Simultaneous Area Minimization and Decaps Insertion for Power Delivery Network Using Adjoint Sensitivity Analysis with IEKS Method Pei-Yu Huang 1 , Yu-Min Lee 1 , Jeng-Liang Tsai 2 , and Charlie Chung-Ping Chen 3 1 Department of Communication Engineering, National Chiao Tung University Hsinchu 300, Taiwan 2 Electrical and Computer Engineering, University of Wisconsin-Madison Madison, WI 53706, USA 3 Graduate Institute of Electronics Engineering and Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan pay youyi@hotmail.com, yu-min@cae.wisc.edu, jltsai@cae.wisc.edu, and cchen@cc.ee.ntu.edu.tw Abstract—The soaring clocking frequency and integration density de- mand robust and stable power delivery to support tens of millions of tran- sistor switching. In this paper, we consider the problem of minimizing the area of wires and decoupling capacitors (decaps) for a power delivery net- work, subject to the limit on integral of voltage drops. First, we derive the gradients of constraint function without Tellegen’s theorem. This greatly simplifies the discuss of adjoint sensitivity analysis. Then, we apply the IEKS method to speed up the sensitivity analysis over 3 times. Finally, this efficient analyzer is incorporated with the state-of-the-art nonlinear programming package, SNOPT, to perform the optimization. Extensive experimental results show that the proposed method can work efficiently for large power delivery networks. I. INTRODUCTION With the ever-increasing clock frequency and the aggres- sively shrinking feature sizes of high speed electronic circuits, power delivery is becoming a critical design issue. The im- proper design of power distribution system can degrade the circuit performance, and the reliability. Two basic problems are the narrowing noise margins caused by voltage drops, and the undesirable wear-out of metal wiring caused by electromi- gration. Given a topology of power delivery network, several techniques may be used for improving the quality of power delivery system: varying widths of wire segments, and adding decoupling capacitors. Wire-sizing has been shown to be an effective way to reduce the power dip/ground bounces as well as improving the electromigration. However, it is too expen- sive to use wiring sources freely. Consequently, it is necessary to minimize the area of power grid network [1], [2], [3], [4]. Most of the existing methods [1], [2] modeled the network as a resistive mesh with different constant currents consumed by different blocks. The design under constant currents consump- tion is not reliable with respect to current variations caused by time-variant current waveforms. Those variations can in- duce higher voltage drops than the expected. This problem can be remedied by over designing the power delivery net- work. While, the wiring sources will be wasted. Although [3] modeled block currents as random variables to take into account current variations. They still did not consider the dy- namic effects, capacitive or inductive, which is significant in high performance circuits. [4] included the dynamic effect and considered the structure of power delivery circuit as a global mesh feeding local trees. They applied PRIMA [5] to calcu- late the transient adjoint sensitivity over multiple intervals, and a proposed heuristic optimizer to minimize the area. In this paper, we first model the power delivery network as a lumped RC equivalent mesh circuit and attach a worst case time-variant current profile at each node. Those current profiles can be estimated by several current extraction meth- ods [6], [7]. Then, we use the PWL (piece wise linear) func- tions to approximate those profiles. After that, we use the integral of voltage drop below a specific noise margin [8] as the noise metric function for each node. Later on, we use the sum of metric function at each node as the measure function, and develop an efficient adjoint sensitivity analyzer with suit- able model order reduction techniques to calculate the gradi- ents of measure function with respect to each wire width, and each decoupling capacitor. Finally, we incorporate the above sensitivity calculation method with the state-of-the-art nonlin- ear programming algorithm, SQP (sequential quadratic pro- gramming) with SNOPT [9], to minimize the occupied area of power delivery network. The rest of the paper is organized as follows. First, the equivalent circuit model we use for the power delivery network will be introduced, and the adjoint sensitivity of voltage drop integral with model order reduction techniques will be derived in Section II. Then, the formulation of the tuning problem and optimization method will be presented in Section III. Fi- nally, the numerical experiments and conclusion will be given in Section IV, and V. II. POWER DELIVERY CIRCUIT AND ITS SENSITIV- ITY COMPUTATION Fig. 1. Equivalent Circuit Model of Power Delivery Network