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Microwave Ring Circuits and Related Structures phần 2

hoặc truyền đi dưới bất kỳ hình thức nào hoặc bằng bất kỳ phương tiện, điện tử, cơ khí, photocopy, ghi âm, quét, hoặc nếu không, trừ khi được cho phép theo Mục 107 hoặc 108 của Hoa 1976 Hoa Đạo luật Bản quyền, mà không cần sự cho phép trước bằng văn bản của Nhà xuất bản, hoặc uỷ quyền thông qua việc thanh toán phí cho mỗi bản sao thích hợp để giải phóng mặt bằng | 30 ANALYSIS AND MODELING OF RING RESONATORS TABLE 2.4 A Comparison of Table 2.3 and the Theoretical Results from upper the Transmission Line Method and lower the Magnetic-Wall Model Frequency Error Circuit n 1 n 2 1 0.79 0.60 2 0.28 0.49 3 0.56 0.28 Frequency Error Circuit n 1 n 2 1 0.78 0.89 2 0.07 0.37 2 0.63 0.38 b FIGURE 2.15 The configurations of one-port a square and b annular ring resonators 10 . TRANSMISSION-LINE MODEL 31 is considered to be a transmission line. Z1 and Z2 are the coordinates corresponding to sections 11 and 12 respectively. The ring is fed by the source voltage V at somewhere with Z1 2 0. The positions of the zero point of Z1 2 and the voltage V are arbitrarily chosen on the ring. For a lossless transmission line the voltages and currents for the two sections are given as follows V 2 Z1 2 Vo e - ibz1 2 T1 2 0 e jbZ1 2 2.65a I1 2 Z1 2 V e-ibz1 2 -T1 2We 2.65b Z o where V oe-bi is the incident wave propagating in the Z1 2 direction VT2T0 eZ is the reflected wave propagating in the -Z1 2 direction r1 2 0 is the reflection coefficient at Z1 2 0 and Z0 is the characteristic impedance of the ring. When a resonance occurs standing waves set up on the ring. The shortest length of the ring resonator that supports these standing waves can be obtained from the positions of the maximum values of these standing waves. These positions can be calculated from the derivatives of the voltages and currents in Equation 2.65 . The derivatives of the voltages are dV 2 Z1 2 -jpVo e jbZ1 2 - T1 2 0 e jbZ1 2 2.66 d Z1 2 Letting d V1-2 z1-2 0 the reflection coefficients can be found as d Z1 2 Z1 2 0 T1 2 0 1 2.67 Substituting r1j2 0 1 into Equation 2.65 the voltages and currents can be obtained as V1 2 z1 2 2V0 cos bZ1 2 2 .68a 11.2 Z1 2 - 2V sin bZ1 2 2.68b Zo Based on Equation 2.68 the absolute values of voltage and current standing waves on each section 11 and 12 are shown in Figure 2.16. Inspecting Figure 2.16 the standing waves repeat for multiples of lg 2 on .