Miniature inductors and related circuit components and methods of making same

Abstract

New types of circuit elements for integrated circuits include structures wherein a thickness dimension is much greater than a width dimension and is more closely spaced than the width dimension in order to attain a tight coupling condition. The structure is suitable to form inductors, capacitors, transmission lines and low impedance power distribution networks in integrated circuits. The width dimension is on the same order of magnitude as skin depth. Embodiments include a spiral winding disposed in a silicon substrate formed of a deep, narrow, conductor-covered spiral ridge separated by a narrow spiral trench. Other embodiments include a wide, thin conductor formed in or on a flexible insulative ribbon and wound with turns adjacent one another, or a conductor in or on a flexible insulative sheet folded into layers with windings adjacent one another Further, a method of manufacture includes directional etching of the deep, narrow spiral trench to form a winding in silicon.

Claims

1. A miniature inductor comprising: a semiconductor substrate constraining a maximum size of the inductor; a coil of the miniature inductor overlying the semiconductor substrate, said coil shaped in a spiral as a multiple-turn winding with spacing between adjacent turns, said coil having a silicon core extending along a length of the coil and metal extending along both sidewalls of the silicon core, wherein immediately adjacent turns of the coil include a first turn and a second turn, the first turn and the second turn separated by a first metal extending along a first sidewall of the silicon core of the first turn and a second metal extending along a second sidewall of the silicon core of the second turn, the first metal and the second metal separated by a material having a composition that is different from that of the silicon core and different from that of the first metal and the second metal; the multiple-turn winding being configured with a high aspect ratio defined as a thickness dimension much greater than a width dimension, and wherein adjacent turns of the winding are closely spaced on a scale comparable to the width dimension to attain a tight coupling condition defined as a high coupling coefficient across multiple turns.

2. The inductor of claim 1, the multiple-turn winding having a thickness-to-width ratio of at least ten and a width-to-spacing ratio between adjacent turns of at least five such that the tight coupling condition is attained with a coupling coefficient of at least 0.5 at the second most adjacent turn.

3. The inductor of claim 1 wherein the width dimension is on the same order of magnitude as skin depth.

4. The inductor of claim 1 wherein the multiple-turn winding has a straight innermost segment with a Q that is greater than one at the design frequency of the inductor.

5. The inductor of claim 1 wherein the space between the adjacent turns of the coil comprises a dielectric material.

6. The inductor of claim 1 wherein the metal extending along the sidewalls of the silicon core includes a first metal layer and a second metal layer that is different from the first metal layer, and the space between the adjacent turns of the coil comprises a dielectric material.

7. The inductor of claim 1 wherein the semiconductor substrate is an interposer, further including terminals of the multiple-turn winding; and pads coupled to the multiple-turn winding through the terminals to provide for external electrical connections.

8. The inductor of claim 7 further including solder bumps juxtaposed to the pads to establish the electrical connections.

9. The inductor according to claim 8 wherein at the terminals of the multiple-turn winding, a plurality of silicon columns disposed in parallel in the substrate are provided, the silicon columns being sufficiently closely spaced to produce an electrically conductive array adjacent the pads to enhance electrical conductivity and reduce electrical resistance to the pads.

10. The inductor according to claim 9 wherein the silicon columns are of a cross-section selected from square, rectangular, triangular, circular, oblong, combination thereof and interconnections among combinations.

11. The inductor according to claim 8 wherein the pads are arranged in the semiconductor substrate in a regular rectangular array.

12. The inductor according to claim 11 wherein the pads include an input pad and an output pad and wherein the input pad is aligned with the output pad on a common row of the rectangular array.

13. The inductor of claim 1 wherein the semiconductor substrate includes a first side and a second side, the first side being separated from the second side by an insulative layer, and wherein a semiconductor circuit is provided on the first side and the multiple-turn winding is provided on the second side.

14. The inductor of claim 13 wherein the coil is formed after the semiconductor circuit is formed, the coil further including terminals extending from the second side through the insulative layer to the first side.

15. The inductor of claim 1 wherein the metal extending along the sidewalls of the silicon core includes a first metal layer and a second metal layer that is a different from the first metal layer.

16. The inductor of claim 15 wherein the first metal layer comprises tantalum, and the second metal layer comprises at least one of copper or aluminum.

17. A miniature inductor comprising: a semiconductor substrate; and a coil of the miniature inductor on the semiconductor substrate, said coil shaped in a spiral as a multiple-turn winding with spacing between adjacent turns, said coil including a silicon core extending along a length of the coil with metal extending along both sidewalls of the silicon core, wherein immediately adjacent turns of the coil include a first turn and a second turn, the first turn and the second turn separated by a first metal extending along a first sidewall of the silicon core of the first turn and a second metal extending along a second sidewall of the silicon core of the second turn, the first metal and the second metal separated by a material having a composition that is different from that of the silicon core and different from that of the first metal and the second metal; wherein the multiple-turn winding is configured with a high aspect ratio defined as a thickness dimension much greater than a width dimension, and wherein adjacent turns of the winding are closely spaced on a scale comparable to the width dimension to attain a tight coupling condition defined as a high coupling coefficient across multiple turns.

18. The miniature inductor of claim 17 wherein the high aspect ratio is 10 or more.

19. The miniature inductor of claim 17 wherein the high coupling coefficient is at least 0.5.

20. A miniature inductor comprising: a substrate comprising a silicon and oxygen layer; and a coil of the miniature inductor disposed on the silicon and oxygen layer of the substrate, the coil shaped as a multiple-turn winding with spacing between adjacent turns, the coil comprising a silicon core with metal extending along sidewalls of the silicon core, each turn of the coil comprising the silicon core with the metal on each side, wherein immediately adjacent turns of the coil include a first turn and a second turn, the first turn and the second turn separated by a first metal extending along a first sidewall of the silicon core of the first turn and a second metal extending along a second sidewall of the silicon core of the second turn, the first metal and the second metal separated by a material having a composition that is different from that of the silicon core and different from that of the first and the second metal.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 0.1 is a top view of a spiral winding inductor according to a first embodiment of the invention.

(2) FIG. 0.2 is a perspective view of a spiral winding inductor of FIG. 0.1.

(3) FIG. 0.3 is a top view of a flex spiral winding inductor according to a second embodiment of the invention.

(4) FIG. 0.4 is a perspective view of the inductor of FIG. 0.3.

(5) FIG. 0.5 is a perspective view of a third embodiment of the invention illustrating a flex-formed folded coil.

(6) FIG. 1.1 is a diagram for illustrating parameters for analyzing an inductor.

(7) FIG. 1.2 is a graph for illustrating how width and thickness of a conductive winding relate.

(8) FIG. 1.3 is a first diagram for defining the relationship of mutual inductance.

(9) FIG. 1.3.1 is a diagram for defining length and spacing.

(10) FIG. 1.3.2 is a second diagram for defining the relationship of mutual inductance.

(11) FIG. 1.4 is a graph showing the relationship of thickness to cross-sectional area of a conductive winding.

(12) FIG. 1.5.1 is a graph relating inductance to length of an inductor to cross-sectional area and thickness.

(13) FIG. 1.6.1 is a top view of a square spiral-type inductor according to the invention to be built into an interposer structure illustrating analysis parameters according to the invention.

(14) FIG. 1.7.1 is a cross-sectional side view of a first interposer structure-built spiral inductor showing C4 bump terminals on top and bottom of the interposer for external connection and IC connection.

(15) FIG. 1.7.2 is a cross-sectional side view of a first coupling between an interposer structure and an integrated circuit.

(16) FIG. 1.8.1 is a cross-sectional side view of a second interposer structure-built spiral inductor with C4 bump terminals connected on one side of the device.

(17) FIG. 1.8.2 is a cross-sectional side view of a second coupling between an interposer structure (inverted as compared to FIG. 1.8.1) and an integrated circuit.

(18) FIG. 1.9.1 is a cross-sectional side view with parameters for analyzing an interposer structure inductor.

(19) FIG. 1.9.2 is a cross-sectional side view with parameters for analyzing a further interposer structure inductor in accordance with the invention.

(20) FIG. 1.10 is a graph illustrating skin depth as a function of frequency.

(21) FIG. 1.11.1 is a graph illustrating the Bernoulli Function.

(22) FIG. 1.11.2 is a graph illustrating a detail of the Bernoulli Function.

(23) FIG. 1.12 is a diagram in side cross-sectional view of a multi-layer inductor with parameters marked as used for analysis.

(24) FIG. 1.13.1 is a graph illustrating a solution for optimal thickness in an inductor according to the invention.

(25) FIG. 1.14.1 is a graph illustrating a range of solutions for optimal thickness in an inductor according to the invention.

(26) FIG. 1.14.2 is a graph illustrating a detail of FIG. 1.14.1.

(27) FIG. 1.15.1 is a graph illustrating a range of solutions for optimal thickness.

(28) FIG. 1.15.2 is a graph illustrating a detail of FIG. 1.15.1.

(29) FIG. 1.16.1 is a graph illustrating a range of solutions for optimal thickness.

(30) FIG. 1.16.2 is a graph illustrating a detail of FIG. 1.16.1.

(31) FIG. 1.17.1 is a graph illustrating a range of solutions for optimal thickness.

(32) FIG. 1.17.2 is a graph illustrating a detail of FIG. 1.17.1.

(33) FIG. 1.18.1 is a graph to show the magnitude of Q.sub.PEAK variation as a function of frequencies for various inductors 20, 40, 60 and 80 nH

(34) FIG. 1.18.2 is a graph to show the uniform current density and current density distribution with “single sided solution” approximation.

(35) FIG. 1.19 is a graph showing physically practical solutions at various thicknesses and Q.

(36) FIG. 1.20.1 is a graph showing normalized magnetic fields across a four-winding inductor.

(37) FIG. 1.20.2 is a graph showing relative magnetic field in the innermost winding.

(38) FIG. 1.21.1 is a graph illustrating a single sided solution vs. a complete solution for various widths.

(39) FIG. 1.21.2 is a graph showing a detail of FIG. 1.21.1.

(40) FIG. 1.22 is a graph showing a relationship between Q and length.

(41) FIG. 1.23 is a top plan view drawing of a spiral inductor constructed according to the invention.

(42) FIG. 1.24 is a portion of a perspective view of a spiral inductor according to the invention.

(43) FIG. 1.25 is side cross-sectional view of a spiral inductor according to the invention with the mathematical matrix equivalent of the inductor.

(44) FIG. 1.26 is a plot of a Table from Grover of Log(k) used in defining a geometric mean distance (g.m.d.) as a function of the ratio of w vs. d between two rectangles.

(45) FIG. 1.27 is a plot of a further Table from Grover of Log(k) used in defining a geometric mean distance (g.m.d.) as a function of the ratio oft vs. d between two rectangles.

(46) FIG. 1.28 is a plot of a further Table from Grover showing ranges of Log(k) and Log(k) about very thin structures.

(47) FIG. 1.29 is a graph illustrating g.m.d. for adjacent rectangles having narrowest sides facing one another.

(48) FIG. 1.30 is a first combined graph to illustrate the differences in two cases of adjacent rectangles, where Case 2 is a preferable embodiment for an inductor having a winding cross section according to the invention.

(49) FIG. 1.31.1 is a second combined graph to illustrate the differences in two cases of mutual inductance and self-inductance, where Case 2 is a preferable embodiment for the cross section and spacing of a winding of an inductor according to the invention.

(50) FIG. 1.31.2 is a log scale graph of FIG. 1.31.1.

(51) FIG. 1.32 is a third combined graph to illustrate the differences in two cases, where Case 2 is a preferable embodiment of spacing of rectangular windings of an inductor according to the invention as based on coupling coefficients.

(52) FIG. 1.33 is a schematic representation of windings to illustrate tight coupling aspect ratio relations vs. adjacent spacing width and number of turns.

(53) FIG. 1.34 is a graph illustrating the relation between number of tightly coupled turns and metal aspect ratio and spacing aspect ratio to determine coupling to non-adjacent turns beyond 1.

(54) FIG. 1.35 is a graph illustrating the relation between number of tightly coupled turns and metal aspect ratio and spacing aspect ratio to determine coupling to non-adjacent turns beyond 2.

(55) FIG. 1.36 is a graph illustrating the relation between number of tightly coupled turns and metal aspect ratio showing impact on total inductance at 50μ metal thickness conditions for 500μ nominal length of four turns according to the invention.

(56) FIG. 1.37 is a graph illustrating the relation between number of tightly coupled turns and metal aspect ratio showing impact on total inductance under 100μ metal thickness conditions for 500μ nominal length of four turns according to the invention.

(57) FIG. 1.38 is a graph illustrating the relation between number of tightly coupled turns and metal aspect ratio showing impact on total inductance under 200μ metal thickness conditions for 500μ nominal length of four turns according to the invention.

(58) FIG. 1.39 is a graph illustrating the relation between number of tightly coupled turns and metal aspect ratio showing impact on total inductance under 300μ metal thickness conditions for 500μ nominal length of four turns according to the invention.

(59) FIG. 1.40 is a graph illustrating the relation between number of tightly coupled turns and metal aspect ratio showing impact on total inductance under 50, 100 and 200μ metal thickness conditions for a nominal length of 600μ according to the invention.

(60) FIG. 1.41 is a graph illustrating Q as a function of metal aspect ratio for 600μ nominal length of four turns according to the invention.

(61) FIG. 1.42 is a graph illustrating inner to outer winding width a function of metal aspect ratio for five turns allowing small construction according to the invention.

(62) FIG. 1.43 is a graph in log scale illustrating inner to outer winding width a function of metal aspect ratio for five turns allowing small construction according to the invention.

(63) FIG. 1.44 is a top plan view of a segment of an inductor according to the invention.

(64) FIG. 1.45 is a perspective view of a segment of an inductor according to the invention.

(65) FIG. 1.46 is a top plan view of two adjacent segments of an inductor according to the invention.

(66) FIG. 2.1 through-FIG. 2.10 show first processing steps of the HARMS process according to the invention.

(67) FIG. 2.11 is a SEM image of a grid of test structures illustrating silicon core encapsulated by tantalum.

(68) FIG. 2.12 is an expanded SEM image of a grid of test structures illustrating silicon core encapsulated by tantalum.

(69) FIG. 2.13 is a schematic of a spiral structure according to the invention illustrating a preferred alignment of pad layout for interconnection of an interposer-type structure.

(70) FIG. 2.14 is a top plan view of a pad for an interposer structure.

(71) FIGS. 2.15 through FIG. 2.19 are illustrations of the result of process steps for interconnection of interposer C4 bumps.

(72) FIG. 2.20 is a schematic top plan view illustrating wafer scale electroplating of interposer structure devices according to the invention.

(73) FIGS. 2.21 through FIG. 2.26 are illustrations of the result of processing steps involved for backside C4 bumping for SiO wafers.

(74) FIG. 2.26.1 through FIG. 2.26.4 are illustrations of the result of processing steps involved in making fully integrated structures according to the invention for buried oxide thicknesses between one and two microns and active IC layer between one and three microns, thus eliminating the interposer structure.

(75) FIG. 2.26.5 through FIG. 2.26.11 are illustrations of the result of processing steps involved in making fully integrated structures according to the invention for buried oxide thicknesses between one and two microns and active IC layer greater than three microns, thus eliminating the interposer structure.

(76) FIG. 2.26.12 through FIG. 2.26.16 are illustrations of the result of processing steps involved in making fully integrated structures according to the invention for a buried oxide thickness of about 20 nm according to the SIMOX process and an active layer of less than about 200 nm, thus eliminating the interposer structure.

(77) FIG. 2.27 is a first schematic side view of adjacent windings illustrating critical dimensions of a first size.

(78) FIG. 2.28 is a second schematic side view of adjacent windings illustrating critical dimensions of a second size.

(79) FIG. 2.28.1 is a graph illustrating silicon core and total winding width for different silicon core thicknesses.

(80) FIG. 2.28.2 is a graph illustrating silicon to silicon core spacing for different silicon core thicknesses.

(81) FIG. 2.29 is a graph illustration needed Si Core thickness t.sub.3Si, to achieve a desired Δ.sub.w values for δ.sub.EM=2, 4, 6 and 8μ.

(82) FIG. 2.30 is a graph illustrating the cross-sectional area of electro-plated metal area that determines the resistance of a spiral inductor.

(83) FIG. 2.31.1 is a graph illustrating the amount of current capacity as a function of silicon core thickness for different thicknesses of electroplating of a spiral inductor

(84) FIG. 2.31.2 is a graph illustrating the resistance vs. silicon core thickness of a 5 mm long line for different thicknesses of electroplating of a spiral inductor.

(85) FIG. 2.32 is a schematic side cross-sectional view illustrating C4 bumping on both sides of an interposer structure of a spiral inductor according to the invention.

(86) FIG. 2.33 is a graph illustrating maximum allowable DC resistances for DC/DC converter inductors.

(87) FIG. 2.34.1 is a graph illustrating the relationship between core column numbers and silicon and electroplating thickness in FIG. 2.17.1.

(88) FIG. 2.34.2 is a graph illustrating the relationship between “silicon through via” (STV) resistance and inductor contact resistance vs. silicon core thickness and electroplating thickness.

(89) FIG. 2.35 is a top pan view of an on-chip capacitor according to the invention.

(90) FIG. 3.1 is a diagram illustrating prior art standard parameters of a Flex PCB structure.

(91) FIG. 3.2 is a diagram illustrating prior art standard dimensions of a Flex PCB structure.

(92) FIG. 3.3 is a diagram of a prior art standard Flex to PCB connector.

(93) FIG. 3.4 is a diagram of a test structure for three different widths of Flex PCB traces as may be implemented according to the invention.

(94) FIG. 3.5 is a schematic top plan view of a portion of a tested structure with a standard connector.

(95) FIG. 3.6 is a schematic top plan view of a Flex PCB inductor according to the invention showing Flex to PCB connector mounting.

(96) FIG. 3.7 is a schematic side cross-sectional view of a Flex PCB inductor according to the invention showing Flex to PCB connector mounting.

(97) FIG. 3.8 is a schematic side cross-sectional view of a prior art two-layer Flex PCB ribbon illustrating dimensions.

(98) FIG. 3.9.1 is a schematic top plan view of a two-port transformer.

(99) FIG. 3.9.2 is a schematic top plan view of a transformer with a secondary with center tap.

(100) FIG. 3.10 is a schematic top plan view of a balun with voltage gain.

(101) FIG. 3.11 is a schematic top plan view of a balun with voltage drop.

(102) FIG. 4.1 is schematic top plan view of a folded stacked Flex structure.

(103) FIG. 4.2 is a schematic top plan view for defining critical dimensions of a folded stacked Flex structure.

(104) FIG. 4.3 is a schematic top plan view of a folded stacked Flex structure illustrating that the segments between folds need not be straight.

DETAILED DESCRIPTION OF THE INVENTION

(105) Referring to FIG. 0.1, there is shown a top view of a spiral inductor 10 embedded in a silicon substrate 11 according to a first embodiment of the invention, wherein the winding 12 is laid out in a rectangular pattern, width w 18 being comparable to the spacing s 16 between segments of the winding, and the depth or thickness t 14 of the winding 12, not shown in FIG. 0.1 but as shown in the perspective view of FIG. 0.2, being substantially greater than the width w 14, in accordance with the invention. The ratio of thickness to width may be on the order of 5, 10, 15 or even 20 to 1, all within the contemplation of the invention based on specific design criteria for inductance, current-carrying capacity and maximum operational frequency. Terminals 20, 22 connect via interposers 24, 26. Dimensions of this structure for an 80 nH inductor are typically less than one mm across and in certain applications are capable of inductance in excess of 100 nH while sustaining high currents (in excess of several Amperes) and/or high frequencies (in excess of 100 MHz) with good Q (in excess of 10), all depending upon certain design considerations, such as winding cross-sectional area (w×t), inter-winding spacing, conductor thickness, conductor width and material resistivity, as well as winding length, as hereinafter defined and shown by way of example. The element 10 may be fabricated in a silicon substrate and mounted on a silicon circuit (not shown) within a restricted pad area. (This disclosure does not teach how to form an inductor of this design in a semiconductor circuit, as that is beyond the scope of this disclosure.)

(106) FIG. 0.2 illustrates the device 10 of FIG. 0.1 showing that the thickness t 14 to be substantially greater than either width w 18 or spacing s 16. A novel manufacturing process is disclosed hereinafter.

(107) Alternative designs, are shown in FIG. 0.3, FIG. 0.4 and FIG. 0.5. As hereinafter explained, is fabricated of a conductive winding in a ribbon manufactured according to Flex PCB Technology adapted to miniaturized application. It is wound tightly around a central air core forming a web. The core may be circular or rectangular. Accordingly the thickness dimension is on a curved plane or series of flat planes and the width dimensions of the winding are juxtaposed and spaced apart only by minimal space and the thickness of the ribbon.

(108) Device 310 of FIG. 0.5 may likewise be formed according to Flex PCB technology. However, the winding is embedded in a ribbon laid out in a serpentine pattern and the ribbon is folded against itself in a zip-zag pattern so as to separate winding segments by the substrate. The winding segments are arranged to be at least partially juxtaposed and separated only by the thickness of the substrate and any air gap.

(109) Only a few windings are shown for illustrative purposes, but there are normally four or more, up to about ten windings. Various patterns of serpentine traces embedded in ribbon may be realized, differing from layer to layer, although a circular or rectangular spiral is likely simpler to design, to fabricate, to analyze and to test. The embodiment of FIG. 0.5 may be mounted either horizontally (flat) on a semiconductor substrate, or vertically, with terminals A and B. The Flex PCB embodiments are specifically useful for circuit verification purposes. Due to the manufacturing processes, air gaps, winding tightness and alignment may be inconsistent, but the devices are useful as inexpensive proof-of-concept and breadboard components.

(110) More precise tolerances may be achieved where the devices are manufactured in a semiconductor substrate, particularly as herein disclosed. The steps are explained in more detail hereinbelow with respect to actual structures.

(111) The basic steps are as follows:

(112) Step A, a semiconductor substrate of silicon over silicon oxide of suitable thickness is provided. If an interposer design is contemplated, then the substrate is stock material. If the inductor is to be integrated into the same chip with a semiconductor circuit, the circuit is formed first and is usually in the silicon layer the surface of the chip or wafer opposite from the inductor.
Step B, a spiral channel 602 is etched into the silicon substrate that is of sufficient width to leave a spiral ridge of width less than winding width w and of sufficient separation for a winding spacing s to yield a continuous ridge in a spiral surface pattern of a design length, wherein height of the ridge is sufficient to establish winding thickness t. Thus the channel is of a depth corresponding to the design thickness t and spacing s+w in the substrate to yield a continuous ridge in a spiral surface pattern (cf. FIG. 0.1 or FIG. 0.2) and of a design length. The height of the ridge 604 is less than or equal to thickness t as hereinafter explained in connection with the design parameters.
Step C, a binding material 606 is applied to the surface of the ridged semiconductor material, for example titanium nitride, tantalum or like common semiconductor-to-metal adhesive. Application is by means of conventional processing, and it should at least cover the length, width and depth of the ridge 604.
Step D, a conductor 608, such as copper, aluminum or gold metal, is plated onto the entire ridge 604, including all tops and sidewalls, and bound to the ridge 604 by the binder. For this purpose, electroplating is suitable, since it is capable of adhering to sidewalls. The build-up depth of the plating process is determined by the intended spacing s between plated facing walls of the ridge 604. The width of the ridge plus the combined thickness of the plating corresponds to the intended width of the winding w.
Step E, the bottom 610 of the channel 602 is etched away, thus forming an air gap between walls of the ridge 604 along its length and forming the winding of thickness t, width w separated by spacing s between windings. Both conductive faces of the ridge 602 are conductively couple across the top. This selective etching process is realized by the technique of directional etching 612. Directional etching includes for example deep silicon reactive etching (the Bosch process), plasma etching or possibly ion beam etching, along a spiral path tracing the bottom 620 of the channel 602. To the extent that side walls may be etched in the process, the electroplating step may include adding sufficient material to the side walls to compensate for residual etching.

(113) Alternative etching techniques, such as use of photoresist patterns are also within the contemplation of the invention. Moreover, the technology to perform the semiconductor fabrication process may be an older, larger-spacing processing technology than is used for fabrication of the associated semiconductor chip. It can now be seen how a high inductance, high current capability, high frequency, high Q, and closely spaced, high-aspect-ratio-winding inductor can be realized that is suitable for embedding on (and eventually in) a semiconductor chip.

(114) The following is a design tutorial leading to an understanding of the parameters employed according to the present invention, plus a description of specific embodiments of the invention made according to these design criteria.

(115) Self and Mutual Inductance Calculations

(116) An excellent source of inductance and mutual inductance calculations can be found in the classic book Inductance Calculations by Frederick Grover, first published in 1946 [1]. Due to demand by those working in the field it has been re-printed many times and remains a valuable source. In that work, numerical calculations of any arbitrary cross-sections for self inductance and mutual inductance of their arbitrary arrangements in spatial coordinates are used. What follows here is a practical guide for design with respect to self-inductance and mutual inductance considerations.

(117) The general formula of the DC self inductance (internal inductance for uniform current density distribution) of a prism geometry with any given cross-section can be given as the general inductance formula [1]:

(118) $\begin{array}{cc}{L}_{\mathrm{GENERAL}}=0.002.Math.{10}^{-6}l.Math.\left[\ln \left(\frac{2l}{r}\right)-1+\frac{{\delta }_1}{l}\right]& \left(1.1\right)\end{array}$
where l, r, δ.sub.t are in cm and (1.1) gives L.sub.GENERAL in Henry units, r and δ.sub.1 being geometric- and arithmetic-mean-distance-related quantities, which are related to the cross-sectional geometry of the prism, which is assumed to have a uniform current distribution flowing through its cross-section. Geometric Mean Distance (g.m.d) and Arithmetic Mean Distance (a.m.d) are important concepts in inductance and mutual inductance calculations [1,2]. The “geometric mean distance” (g.m.d.) between any pair of n points can be expressed as,

(119) $\begin{array}{cc}g.m.d.={\left(\underset{i=1}{\overset{n}{.Math.}}{r}_i\right)}^{\frac{1}{n}}=\exp \left[\frac{1}{n}\underset{i=1}{\overset{n}{.Math.}}\ln \left(r_i\right)\right]& \left(1.2\right)\end{array}$

(120) Similarly, “arithmetic mean distance” (a.m.d.) can be expressed as,

(121) $\begin{array}{cc}a.m.d.=\frac{1}{n}\underset{i=1}{\overset{n}{.Math.}}{d}_i& \left(1.3\right)\end{array}$
where r.sub.i and d.sub.i are distances between pairs of points in selected regions as can be shown in FIG. 1.1, where P(i).sub.u, and P(i).sub.v represent cross sections of two adjacent conductive structures through which current may flow, commonly corresponding with the cross section of windings of an inductor, which are generally NOT rectangular in cross section. As will be explained hereinafter, rectangular cross sections are an important design feature of the invention hereafter explained. The summations (1.2) and (1.3) for very large n can be generalized using multiple integrals performed in closed areas which have very interesting applications in many partial differential equation problems encountered in electromagnetics and quantum mechanics [2]. These integrals take on the form of quadruple integrals, which also can be calculated by numerical integration with some care in Cartesian coordinates.

(122) The general inductance formula (1.1) can be written explicitly for specific cross-sectional geometries. For rectangular cross-sections, of the type upon which this invention is focused, Equation (1.1) becomes.

(123) $\begin{array}{cc}{L}_{\mathrm{RECT}}=0.002.Math.{10}^{-6}l.Math.\left[\ln \left(\frac{2l}{w+t}\right)+0.5-f_1\left(w,t\right)\right]& \left(1.4\right)\\ \mathrm{where}& \\ 0\le f_1\left(w,t\right)\le 0.00249l,w,t.fwdarw.\mathrm{cm}.fwdarw.L_{\mathrm{RECT}}\left[H\right]& \end{array}$
where, l, w, and t are the length, width and thickness, respectively.

(124) For a circular cross-section the general formula Equation (1.1) becomes,

(125) $\begin{array}{cc}{L}_{\mathrm{CIRC}}=0.002.Math.{10}^{-6}l.Math.\left[\ln \left(\frac{2l}{r}\right)-0.75\right]l,r.fwdarw.\mathrm{cm}.fwdarw.L_{\mathrm{CIRC}}\left[H\right]& \left(1.5\right)\end{array}$
where, r is the radius of the circular cross-section.

(126) For a coaxial or ring cross-section which is also named as a “hollow cylinder” Equation (1.1) becomes,

(127) $\begin{array}{cc}{L}_{\mathrm{RING}}=0.002.Math.{10}^{-6}l.Math.\left[\ln \left(\frac{2l}{r_1}\right)+f_2\left(r_1,r_2\right)-1\right]& \left(1.6\right)\\ \mathrm{where}& \\ 0\le f_2\left(r_1,r_2\right)\le 0.25l,r_1,r_2.fwdarw.\mathrm{cm}.fwdarw.L_{\mathrm{RING}}\left[H\right]& \end{array}$
where, r.sub.1 and r.sub.2 are the outer and inner radiuses of the coax, ring or the hollow cylinder.

(128) For elliptical cross-section where a and b are the major and minor axes of the ellipse, (1.1) becomes,

(129) $\begin{array}{cc}{L}_{\mathrm{ELLIPS}}=0.002.Math.{10}^{-6}l.Math.\left[\ln \left(\frac{2l}{a+b}\right)-0.75\right]l,a,b.fwdarw.\mathrm{cm}.fwdarw.L_{\mathrm{ELLIPS}}\left[H\right]& \left(1.7\right)\end{array}$

(130) A closer look at relations (1.4-1.7) shows that there is a minimum length l.sub.MIN as a function of the cross-sectional dimensions where these formulas apply. For the rectangular cross-section, which is the main focus in this invention, l.sub.MIN can be expressed as a function of w and t by solving,

(131) $\begin{array}{cc}\ln \left(\frac{2l_{\mathrm{MIN}}}{w+t}\right)+0.5-f_1\left(w,t\right)=0& \left(1.8\right)\end{array}$

(132) The function ƒ.sub.1(w,t) is plotted in FIG. 1.2 using the same notation as given in Grover [1]. Being an older work, the notation used is no longer conventional. The reason for using Grover's notation is to retain the expressions as original as possible. As can be seen the maximum value for ƒ.sub.1(w,t) is 0.00249, so we can write,

(133) $\begin{array}{cc}\ln \left(\frac{2l_{\mathrm{MIN}}}{w+t}\right)=0.00249-0.5=-0.49751e^{-0.49751}=0.608e^{-0.5}=0.60653& \left(1.9\right)\end{array}$

(134) A solution of (1.9) for l.sub.MIN will give the minimum value for l for which (1.4) will be valid. Below this l.sub.MIN value Equation (1.4) will give a negative inductance value, which is not physical. Therefore (1.4) is valid only for l>l.sub.MIN. The value of l.sub.MIN depends on the sum of width w and the thickness t and can be given as,

(135) 0$\begin{array}{cc}{l}_{\mathrm{MIN}}=\frac{w+t}{2}{e}^{f_1\left(w,t\right)-0.5}\cong \frac{w+t}{2}{e}^{-0.5}=0.30326.Math.\left(w+t\right).& \left(1.10\right)\end{array}$
Similar calculations can be done to obtain l.sub.MIN for other cross-section geometries.

(136) The Neumann integral formulation of mutual inductance for straight infinitely thin filaments (the “Filament Method”) as shown in FIG. 1.3 is,

(137) $\begin{array}{cc}M={\int }_{P_{u1}}^{P_{u2}}{\int }_{P_{v1}}^{P_{v2}}\frac{\stackrel{.fwdarw.}{u}.Math.\stackrel{.fwdarw.}{v}}{r}{\mathrm{ds}}_u{\mathrm{ds}}_v& \left(1.11\right)\end{array}$
where, u and v are unit vectors on each filament.

(138) Performing the integral (1.11) for two parallel lines with a length of l and spacing d as shown in FIG. 1.3.1 leads to,

(139) $\begin{array}{cc}M=0.002.Math.{10}^{-6}.Math.l.Math.\left[\ln \left(\frac{l}{d}+\sqrt{1+\frac{l^2}{d^2}}\right)-\sqrt{1+\frac{d^2}{l^2}}+\frac{d}{l}\right].& \left(1.12\right)\\ l,d.fwdarw.\mathrm{cm}.fwdarw.M\left[H\right]& \end{array}$

(140) To calculate the mutual inductance between any parallel two arbitrary cross-sectional geometries with the same length l—under the assumption of uniform current density distribution—one calculates the (g.m.d) between the two cross-sectional geometries and substitutes the g.m.d. value as the variable d in the filament formula in (1.12).

(141) The “Filament Method” can be generalized for any filament arrangement and placement in three-dimensional space by double integrals in the Neumann integral formulation given by (1.11), again under the assumption of constant current density distribution. For n coupled inductors, where n>1, given as a multi-inductor system shown in FIG. 1.3.2, one can define an (n×n) “Inductance Matrix” L as,

(142) $\begin{array}{cc}L\left(\mathrm{nxn}\right)=\left[\begin{array}{cc}{L}_{1,1}{L}_{1,2}{L}_{1,3}\mathrm{.Math.}& L_{.,.}{L}_{1,n-1}{L}_{1,n}\\ L_{2,1}{L}_{2,2}{L}_{2,3}\mathrm{.Math.}& L_{.,.}{L}_{2,n-1}{L}_{2,n}\\ L_{3,1}{L}_{3,2}{L}_{3,3}\mathrm{.Math.}& L_{.,.}{L}_{3,n-1}{L}_{3,n}\\ \mathrm{.Math.}& \\ \mathrm{.Math.}& \\ L_{n-1,1}{L}_{n-1,2}{L}_{n-1,3}\mathrm{.Math.}& L_{.,.}{L}_{n-1,n-1}{L}_{n-1,n}\\ L_{n1}{L}_{n,2}{L}_{n,3}\mathrm{.Math.}& L_{.,.}{L}_{n,n-1}{L}_{n,n}\end{array}\right]L_{i,j}=L_{j,i}\mathrm{and}{L}_{i,j}>0& \left(1.13\right)\end{array}$

(143) The diagonal matrix entries of (1.13) are self-inductance values, which are always positive. The off-diagonal entries can be positive, negative or zero. The Inductance Matrix L is ALWAYS symmetric as shown.

(144) A quantity that is very important in a mutual inductance matrix for satisfying the passivity requirement of the system is the coupling ratio K, which is expressed as,

(145) $\begin{array}{cc}{K}_{i,j}=\frac{L_{i,j}}{\sqrt{L_{i,i}.Math.L_{j,j}}}\mathrm{and}.Math.K_{i,j}.Math.<1& \left(1.14\right)\end{array}$
where K.sub.i,j, L.sub.i,j, L.sub.i,i and L.sub.j,j are the coupling ratio, self inductances of elements i and j, the mutual inductances between inductances i and j respectively.

(146) For an inductance matrix L resulting from same cross-sectional geometry and lengths will give equal self inductances at the diagonals. Having the same values in the diagonal of the inductance matrix (1.14) states that,
|K.sub.i,j|<1.fwdarw.L.sub.i,j≤L.sub.i,i.  (1.15)
In other words, for this case the mutual inductances in the inductance matrix can never exceed the self-inductance values! This is a very important concept that allows for the derivation and realization of the invention herein disclosed.
Inductance for a Rectangular Cross-Section Having a Constant Area S and Under Constant Current Density J.sub.MAX Across its Cross-Section

(147) As will be explained hereinafter, embodiments of the invention employ a rectangular cross-section. One must always comply with minimum cross-section requirements when designing an inductor for any application. Sources for this minimum cross-sectional area can be many and are listed below, depending on the application;

(148) i) For on-chip power converter applications [3-9], a large DC current must flow through the inductor, at least in the order of several Amperes (1-50 A) [3-9]. This imposes a minimum cross-sectional area S.sub.MIN requirement on the inductor design due to the electro-migration current density limit that cannot be exceeded. This electro-migration current density limit J.sub.EM is a function of the chip metallization process. As an example, for a typical Aluminum/Si alloy metallization, J.sub.EM is in the order of 10.sup.5 to 2.Math.10.sup.6 A/cm.sup.2. For a given maximum DC current specification, one cannot exceed the current density imposed by the electro-migration current density limit, therefore one cannot make the inductor cross-sectional area smaller than S.sub.MIN. S.sub.MIN can be calculated as,

(149) $\begin{array}{cc}{S}_{\mathrm{MIN}}=\frac{I_{\mathrm{DCMAX}}}{J_{\mathrm{EM}}}& \left(1.16\right)\end{array}$
where I.sub.DCMAX is the maximum DC current determined by the design specification.

(150) Table [1] shows the typical cross-sectional areas that need to be satisfied for some practical I.sub.DCMAX values common in today's power management and FIVR work.

(151) TABLE-US-00001 TABLE 1 Inductor Current Determined Cross-Sectional Area, t.sub.OPT0 and t.sub.OPT for 100 MHz and 200 MHz for Inductor Values of 20, 40, 60 and 80 nH. I, Current (A) Cross-Section t.sub.OPT (μ) t.sub.OPT (μ) t.sub.OPT (μ) t.sub.OPT (μ) J.sub.max = 2.10.sup.6 Area L = 20 nH L = 40 nH L = 60 nH L = 80 nH A/cm.sup.2 S (μ.sup.2), t.sub.OPT0 (μ) f = 100 MHz f = 100 MHz f = 100 MHz f = 100 MHz 1 50, 7.07 27.63 28.83 30.08 30.08 5 250, 15.81 107.9 117.5 122.6 122.6 10 500, 22.36 195.9 213.3 222.6 222.6 20 1,000, 31.62  355.6 387.3 404.1 421.7

(152) ii) When designing an inductor, one always has a resistance R.sub.IND in series with the inductor. Again, in power converter applications, the efficiency of a step-down or buck converter is closely related to the DC resistance of the inductor R.sub.IND and cannot exceed an R.sub.IND_MAX value that is a small load-dependent quantity which can be given as,

(153) $\begin{array}{cc}{R}_{\mathrm{IND}\_\mathrm{MAX}}=\frac{V.Math.\left(1-\eta \right)}{I.Math.\eta }& \left(1.17\right)\end{array}$
where V, η and I are the DC voltage output, power conversion efficiency and load current, respectively. As an example, a 1V DC supply with a 1 A load is equivalent to 1Ω load resistance. For a 90% efficiency goal in a buck converter design with no switching losses, one has to keep the DC resistance R.sub.IND_MAX of the inductor less than 0.11Ω. A typical “conservative” FIVR on-chip buck converter specification is 10 A at 1V. As can be seen for this case the DC resistance of the inductor R.sub.IND_MAX has to be less than 0.011Ω regardless of the inductor value. This can be a major challenge. FIG. 2.33 shows a graph of (1.17) illustrating the allowable maximum DC resistance R.sub.IND_MAX as a function of output DC current for 1 VDC output for 95, 90, 85 and 80% theoretical maximum efficiencies in any type of DC/DC converter topology regardless of the inductor values.

(154) iii) The circuit topology of LNA (Low Noise Amplifier) IC designs can employ several on-chip inductors, transformers or Baluns (Balanced-un-Balanced). Since the thermal noise voltage generated in a resistor is proportional to the square root of its resistance at the operating frequency, the resistance value is an important contributor to the overall noise figure of the LNA [31]. Therefore one needs to keep the resistance of these passive elements less than an R.sub.MAX value at the operating frequency. The R.sub.MAX value at a given frequency for this case is not straight-forward to calculate, but it is related to its DC resistance, and the “smaller the better” principal always applies. The AC-resistance-to-DC-resistance relation is given for rectangular cross-sections hereinafter.

(155) iv) For VCO/PLL (Voltage Controlled Oscillators/Phase Lock Loop) applications the phase noise is inversely proportional to the square of the Q value of the inductor. Q of an inductor. It is a complex quantity to calculate as a function of frequency, but it is very closely related to the AC resistance of the inductor. AC resistance is related to the DC resistance R.sub.DC, so again the “smaller the better” rule applies for R.sub.MAX!

(156) v) The power consumption limit on the inductor is another design factor in determining the R.sub.MAX value at the frequency of operation. It is given by relation,

(157) $\begin{array}{cc}{R}_{\mathrm{MAX}}=\frac{P_{\mathrm{MAX}}}{I_{\mathrm{MAX}}^2}& \left(1.18\right)\end{array}$
where P.sub.MAX is the maximum power consumption specification at a given operating frequency on the inductor, where R.sub.MAX is the frequency dependent resistance of the inductor. Again R.sub.MAX is related to R.sub.DC.

(158) As can be seen, there is one factor that directly defines the S.sub.MIN of the inductor design and there are an additional four factors that are indirectly related to it through the DC resistance formula of a prism, given as,

(159) $\begin{array}{cc}{R}_{\mathrm{DC}}=\rho \frac{l}{S}=\rho \frac{l}{w.Math.t}& \left(1.19\right)\end{array}$
where l, ρ, S, w and t are the length, resistivity, cross-sectional area, width and thickness for a rectangular cross-section straight conductor.

(160) In any kind of inductor design one starts with a straight wire and generates geometries that give mutual inductances such that their presence increases the inductance compared to its straight conductor inductance value. Therefore, there is a need to formulate a geometrical configuration that provides for high mutual couplings with the proper sign in the designed structure, such as in coils and spiral inductors. Calculating even R.sub.DC for any kind of inductor design such as spiral or coil inductors is not straightforward, but it is straightforward to calculate it for a straight conductor. That is a good point to start the analysis.

(161) Optimum Width/Thickness Relation for a Desired Inductor Value L for a Given Cross-Sectional Area S for a Rectangular Cross-Section Straight Conductor Giving Highest Possible Q for Uniform Current Distribution Assumption Across its Cross-Section

(162) The “exact” result of the starting analysis under this model is incorrect, but it is a suitable starting point for the optimization of an inductor, and it conveys the key points and the thought process of the invention very quickly and clearly, as hereinafter shown.

(163) The problem at this stage is to find out if one can design an inductor with a desired value L with minimum resistance R for a given cross-sectional area S. If there is such a width/thickness combination, using this combination will give the highest Q possible for a straight inductor. Ignoring capacitive and high frequency effects on the quality factor Q, the simple case is given as,

(164) $\begin{array}{cc}{Q}_{\mathrm{LRDC}}=\frac{L\omega }{R}& \left(1.20\right)\end{array}$

(165) As can be seen, the Q formulation (1.20) assumes that both the resistance and the inductance of the straight wire is frequency independent. Therefore the subscript DC in (1.20) indicates the low frequency Q, where L and R are not taken as functions of frequency and where there are no capacitance effects. These effects will be brought in to the analysis later.

(166) First, inductor relation for a rectangular cross-section, as given in (1.4) as a function of given area S is,

(167) 0$\begin{array}{cc}{L}_{\mathrm{RECT}}=0.002.Math. {10}^{-6}l.Math.\left[\ln \left(\frac{2l}{\frac{S}{t}+t}\right)+0.5-f_1\left(w,t\right)\right].& \left(1.21\right)\end{array}$

(168) As can be seen in (1.21) the width w, in (1.4) is just replaced by S/t. Close examination of (1.21) suggest that one can expect a thickness t, which gives the shortest length l for a desired inductance value L for any given cross-sectional area S. Ideally one needs to differentiate (1.21) with respect to t, equate it to zero and solve t to perform this task. This requires several intermediate differentiation steps in the process. It can be done with simpler and shorter intermediate math and will be more conclusive, as well as get the point across more quickly and clearly. (We will have to take the longer approach for Q optimization later, so no need to complicate the matter at this very first stage!)

(169) Let the u(t,S) function, which is the denominator term in the log function given in (1.21) be defined as,

(170) $\begin{array}{cc}u\left(t,S\right)=w+t=\frac{S}{t}+t.& \left(1.22\right)\end{array}$

(171) The t value, which gives the minimum for u(t,S), will satisfy this “minimum length for a given inductance value of L for any given S” condition by differentiating it with respect to t as,

(172) $\begin{array}{cc}\frac{\mathrm{du}}{\mathrm{dt}}=1-\frac{S}{t^2}& \left(1.23\right)\end{array}$
and equating (1.23) to zero and solving gives,

(173) $\begin{array}{cc}\frac{\mathrm{du}}{\mathrm{dt}}=0.fwdarw.t=\sqrt{S}.& \left(1.24\right)\end{array}$

(174) As can be seen in (1.24), there is an optimal thickness t, for a given S and moreover the width w and thickness t are equal, which is directly the result of area relation, giving,
w=t=√{square root over (S)}.  (1.25)

(175) The advances in the art that form the basis of the present invention can now be identified. One conclusion is this: To minimize the length of a conductor for a desired inductor value of L for a given cross-sectional area S in a rectangular cross-sectional geometry, one would need a square cross-section with sides given as (1.25)!

(176) FIG. 1.4 shows the u(t,S) for several cross-sectional areas where S=50, 250, 500 and 1000μ.sup.2, which are calculated from I.sub.MAX=1, 5, 10 and 20. A design specification is contemplated using J.sub.EM=2.Math.10.sup.6 A/cm.sup.2 to have some “real” cases related to FIVR (Fully Integrated Voltage Regulator) cases in the analysis. As can be seen in FIG. 1.4 the u(t,S) function has an asymptote at t=0 and starts from infinity, goes through a minimum as calculated in (1.25) and approaches the u=t function, again asymptotically, with increasing t from zero to infinity. With the uniform current density distribution approximation, designing the inductor with a square cross-section and sides given as (1.25) will give the highest possible Q because it will have the shortest length giving minimum resistance R given by (1.18) for the desired L and given S! This “minimum length” result is shown in FIG. 1.5.1 which are plots of straight conductor inductor length versus thickness having constant cross-sectional areas S=50, 250, 500 and 1000μ.sup.2. The target inductance values in the FIG. 1.5.1 are 20, 40, 60 and 80 nH, which are considered “very large” value inductors for any on-chip inductor designs. For high frequency LNA/VCO designs, with operating frequencies greater than 900 MHz, the typical targeted on-chip inductor values are between 1 to 5 nH. On the other hand, in studies of the current hot topic of integrating the buck converter on a processor, or FIVR (Fully Integrated Voltage Regulator), the investigated low DC resistance and high Q inductor values are in 10-60 nH range, operating at 20-200 MHz [3-9]. As can be seen, in the simplest Q relation given in (1.20), designing high Q inductors for operating frequencies in the order of 20-200 MHz, along with DC resistances in the order of 0.020 ohms, such as in FIVR work, makes the challenge much greater than for higher frequency RFIC inductor designs.

(177) The typical metal thickness in IC (Integrated Circuit) technology is less than a micron. Very few IC processes provide higher than about 3μ to 4μ metal thickness and only at the top metal layer (M5) with 2.8μ width and spacing. As can be seen, the thicknesses obtainable from this very first simplified analysis are much larger than any metal thicknesses provided in any known IC process technology, which are shown in Table [1] as t.sub.OPT0.

(178) Combining the resistance formulation given by (1.19) for uniform current density distribution, one can plot the Q versus thickness for given L and S values as well and see the peak Q being at the calculated thicknesses. As noted above, the simplified analysis shown here is very important in showing that there is an optimal w and t as a function of a given cross-sectional area S for maximizing Q and it is also independent from the targeted inductance value. Closer examination however demonstrates that this “qualitative” result is misleading. It is misleading in that the resulting cross-sectional geometry, which is said to be a square and is dependent on the given area S, is given as in (1.25). The misleading result comes from the “constant current density” assumption given by (1.19) for the resistance which cannot be satisfied at operating frequencies of the inductors under consideration due to skin and proximity effects. The following section includes an explanation of the non-uniform current density distribution effects in a rectangular cross-section with some detail, taking the skin effect into consideration.

(179) Optimum Width/Thickness Relation for a Desired Inductor Value L for a Given Cross-Sectional Area S for a Rectangular Cross-Section Straight Conductor Giving Highest Possible Q for Non-Uniform Current Distribution Assumption Across its Cross-Section

(180) The general analysis of this problem can be tackled by solving Maxwell's equations for a sinusoidal wave assumption which can be reduced to the solution of the Helmholtz's equation in non-uniform media. Just for the electric field E, Helmholtz wave equation in complex form can be written as [24, 25],

(181) $\begin{array}{cc}{▽}^{2}E+{\omega }^2\mu \mathrm{.Math.}\left(1-j\frac{\sigma }{\omega \mathrm{.Math.}}\right)E=0\mathrm{where}j=\sqrt{-1}\mathrm{and}\omega =2\pi f& \left(1.26\right)\end{array}$
where μ, ε, σ, ω and f are the magnetic permeability, dielectric constant, conductivity, angular frequency and frequency, respectively. A similar equation can be written for the magnetic fields H and it can be shown that they are related [24,25]. A complete spiral inductor analysis can be done by solving (1.26) for the three dimensional spiral geometry (which is possible only numerically). Employing Maxwell's equation which leads to Helmholtz wave equation (1.26) in three dimensions, one can calculate electric and magnetic fields and the non-uniform current density distribution in any point of any of the winding regions [10-23]. This is a complex analysis, but looking at the results of the complex simulations and using some common electromagnetic sense, the analysis can be simplified. Here we will first focus on the principles and the important analytical results of the solution of the simplified Helmholtz wave equation for plane waves, which is applicable to the spiral inductor regions in the interior of its winding as shown in a very simple spiral inductor geometry shown in FIG. 1.6.1. There are several choices of forcing the current I in and getting it out of the spiral inductor. Several of many choices are shown with pad locations at IN.sub.CENTER, OUT.sub.CENTER, IN.sub.SIDE and OUT.sub.SIDE locations in FIG. 1.6.1.

(182) The cross-sectional geometry of the spiral inductor structure shown in FIG. 1.6.1 is given in FIG. 1.7.1. The cross-section geometry is obtained by having cut planes passing from x=x.sub.CUT or y=y.sub.CUT lines perpendicular to the (x,y) plane as shown in FIG. 1.6.1. For simplicity the shown spiral structure has a square inner space with a dimension of d.sub.IN and has its windings have a constant width w and spacing s. Since this invention defines a spiral inductor with a large aspect ratio metal and metal spacing, the FIG. 1.7.1 points to this feature of the invention as is shown very clearly. Known spiral inductor structures have metal winding widths much larger than their thicknesses as shown in FIG. 1.9.1 due to the metallization achievable in any IC, PCB or ceramic processes. If one can build an interposer structure with metallization rules based on what is required for these inductors and place it on top of the IC or under it with the shortest connections possible to the IC, this will give a very practical solution to the problem.

(183) FIG. 1.7.1 also shows another key aspect of the invention in the interposer structure having very thick metal along with very high aspect ratios for metal lines and spacing, which has its connections to the outside circuit from both sides of the structure. FIG. 1.7.1 shows this very preferable capability for power management IC applications. In these circuits, as in Buck converters, the inductor is placed between the supply and the switching network [3-9,30]. The switching network is built into the IC and the inductor is placed between the switch and the supply pin [3-9, 30]. According to this invention, however, the inductor is built on the interposer structure and is connected to the IC from the bottom of the interposer as shown in FIG. 1.7.2 with a ball, a standard technique used in ball grid arrays, and it will connect to the supply pin from the top. As can be seen having the interposer right on top of the IC with connections available on both sides of the interposer saves very valuable space. On the other hand if the interposer access were only possible from one side, a very valuable area will be lost in packaging. In fact for this case, placing the interposer right on top of the chip will not even give any area saving advantage for any power management IC applications.

(184) In some other applications, such as in RFIC, PLL and VCO's, both of the pins of the inductor have to be connected to the IC. This is illustrated in FIG. 1.8.1 and FIG. 1.8.2. In this case the high value, high Q inductor is built on the interposer and, through the two balls on the bottom of the interposer, it is connected to the desired pads in the IC. Since there can be multiple inductors built on the interposer for both types of circuits, both sides of the interposer can be used to connect the IC through the interposer to the outside world with additional passive circuits on the interposer with improvements in performance and packing density without having any area penalty.

(185) FIG. 1.9.1 and FIG. 1.9.2 show, according to the invention, two kinds of rectangular cross-section spiral winding arrangements, both having the same cross-sectional areas S, but placed differently. The top arrangement shown in FIG. 1.9.1, which is noted as subscript 1, represents the typical arrangement in prior art spiral inductor designs having width w larger than thickness t or w>>t. This choice is dictated by the need for a small R.sub.DC for a given thin metal thickness t, so it is something one cannot avoid for a practical design specification. The bottom arrangement, shown in FIG. 1.9.2, noted with the subscript 2, and made according to this invention, shows conductors having thickness t larger than the width w, written as t>>w. To be able to realize this structure, one must employ a process with metal thicknesses fairly large compared to what is available today as shown in Table [1]. As can be seen in this arrangement the conductors are intentionally placed with longer dimensions facing each other and shorter dimension facing toward the ground plane or the substrate. Width w and the thickness t are dimensions defined along x and z directions of the coordinate system. The electromagnetic field propagation in the region is assumed to be along y direction, coming out of the plane of FIG. 1.9.1 and FIG. 1.9.2 and with no electromagnetic field components along the propagation direction, which is known as TEM mode. As a result the electric and magnetic waves only have x and z components as shown in FIG. 1.9.1 and FIG. 1.9.2 which converts the problem into a two-dimensional problem.

(186) When an electromagnetic wave enters a conductive region, such as into the windings of the spiral cross-section shown in FIG. 1.9.1 and FIG. 1.9.2 from the lossless dielectric region between the windings and between the conductors and the ground plane, it will attenuate exponentially with an attenuation constant α along its propagation direction into the conductor as a result of the solution of the Helmholtz wave equation (1.26). Solving (1.26) for plane waves, this attenuation constant α for electric and magnetic fields into the conductors will be [24, 25],

(187) $\begin{array}{cc}\alpha =\omega \sqrt{\frac{\mathrm{.Math.}\mu }{2}\left(\sqrt{1+{\left(\frac{\sigma }{\omega \mathrm{.Math.}}\right)}^2}-1\right)}& \left(1.27\right)\end{array}$
under a “good conductor” approximation which is defined as,

(188) $\begin{array}{cc}\frac{\sigma }{\omega \mathrm{.Math.}}⪢1.& \left(1.28\right)\end{array}$

(189) Equation (1.27) simplifies to the well-known “skin depth” [24, 25] given as,

(190) $\begin{array}{cc}\delta =\frac{1}{\alpha }=\sqrt{\frac{2}{\omega \sigma \mu }}=\sqrt{\frac{\rho }{\pi \sigma \mu f}}.& \left(1.29\right)\end{array}$

(191) FIG. 1.10 show the skin depth as a function of frequency between 10 Hz-10 GHz for Cu, Al and Si having 1 and 10 ohm-cm resistivity. Since the trend and the desire is to integrate inductors on a silicon base material as the substrate, FIG. 1.10 includes the skin depth in typical Si substrate resistivity ranges of silicon. In all cases, relative magnetic permeability is taken as μ.sub.r=1 giving μ=1.25663.10.sup.−8H/cm in (1.29). In (1.29) the resistivity of Cu and Al is taken as 1.75.Math.10.sup.−6 and 2.73.Math.10.sup.−6 ohm-cm respectively, and the skin depth as a function of frequency is plotted and shown in FIG. 1.10. As can be seen, the lower three curves (Cu, Al, 1 ohm-cm Si) skin depth varies inversely proportional with the square root of the frequency in the log scale linearly at the entire frequency range, which tells us that the “good conductor” approximation (1.28) is valid for these cases. On the other hand, at frequencies higher than 2 GHz the 10 ohm-cm resistivity Si material shows some small amounts of flattening. This can be seen at much lower frequencies for higher resistivity Si. In other words the “good conductor” approximation holds very well for Cu, Al and for 1 ohm-cm resistivity Si in the entire frequency range, but not as good for 10 ohm-cm and higher resistivity Si above 2 GHz. Therefore it is always a good practice to check if the “good conductor” approximation holds in the frequency range of interest.

(192) From the calculated electric and magnetic field distributions in the conductor regions of FIG. 1.9.1 and FIG. 1.9.2 one approximates the non-uniform current density distributions in the spiral windings, which will result in they direction current flow as observed in spiral inductor structures.

(193) One can speculate on three possible forms of solutions for the non-uniform current density distribution in the spiral windings:

(194) i) Current Density Distribution is Uniform at the Surface of the Conductors.

(195) This solution cannot apply to the majority of the known spirals which have w>>t. If the width and thicknesses (w,t) are not in the same order and if the distance d.sub.G between the ground plane and winding is not on the same order of magnitude as the spacing s, the external field distribution on the conductor surface will be far from uniform and therefore cannot support this boundary condition. Therefore it only has its merit for square cross-sections in conductors having similar s and d.sub.G. This mode has an analytical solution for circular cross-sections [24] using Bessel functions (Ber, Bei), but for a rectangular cross-section, one needs a numerical solution of the Helmholtz wave equation where an analytical solution does not exist.

(196) On the other hand, if t>>w and s<<d, and there are vertically stacked inductor winding structures which have w>>t, d (as shown for illustrative purposes in FIG. 1.12), in a structure formed according to the invention, then numerical simulations suggest that this boundary condition—namely, uniform surface current density distribution—becomes a fairly good approximation of the actual current density distribution. FIG. 1.12 only shows one side of the vertically stacked inductor winding structures where the layers are connected through vias upward and downward, which is basically a structure of coils made in a planar process with a large number of metal layers.

(197) ii) Majority of the Fields are Confined Between the Bottom of the Conductors and the Ground Plane.

(198) This solution applies fairly well for PCB (Printed Circuit Boards), ribbons and the majority of the on-chip spiral inductors where w>>t and s>d.sub.G. For this case a fairly good analytical derivation of the current density distribution can be given, which is once again in line with the solution of the Helmholtz wave equation for the exact structures. This type of current density distribution in the windings is referred as the “single sided solution” in this disclosure. Therefore the straightforward analytical analysis will be given once and for the case related to the invention below.

(199) iii) Majority of the External Fields are Between the Windings.

(200) The solution in this case is applicable to one of the embodiments of the invention explained in this work given as, t>>w, d.sub.G<<s. This approximation also applies to the structure where longer sides of the windings are along the x axis, but are stacked as shown in FIG. 1.12, a complete 90° flip of the structure of a preferred embodiment of the invention.

(201) Single Sided Current Density Distribution Assumption and its Impact on R.sub.AC/R.sub.DC

(202) Consider an analytical derivation for the conductor cross-section shown on the right of FIG. 1.9.2, which is referred as “single sided solution”. For now assume the metal width of the windings shown in FIG. 1.9.2 is infinite, in other words w.sub.2.fwdarw.∞ and the current density at x=0 is J.sub.0. For the given current density J.sub.0 at the conductor surface (x=0) the solution will give [26-29],

(203) $\begin{array}{cc}{J}_{\mathrm{AC}}\left(x\right)=J_0{e}^{-\frac{x}{\delta }}\mathrm{where}\delta =\sqrt{\frac{2}{\sigma \omega \mu }}.& \left(1.30\right)\end{array}$

(204) In reality the electric and magnetic fields can enter from both sides of the spiral windings. For this case the problem becomes more complex for spiral inductor current density calculations in its windings and it is referred as the “complete solution” and will be discussed herein below.

(205) Integrating the current density relation (1.30) from 0 to the width w gives the total current flowing in the conductor and is formulated as,

(206) $\begin{array}{cc}{I}_{\mathrm{AC}}={\int }_0^w{J}_0{e}^{-\frac{x}{\delta }}\mathrm{dx}=J_0\delta \left(1-e^{-\frac{w}{\delta }}\right).& \left(1.31\right)\end{array}$

(207) On the other hand the DC current, where the current density is uniform, is given as,
I.sub.DC=J.sub.0w.  (1.32)

(208) If the windings were infinitely wide, then the total AC current as a result of (1.31) becomes,
lim.sub.w.fwdarw.∞(I.sub.AC)=J.sub.0δ.  (1.33)

(209) As can be seen in (1.33), the AC current does not go to infinity as it would in the DC current case! We can ratio the AC current expression (1.31) to the DC current relation (1.32) and get the frequency-dependent AC resistance as,

(210) 0$\begin{array}{cc}\frac{R_{\mathrm{AC}}}{R_{\mathrm{DC}}}=\frac{I_{\mathrm{DC}}}{I_{\mathrm{AC}}}=\frac{w}{\delta .Math.\left(1-e^{-\frac{w}{\delta }}\right)}.& \left(1.34\right)\end{array}$

(211) Rearranging variables in (1.34) gives,

(212) $\begin{array}{cc}\frac{R_{\mathrm{AC}}}{R_{\mathrm{DC}}}=\frac{u}{\left(1-e^{-u}\right)}\mathrm{where}u=\frac{w}{\delta }.& \left(1.35\right)\end{array}$

(213) The expression (1.35) is the Bernoulli Generation Function [27] for n=1 defined as,

(214) $\begin{array}{cc}n=1.fwdarw.\frac{t}{\left(1-e^{-t}\right)}=\underset{m=0}{\overset{\infty }{.Math.}}{B}_m\frac{{\left(-t\right)}^m}{m!}.& \left(1.36\right)\end{array}$

(215) In shorthand, we refer to (1.36) as B(u). B(u) is a Bernoulli function that will give only positive values for −∞<u<∞ and 1 at u=0 as shown in FIG. 1.11.1. The region of interest according to this invention is the u>0 region and as can be seen, (1.35) becomes linearly dependent on the w/δ ratio after w/δ>5, giving,

(216) $\begin{array}{cc}u=\frac{w}{\delta }>5.fwdarw.R_{\mathrm{AC}}\approx R_{\mathrm{DC}}\frac{w}{\delta }.& \left(1.37\right)\end{array}$

(217) As an example for u=w/δ=10, one can right away say R.sub.AC/R.sub.DC will be very close to 10 without any calculations. If the w/δ ratio is small, by solving the nonlinear equation (1.35) for a desired acceptable R.sub.AC/R.sub.DC value at a given frequency one can determine the width w as a function of δ. FIG. 1.11.2 shows this process graphically very clearly. FIG. 1.11.2 shows that to have R.sub.AC/R.sub.DC=2 the width w has to be 1.6×δ. If the width is set equal to δ, the R.sub.AC/R.sub.DC will be 1.6 at that frequency, which is very suitable for an inductor design.

(218) The top curve in FIG. 1.21.1 shows R.sub.AC/R.sub.DC as a function of u, which is the w/δ ratio. FIG. 1.21.2 shows R.sub.AC/R.sub.DC as a function of width at 100 MHz the for Cu. As can be seen from a skin-depth-versus-frequency curve shown in FIG. 1.10 and its relation to R.sub.AC/R.sub.DC as shown in FIG. 1.21.1 and FIG. 1.21.2, one has to keep the width of the windings not much larger than the skin depth. These correspond to fairly small widths for the vertical metal arrangement shown in FIG. 1.9.2. The same is true for the thickness for a horizontal metal arrangement as shown in FIG. 1.9.1. In this case having a thick metal compared to skin depth at that frequency will not improve the R.sub.AC/R.sub.DC significantly but will only help to reduce the DC resistance. Since we need to maintain a given cross-sectional area S, the only solution is to increase the thickness t of the vertical metal arrangement as shown in FIG. 1.9.2.

(219) Table [2.0] shows skin depth as a function of typical frequencies for copper and aluminum. As will be noted the skin depth is on the same order of magnitude as the width dimension of the inductor.

(220) TABLE-US-00002 TABLE 2 Skin Depth (δ) as a Function of Selected Typical Frequencies for Copper (1.75 .Math. 10.sup.−6 Ω .Math. cm) and Aluminum (2.73 .Math. 10.sup.−6 Ω .Math. cm) and Critical Width or Thicknesses to Maintain R.sub.AC(f)/R.sub.DC = 2. δ(Al) × 1.6 [μ] δ(Al) × 3.85 [μ] δ(Cu) × 1.6 [μ] δ(Cu) × 3.85 [μ] R.sub.AC/R.sub.DC = 2 R.sub.AC/R.sub.DC = 2 R.sub.AC/R.sub.DC = 2 R.sub.AC/R.sub.DC = 2 “Single Sided” “Complete” “Single Sided” “Complete” f [MHz] δ(Al) [μ] Solution Solution δ(Cu) [μ] Solution Solution 25 16.63 26.608 64.025 13.22 21.152 50.897 50 11.76 18.816 45.276 9.348 14.957 35.989 100 8.316 13.306 32.017 6.610 10.576 25.449 200 5.880 9.408 22.638 4.674 7.479 17.995 900 2.772 4.435 10.672 2.203 3.525 8.483 1,200 2.401 3.841 9.242 1.908 3.053 7.347 1,575 2.095 3.353 8.067 1.666 2.665 6.414 2,400 1.697 2.716 6.533 1.349 2.159 5.195 5,200 1.153 1.845 4.439 0.916 1.467 3.529
Single Sided Current Density Distribution Assumption and its Impact on Q

(221) In designing a high Q inductor, it is important to calculate the R.sub.AC besides the R.sub.AC/R.sub.DC. The AC resistance can be calculated as,

(222) $\begin{array}{cc}{R}_{\mathrm{AC}}=R_{\mathrm{DC}}\frac{u}{\left(1-e^{-u}\right)}\mathrm{where}u=\frac{w}{\delta }.& \left(1.38\right)\end{array}$

(223) For a given length l, substituting the DC resistance relation (1.18) in (1.38) the AC resistance becomes,

(224) $\begin{array}{cc}{R}_{\mathrm{AC}}=\rho \frac{l}{w.Math.t}\frac{u}{\left(1-e^{-u}\right)}\mathrm{where}u=\frac{w}{\delta }=w\sqrt{\frac{\mu \omega }{2\rho }}.& \left(1.39\right)\end{array}$

(225) With some arithmetic manipulation, (1.39) gives,

(226) $\begin{array}{cc}{R}_{\mathrm{AC}}=\frac{1}{t}\sqrt{\frac{\rho \mu \omega }{2}}\frac{1}{\left(1-e^{-u}\right)}\mathrm{where}u=\frac{w}{\delta }=w\sqrt{\frac{\mu \omega }{2\rho }}.& \left(1.40\right)\end{array}$

(227) As can be seen in (1.40) the width dependency AC resistance is very weak compared to the DC resistance for large values of u!

(228) Another interesting result is that the R.sub.AC is no longer directly proportional to resistivity ρ as in R.sub.DC. It becomes linearly proportional to the square root of the resistivity and frequency.

(229) Rearranging (1.40) gives,

(230) $\begin{array}{cc}{R}_{\mathrm{AC}}\left(\omega ,l,t,w,\rho ,\mu \right)=\frac{l}{t}\sqrt{\frac{\rho \mu \omega }{2}}\frac{1}{\left(1-e^{-u}\right)}\mathrm{where}u=\frac{w}{\delta }=w\sqrt{\frac{\mu \omega }{2\rho }}.& \left(1.41\right)\end{array}$

(231) As can be seen, making width w much wider than the skin depth δ increases the R.sub.AC/R.sub.DC ratio, but still helps to reduce the R.sub.AC.

(232) Thus, making the width w wider does not give much reduction in the AC resistance. Rather it just wastes area and adds more capacitance to the inductor if not carefully calculated (as shown herein below)! It is a better practice to increase the thickness rather than to increase the width for maintaining the same cross-sectional area S!

(233) Ultimately we are interested in a width w of a conductor giving highest possible Q at a given frequency while maintaining the same cross-sectional area S for a desired inductance value of L.

(234) First one calculates the Q of a straight inductor, ignoring all the capacitance effects, under the uniform current density distribution and names it Q.sub.LRDC, which actually represents a very low frequency case. Applying Q relation given at (1.20) gives,

(235) $\begin{array}{cc}{Q}_{\mathrm{LRDC}}=\frac{0.002.Math.{10}^{-6}l.Math.\left[\ln \left(\frac{2l}{w+t}\right)+0.5-f_1\left(w,t\right)\right]}{\rho \frac{l}{w.Math.t}}.Math.\omega .& \left(1.42\right)\end{array}$

(236) Doing some arithmetic manipulation gives,

(237) $\begin{array}{cc}{Q}_{\mathrm{LRDC}}=\frac{0.002.Math.{10}^{-6}.Math.w.Math.t}{\rho }.Math.\left[\ln \left(\frac{2l}{w+t}\right)+0.5-f_1\left(w,t\right)\right].Math.\omega .& \left(1.43\right)\end{array}$

(238) As can be seen, the length l in the front of (1.42) cancels and (1.43) can be written in terms of S as,

(239) 0$\begin{array}{cc}{Q}_{\mathrm{LRDC}}=\frac{0.002.Math.{10}^{-6}.Math.S}{\rho }.Math.\left[\ln \left(\frac{2l}{w+t}\right)+0.5-f_1\left(w,t\right)\right].Math.\omega .& \left(1.44\right)\end{array}$

(240) Forcing the constant S condition to (1.44) as done earlier allows (1.44) to be written as,

(241) $\begin{array}{cc}{Q}_{\mathrm{LRDC}}=\frac{0.002.Math.{10}^{-6}.Math.S}{\rho }.Math.\left[\ln \left(\frac{2l}{\frac{S}{t}+t}\right)+0.5-f_1\left(w,t\right)\right].Math.\omega .& \left(1.45\right)\end{array}$

(242) Doing some arithmetic manipulation in the log expression as done earlier gives,

(243) $\begin{array}{cc}{Q}_{\mathrm{LRDC}}=\frac{0.002.Math.{10}^{-6}.Math.S}{\rho }.Math.\left[\ln \left(\frac{2.Math.l.Math.t}{S+t^2}\right)+0.5-f_1\left(w,t\right)\right].Math.\omega .& \left(1.46\right)\end{array}$

(244) Ignoring the terms after the natural log expression in the bracket approximates (1.46) as,

(245) $\begin{array}{cc}{Q}_{\mathrm{LRDC}}\cong \frac{0.002.Math.{10}^{-6}.Math.S}{\rho }.Math.\left[\ln \left(\frac{2.Math.l.Math.t}{S+t^2}\right)\right].Math.\omega .& \left(1.47\right)\end{array}$

(246) Finding the maximum or minimum of (1.47) with respect to t requires differentiation of (1.47) as before. With the help of the following variable transformation,

(247) $\begin{array}{cc}u=\ln \left(\frac{2.Math.l.Math.t}{S+t^2}\right)=\ln \left(v\right)& \left(1.48\right)\end{array}$
and using the basic differentiation rule [26-29],

(248) $\begin{array}{cc}\frac{\mathrm{du}}{\mathrm{dt}}=\frac{1}{v}.Math.\frac{\mathrm{dv}}{\mathrm{dt}}\mathrm{where}v=\frac{2.Math.l.Math.t}{S+t^2}\mathrm{gives},& \left(1.49\right)\\ \frac{\partial Q_{\mathrm{LRDC}}}{\partial t}\cong \frac{0.002.Math.{10}^{-6}.Math.S}{\rho }.Math.\omega .Math.\left[\left(\frac{S+t^2}{2.Math.l.Math.t}\right).Math.\frac{\mathrm{dv}}{\mathrm{dt}}\right]\mathrm{where},& \left(1.50\right)\\ \frac{\mathrm{dv}}{\mathrm{dt}}=\frac{2.Math.l.Math.\left(S+t^2\right)-\left(2.Math.t.Math.l\right).Math.\left(2.Math.t\right)}{\left(S+t^2\right)}.& \left(1.51\right)\end{array}$

(249) With further arithmetic manipulation,

(250) $\begin{array}{cc}\frac{\mathrm{dv}}{\mathrm{dt}}=\frac{2.Math.l.Math.\left(S+t^2\right)-4.Math.l.Math.t^2}{{\left(S+t^2\right)}^2}=\frac{2.Math.l.Math.S-2.Math.l.Math.t^2}{{\left(S+t^2\right)}^2}.& \left(1.52\right)\end{array}$

(251) Simplification of (1.52) gives.

(252) $\begin{array}{cc}\frac{\mathrm{dv}}{\mathrm{dt}}=\frac{2.Math.l.Math.\left(S-t^2\right)}{{\left(S+t^2\right)}^2}.& \left(1.53\right)\end{array}$

(253) Substituting (1.53) in (1.50) gives,

(254) $\begin{array}{cc}\frac{\partial Q_{\mathrm{LRDC}}}{\partial t}\cong \frac{0.002.Math.{10}^{-6}.Math.S}{\rho }.Math.\omega .Math.\left[\left(\frac{S+t^2}{2.Math.l.Math.t}\right).Math.\frac{2.Math.l.Math.S-2.Math.l.Math.t^2}{{\left(S+t^2\right)}^2}\right].& \left(1.54\right)\end{array}$

(255) Straight forward arithmetic manipulation on (1.54) results in,

(256) $\begin{array}{cc}\frac{\partial Q_{\mathrm{LRDC}}}{\partial t}\cong \frac{0.002.Math.{10}^{-6}.Math.S}{\rho }.Math.\omega .Math.\frac{1}{\left(S+t^2\right).Math.t}.Math.\left(S-t^2\right).& \left(1.55\right)\end{array}$

(257) Equating (1.55) to zero and solving it,

(258) 0$\begin{array}{cc}\frac{\partial Q_{\mathrm{LRDC}}}{\partial t}\cong \frac{0.002.Math.{10}^{-6}.Math.S}{\rho }.Math.\omega .Math.\frac{1}{\left(S+t^2\right).Math.t}.Math.\left(S-t^2\right)=0& \left(1.56\right)\end{array}$
which gives the same result for t as done before with much shorter intermediate math giving,
t=√{square root over (S)}w=t=√{square root over (S)}.  (1.57)

(259) Let us denote the optimal thickness given in (1.57) as t.sub.OPT0. The advantage of taking this approach is to be sure of the result, as well as finding the peak Q at (1.57), giving,

(260) $\begin{array}{cc}{Q}_{\mathrm{LRDC}}\left(\mathrm{MAX}\right)\cong \frac{0.002.Math.{10}^{-6}.Math.S}{\rho }.Math.\left[\ln \left(\frac{2.Math.l.Math.\sqrt{S}}{S+S}\right)\right].Math.\omega .& \left(1.58\right)\end{array}$

(261) Doing the arithmetic in the natural log expression gives this very interesting result,

(262) $\begin{array}{cc}{Q}_{\mathrm{LRDC}}\left(\mathrm{MAX}\right)\cong \frac{0.002.Math.{10}^{-6}.Math.S}{\rho }.Math.\left[\ln \left(\frac{l}{\sqrt{S}}\right)\right].Math.\omega .& \left(1.59\right)\end{array}$

(263) Table [1] above shows t.sub.OPT0 for cross-sectional areas determined by desired current values I without violating Al electro-migration current density rules for various large value inductors at 100 MHz. As can be seen, even with the uniform current density assumption, which it will be shown to be incorrect, the required metal thickness values are very large compared to IC process metal thicknesses!

(264) Applying the “Single Sided Current Density Distribution Assumption” in the analysis done earlier by substituting the R.sub.AC term (1.41) in (1.42) gives,

(265) $\begin{array}{cc}{Q}_{\mathrm{LRAC}}=\frac{0.002.Math.{10}^{-6}l.Math.\left[\ln \left(\frac{2l}{w+t}\right)+0.5-f_1\left(w,t\right)\right]}{\frac{l}{t}\sqrt{\frac{\rho \mu \omega }{2}}\left(\frac{1}{1-e^{-u}}\right)}.Math.\omega .& \left(1.60\right)\end{array}$

(266) Similar arithmetic gives,

(267) $\begin{array}{cc}{Q}_{\mathrm{LRAC}}=0.002.Math.{10}^{-6}.Math.t.Math.\left(1-e^{-u}\right).Math.\sqrt{\frac{2}{\rho \mu \omega }}.Math.\left[\ln \left(\frac{2l}{w+t}\right)+0.5-f_1\left(w,t\right)\right].Math.\omega .& \left(1.62\right)\end{array}$

(268) Applying the same approximation to (1.61) gives,

(269) $\begin{array}{cc}{Q}_{\mathrm{LRAC}}=0.002.Math.{10}^{-6}.Math.\sqrt{\frac{2}{\rho \mu \omega }}.Math.t.Math.\left(1-e^{-v}\right).Math.\left[\ln \left(\frac{2l}{\frac{S}{t}+t}\right)\right].Math.\omega & \left(1.62\right)\end{array}$
where v that appears on the exponent as a function of S becomes,

(270) $\begin{array}{cc}v=\frac{w}{\delta }=\frac{S}{t}\sqrt{\frac{\mu \omega }{2\rho }}.& \left(1.63\right)\end{array}$

(271) Differentiation of (1.62) requires longer work, but it will not yield the result as earlier. To apply the chain rule [26-29] for differentiation easily let the following functions be defined as,

(272) $\begin{array}{cc}{g}_1\left(t\right)=\left(1-e^{-v}\right)\mathrm{and}{g}_2\left(t\right)=\ln \left(\frac{2l}{\frac{S}{t}+t}\right).& \left(1.64\right)\end{array}$

(273) Substituting the g.sub.1(t) and g.sub.2(t) in (1.62) gives,

(274) $\begin{array}{cc}{Q}_{\mathrm{LRAC}}=a.Math.t.Math.g_1\left(t\right).Math.g_2\left(t\right)\mathrm{where},& \left(1.65\right)\\ a=0.002.Math.{10}^{-6}.Math.\omega .Math.\sqrt{\frac{2}{\rho \mu \omega }}=0.002.Math.{10}^{-6}\sqrt{\frac{2\omega }{\rho \mu }}.& \left(1.66\right)\end{array}$

(275) The derivatives of g.sub.1(t) and g.sub.2(t) functions with respect to t are,

(276) $\begin{array}{cc}\frac{\partial g_1}{\partial t}=e^{-v}\frac{\partial v}{\partial t}=e^{-v}\left(-\frac{S}{t^2}\sqrt{\frac{\mu \omega }{2\rho }}\right)\mathrm{and},& \left(1.67\right)\\ \frac{\partial g_2}{\partial t}=\frac{1}{\left(S+t^2\right).Math.t}.Math.\left(S-t^2\right).& \left(1.68\right)\end{array}$

(277) The chain rule applied to (1.65) gives,

(278) 0$\begin{array}{cc}\frac{\partial Q_{\mathrm{LRAC}}}{\partial t}=a.Math.\left[g_1\left(t\right).Math.g_2\left(t\right)+t.Math.\left(\frac{\partial g_1}{\partial t}{g}_2\left(t\right)+g_1\left(t\right)\frac{\partial g_2}{\partial t}\right)\right].& \left(1.69\right)\end{array}$

(279) To shorten Equation (1.69) to a manageable representation to perform the analysis more clearly, (1.69) can be written as the sum of y(t) and z(t) functions given as,

(280) $\begin{array}{cc}\frac{\partial Q_{\mathrm{LRAC}}}{\partial t}=a.Math.\left[z\left(t\right)+y\left(t\right)\right]\mathrm{where},& \left(1.70\right)\\ y\left(t\right)=t.Math.\left(\frac{\partial g_1}{\partial t}{g}_2\left(t\right)+g_1\left(t\right)\frac{\partial g_2}{\partial t}\right).& \left(1.71\right)\end{array}$

(281) The first function y(t) can be written explicitly as,

(282) $\begin{array}{cc}y\left(t\right)=t.Math.\left\{e^{-v}\left(-\frac{S}{t^2}\sqrt{\frac{\mu \omega }{2\rho }}\right).Math.\ln \left(\frac{2\mathrm{lt}}{S+t^2}\right)+\frac{\left(S-t^2\right)}{\left(S+t^2\right).Math.t}\left(\frac{S}{t}\sqrt{\frac{\mu \omega }{2\rho }}\right)\right\}.& \left(1.72\right)\end{array}$

(283) With some arithmetic manipulation, (1.70) becomes,

(284) $\begin{array}{cc}y\left(t\right)=\frac{S}{t}\sqrt{\frac{\mu \omega }{2\rho }}\left[-e^{-v}.Math.\ln \left(\frac{2\mathrm{lt}}{S+t^2}\right)+\frac{\left(S-t^2\right)}{\left(S+t^2\right)}\right]\mathrm{and},& \left(1.73\right)\\ z\left(t\right)=\left(1-e^{-v}\right).Math.\ln \left(\frac{2\mathrm{lt}}{S+t^2}\right).& \left(1.74\right)\end{array}$

(285) After some simple and straightforward arithmetic manipulations, (1.72) becomes simpler. Equating it to zero gives,

(286) $\begin{array}{cc}\left[1-\left(1+\frac{b}{t}\right).Math.e^{-v}\right]\ln \left(\frac{2\mathrm{lt}}{S+t^2}\right)+\frac{b}{t}\frac{\left(S-t^2\right)}{\left(S+t^2\right)}=0\mathrm{where},& \left(1.75\right)\\ b=S\sqrt{\frac{\mu \omega }{2\rho }}=v.Math.t& \left(1.76\right)\end{array}$

(287) By using the relation (1.63) for v, (1.75) can be rewritten using “only” the variable v instead of variables of v and b as,

(288) $\begin{array}{cc}\left[1-\left(1+v\right).Math.e^{-v}\right]\ln \left(\frac{2\mathrm{lt}}{S+t^2}\right)+v\frac{\left(S-t^2\right)}{\left(S+t^2\right)}=0.& \left(1.77\right)\end{array}$

(289) As can be seen, (1.77) is a nonlinear equation in t and can be solved “only” numerically. Before going into the solution of (1.77) it is useful to see the functional behavior of v, which is an important argument of (1.77) as a function of frequency f. This analytical investigation will also lead to very important results which are the basis of claims of this application without even solving (1.77).

(290) The first thing to note is that v will be “always” a positive number. In addition to that for any material when the signal in the medium is at zero frequency (ω=0), v will become zero (v= 0). If we substitute some real values into the ν expression given in (1.63), such as for copper (Cu) resistivity, along with the numerical value of μ, v becomes,

(291) $\begin{array}{cc}{v}_{\mathrm{Cu}}=\frac{S}{t}\sqrt{\frac{1.25.Math.{10}^{-8}.Math.\omega }{2.Math.1.68.Math.{10}^{-6}}}=\frac{S}{t}.Math.0.6099.Math.{10}^{-1}\sqrt{\omega }.& \left(1.78\right)\end{array}$

(292) It can be shown that (1.78) for Cu, frequencies above 42.7 Hz, the square root term exceeds 1 and increases with the square root of the frequency. Now let's investigate the properties of the solution of (1.77) without solving it. It will show that the uniform current distribution solution is incorrect for frequencies f>0!

(293) i) The Optimal Thickness t.sub.OPT Satisfying the Solution of (1.80) is Greater than S.sup.0.5, Giving High Aspect Ratio Conductor Cross-Section!

(294) As can be seen, equation (1.77) is a summation of two terms. If we divide both sides of (1.77) by v we get,

(295) $\begin{array}{cc}\frac{\left[1-\left(1+v\right).Math.e^{-v}\right]}{v}\ln \left(\frac{2\mathrm{lt}}{S+t^2}\right)+\frac{\left(S-t^2\right)}{\left(S+t^2\right)}=0.& \left(1.79\right)\end{array}$

(296) As can be seen, the second term in (1.79) is,

(297) $\begin{array}{cc}\frac{\left(S-t^2\right)}{\left(S+t^2\right)}.& \left(1.80\right)\end{array}$

(298) Equating (1.80) to zero and solving t will give the uniform current expression derived earlier, which will give the t.sub.OPT0=S.sup.0.5 solution. It should be noted that (1.80) will be negative for t>S.sup.0.5 and positive for t<S.sup.0.5. Moreover (1.80) is zero for t=S.sup.0.5. The first term in (1.79) is product of two functions,

(299) $\begin{array}{cc}\frac{\left[1-\left(1+v\right).Math.e^{-v}\right]}{v}\ln \left(\frac{2\mathrm{lt}}{S+t^2}\right)>0\mathrm{for}\ln \left(\frac{2\mathrm{lt}}{S+t^2}\right)>0.& \left(1.81\right)\end{array}$

(300) Since the second logarithmic term in (1.81) is always a positive number for any practical length l, all we have to do is find the behavior of the first term of (1.81) with respect to v to determine the sign of (1.81). In other words, if the first term in (1.81) is proven to be always positive for any v>0, the relation (1.81) will also become always positive for any length l where we can apply the inductor formula (1.4). This can be done graphically very easily. FIG. 1.13.1 shows the (1.82) u.sub.1, as a function of v. As can be seen for all v>0,

(301) 0$\begin{array}{cc}{u}_1\left(\rho ,\omega ,t\right)=\frac{\left[1-\left(1+v\right).Math.e^{-v}\right]}{v}\ge 0.& \left(1.82\right)\end{array}$

(302) For v=0 (ω=f=0), (1.82) becomes a 0/0 type uncertainty, but it can be resolved by using L'Hospital's rule [26, 27]. Differentiating the dominator and denominator of it with respect to v and substituting v=0 in the expression, gives 0 for v=0. It can also be shown that (1.82) has a maximum for v>1 giving 0.2983 as its maximum value at v=1.8. Since it is proven that the first term of (1.79) is always positive for all v>0 values, (1.81) is also positive for any v, l, S and t. In this case the equation (1.79) can only be satisfied if its second term (1.80) is negative. This condition can mathematically given as,

(303) $\begin{array}{cc}\frac{\left[1-\left(1+v\right).Math.e^{-v}\right]}{v}\ln \left(\frac{2\mathrm{lt}}{S+t^2}\right)+\frac{\left(S-t^2\right)}{\left(S+t^2\right)}=0\mathrm{only}\mathrm{if}\frac{\left(S-t^2\right)}{\left(S+t^2\right)}<0& \left(1.83\right)\end{array}$

(304) If we rewrite (1.79) as,

(305) $\begin{array}{cc}a+\frac{\left(S-t^2\right)}{\left(S+t^2\right)}=0\mathrm{where},& \left(1.84\right)\\ a=\frac{\left[1-\left(1+v\right).Math.e^{-v}\right]}{v}\ln \left(\frac{2\mathrm{lt}}{S+t^2}\right)\ge 0& \left(1.85\right)\end{array}$
and some arithmetic manipulation on (1.84) gives,
a(S+t.sup.2)+(S−t.sup.2)=0.  (1.86)

(306) Finally, equation (1.84) becomes,
(a−1)t.sup.2+S(1+a)=0.  (1.87)

(307) Solving t from (1.87) gives,

(308) $\begin{array}{cc}{t}^2=\frac{S\left(1+a\right)}{\left(1-a\right)}.& \left(1.88\right)\end{array}$

(309) As can be seen, a real solution to (1.88) is only possible if a<1 and can be given as,

(310) $\begin{array}{cc}{t}_{\mathrm{OPT}}=\sqrt{S}\sqrt{\frac{\left(1+a\right)}{\left(1-a\right)}}\mathrm{for}a<1.& \left(1.89\right)\end{array}$

(311) The direct result of (1.89) gives,

(312) $\begin{array}{cc}{t}_{\mathrm{OPT}}=b\sqrt{S}\mathrm{where}b=\sqrt{\frac{\left(1+a\right)}{\left(1-a\right)}}>1\mathrm{for}a<1& \left(1.90\right)\end{array}$
which, proves our case without even solving the non-linear equation (1.77) at all! Since b>1 for a<1,
t.sub.opt>√{square root over (S)}.  (1.91)

(313) It also states that since a is a function of frequency f, length l, which defines the desired inductance value L and resistivity ρ. As a result, the optimal thickness t.sub.OPT is a function of these three additional parameters, not a just a function of t.sub.OPT=S.sup.0.5 as derived earlier for the constant current density assumption. It also has to be noted that (1.90) gives the same result for a=0 which corresponds to zero frequency and for this case b=1 and satisfies the t.sub.OPT0=S.sup.0.5 result!

(314) Since the cross-sectional area is a given value of S, for t.sub.OPT we can also define the optimal width w.sub.OPT that satisfies,

(315) $\begin{array}{cc}S=w_{\mathrm{OPT}}.Math.t_{\mathrm{OPT}}\mathrm{giving},& \left(1.92\right)\\ w_{\mathrm{OPT}}=\frac{S}{b\sqrt{S}}=\frac{\sqrt{S}}{b}\mathrm{where}b>1.& \left(1.93\right)\end{array}$

(316) As a result of (1.90) and (1.93), the high-Q inductor rectangular cross-section has to be a high aspect ratio cross-section having larger thickness than width, or in other words t>w. This is one of the key insights of this invention.

(317) The aspect ratio Δ.sub.OPT=t/w can be defined as,

(318) $\begin{array}{cc}{\Delta }_{\mathrm{OPT}}=\frac{t_{\mathrm{OPT}}}{w_{\mathrm{OPT}}}=\frac{b\sqrt{S}}{\frac{\sqrt{S}}{b}}=b^2>1.& \left(1.94\right)\end{array}$

(319) The same fact could have been proven by writing (1.79) as a function of w instead of t giving,

(320) $\begin{array}{cc}\frac{\left[1-\left(1+v\right).Math.e^{-v}\right]}{v}\ln \left(\frac{2\mathrm{lw}}{w^2+S}\right)+\frac{\left(w^2-S\right)}{\left(w^2+S\right)}=0.& \left(1.95\right)\end{array}$

(321) As can be seen, equation (1.95) is a non-linear function of w. In this case (1.84) becomes,

(322) $\begin{array}{cc}a+\frac{\left(w^2-S\right)}{\left(w^2+S\right)}=0.& \left(1.96\right)\end{array}$

(323) The solution this time leads to the solution for w of,

(324) 0$\begin{array}{cc}{w}^2=S\frac{\left(1-a\right)}{\left(a+1\right)}.\mathrm{giving},& \left(1.97\right)\\ w_{\mathrm{OPT}}=\sqrt{S}.Math.\sqrt{\frac{\left(1-a\right)}{\left(a+1\right)}}\mathrm{for}a<1& \left(1.98\right)\end{array}$
which (1.98) gives the identical result of (1.93) satisfying S=w.sub.OPT.Math.t.sub.OPT. Furthermore it can also be proven that w.sub.OPT<δ, or in other words the shorter dimension of the rectangular cross-sections will be less than the skin depth δ for all of the typical inductor values. Thus (1.97) gives the identical result of (1.93).

(325) As a conclusion of this analysis we can state:

(326) The Q.sub.MAX for a rectangular cross-section by enforcing the constant area constraint S is a rectangular cross-section having a larger dimension along z axes, which is the thickness t, compared to its width (w). It is NOT a square. Furthermore, there is an optimal thickness t.sub.OPT that can be found exactly by solving (1.79) for any given frequency, inductor value and cross-sectional area of the spiral inductor windings. The exact thickness t.sub.OPT which is the solution of (1.79) has a weak dependency to resistivity, frequency of operation and the desired inductance value L due to the l dependency in (1.77), but it is a function of these parameters and does not appear in the previously derived t.sub.OPT0.

(327) This conclusion is verified in FIG. 1.14.1 to FIG. 1.17.2 which show the Q versus Al metal thickness t, varying between 0.1 to 500μ, for 20, 40, 60 and 80 nH inductors for constant area S constraints of 50, 250, 500 and 1,000μ.sup.2 for 2 different current boundary conditions mentioned earlier along with the constant current density approximation at a frequency f at 100 MHz.

(328) The top set of curves in each plot of these figures corresponds to the uniform current density assumption. As can be seen, the Q peaks are corresponding to thicknesses of the square root of S at 7.07μ, 15.81μ, 22.36μ and 31.62μ for S=50, 250, 500 and 1,000μ.sup.2, respectively, as derived earlier. These solutions are marked with a vertical dotted line in each plot. As can be seen, the thickness for the Q peaks are independent of the L values, but the peak Q value is a function of the desired inductance values taken as 20, 40, 60 and 80 nH as predicted analytically.

(329) The case of forced uniform current density at the surface of the conductor gives two peaks, one below and the other one after t.sub.OPT0. It is noteworthy that in this case there is a double peak with a minimum at t.sub.OPT0. The reason for a double peak is that a scan of the thickness from 0.1μ to 500μ till t<S.sup.0.5 yields a width w that is higher than the thickness t, to satisfy the constant area S condition imposed, and after t>S.sup.0.5 the thickness t becomes larger than the width w. Forcing a uniform current density boundary condition at the surface of the conductor gives symmetric results with respect to S.sup.0.5.

(330) In the case where constant current density along the thickness (along z axes) is assumed, the curves start with very low Q's for small thicknesses. This is due to the fact that for very small thickness t, will give very large widths, w to satisfy the constant area S condition. All the curves have single peaks, which matches the second peak in the earlier case, because as the thickness increases the surface current density at the surface of the conductor case applicable to the metal geometry. The optimal thicknesses t.sub.OPT where the Q.sub.PEAK for all cases for S=50, 250, 500 and 1,000μ.sup.2, at 100 MHz are significantly higher than t.sub.OPT0 as predicted by the earlier analysis. However, as can be seen, none of these metal thicknesses shown in FIG. 1.14.1 to FIG. 1.17.2 can be provided by any known IC process. This limitation is addressed herein below.

(331) FIG. 1.18.1 show the magnitude of Q.sub.PEAK variation as a function of frequencies for the inductors 20, 40, 60 and 80 nH for 100 and 200 MHz. As can be seen, the peaks are at the same t.sub.OPT0 giving linearly scaling peak values with frequency as predicted. FIG. 1.18.2 show the uniform current density and current density distribution with “single sided solution” approximation together to show the relative differences between two assumptions. As can be seen, the Q(t) variations in the “single sided solution” approximation gives much more realistic results. To have a better representation of the realistic “single sided solution” approximation they are plotted alone in FIG. 1.19.

(332) The “Complete Solution Current Density Distribution” for Spiral Windings

(333) In spiral inductor windings the current density distribution in the windings is different than the straight wire with a rectangular cross-section. The magnetic fields in the windings are also a function of the distance of the winding from the center (air core) of the spiral inductor, and also vary along the winding and its z coordinate. One can see this very easily by applying Ampere's law to the spiral cross-section. The magnetic field distribution in the winding is the cause of the all the combined eddy currents, skin and proximity effects. The solution requires three dimensional numerical simulation of the entire spiral inductor. The normalized magnitudes of the magnetic field in the windings for widths of 0.5δ, δ, 2δ, 3δ, 4δ, 5δ and 10δ are shown in FIG. 1.20.1 for a spiral inductor having four windings, assuming that each winding is carrying the same current I. As can be seen, the B field enters from both sides of the windings, with the exception of the inner most winding and will create similar current density distributions in the windings. The innermost winding normalized B-field distribution is shown in FIG. 1.20.2 and it looks very similar to the solution obtained with “single sided solution”. Looking at the normalized B-field distributions in FIG. 1.20.1 one can also see that these distributions satisfy Ampere's law. FIG. 1.21.1 show how R.sub.AC/R.sub.DC compare to the “single sided” and “complete solution” results, the x axis being winding-width normalized with the skin depth, w/δ. FIG. 1.21.2 show R.sub.AC/R.sub.DC comparing the “single sided” and “complete solution” results, the x axis being the winding width w. One interesting property that is fairly clear in FIG. 1.21.1 and FIG. 1.21.2 is that assuming a “single sided solution,” estimates R.sub.AC/R.sub.DC can be corrected based on a “complete solution.”

(334) The short summary of the curves related to Q.sub.PEAK vs t.sub.OPT at 100 MHz for different cross-section areas and inductor values is given in Table [2]. More detailed explanations and critical parameters for different inductor values of 20, 40, 60 and 80 nH at 100 and 200 MHz are shown in Tables [2.1-2.4] below.

(335) TABLE-US-00003 TABLE 2.1 f = 100 MHz f = 200 MHz t.sub.OPT (μ) 195.9 264 w.sub.OPT (μ) [S/t.sub.OPT] 2.552 1.894 Q.sub.max 11.26 10.69 Δ(t.sub.OPT/w.sub.OPT) 76.77 139.4 Δ.sub.δ (w.sub.OPT/δ) 0.307 0.322 Length (μ) 17,600 18,400 R.sub.DC (Ω) 0.91612 1.005 R.sub.AC (Ω) 1.116 1.175 L = 20 nH, S = 500μ.sup.2, w.sub.OPT0 = t.sub.OPT0 = S.sup.1/2 = 22.36μ, Length = 14,360μ δ = 8.316μ@100 MHz, δ = 5.880μ@200 MHz R.sub.DC = 0.7841 Ω, R.sub.AC@100 MHz = 2.345 Ω, R.sub.AC@200 MHz = 3.176 Ω Δ(t.sub.OPT0/w) = 1, Δ.sub.δ(w.sub.OPT0/δ) = 2.693@100 MHz, Δ.sub.δ(w.sub.OPT0/δ) = 3.803@200 MHz

(336) TABLE-US-00004 TABLE 2.2 f = 100 MHz f = 200 MHz t.sub.OPT (μ) 213.3 287.4 w.sub.OPT (μ) [S/t.sub.OPT] 2.334 1.74 Q.sub.max 12.44 11.86 Δ(t.sub.OPT/w.sub.OPT) 91.03 165.2 Δ.sub.δ (w/δ) 0.2818 0.2958 Length (μ) 32,250 33,610 R.sub.DC (Ω) 1.761 1.835 R.sub.AC (Ω) 2.021 2.12 L = 40 nH, S = 500μ.sup.2, w.sub.OPT0 = t.sub.OPT0 = S.sup.1/2 = 22.36μ, Length = 26,410μ δ = 8.316μ@100 MHz, δ = 5.880μ@200 MHz R.sub.DC = 1.442 Ω, R.sub.AC@100 MHz = 4.312 Ω, R.sub.AC@200 MHz = 5.841 Ω Δ(t.sub.OPT0/w.sub.OPT0) = 1, Δ.sub.δ(w.sub.OPT0/δ) = 2.693@100 MHz, Δ.sub.δ(w.sub.OPT0/δ) = 3.803@200 MHz

(337) TABLE-US-00005 TABLE 2.3 f = 100 MHz f = 200 MHz t.sub.OPT (μ) 222.6 299.9 w.sub.OPT (μ) [S/t.sub.OPT] 2.246 1.667 Q.sub.max 13.14 12.55 Δ(t.sub.OPT/w.sub.OPT) 99.12 179.9 Δ.sub.δ (w.sub.OPT/δ) 0.2701 0.2835 Length (μ) 46,040 47,910 R.sub.DC (Ω) 2.514 2.616 R.sub.AC (Ω) 2.869 3.004 L = 60 nH, S = 500μ.sup.2, w.sub.OPT0 = t.sub.OPT0 = S.sup.1/2 = 22.36μ, Length = 27.820μ δ = 8.316μ@100 MHz, δ = 5.880μ@200 MHz R.sub.DC = 2.065 Ω R.sub.AC@100 MHz = 6.175 Ω, R.sub.AC@200 MHz = 8.365 Ω Δ(t.sub.OPT0/w.sub.OPT0) = 1, Δ.sub.δ(w.sub.OPT0/δ) = 2.693@100 MHz, Δ.sub.δ(w.sub.OPT0/δ) = 3.803@200 MHz

(338) TABLE-US-00006 TABLE 2.4 f = 100 MHz f = 200 MHz t.sub.OPT (μ) 222.6 313 w.sub.OPT (μ) [S/t.sub.OPT] 2.246 1.598 Q.sub.max 13.65 13.05 Δ(t.sub.OPT/w.sub.OPT) 99.12 195.9 Δ.sub.δ (w.sub.OPT/δ) 0.2701 0.2717 Length (μ) 59,120 61,780 R.sub.DC (Ω) 3.228 3.373 R.sub.AC (Ω) 3.684 3.852 L = 80 nH, S = 500μ.sup.2, w.sub.OPT0 = t.sub.OPT0 = S.sup.1/2 = 22.36μ, Length = 48,850μ δ = 8.316μ@100 MHz, δ = 5.880μ@200 MHz R.sub.DC = 2.667 Ω R.sub.AC@100 MHz = 7.977 Ω, R.sub.AC@200 MHz = 10.80 Ω Δ(t.sub.OPT0/w.sub.OPT0) = 1, Δ.sub.δ(w.sub.OPT0/δ) = 2.693@100 MHz, Δ.sub.δ(w.sub.OPT/δ) = 3.803@200 MHz

(339) Performing the analysis at any frequency gives similar results, indicating that a thick metal with widths in the order of the skin depth is needed to achieve a high-Q straight inductor. Spiral inductors can be thought of as straight lines which are coupled with desirable sign and highest possible mutual inductances. The following section is related to pointing out the inherent advantage of spacing the high aspect ratio thick metals, again with high aspect ratio spacing rules for making very high performance spiral inductors.

(340) Since thick metal processes on the order of metal thicknesses shown in FIG. 1.14.1-FIG. 1.17.2 and summarized in Tables [2.1-2.4] are not available in standard IC processes, one needs to explore processes with metal thickness of this order that can be realized, along with metal widths giving high aspect ratios—in other words, widths on the order of skin depth dimensions with comparable metal-to-metal spacing processing capabilities. The initial results suggest that if there is a high-Q inductor requirement, these inductors must be designed and built using a different process, generally resulting what are called interposer structures, built with IC processing techniques and connected to a traditional IC with 3D stacking methods. On the other hand, for low current density (small S) applications, if the IC process allows 4μ metal thicknesses with 2.8μ width and spacing, giving a metal aspect ratio of Δ.sub.M (t/w=1.43) that type of IC process might be enough for many RFIC inductors for f>900 MHz applications. Known IC designs using 4μ metal thickness do not employ the high aspect ratio rule as given in this disclosure, however. Before proposing an interposer process requirement, one needs to also explore the geometrical size of these high value inductors to find out if they are within feasible dimensions that can be connected to a conventional integrated circuit. Hence, the foregoing proofs and formulae become useful analytic tools.

(341) Inner Space Dimension Optimization of the Spiral Inductor

(342) Spiral air-core inductors have an inner space where there are no windings. The geometry of inner space can be almost anything, but in most practical applications they are circular, rectangular, square or octagonal areas. Among these four most commonly seen inner space geometries, probably the square inner space geometry is the most popular geometry due to many reasons, such as better packing density, ease of design and better performance compared to rectangular geometry from any design specification. The most critical part in area reduction in spiral inductor having a constant spacing s between windings is winding width w and the inner dimension d.sub.IN, where the spiral inductor windings start and winds outward from this square shaped inner geometry. Having thick metal and using the high aspect ratio metal rule as herein taught minimizes the spiral area very significantly and improves high frequency performance as shown above. Similar optimization is needed for optimizing the inner dimension of a square inner space, which can be extended to any type of inner space geometry. The present invention reveals that d.sub.IN optimization should have a minimum inner dimension greater than a mathematically defined d.sub.INMIN.

(343) To make highly efficient high coupling metal structures, one needs to consider also the Q value of the individual structures, such as each leg, besides their highest possible mutual inductive coupling ratios with other legs, as explained above. The use of high coupling ratio legs with individually low Q structures increases the loss and does not contribute to the inductance value at elevated operating frequencies. The analytical expression for Q.sub.LRAC (1.62) shows that Q.sub.LRAC (l, S, t, f) has a logarithmic length dependency. This is clearly shown in the Q.sub.LRAC (l, S, t, f) plot in FIG. 1.22 for S=50, 250, 500 and 1,000μ.sup.2, at 100 MHz for Al metal width of w=2μ.

(344) According to the invention there should be no inductive structure, such as any leg in a spiral inductor, having its individual Q.sub.LRAC<1 for the operating frequency of the spiral inductor. The smallest leg length in a spiral inductor is determined by the inner dimension d.sub.IN as shown in FIG. 1.6.1 in a square spiral. The d.sub.IN is an important spiral inductor design parameter that greatly affects the overall Q of the spiral inductor. According to the invention d.sub.IN is calculated such that Q(d.sub.INMIN, S, t, f)>1 from the analytical expression (1.62). (By contrast, most known spiral inductors have a very small inner dimension, teaching away from the present invention, but which increases the loss and reduces the Q of the spiral inductor, a characteristic evidently not recognized in the prior art as a factor in classical books in the field [31]. In practice there are properly designed inductors which might have a larger inner dimension, but there is no rule that determines the selected inner dimension of any inner space geometry. In the past, the selection was based on experimental measurement data collected from many spiral inductor designs for that particular process without knowing the reason.)

(345) According to this invention, a minimum inner dimension d.sub.INMIN value is calculated with the analytical relation (1.62) having lengths supporting Q(d.sub.INMIN, S, t, J)>1 for a selected cross-sectional area S and thickness t at the operating frequency. As can be seen in FIG. 1.22 the bottom curve, which corresponds to S=50μ.sup.2 should have a length larger than 800μ to meet this condition at 100 MHz for Al metal width of w=2μ. The remaining l.sub.MIN values for S=250, 500 and 1,000μ.sup.2 correspond to the l.sub.MIN of each case having Q(l.sub.MIN, S, t, f)>1 are met at smaller lengths of 60-80μ.

(346) Another advantage of a larger inner dimension d.sub.IN is seen in the reduction of mutual inductive coupling between inductor legs on both sides of the inner space of the spiral inductor. Since these inductors have coupling ratios of opposite signs, minimizing the coupling ratio will increase the inductance L and therefore increase the Q of the spiral inductor. Mathematically stating, the s<<d.sub.IN condition must also be satisfied! On the other hand increasing the inductive coupling between inductor legs on the same side of the inner empty space increases the inductance value L, thereby increasing the Q of the inductor. Since one generally is interested designing a specific L, one will achieve this design goal with shorter wire length, giving less resistance and capacitance resulting in higher Q.

(347) Increase of Mutual Inductance by Using High Aspect Ratio Metal Giving Higher Q and Smaller Area for a Desired Inductance Value

(348) It has now been mathematically proven that high aspect ratio rectangular cross-sectional geometry in the z dimension, facing their larger dimensional sides to each other, yields a definitive advantage in giving high Q spiral inductors, due to giving smaller R.sub.AC at any frequency of operation compared to placement as in prior art placement with the smaller dimension facing each other. Having rectangular cross-section conductor windings facing their larger dimensional sides to each other unexpectedly yields much higher mutual inductive coupling compared to facing the smaller sides together for the same spacing s. Having high inductive coupling between the legs increases the inductor value of the spiral inductor reducing the total length of the winding for a desired inductance value L and consequently reducing the resistance and giving higher Q. This fact is due to the geometric mean distance (g.m.d) reduction of the geometries if arranged as herein disclosed. The leg structures for spiral inductors according to the invention are shown in FIG. 1.24 and FIG. 1.25 as perspective and cross-sectional plots differ very clearly from the structures shown in FIG. 1.9.1 and FIG. 1.9.2, having the same cross-sectional winding areas as shown earlier. FIG. 1.25 is the mathematical matrix equivalent of the geometry shown in FIG. 1.25. This arrangement was previously shown in FIGS. 1.7.1, 1.81 and 1.9.2 where the non-uniform current distribution analysis was performed which showed a significant advantage on the reduction of R.sub.AC by the metal arrangement configuration as herein explained. As a result, how one configures the rectangles together, in other words the orientation of metal windings, greatly affects the inductor performance and the total area for a spiral inductance having the same inductance value L.

(349) Total Inductance Calculation of a Spiral Inductor Using Circuit Theory (Partial Inductance)

(350) An easier way of explaining the total inductance calculation of a spiral inductor from its metal geometry is through circuit theory, rather than electromagnetics theory through the “Partial Inductance” concept [1, 10-24]. FIG. 1.23 shows the cross-sectional geometry obtained by slicing the top view of a square spiral inductor along the marked “y cut” plane. As can be seen, the metal aspect ratio Δ.sub.M, is a large number according to the invention, but the formulation given here applies for any cross-sectional metal winding geometry. The three-dimensional perspective view is also shown in FIG. 1.24 for adjacent three windings in one side of the “inner empty area” of the spiral inductor, where the windings are separated by the inner dimension of the spiral inductor noted as d.sub.IN. FIG. 1.25 also shows the leg numbering for the inductance matrix generation. The p, C, B notation on the top of FIG. 1.25 are the Grover [1] notations for mutual inductance calculation tables and their equivalent variables used in this work are shown with arrows as w, s and t.

(351) We can represent the width and spacing of windings as shown in FIG. 1.26 (which is extracted from an analysis of Grover (Ref. [1]) p. 19, Table[1], in terms of metal thickness t and aspect ratios Δ.sub.M and Δ.sub.S that can be calculated from the notations in this figure.

(352) It is a good practice to have a quick method of estimating the number of turns needed to design for an inductor value L for a given t, w and s rule having the d.sub.IN as a parameter, assuming that each side will have the same number of turns. Although the final structure of a computed designed will end up different, this assumption will give a good initial approximation of the spiral size and idea of the inductance matrix terms, which is very important.

(353) As can be seen the distance d, in the mutual inductances formulation given in (1.12) will be replaced by the geometric mean distances (g.m.d) between the cross-sectional metal windings. For any cross-sectional geometry and the numbering given in FIG. 1.25 the mutual inductances given in the inductance matrix and their signs will be as,

(354) $\begin{array}{cc}\left[\begin{array}{cc}{L}_{11}{L}_{12}{L}_{13}& L_{14}{L}_{15}{L}_{16}\\ L_{21}{L}_{22}{L}_{23}& L_{24}{L}_{25}{L}_{26}\\ L_{31}{L}_{32}{L}_{33}& L_{34}{L}_{35}{L}_{36}\\ L_{41}{L}_{42}{L}_{43}& L_{44}{L}_{45}{L}_{46}\\ L_{51}{L}_{52}{L}_{53}& L_{54}{L}_{55}{L}_{56}\\ L_{61}{L}_{62}{L}_{63}& L_{64}{L}_{65}{L}_{66}\end{array}\right]& \left(1.99\right)\\ \left[\begin{array}{cc}+++& ---\\ +++& ---\\ +++& ---\\ ---& +++\\ ---& +++\\ ---& +++\end{array}\right]& \end{array}$

(355) The negative signs in the inductance matrix (1.99), which are for the leg inductors in the opposite sides of the inner space, originate from the current direction going in or out of the reference bubble placement on each mutually coupled inductor of the spiral inductor. In any case the inductance matrix will be a symmetric matrix. Since the leg width w and spacing between the legs s in each side are kept constant and d.sub.IN>s, the following relations between the magnitudes inductance matrix elements also holds for any winding aspect ratio having the same numbering as given in FIG. 1.25,

(356) $\begin{array}{cc}{L}_{1,1}>.Math.L_{1,2}.Math.>.Math.L_{1,3}.Math.>.Math.L_{1,4}.Math.>.Math.L_{1,5}.Math.>.Math.L_{1,6}.Math.L_{2,2}>.Math.L_{2,1}.Math.=.Math.L_{2,3}.Math.>.Math.L_{2,4}.Math.>.Math.L_{2,5}.Math.>.Math.L_{2,6}.Math.& \left(1.100\right)\\ L_{3,3}>.Math.L_{3,2}.Math.>.Math.L_{3,1}.Math.>.Math.L_{3,4}.Math.>.Math.L_{3,5}.Math.>.Math.L_{3,6}.Math.& \\ L_{4,4}>.Math.L_{4,5}.Math.>.Math.L_{4,6}.Math.>.Math.L_{4,1}.Math.>.Math.L_{4,2}.Math.>.Math.L_{4,3}.Math.L_{5,5}>.Math.L_{5,4}.Math.=.Math.L_{5,6}.Math.>.Math.L_{5,1}.Math.>.Math.L_{5,2}.Math.>.Math.L_{5,3}.Math.L_{6,6}>.Math.L_{6,5}.Math.>.Math.L_{6,4}.Math.>.Math.L_{6,1}.Math.>.Math.L_{6,2}.Math.>.Math.L_{6,3}.Math.\mathrm{and}{L}_{i,i}>0L_{i,j}=L_{j,i}& \end{array}$

(357) Since the length of each leg (l) is greater than the inner lengths, one can also indicate that,
l.sub.3=l.sub.6>l.sub.2=l.sub.5>l.sub.1=l.sub.4=d.sub.IN
L.sub.6,6=L.sub.3,3>L.sub.2,2=L.sub.5,5>L.sub.1,1=L.sub.4,4  (1.101)

(358) The passivity requirement as given in (1.14) also holds between the diagonal and off-diagonals of the inductance matrix having coupling ratio K<1 between the diagonals and off-diagonals as stated before. For an equal length assumption, this will result in diagonal entries of the inductance matrix being always larger than any of the magnitude of the off-diagonals.

(359) Identical relations can be given for the “x cut” of the spiral inductor in FIG. 1.23, which have zero mutual couplings between the “y cut” matrix terms due to the fact that “x cut” and “y cut” elements are perpendicular and have zero inductive coupling as shown by the Neumann mutual inductance formula given in (1.11) for a rectangular inner hole. As a result, the complete spiral inductor inductance matrix for the “equal number of legs in each side” assumption can be written with the use of sub-matrix notation as,

(360) $\begin{array}{cc}{L\left(\mathrm{nxn}\right)}_{\mathrm{SPIRAL}}=\left[\begin{array}{cc}{L}_{\mathrm{xcut}}& 0\\ 0& L_{\mathrm{ycut}}\end{array}\right]->n=12& \left(1.102\right)\end{array}$
where the L.sub.xcut and L.sub.ycut are (6×6) sub-matrices having properties as given in (1.100)-(1.101).

(361) The total effective inductance of the spiral inductor under the assumption of having the same current passing through each leg of it then becomes the sum of all terms of the (1.102), which is a (12×12) matrix. Since L.sub.xcut and L.sub.ycut (6×6) are identical sub-matrices, the whole operation for calculating the effective spiral inductance of a spiral inductor having three complete turns can be reduced to,

(362) $\begin{array}{cc}{L}_{\mathrm{SPIRAL}}=2.Math.\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{n}{.Math.}}{L}_{i,j}& \left(1.103\right)\end{array}$
where n=6 and L.sub.i,j's are the L.sub.xcut or L.sub.ycut are the (6×6) sub-matrix terms in (1.103). To increase the inductance of the spiral inductor then one must increase the positive coupling and reduce the negative coupling of the inductance matrix as given in (1.103) which is basically with properties given in (1.100) and (1.101) for a 3 turn spiral inductor. The negative couplings can be reduced as one increase the d.sub.IN and the positive couplings can be increased by reducing the spacing s, between the windings.

(363) The geometric mean distance (g.m.d) between two physical rectangles, which has w>>t, increases with width w and is a weak function of spacing s, even for zero spacing s between them [1]. Therefore putting wide and thin rectangles closer does not increase the mutual inductive coupling between them significantly, which is not very obvious. This is illustrated in FIG. 1.29 where (g.m.d) of infinitely thin rectangles (w=0) with spacing of 0.5, 1, 5, 10, 40 and 100μ are plotted versus metal width ranging from 1μ to 500μ. As can be seen the mutual inductance after 10μ of width becomes almost independent of their spacing and becomes very close to a linear function of the width w, not the spacing s [1]. This property works against the objective, when one has to make wider metals to increase Q, and it also increases (g.m.d) between them and as a result reduces the mutual inductances between legs separated by the distance s. The spiral inductors in use today fall into this “small inductive coupling” category.

(364) The reason for selecting (for analysis) infinitely thin rectangles having w=0 is to show the (g.m.d) versus thickness relation to be reconstructed very easily by anyone by the use of tables given in [1]. The plot of the related table for calculating g.m.d as given in [1] is given in FIG. 1.26.2, based on parameters shown in FIG. 1.26.1 with the exact notation given. FIG. 1.27 is a graph that illustrates t vs. d values for an inductor and FIG. 1.28 is a combined plot.

(365) The insight is reducing the geometric mean distance (g.m.d) between rectangular cross-sections of the windings by arranging them such that they have high aspect ratios and facing larger dimensions toward each other, as according to this invention. As shown in [1] the (g.m.d) versus thickness for high aspect ratio rectangles arranged as said will be a function of the aspect ratio of the metals themselves and weakly dependent to the spacing s between them. This is illustrated in FIG. 1.30 where the FIG. 1.29 data is superimposed to show the significant differences very clearly. The data is shown in FIG. 1.29 is marked by loops at both ends in FIG. 1.30. As can be seen the geometric distance (g.m.d) between the infinitely thin two rectangles placed according to this invention for spacing of 0.5 and 1μ gives much smaller (g.m.d) compared to prior-art placement for thicknesses varying from 0.5 to 500μ. That would result in higher mutual inductances between them. The (g.m.d) versus thickness curves for even very large spacing range of 5, 10, 40 and 100μ starts flat and, larger than the prior-art arrangement, but as the thickness becomes on the order of spacing, it still gives much smaller (g.m.d) compared to prior-art placement. (g.m.d) versus thickness but still increases with thickness, and remains an order of magnitude less compared to prior-art placement. This property allows for large cross-sectional area rectangles to have large mutual inductances between them by placing them with the preferred orientation according to this invention. The mutual inductance along with very high inductive coupling K values that are obtained with according to this invention is theoretically and practically impossible following the prior-art placement protocols for spiral windings.

(366) In summary, the advantage of placement according to the invention lies in optimizing mutual inductance and self inductance. One needs to plot the mutual and self inductances as a function of thickness. The (g.m.d) versus thickness plot explains the reason for the increase in mutual inductance that is shown; therefore it is important to have them presented before.

(367) FIG. 1.31.1 shows mutual and self inductances as a function of thickness for a 2μ wide 4,000μ long rectangular cross-section structure spiral winding separated by again 0.5, 1, 5, 10, 40 and 100μ compared to prior-structure spaced 0.5μ apart and having thickness of 2μ plotted as a function of width. As can be seen both arrangements have the same cross-sectional areas equal to each other as in FIG. 1.31.1. As expected, both the self and mutual inductances decrease as the thickness increases, but the mutual inductance remains much higher for 0.5, 1, 5, 10, 40 and 100μ spacing compared to the prior-art placement. FIG. 1.31.2 is the same data but shown having thickness in logarithmic scale to show the small thickness dependency better.

(368) As a result the proposed metallization process in this invention is High Aspect Ratio Metallization along with High Aspect Ratio Metal Spacing, “HARMS.” This results in a “Tight Coupling Condition” between the windings which can be expressed only with an aspect ratio of metal Δ.sub.M and the adjacent winding aspect ratio between the spacing Δ.sub.S independent of metal thickness!

(369) “Tight Coupling Condition”

(370) The definition of “Tight Coupling Condition” as herein given according to the invention is such that not only the adjacent legs are tightly coupled with large coupling coefficients K, but also, where there is more than two legs or segments of windings, a number of the next closest legs are also coupled strongly with large coupling coefficients K. The advantage of this is clear from the inductance matrix and the total inductance formulation given in (1.99-1.103). Since the effective inductance of the spiral inductor is the sum of all of the inductances in the inductance matrix, increasing their coupling will increase the inductance value. Due to the passivity requirement, one can never increase the maximum mutual inductance value between any legs more than the value as given in (1.15).

(371) FIG. 1.31.1 and its logarithmic plot of FIG. 1.31.2 show self and mutual inductances. Another way of looking at this is by inspecting the mutual inductance coupling ratios K, that has been shown in FIG. 1.32 which never exceeds 1, no matter how close the adjacent windings are placed. The only way one can achieve a significant increase in the effective inductance of the spiral inductor is by increasing the mutual inductance between a number of adjacent legs of each side of the spiral inductor. This can be achieved only if the aspect ratio of the spacing between the adjacent windings Δ.sub.S remains a large value, on the order of 5 to 10 along with a winding metal aspect ratio Δ.sub.M of 10. This tight coupling condition can be expressed independent of the metal thickness and with only two target processing metallization parameters, an aspect ratio of metal Δ.sub.M and an aspect ratio of the adjacent spacing between windings represented as Δ.sub.S. A thickness-to-width ratio of at least ten and a width-to-spacing ratio between adjacent turns of at least five attains a tight coupling condition where the coupling coefficient is well above 0.5 and typically above 0.6 at the second third and fourth adjacent windings.

(372) One can represent the width and spacing of windings as shown in FIG. 1.33 in terms of metal thickness t and aspect ratios Δ.sub.M and Δ.sub.S as,

(373) $\begin{array}{cc}w=\frac{t}{{\Delta }_M}\mathrm{and}& \left(2.1\right)\\ s=\frac{t}{{\Delta }_s}& \left(2.2\right)\end{array}$
where Δ.sub.M and Δ.sub.S are both significantly larger than 1. It is clear that the spacing aspect ratio of winding number 0 to 1 is,

(374) $\begin{array}{cc}{\Delta }_s\left(1\right)=\frac{t}{s}.& \left(2.3\right)\end{array}$

(375) The spacing aspect ratio of winding number 0 to 2 also becomes a function of the metal width w of the winding number 1

(376) $\begin{array}{cc}{\Delta }_s\left(2\right)=\frac{t}{w+2.Math.s}.& \left(2.4\right)\end{array}$

(377) One point clearly evident from (2.4) is that the Δ.sub.S(2) can never be larger than Δ.sub.M! Therefore as long as Δ.sub.M<1, as in most of the prior art spiral inductors, Δ.sub.S(2) will always remain less than 1 for any spacing s>0 value. Therefore it is proper to call this condition of having Δ.sub.M>1, the “necessary condition of tight coupling.” The spacing aspect ratio of winding number 0 to 3 similarly becomes,

(378) $\begin{array}{cc}{\Delta }_s\left(3\right)=\frac{t}{2.Math.w+3.Math.s}.& \left(2.5\right)\end{array}$

(379) Recursively one can formulate that the spacing aspect ratio of winding number 0 to n becomes,

(380) $\begin{array}{cc}{\Delta }_s\left(n\right)=\frac{t}{\left(n-1\right).Math.w+n.Math.s}\mathrm{for}n=1,2,3,\mathrm{.Math.}.& \left(2.6\right)\end{array}$

(381) As can be seen in (2.6) Δ.sub.S(n) will get smaller with increasing n and again it will always be smaller than Δ.sub.M/(n−1) regardless how small s is taken! Substituting (2.1) and (2.2) in relation (2.6) can be written as,

(382) 0$\begin{array}{cc}{\Delta }_s\left(n\right)=\frac{t}{\left(n-1\right).Math.\left(\frac{t}{{\Delta }_M}\right)+n.Math.\left(\frac{t}{{\Delta }_S}\right)}.& \left(2.7\right)\end{array}$

(383) Eliminating t from (2.7) gives,

(384) $\begin{array}{cc}{\Delta }_s\left(n\right)=\frac{1}{\left(\frac{n-1}{{\Delta }_M}\right)+\left(\frac{n}{{\Delta }_S}\right)}.& \left(2.8\right)\end{array}$

(385) As can be seen, (2.8) is only functions of metallization parameters Δ.sub.M and Δ.sub.S and can be simplified to,

(386) $\begin{array}{cc}{\Delta }_s\left(n\right)=\frac{{\Delta }_M.Math.{\Delta }_S}{{\Delta }_S.Math.\left(n-1\right)+n.Math.{\Delta }_M}.& \left(2.9\right)\end{array}$

(387) The goal then becomes to find which combination of Δ.sub.M and Δ.sub.S values can give Δ.sub.S(n)>1 for n>1. With some simple arithmetic, (2.9) becomes,

(388) $\begin{array}{cc}{\Delta }_S.Math.\left(n-1\right)+n.Math.{\Delta }_M=\frac{{\Delta }_M.Math.{\Delta }_S}{{\Delta }_S\left(n\right)}.& \left(2.10\right)\end{array}$

(389) One is interested in representing n which becomes larger than 1 from (2.10) having Δ.sub.M and Δ.sub.S as independent process parameters that we need to enforce. This requires the solution of n from (2.10) as,

(390) $\begin{array}{cc}{\Delta }_S.Math.n-{\Delta }_S+n.Math.{\Delta }_M=\frac{{\Delta }_M.Math.{\Delta }_S}{{\Delta }_S\left(n\right)}.& \left(2.11\right)\end{array}$