Multi-Aggressor Capacitive and Inductive Coupling Noise - Modeling and Mitigation

Transcription

1 Mult-Aggressor apactve and Inductve ouplng ose - Modelng and Mtgaton Vctora Vshnyakov, Eby G. Fredman*, and Avnoam Kolodny Electrcal Engneerng Department, Technon, Hafa, Israel *Department of Electrcal and omputer Engneerng, Unversty of Rochester, Rochester, Y 14627, USA Abstract rosstalk nose n on-chp nterconnect plays a major role n the performance of modern ntegrated crcuts. Mult-aggressor capactve and nductve couplng complcates both the modelng and mtgaton of the nose. A novel method to model and analyze nose n RL mult-lne structures s proposed n ths paper, exhbtng an error of up to 9% as compared to SPIE. Ths method s physcally ntutve snce t decomposes the nose produced by each of the aggressors nto ndvdual capactve and nductve nose sources. The proposed model and related layout nose mtgaton gudelnes are appled to crosstalk nose reducton n mult-lne structures. 1. Introducton Global and sem-global nterconnect do not scale wth feature sze due to ncreased desgn complexty, demand for greater ntegraton, and technology constrants. As technology progresses, the effect of the nterconnect on the performance of hgh speed and hgh densty ntegrated crcuts has greatly ncreased. Wth shorter transton tmes and the nablty to scale the global wres, nductve effects exhbted n the upper metal layers cannot be neglected. onsequently, long range nductve couplng should be ncluded wth the already sgnfcant capactve couplng n global nterconnect lnes snce nose analyss and mtgaton s not lmted to only the nearest neghbors. Smultaneous capactve and nductve couplng together wth multple aggressors are sgnfcant rsks to the sgnal ntegrty of the global nterconnects. The prmary nterconnect structures n the upper metal layers are the clock and power/ground (P/G) dstrbuton networks and wde data busses. The global clock network s usually hghly shelded and affected by the self-nductance rather than mutual nductve effects, whch greatly smplfes the analyss and optmzaton process. In P/G networks, the nose s caused by both the self-nductance and mutual nductance. These networks however generally exhbt unform structures. Random data sgnals, unshelded clock sgnals, and wde data busses, as llustrated n Fg. 1, suffer from both selfnductance and mutual nductve couplng and can be affected by numerous aggressors due to the long range nature of nductve couplng. Smultaneous capactve and nductve couplng and dfferent swtchng patterns further complcate the modelng and analyss process. Several authors have addressed modelng and behavoral analyss of nose n multlne structures n the presence of nductance. In [1], a two lne decouplng technque s extended by applyng superposton of the fundamental modes to three lnes and proposes ths technque for coupled lnes. The model s rather general (no lmtatons on the lne parameters) but s complex and requres adjustment for dfferent bus szes. The use of modal analyss to decouple multple transmsson lne (TL) systems s descrbed 1

2 n [2], [3]. These models are vald for dentcal lnes wth an dentcal drver and loads assumng deal transmsson lnes and are computatonally complex. A TL based model s used n [4], but assumes no capactve couplng and low loss, whch s also assumed n [5]. The TWA method [6] s extended to mult-coupled transmsson lnes n [7]. The concept of an effectve swtchng factor for mult-lne systems s presented n [8] and the dfferences between mult-lne worst case nose patterns for R and RL lnes are dscussed n [9]. Fg. 1. Typcal upper metal routng structure These models do not descrbe the ndvdual nose components the nose caused by each aggressor and the nose due to both capactve and nductve couplng. Snce methods to mtgate each nose source (nductve, capactve, dfferent aggressors) can be dfferent and sometmes contradctory, dentfcaton of the most crtcal nose sources and the preferable mtgaton method for a specfc physcal layout and swtchng pattern are necessary. The condtons and methods to model nose as a combnaton of the nose caused by each aggressor and nose source (capactve and nductve), as llustrated n Fg. 2, are descrbed n ths paper based on the addtvty propertes of capactve and nductve couplng as well as mult-lne system behavor n the presence of multple nose sources. Fg. 2. Types of nose sources and mechansms The paper s organzed as follows. In secton 2, the addtvty propertes of nductve and capactve nose for a two lne coupled system wth smultaneous nductve and 2

3 capactve couplng are examned. ondtons under whch the addtvty of the two couplng nose sources can be appled are formulated. The behavor of mult-lne systems s examned n the presence of capactve couplng, nductve couplng, and smultaneous capactve and nductve couplng n secton 3. The condtons whch represent the sum of the nose caused by each aggressor are also descrbed. In secton 4, nose models of mult-lne systems are descrbed. In Secton 5, layout-based nose mtgaton gudelnes are presented and together wth the proposed models are used to reduce nose n multlne systems. The results are summarzed n secton Addtvty of capactve and nductve couplng nose Smultaneous capactve and nductve couplng can occur between adjacent RL lnes. Accordng to [10], capactve and nductve couplng nose s addtve under the low loss approxmaton, Rlne ( Lself M ), where R lne and L self are, respectvely, the self-resstance and nductance of a lne, M s the mutual nductance between the lnes, and s the swtchng frequency, 2/ t r, where t r s the sgnal transton tme. ote however that the low loss approxmaton cannot always be assumed n modern ntegrated crcuts. Therefore, addtvty of the two nose sources cannot be assumed n modern ntegrated crcuts. By performng an analyss smlar to [10], those regmes where addtvty of capactve and nductve couplng can be assumed, as llustrated n Fg. 3 for a two coupled lne system, are establshed. The addtvty property permts the nose components to be broken nto nose sources. As a result, the domnant nose source can be dentfed and a sutable nose mtgaton technque can be chosen. Ths dstncton s mportant snce nose reducton technques for nductve and capactve couplng are not only dfferent, but often n conflct. Fg. 3. Decouplng capactve and nductve nose usng addtvty Accordng to transmsson lne theory [18], coupled lnes exhbt two modes of propagaton wth two dfferent propagaton constants and two dfferent lne mpedances. The even mode represents the case of same drecton swtchng and the odd mode represents opposte drecton swtchng. Any sgnal n the system can be expressed as the sum of these modes. The characterstc mpedance of the even and odd modes for two dentcal lnes s presented n (2.1) - (2.2) where R11, 11, and L11are, respectvely, the 3

4 resstance, capactance, and nductance of the lne, and and 12 L are, respectvely, the 12 mutual capactance and nductance. s the swtchng frequency, defned by 2/ t r, where t r s the sgnal transton tme. Z ( ) RL even Z RL ( odd) R j( L L ) j (2.1) 11 R11 j( L11 L12) j( 2 ) (2.2) ote that the characterstc mpedance of the even mode s not dependent on the capactve couplng between the lnes ( 12 ), and s the same as the characterstc mpedance of a system wthout capactve couplng, Z ( even, 0) Z ( even, =0). (2.3) RL 12 RL 12 Thus, when the lnes swtch n the same drecton, capactve couplng between the lnes does not affect the sgnal propagaton characterstcs. The odd mode characterstc mpedance, however, s the same as that of a system wth no nductve couplng only f the dependence of the mpedance on nductve effects can be neglected, R ( L L ). (2.4) In ths case, Z ( odd, L 0) Z ( odd), (2.5) RL 12 R where ZR ( odd ) s the mpedance of two coupled lnes wth L11 L12 0. When both (2.3) and (2.5) are assumed, the total nose can be expressed as the sum of the nose caused by nductve couplng and the nose caused by capactve couplng, as shown n (2.6), Vnose( Total) Vnose( even, =0) Vnose( odd, L L =0). (2.6) When nductve couplng s strong ( L 11 L 12 ), (2.4) s satsfed. Another case where (2.4) s satsfed s the low loss approxmaton [4], [10]. Ths approxmaton, however, s not always approprate n modern semconductor processes. Another condton that supports approxmatng the nose as the sum of the nductve and capactve couplng s weak capactve couplng,.e., 12 11, or same drecton swtchng. In ths case, most of the nose s due to nductve couplng, and Vnose ( odd ) s small or non-exstent and can be neglected. To summarze, there are three condtons n whch addtvty of nductve and capactve couplng can be assumed: 4

5 L11 L R ( L L ) - low loss approxmaton. - strong nductve couplng case - weak capactve couplng or same drecton swtchng 3. ouplng nose n mult-lne structures apactve couplng, nductve couplng, and smultaneous nductve and capactve couplng exhbt dfferent behavor n mult-lne structures (see Fg. 4). Whle capactve couplng prmarly exsts between nearest neghbors, nductve couplng s a long range phenomenon and can affect dstant wres as well. Furthermore, t s shown n ths secton that whle both capactve and nductve couplng nose can be expressed as the sum of the nose caused by each aggressor, for smultaneous capactve and nductve couplng, ths s not the case. The ablty to separate the nose caused by each aggressor s however mportant for nose analyss and mtgaton. Therefore, n ths secton, those condtons that express the total nose as the sum of the nose caused by each aggressor are descrbed. Fg. 4. coupled lnes: (a) capactve couplng only, (b) nductve couplng only, and (c) capactve and nductve couplng 3.1. Inductve couplng n mult-lne structures apactve couplng nose (see Fg. 4a) s caused prmarly by nearby neghbors and can be expressed as the sum of the nose caused by each of the adjacent aggressors. Inductve couplng, however, s a long range effect, thus non-adjacent neghbors can also couple nose and cannot be neglected. In mult-lne structures, nductve couplng among all of the lnes may occur (see Fg. 4b). To analyze nductve couplng nose n mult-lne systems, the lnes are represented as a lumped RL model, as descrbed n (3.1.1), where V n,, R,, and L are, respectvely, the nput waveform, resstance, self-capactance, and self-nductance of each lne and M s the mutual nductance between lnes and j. j 5

6 1 V ' R * I ' L * I '' M * I '' * I. (3.1.1) n, j j j The matrx representaton of (3.1.1) s d d d d R1 * L1 * M 2 1, j* M 2 1, * 2 1 dt dt dt dt Vn,1 ' I d 1 d d d Vn, ' M,1 * R * * 2 L M 2, * * 2 dt dt dt dt I Vn, ' I d d 1 d d M,1 * M 2, j * R * * 2 L 2 dt dt dt dt = Vn ' 1X AX * In1X (3.1.2) The current per lne can be expressed as a functon of the nput swtchng pattern by nvertng A, as shown n (3.1.3), X 1 I A * Vn '. (3.1.3) 1X X 1X By extendng (3.1.3) and assumng Vn, ' 0 for the non-swtchng neghbors, t can be observed that the soluton for multple swtchng aggressors s the sum of the ndvdual solutons wth each aggressor swtchng, as descrbed by (3.1.4). I1 Vn,1 ' V 0 n,1 ' I * Vn, ' * 0 * 0 A A A Vn, ' A * I V ' 0 0 V ' n, n, (3.1.4) The current per lne can be determned by solvng (3.1.2), whch can be rewrtten n the followng form, '' ' 1 L1 M1, j M1, I 1 R1 0 0 I 1 I1 Vn,1 ' 1 '' ' M,1 L M, * I 0 R 0 * 0 0 I * I, ' V n '' ' M,1 M, j L I 0 0 R I I Vn, '

7 '' ' 1 = L* I R* I * I V, (3.1.5) where L, R, and are, respectvely, x matrces of the nductance, resstance, and capactance, V s an x 1 vector of the dervatve of the nput voltages V, and I s the x 1 vector of the currents. Expresson (3.1.5) s a second order dfferental equaton wth a well-known soluton, as presented n (3.1.6), where y 0 and y 1 are determned by the boundary condtons and,, 0 and d are descrbed, respectvely, n (3.1.7) - (3.1.9). ' n, t V * * 1 e *snh( dt) cosh( dt) t Tr d It () t e y1 y0 *snh( dt) y0 d cosh( dt) t Tr d (3.1.6) 1 R L (3.1.7) 0.5* * j L (3.1.8) j (3.1.9) d From the form of the soluton of (3.1.6), the nose on each lne s the sum of the nose caused by each aggressor, snce, for non-swtchng aggressors, V ( ) V n, ' Smultaneous capactve and nductve couplng When only one couplng effect exsts (nductve or capactve), the total nose for any swtchng pattern s the sum of the nose caused by each aggressor, as descrbed n Secton 3.1. In a mult-lne structure, capactve nose wll couple to the nearest neghbors and nductve nose may couple among all of the lnes (Fg. 4c). Therefore, for a gven vctm, nteractons wth the nearest aggressors wll be both capactve and nductve, and, wth the more dstant lnes, nductve nose wll be the prmary source of couplng. However, the farther lnes whch are nductvely coupled to the vctm are capactvely coupled to the nearest neghbors. ouplng nose on these farther lnes are coupled to the far vctm by the long range nductve effect. As a result, the overall nose on the vctm s affected also by the capactve couplng nose experenced by the far aggressor and not just nductve couplng. Therefore, smultaneous capactve and nductve couplng n a mult-lne system s not the sum of the nose caused by each aggressor, whch sgnfcantly ncreases the complexty of the nose analyss and mtgaton process n these structures. To demonstrate the problem, an example three lne system s descrbed n secton and extended to a general coupled lne case n secton The condtons that allow the nose sources to be summed despte the aforementoned phenomenon are presented n secton Example - three coupled lnes 7

8 To demonstrate ths phenomenon, the behavor of three dfferent confguratons, as llustrated n Fg. 5, s compared: Three lnes wth capactve couplng only, nductve couplng only, and both capactve and nductve couplng. Fg. 5 Three coupled lne confguraton (a) capactve couplng only, (b) nductve couplng only, and (c) both capactve and nductve couplng Lne 1 s the vctm and lnes 2 and 3 are the aggressors. For smplcty, dentcal lnes wth dentcal couplng are assumed,.e., R1 R2 R3, 1 2 3, and L1 L2 L3, Xcap1 Xcap2 and M12 M 23 M13. The couplng capactance and mutual nductance are chosen to mantan the addtvty approxmaton, as descrbed n secton 2. onsder the nose on the vctm (lne 1) for three dfferent swtchng patterns, as lsted n Table 1. In pattern # 1, only the nearest neghbor s swtchng and n patterns # 2 and # 3 both the nearest and the dstant neghbors are swtchng. In pattern # 2, the lnes swtch n the same drecton and n pattern # 3 the lnes swtch n opposte drectons. For the confguraton shown n Fg. 5a, the nose on the vctm s approxmately the same for all three swtchng patterns snce the effect of the dstant neghbor s neglgble n the case of only capactve couplng. For the confguratons shown n Fg. 5b, accordng to secton 3.1, the total nose s the sum of the nose caused by each aggressor. Thus, the nose on the vctm n pattern # 2 s twce the nose on the vctm n pattern # 1. The nose on the vctm of pattern # 3 s zero snce aggressors # 2 and # 3 cancel. Table 1 Swtchng patterns defnton for three coupled lnes Pattern #1 Pattern #2 Pattern #3 Vn Vn2 Vn3 0 For the confguratons shown n Fg. 5c, based on the addtvty propertes, the nose on the vctm lne s equal to the sum of the nose caused by only capactve couplng, as n the confguraton shown n Fg. 5a, and only nductve couplng, as n the confguraton shown n Fg. 5b. However, as llustrated n Fg. 6, t s not always the case due to capactve couplng between dstant lnes. For pattern # 2 (Fg. 6a), the actual nose, as smulated n SPIE, matches the nose calculated usng the addtvty propertes. For pattern # 3 (Fg. 6b), the nose calculated usng the addtvty propertes s smaller than the actual nose as smulated n SPIE. The reason for the dfferent behavor s that for pattern # 3 there s capactve couplng between lnes 2 and 3, whch does not exst n 8

9 pattern # 2 due to same drecton swtchng. The capactve couplng nose between lnes 2 and 3 s coupled to the vctm lne through the nductve effect and s added to the total nose. Fg. 6 SPIE vs. addtve nose waveform for the confguraton shown n Fg. 5c for (a) pattern # 2 and (b) pattern # 3 Several conclusons can be derved from ths example that are generalzed n the followng secton. In the case where both capactve and nductve couplng exsts, the total nose cannot always be expressed as the sum of the nose from each aggressor. However, there are cases where ths summaton s possble; for example, when the capactve couplng s suffcently small to be neglected or the lnes swtch n the same drecton General case - coupled lnes Smlar to the analyss descrbed n secton 3.1, a lumped RL model s used to represent a mult-lne system n the case of smultaneous capactve and nductve couplng, as shown n (3.2.1), where V n,, R,, L, and M j are defned n secton Vn, ' R * I ' L * I '' Mj * I j '' *( I I 1, I, 1) (3.2.1) j The matrx representaton of (3.2.1) s Vn' A * In B * Ij (3.2.2) 1X X 1 X ( 1) X 1 X ( 1) where A and B( 1) X are descrbed by (3.2.3) - (3.2.4), X 9

10 A X d d d d R1 * L1 * M 2 1, j* M 2 1, * 2 1 dt dt dt dt d 1 d d d M,1 * R * * 2 L M 2, * 2 dt dt dt dt d d 1 d d M * M * R * L * 2 2 2,1 2, j 2 2 dt dt dt dt (3.2.3) B ( 1) X (3.2.4) The current per lne can be expressed as a functon of the nput voltages (.e., swtchng pattern) by nversng A, as shown n (3.2.5), 1 X I A *( Vn ' B * Ij ). (3.2.5) 1X X 1 X ( 1) X 1X From (3.2.5), observe that the soluton for multple swtchng aggressors s not the sum of the soluton of each aggressor swtchng by tself snce the current of each lne s dependent not only on the nput voltage, but also on the currents from the couplng capactors, as explaned n secton However, by examnng (3.2.5), note that there are cases where the total nose can be expressed as the sum of the nose caused by each aggressor, when B( 1) X approaches zero or A B. These condtons translate to ether weak capactve couplng, same drecton swtchng, or strong nductve couplng, accordng to (3.2.3) - (3.2.4). These condtons correspond to the condtons on the addtvty of capactve and nductve couplng, as presented n secton 2, L 1. j 2. j M - strong nductve couplng case - weak capactve couplng 3. same drecton swtchng To determne the current wthn the lnes, the same method as descrbed n secton 3.1 s appled. Frst, (3.2.2) s rewrtten n the followng form, 10

11 L I R I I V (3.2.6) '' ' 1 j * * * where L j, R,, and V are descrbed, respectvely, by (3.2.7) - (3.2.10), M1,1 (1 ) M 2,1 M1, j (1 ) M 2, j M1, (1 ) M 2, L = M (1 ) M M M (1 ) M M M (1 ) M M j 1, 1, 1, 1, 1, 1, M,1(1 ) M 1,1 M, j (1 ) M 1, j M, (1 ) M 1, 1,, 1 1, 1, 1,, 1 1, 1, 1,, 1 1, 1,,1 1,1 1,1, j 1, j 1, j, 1, 1, (3.2.7) R1(1 ) R , 1,, 1, 1 R= 0 R 1 R (1 ) R , 1, 0 0 R 1 R(1 ) (3.2.8) (3.2.9) Vn,1 '(1 ) Vn,2 ' 1 1 V V '(1 ) V ' V ' 1,, 1 1,, 1 n, n, 1 n, 1 V '(1 ) V ' 1, 1, n, n, 1 (3.2.10) Expresson (3.2.6) s a second order dfferental equaton, where the soluton s presented n (3.2.11), y 0 and y 1 are determned by the boundary condtons, and,, 0 and d are matrces descrbed, respectvely, by (3.2.12), (3.2.13), and (3.2.14). 11

12 t V * * 1 e *snh( dt) cosh( dt) t Tr d It () t e y1 y0 *snh( dt) y0 d cosh( dt) t Tr d (3.2.11) 1 R L (3.2.12) 0.5* * j L (3.2.13) j (3.2.14) d From the form of (3.2.9) and vector V (see ), note that the nose on each lne cannot be expressed as the sum of the nose caused by each aggressor as n the purely nductve case. ote that matrx d can be ether real, equal to zero, or complex dependng upon the lne parameters. When d s real, (3.2.11) mantans the orgnal form and ths regme s referred to as overdamped, where the transent response s a decayed current wthout oscllaton, smlar to the waveform of pattern # 3 shown n Fg. 6b. The crtcally damped response ( d 0 ) represents the crcut response that decays n the fastest possble tme wthout oscllatng. For complex values of d, or the underdamped regme, (3.2.11) ncludes snusodal functons rather than hyperbolc functons and the waveform s n the form of decayng oscllatons, smlar to the waveform of pattern # 2 shown n Fg. 6a. 4. Modelng mult-aggressor nose In secton 3, closed-form expressons, (3.2.11) - (3.2.14), for modelng nose n capactvely and nductvely coupled mult-lne structures are presented. These expressons, however, do not provde physcal ntuton descrbng the nose sources (capactve or nductve, dfferent aggressors). In ths secton, an alternatve model that enables decomposng the nose sources s presented, permttng the nose n coupled mult-lne systems to be estmated. In prevous sectons, the condtons that express the total nose as a sum of the nose caused by each aggressor and each couplng effect (capactve and nductve) are presented. In the case of strong nductve couplng ( L Mj ), weak capactve couplng ), or same drecton swtchng, t s possble to express the nose of a vctm as (( j a sum of the nose caused by each aggressor. The regme that satsfes these condtons s referred to here as the Lnear Regme. If none of these condtons s satsfed, the regme s referred to here as the on-lnear Regme, and the addtvty approxmaton cannot be made. In the mult-lne scenaro, where the prmary concern s the long range nductve couplng effect, the lnear regme s also the worst case nose scenaro. V nose s the peak total nose of the vctm and Vnose ( aggressor ) s the peak nose of the vctm caused by aggressor. In the lnear regme, the total nose s 12

13 V V ( aggressor ). (4.1) nose nose 1 For adjacent aggressors, the nose s caused by both capactve couplng ( V nose, xcap ) and nductve couplng ( V nose, M ). For dstant aggressors, the nose s caused prmarly by nductve couplng. The nose caused by capactve couplng from the non-adjacent aggressors s small and s assumed to be neglgble to mantan the smplcty of the model and analyss. Thus, (4.1) can be rewrtten as 1 2. (4.2) V V ( aggressor ) V ( aggressor ) nose nose, M nose, xcap j 1 j1 To valdate ths model, the nose, as calculated from (4.2) usng addtvty, s compared to the nose as smulated by SPIE for a varety of swtchng patterns and crcut parameters wthn an eght lne structure. As lsted n Table 2, the nose exhbts a maxmum error of 9% as compared to SPIE. By analyzng the range of error of the dfferent experments lsted n Table 2, t can be seen that for stronger nductve couplng as compared to capactve couplng, the smaller s the error. For any swtchng pattern and lnearty condton, the error decreases wth ncreasng nductve couplng coeffcent (K, experments 1 vs. 2 and 4 vs. 5 n Table 2). A decrease n the rato of couplng capactance vs. self-capactance (Xcap/lne) lowers the error (experments 5 vs. 6, 12 vs. 13, and 25 vs. 26 n Table 2). The resstance satsfes the frst lnearty condton (low loss approxmaton) - the larger the resstance, the smaller the error due to suppresson of rngng (experment 14 vs. 15 n Table 2). These observatons are n agreement wth the lnearty condtons defnton. 13

14 Table 2 Model verfcaton for the lnear regme Pattern: R - swtchng 0 -> 1 ; F - swtchng 1 -> 0 ; 0/1- constant 0 or 1 Rlne, lne, Xcap - Lne resstance, lne capactance, and couplng capactance K - Inductve couplng coeffcent L / L Lnearty condton Strong nductve couplng Weak capactve couplng Same drecton swtchng Exp. # Swtchng Pattern j rcut Parameters ose Peak for lne #1 Rlne lne Xcap K Exact Usng Error [Ω] [pf] [pf] (Spce) Addtvty [%] [V] [V] 1 00R0R0RR % 2 00R0R0RR % 3 00R0R0RR % 4 100FRF0R % 5 100FRF0R % 6 100FRF0R % 7 110FR % 8 110FR % 9 RRFFR % 10 RRFFR % 11 00R0R0RR % 12 00R0R0RR % 13 00R0R0RR % FRF0R % FRF0R % FR % FR % 18 RRFFR % 19 RRFFR % 20 RRRRRRRR % 21 RRRRRRRR % 22 RRRRRRRR % 23 FFFFFFFF % 24 FFFFFFFF % 25 RRRRRRRR % 26 RRRRRRRR % 5. Applcaton to nose mtgaton through layout optmzaton The model presented n secton 4 can be used to evaluate the crtcalty of each nose source (capactve vs. nductve, nose caused by dfferent aggressors), permttng the most effectve nose mtgaton method to be appled. In ths secton, analyss of the dependence of the nose on the physcal layout parameters s presented, followed by a formulaton of layout based mtgaton gudelnes. The nose mtgaton gudelnes are demonstrated on a case study usng the model descrbed n secton Dependence of couplng nose on layout parameters 14

15 The topology of global and sem-global nterconnects s generally determned by floorplan constrants. onsequently, the prmary nose mtgaton technques, n addton to sheld lnes, modfy layout propertes such as the lne wdth and spacng between adjacent lnes. apactve and nductve couplng nose typcally exhbts dfferent and sometmes contradctory layout dependences [13]- [14], whch drectly affect the nose mtgaton technques for capactve couplng, nductve couplng, and smultaneous capactve and nductve couplng. To better understand the dependence of each couplng effect on the layout parameters, the lne parameters as a functon of lne wdth and spacng are examned, as llustrated n Fg. 7, based on closed-form expressons for the lne resstance, capactance, and nductance [11], [12]. Fg. 7 Lne parameters as a functon of lne wdth and spacng (a) resstance as a functon of lne wdth and spacng, (b) self-capactance and crosscapactance as a functon of lne wdth and spacng, and (c) self-nductance and mutual nductance as a functon of lne wdth and spacng The crtcalty of the capactve couplng s determned by the rato of the couplng and self-capactance, where an ncrease n the rato results n ncreased capactve couplng nose [15]. The maxmum rato occurs for narrow and close lnes (see Fg. 7b). For nductve couplng, the nose peak s determned by the rato of the mutual and selfnductance. Snce the senstvty of the mutual nductance to the lne wdth and spacng s low, ths rato remans approxmately constant over a wde range of layout confguratons (see Fg. 7c). However, the lne resstance whch exhbts hgh senstvty to the lne wdth (see Fg. 7a) has a substantal effect on the nose caused by the couplng effects, where an ncrease n the resstance results n a lower peak nose. To analyze the dependence of the couplng nose on layout parameters, the analytc soluton (wthout the addtvty approxmaton), as descrbed n (3.2.11), s combned wth the closed-form expressons of the lne resstance, capactance, and nductance [12] for a two lne coupled system. Inductve and capactve couplng nose as a functon of lne wdth and spacng s llustrated n Fg. 8. ote that the capactve couplng nose s affected by both the lne wdth and spacng due to the effect of the lne wdth on the selfcapactance, whle nductve couplng nose s prmarly dependent on the lne wdth due to the strong effect dependence on the lne resstance. 15

16 Fg. 8 ouplng nose as a functon of lne wdth and spacng (a) peak nose due to capactve couplng, and (b) peak nose due to nductve couplng To reduce capactve couplng nose, ncreasng ether or both the space or the lne wdth s effectve [16], [17]. Spacng the lnes results n lower couplng capactance and wdenng the lnes ncreases the self-capactance whch decreases the nose. Both technques are smlarly effectve, as shown n Fg. 8a, however note that both spacng and wdenng are effectve only over a lmted range. For lnes wder than twce the mnmum wdth and farther than twce the mnmum space, nether technque reduces the nose. To mtgate nductve couplng, narrow lnes are more effectve. Ths result s expected snce ncreasng the space between the lnes decreases the couplng capactance and has lttle effect on the mutual nductance; thus, ths method s only effectve for mtgatng capactve couplng nose. Wdenng the lnes, however, reduces the lne resstance, ncreases the self-capactance of the lne, and has a small effect on the nductance. As a result, there s a contradctory effect on the capactve and nductve couplng nose. Inductve couplng nose wll ncrease due to a decrease n the lne resstance, whle capactve couplng nose wll decrease due to an ncrease n the lne self-capactance. The mplcaton s that n the presence of smultaneous nductve and capactve couplng, the approprate nose mtgaton method should be chosen based on the crtcalty of the couplng effect as well as tmng and power constrants. If capactve couplng s more crtcal, ncreasng the space between the lnes should be consdered. Wdenng the lnes should be consdered only f the nductve couplng s neglgble, snce whle the capactve couplng nose wll decrease, the nductve couplng nose wll ncrease. However, f nductve couplng s domnant, ncreasng the space between the lnes wll have a neglgble effect on the nose and the lnes should be made more narrow [14]. Domnant capactve couplng s expected between narrow and close lnes snce the small dstance between the lnes ncreases the couplng capactance, and a narrow wdth results n hgh resstvty whch decreases the nductve effects. Due to the strong dependence of the lne resstance and nductve effects on the wdth, for wde, low resstance lnes, nductve couplng s expected to domnate. 16

17 5.2. ase Study To demonstrate the model descrbed n secton 4 for nose analyss and mtgaton, a bus composed of eght parallel lnes n a 32 nm MOS process [11] s examned. The layout parameters are chosen to concde wth the lnear regme, thus the model proposed n the prevous secton can be appled. The nose on lne 3 of an eght bt wde bus caused by each aggressor as well as the nterconnect mpedances are lsted n Table 3, where lne 3 s a quet vctm and four dfferent swtchng neghbors (lnes 2, 4, 6, and 8) are aggressors. Based on (4.2), the peak nose of the vctm lne s. (5.1) V ( lne #3) V ( aggressor ) V ( aggressor ) nose nose, M nose, xcap j 2,4,6,8 j2,4 ote the contrbuton of each aggressor to the total nose s almost dentcal. The concluson of an analyss of the nose components s that nductve effects are more domnant than capactve couplng and nearest neghbors are not more dangerous than a dstant aggressor. In addton, long range nductve couplng cannot be neglected for any bt wthn the bus. The most effectve nose mtgaton technque s to not ncrease the space between the lnes, but rather the lnes should be made more narrow to ncrease the lne resstance and, as a result, decrease the peak nose. Table 3 ose breakdown for a typcal 8 bt bus Aggressor ose V V V V All (Usng Addtvty) 0.398V All (Exact - SPIE) 0.372V Swtchng Pattern: 0R0R0R0R, R - swtchng 0 -> 1 ; 0 - constant 0 Lne parameters: length = 1000 µm, wdth = 0.79 µm, space = µm Extracted crcut parameters: resstance Ω, lne capactance 76.2 ff, couplng capactance 55.6 ff, and nductve couplng coeffcent K between 0.6 and onclusons A method based on modelng the addtvty propertes of capactve and nductve couplng nose s proposed for analyzng multple lnes systems n the presence of smultaneous nductve and capactve couplng. The method can be used to estmate the crtcalty of each nose source. It s shown that capactve and nductve couplng nose s not always addtve and that although each of the couplng effects can be modeled as the sum of the nose caused by each aggressor, when both effects exst ths approach s not always accurate. The condtons that allow expressng the total nose as the sum of the nose caused by each aggressor and nose source are based on an analytc analyss of a 17

18 multple lne system. These condtons concde wth the crtcal nose scenaros of a mult-lne structure n the presence of smultaneous nductve and capactve couplng. The analytc method s compared to SPIE and a maxmum error of 9% s demonstrated for a varety of swtchng patterns and crcut parameters. Layout gudelnes are provded for reducng nose n coupled RL nterconnects and, together wth the proposed models, nose reducton n mult-lne structures s demonstrated. 18

19 References [1] T. Km and Y. Eo, Analytc AD models for the sgnal transents and crosstalk nose of nductance-effect-promnent multcoupled RL nterconnect lnes, IEEE Transactons on omputer-aded Desgn of Integrated rcuts and Systems, Vol. 27, o. 7, pp , July [2] G. hen and E. G. Fredman, An RL nterconnect model based on Fourer analyss, IEEE Transactons on omputer-aded Desgn of Integrated rcuts and Systems, Vol. 24, o. 2, pp , February 2005 [3] S. Tuuna, J. Isoaho, and H. Tenhunen, Analytcal model for crosstalk and ntersymbol nterference n pont-to-pont buses, IEEE Transactons on omputer- Aded Desgn of Integrated rcuts and Systems, Vol. 25, o. 7, pp , July 2006 [4] A. aeem, J. A. Davs, and J. D. Mendl, ompact physcal models for multlevel nterconnect crosstalk n ggascale ntegraton (GSI), IEEE Transactons on Electron Devces, Vol. 51, o. 11, pp , ovember 2004 [5] S. H. ho, B.. Paul, and K. Roy, Dynamc nose analyss wth capactve and nductve couplng, Proceedngs of the IEEE Asa and South Pacfc Desgn Automaton onference, pp , January 2002 [6] Y. Eo, J. Shm, and W. R. Esenstadt, A travelng-wave-based waveform approxmaton technque for the tmng verfcaton of sngle transmsson lnes, IEEE Transactons on omputer-aded Desgn of Integrated rcuts and Systems, Vol. 21, o. 6, pp , June [7] S. Shn, Y. Eo, and W. R. Esenstadt, Analytc models and algorthms for the effcent sgnal ntegrty verfcaton of nductance-effect-promnent multcoupled VLSI crcut nterconnects, IEEE Transactons on Very Large Scale Integraton (VLSI) Systems, Vol. 12, o. 4, pp , Aprl [8] Y. ao, X. Yang, X. Huang, and D. Sylvester, Swtch-factor-based loop RL modelng for effcent tmng analyss, IEEE Transactons on Very Large Scale Integraton (VLSI) Systems, Vol. 13, o. 9, pp , September [9] J. hen and L. He, Worst-ase rosstalk ose for onswtchng Vctms n Hgh- Speed Buses, IEEE Transactons on omputer-aded Desgn of Integrated rcuts and Systems, Vol. 24, o. 8, pp , August 2005 [10] J. E. Lorval, D. Deschacht, Y. Quere, T. Le Gouguec, and F. Huret, Addtvty of capactve and nductve couplng n submcronc nterconnects, Proceedngs of the IEEE Desgn & Test of Integrated Systems n anoscale Technology, pp , September [11] Internatonal Technology Roadmap for Semconductors, 2010, [12] Predctve Technology Model - [13] K. Agarwal, D. Sylvester, and D. Blaauw, Modelng and analyss of crosstalk nose n coupled RL nterconnects, IEEE Transactons on omputer-aded Desgn of Integrated rcuts and Systems, Vol. 25, o. 5, pp , May 2006 [14] Y. Massoud, S. Majors, J. Kawa, T. Bustam, D. MacMllen, and J. Whte, Managng on-chp nductve effects, IEEE Transactons on Very Large Scale Integraton (VLSI) Systems, Vol. 10, o. 6, pp , December

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