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Classical Electrodynamics. Francesco Lacava. Ultra-Wideband, Short-Pulse Electromagnetics Frank Sabath. Powering Autonomous Sensors. When the current flow is in opposite direction, the mutual inductance is a minimum and negative; while, if the current elements are in orthogonal directions, then, the mutual inductance is zero.
Likewise for electric field we obtain :. In 13 we can identify the real part as due to the finite conductivity of the substrate, in fact apply the charge conservation equation at each segment we obtain:. The inversion of the matrix [P] is neither simple nor quickly solved; however, the use of techniques such as Cholesky decomposition or DCT Discrete Cosine Transform methods help the development of efficient algorithms making calculations .
For evaluation 8 and 12 we introduce the GMD AMD and AMSD concept, yielding the following general approximate expression  for two parallel conductor i and j straight with rectangular cross-section :. Equations 16 are expressed in nH. There are four cases possible for calculating the mutual inductance;. The index i and j are conductors; p and q are the index of the length for the difference in the length of two conductors and its angle tilt. A matrix representation allows one to visualize the terms better.
Using symmetry we can write the total inductance as. As well from trace of matrix 17 the coefficient of self-induction of the coil. All integrated passive devices suffer from substrate effects, in fact the spiral inductor fabricated on substrate experienced significant losses due to the magnetically and electrically induced Eddy current , ,  and . The objective of the parasitic extraction is to compute the capacitance matrix C for a multi conductor geometry . Analysis shows that the magnetic field and thus inductance of one conducting loop is dramatically affected by eddy currents distributed within a nearby conducting ground plane.
The Green function has been previously computed in analytical form from .
Replacing this in 6 , 11 and solving for the source leads to the finding in . Follow immediately from here:. With equation 19 , 20 and 21 we account for all loss mechanisms. Here ohmic loss is a factor limiting the inductor performance. For spiral inductors operating at high frequencies, the series resistance is frequency dependent; substrate parasitics result from the electrical coupling between the metal track and substrate, as the metal track of a spiral inductor can be considered as a microstrip on substrate with waves passing through it.
Some studies have been conducted to improve the accuracy of the simple lumped models. Though Cao et al. The equivalent circuit and results of Q factor are given in Figure 3 and Figure 7. A symmetrical spiral inductor fabricated with the 0. The inductor was built on a 9. Two-port parameters were measured, and the inductance, resistance and quality factor of the inductor were extracted.
In addition to the present model, two existing inductor models Yue and Wong , Mohan et al. The topology of the inductors and its layers are shown in Figure 4. The inductors were measured using an Agilent network analyzer and properly de-embedded see Figure 8. We carry out the simulated and measured data from Figure 5 to Figure 7 and in Table 1.
The present model demonstrates a better accuracy over the existing models for a wide range of operating frequencies. Thus, our results suggested that it is erroneous and impractical to use the inductor model developed intended for asymmetrical inductors for predicting the characteristics of symmetrical inductors. The Sonnet 2,5D EM simulator tool was used to numerical model. Unlike the existing inductor models which were developed intended only for asymmetrical inductors, the present model is shown capable of predicting accurately both the symmetrical and asymmetrical inductors.
The concept of the effective line width was introduced to account for the effect of non-uniform current distribution in the metal lines, and overlap parasitics and geometry factors have also been included. Comparisons among the present model, existing models, and measured data were presented to illustrate the usefulness of this work.
Dottorato was born in November 23, He is a Sr. Tegopoulos and E.
High performance on-chip inductors have become increasingly important and with the increasing frequencies of operation of the circuits, the on-chip inductors have gained even more importance . Complementary metal oxide semiconductor CMOS technology has been widely adopted for its mature and mass productivity [2, 3]. Steady improvements in the radio frequency characteristics of CMOS devices via scaling is driven by advancement in lithography.
It has enabled increased integration of RF functions. Spiral inductors are widely used even at microwave frequencies and their applications in millimeter-wave circuits are investigated . In this chapter a brief summary of the silicon integrated passive devices is given in Sect.
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- Analysis , Design , and Optimization of Spiral Inductors and Transformers for Si RF IC ’ s.
An introduction to on-chip inductor is presented in Sect. The losses in the conductor and the substrate are also explained. An overview of the evolution and progress of the on-chip inductor with a review on the integrated inductor design is presented in Sect.
The design complexity and performance issues are also discussed. A typical spiral inductor design problem is to determine its optimum layout parameters for a given inductance that will result in the highest quality factor at desired frequency.
Analysis , Design , and Optimization of Spiral Inductors and Transformers for Si RF IC ’ s
This chapter discusses a new approach for spiral inductor design and its optimization. In most of the integrated circuits like amplifiers, mixers, oscillators, etc. There are mainly two categories of differential inductor design found in the literature. The first one is a pair of asymmetric planar inductors connected together in series  as shown in Fig. Since the currents always flow in opposite direction in these two inductors, there must be enough spacing between them to minimize electromagnetic coupling.