Compact Modeling of Spiral Inductors for RF Applications

Transcription

1 Compact Modeing of Spira Inductors for RF Appications Ji Chen and Juin J. Liou Eectrica and Computer Engineering Dept. University of Centra Forida, Orando, FL 816 Phone: , Fax: , E-mai: ABSTRACT Recent growth in RF appications has increased the use of spira inductors and thus demanded a more accurate mode for such devices. In this paper, we focus on the mode deveopment of spira inductors having symmetrica terminas, but the same approach can be appied readiy to asymmetrica inductors. Reevant and important physics such as the current crowding in meta ine, frequency-dependent permittivity in oxide, and overap parasitics are accounted for. Experimenta data and resuts cacuated from the existing inductor modes are incuded in support of the mode deveopment. Keywords: spira inductor, compact mode, RF appication, nonuniform current distribution 1. INTRODUCTION Wireess communications is aready part of our daiy ife. To reduce the cost of monoithic microwave integrated circuits (MMICs), passive devices are frequenty integrated with active components on the same chip. Spira inductors are particuary important and widey used in MMICs such as ow-noise ampifiers, osciators, and mixers [1]. Many spira inductors modes have been reported in the iterature []-[16], and a these modes were deveoped intended for asymmetrica inductors but nonetheess sometimes used for symmetrica inductors. To the best of our knowedge, an accurate and compact symmetrica inductor mode is not yet avaiabe and urgenty needed. Empirica technique based on curve fitting for symmetrica inductors has been reported in [7], but modes derived this way cannot be scaed to refect changes in the inductor s ayout or fabrication technoogy. Commercia eectromagnetic fied sover may aso be used to predict the inductor s performance accuratey, but the computation time can be too extensive to be practica. In this paper, a physics-based mode appicabe for both symmetrica and asymmetrica inductors wi be deveoped. Mode deveopment and the proposed equivaent circuit for symmetrica inductors wi first be given in Section. In Section, resuts obtained from the present mode, existing modes, and ments are compared. Finay, concusions are given in Section 4.. MODEL DEVELOPMENT OF SYMMETRICAL INDUCTOR Our mode deveopment wi first focus on symmetrica inductors. In addition, an octagona spira pattern wi be considered, but the approach appies generay to other non-circuar patterns. It has been suggested that the octagona spira provides a higher Q factor and ower series resistance than the square pattern [1] and is more area efficient and easier reaized than the circuar spira [4]. The consideration of the octagona in fact makes the mode more comprehensive than most existing modes which consider ony square or hexagona patterns. Fig. 1(a) shows a symmetrica, octagon spira inductor with turns. For such an inductor, the meta track can be divided into 5 segments and overaps (see Fig. (a)), and the improved equivaent circuit for the inductor is given in Fig. 1(b). In Fig. 1(b), each segment box is represented by a umped mode shown in Fig. 1(c). In addition, couping capacitances between the meta ines and parasitics associated with the overaps need to be considered. These are accounted for with a the other components besides the segment boxes in Fig. 1(b), where C C,ij is the couping capacitance between two particuar meta ines i and j, C mm represents the capacitance associated with the overap, C ox_up and C sub_up mode the capacitances associated with the oxide and substrate of the underpass meta, respectivey, and G sub_up modes the substrate conductance of the underpass meta. Note that there are two sets of the overap parasitic components, and the subscripts 1 and denote the components pertinent to overaps 1 and, respectivey. It is worth mentioning that the umped equivaent circuit in Fig. 1(c) is the soe framework used in the conventiona modeing of spira inductors..1 Nonuniform Current Distribution in Meta Lines A difficut issue in modeing the spira inductor is the fact that the current distribution in a meta ine is not uniform and is a strong function of its ocation and operating frequency. Such a nonuniform current distribution is an important 74 NSTI-Nanotech 6, ISBN Vo., 6

2 mechanism affecting the inductor performance. Traditionay, the current density in a meta ine is considered to be governed by the skin and proximity effects [7-9]. It is more reaistic, however, to consider the subject meta ine ies in midst of eectromagnetic fied generated by a the other meta ines. According to the partia-eement-equivaent-circuit (PEEC) simuation [1] and -D eectromagnetic simuation [11], the current distribution in a meta ine in genera exhibits an exponentia decay from the inner edge (side of meta ine coser to the center of spira) to the outer edge (side of meta ine farther away from the center of spira). Furthermore, this exponentia-decay distribution is more prominent in the inner turns as the frequency is increased. Fig. shows the current density distributions in segments 1, 4, and (circed in Fig. ) simuated from an EM simuator. The frequency and ocation dependencies of the current distribution in the meta ines are ceary iustrated. Outer Diameter Overap1 Space 5 Overap Width We now introduce the concept of the effective ine width W eff in which the majority of the current density exists (the region where the first exponent of current density exists). Once W eff is in pace, then the nonuniform current distribution effect can be accounted for by repacing a the physica ine widths in the mode parameters with W eff. The foowing expressions are proposed to describe the effective ine width as a function of the frequency and the segment number (i.e., ocation of the meta ine): (a) C MM C MM w W eff, i w W, i 1 exp (1a) W, = i i 1 1 = c1 c (1b) f C MM 14 CMM C MM 4 C MM 5 Seg. 1 Seg. Seg. Seg. 4 Seg. 5 where w is the physica width of meta ine, f is frequency in Hz, i is turn index (i.e., for segments 1 and 5, i=1; segments and 4, i=; segments, i=), and c1 and c are fitting parameters. C ox_up1 G C Sub _up 1 Sub _ up GSub Ls C ox CSub R s C ox _ up G Sub _ up C Sub _up (b) Capacitive Couping to Neighbor Meta Fig. 1: (a) Schematic of an octagon symmetrica inductor of turns, (b) overa equivaent circuit for the inductor incuding the segment, capacitive couping, and overap components, and (c) equivaent circuit for the segment box in (b). (c) R s L s GSub C ox CSub Fig. shows the normaized current distributions in segments,, and 5 obtained from an EM simuator and the effective ine width mode. The mode resuts were cacuated by first integrating the current distribution simuated from the EM simuator over the physica ine width, normaizing it with the current integrated over the effective ine width, and then using it as the peak vaue foowed by an exponentia decay function. The good agreement demonstrates the soundness of using the effective ine width for modeing the frequency- and ocation-dependent natures of the spira inductors.. Modeing the Segment Box As shown in Fig. 1(c), the mode components in the segment box incude the series inductance L S, series resistance R S, and substrate parasitics. The meta track in each segment can be further divided into severa straight meta ines (for exampe, 5 straight meta ines for segment 4, see Fig. 1(a)), so that the inductance L S_ines of each straight meta ine can be expressed as the sef inductance L ine_sef pus the mutua inductance M from a other meta ines [1]: NSTI-Nanotech 6, ISBN Vo., 6 75

3 LS _ ine = Line _ sef + M () Outer Diameter Space Width Normaized Current Density at.9ghz in this mode simuated in HFSS Normaized Current Density at 1.8GHz in this mode simuated in HFSS Normaized Current Density at.4ghz in this mode simuated in HFSS Fig. : Current density contours in the three meta ines at three different frequencies simuated from an EM simuator. As the current redistribute in the meta ine at high frequencies, the corresponding inductance and resistance become frequency-dependent. The sef-inductance and mutua inductance in () were cacuated using the method in [1], but with the meta width repaced with the effective ine width stated above. Substrate parasitics resut from the eectrica couping between the meta track and substrate, as the meta track of a spira inductor can be considered as a microstrip on substrate with waves passing through it [1]. Three eements, C ox, G sub, and C sub, are used to mode the substrate parasitics (see Fig. 1(c)) Fig. : Comparison of current density distributions in the three meta ines cacuated from the present mode (ines) and obtained from EM simuations (symbos).. Modeing Components Outside Segment Box In Fig. 1(b), C C,ij describes the couping capacitance between meta ine segments i and j, and the approach in [15] was used to cacuate such a capacitance. Couping at the underpass was modeed using a parae-pate capacitor.. MODEL VERIFICATION A symmetrica, octagon inductor fabricated with the.5 µm CMOS technoogy was considered and d to verify the mode deveoped. The inductor was buit on a 9.59 µm oxide and 5 µm siicon substrate. The inductor has turns, meta width of 15 µm, meta thickness of µm, and spacing of 8 µm. Two-port parameters were d, and the inductance, resistance and quaity factor of the inductor were extracted. In addition to the present mode, two existing 76 NSTI-Nanotech 6, ISBN Vo., 6

4 inductor modes deveoped intended for the asymmetrica inductors were considered and compared. A systematic method to determine the vaues of the fitting parameters (c1 and c) in () is needed. To this end, the foowing function was deveoped: D = Q ( f ) Q Q ( f ) mode ( f ) + L ( f ) Lmod L ( f ) e ( f ) () where D describes the averaged error associated with fitting the mode to the d Q factor and inductance using different fitting parameter vaues at severa different frequencies. The parameter vaue that yieds the smaest D is the one to use, and c1 =.65 and c=.5 were obtained from this approach for the spira inductor considered. Figs. 4(a)-(c) show the quaity factor, inductance, and resistance, respectivey, cacuated from the present mode, cacuated from the two existing modes, and obtained from ments. The present mode demonstrates a better accuracy over the existing modes for a wide range of operating frequencies. Thus, our resuts suggested that it is erroneous and impractica to use the inductor mode deveoped intended for asymmetrica inductors for predicting the characteristics of symmetrica inductors. The phase and magnitude of S 1 parameter cacuated from the modes and obtained from ments are compared in Figs. 4(d) and (e), respectivey. 4. CONCLUSIONS A compact and accurate mode for spira inductors has been deveoped. Unike the existing modes which were deveoped intended ony for asymmetrica inductors, the present mode was shown capabe of predicting the characteristics of both the symmetrica and asymmetrica inductors accuratey. Reevant device physics, such as the current crowding, overap parasitics and geometry effect, have been taking into account. Quaity Factor Proposed Mode Yue's Mode [6] Mohan's Mode [16] 1 1 (a) Inductance (nh) Resistance (Ω) Mag. S 1 (db) Phase S 1 (rad) Proposed Mode Yue's Mode [6] Mohan's Mode [16] 1 1 (b) Proposed Mode Yue's Mode [6] Mohan's Mode [16] 1 1 (c) Proposed Mode Yue's Mode [6] Mohan's Mode [16] 1 1 (d) Proposed Mode Yue's Mode [6] Mohan's Mode [16] 1 1 (e) Fig. 4: Comparisons of the present mode, existing modes, and ments of (a) quaity factor (b) inductance, (c) series resistance, (d) S1 magnitude and (e) S1 phase for the -turn symmetrica inductor. NSTI-Nanotech 6, ISBN Vo., 6 77

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