Published in the Proceedings of 49th IEEE Electronic Components and Technology Conference, June 1-4, San Diego, CA.

Electrical Characteristics of Planar Spiral Inductors Ronald T. Anderson and Benjamin Beker Department of Electrical and Computer Engineering University of South Carolina, Columbia, SC 29208 Ph. (803) 777-3469 fax (803) 777-8045 Email: [email protected] Abstract This paper describes two types of PEEC-based methods for the analysis of planar spiral inductors. One approach is a high-level distributed network method (DNM), while the second is based on a relatively simple equivalent SPICE circuit. The inductors are printed on lossless, grounded ceramic substrates and the metallization is assumed to have finite conductivity, but is infinitesimally thin, so that the formulation does not take in to account the skin effect. The methodology is applied to analyze the electrical characteristics of several different inductors. Introduction Recently, the use of spiral inductors has increased significantly in wireless [1] and some high-speed digital applications. Factors such as product miniaturization, reliability, ease of assembly, and reduction of component count have paved the way for embedding passive components directly into the substrate. Unlike LTCC technology [2], where spiral inductors may be printed on several metallization layers, there are applications where the inductor metallization is confined to single layer only. The electrical characteristics of planar passive components, such as inductors, can be assessed using several techniques, which range from carrying out purely static to rigorous full-wave computations. The static methods typically involve the calculation of lumped inductance, capacitance, and resis tance values required to build a simple equivalent circuit [3]. The component values are commonly calculated from formu las for simplified goemetries approximating the actual structure. Or, they can be extracted from static field solvers for arbitrary three-dimensional geometries. Unfortunately, the validity of this approach is restricted to low frequencies only. To extend the frequency range of the model for planar inductors, the resistive partial electrical equivalent circuit (PEEC) method can also be used [4]. This technique is based on building a distributed equivalent circuit for the inductor. It incorporates the frequency dependence into the model by way of coupling between partial inductances and capacitances through the network equations. The attractive feature of this technique is that a complex electromagnetic field boundary value problem can be cast in terms of more intuitive and simpler circuit concepts. As the frequency increases, even the PEEC method encounters limitations and, at that point, integral or differential equation based full-wave methods must be used instead. Fullwave techniques ranging from time-domain partial differential equation (PDE) solvers such as FD-TD [5], TLM [6], and ESN [7], have been successfully employed to model passive structures in the past. In addition, integral equation methods

such as the method of moments (MoM) in space [8] and spectral [9] domains, as well as the finite element approach [10] still enjoy wide popularity to date. In as much as rigorous, powerful, and widely applicable these full-wave techniques are, they require significant computational resources for the analysis of large scale problems that often occur in practice. The other drawback of full-wave techniques is that the de-embedding of equivalent circuits from the data they produce is not straightforward. For modeling of planar spiral inductors, the dynamic network method (DNM), presented in this paper, bridges the gap between comprehensive full-wave techniques and simple static methods. Unlike the time-domain PDE solvers, it requires only modest computer resources and is amenable to adaptive non-uniform meshing of irregular geometries. In contrast to full-wave spectral-domain integral equation (IE) techniques, DNM is formulated directly in the space-domain, leading to an intuitive equivalent multi-port network representation of the problem. It is faster than the full-wave space domain IE and FEM formulations, since most integrals needed to generate the elements of the network matrices can be evaluated in closed form. Moreover, they are only calculated once, as they are independent of the frequency, provided there is no skin effect. In addition, as opposed to the purely static methods which lead to lumped-circuit models of passive components, DNM deals with a distributed network, wherein the frequency dependence in built-in directly through network equations. Furthermore, in contrast to the conventional PEEC formulation, the DNM also allows to automatically incorporate the effects of the semi-infinite input and output microstrips (or ports) that are attached to the inductor. This easily permits for the electrical characterization of inductors (or other passive elements) through the admittance matrix formulation of the linear network theory. With all of its advantages, DNM has obvious limitations. Since the wave effects are simulated by the coupling between capacitive and inductive parts of the equivalent network, DNM can not be used to accurately predict the frequency response past the first couple of resonances of passive components. However, since the useful frequency range of operation for most such devices is up to the first resonance, DNM offers an effective alternative for their analysis. In addition, the fact that DNM is based on integral equations, different Green’s functions are required for different structural profiles (mostly material variation) in the transverse plane. Nonetheless, this is not a significant drawback, as the variety of transverse profiles of many practical structures is rather limited. Hence, the knowledge of a few Green’s functions will permit characterization of many families of passive circuit

Published in the Proceedings of 49th IEEE Electronic Components and Technology Conference, June 1-4, San Diego, CA.

components, including spiral inductors. Moreover, Green’s functions eliminate the need for discretizing large metal reference planes, which considerably reduces the computational size of the problem. The proposed DNM approach is validated, and, subsequently, employed to compute the reflection and transmission characteristics (S-parameters) of the inductor as functions of the frequency. The substrate material on which the spiral pattern is printed is assumed to be a lossless ceramic. The ground plane and the spiral trace are assumed to be very good conductors, but the trace is electrically thin. As a result, while the current on the ground plane is assumed to flow on the conductor surface, it completely penetrates the entire cross sectional area of the spiral trace. Moreover, since the dimensions of the inductor are small compared to the wavelength of the highest operational frequency of the component, the retardation effects are neglected as well. In addition to DNM, another model for the inductor is presented and its effectiveness is discussed. The model includes a simple lumped element circuit, whose RLC element values are obtained from purely static concepts. The results of this PEEC-based distributed model, consisting of a “coarsely” partitioned structure, are also presented for comparison with DNM. Ample data are given to indicate the range of validity and to show effectiveness of each aforementioned model in predicting the electrical characteristics of planar inductors. In addition, the performance of several spiral geometries is examined as functions of the geometry and various feeding schemes. Overview of DNM Since the emphasis of the paper is on the electrical characteristics of spiral inductors and the literature dealing with PEEC-type techniques is abundant, only the salient features of the DNM will be provided below. To that end, consider a one turn spiral inductor shown in Fig. 1.

The standard approach to obtaining the numerical solution to the above equation is to approximate (or expand) the surface current and charge densities on the metallic strip as a sum of weighted, two-dimensional pulse functions

K x , y ( x, y ) =

N x ,N y

∑ n =1

I( x,y) n ∆y nI , ∆x nI

P( x , y ) n ( x , y )

(A/m) (2)

NQ

Qn P ( x, y ) c c cn n =1 ∆x n ∆y n

ρ s ( x, y ) = ∑

(C/m2) (3)

and then to enforce the equatily in (1) (or test Eq. (1)) at M points on the metallization, i.e., at the center of each current patch (or pulse). The results of the dicretization process are illustrated graphically for a simpler geometry—namely, just a simple T-junction, as shown in Figs. 2 and 3. output port terminal planes input port

z y x

substrate

ground plane

Figure 2: T-junction with discretization for charge density. x-directed current patch

y-directed current patch

input port

ground plane substrate

output port

Figure 3: T-junction with discretization for current density.

Figure1: Geometry of a planar spiral inductor. The electrical characteristics of the inductor can be determined from the following E-field integral equation:

r r r r r r J (r ) / σ (r ) = − jωA(r ) − ∇ φ(r ) .

voltage node (center of charge patch)

(V/m) (1)

where A and φ are the magnetic vector and electric scalar potantials, respectively.

Notice that irregularly shaped pulse functions are used to expand the charge density in order to account for bends in the metallization. Form circuit point of view, this also represents an equipotential sufrace, whose center is defined as a voltage node. To quickly point out how the above discretization scheme leads to an equivalent network interpretation, consider just the x-component of Eq. (1):

1 ∂φ K x ( x, y ) = Rs K x ( x , y ) = − j ωAx ( x, y ) − σ∆z ∂x which after expansion and testing at point m becomes:

Published in the Proceedings of 49th IEEE Electronic Components and Technology Conference, June 1-4, San Diego, CA.

R K (x , y )P ( x , y ) I dam = ∫AmI s x ∆y mI xm A ( x , y ) Pxm ( x , y ) I − jω∫ I x da m . Am ∆y mI ∂φ( x, y ) Pxm ( x , y ) I I −∫ I ∫ I dxm dy m ∆ym ∆xm ∂x ∆y mI

where

(3)

Pictorially, the above result can be interpreted as the KVL around the following circuit: j ωLx I x =

Rx I x

+

+

1 wy

r

∫∫ A( x, y ) ⋅ xˆdxdy Ax

+ V (x −) =

r rm ±1 = x m±1 xˆ + yyˆ , GA and Gφ are the vector and

scalar potential Green’s functions, whose expressions are available elsewhere [11]. The same network interpratation can be given to all other components of Eq. (1). Once the discretization is complete and Eq. (1) has been tested at all M points of the discretized geometry, it can be written in a matrix form:

[Z b ][I b ] = [Vb ]

.

(5)

The branch impedances, voltages and currents correspond to the following equivalent network representation, when associated with the inductor shown in Fig. 1:

+

1 φ( x −) dy wy w∫y

V (x + ) =

1 φ( x + )dy w y w∫y

ground plane (reference voltage)

Figure 4: Circuit interpretation of Eq. (3). This, in essesence, corresponds to the physical geometry of the structure (centered at point m) as illustrated below: Figure 6: Equivalent network of inductor shown in Fig. 1. V(x+)

V(x-) Ix

wy Ax

ground plane (reference voltage)

Figure 5: Relation of circuit in Fig.4 to physical geometry. Once Eq. (3) is tested at every mth voltage node (between xdirected currents), it leads to the following set of equations: Nx

Rm I xm = − j ω∑ Lxmn I xn n =1

NQ  NQ  −  ∑ pmx +1, nQ n − ∑ p mx −1, nQ n    n =1  n=1 

(4)

The R’s, L’s, and p’s in Eq. (4) are the patch (dc) resistances, partial inductances, and electrostatic potential coefficients, respectively, which are given by:

1 ∆xmI Rm = σ m ∆z ∆y mI Lxmn =

p

x m±1 ,n

1 I ∆y ∆y n I m

=

∫ ∫

∆y Im Anc

∫ ∫ AmI

AnI

r r r r Pxm ( r )G A ( r , r ' ) Pxn ( r ' ) da nI da mI

r r Pxm ±1 ( r ) r r Pcn ( r ' ) Gφ ( rm ±1 , r ' ) c c da nI dymI ∆y mI ∆xn ∆y n

It is important to add that the resistances and the mutual capacitances are omitted from Fig. 6 for clarity. Also, it should noted that the inductors connected to the sources and terminations correspond to the semi-infinite sections of the microstrips attached to the inductor. Their special treatment is beyond the scope of this paper, but is available in [11]. Finally, for the frequency-domain solution of the network equations, Eq. (5) is transformed to an admittance matrix form:

[Yn ][Vn ] = {jω[C ] + [ A]([R] + jω[L])−1 [ A]T }[Vn ] = [I n ]

, (5)

where the capacitance matrix is the inverse of the potential coefficient matrix [p] defined above, and [A] is the connectivity matrix [12]. “Coarse” PEEC model From the above discussion, it is evident that the development and implementation of DNM involves considerable effort. Therefore, it is instructive to examine a simpler, but still a PEEC-based, model for the inductor. This model involves a coarse segmentation of the inductor geometry and extraction of the partial RLCs from 3-D field solvers. Furthermore, it neglects the coupling between the inductor (spiral section) and the semi-infinite microstrip lines attached to its input and output ports. As an illustration, consider a two turn spiral inductor shown in Fig. 7, which is segmented into 3 and 5 charge patches (equipotential surfaces or voltage nodes). Note that such discretization requires 2 and 4 corresponding current (or inductance) patches to simulate the current flow between the voltage nodes. The equivalent SPICE circuit for the three patch model is shown in Fig. 8, whose component values were

Published in the Proceedings of 49th IEEE Electronic Components and Technology Conference, June 1-4, San Diego, CA.

calculated using 3-D quasi-static field solvers [13], [14]. The capacitances were computed with respect to ground, while the lumped inductance values include the contributions of the image spiral beneath the ground plane. The resistance values were calculated using the expression for Rm given in the previous section. charge patches

1

inductance patches 2 3

4w

1

0.5w

Higher order equivalent PEEC-based, SPICE circuits can also be constructed by utilizing finer discretization (e.g., 5 segments as shown in Fig. 7). These circuits can be used to calculate the input impedance of the inductor, as well as its insertion and transmission properties (S11 and S12 parameters), as will be illustrated in the following section. Numerical results The numerical implementation of DNM was validated against the full-wave method reported in [15]. The S11 parameter for the 2-loop spiral inductor (see Fig. 7), that was obtained with DMN, is plotted in Fig. 9.

3 2 2.5w

5 4

Figure 7: 2 segmentation levels for PEEC model of inductor. In the above geometry, the ceramic (εr = 9.978) substrate is 25 mils thick, as is the width of the metallization.

Figure 8: Equivalent SPICE circuit for the 3 section model.

Figure 9: S11 parameter of two loop spiral inductor. The spiral winding was discretized into 102 segments, each as wide as the metallization of the strip. Comparison of data in Fig. 9 and Fig. 5 in reference [15] indicates that DMN and the full-wave solution agree up to 20 GHz. In order to compare the range of validity of the “coarse” PEEC models to that of DNM, the S11 parameters extracted from the SPICE circuits based on the 3 and 5 section models (see Figs. 8 and 10, respectively) are shown in Fig. 11. Note that the results generated from the simple PEEC models compare quite well to DMN and full-wave data up to the first resonance of the inductor. Naturally, the 5 section model is considerably better near and at the first resonance than its 3 section counterpart.

Figure 10: Equivalent SPICE circuit for the 5 section model.

Published in the Proceedings of 49th IEEE Electronic Components and Technology Conference, June 1-4, San Diego, CA.

Figure 11: Comparison between the S11 parameter of the 3 and 5 section models. Next, DMN was applied to simulate the response of several shapes that are commonly used to implement planar spiral inductors—specifically, rectangular and circular loops. Figs. 12 and 13 show the effects on the frequency characteristics of the passive component due to the addition of loops. In this case, the dielectric substrate (εr = 9.1) is assumed to be 0.1 mm thick, while the metallization is 0.2 mm wide. The loops are formed of 3 mm long legs, separated by 0.2 mm, with “port” ends being 0.6 mm long. The discretization consists of 351 and 657 square patches (0.067 x 0.067 mm) for the 1- and 2-loop inductors, respectively. As expected, the addition of the second loop increases the total inductance and capacitance of the passive component, resulting in lower resonant frequency.

Figure 12: S11 parameter of one loop spiral inductor.

Figure 13: S11 parameter of two loop spiral inductor. It is important to point out that for this structure similar results can also be obtained with the simple coarse PEEC model, as shown earlier. The lumped partial RLC element values, in this case, can be computed from the existing formulas or static field solvers. However, if the metallization pattern of the spiral does not conform to the canonical rectangular shapes, then 3-D field solvers should be used to extract the equivalent circuit parameters. To demonstrate the application of DNM to non-rectangular spirals, the S-parameters on an “Ohm-shaped” inductor were calculated. For this structure, the S11 and S21 parameters are displayed in Figs. 14 and 15, respectively. The inductor is printed on a 0.3 mm thick, grounded dielectric substrate (εr = 9.1). The width of the metallization is 0.6 mm and the

Published in the Proceedings of 49th IEEE Electronic Components and Technology Conference, June 1-4, San Diego, CA.

approximate loop radius is 3 mm, while the “port” ends are 2.4 mm long. For this structure the discretization was formed of 351, 0.6 x 0.6 mm square patches.

Figure 14: S11 parameter of “Ohm-shaped” spiral inductor.

Figure 15: S21 parameter of “Ohm-shaped” spiral inductor. Finally, it should be added that all computations were carried out on a 766 MHz Alpha based system [16]. Typical computation times, required to perform the frequency sweep (40 points) for an individual inductor with 351 segments, took 368 minutes, i.e., about 9 minutes per frequency point. Conclusions This paper presented an accurate PEEC-type, Green’s function based approach, called DNM, for characterizing passive comp onents. This method was applied to examine the frequency-dependent electrical characteristics of spiral inductors. The results obtained from DNM were validated against full-wave techniques and were compared to ones obtained from simple PEEC models and found to be in good agreement up to the first resonance, showing the utility of simple models at low frequencies. Acknowledgments This work was supported by the US Office of Naval Research under grant N00014-96-0926. References 1. R. A. Johnson, et. al, “A 2.4-Ghz Silicon-on-Sapphire CMOS Low-Noise Amplifier,” IEEE Microwave Guided Wave Lett., vol. 7, no. 10, pp. 350-352, Oct. 1997. 2. J. Mueller, H. Thust, and K. H. Drue, “RF-Design Considerations for Passive Elements in LTCC Material Systems,” Int. J. Microcircuits Electron. Packaging, vol. 18, no. 3, pp. 200-206, 1995.

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G. Avitabile, A. Cidronali, and C. Calvador, “Equivalent Circuit Model of GaAs MMIC-Coupled Planar Spiral Inductors,” Int. J. Microwave Millimeter-Wave CAE, vol. 7, pp. 318-326, 1997. 4. H. Heeb and A. E. Ruehli, “Three-Dimensional Interconnect Analysis Using Partial Element Equivalent Circuits,” IEEE Trans. Circuits Sys.—I: Fund. Theory Applications, vol. 39, no. 11, pp. 974-982, Nov. 1992. 5. X. Zhang and K. K. Mei, “Time-domain finite difference approach to the calculation of frequency-dependent characteristics of microstrip discontinuities,” IEEE Trans. Microwave Theory Tech., vol. 36, no. 12, pp. 1775-1787, Dec. 1988. 6. M. Righi, G. Tardioli, L. Cascio, and W. J. R. Hoefer, “Time-domain characterization of packaging effects via segmentation technique,” IEEE Trans. Microwave Theory Tech., vol. 45, no. 10, pp. 1905-1910, Oct. 1997. 7. D. M. Bica and B. Beker, “Enhanced spatial network method for the analysis of open microstrip discontinuities,” IEEE Trans. Microwave Theory Tech., vol. 45, no. 6, pp. 905-910, June 1997. 8. R. Bunder and F. Arndt, “Efficient MPIE approach for the analysis of three-dimensional microstrip structures in layered media,” IEEE Trans. Microwave Theory Tech., vol. 45, no. 8, pp. 1141-1153, Aug. 1997. 9. L. Zhu and K. Wu, “Characterization of unbounded multiport microstrip passive circuits using an explicit networkbased method of moments,” IEEE Trans. Microwave Theory Tech., vol. 45, no.12, pp. 2114-2124, Dec. 1997. 10. A. C. Polycarpou, P. A. Tirkas, and C. A. Balanis, “The finite element method for modeling circuits and interconnects for electronic packaging,” IEEE Trans. Microwave Theory Tech. , vol. 45, no. 10, pp. 1808-1874, Oct. 1997. 11. R. T. Anderson, Extension of Quasi-Static Methods for the Analysis of Microstrip Discontinuities, M.S. Thesis, University of South Carolina, Columbia, SC, 1997. 12. J. Vlach and K. Singhal, Computer Methods for Circuit Analysis and Design, New York: Van Nostrand Reinhold: 1983. 13. B. Beker, G. Cokkinides, and A. Templeton, “Analysis of microwave capacitors and IC packages,” IEEE Trans. Microwave Theory Tech., vol. MTT-42, no. 9, pp. 17591764, Sept. 1994. 14. D. Jatkar and B. Beker, “Effects of package parasitics on the performance of SAW filters,” IEEE Trans. Ultrason. Ferroelect. Freq. Control, vol. 43, no. 6, pp. 1187-1194, Nov. 1996. 15. W. P. Harakopus and L. P. B. Katehi, “Electromagnetic coupling and radiation loss considerations in Microstrip (M)MIC Design,” IEEE Trans. Microwave Theory Tech., vol. 39, no. 3, pp. 413-421, March 1991. 16. KryoTech/Digital 767 Personal Supercomputer™. For more information see http://www.kryotech.com