Analytical evaluation of inductance of spiral inductors using partial by rbb25794


									           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,

     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

     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
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:

                                       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 , =
I2 = . . . = I. The total impedance of the inductor can be calculated as

where Z,, is the element of ZNx,v R, is the resistance of ith segment. Obviously,

  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   -


                                                             .....    . .

                  -      3-
                                           ...--     - -

                         0            05                1                 1.5            2
              Fig. 2. Values of the series inductance o f a one- and two-tum inductor versus the h

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
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
 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.


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-


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