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 F´abio 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. INTRODUCTION THE 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: [email protected]t.unl.pt; [email protected]). E. R. Moreno is with the Institute of Microelectronics of Seville, CNM, CSIC and University of Seville, 41092 Seville, Spain, (e-mail: [email protected]). 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. Brought to you by | CSIC - Unidad Coordinacion Bibliotecas Authenticated Download Date | 10/29/14 12:26 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. SERIES INDUCTANCE 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) III. POLYGONAL SERIES INDUCTANCE CALCULATION In this section the evaluation of the series inductance, Ls, for any n-side inductor is presented. For illustrative purposes, the particular case of an octagonal inductor is considered. For an octagonal layout, such as the one in Fig. 3, the series inductance can be calculateb by (2). Ls=L0+Mp+−Mp−−Mlm (2) where Lsis the total series inductance of the inductor, L0 is the self inductance of each segment, Mp+is the parallel mutual inductance between segments where the current flows in the same direction whereas Mp−accounts for parallel mutual inductance where current flows in opposite directions, Mlm accounts for all the different types of mutual inductances Fig. 3. Twenty-five section octagonal spiral inductor layout. resulting from non-parallel segments. These mutual inductances should be added or subtracted depending on the relative flow of the current. The series inductance of the octagonal inductor depicted in Fig. 3 may be calculated with (3). Ls=L1+L2+... +L25 (Self inductance) +2(M1,9+M2,10+M3,11+M4,12+M5,13+M6,14+...+M17,25) (Positive mutual inductances) −2(M1,5+M2,6+M3,7+M4,8+M5,9+M10,6+...+M25,21) (Negative mutual inductances) −2(M1,2+M2,3+M24,25 +... +M1,3+M3,5+M23,25+ ... +M1,11 +M2,12 +... +M16,25) (Different mutual inductances) (3) where to calculate the self inductance of each segment, Li, the following formula should be used, Li= 2lln 2l w+t+ 0.50049 + w+t 3l(nH)(4) where lis the segment length, wis the segment width and tis the segment thickness. This formula was proposed by Greenhouse and is suitable for integrated inductors [14]. For the evaluation of the mutual inductances, Mi,j, the relative positions of the segments must be taken into account. Due to the magnitude and phase of the currents, the mutual inductances are assumed identical in all sections, hence Mi,j=Mj,i. For the evaluation of the mutual inductances, four different cases reflecting the relative positions of the segments must be considered: 1) between parallel segments, 2) between segments that are connected at one end, 3) between segments where the intersection point is lying outside the two segments and 4) between segments where the intersection point lies upon one segment. All these cases can be calculated with the formulas provided by Groover [13], as presented in the following sub-sections. A. Parallel Segments An example of two parallel segments is illustrated in Fig. 4 where liand ljrepresent the segments lengths, pis the pitch Brought to you by | CSIC - Unidad Coordinacion Bibliotecas Authenticated Download Date | 10/29/14 12:26 PM FULLY ANALYTICAL CHARACTERIZATION OF THE SERIES INDUCTANCE OF TAPERED INTEGRATED INDUCTORS 75 li p r lj q Fig. 4. Parallel segments. li lj R ε Fig. 5. Inductor segments connected at one end. of the two wires. The pitch, p, of the inductor, is the distance between the center of two parallel segments as can be observed in Fig. 3. On the other hand, rand qare the difference between the length of the segments. For the case of two parallel segments, as depicted in Fig. 4, the mutual inductances between two segments are calculated through the following equation. Mli,lj= 0.5·(mlj+r+mlj+q)−(mr+mq)(5) where each mlx, is given by, mlx= 2 ·lx·Ux(6) When calculating mlj+pfor example, in (6) the segment length lx, should be equal to lj+p. The mutual inductance factor, U, and can be calculated through (7). U=ln   lx dx +s1 + lx dx2 −s1 + dx lx2 +dx lx (7) where the distance between segments, dx, is considered as the geometric mean distance (GMD), between segments and calculated by (8). ln(dx) = ln(p)−w2 12p2−w4 60p4−w6 168p6−w8 360p8−w10 660p10 (8) where wis the width of the segments in study. B. Segments which are Connected at One End An example for mutual inductance for segments which are connected at one end, such as M4,3, is presented in Fig. 5. These types of mutual inductance are calculated through (9), Mli,lj= 2cos(ε)· litanh−1lj li+R+ljtanh−1li lj+R (9) µ li ν lj ε Fig. 6. Inductor segment intersection outside the two segments. µ li r lj ε Fig. 7. Case for when the intersection point lies upon one segment. where liand ljare the lengths of the segments and Ris the distance between the segment ends, that can be calculated by (10). R2= 2l2 i(li−cos(ε)) (10) It is also possible to use one of the following relations to obtain either Ror ε. cos(ε) = l2 i+l2 j−R2 2lilj (11) R2 l2 i = 1 + l2 j l2 i −2lj li cos(ε)(12) C. Segments where the Intersection Point is Lying Outside the Two Segments The case of mutual inductance where the intersection point is lying outside the two segments, for example, M3,5(in Fig. 3), is given in Fig. 6. The mutual inductances in Fig. 6 are calculated by the following equation, Mli,lj= 2cos(ε) (Mµ+li,ν+lj+Mµ,ν )−(Mµ+li,ν +Mν+lj,µ)(13) where each one of the mutual inductances inside the parenthesis are calculated using (9). D. Segments where the Intersection Point Lies upon One Segment The case where the intersection point lies upon one segment, is the most complex case. An example of this type of mutual inductances is for example M5,14 (in Fig. 3). This mutual inductance is presented in Fig. 7. It is important to state that ljis the entire segment. The mutual inductances in Fig. 7 are calculated by the following equation, Mli,lj= 2cos(ε) (Mli+µ,lj−r+Mlj−r,µ)−(Mli+µ,r +Mr,µ)(14) Brought to you by | CSIC - Unidad Coordinacion Bibliotecas Authenticated Download Date | 10/29/14 12:26 PM 76 F. PASSOS, M. H. FINO, E. R. MORENO where again, each one of the mutual inductances inside the parenthesis are calculated using (9). For the evaluation of the mutual inductances between segment liand r, and between li+µand ran angle of π−εmust be considered. IV. TAPERED SQUARE INDUCTOR The design of an optimal spiral is highly frequency dependent. This is due to the multiple loss mechanisms that appear from the distributed effects in the structure. In general, for a fixed area, we can design an inductor with many different values of metal width w, spacing s, and turns n, to achieve the same value of inductance. As we increase wfor instance, the resistance drops. This drop is proportional to 1/W at low frequencies but much more gradual at higher frequencies, due to the skin effect, especially with w≫δ, where δis the effective skin width [16]. The substrate losses, though, tend to increase with w, since this increases the parasitics. At the same time, we observe that structures with more turns, n, tend to have higher resistive losses in the inner turns. The origin of these enhanced losses is due to the fact that the magnetic flux increases as we move towards the center of the spiral, due to the additive nature of the flux from each successive loop of the spiral, for this reason hollow spirals are preferred [17]. If the width of the structure is tapered, as shown in Fig. 8, then the performance of the spiral can be improved [18]. Since the wide inner turns do not lower the resistance (due to current constriction), it is better to transfer the width to the outer turns, while keeping the total area of the spiral constant. For the optimum design of tapered inductors a general method for the characterizing its behaviour is needed. The method proposed to calculate the series inductance of this tapered layout is exactly the same as it was for non tapered. A simple expression can be used to increase the width of each segment proportionately. This formula is the same one that ASITIC uses for the design the layout of tapered integrated inductors and is given by, w2 x=w0+ (wi−w0)·x Nsides ·n(15) where w0is the width of the last segment of the outer turn, whereas wistands for the first segment of the inner turn, xis the number of the segment, nis the number of turns and Nsides is the number of sides of the polygonal structure. However the width of each turn can be completely independent of any equation. However, a small consideration has to be taken into account. For tapered inductors, the pitch, p, in (8), is no longer constant. A tapered integrated inductor is depicted in Fig. 8. V. RESULTS AND DISCUSSION In this section we present the results obtained for the evaluation of the series inductance with the proposed method. For the validation of the model we present results for square, hexagonal and octagonal inductors with non-variable width. The obtained results were compared against EM simulation with ADS Momentum [19]. The technology used for the Fig. 8. A square spiral inductor with tapered trace width. simulation was UMC 130 nm. To reduce the losses, all inductors were implemented in the top most available layer, so in UMC 130 nm, metal 8 should be used, which has thickness, tof 2 um. The square inductors have an area of 350 ×350 µm2and the hexagonal and octagonal inductors have an area of 400 ×400 µm2. All the inductors have a metal width, w= 10µm and a spacing between metals, s= 2.5µm. The error presented as εM, is the error of the model compared to EM simulation. TABLE I LsRESULTS WITH THE MODEL COMPARED TO EM ADS MOMENTUM SIMULATIONS FOR EQUAL WIDTH INDUCTORS. Sides n LMLADS εM(%) 4 2 3.48 3.45 0.99 4 3 6.55 6.46 1.39 4 4 9.89 9.77 1.23 4 5 13.25 13.24 0.08 6 2 3.05 3.00 3.15 6 3 5.90 5.65 4.44 6 4 9.20 8.60 6.98 6 5 12.24 11.68 7.96 8 2 3.30 3.17 4.10 8 3 6.40 6.00 6.67 8 4 9.91 9.25 7.14 8 5 13.40 12.60 6.35 For square inductors we have typical errors of 2-3%. On the contrary of what was stated in other papers [12], it was possible to design hexagonal and octagonal inductors, with errors typically smaller than 2-3%. For smaller inductors the error percentage is slightly higher and the reason may be that for smaller inductors and for higher number of turns, the inductor stops being hollow, increasing the eddy currents which are not considered in this model. For the tapered inductors, several simulations have been made for octagonal tapered width inductors. The results are presented in Tab. II. In this table the inductors simulated go from n= 2 until n= 6. Every inductor has the first turn width (w1) equal to 14 µm and the last turn width equal to 20 µm. The variation between each turn is independent of any equation, but a ratio between each turn width was maintained and the area of all the inductors is the same. The results presented for the tapered octagonal inductors have an average error around 2.5% which proves the validity of the proposed method even for the most difficult layout used nowadays. Square and hexagonal layouts have much less segments, therefore decreasing the complexity of the Brought to you by | CSIC - Unidad Coordinacion Bibliotecas Authenticated Download Date | 10/29/14 12:26 PM FULLY ANALYTICAL CHARACTERIZATION OF THE SERIES INDUCTANCE OF TAPERED INTEGRATED INDUCTORS 77 TABLE II LsRESULTS WITH THE MODEL COMPARED TO EM ADS MOMENTUM SIMULATIONS FOR TAPERED WIDTH INDUCTORS. n Din w1w2w3w4w5w6Area LMLEM Error 2 227 14 20 - - - - 300 1.84 1.97 6.41 3 188 14 17 20 - - - 300 3.43 3.46 0.84 4 149 14 16 18 20 - - 300 4.92 4.88 0.84 5 110 14 15.5 17 18.5 20 - 300 6.18 6.08 1.66 6 74 14 15 16 17 18.5 20 300 7.26 7.07 2.69 calculations, which is an indicator that the results would be even more accurate. VI. CONCLUSION In this paper we have introduced an efficient integrated inductor model for the prediction of the inductance for several integrated inductor layouts. The results have been compared with EM simulations with ADS Momentum where the model has proven its validity for the different layouts, with all results showing low deviations with respect to the EM simulations. It was also proven that this model is suitable for its usage with tapered inductors of different layouts. It may be observed that generally the model achieves average errors of 3 %, thus making this model a candidate for the design of integrated inductors. As future work, the idea is to integrate this inductance model into a pi lumped element model as a way of evaluating the other performance parameters related with integrated inductors, such as quality factor and self-resonance frequency. 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