Electrical Characteristics of Planar Spiral Inductors

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 49 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.