Equivalent Circuit Model Overview of Chip Spiral Inductors The applications of the chip Spiral Inductors have been widely used in telecommunication products as wireless LAN cards, Mobile Phone and so on. The chip spiral inductors are widely used for bias injection into oscillators, amplifiers, microwave switches, bias tuning varactors, PIN diodes, transistors and monolithic circuits. The parameters of interest for the chip inductors are its inductance (L), quality factor (Q), direct current resistance (R dc ) and self-resonant frequency (SRF). Unfortunately there are no simple formulas to determine them accurately. That has to be through simulations and modeling to determine these parameters. In present market, the software that could effectively help to build model and simulate the chip spiral inductors mostly relies on electromagnetic simulation software. Thus, the design of chip inductors can be optimized using a combination of full-wave electromagnetic solvers such as method-of-moment based tools and quasi-tem approaches. Equivalent Circuit Model Overview When a voltage is applied to the chip inductor terminal pads, it will cause one magnetic and three electric fields appear. The magnetic field caused by the AC current which flow through the tracks of the spiral and it also induces inductive behavior as well as parasitic currents in the tracks and in the substrate; Because of the changing B field, eddy currents are formed which flow in the substrate. This can be modeled as an inductor and resistor loop with appropriate coupling coefficients between the main turn inductor and the eddy inductor. Due to the voltage difference between the spiral connections, an electric field is caused. However, the field will induces ohmic losses because of current
flowing along the tracks of the spiral. An electric field caused by the voltage difference between the turns in the metal that forms the spiral. This induces capacitive coupling between the tracks (interwinding capacitance) because of the dielectric layer. An electric field, due to the voltage difference that appears between the metal of the spiral and the substrate, when the substrate is grounded as usual. It induces a capacitive coupling between the inductance and the substrate, as well as ohmic losses since the electric field penetrates into the conductive substrate. The spiral inductor is, in the most general terms, a distributed structure. There is capacitive and inductive coupling between each of the metal strips, and the series resistance is distributed over the entire circuit [2]. The long length of the spiral coils may render it susceptible to transmission-line effects. In general though, these complicated effects can be ignored for typically-sized inductors up to frequencies well above tens of GHz. Fig.1 The equivalent circuit model of a chip spiral inductor
Figure 1 shows the detailed equivalent circuit model for a chip spiral inductor. It is basically identical to the model for a thin film resistor, and in fact its low-frequency resistance can be calculated using the resistor methodology. The key differences in the design of the two structures are that inductors are usually formed in a spiral pattern, whereas resistors are generally formed in a meander pattern. The spiral pattern maximizes the proximity of lines in which current flows in the same direction while minimizing proximity of lines in which current flows in the opposite direction. This results in a positive mutual inductance that reinforces the self-inductance of the device. The meander pattern of a resistor, on the other hand, places opposing current flows next to each other to obtain a negative mutual inductance that helps to offset the self -inductance of the device. The general definition of quality factor for inductors is Q = Im(Z11)/Re(Z11) ------------------ (1) where Z11 is the device impedance. Usually, the most important contribution to the real part of the impedance come from R, and to the imaginary part from L, in which case Equation (1) becomes the more familiar form, Q= (jωl)/rs --------------------(2) The series resistance, Rs, takes into account the skin depth of a conductor with finite thickness [3] and the current distribution in a microstrip conductor [4] The series resistance, Rs, can be expressed as ----------------------(3) where ρ m and L represent the resistivity and length of the wire. If t is the physical thickness of the wire, an effective thickness t eff can be defined as -------------------- (4) Skin effects can be significant for inductors, and can have a detrimental effect
on the quality factor. Current density at a frequency f decays exponentially inward from the surface of a conductor with a 1/e decay distance(skin depth), δ, given by δ= ρ/(πfu), -------------------------(5) where ρis the metal resistivity and u its permeability. This effect leads to an apparent rise in metal resistance and a slight decrease in inductance a higher frequencies. Generally, when metal film thickness is less than the skin depth, current flow is nearly uniform and the effect is negligible. In spiral inductors, however, a related current crowding effect occurs across the width of the inductor line, leading to a rise in resistance and to a minor decrease in the inductance at higher frequencies. Spiral inductors are primarily used in high frequency applications. Realistic values for spiral inductors range from about 1 to 100nH. Values smaller than 1nH can be accomplished by simple, straight-line wire segments and values much greater than 100nH are physically too large, and are better made using permeable cores. An important consideration in the design of these components is their self-resonant frequency. In the model shown in Figure 1, this is given by F res = 1/(2Π LC p ) Unfortunately, it is not generally possible to do a straightforward analytical calculation of Cp. The two most common methods are to rely on empirical formulae, which are necessarily limited in their generality, or to use finite-element numerical methods, which are slow and require large amounts of computation and memory. Like the other passive components, the model shown in Figure 1 includes coupling to ground. Lossy dielectric materials or significant capacitive coupling to ground contributes to these elements, whose main effects are to degrade Q (through Equation 22) and to lower the self-resonant frequency. Inductance calculations are straightforward for inductors that are completely isolated from ground and also for inductors that lie above a good conducting ground plane. The intermediate case, commonly encountered in inductors integrated directly onto a chip, poses an especially difficult analytical problem, since the method of images will not work. In these cases, the resistive coupling to ground becomes an important factor in the overall
device losses, and the inductance is reduced by image currents induced in the substrate. The overall effect is significant reduction in the inductor quality factor. References [1] H. M. Greenhouse, "Design of Planar Rectangular Microelectronic Inductors," IEEE Trans. Parts, Hybrids, and Packaging, v. PHP-10, no.2, June 1974, pp. 101-109 [2] C.P. Yue, "On-Chip Spiral Inductors for Silicon-Based Radio-Frequency Integrated Circuits," Ph.D. thesis, Integrated Circuits Laboratory, Stanford University, 1998. [3] H.-S. Tsai, J.Lin, R.C.Frye, K.L.Tai, M.Y.Lau, D. P. Kossives, F. Hrycenko and Y.K.Chen, Investigation of current crowding effect on spiral inductors, Proc. IEEE MTT-S Int Topical Symposium on Techologies for Wireless Applications, Vancouver, BC, Canada. [4] J.Zhao, W.Dai, R.C.Frye and K.L.Tai, Modeling and Design Considerations for Embedded Inductors in MCM-D, Poc.Int. Conf. Multichip Modules, 340-344, Denver,CO, April 2-4, 1997.