Optimal design of an inductor and inductor circuit

[0002] This application is related to U.S. co-pending patent application Ser. No. 09/123,129, filed Jul. 28, 1998, and is entitled “System and Method for Designing Integrated Circuits,” which is assigned to the assignee of the present invention.

[0003] 1. Field of the Invention

[0004] The invention relates to the field of designing inductors and inductor circuits particularly for integrated circuits.

[0005] 2. Prior Art

[0006] There is a growing need for optimally designed inductors and integrated circuit, as radio-frequency integrated circuits (RFICs) become more widely used in low-cost communications. This, for instance, has created great demand and interest in on-chips spiral inductors. The parasitic resistance and capacitance associated with these spiral inductors result in several engineering trade-offs. Unfortunately, no inductor optimization tool exists to aid in circuit design. Currently most designers are limited to use of a library of previously fabricated inductors or generating a large database of inductors using a 3-D field solver. While the former option severely constrains the available design space, the latter one requires a sophisticated search engine and is numerically expensive, especially when process parameters change. Moreover, neither approach is amenable to the application dependent nature of inductor design. For example, while a resonator requires an inductor that maximizes the magnitude of the impedance at the resonant frequency (subject to a specific load capacitance), a shunt-peaked amplifier requires one that maximizes bandwidth. Consequently, the optimal layout of these inductors is determined by the design goals.

[0007] A variety of limited computer-aided-design (CAD) tools have been developed for analog circuit design, including optimization tools. In general, these tools do not provide good results for inductors in integrated circuits.

[0008] The TILOS optimization system applies geometric programming to digital circuit design, and more specifically to transistor and wire sizing for Elmore delay minimization, as described in U.S. Pat. No. 4,827,428. The geometric programs that arise in Elmore delay minimization are very specialized (the only exponents that arise are 0 and ±1). Furthermore, the representation of the problem as a geometric program is only approximate (since the actual circuits are nonlinear, and the threshold delay, not Elmore delay, is the true objective).

[0009] Thus, an improved optimization method that quickly generates globally optimal designs for an inductor and inductor circuits is needed.

[0010] A method for optimally designing an inductor or inductor circuit is disclosed. An equivalent circuit of the inductor or inductor circuit using lumped parameters is first selected. Then for each lumped parameter, a posynomial expression is obtained. Next inductor performance specifications such as quality factor or self-resonance frequency are written in posynomial form. Then a problem in which performance specification is optimized subject to constraints on other performance specifications is written as a geometric program in posynomial form. This method may be used to design inductors for low noise amplifiers, matching networks, resonators, LC resonators, shunt-peaked amplifiers and the like.

[0011] FIG. 1 is a block diagram illustrating the method of the present invention.

[0012] FIG. 2 is a plan view of a spiral inductor.

[0013] FIG. 3 is a plan view of a spiral inductor fabricated as an integrated circuit with a patterned ground shield (PGS).

[0014] FIG. 4 is a two port equivalent circuit for an inductor.

[0015] FIG. 5 is a simplified two port equivalent circuit for an inductor.

[0016] FIG. 6 is a single port equivalent circuit for an inductor.

[0017] FIG. 7 is a circuit for a low-noise amplifier.

[0018] FIG. 8 is a circuit for an LC resonator.

[0019] FIG. 9ais a small-signal equivalent circuit for the LC resonator of FIG. 8.

[0020] FIG. 9bis another equivalent circuit for the circuit of FIG. 8.

[0021] FIG. 10 is an equivalent circuit for a tunable LC resonator.

[0022] FIG. 11ais a small-signal equivalent circuit for the circuit of FIG. 10.

[0023] FIG. 11bis another equivalent circuit for the circuit of FIG. 10.

[0024] FIG. 12ais a single-pole common source amplifier.

[0025] FIG. 12billustrates the circuit of 12awith an inductor in series with the resistor, this circuit having a double-pole single-zero system.

[0026] FIG. 13 is a graph illustrating the equality factor versus inductance at 2.5 GHz.

[0027] FIG. 14 is a graph illustrating the performance of an inductor without PGS.

[0028] FIG. 15 is a graph illustrating the performance of the LC tunable oscillator of FIG. 10 where maximum parallel tank impedance is plotted against tuning range.

[0029] A method is described for optimizing inductors and inductor circuits and more particularly useful in integrated circuits to optimize RF inductors and inductor circuits. In the following description numerous specific details will be set forth in order to provide a thorough understanding of the present invention. It will be apparent to one skilled in the art that the present invention may be practiced without these specific details. In other instances, well-known mathematical techniques have not been described in detail in order not to obscure the present invention.

[0030] In co-pending application patent application Ser. No. 09/123,129, filed Jul. 28, 1998, entitled “System and Method for Designing Integrated Circuits,” computer aided design tools for designing globally optimal integrated circuits using geometric programs is described. As described in automated synthesis of globally optimal circuit designs for a given circuit topology library is obtained directly from a user defined specification. The present invention is an extension of this technology to the design of an inductor and inductor circuits. A simple and efficient CAD tool for designing an on-chip spiral inductor circuit is described in detail. The tool is based on geometric programming and a relatively simple well-known inductor model described in terms of lumped parameters where each of the parameters is represented by a posynomial expression.

[0031] Referring first to FIG. 1, the method of the present invention first chains an equivalent circuit of an inductor or inductor circuit where each of the elements of the circuit are represented as lumped parameters. Each lumped parameter is then expressed as a posynomial expression. A term for inductance is also used which includes only layout dependent variables such as the number of turns and the dimensions of the spiral inductor. This is shown by step 12 of FIG. 1. Following this as shown by step 13, a characteristic of the inductor or circuit which is to be optimized is selected. For instance, the quality factor for an inductor is often an important consideration in the design if an integrated circuit inductor. The characteristic to be optimized is expressed as a inequality, many of the examples discussed below, thereby providing constraints needed to express the characteristic as a posynomial. In particular, as shown in steps 14 and 15, the optimization problem with objective and constraints in posynomial form is expressed as a geometric program in convex form that can be solved globally and efficiently.

[0032] If f is a real-valued function of n real, positive variables x₁, x₂. . . , x_n, is referred to as a posynomial function if it has the form $f\left(x_1,\dots ,x_n\right)=\sum _{k=1}^t{c}_k{x}_1^{{\alpha }_{1k}}{x}_2^{{\alpha }_{2k}}\cdots x_n^{{\alpha }_{\mathrm{nk}}}$

[0033] where c_j≧0 and a_ijεR. When t=1, f is called a monomial function. Thus, for example, 0.7+2x₁/x₃²+x₂^0.3is posynomial and 2.3(x₁/x₂)^1.5is a monomial. Posynomials are closed under sums, products and nonnegative scaling.

[0034] A geometric program (GP) has the form $\begin{array}{cc}\begin{array}{ccc}\mathrm{minimize}& f_0\left(x\right)& \\ \mathrm{subject}\mathrm{to}& f_i\left(x\right)\le 1,& i=1,2,\dots ,m\\ & g_i\left(x\right)=1,& i=1,2,\dots ,p\\ & x_i>0,& i=1,2,\dots ,n,\end{array}& \left(1\right)\end{array}$

[0035] where f_iare posynomial functions and g_iare monomial functions. If f is a posynomial and g is a monomial, then the constraint f(x)≦g (x) can be expressed as f(x)/g(x)≦1 (since f/g is posynomial). From closure under non-negativity, constraints of the form f(x)≦a, where a>0 can also be used. Similarly, if g₁and g₂are both monomial functions, the constraint g₁(x)=g₂(x) can be expressed as g₁(x)=g₂(x)=1 (since g₁and g₂is monomial).

[0036] In general, posynomial functions are not necessarily convex. A simple change of variables converts the posynomial objective functions and constraints into convex functions. If y₁=log x₁, and take the logarithm of a posynomial f yields $h\left(y\right)=\log \left(f\left(e^{\mathrm{y1}},\dots ,e^{y_n}\right)\right)=\log \left(\sum _k^t{e}^{a_k^{T}y+b_k}\right)$

[0037] where α_k^T=[α_1k. . . α_nk] and b_k=log c_k. It can be shown that h is a convex function of y. This transformation converts the standard geometric program (1) into the convex program: $\begin{array}{cc}\begin{array}{ccc}\mathrm{minimize}& \log f_0\left(e^{\mathrm{y1}},\dots ,e^{y_n}\right)& \\ \mathrm{subject}\mathrm{to}& \log f_i\left(e^{\mathrm{y1}},\dots ,e^{y_n}\right)\le 0,& i=1,\dots ,m\\ & \log g_i\left(e^{\mathrm{y1}},\dots ,e^{y_n}\right)\le 0,& i=1,\dots ,p.\end{array}& \left(2\right)\end{array}$

[0038] This is the exponential form of the geometric program. Since this problem is convex, an efficient interior-point method can be used to solve it. The most important feature of geometric programs is that they can be globally solved with great efficiency. The algorithm determines whether the problem is infeasible (i.e., no design can meet all constraints). Also, the starting point for the optimization algorithm does not have any effect on the solution.

[0039] Efficient interior point methods to solve geometric programs have recently been developed. One such method is described by K. O. Kortanek, X. Xu, and Y. Ye, “An Infeasible Interior-point Algorithm for Solving Primal and Dual Geometric Programs,” MathProgramming,76:155-181, 1996, which is hereby incorporated by reference. This method has the desirable feature of exploiting sparsity in the problem, i.e., efficiently handling problems in which each variable appears in only a few constraints. Additional methods of solving geometric programs are described in: Y. Nesterov and A. Nemirovsky, “Interior-Point Polynomial Methods in Convex Programming,” Volume 13 of Studies in Applied Mathematics, SIAM, Philadelphia, Pa., 1994; and A. Fiacco and G. McCormick, NonlinearProgramming: Sequential Unconstrained Minimization Techniques, Wiley 1968 (Reprinted in SIAM Classics in Applied Mathematics series 1990) which are hereby incorporated by reference.

[0040] For the purposes of the present invention, the most important feature of geometric programs is that they can be globally solved with great efficiency. Perhaps even more important than the great efficiency is the fact that the present invention geometric programming method always obtains the global minimum. Infeasibility is unambiguously detected: if the problem is infeasible, then the method determines this fact. This is in contrast to conventional optimization systems that just fail to find a feasible point. Another benefit of the global solution is that the initial starting point is irrelevant, the same global solution is found no matter what the initial starting point is.

[0041] A primal barrier method can be used to solve the exponential form problem. In one embodiment, using a primal only method without exploiting sparsity, and with the overhead of an interpreted language, the geometric programs arising in an op-amp design example discussed below were all solved in approximately one to two seconds on an ULTRA SPARCI1, 170 MHz. The efficiency of the method can be increased by using a C implementation of a primal-dual method.

[0042] FIG. 2 is a plan view of a spiral conductor of the type typically fabricated in an integrated circuit using ordinary photolithographic techniques. The inductor may be fabricated from metal or polysilicon and in some cases as will be discussed, is fabricated on a grounded shield. The ends of the inductor 20 and 21 are not shown connected in FIG. 2. As will be discussed, a monomial expression is used to represent the inductance (equation 4) in an equivalent circuit. This expression is independent of process variables and parameters and uses only the physical layout of the inductor. More specifically, the square, spiral inductor of FIG. 2 is defined by the number of turns (n) and the width of each turn (w) shown by dimension 24; the distance between each turn (s) shown by dimension 25, an outer diameter (d_out) shown by dimension 23 and an average diameter given by the equation

[0043] (d_av=0.5(d_out+d_in)).

[0044] The average diameter makes use of the inner diameter shown as dimension 22. The variables w, s and d_outare restricted to take values on a discreet grid typically defined by the process used to fabricate the inductor. The number of turns is restricted to take values that are integral multiples of 0.25. However, for purposes of the explanation below, continuous variables are used. Other geometric parameters that can be used are inductor length and inductor area.

[0045] Another inductor 28 is shown in plan view in FIG. 3. The ends of the inductor are connected to the pads 26 and 27 to permit testing of the inductor. This inductor uses a grounded polysilicon shield 29. This shield resembles a brick-like pattern which is broken regularly in the direction perpendicular to the current flow within the inductor. This shield is sometimes referred to as a patterned ground shield (PGS)

[0046] The inductor of FIG. 2 may be represented as an equivalent circuit (two port model) such as shown in FIG. 4. As can be seen each of the elements in the equivalent circuit are lumped parameters which may be represented by posynomials and more specifically, in the case of FIG. 4 by monomial expressions.

[0047] The pure inductor 30 (L_s) is connected in series with a series resistance 31 (R_s) in the equivalent circuit. There is a feed-forward capacitance represented by capacitor 32 (C_s). This capacitance is due to the coupling between the spiral and the metal underpass required to connect the inner end of the spiral inductor to external circuitry. The capacitors 33 and 34 (C_ox) results from the spiral to substrate oxide capacitance. The resistor 35 (R_si) and the capacitor 36 (C_si) result from the substrate capacitance and the substrate spreading resistance. In the equivalent circuit this parallel combination of resistance 38 and capacitance 37 occurs also at the second port.

[0048] Relatively accurate closed-form expressions exist for each of the circuit elements of FIG. 4 as follows:

[0049] R_s: The series resistance is given by $\begin{array}{cc}{R}_s=\frac{l}{\sigma w\delta \left(1-e\left(\frac{-t}{\delta }\right)\right)}=k_1\left(\omega \right)\frac{l}{w},& \left(3\right)\end{array}$

[0050] where σ is the conductivity, and t is the turn thickness and δ, the skin length, is given by $\delta =\sqrt{\frac{2}{\omega {\mu }_o\sigma }},$

[0051] where ω is the frequency, is the magnetic permeability of free space (μ=4π10⁻⁷H/m) and k₁, (ω) is a constant that depends on frequency and technology parameters. Since the length is given by a monomial expression (l=4nd_avg), expression (3) is also a monomial.

[0052] L_s: An accurate monomial expression for the inductance is

[0053] L_s=βd_out^αw^αd_avg^αn^αs^α,  (4)

[0054] where the coefficients β and α₁are only layout dependent and do not depend on the technology

[0055] (β=1.66·10⁻³, α₁=−1.33,α₂=−0.125,α₃=2.50, α₄=1.83,α₅=−0.022).

[0056] This expression exhibits a maximum error of 8% over a broad design space.

[0057] C_cx: The spiral-substrate oxide capacitance accounts for most of the inductor's parasitic capacitance. It can be approximated by a monomial expression $\begin{array}{cc}{C}_{\mathrm{ox}}=\frac{1}{2}\frac{{\epsilon }_{\mathrm{ox}}}{t_{\mathrm{ox}}}\mathrm{lw}=k_2\mathrm{lw},& \left(5\right)\end{array}$

[0058] where ε_oxis the oxide permitivity. (ε_ox=3.4510⁻¹³F/cm), t_oxis the oxide thickness between the spiral and the substrate and $k_2=\frac{1}{2}\frac{{\epsilon }_{\mathrm{ox}}}{t_{\mathrm{ox}}}$

[0059] is a technology dependent constant.

[0060] C_s: This capacitance is mainly due to the capacitance between the spiral and the metal under-pass required to connect the inner end of the spiral inductor to external circuitry. It is modeled by the monomial expression, $\begin{array}{cc}{C}_s=\frac{\epsilon }{t_{\mathrm{ox},\mathrm{M1}-\mathrm{M2}}}{\mathrm{nw}}^2=k_3{\mathrm{nw}}^2,& \left(6\right)\end{array}$

[0061] where t_ox,M1−M2is the oxide thickness between the spiral and the under-pass, and $k_3=\frac{{\epsilon }_{\mathrm{ox}}}{t_{\mathrm{ox},\mathrm{M1}-\mathrm{M2}}}$

[0062] is a technology dependent constant.

[0063] C_si: The substrate capacitance is given by the following monomial expression,

[0064] C_si=½C_sublw=k₄lw,  (7)

[0065] where C_subis the substrate capacitance per unit area and $k_4=\frac{C_{\mathrm{sub}}}{2}$

[0066] is a technology dependent constant.

[0067] R_si: The substrate resistance can be expressed as $\begin{array}{cc}{R}_{\mathrm{si}}=\frac{2}{G_{\mathrm{sub}}\mathrm{lw}}=k_5\frac{1}{\mathrm{lw}},& \left(8\right)\end{array}$

[0068] where G_subis the substrate conductance per unit area and $k_5=\frac{2}{G_{\mathrm{sub}}}$

[0069] is a technology dependent constant. Note that equation (8) is monomial.

[0070] A somewhat simpler equivalent circuit for the inductor of FIG. 2 is shown in FIG. 5, again as a two-port model. In addition to C_s, L_sand R_spreviously discussed, this equivalent circuit has the parallel combination of a resistor 33 and a capacitor 44 at one port and a resistor 45 and a capacitor 46 at the other port. In this equivalent circuit, R_pand Cp are frequency dependent and have the following expressions:

[0071] R_p: The shunt resistance is given by the monomial expression, $\begin{array}{cc}{R}_p=\frac{1+{\left[\omega R_{\mathrm{si}}\left(C_{\mathrm{si}}+C_{\mathrm{ox}}\right)\right]}^2}{{\omega }^2{R}_{\mathrm{si}}{C}_{\mathrm{ox}}^2}=k_6\left(\omega \right)\frac{1}{\mathrm{lw}},& \left(9\right)\end{array}$

[0072] where the technology and frequency dependent constant k₆(ω) is given by $k_6\left(\omega \right)=\frac{2G_{\mathrm{sub}}{t}_{\mathrm{ox}}^2}{{\omega }^2{\epsilon }_{\mathrm{ox}}^2}+\frac{2}{G_{\mathrm{sub}}}\left(1+\frac{C_{\mathrm{sub}}{t}_{\mathrm{ox}}}{{\epsilon }_{\mathrm{ox}}}\right)$

[0073] C_p: The shunt capacitance is given by the posynomial expression, $\begin{array}{cc}\begin{array}{c}{C}_p=\frac{C_{\mathrm{ox}}+{\omega }^2{R}_{\mathrm{si}}\left(C_{\mathrm{si}}+C_{\mathrm{ox}}\right)C_{\mathrm{si}}{C}_{\mathrm{ox}}}{1+{\left[\omega R_{\mathrm{si}}\left(C_{\mathrm{si}}+C_{\mathrm{ox}}\right)\right]}^2}\\ =k_7\left(\omega \right)\mathrm{lw}+k_8\left(\omega \right){\left(\mathrm{lw}\right)}^2,\end{array}& \left(10\right)\end{array}$

[0074] where k₇(ω) and k₈(ω) are technology and frequency dependent constants given by $\begin{array}{cc}{k}_7\left(\omega \right)=\frac{{\epsilon }_{\mathrm{ox}}}{2t_{\mathrm{ox}}+2{t_{\mathrm{ox}}\left[\frac{\omega }{G_{\mathrm{sub}}}\left(\frac{{\epsilon }_{\mathrm{ox}}}{t_{\mathrm{ox}}}+C_{\mathrm{sub}}\right)\right]}^2}& \text{(10a)}\\ k_7\left(\omega \right)=\frac{{\epsilon }_{\mathrm{ox}}}{2t_{\mathrm{ox}}+2{t_{\mathrm{ox}}\left[\frac{\omega }{G_{\mathrm{sub}}}\left(\frac{{\epsilon }_{\mathrm{ox}}}{t_{\mathrm{ox}}}+C_{\mathrm{sub}}\right)\right]}^2}& \text{(10b)}\end{array}$

[0075] In some cases, the placement of a PGS beneath the inductor improves performance. The PGS eliminates the resistive and capacitive coupling to the substrate at the expense of the increased oxide capacitance. The expressions for C_ox, R_pand C_pare given by: $\begin{array}{cc}\begin{array}{c}{R}_p=\infty \\ C_{\mathrm{ox}}=\frac{1}{2}\frac{{\epsilon }_{\mathrm{ox}}}{t_{\mathrm{ox},\mathrm{po}}}\mathrm{lw}=k_2^{*}\mathrm{lw},\\ C_p=C_{\mathrm{ox}},\end{array}& \text{(10c)}\end{array}$

[0076] where t_ox,pois the oxide thickness between the spiral and the polysilicon layer, and k₂* is a technology dependent constant. The expressions for L_sand R_sare not altered by the PGS.

[0077] In FIG. 6 a one-port equivalent circuit for the inductor is shown. It includes the series inductor 50 and series resistor 51 (L_s, R_s) and the parallel resistor 53 (R_p) and parallel capacitor 52 (C_tot). The total shunt capacitance (C_tot) is equal to C_splus C_pand is a posynomial since both C_sand C_pare given by monomial expressions. As will be discussed below, a wide variety of inductor design specifications which typically optimize performance result in posynomial restraints.

[0078] Assume first one seeks the minimum quality factor Q≦Q_min. The quality factor can be defined as the ratio of peak magnetic energy minus peak electric energy to energy loss in one cycle. This results in the expression $\begin{array}{cc}{Q}_{\mathrm{ME}}=\frac{\omega L_s}{R_s}·\frac{\stackrel{\_}{R_p}}{\stackrel{\_}{R_p}+\left[{\left(\frac{\omega L_s}{R_s}\right)}^2+1\right]R_s}·\left(1-\frac{R_s^2\stackrel{\_}{C_{\mathrm{tot}}}}{L_s}-{\omega }^2{L}_s\stackrel{\_}{C_{\mathrm{tot}}}\right),& \left(11\right)\end{array}$

[0079] where {overscore (R_p+L )}=2R_pand {overscore (C_tot+L )}=C_tot/2 for two-port devices and {overscore (R_p+L )}=R_pand {overscore (C_tot+L )}=C_totfor one-port devices. Often, circuit designers define the quality factor as the ratio of peak magnetic energy to energy loss in one cycle (for example, in resonator designs). In this case the quality factor is given by, $\begin{array}{cc}{Q}_M=\frac{\omega L_s}{R_s}·\frac{\stackrel{\_}{R_p}}{\stackrel{\_}{R_p}+\left[{\left(\frac{\omega L_s}{R_s}\right)}^2+1\right]R_s}.& \left(12\right)\end{array}$

[0080] Equations (11) and (12) are not posynomial in nature. However, the specification for minimum quality factor can be written as $\begin{array}{cc}{Q}_{\mathrm{ME},\min }\frac{R_s}{\omega L_s}+Q_{\mathrm{ME},\min }\frac{R_s^2}{\omega L_s\stackrel{\_}{R_p}}·\left[{\left(\frac{\omega L_s}{R_s}\right)}^2+1\right]+\frac{R_s^2\stackrel{\_}{C_{\mathrm{tot}}}}{L_s}-{\omega }^2{L}_s\stackrel{\_}{C_{\mathrm{tot}}}\le 1,\mathrm{or}& \left(13\right)\\ Q_{E,\min }\frac{R_s}{\omega L_s}+Q_{\mathrm{ME},\min }\frac{R_s^2}{\omega L_s\stackrel{\_}{R_p}}·\left[{\left(\frac{\omega L_s}{R_s}\right)}^2+1\right]\le 1.& \left(14\right)\end{array}$

[0081] Equations (13) and (14) are posynomial expressions in the design variables and in Q_ME,_min(or Q_E,_min). Therefore a minimum required quality factor can be specified. Also the quality factor can be maximized by maximizing Q_ME,_min(or Q_E,_min) subject to the constraints (13) or (14).

[0082] Assume for another restraint that the minimum resonance frequency is sought. The self-resonance frequency is the frequency at which the quality factor Q_MEis zero. It is given by $\begin{array}{cc}{\omega }_{\mathrm{sr}}^2=\frac{1}{L_s\stackrel{\_}{C_{\mathrm{tot}}}}-\frac{R_s^2}{L_s^2}.& \left(15\right)\end{array}$

[0083] The specification of minimum self-resonance frequency results in a posynomial constraint in the design variables and ω_sr,min, $\begin{array}{cc}{\omega }_{\mathrm{sr},\min }^2{L}_s\stackrel{\_}{C_{\mathrm{tot}}}+\frac{R_s^2\stackrel{\_}{C_{\mathrm{tot}}}}{L_s}\le 1.& \left(16\right)\end{array}$

[0084] Therefore a specification on minimum self-resonance frequency the self-resonance frequency can be and maximized (by maximizing ω_sr,min, subject to constraint (16)).

[0085] A simple capacitor constraint would be use of the posynomial constraint.

[0086] C_tot≦C_tot,max.  (17)

[0087] Other inductor constraints can be imposed. For instance, the inductors can be made equal to some value or be written as some range. $\begin{array}{cc}\begin{array}{cc}{L}_s=L& L_{s,\min }\le L_s\le L_{s,\max }.\end{array}& \left(18\right)\end{array}$

[0088] Typically the process determines the minimum geometries (e.g., turn spacing and width) and these constraints are monomials which can be dealt with readily by geometry processing $\begin{array}{cc}\begin{array}{cc}w\ge w_{\min }& s\ge s_{\min }.\end{array}& \left(19\right)\end{array}$

[0089] The maximum inductor area is also a monomial constraint represented by

[0090] d_out²≦Area_max  (20)

[0091] Additionally the area can be minimized if needed.

[0092] The average radius d_avgis determined by the choice of d_out, n, w and s. They are related by the expression,

[0093] d_avg+(n−1)s+nw=d_out  (21)

[0094] Since the spacing s is typically compared to d_out, and w, equation 21 can be approximated by

[0095] d_avg+ns+nw=d_out  (22)

[0096] The following posynomial constraint can be imposed

[0097] d_avg+ns+nw≦d_out.  (23)

[0098] In many design problems the constraint (23) will be tight so equation (22) will hold. For all the design examples shown in this application constraint (23) is always tight.

[0099] A common problem in inductor design is to maximize the quality factor for a given inductance value and for a minimum self-resonance frequency. For example, in a narrowband low noise amplifier (LNA) such as the one shown in FIG. 7, the matching inductor 61 (L_s) is required to take a value L_s=R_sC_gs/g_mwhere C_gsis the gate to source capacitance and g_mis the transconductance of transistor 60, both of which are determined once the size and/or power of transistor 60 is determined. In the circuit of FIG. 7 the off-chip components associated with the input circuit are a resistor 64, a capacitor 63 and an inductor 62 (R_s, C_Band L_g, respectively). Ideally, the inductor 61 must have a high quality factor Q. The design problem can be expressed as follows $\begin{array}{cc}\begin{array}{cc}\mathrm{maximize}& Q_{\min }\\ \mathrm{subject}\mathrm{to}& Q\ge Q_{\min }\\ & L=L_{\mathrm{req}}\\ & {\omega }_{\mathrm{sr}}\ge {\omega }_{\mathrm{sr},\min }\\ & \dots \end{array}& \left(24\right)\end{array}$

[0100] For a particular input impedance, say 50 ohms, and a selected frequency ω_{sr ,}Q_MINcan be expressed by the equations (13) or (14).

[0101] Other constraints that can be considered are, for example, the minimum spacing, turn width, maximum area, minimum parallel capacitance, etc. In any event, the problem can be formulated as a geometric program. By repeatedly solving optimal design considerations for different inductor circuit constraints, globally optimal trade-off curves, that helps in the choice of the optimal inductor, are easily obtained. The resultant curve of FIG. 13 shows the globally optimal trade-offs between quality factor and inductance value. This is shown for the minimum quality factor at 2.5 GHz versus the inductance value, in this case with PGS. The selected design restraints are shown below in Table 1. Note that when the self-resonance frequency is not constrained a higher Q_MEresults. Details of test indicators built are shown in Table 2 below. Note that the closeness between the predicted and measured data authenticates the inductor model used.

[0102] Test inductors (dimensions are in μm, t_ox,m5-poly=5.2 μm, t_M5=0.9 μm, σ_M5=3.10⁵(Ωcm)⁻¹).

[0103] In FIG. 14 the performance of inductors with and without PGS is shown. In both cases Q_MEis maximized at an operating frequency of 1.5 GHz while meeting the specifications shown in Table 1 and guaranteeing a self-resonance frequency above 5 GHz. The PGS option is obviously preferred for small inductors because of the increase in Q_ME. This results from the elimination of the substrate loss which more than offsets the degradation due to the increase in oxide capacitance. This does not mean that PGS provides a benefit in every case and each example should be considered separately. The computations required for FIGS. 12 and 13, including the trade-off curves, can be done in real-time.

[0104] In FIG. 8 a simple tuned amplifier is shown which provides gain over selected frequencies. It includes a simple tank circuit comprising the inductor 71 and capacitor 70 and a transistor 72. A simplified small-signal equivalent circuit is shown in FIG. 9a. Here the transistor 72 of FIG. 8 has been replaced by an ideal transconductance amplifier 75. The inductor 71 has been replaced by an ideal inductor 76 (L_s) and includes a series resistor 77 (R_s), C_totand R_pare described in connection with the single port equivalent circuit of FIG. 6. The capacitor 78 (C_ext) represents the additional load capacitance. A simplified small signal equivalent circuit is shown in FIG. 9bas the tank inductance and capacitance in parallel resistance (L_tank, R_tankand d C_tank).

[0105] If the objective is to maximize the total parallel tank impedance for a given frequency (ω_res) then for a practical tank quality factor $\left(Q_{\mathrm{tank}}=\frac{R_{\mathrm{tank}}}{{\omega }_{\mathrm{res}}{L}_{\mathrm{tank}}}>1.5\right),$

[0106] then the tank elements can be expressed as $L_{\mathrm{tank}}=\left[1+{\left(\frac{R_s}{L_s\omega }\right)}^2\right]L_s$ $R_{\mathrm{tank}}=R_p\Vert R_{s,p}={\left(\frac{1}{R_p}+\frac{1}{R_{s,p}}\right)}^{-1}$

[0107] and where Q_tank>1.5, R_s,pis approximately given for a monomial, $R_{s,p}=\left[(1+{\left(\frac{L_s\omega }{R_s}\right)}^2\right]R_s\approx {\left(L_s\omega \right)}^2{R}_s.$

[0108] Since both R_pand R_s,pare given by monomial expressions, R_tankis an inverse-posynomial function of the design variables and therefore R_tankcan be maximized. A typical design problem can then be posed as a geometric program, $\begin{array}{cc}\begin{array}{cc}\mathrm{maximize}& R_{\mathrm{tank}}\\ \mathrm{subject}\mathrm{to}& L_{\mathrm{tank}}{C}_{\mathrm{tank}}\le \frac{1}{{\omega }_{\mathrm{res}}^2}\\ & Q_{\mathrm{tank}}\ge Q_{\mathrm{tank},\min }\\ & C_{\mathrm{ext}}\le C_{\mathrm{ext},\max }\\ & \dots \end{array}& \left(25\right)\end{array}$

[0109] Other design constraints such as inductor area, minimizing the turn spacing and width, etc., can be considered can still be solved as a geometric program.

[0110] The example of FIGS. 8 and 9 can be extended to a tuned resonator such as shown in FIG. 10. Note that FIG. 8 is the differential half circuit of the oscillator of FIG. 10 with the addition of the variable capacitors 80 and 81. While the value of the capacitors 80 and 81 are not limited, the ratio of the maximum to minimum capacitance is limited to approximately 2. Capacitors 82 and 38 (C_ext) again represent the additional capacitance loading. FIG. 11acorresponds to FIG. 9aand represents the simplified small signal half circuit of the oscillator of FIG. 10. Similarly, FIG. 11bcorresponds to FIG. 9bexcept that the tank capacitor of FIG. 11bincludes the variable capacitance component.

[0111] Typically, the design goal is to maximize the power resistance for parallel resistance for a given tuning range. The tuning range is specified with two constraints $\begin{array}{cc}{L}_{\mathrm{tank}}\left(C_{\mathrm{ext}}+C_{\mathrm{tot}}+C_{\mathrm{var},\min }\right)\le \frac{1}{{\omega }_{\mathrm{res},\max }^2}& \left(26\right)\\ L_{\mathrm{tank}}\left(C_{\mathrm{ext}}+C_{\mathrm{tot}}+C_{\mathrm{var},\max }\right)\ge \frac{1}{{\omega }_{\mathrm{res},\min }^2}.& \left(27\right)\end{array}$

[0112] Constraint 27 is not posynomial and cannot be directly solved by a geometric program. However, Constraint 26 is generally tight (if it were loose it would mean that an inductor with wider turns could be used and a better R_tankcould be obtained. This fact allows the handling of constraint (27) which can be written as)

[0113] ω_res,min²(C_ext+C_tot+C_var,max)≧ω_res,max²(C_ext+C_tot+C_var,min).  (27a)

[0114] If we let $r=\frac{{\omega }_{\mathrm{res},\max }^2}{{\omega }_{\mathrm{res},\min }^2},$

[0115] then $\begin{array}{cc}\left(r-1\right)\frac{C_p}{C_{\mathrm{var},\max }}+\left(r-1\right)C_{\mathrm{ext}}+{\mathrm{rC}}_{\mathrm{var},\min }\le 1.& \left(28\right)\end{array}$

[0116] Substituting the constraint 27 by the posynomial constraint 28, the design can probably be written as, $\begin{array}{cc}\begin{array}{cc}\mathrm{maximize}& R_{\mathrm{tank}}\\ \mathrm{subject}\mathrm{to}& L\left(C_{\mathrm{ext}}+C_{\mathrm{tot}}+C_{\mathrm{var},\min }\right)\le \frac{1}{{\omega }_{\mathrm{res},\max }^2}\\ & \left(r-1\right)\frac{C_p}{C_{\mathrm{var},\max }}+\left(r-1\right)C_{\mathrm{ext}}+{\mathrm{rC}}_{\mathrm{var},\min }\le 1\\ & C_{\mathrm{var},\min }\ge \alpha C_{\mathrm{var},\max }\\ & Q\ge Q_{\min }\\ & \dots \end{array}& \left(29\right)\end{array}$

[0117] FIG. 15 shows the maximum parallel tank impedance versus the tuning range for the oscillator when built with PGS. The design constraints used are shown in Table 3 below. Note that the maximum parallel impedance is high when the tuning ratio is small. When the desired tuning ratio is high, only a small inductance can be used since it contributes less parasitic capacitance and the maximum parallel impedance decreases.

[0118] The discussion above can also be used in the design of a shunt-peaked amplifier. As shown in FIG. 12athe amplifier has the parallel combination of resistor 80 and capacitor 81 coupled in series with the transistor 82. This single-pole common source amplifier has a 3 dB frequency of approximately 2π/RC. The introduction of an inductor 84, as shown in FIG. 12b, provides a double-pole, single-zero system with a frequency response which is determined by the time constant ratio (m), $\begin{array}{cc}m=\frac{\mathrm{CR}}{L/R}.& \text{(29a)}\end{array}$

[0119] The optimal value of m is determined by whether the design goal is to maximize bandwidth, minimize group delay or achieve maximally flat response.

[0120] In an integrated circuit implementation, the series resistance of the spiral inductor is absorbed in the resistor 80 and the capacitance C_totis added to the capacitor 81. The design goal is to minimize the capacitance added by the inductor so that a large bandwidth can be obtained. $\begin{array}{cc}\begin{array}{cc}\mathrm{minimize}& C_{\mathrm{tank}}\\ \mathrm{subject}\mathrm{to}& L_{\mathrm{tank}}=\frac{R^2{C}_{\mathrm{tank}}}{m}\\ & w>w_{\min }.\\ & C_{\mathrm{tank}}\ge C+C_{\mathrm{tot}}\\ & \dots \end{array}& \left(30\right)\end{array}$

[0121] Note that the last constraint is always tight. If it were not, one could obtain a smaller value for C_tank, and the result would not be optimal. This geometric program can be resolved as discussed above and additionally, other design constraints can be considered.

[0122] Thus, the design specifications for several different inductor circuits can be represented by posynomial expressions as described above. This enables the translation of the expression into one solvable via geometric programs and permits the circuits to be optimized efficiently and globally.