WITH high operating frequencies and scaled geometries,

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1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS BRIEFS, VOL. 56, NO. 7, JULY Inuctance Moel of Interigitate Power an Groun Distribution Networks Renatas Jakushokas, Stuent Member, IEEE, anebyg.frieman,fellow, IEEE Abstract A close-form expression is presente in this brief to accurately estimate the effective inuctance of a single layer within an interigitate power an groun P/G istribution network. Due to the large number of P/G lines in these networks, excessive time is require to calculate the inuctance using 3-D simulation tools. The propose expression is favorably compare with previous moels an FastHenry, exhibiting accuracy an computational efficiency. The inuctance of a single layer within an interigitate P/G istribution network is boune for any number of lines. The error of the propose expression rapily ecreases with an increasing number of pairs within the network. The upper boun for the error of the propose moel is also etermine. Inex Terms Interigitate structure, power an groun P/G network, mutual inuctance, self-inuctance. I. INTRODUCTION WITH high operating frequencies an scale geometries, power an groun P/G istribution networks require greater esign optimization to effectively provie higher current flow. Low supply voltages an high currents in ICs place stringent constraints on P/G istribution networks. Higher frequencies an smaller transistors prouce shorter transition times, such that Li/t voltage rops excee IR voltage rops. All of these factors require the inuctance to be consiere in the esign of on-chip P/G istribution networks. To optimize these large-scale P/G istribution networks, the inuctance nees to be accurately an efficiently calculate. An interigitate P/G istribution network structure, where a few wie lines are replace by a large number of narrow lines, is often use to reuce the inuctance effect 1], 2]. Different P/G structures have been compare in 3], where the interigitate structure is shown to achieve the greatest reuction in inuctive effects. An interigitate structure has also been applie to clock networks in 4]. Each layer of a multiplane P/G istribution network consists of interigitate P/G lines. The irection of the wires is perpenicular to the irection of the wires in the previous layer, as shown in Fig. 1, ensuring no inuctive coupling between the layers. The avantages of an interigitate structure are in- Manuscript receive January 14, 2009; revise March 27, First publishe June 16, 2009; current version publishe July 17, This work was supporte in part by the National Science Founation uner Grant CCF , Grant CCF , an Grant CCF , by grants from the New York State Office of Science, Technology an Acaemic Research to the Center for Avance Technology in Electronic Imaging Systems, an by grants from Intel Corporation, Eastman Koak Company, an Freescale Semiconuctor Corporation. This paper was recommene by Associate Eitor P. Li. The authors are with the Department of Electrical Engineering, University of Rochester, Rochester, NY USA jakushok@ece. rochester.eu; frieman@ece.rochester.eu. Digital Object Ientifier /TCSII Fig. 1. Interigitate P/G istribution network. The arker an lighter lines represent the power an groun lines, respectively. Fig. 2. Four pairs of a single layer within an interigitate P/G istribution network. crease routing flexibility an reuce inuctance effects. The current flow of the P/G lines within a layer is assume to flow in opposite irections, thereby reucing the loop inuctance of the network 5]. The loop inuctance of two parallel wires with opposite current flow is L loop = L 11 + L 22 2M 12 1 where L 11, L 22, an M 12 are the self-inuctance of the P/G lines an the mutual inuctance between these two wires, respectively. The process of estimating the inuctance becomes problematic with a large number of wires. To calculate the loop inuctance, the mutual inuctance terms among all of the wires nee to be iniviually etermine, which is a computationally expensive process. A close-form expression characterizing this inuctance woul therefore be useful. The inuctance of a single layer within an interigitate P/G istribution network structure with four pairs eight wires, as showninfig.2,is 1 = L eff L 1 L 2 L 3 L 4 where L 1, L 2, L 3, an L 4 are the inuctance of the first, secon, thir, an fourth pairs of a single layer within a P/G istribution network, respectively. The inuctance of the first pair is L 1 = L 1p + L 1g + 4 M 1p i p + M 1g i g i=2 4 M 1p i g + M 1g i p /$ IEEE

2 586 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS BRIEFS, VOL. 56, NO. 7, JULY 2009 Fig. 3. n pairs of a P/G istribution network. The focus of 10 is on the effective inuctance of pair m. where subscripts p an g represent power an groun, respectively. In this case, the overall inuctance requires sixteen terms to etermine each pair. For n pairs of P/G istribution networks, 2 2n =4n terms are require to characterize each pair, making complexity On for a single pair. For n pairs, the complexity in estimating the inuctance of a single layer within a P/G network is On 2. The brief is organize as follows: A close-form expression of the self- an mutual inuctance of an interigitate P/G istribution network is presente in Section II. The accuracy of this expression an a comparison to other moels are provie in Section III. The upper boun for the error of the propose moel is also provie. The brief is conclue in Section IV. II. INDUCTANCE OF P/G DISTRIBUTION NETWORK The efinition of the inuctance between two loops i an j for a uniform current ensity is presente by the Neumann equation L ij μ 0μ r 4 C i C j s i s j R ij where μ 0, μ r, an R ij are the vacuum an relative permeability, an the istance between two loops, respectively. From 6], the mutual inuctance between a pair of two rectangular conuctors is μ 0l ] l + 1+ l l 2 + l where l an are the length of the wire an the pitch of two wires, respectively. If l, an approximate expression base on a Taylor series expansion is 7] μ 0l 1+ l 4 5 ]. 6 The self-inuctance is erive in a similar way. For those cases where the length is larger than the with 8] L s = μ 0l + 1 ] k l where w, t, an k are the wire with, wire thickness, an fitting parameter k 0.22 for smaller length wires, respectively. In P/G istribution networks where l an l, the last term characterizing the ege effect of the self- an mutual inuctance can be neglecte, simplifying 6 an 7 to μ ] 0l 1 8 L s = μ 0l + 1 ] 9 2 respectively. The mutual component of the inuctance within an interigitate P/G istribution network ecreases with increasing istance between the wires an can be treate as a local effect, accoring to 5]. In this case, the effective inuctance of each pair is the sum of the self-inuctances an a single mutual inuctance between the two wires in the pair. This approach supports fast estimation of the effective inuctance of a P/G istribution network but suffers in accuracy since the mutual inuctance terms between all other parallel wires are neglecte. Enhance accuracy in estimating the mutual inuctance terms is require. The effective inuctance of an arbitrary pair of P/G lines m within an interigitate P/G istribution network is shown in Fig. 3 an is L m =2L ms 2M mp m g + M mp i p M mp i g M mg i p + M mg i g. 10 The terms M mp i p = M mg i g are equal for any i in 10 since the istance between the power lines of pair m an i an the groun lines of pair m an i is the same. In aition, 10 can be rewritten as a function of istance = w + s, where s is the spacing L m =2L ms 2M + 2M M M+] 11 where Mx = μ 0l ] x Equation 11 consists of three terms: 1 the self-inuctance of two wires; 2 the mutual inuctance between these two wires; an 3 the sum of the mutual inuctance between all of the other

3 JAKUSHOKAS AND FRIEDMAN: INDUCTANCE MODEL OF INTERDIGITATED P/G DISTRIBUTION NETWORKS 587 The limit of the prouct can be solve using the Wallis formula 9] sinx x = 1 x2 n=1 2 n 2 17 Fig. 4. Terms of 14. The values quickly ecline in magnitue to zero. wires. The thir term is neglecte in 5]. Substituting 12 into 11, the summation term is n 2 ] The sum of the logarithmic terms is the prouct of a single logarithm, permitting 13 to be expresse as = n = μ 0l P/G istribution networks typically consist of a large number of interigitate pairs, an, as shown in Fig. 4, the terms of 14 quickly ecline in magnitue to zero. The number of pairs on the left an right is therefore assume to be infinite, permitting 14 to be formulate as lim n 1 2 ] The factor of two originates from the two sies of the target pair. The infinite sum of 15 is presente as an infinite prouct ] lim n =2 μ 0l lim 1 1 ] n at x = /2, leaing to the equality lim n = Base on 18, 10 may be presente in close form as L m =2 μ 0l =2 μ 0l ] ]. 19 To estimate the overall inuctance of a structure with N P/G line pairs, the inuctance of each pair is assume to be equal. The mutual inuctance between all of the other P/G pairs converges to a constant, making the inuctance inepenent of the number of P/G pairs. The error is greatest in those cases where the number of pairs is smallest; however, in these cases, the effective inuctance can quickly be etermine with no approximation ue to the small number of pairs. For those cases where the number of pairs is sufficiently large eight pairs prouce less than 10% error, the effective inuctance is L eff = 2 N ]. 20 Note than the effective inuctance is escribe for a single layer within an interigitate P/G network, where it is assume that no metal layers are above or below the structure. In practical cases, the existence of ifferent interconnect structures above or below the structure may reuce the accuracy of the propose moel. For structures with large spacing, an interconnect structure below the target structure reuces the accuracy of the estimate inuctance 10]. Interigitate P/G networks, however, are esigne with small spacing to exploit the available metal resources; therefore, the accuracy of the effective inuctance moel is maintaine. Aitionally, since the current is assume to flow throughout the entire interigitate structure, the inuctance etermine in 20 represents the worst case effective inuctance. Assuming that the current is uniformly istribute throughout the interigitate structure, the worst case effective inuctance prouces the largest voltage rop over the P/G istribution network. III. COMPARISON AND DISCUSSION Three ifferent moels are compare in this section. The Grover moel escribes the inuctance of each pair base on 10, where every mutual component is iniviually calculate 7]. While the iniviual inuctance of each pair is etermine, the effective inuctance of a single layer within an interigitate P/G network structure is estimate, assuming that the iniviual inuctive lines are in parallel. Hence, the Grover moel refers to the evaluation of every mutual term among all of the wires

4 588 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS BRIEFS, VOL. 56, NO. 7, JULY 2009 Fig. 5. Comparison of FastHenry an the Grover, Mezhiba, an propose moels for two ifferent esign cases. in a system. In 5], the effective inuctance is etermine base on an approximation, where the inuctance is treate as a local effect, an the mutual inuctance between other pairs is neglecte. This moel is calle the Mezhiba moel. The propose moel, as represente by 20, etermines the effective inuctance, assuming that the number of P/G pairs is infinite. Since the magnitue of the mutual terms quickly eclines to zero as a function of istance, this assumption is highly accurate. A comparison among FastHenry 11] which is a multipole 3-D inuctance extraction program an the Grover, Mezhiba, an propose moels is summarize in this section. In aition, the complexity an accuracy of the propose, Grover, an Mezhiba moels are compare. Two ifferent structures of an interigitate P/G istribution network are evaluate. For both structures, the with an spacing are maintaine constant, i.e., w =1μm an s =1μm; however, the length an thickness are ifferent, i.e., l 1 =1mm, t 1 =0.975 μm, an l 2 = 100 μm, t 2 =0.17 μm. The thickness values are base on a 65-nm CMOS technology 12] for the top M8 an bottom M1 metal layers, respectively. Both cases represent a single layer within a P/G istribution network. In Fig. 5, two structures are extracte using FastHenry an compare with the Grover, Mezhiba, an propose moels. The propose an Grover moels exhibit enhance accuracy as compare with the Mezhiba moel. In Figs. 6 an 7, the accuracy an complexity are evaluate, respectively. The accuracy is evaluate by comparing the results with FastHenry. The complexity of FastHenry is, however, not evaluate since the require number of terms for the simulator is excessively large as compare with the analytic moel. The complexity an error of the Grover, Mezhiba, an propose moels relative to FastHenry are evaluate. The Grover moel consiers all of the mutual terms an exhibits the lowest error less than 1% error; however, the complexity of the Grover moel rastically increases for a large number of pairs. The complexity of the propose an Mezhiba moels is inepenent of the number of P/G pairs. The error of the propose moel ecreases with a larger number of P/G pairs, whereas the error of the Mezhiba moel increases. The highest error 30% of the propose moel occurs with the fewest number of pairs, whereas the error of the Mezhiba moel is Fig. 6. Error comparison for the Grover, Mezhiba, an propose moels. All of the moels are compare with FastHenry. Fig. 7. Comparison of complexity of the Grover, Mezhiba, an propose moels. highest with the greatest number of P/G pairs. Hence, the error of the propose moel can be reuce using the Grover moel, which is only computationally efficient for a few P/G pairs. Assuming that the number of P/G pairs is infinite, the effect of the mutual inuctance terms is greater; therefore, the propose moel unerestimates the effective inuctance. The mutual inuctance of only a single pair is consiere by the Mezhiba moel, overestimating the inuctance. The bounary conitions of the effective inuctance are etermine from the propose an Mezhiba moels. These conitions permit the effective inuctance of a single layer within an interigitate P/G istribution network structure to be etermine for any number of P/G line pairs. The bounary conitions for the effective inuctance of an interigitate P/G istribution network structure are therefore etermine by the propose an Mezhiba moels 2 N L eff x 2 N ] + 3 ] 2 21

5 JAKUSHOKAS AND FRIEDMAN: INDUCTANCE MODEL OF INTERDIGITATED P/G DISTRIBUTION NETWORKS 589 where x represents any number of pairs within a single layer of an interigitate P/G istribution network. An expression is erive for the error between the propose an Grover moels. The normalize error is error = L grover L propose L grover = 1 L propose. 22 L grover Since the Grover moel cannot be expresse by a single equation, only the worst case error is etermine. An assumption in the propose moel is that the number of interigitate pairs is infinite; therefore, the error is highest when only a single pair N =1 is present, expressing error N=n error N=1. For this case, the inuctance base on the Grover moel is L grovern=1 =2L s 2 2μ 0l The inuctance base on the propose moel for N =1is L proposen=1 = 2μ 0l + 3 ] Substituting 23 an 24 into 22, the error boun is 2 ErrorBoun N 1 =. 25 w+t Base on the parameters of with, spacing, an thickness provie earlier in this brief, the ErrorBoun is less than 0.3 or 30%. The error of the propose moel rastically ecreases with a higher number of pairs, as shown in Fig. 6. Similarly, the ErrorBoun can be expresse for those cases where N 2 ErrorBoun N 2 = w+t ] The ErrorBoun is less than 0.23 or 23% for those cases where N 2 with the aforementione parameters of with, spacing, an thickness. IV. CONCLUSIONS 2 Due to the higher operating frequencies, the role of inuctance in the IC esign process has become significant. Inuctance estimation is a complicate task since every mutual inuctance within a system must be consiere. An interigitate P/G istribution network reuces the effective inuctance as compare with other P/G istribution network structures. An accurate yet computationally efficient moel of this system is esirable to enhance the P/G network esign process. The symmetric structure of an interigitate P/G istribution network supports the evelopment of a close-form expression to moel the effective inuctance. A close-form expression characterizing the inuctance of a single layer within a P/G istribution network has been presente. The solution is compare with previous work an FastHenry, exhibiting goo accuracy. The error for the propose moel is highest for a few P/G pairs; however, ue to the small number of lines, the Grover moel can be use in these cases. With an increasing number of P/G network pairs, the error of the propose moel rapily ecreases, permitting the effective inuctance of a P/G istribution network to be accurately an efficiently estimate. The magnitue of the effective inuctance is boune by the values etermine from the propose an Mezhiba moels. The boun ramatically ecreases with an increasing number of pairs. The effective inuctance of a single layer within an interigitate P/G istribution network structure can therefore be etermine for any number of P/G line pairs. REFERENCES 1] D. A. Priore, Inuctance on silicon for sub-micron CMOS VLSI, in Proc. IEEE Symp. VLSI Circuits, May 1993, pp ] L.-R. Zheng an H. Tenhunen, Effective power an groun istribution scheme for eep submicron high spee VLSI circuits, in Proc. IEEE Int. Symp. Circuit Syst., May 1999, vol. I, pp ] M. Popovich, A. V. Mezhiba, an E. G. Frieman, Power Distribution Networks With on-chip Decoupling Capacitors. New York: Springer- Verlag, ] Y. Massau, S. Majors, T. Bustami, an J. White, Layout techniques for minimizing on-chip interconnect self inuctance, in Proc. IEEE/ACM Des. Autom. Conf., Jun. 1998, pp ] A. V. Mezhiba an E. G. Frieman, Inuctance properties of highperformance power istribution gris, IEEE Trans. Very Large Scale Integr. VLSI Syst., vol. 10, no. 6, pp , Dec ] E. B. Rosa, The self an mutual inuctances of linear conuctors, Bull. Bureau Stanars, vol. 4, no. 2, pp , ] F. Grover, Inuctance Calculation: Working Formulas an Tables. New York: Dover, ] E. B. Rosa an F. W. Grover, Formulas an Tables for Calculation of Mutual an Self-Inuctance. Washington, DC: Gov. Printing Office, ] J. Wallis, Opera Mathematica, Oxonii, Leon: Lichfiel Acaemiæ Typographi, ] B. Klevelan, X. Qi, L. Maen, T. Furusawa, R. W. Dutton, M. A. Horowitz, an S. S. Wong, High-frequency characterization of on-chip igital interconnects, IEEE J. Soli-State Circuits, vol.37,no.6, pp , Jun ] M. Kamon, M. J. Tsuk, an J. K. White, FASTHENRY: A multipoleaccelerate 3-D inuctance extraction program, IEEE Trans. Microw. Theory Tech., vol. 42, no. 9, pp , Sep ] P. Bai, C. Auth, S. Balakrishnan, M. Bost, R. Brain, V. Chikarmane, R. Heussner, M. Hussein, J. Hwang, D. Ingerly, R. James, J. Jeong, C. Kenyon, E. Lee, S.-H. Lee, N. Linert, M. Liu, Z. Ma, T. Marieb, A. Murthy, R. Nagisetty, S. Natarajan, J. Neirynck, A. Ott, C. Parker, J. Sebastian, R. Shahee, S. Sivakumar, J. Steigerwal, S. Tyagi, C. Weber, B. Woolery, A. Yeoh, K. Zhang, an M. Bohr, A 65 nm logic technology featuring 35 nm gate lengths, enhance channel strain, 8 Cu interconnect layers, low-k ILD an 0.57 μm 2 SRAM cell, in IEDM Tech. Dig., Dec. 2004, pp