Optimal Design of a CMOS Op-amp via Geometric Programming Maria del Mar Hershenson, Stephen P. Boyd, Thomas H. Lee Electrical Engineering Department Stanford University Stanford CA 94305 [email protected], [email protected] [email protected] Submitted to Transactions on Computer-Aided Design, November 1997 Abstract We describe a new method for determining component values and transistor dimensions for CMOS operational ampli(cid:12)ers (op-amps). We observe that a wide variety of design objec- tives and constraints have a special form, i.e., they are posynomial functions of the design variables. As a result the ampli(cid:12)er design problem can be expressed as a special form of optimizationproblemcalledgeometric programming,for which very e(cid:14)cient global optimiza- tion methods have been developed. As a consequence we can e(cid:14)ciently determine globally optimal ampli(cid:12)erdesigns, or globallyoptimaltrade-o(cid:11)s amongcompeting performance mea- suressuchaspower, open-loopgain,andbandwidth. Ourmethodthereforeyieldscompletely automated synthesis of (globally) optimal CMOS ampli(cid:12)ers, directly from speci(cid:12)cations. In this paper we apply this method to a speci(cid:12)c, widely used operational ampli(cid:12)er ar- chitecture, showing in detail how to formulate the design problem as a geometric program. We compute globally optimal trade-o(cid:11) curves relating performance measures such as power dissipation, unity-gain bandwidth, and open-loop gain. We show how the method can be used to synthesize robust designs, i.e., designs guaranteed to meet the speci(cid:12)cations for a variety of process conditions and parameters. Contents 1 Introduction 3 1.1 The two-stage ampli(cid:12)er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Other approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Outline of paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Geometric programming 8 2.1 Geometric programming in convex form . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Solving geometric programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Dimension constraints 12 3.1 Symmetry and matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Limits on device sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Systematic input o(cid:11)set voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Bias conditions, signal swing, and power constraints 13 4.1 Bias conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Gate overdrive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 Quiescent power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5 Small signal transfer function constraints 15 5.1 Small signal transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.2 Open-loop gain constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.3 Minimum gain at a frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.4 3dB bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.5 Dominant pole conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.6 Unity-gain bandwidth and phase margin . . . . . . . . . . . . . . . . . . . . . . . . . 18 6 Other constraints 20 6.1 Slew rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.2 Common-mode rejection ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.3 Power supply rejection ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.4 Noise performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7 Optimal design problems and examples 22 7.1 Summary of constraints and speci(cid:12)cations . . . . . . . . . . . . . . . . . . . . . . . . 22 7.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 7.3 Trade-o(cid:11) analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 7.4 Sensitivity analysis example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.5 Design veri(cid:12)cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 8 Design for process robustness 34 9 Discussions and conclusions 37 1 A MOSFET models 39 A.1 Large signal models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 A.2 Small signal models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 B Derivation of bias conditions 42 C Technology parameters 44 2 1 Introduction As the demand for mixed mode integrated circuits increases, the design of analog circuits such as operational ampli(cid:12)ers (op-amps) in CMOS technology becomes more critical. Many authors have noted the disproportionately large design time devoted to the analog circuitry in mixed mode integrated circuits. In this paper we introduce a new method for determining the component values and transistor dimensions for CMOS op-amps. The method handles a very wide variety of speci(cid:12)cations and constraints, is extremely fast, and results in globally optimal designs. The performance of an op-amp is characterized by a number of performance measures such as open-loop voltage gain, quiescent power, input-referred noise, output voltage swing, unity-gain bandwidth, input o(cid:11)set voltage, common-mode rejection ratio, slew rate, die area, and so on. These performance measures are determined by the design parameters, e.g., transistor dimensions, bias currents, and other component values. The CMOS ampli(cid:12)er design problem we consider in this paper is to determine values of the design parameters that optimize an objective measure while satisfying speci(cid:12)cations or constraints on the other performance measures. This design problem can be approached in several ways, for example by hand or a variety of computer-aideddesign methods, e.g., classicaloptimizationmethods, knowledge-based methods, or simulated annealing. (These methods are described more fully below). In this paper, we introduce a new method that has a number of important advantages over current methods. We formulate the CMOS op-amp design problem as a very special type of optimizationproblem called a geometric program. The most important feature of ge- ometricprograms is that the globally optimal solutioncan be computed with great e(cid:14)ciency, even for problems with hundreds of variables and thousands of constraints, using recently developed interior-point algorithms. Thus, even challenging ampli(cid:12)er design problems with many variables and constraints can be (globally) solved. The fact thatgeometric programs(and hence, CMOS op-ampdesign problemscast as ge- ometric programs) can be globallysolved has a number of important practical consequences. The(cid:12)rstisthatsetsofinfeasiblespeci(cid:12)cationsareunambiguouslyrecognized: thealgorithms either produce a feasible point or a proof that the set of speci(cid:12)cations is infeasible. Indeed, the choice of initial design for the optimization procedure is completely irrelevant (and can even be infeasible); it has no e(cid:11)ect on the (cid:12)nal design obtained. Since the global optimum is found, the op-amps obtained are not just the best our method can design, but in fact the best any method can design (with the same speci(cid:12)cations). In particular, our method computes the absolute limit of performance for a given ampli(cid:12)er and technology parameters. The fact that geometric programs can be solved very e(cid:14)ciently has a number of practical consequences. For example, the method can be used to simultaneously optimize the design of a largenumber of op-ampsin a singlelargemixedmode integrated circuit. In thiscase the designs of the individual op-amps are coupled by constraints on total power and area, and by various parameters that a(cid:11)ect the ampli(cid:12)er coupling such as input capacitance, output resistance, etc. Another application is to use the e(cid:14)ciency to obtain robust designs, i.e., designs that are guaranteed to meet a set of speci(cid:12)cations over a variety of processes or technology parameter values. This is done by simply replicating the speci(cid:12)cations with a 3 VDD M8 M7 M5 Vin+ M1 M2 Vin(cid:0) Rc Cc CL Ibias M6 M3 M4 VSS Figure 1: Two stage op-amp considered in this paper. (possiblylarge)number ofrepresentative process parameters, which ispracticalonly because geometric programs with thousands of constraints are readily solved. The method we present can be applied to a wide variety of ampli(cid:12)er architectures, but in this paper we apply the method to a speci(cid:12)c two-stage CMOS op-amp. The authors show how the method extends to other architectures in another paper [1]. 1.1 The two-stage ampli(cid:12)er The speci(cid:12)c two-stage CMOS op-amp we consider is shown in Figure 1. The circuit consists of an input di(cid:11)erential stage with active load followed by a common-source stage also with active load. An output bu(cid:11)er is not used; this ampli(cid:12)er is assumed to be part of a VLSI system and is only required to drive a (cid:12)xed on-chip capacitive load of a few picofarads. This op-amp architecture has many advantages: high open-loop voltage gain, rail-to-rail output swing, large common-mode input range, only one frequency compensation capacitor, and a small number of transistors. Its main drawback is the nondominant pole formed by the load capacitance and the output impedance of the second stage, which reduces the achievable bandwidth. Another potential disadvantage is the right half plane zero that arises from the feedforward signal path through the compensating capacitor. Fortunately the zero is easily removed by a suitable choice for the compensation resistor Rc (see [2]). Thisop-ampisawidelyusedgeneralpurposeop-amp[3]; it(cid:12)ndsapplicationsforexample in switched capacitor (cid:12)lters [4], analog to digital converters [5, 6], and sensing circuits [7]. There are 18 design parameters for the two-stage op-amp: The widths and lengths of all transistors, i.e., W1;:::;W8 and L1;:::;L8. (cid:15) The bias current Ibias. (cid:15) 4 The value of the compensation capacitor Cc. (cid:15) The compensation resistor Rc is chosen in a speci(cid:12)c way that is dependent on the design parameters listed above (and described in 5). There are also a number of parameters that x we consider (cid:12)xed, e.g., the supply voltages VDD and VSS, the capacitive load CL, and the various process and technology parameters associated with the MOS models. 1.2 Other approaches Thereisahugeliterature,whichgoesbackmorethantwentyyears, oncomputer-aideddesign of analogcircuits. Agoodsurvey ofearlyresearch canbe foundinthe survey [8]; morerecent papers on analogcircuit CAD tools include, e.g., [9, 10, 11]. The problem we consider in this paper, i.e., selection of component values and transition dimensions, is only a part of a com- plete analog circuit CAD tool. Other parts, that we do not consider here, include topology selection (see, e.g., [12]) and actual circuit layout (see, e.g., ILAC [13], KOAN/ANAGRAM II [14]). The part of the CAD process that we consider lies in between these two tasks; the remainder of the discussion is restricted to methods dealing with component and transistor sizing. Classical optimization methods Generalpurposeclassicaloptimizationmethods,suchassteepestdescent, sequentialquadratic programming and Lagrange multiplier methods, have been widely used in analog circuit CAD. These methods can be traced back to the survey paper [8]. The widely used gen- eral purpose optimization codes NPSOL [15] and MINOS [16] are used in, e.g., [17, 18, 19]. Other CAD methods based on classical optimization methods, and extensions such as a minimax formulation, include the one described in [20, 21, 22], OAC [23], OPASYN [24], CADICS [25], WAPOPT [26], and STAIC [27]. The classical methods can be used with more complicated circuit models, including even full SPICE simulations in each iteration, as in DELIGHT.SPICE [28] (which uses the general purpose optimizer DELIGHT [29]) and ECSTASY [30]. The main advantage of these methods is the wide variety of problems they can handle; the only requirement is that the performance measures, along with one or more derivatives, can be computed. The main disadvantage of the classical optimization methods is they only (cid:12)nd locally optimal designs. This means that the design is at least as good as neighboring designs, i.e., smallvariationsofany ofthe design parametersresults ina worse (or infeasible) design. Unfortunately this does not mean the design is the best that can be achieved, i.e., globallyoptimal; it is possible (and often happens) that some other set of design parameters, far away from the one found, is better. The same problem arises in determining feasibility: a classical (local) optimization method can fail to (cid:12)nd a feasible design, even though one exists. Roughlyspeaking, classicalmethods can get stuck atlocalminima. This shortcoming is so well known that it is often not even mentioned in papers; it is taken as understood. The problemofnonglobalsolutionsfromclassicaloptimizationmethods canbe treated in several ways. The usual approach is to start the minimization method from many di(cid:11)erent initial designs, and to take the best (cid:12)nal design found. Of course there are no guarantees 5 that the globally optimaldesign has been found; this method merely increases the likelihood of (cid:12)nding the globally optimal design. This method also destroys one of the advantages of classical methods, i.e., speed, since the computation e(cid:11)ort is multiplied by the number of di(cid:11)erent initials designs that are tried. This method also requires human intervention (to give \good" initial designs), which makes the method less automated. The classical methods become slow if complex models are used, as in DELIGHT.SPICE, which requires more than a complete SPICE run at each iteration (\more than" since the gradients and second derivatives must also be computed). Knowledge-based methods Knowledge-based and expert-systems methods have also been widely used in analog cir- cuit CAD. Examples include genetic algorithms or evolution systems like SEAS [31], DAR- WIN [32, 33]; systems based on fuzzy logic like FASY [34] and [35]; special heuristics based systems like IDAC [36, 37], OASYS [38], BLADES [39], and KANSYS [40]. Oneadvantageofthesemethodsisthattherearefewlimitationsonthetypesofproblems, speci(cid:12)cations, and performance measures that can be considered. Indeed, there are even fewer limitations than for classical optimization methods since many of these methods do not require the computation of derivatives. These methods have several disadvantages. They (cid:12)nd a locally optimal design (or, even just a \good" or \reasonable" design) instead of a globally optimal design. The (cid:12)nal design depends on the initial design chosen and the algorithm parameters. As with classical opti- mization methods, infeasibility is not unambiguously detected; the method simply fails to (cid:12)nd a feasible design (even when one may exist). These methods require substantial human intervention either during the design process, or during the training process. Global optimization methods Optimization methods that are guaranteed to (cid:12)nd the globally optimal design have also been used in analog circuit design. The most widely known global optimizationmethods are branch and bound [41] and simulated annealing [42, 43]. A branch and bound method is used, for example, in [12]. Branch and bound meth- ods unambiguously determine the global optimal design: at each iteration they maintain a suboptimal feasible design and also a lower bound on the achievable performance. This enables the algorithm to terminate nonheuristically, i.e., with complete con(cid:12)dence that the global design has been found within a given tolerance. The disadvantage of branch and bound methods is that they are extremely slow, with computation growing exponentially with problem size. Even problems with ten variables can be extremely challenging. Simulated annealing (SA) is another very popular method that can avoid becoming trapped in a locally optimal design. In principle it can compute the globally optimal so- lution, but in implementations there is no guarantee at all, since, for example, the cooling schedules called for in the theoretical treatments are not used in practice. Moreover, no real-time lower bound is available, so termination is heuristic. Like classical and knowledge- based methods, SA allows a very wide variety of performance measures and objectives to be handled. Indeed, SA is extremely e(cid:11)ective for problems involving continuous variables and 6 discrete variables, as in, e.g., simultaneous ampli(cid:12)er topology and sizing problems. Simu- lated annealing has been used in several tools such as ASTR/OBLX [44], OPTIMAN [45] , FRIDGE [47], SAMM [48] and [49]. The mainadvantagesof SAarethatithandles discretevariableswell, andgreatlyreduces the chances of (cid:12)nding a nonglobally optimal design. (Practical implementations do not reduce the chance to zero, however.) The main disadvantage is that it can be very slow, and cannot (in practice) guarantee a global optimal solution. Convex optimization and geometric programming methods In this section we describe the general optimizationmethod we employ in this paper: convex optimization. These are special optimizationproblems in which the objective and constraint functions are all convex. While the theoretical properties of convex optimization problems have been appreciated for many years, the advantages in practice are only beginning to be appreciated now. The main reason is the development of extremely powerful interior-point methods for general convex optimization problems in the last (cid:12)ve years ( e.g., [51, 52]). These methods can solve large problems, with thousands of variables and tens of thousands of constraints, very e(cid:14)ciently (say, in minutes on a small workstation). Problems involving tens of variables and hundreds of constraints (such as the ones we encounter in this paper) are considered small, and can be solved on a small current workstation in less than one second. The extreme e(cid:14)ciency of these methods is one of their great advantage. The other main advantage is that the methods are truly global, i.e., the global solution is always found, regardless of the starting point (which, indeed, need not be feasible). In- feasibility is unambiguously detected, i.e., if the methods do not produce a feasible point they produce a certi(cid:12)cate that proves the problem is infeasible. Also, the stopping criteria are completely nonheuristic: at each iteration a lower bound on the achievable performance is given. One of the disadvantages is that the types of problems, performance speci(cid:12)cations, and objectives that can be handled are far more restricted than any of the methods described above. This is the price that is paid for the advantages of extreme e(cid:14)ciency and global solutions. (For more on convex optimization, and the implications for engineering design, see [53].) The contribution of this paper is to show how to formulate the analog ampli(cid:12)er design problemasacertaintypeofconvex problemcalledgeometricprogramming. Theadvantages, compared to the approaches described above, are extreme e(cid:14)ciency and global optimality. The disadvantage is less (cid:13)exibility in the types of constraints we can handle, and the types of circuit models we can employ. As far as we know, the only other application of geometric programming to circuit de- sign is in transistor and wire sizing for Elmore delay minimization in digital circuits, as in TILOS [54] and other programs [55, 56, 57]. Their use of geometric programming can be distinguished from ours in several ways. First of all, the geometric programs that arise in Elmore delay minimization are very specialized (the only exponents that arise are 0 and 1). Second, the problems they encounter in practice are extremely large, involving up to (cid:6) 7 hundreds of thousands of variables. Third, their representation of the problem as a geomet- ric program is only approximate (since the actual circuits are nonlinear, and the threshold delay, not Elmore delay, is the true objective). Convex optimization is mentioned in several papers on analog circuit CAD. The ad- vantages of convex optimization are mentioned in [12, 58]. In [19, 59] the authors use a supporting hyperplane method, which they point out provides the global optimum if the feasible set is convex. In [60] the authors optimize a few design variables in an op-amp using a Lagrange multipliermethod, which yields the global optimum since the small subproblems considered are convex. 1.3 Outline of paper In 2, we brie(cid:13)y describe geometric programming, the special type of optimization problem x attheheartofthemethod,andshow howitcanbecastasaconvex optimizationproblem. In 3{6we describeavarietyofconstraintsandperformancemeasures, and show thatthey have x the special form required for geometric programming. In 7 we give numerical examples of x the design method, showing globally optimal trade-o(cid:11) curves between various performance measures such as bandwidth, power, and area. We also verify some of our designs using high (cid:12)delity SPICE models, and brie(cid:13)y discuss how our method can be extended to handle short-channel e(cid:11)ects. In 8 we discuss robust design, i.e., how to use the methods to ensure x proper circuit operation under various processing conditions. In 9 we give our concluding x remarks. 2 Geometric programming Let x1;:::;xn be n real, positive variables. We will denote the vector (x1;:::;xn) of these variables as x. A function f is called a posynomial function of x if it has the form t (cid:11)1k (cid:11)2k (cid:11)nk f(x1;:::;xn) = ckx1 x2 xn k=1 (cid:1)(cid:1)(cid:1) X where cj 0 and (cid:11)ij R. Note that the coe(cid:14)cients ck must be nonnegative, but the (cid:21) 2 exponents (cid:11)ij can be any real numbers, including negative or fractional. When there is only one term in the sum, i.e., t = 1, we call f a monomial function. (This terminology is not consistent with the standard de(cid:12)nition of a monomial in algebra, but it should not cause 2 0:3 any confusion.) Thus, for example, 0:7+2x1=x3 +x2 is posynomial (but not monomial); 1:5 2 0:3 2:3(x1=x2) isamonomial(and, therefore, alsoaposynomial); while2x1=x3 x2 isneither. (cid:0) Note that posynomials are closed under addition, multiplication, and nonnegative scaling. Monomials are closed under multiplication and division. A geometric program is an optimization problem of the form minimize f0(x) subject to fi(x) 1; i = 1;:::;m; (cid:20) (1) gi(x) = 1; i = 1;:::;p; xi > 0; i = 1;:::;n; 8