1. CMOS Op-Amp Power Optimization in All Regions of Inversion Using Geometric Programming Pablo Aguirre and Fernando Silveira IIE – FING Universidad de la República Montevideo, Uruguay. paguirre@fing.edu.uy

2. SBCCI 2008Pablo Aguirre - Fernando Silveira2 Outline Introduction Geometric Programming (GP) and the gm/ID ratio Relaxed GP Proof of concept: The intrinsic Amplifier Design Example: The Miller Amplifier Conclusions

3. SBCCI 2008Pablo Aguirre - Fernando Silveira3 Introduction Automatic analog circuit synthesis: since 1980's Broad classification of techniques: Simulation-based or Equation-based. Optimization methods: we want to assure a global optimum. Three main methods: Branch and bound (B&B), Simulated annealing (SA) and Convex optimization B&B and SA: slow, computational effort grows exponentially with problem size. SA does not guarantee global optimum in practice. Convex optimization: Pros: extremely efficient, can cope with 100’s of vars. and 1000’s of constraints in seconds. Cons: Eqs. must be formulated in posynomial form.

4. SBCCI 2008Pablo Aguirre - Fernando Silveira4 GP and the gm/ID ratio GP Problem: Monomials: Posynomials: x: vector real positive vars c,c k : ≥0; a ik real GP problems can be cast into convex form. Therefore, convex optimization methods guarantee a global solution or will unambiguosly detect that the GP problem is unfeasible.

5. SBCCI 2008Pablo Aguirre - Fernando Silveira5 GP and the gm/ID ratio ACM [Cu98]: Inv. coeff: gm/ID ratio: KEY PARAMETER IN ANY ANALOG DESIGN Present on most measurements of analog performance [Si04]: Gain Speed Noise Offset … The larger the gm/ID ratio, the more power efficient the transistor is [Si96]. Power efficiency

6. SBCCI 2008Pablo Aguirre - Fernando Silveira6 GP and the gm/ID ratio Posynomial equality: NOT VALID IN GP CONVEX SET CONCAVE SET (≤ 1) (≥ 1) Let’s pose the relation between gm/ID ratio and i f as a posynomial:

7. Where are we? Where are we heading? Prior ART: Hershenson et al. [He01] and Mandal et al. [Ma01]: First report on GP applied to analog CMOS design. STRONG INVERSION ONLY MODELS, therefore not true global optimum. Brodersen et al. [Va04]: GP on all regions. Uses branch and bound algorithm on top of GP. Looses the computational efficiency of GP. This work: We look for an efficient solution to the posynomial equality of the gm/ID ratio in the case of POWER OPTIMIZATION. This problem is intrinsic to the physics of the MOS transistor. Does not depends on the model used.

8. SBCCI 2008Pablo Aguirre - Fernando Silveira8 Relaxed GP Original Problem: Global optimum: x s if h i (x s )=1 for all i, then x s is also an optimum for the original GP monomials posynomials Relaxed GP: Valid Not valid

9. SBCCI 2008Pablo Aguirre - Fernando Silveira9 Relaxed GP What does this mean? Algorthim side: We relaxed the original GP, now we can solve it as any other GP. Circuit side: We artifically increased the set of feasible transistors (shaded area). How do we use this on our problem?

10. SBCCI 2008Pablo Aguirre - Fernando Silveira10 Relaxed GP How do I know that a power optimization algorithm will not use transistors that don’t really exist? Because we know that the larger the gm/ID ratio, the more power efficient a transistor will be. Therefore, it is reasonable to assume that the power optimization will always find its solution on the curve How do we use this on our problem?

11. SBCCI 2008Pablo Aguirre - Fernando Silveira11 Proof of Concept: The Intrinsic Amplifier The most simple amplifier We don’t need GP to solve it, but its simplicity gives a clear picture on how the idea works. Goal: Synthesize an intrinsic amp. with optimal power consumption that complies with a GBW constraint for a given C L. Design eqs.: Design vars.: ID, W, L, i f, gm/ID

12. SBCCI 2008Pablo Aguirre - Fernando Silveira12 Proof of Concept: The Intrinsic Amplifier Synthesis results for 4 different cases. GP assures that these solutions are the global optimal solutions to the relaxed problem. They all comply with the gm/ID ratio equality constraint (they are on the curve). Therefore, these solutions are also the global optimal solutions to the original problem: Those are real transistors, we are done. Are these four cases just a coincidence? CL=1pF

13. The Intrinsic Amplifier: Method Scope Are these four cases just a coincidence? No, formal proof on the paper. Which is the scope of this approach? Power consumption dictated by gm requirements. Out of scope example: Consumption imposed by Slew-rate In these cases: The exception is easily noticed. Increasing the computational effort and complexity around the initial non-valid solution, we can find a valid global optimum. Ex.: Branch and bound [Va04], Tightening [Bo05].

14. SBCCI 2008Pablo Aguirre - Fernando Silveira14 Design Example: The Miller Amplifier Two stage Miller Amplifier Goal: Optimize power consumption for a given GBW, PM and CL Other constrains included: Gain, Output swing, Area, etc. Hershenson [He01] deeply analize how to pose all the equations in monomial/posynomial form. We used expressions valid in all regions of operation (e.g. intrinsic capacitances). Design variables: W 1,W 2,W 3, W 4, W 5 =W 6, L 1, L 2, L 3, L 4 =L 5 =L 6, ID 1, ID 2, (gm/ID) 1, (gm/ID) 2=3, i f1, i f2 =i f3, i f4 =i f5 =i f6, Cm

15. SBCCI 2008Pablo Aguirre - Fernando Silveira15 Design Example: The Miller Amplifier Two relaxed constraints: (gm/ID) 1 and (gm/ID) 2 All the solutions on the curve, i.e. the idea works in more complex designs. Most optimums on moderate inversion: Expected result. Nevertheless it is important to confirm the need for a model valid in all regions of inversion.

16. SBCCI 2008Pablo Aguirre - Fernando Silveira16 Conclusions GP applied in CMOS power optimization with true (all regions) global optimum. Posynomial characteristic of the gm/ID curve efficiently handled. This work confirms that power optimums usually lie on moderate inversion region. Future work: Work on the exceptions to handle them efficiently. Inclusion of short channel effects (this work: long channel models only).

17. SBCCI 2008Pablo Aguirre - Fernando Silveira17 Acknowledgments PDT - Programa de Desarrollo Tecnológico - Uruguay

18. References [He01] M. Hershenson, S. Boyd, and T. Lee, “Optimal design of a cmos op-amp via geometric programming," IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 20, no. 1, pp. 1- 21, Jan. 2001. [Ma01] P. Mandal and V. Visvanathan, “Cmos op-amp sizing using a geometric programming formulation," IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 20, no. 1, pp. 22- 38, Jan. 2001. [Va04] J. Vanderhaegen and R. Brodersen, “Automated design of operational transconductance ampliers using reversed geometric programming," in Proceedings on 41st Design Automation Conference, vol. I, Jun. 2004, pp. 133-138. [Si96] F. Silveira, D. Flandre, and P. G. Jespers, “A (gm/ID) based methodology for the design of cmos analog ircuits and its application to the synthesis of a silicon-on-insulator micropower OTA," IEEE J. Solid-State Circuits, vol. 31, no. 9, pp. 1314-1319, Sep. 1996. [Cu98] A. Cunha, M. Schneider, and C. Galup-Montoro, “An MOS transisotr model for analog circuit design," IEEE J. Solid-State Circuits, vol. 33, no. 10, pp. 1510-1519, Oct. 1998. [Bo05] S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, “A tutorial on geometric programming," Stanford University and University of California Los Angeles, Tech. Rep., 2005. [Online]. Available: http://www.stanford.edu/~boyd/gptutorial.html [Si04] F. Silveira and D. Flandre, Low Power Analog CMOS for Cardiac Pacemakers Design and Optimization in Bulk and SOI Technologies, ser. The Kluwer International Series In Engineering And Computer Science. Boston: Kluwer Academic Publishers, Jan. 2004, vol. 758, ISBN 1-4020-7719-X.

19. CMOS Op-Amp Power Optimization in All Regions of Inversion Using Geometric Programming Pablo Aguirre and Fernando Silveira IIE – FING Universidad de la República Montevideo, Uruguay. paguirre@fing.edu.uy