A New Analytical Model for Square Spiral Inductors Incorporating a Magnetic Layer Cevdet Akyel #1 , Slobodan Babic #2 , Edvin Skaljo #3 #1 École Polytechnique, Département de Génie Électrique, H3C 3A7, Montréal, Succ., Centre-Ville, Québec, Canada #2 École Polytechnique, Département de Génie Physique, H3C 3A7, Montréal, Succ., Centre-ville, Québec, Canada #3 BH Telecom, Sarajevo, Zmaja od Bosne 88, Bosnia and Herzegovina cevdet.akyel@polymtl.ca, slobodan.babic@polymtl.ca, skaljo@bhtelecom.ba Abstract — Embedding magnetic layer in inductors is an attractive option for increasing inductance density, which is a critical issue for radiofrequency applications. In this work, a magnetostatic model for square spiral inductors incorporating a magnetic layer is developed. This analytical model provides a fast and accurate calculation of inductance for integrated inductors with embedded magnetic layers either for regular cases or a singular case. In the presented model we replaced square spiral inductors with by square rings and we shown that this replacement gives accurately alike results. The results of the presented approach have been confirmed by already published data. Index Terms — Inductors, inductance, magnetic layer, square spiral, rings. I. INTRODUCTION An on-chip spiral inductor is an important component for radio frequency integrated circuits (RFICs) such as low-noise amplifiers, voltage-controlled oscillators, impedance matching networks, inductive coils, apparatus for bio-medical telemetry. Many works have been done on inductor design and modeling. Accuracy in the inductor model is an important part of RFIC design. An inductor typically has one or more "turns" that concentrate the magnetic field flux induced by current flowing through each turn of the conductor in an "inductive" area defined within the inductor turns. In some cases the aspect ratio of the turn can be large so that the turn forms an ellipse or a rectangle. There is a growing need for increased inductance density as the inductors occupy increasingly larger portion of the total circuit area. While the initial efforts were focused on the increased quality factor of inductors, there is a growing need for increased inductance density as the inductors occupy increasingly larger portion of the total circuit area. This is because the scaling, which progressively reduces the area occupied by active devices, does not quite shrink the area taken by passive devices, which is dominated by inductors. The most promising approach to increase the inductance density has been the incorporation of ferromagnetic material into the inductors as a part of their structure [1-10]. In [1] a magnetostatic model for square- shaped spiral inductors above a magnetic layer is developed to calculate the inductance. This method, with some assumptions, has been obtained in the form of a double integral where the double numerical integration is required. In this paper we give the analytical solution of this integral which appears in the form of elementary analytical expressions. This expressions permit fast, easy and accurate calculation of inductance either for regular cases or a singular case. We confirmed the presented formula by results given in [1]. II. BASIC EXPRESSIONS S. Pinhas at al. treated a basic configuration of a square spiral inductor above a magnetic layer, [1], (See Fig.1). The metallic square-spiral is assumed to have zero thickness, and to be located in free space. The configuration is shown in Fig. 1(a). The strip width of the square is denoted by W, its strip spacing by S 0 , its numbers of turns N (N is an integer), and its outer side by a. The distance between the square spiral and the magnetic layer is denoted D 0 . In order to facilitate the calculations, they used the following replacement and additional assumptions, as described below. They replaced the square spiral by concentric square rings with the same strip width W and strip spacing S 0 . In this case the number of the concentric square rings is equal to the number of turns N of the square spiral. Also, the outer side of the outermost square ring is equal to the outer side ‘a’ of the square spiral [3-4]. They assumed that a = 2N (W+ S 0 ). The configuration with this replacement is shown in Fig. 1(b), [1].They calculated the inductance by a magnetostatic model and they expected a good accuracy for sufficiently low frequencies. Also, they assumed that the total current in each square ring was the same for all concentric square rings and that the surface current density was uniform till ‘the diagonal’ of a corner of a square ring. When the current density is rotated by 90 0 it keeps its magnitude. The magnetic layer is infinite in the x and y directions, and the permeability of the magnetic layer is isotropic and infinite. They take the permeability of the half space z < 0 to be isotropic and infinitive so that the magnetic-layer thickness does not appear in their calculations. The permeability of the half-space z > 0 remains the permeability of free space, μ 0 . In the half-space z > 0 the vector potential A is calculated by the method of current image [11]. In this space the vector A is given by [1], Proceedings of the Asia-Pacific Microwave Conference 2011 978-0-85825-974-4 © 2011 Engineers Australia 967