Approximate analytical modeling of current crowding effects in multi-turn spiral inductors

Abstract

The effective series resistance of a multi-turn spiral inductor operating at high frequencies is known to increase dramatically above its DC value due to proxhity-effect or current crowding. This phenomenon is difficult to analyze precisely and has generally required electromagnetic . simulation for quantitative assessment. Current crowding is studied in this work through approximate analytical modeling and fvst order expressions are derived for predicting resistance as a function of frequency. The results are validated through electromagnetic simulation and with measured data taken from a spiral inductor implemented in a Silicon-on-Sapphire process.

A unified analytical and scalable lumped model of RF CMOS spiral inductors based on electromagnetic effects and circuit analysis

“…Considering the layout parameters shown in Fig. 4, the current density is concentrated on the metal edges and can be approximated to be uniformly concentrated in region of w/4 width along the spiral edges [26]. This implies that the effective metal width which is traversed by current is equivalent to half the physical width.…”

Broadband scalable model for Si-RF on-chip spiral inductors with substrate eddy current effect

“…The existing models suffer from broadband accuracy and scalability. Frequency dependent series resistance has been introduced to model the skin and proximity effects on metal resistance in [8,9]. More recently some 1-p and 2-p frequency independent models have been reported in [10][11][12].…”

Spiral inductor performance in deep-submicron bulk-CMOS with copper interconnects

“…The extracted Q and L values for the reference spiral up to SRF are shown in Figure 4, together with the series resistance variation. This series resistance variation is also compared to analytically predicted results using the formulas in [7]. Note that the measured Q has reached a respectable 5 even at 500 MHz, thanks to the low resistance copper metalization and the paralleling of layers 3 through 6.…”

“…The spiral's B-field generated by nearby turns passes perpendicularly through the traces, setting up eddy currents and pushing currents to the trace edges [6]. The result is a quadratic increase in resistance with frequency, with the frequency of on-set determined by trace width, pitch, and sheet resistance [7]. Studies of this problem indicate that narrower traces may be helpful, since the areas of the eddy loops are reduced.…”

“…Raw S11 measurements of the reference spiral from 0 to 6 GHz are shown in Figure 3. S11 measurements of the inductors were then fit to a simple model consisting of a series R and L, with a parallel C placed across this series combination [7]. While more complex 'PI' type models are employed by many authors, these models originated from the desire to model substrate resistance (Rts) effects, and are not needed here due to the use of a ground shield.…”