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Tham khảo tài liệu 'advanced microwave circuits and systems part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 294 Advanced Microwave Circuits and Systems Ls can be derived from Y12. However Ls is usually not utilized to evaluate an on-chip inductor because it is not an effective value used in a circuit. Actually the inductance of on-chip inductor becomes zero at high frequency due to parasitic capacitances. To express this frequency dependence inductance defined by Y11 is commonly employed. The reason is explained as followed. As explained inductance and quality factor depend on each port impedance and inductors are often used at a shunt part as shown in Fig. 6 a . In this case input impedance of the inductor can be calculated by 1 Y11 as shown in Fig. 5 b . input ị input input n nõ r2 L Y11 Ỹ - Y11 Y12 p input ị input Ệ ụ no a n model Fig. 5. Y-parameter calculation. b shunt model input -------1 ĩì1-ĨÌ2 2 1------ 2 c differential model a shunt b differential Fig. 6. Inductor usage. Modeling of Spiral Inductors 295 Lshunt and Qshunt are defined by the following equations. Lshunt Im y w 1 Qshunt Im Y Y 2 Re ỹM Y11 Im Y11 3 Re Y11 This definition 1 3 is widely used because the definition does not depend on equivalent circuits and only Y11 is required to calculate them. In case using the equivalent circuit in Fig. 4 a Y11 can be derived by the following equation. 1 1 Y11 Rjs i j CSiJ 4 and it can be approximated at lower frequency as follows. 1 Y11 Rs jw Ls 5 This means that Lshunt and Oshunt are close to Ls and wLs Rs at lower frequency respectively and they are decreased by the parasitic capacitances at higher frequency. On the other hand on-chip inductors are often used in differential circuits as shown in Fig. 6 b . In this case the inductor has different characteristics from the shunt case shown in Fig. 6 a and the input impedance in differential mode becomes Y Y2 12 Y21 while the input impedance in single-ended mode is 1 Y11. The detailed calculation is explained in Sect. 2.2. Thus effective inductance Ldiff and effective quality factor Qdiff in differential mode can be
