Fully Analytical Characterization of the Series Inductance of Tapered Integrated Inductors

INTL JOURNAL OF ELECTRONICS AND TELECOMMUNICATIONS, 2014, VOL. 60, NO. 1, PP. 73–77 Manuscript received January 22, 2014; revised March, 2014. DOI: 10.2478/eletel-2014-0007 Fully Analytical Characterization of the Series Inductance of Tapered Integrated Inductors Fábio Passos, M. Helena Fino, and Elisenda R. Moreno Abstract—In this paper a general method for the determination of the series inductance of polygonal tapered inductors is presented. The value obtained can be integrated into any integrated inductor lumped element model, thus granting the overall characterization of the device and the evaluation of performance parameters such as the quality factor or the resonance frequency. In this work, the inductor is divided into several segments and the corresponding self and mutual inductances are calculated. In the end, results obtained for several working examples are compared against electromagnetic (EM) simulations are performed in order to check the validity of the model for square, hexagonal, octagonal and tapered inductors. The proposed method depends exclusively on the geometric characteristics of the inductor as well as the technological parameters. This allows its straight forward application to any inductor shape or technology. Keywords—inductor design, variable width integrated spiral inductors, CMOS analog integrated circuits, RF IC design I. I NTRODUCTION T HE BENEFITS of wireless connections through radio frequency (RF), for both communications and data transmission, has been motivating research work in this field ever since Guglielmo Marconi sent the first radio signal across the Atlantic ocean in 1901 [1]. At the time the motivation was the ability of communicate with people at hundreds of kilometres away. Nowadays, the ability to communicate with people is taken for granted, and the main goal is now to increase the amount of information sent. To accomplish this goal, an increasing demand for bandwidth has pushed new standards in the wireless domain. These new standards evolved towards higher operating frequencies. Besides the importance of the increasing bandwidth, wireless transmission allows the elimination of a physical connection between receiver and transmitter, which is a key advantage in modern communication systems. With the explosive growth of the wireless communication market the demand for fully integrated single chip RF transceiver systems also increased. The demand for low-cost RF integrated circuits also increased during the last years and a tremendous interest has been generated in on-chip passive components. During the past few years design efforts were made with the goal of integrating passive components such as resistors, capacitors and inductors. Compared to resistors and capacitors which nowadays have several integrated options, with most implementations being easy to model and F. Passos and M. H. Fino are with the New University of Lisbon, Faculty of Science and Technology, 2829-516 Caparica, Portugal, (e-mail: ; ). E. R. Moreno is with the Institute of Microelectronics of Seville, CNM, CSIC and University of Seville, 41092 Seville, Spain, (e-mail: ). implement, considerable effort is still needed to design and model on-chip inductors. Integrated inductors are commonly used in tuning, filtering and impedance matching. The general lack of accurate Spice like models, leads RF designers to design inductors through a time consuming process of EM simulation and silicon verification [2]–[4]. The design of integrated inductor involves the determination of correlated geometric parameters, thus making this process a candidate for optimization based design methodologies. The integration of electromagnetic simulators into optimization loops in order to calculate the performance parameters of an inductor, such as inductance, quality factor and self-resonance frequency (SRF), is a timely prohibitive solution. To overcome the above mentioned problem, designers usually adopt analytical solutions or inductor lumped-element models to use in Spice like simulations. The first lumped element circuit to model an inductor was used in 1980 [5]. Since that date several authors suggested many different circuits to model an inductor and to incorporate effects such as substrate losses, skin effect, proximity effects and eddy currents [6]– [8]. A survey about integrated inductor state-of-the-art can be found in [9]. It is possible to develop non-intuitive models that integrate several field effects thus providing a more accurate model, however a trade-off between simplicity and accuracy should be maintained, so a simple model such as the well known π-model [10], as shown in Fig. 1, may be used. Our focus in this paper is the determination of an analytical expression for the evaluation of the series inductance, Ls , for integrated planar tapered inductors of any shape (square, hexagonal, octagonal). It should be noted that some developments have been proposed over this methodology but always for non-variable width integrated inductors [4], [11]. The other passive elements presented in the model represent physical effects and may be calculated through a series of formulas given in [11], [12]. The proposed analytical expressions for the series inductance rely exclusively on Fig. 1. Lumped-element inductor π-model. Unauthenticated | 89.73.89.243 Download Date | 5/3/14 2:20 PM 74 F. PASSOS, M. H. FINO, E. R. MORENO Fig. 2. Inductor model description for a one turn inductor. geometric and technological parameters as a way of providing more physical insights into the design key parameters as well as enabling the straight forward application to new topologies and technologies. The method used to characterize inductors in this work is based on the series inductance calculation, which is explained in Section II. Section III describes in detail the proposed modelling technique when applied to square, hexagonal and octogonal inductors. Section IV presents the advantages of using variable width inductors, and how to calculate the series inductance for this type of inductors. Section V present the experimental results against EM simulations, thus proving the validity of the model. Finally in Section VI conclusions and future work is presented. II. S ERIES I NDUCTANCE In 1929, Grover derived formulas for inductance calculation between filaments in several different relative positions [13]. Greenhouse later applied these formulas to calculate the inductance of a square shaped inductor by dividing the inductor into straight-line segments, as ilustrated in Fig. 2, and evaluating the inductance by adding up the self inductance of the individual segment and mutual inductance between segments [14]. Some authors call this method the mutual inductance approach [15]. For the inductor depicted in Fig. 2, the series inductance is given by (1). This specific case is the least complex one, where there are no mutual inductances between segments. Ls = L1 + L2 + L3 + L4 (1) (...truncated)