Latency Insertion Method (LIM) for IR Drop Analysis in Power Grid

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1 Abstract Latency Inserton Method (LIM) for IR Drop Analyss n Power Grd Dmtr Klokotov, and José Schutt-Ané Wth the steadly growng number of transstors on a chp, and constantly tghtenng voltage budgets, the problem of the desgn, analyss, and verfcaton of the power grd of a chp s becomng crtcally mportant and ncreasngly challengng. Due to the large sze of contemporary power dstrbuton networks, applcaton of the tradtonal methods based on assemblng and solvng a lnear system s becomng prohbtve. A number of teratve methods were recently proposed that avod buldng a large lnear system and thus are able to greatly reduce the requred computatonal resources and acheve orders of magntude speed ups. In ths paper, we use a transent analyss method, namely LIM, to solve the DC IR drop analyss problem of large scale power grds. The method s based on a fnte dfference formulaton; t solves crcut equatons n tme doman. The algorthm makes use or ntroduces reactve latency n all branches and nodes of a crcut to generate update algorthms for the voltage and current quanttes. Because of ts lnear numercal complexty t greatly outperforms conventonal matrx-based methods and enables analyss of very large networks. 1. Introducton Process scalng has led to a sgnfcant growth n the number of devces on a chp. Consequently, power grds are requred to carry ncreasng amounts of supply current. At the same tme, power dsspaton ssues n hgh-performance, hgh frequency devces are forcng the use of lower supply voltages [1]. Consequently, large amounts of current are dstrbuted through large scale power grds at low voltages, whch causes sgnfcant voltage drop (IR drop) on the grd, serously compromsng the performance of the chp. Therefore, effcent and accurate analyss of the IR drop n power grd becomes a prorty. Steady state power grd analyss can be formulated as a DC analyss problem. The network conssts of power sources, drans, and nterconnects. Followng [2] the power grd can be represented as a regular network as shown n Fg.1. In ths representaton power sources are modeled as ndependent constant voltage sources, and drans are modeled as constant ndependent current sources. The metal wres and vas can be represented as a regular two-dmensonal grd of lnear constant resstors Fgure 1: A power grd model.

2 For a network of sze N, the soluton va conventonal matrx based methods, would requre assemblng 2 and nvertng a matrx of the sze N. For large power grds such matrx would take a very long tme to compute, whch could easly exceed capabltes of a typcal workstaton. Therefore, dfferent knd of teratve algorthms are requred to analyze problems of such sze. 2. Iteratve Algorthms for Power Grd Analyss A number of recently proposed teratve methods take advantage of the specfc structure of the power grd. These methods avod constructng a large matrx. One such scheme s the Random Walk method [3]. Ths method apples a classcal statstcs problem to the DC analyss of a power grd and s now the stateof-the-art approach for addressng DC power grd analyss. More recently, frst-order node-based and row-based methods were ntroduced n [2], whch have both accuracy and runtme advantages over the Random Walk method. Fnally, second order teratve algorthms were presented n [4], further mprovng the rate of convergence and shortenng the runtme. All these methods explot the specal characterstcs of power grd topology to formulate an algorthm for the DC analyss. The method we propose n ths paper was orgnally developed for the transent smulaton of large networks. It nherently models capactve and nductve propertes of the network elements. However, snce DC analyss s essentally a soluton at zero frequency, we can demonstrate that our method s capable of IR drop analyss n power grd. 3. Method Formulaton The Latency Inserton Method (LIM) [5] treats a network as a grd composed of nodes nterconnected wth branches. Each branch s represented as a combnaton of a voltage source, an nductor, and a resstor n seres. Current I j s assumed to be drected from node at voltage V to node j at potental V j as shown n Fg. 2). Fgure 2: Branch representaton. Each node s modeled as a combnaton of a current source, a conductance, and a capactor to the ground (as n Fg.3). Fgure 3: Node representaton.

3 Solvng the dscrete tme equaton we arrve to the followng equatons for branch currents: I I I I R I E t n+ 1 n n+ 1 n j j n n+ 1/2 j = j + + j j j. (1) The equaton for node voltages reads: V CV M n 1/2 n + H n+ 1/2 t k= 1 = C + G t I n k (2) where M s the number of branches connected to node (excludng connectons to the ground). Superscrpt n n (1) and (2) denotes dscrete tme. Currents are computed n whole tme ntervals, voltages n half ntervals. Ths way, computatons of voltages and currents are alternated n tme. Frst, currents for all branches of a network, then voltages for all nodes of the network are computed, then tme s ncremented and computatons are repeated. Ths scheme s known as the leapfrog algorthm and s smlar to the Yee algorthm used n the fnte-dfference tme-doman (FDTD) method [6]. The presence of latency n the network, whch s generated by reactve elements, s essental for the operaton of the LIM algorthm. If a branch of the network does not contan an nductance, a small nductance must be nserted nto the branch to generate the latency. Smlarly, f some of the nodes of the network do not provde capactve paths to ground, small shunt capactors must be added to generate latency at these nodes. A numercal stablty analyss of the LIM leads to a condton for the maxmum tme step, whch can be used n the smulaton. The condton s analogous to the Courant-Fredrchs-Lewy (CFL) crteron for wave propagaton n a dscrete grd [7]. Ths condton states: t LC Ths nequalty can be vewed as a causalty condton n trackng a propagatng sgnal through a lattce of crcut elements. From (3) t follows that for networks wth no latency very small reactve elements can be nserted, however at a cost of smaller tme steps and consequently longer smulatons. 4. Applcaton of LIM to IR Drop Analyss We can consder a smple example. Fgure 4 shows a basc crcut, whch can be treated as a representaton of a small secton of the power grd [2]. The crcut conssts of 9 nodes and 12 branches. (3)

4 Fgure 4: A basc crcut example. If we set R b = 1 Ω, I D = 0.1 A, and V 0 = 1.8 V, then voltages at nodes 2, 4, 6 and 8 are V ; voltage at node 5 s V. The crcut n Fg. 4 does not contan latency; therefore, we cannot drectly apply LIM to ts analyss. In order to smulate such crcut we need to mplement the augmented network, addng nductor to every branch and capactor at every node excludng the source nodes. Fgure 5 shows a segment of the augmented verson of the crcut n Fg. 4. Fgure 5: A segment of the augmented network used for smulaton wth LIM. Snce we assume that the grd s unform, all nserted nductances have the same value, as well as all nserted capactors. Frst, base values for reactve elements are chosen. These base values are then dvded by a scalng factor to acheve a reasonable compromse between accuracy and the sze of the tme step n the smulaton, satsfyng condton (3) [5]. The augmented network can now be smulated wth LIM. Source nodes 1, 3, 7 and 9 n Fg. 4 are excluded from the update sequence, voltages at these nodes are smply set to V 0 ; Ej n (1) s set to zero; H n (2) s I D, the current dran; and there are no conductve paths to the ground G = 0 n (2). Fgure 6 shows the results of LIM smulaton of the crcut n Fg. 4.

5 The base values for nserted nductances and capactances were chosen arbtrarly: L = 30 nh, and C = 20pF. The base values were then scaled down by a factor of 10 to make them small enough, so that they do not affect the results of the smulaton. Values of the reactve elements nserted to create latency determned the lmt for the tme step of the smulaton. Smulaton s stable as long as t LC. In the actual computer smulaton t = LC s not realzable due to round-off errors. In our smulaton t = 0.65ps was used. max Fgure: Smulaton results for the basc network. In Fg. 6 voltages at the source node (node 1 n partcular) and at node 5 are shown. At frst voltage at node 5 oscllates rapdly n a rather wde range between -10 V and +15 V, then t converges to the steady state and to the correct value of V. To reduce oscllatons and acheve faster convergence, the source voltage can be slowly ncreased resultng n the waveforms shown n Fg. 7. Fgure 7: Smulaton wth ramped sources.

6 As can be seen from Fg. 7, ampltude of the oscllatons dd decrease; however, t took approxmately the same number of steps for the voltage to stablze. To further mprove convergence, an alternatve model for the power supply grd was mplemented. Instead of usng current sources to represent power snks, we nserted conductve paths to the ground. Fgure 8(a) shows the alternatve model for a secton of the power supply grd; a segment of correspondng augmented network s shown n Fg. 8(b). Fgure 8:Alternatve model for power supply network. If we set R b = 1Ω, R n = 18Ω, V 0 = 1.8 V, then voltage at node 5 s V. Fgure 9 demonstrates results of the LIM smulaton for crcut n Fg. 8. Fgure 8: Smulaton wth the alternatve model. Expermental results show that the smulaton wth the model n Fg. 8 converges farly quckly, and does not exhbt oscllatons seen n the results obtaned usng the orgnal model. The smulaton remaned stable wth t = 0.99 t = ps. max

7 5. Summary of Expermental Results We appled our method to several large crcuts modelng power grd network. Results for Random Walk method are taken from [2]; LIM computatons were carred out on a machne wth smlar characterstcs (2.4 GHz CPU, 2-GB of RAM). Results are summarzed n Table 1. Table 1: Runtme Comparson between Random Walk and LIM Methods Number of Nodes Random Walk CPU Tme (mn:sec) LIM (resstve model) CPU Tme (mn:sec) 10K 0:10 0:00 250K 12:21 0:18 1M 43:47 1:16 From Table 1 we observe that LIM outperforms Random Walk algorthm. Fgure 9 shows a plot of runtme vs. number of nodes for LIM smulatons wth resstve model. From Fg. 9 t can be seen that the runtme changes lnearly wth the sze of the network. 6. Conclusons Fgure 9: Runtme vs. number of nodes for LIM smulatons usng alternatve model. A new method for IR drop analyss n power supply network was proposed. The method s based on the latency nserton method (LIM) and essentally apples the technque for transent smulatons to the DC analyss of the power grd. Expermental results show that the LIM based algorthm outperforms the Random Walk method, especally for crcuts wth large numbers of nodes. Runtme of the algorthm demonstrates lnear dependency on the sze of the network under analyss. Another advantage of the latency nserton method s ts wder range of applcatons; capablty of performng transent analyss as well as DC. Also, LIM s not lmted by the specfc structure of a power grd model, snce t s able to smulate networks contanng reactve and nonlnear elements.

8 7. References [1] M. K. Govan, L. L. Bro, and D. B. Jackson. Power consderatons n the desgn of the alpha mcroprocessor, Proceedngs of 35th Desgn Automaton Conference, San Francsco, 1998, pp [2] Y. Zhong and M. D. Wong. Fast algorthms for IR drop Analyss n large power grd, Proceedngs of Internatonal Conference on Computer Aded Desgn, 2005, pp [3] H. Qan, S. R. Nassf, and S. S. Sapatnekar. Random walks n a supply network, Proceedngs of Annual ACM IEEE Desgn Automaton Conference, 2003, pp [4] Y. Zhong and M. D. Wong. Effcent Second-Order Iteratve Methods for IR Drop Analyss n Power Grd, Proceedngs of 12th Asa and South pacfc Desgn Automaton Conference, 2007, pp [5] José E. Shutt-Ané, Latency nserton method (LIM) for the fast transent smulaton of large networks, IEEE Trans. Crcut Syst., vol. 48, Jan. 2001, pp [6] K. S. Yee, Numercal soluton of ntal boundary value problems nvolvng Maxwell s equaton n sotropc meda, IEEE Antennas Propagat., vol. AP-14, May 1996, pp [7] R. Courant, K. O. Fredrchs, and H. Lewy, Uber de Partallen Dfferenzenglechungen der Mathematschen Physk, Math. Ann., vol. 100, no. 32, 1928.

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