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Analytical Evaluation of Inductance of Spiral Inductors using Partial Element Equivalent Circuit (PEEC) Technique 'Ke-Ying Su and Jen-Tsai Kuo Department of Communication Engineering, National Chiao Tung University 1001 Tahsueh Rd., Hsinchu 300 Taiwan TEL: +886-3-571-2121 ext. 54515, FAX: +886-3-571-0116, jtkuo@cc.nctu.edu.tw 1 Introduction Inductors are essential elements in RF circuits and modules. In planar circuits, inductors are in spiral form. Due to geometrical complexity of planar or quasi- planar inductors, accurate computation of the inductance is usually difficult and time consuming. The partial element equivalent circuits (PEEC) technique is found suitable for this purpose. The PEEC technique is a circuit based formulation which is numerically equivalent to a full-wave method of moments solution with Galerkin matching [I, 21. In the technique, the inductance is formulated as a sum of partial self and mutual inductances, which are expressed in terms of six-fold integrals, if each partitioned conductor is considered as a solid rectangular parallelepiped. The partial self inductance can be evaluated using the closed-form expression in [2], and the partial mutual inductance is estimated using the filament approximation which represents a conductor in terms of a set of filaments in the direction of current flow. However, the accuracy of this approximation depends on the distance and relative orientation between the conductors and the number of filaments. If the distances between the conductors that are smaller than the crotis sectional dimensions, a large number of filaments will be required for accurare calculations. In this paper, the thickness of conductor is ignored since it has only a very small effect on the value of inductance [3]. The six-fold integrals are thus reduced lo four-fold integrals. It is found that these integrals can be analytically evaluated [ I ., 51. The analytical expressions not only overcome the singularity of the Green's function in the integral when the source and field points coincide, but also take the effect of ground plane into account. As closed-form results are used, very fast and accurate computation can be achieved. The presentation is organized as follows. The PEEC technique is briefly reviewed and the analytical integration results for the integrals are listed in the next section. In section 3, the inductances of RF-MEMS inductors and spiral inductors placed above the ground plane are evaluated. Numerical results are presented and compared with those in existing literature. Section 4 draws the conclusion. 2 Partial Inductance The spiral inductor can be completely specified by the number of turns n, the turn 3 l p q width w, the turn spacing s, and the outer diameter d. The geometry of a spiral inductor is shown in Fig. 1. In the PEEC method, the *'! inductor is divided into N sections of straight Fig. I Top view of a spiral inductor. 0-7803-733W02$17.0002002 CEEE 364 segments, and the total inductance can be written as the summation of self (&) and mutual (L,,,,) partial inductances [ 2 ] where w,, b,, and c are the width, starting point and end point of segment k, , respectively, r,, = /r, - r,l, and dl, is an element of segment k with the direction + along the axis of the segment. It is found that for XE [xk - AJ2, xk A421 and y € b, - SAL?, + 6J2], the integration result of ( I ) can be analytically evaluated y, as [4]: 0, i = j o r i + J = 5 , s( i,J ) = I, otherwise. a, = X,,,, + AJ2, + AJ2 P, = Y,,, + 6,,,/2 + 6,,12 = X,,, + An/2- A9,123 Pz = Ym,, 6J2 - 6J2 + CC,= X,,,, - AJ2 + AJ2, = Y,,,,6J2 + 6J2 - = X,,, - AJ2 - AJ2, P4= Y,,, - 6J2 - 6,/2 X,,, = xm - x,!, and Y,,, = ym- Y,,. It can be seen that this result applies to infinitely thin conductors of rectangular shape with arbitrary dimensions. As long as the dimensions of each discretized conductor are much smaller than the wavelength, the above results can be used to calculate the partial self and mutual inductance between conductors in the same plane of arbitrary size. If the separation between ground plane and the conductor is h , the inductance can be obtained in a similar fashion, based on image theory, by replacing llr,, with Urk, - 1lrtk,and r',, = [(xk - x,)~ + (yk - yJ2 + 4/1~]''~.The closed-form integration for the integrals with 1Pk, can be found in [ 5 ] . 3 Numerical Results The inductances of RF-MEMS and spiral inductors placed above ground plane are calculated based on the integration formulas described above. Consider a RF- MEMS inductor shown in Fig. 1. If the inductor is divided into N straight segments, then a matrix equation can be derived: 1, VN= Z N ~ N (3) where VNis the voltage vector of which the element V, represents the voltage of segment i, ZNXN the impedance matrix, and is the current vector of which I , = is I2 = . . . = I. The total impedance of the inductor can be calculated as 365 where Z,, is the element of ZNx,v R, is the resistance of ith segment. Obviously, and z,:, E:=, L,, is the total inductance [ 6 ] . Table I Comparison o f measured values with various approximate expressions and calculations Inductor n d w s L,,,> e,),, e n,Yn I 2.75 344 29.7 1.9 3.20 5.2 6.4 3.6 2 3.75 292 130 1.9 6.00 -1.2 -.07 -.04 3 6 50 217 5.4 1.9 12.50 1.4 2.3 4.9 4 2.75 279 18.3 1.9 3.10 2.1 2.8 1.4 5 4.75 206 78 1.9 6.10 -0.7 0.3 2.0 6 7.50 166 3.2 1.9 12.40 2.2 32 5.5 7 9.50 153 1.8 1.9 18.20 0.8 1.9 2.7 8 2.75 277 18.3 1.9 3.10 0.8 1.3 -2.0 -0.3 I 9 2.75 307 183 0.8 290 13.5 14.6 12.4 I2 62 10 3.75 321 16.5 18.0 6.10 0.2 1.1 0.7 II 7.75 225 4.4 I 9 18.10 -0.9 0.0 29 12 3.75 193 9.1 1.9 4.00 6 6 7.5 8.4 7.18 13 5.00 171 5.4 1.9 6.10 3.0 3.8 5.8 4.45 14 3.25 400 31.6 1.9 4.9 7.2 8.3 5.9 15 5.75 339 10.0 1.9 16.20 2.0 2.7 4.5 2.82 In Table I, we list the inductance of the fifteen inductors measured by Mohan e / al. [3]. The results (cal) are also compared with the results obtained by Wheeler formula (Whe), current sheet expression (gmd), and data fitted monomial expression (mon) provided in [3]. L,,,, is the measured value of the inductance in nH, and e is the percent relative errors between L,,,, and L calculated by the four methods. The percent relative error is defined as error = 100(L,mas-L)/L,mas. Only three of our calculated results have error greater than 5%. 4.5 - /:. 4- - ..... . . 3.5 z - 3- ...-- - - 0 05 1 1.5 2 h" ) Fig. 2. Values of the series inductance o f a one- and two-tum inductor versus the h 366 Fig. 2 plots the inductance values of a one-turn and a two-tum spiral inductors as functions of h, the separation between the planar conductor and the ground plane. Referring to the inset of Fig. 2, the structural parameters are m = 2.5 mm, n = 2.0 mm,v=0.3mm,w=0.25mm,ands=0.15mm.Ath=0.lmm,oursolutionsare 1.67 nH and 3.49 nH for the one- and two-turn inductors, respectively. The quasi- static data reported in [7] are 1.7 nH and 3.3 nH. Note that when h is smaller than O" S, both inductance values decrease rapidly when h is decreased. When h is larger than 0.5 mm, both the inductance values tend to he saturated, since the contribution from the integral of Wkm term to the total inductance becomes negligible as compared with that from 1hh. 4 Conclusions Analytical integration expressions of the four-fold integrals encountered in the PEEC method for calculating planar inductors of are presented. Very fast and accurate results for spiral inductors and RF-MEMS inductors are obtained. Acknowledgements This work was supported in part by the National Science Council, TAIWAN, under Grants NSC 89-2213-E-009-193, and in part by the joint program of the Ministry of Education and the National Science Council under the Contract: 89-E- F-A06-24. References [ l ] J. E. Garrett, A. E. Ruehli and C. R. Paul, "Accuracy and stability improvements of integral equation models using the partial element equivalent circuit approach", IEEE Trans. Antennas Propagat., vol. 46, no. 12, pp. 1824- 1832, Dec. 1998. [2] A. E. Ruehli, "Inductance calculations in a complex integrated circuit environment", IEMJ. Res. Develop., vol. 16, pp. 470-481, Sep. 1972. [3] S. S. Mohan, M. d. M. Hershenson, S. P. Boyd and T. H. Lee, "Simple accurate expressions for planar spiral inductances", IEEE J. Solid-state Circuifs,vol. 34, no. 10,pp. 1419-1424, Oct, 1999. [4] J. T. Kuo and K. Y Su, "Analytical evaluation of the MOM matrix elements for the capacitance of a charged plate", to appear on the May 2002 issue of IEEE Trans. Microwave Theory Tech. [5] K.Y Su and J. T. Kuo, "Analytical evaluation of MOM matrix elements for the capacitance of microstrip discontinuities", in 2001 Asia-Pacrfic Microwave Con$, Taipei, TAIWAN, R.O.C., Dec. 2001, pp. 445-448, [6] Z. Yaojiang, L. Haibo and F. Zhenghe, "Calculation of RF-MEMS inductors using partial element equivalent circuits", in 2000 Int. Microwave Millimeter Wave Tech. Proc., Beijing, CHINA, Sep, 2000, pp. 391-394. [7] A. R. Djordjevic, C. K. Allen, T. K. Sarkar, and Z. A. Maricevic, "Inductance of perfectly conducting foils including spiral inductors", IEEE Trans. Microwave Theory Tech.,vol. 38, no. 10, pp. 1407-1414, Oct. 1990. 367