Efficient Signal Integrity Verification of Multi-Coupled Transmission Lines with Asynchronously Switching Non-Linear Drivers

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1 Efficient Signal Integity eification of Multi-Coupled Tanmiion Line with Aynchonouly Switching on-linea Dive Taeyong Je Dept. of Electical and Compute Engineeing, Hanynag Univ., Anan, Kyunggi-do, Koea Yungeon Eo Dept. of Electical and Compute Engineeing, Hanynag Univ., Anan, Kyunggi-do, Koea Abtact A non-linea dive i modeled a a imple equivalent linea cicuit which can be epeented with pechaacteized dive model paamete, i.e., a dive input ie time modulation, a dive witching, ective-dive eitance, and ective tanmiion line capacitance. Theeby, an icient linea-dive-baed imulation algoithm fo multi-coupled inteconnect line can be diectly eploited. In thi wo, a taveling-wavebaed wavefom appoimation technique which i a linea-dive-baed imulation algoithm fo the icient ignal integity veification of multi-coupled tanmiion line i combined with the developed dive model. Then it i hown that the ignal integity of aynchonouly witching inteconnect net with non-linea dive can be accuately a well a iciently veified.. Intoduction With the poibility of nano-cale-baed ulta-high denity and ulta-high peed integated cicuit deign, inteconnect line ae cucial deign iue ince today integated cicuit pefomance i dominated not by gate but by inteconnect net. Much tighte phyical pacing between the line and much longe inteconnect line length mae cicuit timing and noie magin be moe tingent []. ot only doe the capacitive and inductive coupling between neighbo line eult in a ignificant cotal noie, but alo the timing vaiation due to the coupling ect may be lage than conventional RC time. To mae matte woe, aynchonou witching between the line dive mae conventional monotonicwavefom-baed timing model and cotal model be uele ince thee i a ignificant lew ate modulation due to the neighbo line coupling ect []-[5]. In pactice, inteconnect net ae diven by non-linea cicuit. The non-linea dive witching may be the mao ouce of an input ie time modulation, theeby leading to the aynchonou witching between the line. The aynchonou cicuit witching of multi-coupled line lead to non-monotonic output wave-hape that mae the timing and noie analyi of the multi-coupled line be inheently difficult [6][7]. Theefoe, without taing the non-linea dive ect into account, the ignal integity analyi of the tongly coupled inteconnect net may not be aued. = cm R = 3.79 Ω/ cm I CL =.pf L = 4.45 nh / cm C = pf / cm Figue. A tanmiion line diven by a dive uing.8um CMOS poce. Up to date, a non-linea dive i imply modeled a a linea Thevenin equivalent cicuit ( th, Rth that i baed on empiical -facto-baed model [8], loo-up table with ective capacitance [9][], o alpha powe low[]. Howeve, thee model ae not accuate enough to iciently veify the ignal integity of today integated cicuit. In thi wo, at fit, a non-linea dive i modeled with a compact linea cicuit by employing a dive input ie time modulation, dive witching, ectivedive eitance, ective tanmiion line capacitance, and neighbo-line coupling ect. Then, multi-coupled inteconnect net with non-linea dive ae epeented with a compact linea equivalent cicuit. et, the accuacy and iciency of the model i veified by combing the developed non-linea dive model and taveling-wave-baed wavefom appoimation (TWA technique [] which i an icient ignal integity veification algoithm baed on the linea dive cicuit [3][4].. Poblem due to on-linea Dive If an inteconnect line i diven by a non-linea dive cicuit a hown in Fig., the amp input ignal of a dive with the ie time of t i modulated at the output of the dive (i.e., the input ignal of the inteconnect line. The input ignal modulation i implicitly o eplicitly coelated with the dive ize, input lew ate, tanmiion line ect, eceive loading ect, and etc. Among them, the dive ize and loading ect ae moe pominent than othe ect. A illutated in Fig., the Poceeding of the 8th Intenational Sympoium on Quality Electonic Deign (ISQED' /7 $. 7

2 output epone lew ate fo a dive with an input ie time ( t i not a contant but a function of time. The input dive lew ate modulation ha a ubtantial ect on the ignal tanient of the multi-coupled inteconnect line a hown in Fig. 3. Futhemoe, if the input dive of multi-coupled inteconnect line aynchonouly witch with diffeent lew ate, the ignal tanient become much moe deteioated a hown in Fig. 4. Thu, the conventional imple dive cicuit model baed on the ed input ignal and ective dive eitance [3][4] may not be accuate enough fo the timing and noie veification of today integated cicuit which have vey tingent timing and noie magin. Step Input t =p t =5p t =p Figue. Fa end ignal wavefom of Fig. fo vaiou input ignal tanition time. The output epone lew ate fo a dive with an input ie time ( t i not a contant but a function of time. eaend Faend t =5p t =p t =p..4.8 ictim (Cente Line Aggeo (Side Line Figue 4. Signal tanient deteioation due to the co-coupled witching of neighbo line with aynchonouly witching dive. ote, the lew ate of victim line (the cente line and the witching time of the aggeo line (oute line have a tong coelation. 3. on-linea Dive Model The output ignal of a non-linea dive (e.g., an invete can be modeled a a imple linea cicuit by uing due to the dive (both % and 5%, the dive eitance, and the dive loading ect a non linea hown in Fig. 5, whee the non-linea dive, out, can be epeented with a function of model paamete non linea ( (, %, 5% out δ f t t t, R, S C ( ' in % 5% whee t, t t, R S, and C ae input ie time, % of the dive output, 5% of the dive output, ective dive eitance, and ective load capacitance, epectively. The δ i a ign digit (i.e., - fo an inveting logic and + fo a non-inveting logic. In thi ub-ection, uch model paamete detemination method concened with non-linea dive pechaacteization ae decibed in moe detail. R S Real Dive Linea Dive with t I I...4 Figue 3. Input lew ate modulation due to a nonlinea dive when t = p. ote, the non-linea dive ha a conideable ect on the tanient epone. Figue 5. An equivalent cicuit model of a nonlinea dive 3.. Effective Capacitance of a Tanmiion Line The ective capacitance ( C of a tanmiion line can be modeled a Poceeding of the 8th Intenational Sympoium on Quality Electonic Deign (ISQED' /7 $. 7

3 C t Itdt ( =. ( Thu, it can be appoimately detemined by olving the cuent equation of numeato [9][]. The cuent of a tanmiion line i I( = ( Y( = Y( t. (3 eglecting the dielectic lo, the admittance of the tanmiion line can be epeented by C C Y( =, (4 L+ R L + ( R L y whee y =. Then epanding Y( at y = ( = with Taylo eie, Y( can be epeented by C + m + m + m 3 + m 4 + m5 Y ( 5, (5 L whee [ ] d dy + ( R L y m =. (6! Thu, the ective capacitance ( C can be yielded by Thu, C 5 n+ C t mn (7 L n= ( n+! C t R t 3 R t 5 R t = + L L 9 L 9 % t and given dive R t 63 R t L 94 L 5% t can be eadily chaacteized fom a 3.. Effective Dive Reitance Effective dive eitance ( R S of the dive can be appoimately detemined with the tanito I- chaacteitic cuve a hown in Fig. 5. A we can ee, the eitance i not a contant but the vaiable of DS in the ange of RS RS RS dd, (8 whee RS and RS dd ae the eitance at the two eteme value of DS a hown in Fig. 5. That i, RS = RS ( DS = and RS dd= RS( DS=. ote, the dive output epone apidly change nea the dive witching thehold voltage which i appoimately DD. Thu, the dive eitance hould be elected nea the dive witching thehold voltage, RS R S DS. (9 I D (A R S dd R S DS ( R S R S dd Figue 5. Effective dive eitance ( R S deteminant fom MOS I cuve 3.3. Rie Time Modulation due to on-linea Dive A dive input ignal with a ie time ( t can be epeented by t, t < t I ( t = t. (, t t Howeve, ince the input ignal will be modulated with non-linea dive chaacteitic, the modulation ect ha to be taen into account. Once the ective load capacitance of the dive i detemined, both t % and 5% t can be eadily detemined by pefoming a SPICE imulation fo the dive with input ignal time ( t. Since a CMOS dive may apidly witch nea the dive witching thehold voltage, the modulated input ignal ie time ( t due to the dive can be appoimately epeented by 5% % t.7( t t, ( and the dive time due to the dive witching time can be faily well epeented by % t ( th + t τ. ( Thu, with t and τ, the modulated input ignal ( I ( t can be epeented by ut ( τ, t < t I ( t = t. (3, t t The output epone of the model ae compaed in Fig. 6 with the SPICE imulation. They how ecellent ageement. Poceeding of the 8th Intenational Sympoium on Quality Electonic Deign (ISQED' /7 $. 7

4 t =p t =5p t =p t =p ut ( τ τ, t < t + τ I ( t = t. (6, t + τ t Although the model i deived with two-coupled line, it can be eadily etended into n-coupled line....4 SPICE Thi Wo Figue 6. The ignal tanient epone compaion of the popoed model and SPICE imulation.7um.7um 3.4. Co-tal Effect due to eighbo Line Switching In multi-coupled line, ignal coupling between the line (i.e., cotal ha a ignificant ect on the ignal tanient [7]. The amount of the co-tal depend on the witching patten a hown in Fig. 7. Defining the thee witching ymbol a (witching fom logic to logic, (witching fom logic to logic, and (the quiet tate, the witching patten can be epeented with the aow ymbol. In two coupled line, thee ae thee poible witching patten (i.e.,,,. In the witching patten of, the ective load capacitance ( C i eactly equal to the ingle iolated line capacitance. In contat, the ective capacitance of the witching patten of can be eadily detemined with (6 by eplacing the C by C = C + C. Howeve, in the co-coupled witching (, it i not the cae. In thi paticula witching cae, ince the cotal induce the negative notch in a ignal tanient wave-hape, the ect eult in an additional. Thu, the ect can be appoimately modeled with a time ( τ. Auming the capacitive coupling, the amount of cotal voltage ( of two coupled line i given by [5] Thu, C C + C C DD ' ' τ = τ = τ ( C + C. (4 (5 Taing the additional time into account, the modulated input ignal ( I of ( can be modified fo the co-coupled witching cae a..4.8 Figue 7. Co-tal ect of -coupled line fo vaiou witching patten 4. Efficient Signal Integity eification of Multi-Coupled Tanmiion Line In thi ection, it i hown that the ignal tanient wave-hape of multi-coupled tanmiion line with non-linea dive can be vey iciently a well a accuately detemined by combining the developed nonlinea dive model and TWA technique [] which i baed on the linea dive model. In the TWA-technique, the output epone of the i-th line due to the -th input witching of n-coupled tanmiion line i epeented with the following nomenclatue. d : the numbe of the decompoed input ignal fo an aynchonouly witching input ignal m : tep count of the -th mode n : the numbe of ed tep function fo a amp function epeentation uing tep function τ : non-linea dive time- (ee ection 3 t : modulated input ignal tanient time : the numbe of the tep of the -th mode fo a ingle input witching : the numbe of the tep of the -th mode when -th line caue coupling ect i : the i-th ow and the -th column component of nomalized n n voltage eigenmati of the tanmiion line ytem b : the -th modal input voltage Poceeding of the 8th Intenational Sympoium on Quality Electonic Deign (ISQED' /7 $. 7

5 p : the TWA-baed 3-pole epone of the -th mode when input ignal i I : the timing ew of the -th input witching t ew, Fit, in multi-coupled tanmiion line, the time domain tep epone of the i-th line can be epeented by n v ( t b p ( t, (7 i i = whee p ( t i the TWA-baed 3-pole epone of the - th mode which can be eadily detemined []. et, a amp input can be epeented with many ed tep function [6]. n ib mt mt vi( t = p ( t u( t τ, (8 = m= Futhe, the output epone of multi-coupled line which ae diven by aynchonouly witching input ignal can be epeented with the ummation of epective input ignal epone [6]. That i, the output epone of the i- th line due to the -th input witching can be epeented by n b i mt vi ( t = p ( t tew, = m=. (9 mt ut ( tew, τ Thu, the final output epone of the i-th line fo all of the aynchonou witching input can be epeented by D D n b i vi( t = vi( t = X ( t, m,, τ, ( = = = m= whee mt X ( t, m,, τ = p ( t tew, ( mt ut ( tew, τ. ow that the non-linea dive chaacteitic ae combined with the tanmiion line chaacteitic in tem of the mathematical epeion in (, the ignal integity of multi-coupled tanmiion line with nonlinea dive can be vey iciently detemined. In ode to invetigate the accuacy and iciency of the model, 5-coupled tanmiion line with non-linea dive ae teted with the wot cae witching patten of. A hown in Fig. 8. the TWA-baed ignal tanient wave-hape fo the epective line have good ageement with SPICE imulation. In contat, it computation time i eveal thouand time fate than that of the SPICE imulation. In the 5-coupled line, although the CPU time fo the SPICE imulation i 64 econd, the TWA-baed calculation time i ut milliecond that i 3 time fate than that of SPICE imulation..7um.7um..5. (a SPICE (onlinea Dive SPICE (Linea Dive Thi Wo (TWA-baed.7um.7um..5. (b SPICE (onlinea Dive SPICE (Linea Dive Thi Wo (TWA-baed.7um.7um SPICE (onlinea Dive SPICE (Linea Dive Thi Wo (TWA-baed (c Figue 8. Output ignal tanient wavefom of 5- coupled line with dive (.8um poce fo wot cae input witching (a the mot outide line (b two adacent line of cente line (c cente line Poceeding of the 8th Intenational Sympoium on Quality Electonic Deign (ISQED' /7 $. 7

6 5. Concluion In thi pape, an icient ignal integity veification method fo the multi-coupled line with aynchonouly witching non-linea dive wa peented. A non-linea dive i chaacteized with it ize, input ie time, dive, and ective eitance. Taing the tanmiion line ect and eceive loading ect into account, the loading ect of the dive i epeented with ective capacitance, followed by the dive chaacteization. Theeby, the ignal tanient vaiation due to the aynchonou witching multi-coupled tanmiion line with non-linea dive can be vey accuately a well a iciently imulated by uing a implified modal decoupling algoithm, i.e., taveling-wave-baed wavefom appoimation technique (TWA-technique. It wa hown that the ignal tanient wave-hape of multicoupled tanmiion line with non-linea dive can be vey iciently a well a accuately detemined by combining the developed non-linea dive model and TWA technique. The TWA-baed ignal tanient fo the multi-coupled tanmiion line with the non-linea dive have ecellent ageement with SPICE imulation uing egment RLC cicuit model. It computation time i eveal thouand time fate than that of SPICE. Refeence [] Intenational technology oadmap fo emiconducto, SIA Repot, 4. [] A. B. Kahng, S. Muddu, and E. Sato, On witch facto baed analyi of coupled RC inteconnect, in Poc. DAC,, pp [3] T. Sato, Y. Cao, D. Sylvete, and C. Hu, Chaacteization of inteconnect coupling noie uing initu -change cuve meauement, in Poc. ASIC/SOC Conf.,, pp [4] Y. Cao, X. Yang, X, Huang, and D. Sylvete, Switched-facto baed loop RLC modeling fo icient timing analyi, in Poc. ICCAD, 3, pp [5] B. Chen, H. Yang, R. Luo, and H. Wang, A novel method fo wot-cae inteconnect etimation, IEEE Tan. Cicuit Syt. I, vol. 5, pp , June 3. [6] Y. Maoud, S. Mao, J. Kama, T. Butami, D. MacMillen, and J. White, Managing on-chip inductive ect, IEEE Tan. LSI Syt., vol., no. 6, pp [7] Y. Eo, S. Shin, W. R. Eientadt, and J. Shim, A decoupling technique fo icient timing analyi of LSI inteconnect with dynamic cicuit witching, IEEE Tan. Compute-Aided Deign, vol. 3, pp , Sep. 4. [8] Y. Gao and D. F. Wong, A fat and accuate etimation method fo buffeed inteconnect, in Poc. ASPDAC,, pp [9] K. Agawal, D. Sylvete, and D. Blaauw, An ective capacitance baed dive output model fo onchip RLC inteconnect, in Poc. DAC, 3, pp [] L. K. aati and J. Wang, A new multi-amp dive model with RLC inteconnect load, in Poc. ISCAS, 4, vol. 5, pp [] Y. I. Imail and E. G. Fiedman, Optimum epeate inetion baed on a CMOS model fo on-chip RLC inteconnect, in Poc. ASIC, 998, pp [] Y. Eo, J. Shim, and W. R. Eientadt, A tavelingwave-baed wavefom appoimation technique fo the timing veification of ingle tanmiion line, IEEE Tan. Compute-Aided Deign, vol., no. 6, pp , June. [3] W. Chen, S. Gupta, and M. Beue, Analytic model fo cotal and pule analyi unde nonideal input, in Poc. IEEE Int. Tet Conf., 997, pp [4] S. Siichotiyaul, D. Blaauw, C. Oh, R. Levy,. Zolotov, and J. Zuo, Dive modeling and alignment fo wot-cae noie, in Poc. DAC,, pp [5] T. Sauai, "Cloed-fom epeion fo inteconnection, coupling, and cotal in LSI'," IEEE Tan. Electon Device, vol. 4, no., pp. 8-4, Jan [6] T. Je and Y. Eo, Efficient ignal integity veification method of multi-coupled RLC inteconnect line with aynchonou cicuit witching, in Poc. ISQED, 6, pp Poceeding of the 8th Intenational Sympoium on Quality Electonic Deign (ISQED' /7 $. 7