Calculation of Ramp Response of Lossy Transmission Lines Using Two-port Network Functions

Transcription

1 Calculatin f amp espnse f Lssy Transmissin Lines Using Tw-prt Netwrk Functins Payam Heydari and Massud Pedram Dept. f EE-Systems University f Suthern Califrnia Ls Angeles, CA {payam,massud}@zugrs.usc.edu Abstract-In this paper, we present a new analytical apprach fr cmputing the ramp respnse f an LC intercnnect line with a pure capacitive lad. The apprach is based n the tw-prt representatin f the transmissin line and accunts fr the utput resistance f the driver and the line inductance. The results f ur analysis are cmpared with the results f HSPICE simulatins demnstrating the high accuracy f ur slutin under varius values f driver, intercnnect, and lad impedances.. INTODUCTION With the expnential reductin in the feature size, the delays due t intercnnectins have becme the dminating factr in determining the circuit perfrmance. Due t aggressive scaling f intercnnects, even an average length metal line may have significant resistance cmpared t the driver resistance. Thus the distributed nature f the intercnnect must be mdeled. Furthermre, the IC perating frequency nears multi-gigahertz requiring the intercnnect inductance t be prperly mdeled. Apprximatin techniques fr estimating the time dmain respnse f intercnnect structures have been prpsed. AWE [] prvides ne apprximatin f general LC intercnnect mdel and has been successfully applied t analyze n-chip signal prpagatin. AWE begins with the differential state equatins f a lumped linear time-invariant circuit and then btains the Laplace transfrm slutin f the hmgeneus equatin. This slutin is expanded in a McLaurin series, and the time-dmain mments are cmputed frm this series and are matched t an apprximating functin cnsisting f a linear cmbinatin f expnential functins. EX [] is anther apprach fr rapidly estimating the transient respnse f lssy transmissin line which expands the reciprcal f transfer functin f the system. Fr critical underdamped intercnnects, this methd prvides better results cmpared t AWE. Bth f these appraches suffer frm inaccuracy especially in high speed integrated circuits. Lia and Dai [] prpsed using an S-parameter based macrmdel as a tw-prt netwrk fr mdeling the intercnnect structures. Anther way f btaining the time dmain respnse f an intercnnect line is t slve the Telegrapher s equatins. Kahng and Muddu [4] used this apprach fr a distributed C intercnnectin under the ramp excitatin. They assumed that a finite number f reflectins (namely fur is sufficient fr generating a result very clse t SPICE simulatin. The authrs hwever d nt cnsider the inductive effect f intercnnect line in their mdel and assume that the exciting vltage surce has zer valued utput resistance. In this paper, we begin with a tw-prt mdel f the transmis- This wrk was supprted in part by SC cntract #98-DJ-66. Permissin t make digital r hard cpies f all r part f this wrk fr persnal r classrm use is granted withut fee prvided that cpies are nt made r distributed fr prfit r cmmercial advantage and that cpies bear this ntice and the full citatin n the first page. T cpy therwise, t republish, t pst n servers r t redistribute t lists, requires prir specific permissin and/r a fee. sin line and btain the time-dmain expressin f the ramp respnse fr a finite-length LC lines. The effects f wire inductance and the resistance f CMOS driver f the intercnnect are cnsidered in ur methd. Sectin summarizes the backgrund knwledge abut the Telegrapher s equatins. Sectin presents ur analytical methd fr cmputing the ramp respnse f a lssy transmissin line. We present ur experimental results and cncluding remarks in sectins 4 and 5, respectively.. BACKGOUND We give sme definitins and terminlgy first. A linear circuit belngs t the class f linear time invariant systems. Hence it can be cmpletely characterized by its impulse respnse. The transient behavir f any linear system is cntained in its system functin which is the Laplace transfrm f the impulse respnse. A unifrm transmissin line with capacitive lad is depicted in Fig. (. The transmissin line has the prperty that a signal prpagates ver the intercnnectin medium as a wave. Fig. (.b depicts the electrical mdel f the transmissin line. + - Z s d CL r Fig. (. Unifrm transmissin line. (adistributed transmissin line f length d with a lad. (bthe electrical mdel. Let r, l, c, g be the resistance, inductance, capacitance, and cnductance values per unit length f a unifrm transmissin line. The Telegrapher s equatins fr such a transmissin line is [5]: v( x, t v( x, t ( x lc t + ( lg + rc v( x, t + t rgv ( x, t Eq. ( is the fundamental relatinship gverning wave prpagatin alng a unifrm transmissin line. The shunt cnductance is ften negligible, hence we set g. The bundary and initial cnditins fr Eq. ( are: Bundary Cnditin: v(, t et ( Initial Cnditins: v x v( x, (,, t At each pint x n the transmissin line, the vltage is the sum f incident and reflected cmpnents f the wave. In the subsequent analysis, we mdel the input vltage as a ramp in the [,t rise ] interval and a delayed step functin in the [t rise, interval. We btain the utput respnse fr each f the inputs separately. Hwever when btaining the utput respnse fr secnd part we use the initial cnditin impsed by the first part f the input wavefrm. l c g ISPD 98, April 6-8, 998, Mnterey, CA USA ACM ISBN -58--x/98/4 $5. 5

2 . INTECONNECT TANSFE FUNCTION Since we are interested in calculating the wavefrm at the utput f the intercnnect line, we d nt g thrugh cmplicated details f wave reflectins thrugh the endpints f transmissin line. Instead we resrt t the tw prt representatin f transmissin lines whereby chain parameters are used fr relating the prt variables. Mre precisely we have [5], [6]: V e λ d e λ d e λ d e λ d Y ( s I ( e λ d e λ d Y ( s e λd + e λd sc where λ λ( s sc( sl +, Y ( s , and d dentes sl + the length f the transmissin line. L,, and C dente the ttal line inductance, line resistance, and line capacitance, i.e. Ldl, dr, and Cdc. The vltage at the utput prt (cf. Fig. ( is related t the current at that prt by the lad capacitr equatin. Hence the transfer functin f the intercnnectin laded by a capacitr C L is btained as: H( s where Z ( s Y ( s. The inverse f the first parenthesis in the denminatr term f Eq. (4 is a limit summatin f a pwer series. The transfer functin can thus be written in the fllwing frm: H( s As can be seen frm Eq. (5, λ(s and Z (s depend n the square rt f the frequency variable s. Cnsequently, the inverse Laplace transfrm cnsists f the errr functin which des nt result in a simple frmula fr the time dmain representatin f the utput wavefrm. S we extend the McLaurin series f λ(s and Z (s abut s, and then based n practical values f parameters, truncate the series int the first tw terms f the series. A gd apprximatin fr λ(s and Z (s is then btained as fllws: λ ( s LCs + -- C and (6 --- L Z L ( s C Ls Ntice that neglecting the resistive term in Z (s expressin yields the well knwn characteristic impedance fr a lssless transmissin line and that the prpagatin delay f wave thrugh the intercnnect media is cmpletely captured in the apprximatin t λ(s. Cmbining Eqs (5 and (6, the transfer functin f a lssy transmissin line is btained as: H( s The abve apprximatin fr λ(s causes a large change in the DC value f the transfer functin. We alleviate this errr by adding a gain cmpensatin factr t the transfer functin. T find an effective gain, let H (s be defined as: The utput expressin is then cmpsed f the delayed versins f h (t, the inverse Laplace transfrm f H (s is calculated as: V V V ( e λ d + e λ d ( + Z ( s ( C L s tanh( λd H ( s ( n + ( e n λd n ( + Z ( s ( C L s tanh( λd ( n + ( n -- C --- L ( e e n LCs n ( + Z ( s ( C L s tanh( λd ( + Z ( s ( C L s tanh( λd I ( (4 (5 (7 (8 v ut ( t ( n + ( n -- C L e h [ t ( n T] n where T LC is the time f flight f the wave. Since T is very small cmpared t tempral changes f h (t, we can assume that T is negligible, factrize h (t, and put it utside the summatin. We therefre cme up with the fllwing equatin: ( n -- C L v ut ( t h ( t ( n + e ( n The limit f the pwer series in Eq. ( gives us an estimate f the steady-state value f v ut (t which is interpreted as V ut ( in the s-dmain (the final-value therem [8]. Ding this we btain: -- C L e v ut ( t h ( t ( + e C L The actual steady-state value f h (t is ne. The errr is due t the secnd term in the right hand-side f Eq. (. Cnsidering the practical values f intercnnect parasitics, we see that exp( C L «. Cnsequently the cmpensating gain is set t exp( ( C L. The mdified transfer functin after taking this multiplicative factr int cnsideratin is written as: H( s H ( s ( n + ( n C --- L ( e e n LCs n v ut ( t ( n + e ( n C L h [ t ( n T] n (9 (.a (.b H (s depends upn the Laplace transfrm f vltage at prt (cf. Fig. (. Nw we d further manipulatin t make the analysis mre efficient. We prpse the fllwing piece-wise linear functin as an apprximatin t tanh(.: λd tanh( λl (.5( λd λd We will btain the transfer functin and the ramp respnse f the lssy intercnnect fr each f these cases in the fllwing subsectins. H(s in Eq. ( dentes the relatin between the vltages at the utput prt (i.e. prt and the input prt (i.e. prt f the intercnnect line. If we wish t have the relatin between the utput vltage f the intercnnect and the surce vltage e(t, we have t cnsider the vltage divisin between the driver impedance and the input impedance Z i (s seen by lking int the intercnnect line. We knw frm [6] that Z i (s is: z Z i ( s z ( s z (4 z + Z L where z, z, z, z are the tw-prt pen-circuit impedance parameters and Z L is the lad impedance. By knwing the chain parameters, any f the ther sets f tw-prt parameters, such as the z-parameters, can be cmputed [6]. Hence the input impedance f the intercnnect can be expressed in terms f the parameters f intercnnect as fllws: Z i ( s Z L ( + Z Y L tanh( λd Z L Y tanh( λd (5 We culd use a similar piece-wise linear apprximatin fr tanh(. which was used in Eq. (. Using this apprximatin wuld hwever result in a furth-rder surce t the utput transfer func- 5

3 tin. T avid this cmplexity, we use a different apprximatin. Essentially we ignre the term in cmparisn with the term. In the fllwing we shw that this apprximatin des nt cause a large errr. We can rewrite Eq. (5 in the fllwing frm: Z i ( s (6 where H (s was given in Eq. (8. Since we are cncerned abut the magnitude f errr we can write the magnitude f the frequency respnse f Eq. (6 as: (7 Z L ( j ω is very small in tday s high-speed circuits. Furthermre, as frequency increases Z L ( jω becmes even smaller. Any errr in apprximating H (s is multiplied by this small value. In practice, usually exp( λd is abut times greater than exp( λd. Fr instance, using the intercnnect parameters fr a.8µm CMOS technlgy and assuming a mm f Metal wire, a typical value fr λd at 5MHz clck frequency is arund. [7]. Fr glbal intercnnect lines, this value is even larger. Based n the abve apprximatin, we cme up with the fllwing expressin fr Z i (s: (8 Cnsequently, the utput vltage f the intercnnect line is related t surce vltage, e(t, by a simple vltage divisin made by Z i (s and s. Z i ( s Z ( s V ( s E( s (9 Z i ( s s s + Z ( s E ( s In the fllwing subsectins each f the tw cases fr apprximating tanh(. are cnsidered separately. The utput wavefrm is btained fr each f tw cases. CASE I. tanh( λd : In this case H (s is represented by a first-rder ratinal functin f s as fllws: ( Cmparing Eq. ( with the actual value f H (s, again we see that the actual DC value f H (s differs frm the DC value f the apprximated expressin f H (s. This difference will affect the steady state value f the utput vltage. T vercme this, we can add a cnstant multiplicative gain s that the DC value f Eq. ( becmes unity value. Therefre we btain: H ( s ( τ L ( s+ τ L ( where: Z L Z L Y tanh( λd H ( s Z i ( j ω Z L ( jω H ( jω Z i ( s H ( s Y ( s Z ( s L --- C C C L s L C L L τ L C L C --- L L e λd e λd As stated befre, we break up the input wavefrm int tw parts: (i ramp input e r (t (ii step input e s (t. Hereafter we use the cnventin that any vltage variable with index r is related t the ramp input, and any vltage variable with index s is related t the step sectin f the input. Output respnse is cmputed fr each f these parts. Let s cnsider the first part f the input. Frm Eq. (9 we are able t btain the input vltage t transmissin line: V r ( s where: V s DD L C L t rise L C + s ( s + τ s s τ s L C( ( L L C + s ( We apply partial fractin expansin t V r (s and then, after btaining the respnse t each f the fractinal terms, simply utilize the superpsitin prperty t calculate the final value f the utput respnse as fllws. Eq. ( represents the partial fractin expansin f V r (s: V r ( s V DD t rise s s ( ( L L C s s + τ s ( As can be seen frm the abve equatin, three terms are present in the partial fractin expansin f the vltage at prt. We name each f the terms as V r (s, V r (s, and V r (s, respectively. The Laplace transfrm f the utput vltage at prt is cmpsed f the respnse t each f the three terms. We name each f the utput terms as V r (s, V r (s, and V r (s, respectively. It is well knwn that the Laplace transfrm f the system respnse is a prduct f the system transfer functin and the Laplace transfrm f the input[8]. Cnsequently, we have: V DD τ V r( s H ( sv r ( s L (4 t rise s ( s + τ L Applying partial fractin expansin, then taking inverse Laplace transfrmatin, we cme up with the tempral wavefrm f the utput: v r V DD V ( t DD t τ tu( t τ L t rise t L ( e rise (5 Similarly, we repeat the abve steps t btain the timing wavefrms f v r (t, v r (t as fllws: v r v r V DD ( t s t τ L ( e t rise ( ( L L C V DD L τ ( t s τ L e t τ s e t τ L ( t rise τ L s C L (6 (7 The utput vltage is cmpsed f the algebraic summatin f the three cmpnents, i.e.: v r( t vr ( t + v r ( t + v r ( t (8 Fr the time interval [t rise, the input has a step frm, e s (t. The utput vltage is made up f the step input and the initial cnditin impsed by the ramp input. As a result f the cntinuity cnditin, the intercnnect respnse t the secnd part f the input at time t rise must be equal t the respnse f the intercnnect t the first part at that same time. The same relatinship exists fr vltages at all ther pints especially the vltage at the input prt f the intercnnect, that is: v r ( t t rise v s ( t t rise (9 As defined befre, v r (t is the input vltage f prt when the excitatin is in the frm f a ramp whereas v s (t is the input vltage f prt when the excitatin is a step functin. v r (t is simply determined by taking the inverse Laplace transfrm f Eq. (: v r ( t V DD V DD r( t s t τ s ( e t rise t rise ( ( L L C ( 54

4 V s (s is determined by substituting E(s in Eq. (9 with the Laplace transfrm f the step functin. Furthermre we shuld include an extra term which specifies the respnse under the initial cnditin. After taking the inverse Laplace transfrm, we btain the fllwing expressin: v s ( t V e t τ s V DD s e t τ s + ( s + L C In Eq.( the unknwn variable V is determined by satisfying Eq. (9. We can subsequently write: V DD v s ( t s e t rise τ s ( e t τ s + V ( t DD rise ( ( L L C We take the same steps as was taken fr btaining the respnse fr interval [,t rise ] t derive the respnse fr [t rise,. v s (t is cmpsed f tw terms. One is expnentially rising in time and asympttically ges tward a cnstant value, while the ther is cnstant in time. We name v s (t, v s (t as the system respnses t each f the abve prtins f the v s (t. We have: ( t V DD e t τ L ( ( v s V DD L τ v s ( t L e t rise τ s ( e t τ s e t τ L ( (4 t rise τ L s C L The final expressin f the utput vltage is cmpsed f the tw functins f Eq. (, and Eq. (4: v s( t vs ( t + v s ( t (5 The system respnse is the algebraic summatin f v s (t and v r (t. CASE II. tanh( λd.5( λd : In this case we expect that the equatins becme mre cmplicated since the rder f the transfer functin is increased. As we will shw this cmplexity als appears in the utput vltage wavefrm s that the wavefrm asympttically ges t its final value with ringing. Similar t the previus case, we change the DC value f the transfer functin such that it becmes identical t the DC value f the actual transfer functin: H ( s (6 where ω n and α L LC L 4 L The denminatr has always tw cmplex cnjugate ples, hence the respnse has underdamped behavir and we shuld bserve ringing effect n the utput wavefrm. The methd fr btaining the utput wavefrm is the same as that used fr case I. S we d nt present the details f cmputatins. Fr interval [,t rise ], the utput is cmpsed f three terms: where ω d ω n α α ω, d θ arctan , αω d where τ s ω n s + αs + ω n V DD V v r ( t DD α V r( t DD e α t t rise t ut ( cs( ω rise ω n t rise ω d t θ d v r V DD ( t s ω n e α t cs( ω t rise ( ( L L C ω d t ϕ d α ϕ arc tan ω d, and V DD s v r( t t rise L H e t τs e αt cs( ω τ s d t ξ L C (7 (8 (9 α ( τ where s ξ arc tan ω d The utput vltage is again stated as an algebraic summatin f v r (t, v r (t, v r (t. Similar t case I, the utput fr [t rise, is cmpsed f tw terms, v s (t, v s (t, due t the tw separable parts f the input: v s v s ω n ( t V DD e α t cs( ω ω d t ϕ d ( t V s H e t τ s ( e αt cs( ω τ s d t ξ V where DD V s s e t rise τ s (. Again v. t s( t v s( t + v s( t rise L L C Finally the system respnse is v s+ r ( t v s( t + vr( t. (4 (4 Frm Eq. (.b, the final value f the utput is a summatin f delayed versins f the cmputed utput vltage. Since the attenuatin factr is large, we nly cnsider a few terms f Eq. (.b t calculate the final value f the vltage. Experiments have shwn that fur terms are sufficient t prduce high accuracy. With this apprximatin, the final value f the utput vltage will be: 4 v ut ( t ( n + ( n C --- exp v (4 L s + r [ t ( n LC] n 4. EXPEIMENTAL ESULTS We examined several examples with different values f intercnnect length, technlgy parameters, and lad and surce impedances. Amng these experiments fur representative examples are reprted. Table. cmpares the rise-time f the wavefrms derived by ur analysis with thse derived by HSPICE simulatin fr a number f different transmissin lines. ws thrugh 4 crrespnd t the transmissin lines depicted in Fig. ( thrugh Fig. (5. The remaining rws represent new data pints. In table, WD represents the width f each cnductr, HT is the height f cnductr, and TH is thickness f the cnductr. Figs ( and (4 shw the ramp respnse f tw lssy intercnnects with λ less than tw. The analysis f case II is used fr these tw figures. Fig. (.a depicts the utput wavefrm btained by ur analysis whereas Fig. (.b shws the utput wavefrm f the same cnfiguratin by HSPICE simulatin. Similarly Figs (4.a and (4.b shw the utput respnse f anther intercnnect btained by ur analysis and HSPICE simulatin respectively. Figs ( and (5 shw the ramp respnses f lssy intercnnects with λ greater than tw. The analysis f case I is used fr these tw cases. 5. CONCLUSION In this paper we prpsed a new methd fr btaining the analytical expressin fr the ramp respnse f a lssy intercnnect. The inductive effects f the wire line, and mst imprtantly the utput resistance f the wire driver were cnsidered in ur analysis. We started with the tw-prt representatin f the transmissin line. Amng varius tw-prt parameters the chain matrix was selected. Ntice that this prperty is very useful when analyzing the ramp respnse f a cascade f intercnnect segments with different widths.this kind f tw-prt matrix allws us t btain the tw prt matrix f any number f cascade cnnectins f different wires with different wire sizes very easily by simply multiplying their tw-prt chian matrices. We then btained the ramp respnse f the system by ding sme further simplificatin. The results shw that this methd is able t btain the ramp respnse f the lssy intercnnect with small errr. 55

5 m 6m 4m m Panel n 4n 6n 8n n n Time (lin ( Table : Cmparisn between rise-times f ur analysis and HSPICE WD HT TH s C L C L d t rise (urs t rise (HSPICE (µm (µm (µm kω pf Ω pf nh mm nsec nsec errr % % % % % % % % % Output Vltage f a Lssy Intercnnect t a Flattened amp (7.8, CPF, LnH, Lengthcm, sk, CL.PF.5 Output Vltage (vlts.5 Vltages (lin x 8 (a (b Fig. (. amp respnse f a lssy intercnnect (Lengthcm, 7.8, CpF, LnH excited by a ramp input with s K as the surce resistance and C L.pF as the lad capacitance. (a The result btained by ur methd. (b The result btained frm HSPICE simulatin. Output Vltage f a Lssy Intercnnect t a Flattened amp (68, C.76PF, L.45nH, Lengthmm, sk, CL.PF Panel l Output Vltage (vlts.5 Vltages (lin m 6m.5 4m m x p 4p 6p 8p n Time (lin ( (a (b Fig. (. amp respnse f a lssy intercnnect (Lengthmm, 68, C.76pF, L.45nH excited by a ramp input with s K as the surceresistance and C L.pF as the lad capacitance. (a The result btained by ur methd. (b the result btained frm HSPICE simulatin. 56

6 l l m 6m 4m m m 6m 4m m Panel p p p 4p 5p 6p 7p 8 Time (lin ( Panel p 4p 6p 8p n Time (lin ( Output Vltage f a Lssy Intercnnect t a Flattened amp (7, C.4PF, L.65nH, Lengthmm, sk, CL..5 Output Vltage (vlts.5 Vltages (lin x (a (b Fig. (4. amp respnse f a lssy intercnnect (Lengthmm, 7, C.4PF, L.65nH excited by a ramp input with s K as the surce resistance and C L.pF as the lad capacitance (a The result btained by ur methd. (b the result btained frm HSPICE simulatin. Output Vltage f a Lssy Intercnnect t a Flattened amp (54, C.77PF, L.4nH, Lengthmm, sk, CL.PF.5 Output Vltage (vlts.5 Vltages (lin x 9 (a (b Fig. (5. amp respnse f a lssy intercnnect (Lengthmm, 54, C.77PF, L.4nH excited by a ramp input with sk as the surce resistance and C L.pF as the lad capacitance. (a The results. btained by ur methd. (b the result btained frm HSPICE simulatin. 6. EFEENCES [] L. T. Pillage and. A. hrer, Asympttic Wavefrm Evaluatin fr Timing Analysis, IEEE Trans. Cmp. Aided Design, vl. 9, pp. 5-66, Apr. 99. [] M. Sriram and S. M. Kang, Efficient Apprximatin f the Time Dmain espnse f Lssy Cupled Transmissin Line Trees, IEEE Trans. n Cmputer-Aided Design f Integrated Circuits and Systems, vl. 4, N. 8, Aug [] H. Lia, W. Dai,. Wang, and F. Y. Chang, S-Parameter Based Macr Mdel f Distributed-Lumped Netwrks Using Expnentially Decayed f Plynmial Functin, Prc. f the th ACM/IEEE Design Autmatin Cnf., pp. 76-7, June 99. [4] A. B. Kahng and S. Muddu, "Analysis f C Intercnnectins Under amp Input, ACM Trans. n Design Autmatin f Electrnic Systems, Jan [5] S. Lin and E. S. Kuh, Transient Simulatin f Lssy Intercnnect, Prc. f the 9th ACM/IEEE Design Autmatin Cnf., pp. 8-86, June 99. [6] N. Balabanian, T. A. Bickart, Electrical Netwrk Thery, Jhn Wiley & Sns, INC, Sec..-.4, 969. [7] J. Cng, Z. Pan, H. He, C.-K. Kh, K.-Y. Kh, Intercnnect Design fr Deep Submicrn ICs, Prc. Int. Cnf. n Cmputer Aided Design, pp , 997. [8] A. V. Oppenheim, A. S. Willsky, H. Nawab, Signals and Systems, nd editin, Prentice-Hall,