
3.1005 | 1.8793 | 3.0795 | 1.8688 | 3.0539 | 1.8612 | 3.0391 | 1.8496 | 3.0197 | 1.8376 | |
3.1117 | 1.8987 | 3.1018 | 1.8994 | 3.0935 | 1.8973 | 3.0860 | 1.8964 | 3.0773 | 1.8936 |
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
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:
In the second case, the cutoff wavelength changes little and the single-mode bandwidth decreases. For deformed unilateral finline, the details are as follows:
In the third case, the cutoff wavelength changes little and the single-mode bandwidth increases. For deformed unilateral finline, the details are as follows:
In the fourth case, the cutoff wavelength decreases and the single-mode bandwidth changes little. For deformed unilateral finline, the details are as follows:
Careful analysis of the above results shows that the situation in the third point includes the deformation of the dielectric substrate (includes
The effects of deformation on transmission characteristics in the three types of finlines are approximately the same.
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.
Comparisons of calculation results of cutoff wave numbers of unilateral finline (b/a = 0.5, d/b = 0.25, t/a = 0.01).
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.
( |
( |
( |
|||
---|---|---|---|---|---|
2.9067 | 2.0202 | 2.8902 | 2.0342 | 3.1089 | 1.8939 |
The changes of λc/a and λc1/λc2 in deformed antipodal finline.
2.8774 | 2.0200 | 2.8619 | 2.0086 | 2.8427 | 1.9999 | 2.8263 | 1.9875 | 2.8110 | 1.9757 | |
2.8866 | 2.0199 | 2.8828 | 2.0078 | 2.8767 | 1.9997 | 2.8707 | 1.9896 | 2.8638 | 1.9786 | |
2.8964 | 2.0488 | 2.9006 | 2.0585 | 2.9055 | 2.0649 | 2.9100 | 2.0714 | 2.9176 | 2.0758 | |
2.8860 | 2.0264 | 2.8779 | 2.0212 | 2.8691 | 2.0151 | 2.8587 | 2.0079 | 2.8485 | 2.0030 | |
2.8754 | 2.0232 | 2.8603 | 2.0114 | 2.8453 | 2.0001 | 2.8300 | 1.9883 | 2.8144 | 1.9767 | |
2.8920 | 2.0207 | 2.8876 | 2.0130 | 2.8801 | 2.0062 | 2.8749 | 1.9959 | 2.8671 | 1.9778 | |
2.8969 | 2.0481 | 2.9006 | 2.0571 | 2.9051 | 2.0630 | 2.9101 | 2.0679 | 2.9163 | 2.0733 | |
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.
3.0963 | 1.8855 | 3.0798 | 1.8801 | 3.0642 | 1.8748 | 3.0455 | 1.8695 | 3.0281 | 1.8641 | |
3.1059 | 1.8866 | 3.0970 | 1.8801 | 3.0856 | 1.8768 | 3.0768 | 1.8709 | 3.0660 | 1.8644 | |
3.1164 | 1.8986 | 3.1170 | 1.9010 | 3.1209 | 1.9041 | 3.1246 | 1.9090 | 3.1263 | 1.9090 | |
3.1083 | 1.8846 | 3.0962 | 1.8809 | 3.0880 | 1.8770 | 3.0772 | 1.8710 | 3.0612 | 1.8702 | |
3.1007 | 1.8846 | 3.0833 | 1.880 | 3.0668 | 1.8758 | 3.0506 | 1.8705 | 3.0338 | 1.8653 | |
3.1076 | 1.8856 | 3.0982 | 1.8806 | 3.0870 | 1.8769 | 3.0783 | 1.8712 | 3.0660 | 1.8668 | |
3.1147 | 1.9001 | 3.1180 | 1.9029 | 3.1191 | 1.9055 | 3.1195 | 1.9081 | 3.1261 | 1.9115 | |
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.
2.9196 | 2.0043 | 2.9071 | 1.9942 | 2.8945 | 1.9838 | 2.8817 | 1.9732 | 2.8688 | 1.9621 | |
2.9269 | 2.0104 | 2.9213 | 2.0061 | 2.9158 | 2.0011 | 2.9106 | 1.9944 | 2.8816 | 1.9809 | |
2.9355 | 2.0350 | 2.9391 | 2.0505 | 2.9434 | 2.0620 | 2.9492 | 2.0712 | 2.9558 | 2.0794 | |
2.9210 | 2.0066 | 2.9100 | 1.9992 | 2.8992 | 1.9918 | 2.8882 | 1.9843 | 2.8775 | 1.9771 | |
2.9127 | 2.0010 | 2.8930 | 1.9873 | 2.8735 | 1.9736 | 2.8536 | 1.9596 | 2.8331 | 1.9453 | |
2.9178 | 2.0070 | 2.9074 | 2.0003 | 2.8960 | 1.9926 | 2.8851 | 1.9851 | 2.8765 | 1.9780 | |
2.9350 | 2.0368 | 2.9387 | 2.0537 | 2.9429 | 2.0659 | 2.9474 | 2.0749 | 2.9543 | 2.0833 | |
2.9059 | 2.0100 | 2.8983 | 2.0049 | 2.8944 | 1.9989 | 2.8888 | 1.9924 | 2.8834 | 1.9846 |
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