Representative Frequency for Interconnect R(f)L(f)C Extraction

Transcription

1 Representative Frequency for Interconnect R(f)L(f)C Extraction Akira Tsuchiya Masanori Hashimoto Hidetoshi Onodera Department of Communications and Computer Engineering, Graduate choo of Informatics, Kyoto University Kyoto 66 85, Japan e-mai: Abstract This paper discusses the frequency to extract RLC vaues from interconnects. The frequency used for RLC extraction affects the accuracy of interconnect characterization, and hence carefu determination of extraction frequency is crucia. We propose a representative frequency for RLC extraction based on the interconnect ength. We show that the proposed method enabes accurate anaysis of the waveform at the far-end of interconnects. We verify that the extraction at the proposed frequency provides the most accurate transition waveform against various input signas and interconnect structures in digita circuits. I. INTRODUCTION As increasing operating frequency, frequency-dependence of interconnect characteristics is becoming significant. Interconnect characteristics, especiay resistance and inductance depend on frequency because of skin-effect and proximity effect. In frequency-dependent interconnects, the behavior of interconnects depends on frequency e.g. attenuation and phase veocity dispersion. In digita circuits, common input waveform of interconnects are trapezoida puses. A trapezoida puse contains frequency components from to. Moreover, the input puse pattern is not entirey periodic. The frequency spectrum varies depending on the width of puse and the period. The minimum puse width and period are determined by system cock. But on signa ine, the puse pattern depends on the circuit behavior. To treat frequency-dependent interconnects, severa circuit modes are proposed [ 3]. The frequency-dependent modes improve simuation accuracy [, 4], but in circuit design, frequency-dependent modes are not used so commony. Because most of conventiona design methods are based on the frequency-independent mode. If interconnect characteristics can be modeed we by a singe frequency, we can use the design techniques proposed so far, e.g. circuit reduction, buffer insertion and timing anaysis [5, 6]. Furthermore, frequency-independent RLC vaues can intuitivey predict fundamenta interconnect characteristics such as characteristic impedance. We can aso save the cost to extract RLC vaue from to high frequency. However, determination of a singe extraction frequency is difficut. In Ref. [7], the impact of a frequency-dependent mode is discussed. A frequency-dependent mode is compared with an equivaent circuit extracted at from the viewpoint ona deay, crosstak noise and so on. Ref. [7] reports that a frequency dependent mode is necessary for crosstak noise estimation. However the authors examine ony a extracted mode and frequency dependent mode. Therefore it is not cear whether crosstak can be estimated using a frequencyindependent mode extracted at a certain representative frequency. In this paper, the extraction frequency based on the interconnect ength is proposed. It is commony adopted to determine the representative frequency from the shape of an input signa waveform, especiay from the rise time, focusing on the spectrum of the input signa. This is natura and reasonabe when we anayze the incident waveform to the near-end of the interconnects. On the other hand, our main interest is the anaysis of the waveform at the far-end. As signas are propagating through an interconnect, high-frequency components are easy to attenuate. The dominant frequency components that determine the far-end waveform are different from those for the near-end waveform. We observe that accurate estimation of attenuation behavior is crucia to obtain accurate farend waveforms. Open-ended transmission-ines can be treated as resonator and transmission-ine resonators are used in microwave circuits. An on-chip transmission-ine with CMO receiver can be regarded as a resonator. From the theory of a resonator, the frequency where attenuation becomes minimum is decided by the interconnect ength. We revea that this resonance frequency is the dominant frequency to characterize far-end waveforms, and then propose to adopt it as the representative frequency used for interconnect RLC extraction. We experimentay verify that the most accurate waveform is obtained when the proposed frequency is used for extraction. We show that the maximum errors in our experiments are beow 8% in the votage ampitude, signa deay and the ampitude of crosstak noise. Therefore the proposed frequency enabes accurate transient anaysis using frequency-independent interconnect mode. In ection II, interconnect modeing and its probems are described. We next discuss the extraction frequency in digita circuits. We then show the experimenta resuts in ection IV. ection V concudes the discussion. II. PROBLEM DECRIPTION This section describes the probem discussed in this paper. We first show frequency-dependence of interconnect characteristics and demonstrate its impact on transient anaysis. A. Frequency-Dependence of Interconnect Characteristics Frequency-dependence of interconnect characteristics is mainy caused by skin-effect and proximity effect. o the characteristics variation is strongy reated with the interconnect structure as we as the frequency. kin effect and proximity effect are remarkabe on wide and thick interconnects. Because, skin depth becomes comparabe to the interconnect size in reativey ower frequency. Figure shows an exampe of resistance and inductance characteristics. The resistance and inductance vaues are cacuated by a fied-sover [8]. The assumed interconnect structure is co-panar, and the width of the signa ine is µm, the width of the ground ine is µm and their spacing is µm. In this case, the resistance increases by % from to.ghz, and the inductance decreases by % from to.9ghz. The resistance and the inductance start changing from reativey ow frequency of to GHz, and thus frequency-dependence is not negigibe to mode interconnects in current high-performance circuits any onger. 4 IEEE. Persona use of this materia is permitted. However, permission to reprint/repubish this materia for advertising or promotiona purposes or for creating new coective works for resae or redistribution to servers or ists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. 69

2 Resistance [Ohm/mm] Inductance.4 um.35 G G.95um.3 8 um um Resistance..5.. Frequency [GHz] Fig.. Frequency-dependence of resistance and inductance. (co-panar structure, signa ine width µm, ground ine width µm, spacing µm) Inductance [nh/mm].5.5 f =34GHz sig 3mm R d Z Fig. 3. The impact of frequency-dependence. (interconnect structure is shown in Figure, Z = 55Ω, R d = Ω) Fig.. RLC adder circuit mode. B. Interconnect Modes and their Impact on Waveform Generay, interconnects in VLIs are expressed by umped RLC for circuit design. To mode ong interconnects that have transmission ine characteristics, an RLC adder circuit as Figure is used. This adder mode cannot consider the frequency-dependence of interconnect characteristics. A number of frequency-dependent modes are proposed [ 3]. In this paper, we use the mode of Ref. [3] as a goden frequencydependent mode. It is impemented in HPICE [9] as w- eement mode. In interconnect design, characteristic parameters such as characteristic impedance and attenuation constant are essentia factors for designers. Athough frequencydependent modes such as Ref. [3] can provide accurate waveforms, circuit designers can not know such parameters that shoud be used for circuit design, because such parameters are aso freqency dependent. We therefore have to determine a singe frequency to specify the characteristic impedance, attenuation constant and so on. As mentioned in ection I, the frequency spectrum of propagating signa depends on circuit behavior, so it is difficut to specify the most representative frequency from the frequency spectrum. This paper proposes a method to determine the representative frequency. Figure 3 shows the impact of frequency-dependence on transient anaysis. The simuated circuit is shown in Figure 3. The interconnect shown in Figure is driven by a votage source and a resistor R d that correspond to a CMO driver. Interconnect characteristic impedance Z is 55Ω and the output impedance of the driver R d is Ω. The soid ine abeed shows the votage waveform at the far-end by the frequencydependent mode. In this paper, we use as the abbreviation of Frequency-Dependent mode. The dashed ines abeed and are the resuts of frequency-independent modes. means the RLC adder mode extracted at, and corresponds to RLC extraction at the significant frequency [6]. The number of adder is 5. ignificant frequency is one of a representative frequency defined from the frequency components of a trapezoida puse, and it is expained in the next section. As you see, both waveforms of the conventiona frequency-independent modes ( and ) are far from that of frequency-dependent mode ( ). When R and L are extracted at, the extracted resistance is too ow, and, the resistance extracted at significant frequency is too high. From the above observations, we can expect that a frequency between and significant frequency provides the waveform that is cose to the waveform of the frequency-dependent mode. If the representative frequency can be determined systematicay, we can mode interconnects by a singe frequency. In the foowing section, we discuss the way to determine the representative frequency to mode interconnects at a singe frequency. III. REPREENTATIVE FREQUENCY FOR EXTRACTION In this section, we discuss the representative frequency to extract interconnect RLC. Conventionay, frequency determined from input puse is used for interconnect extraction. We first expain some representative frequencies conventionay used for extraction, and we then propose the representative frequency cacuated from interconnect ength. A. Conventiona Methods In digita circuits, a trapezoida puse that contains mutipe frequency components is a common waveform. In order to derive frequency-independent mode of Figure, we have to choose a singe extraction frequency. There are severa representative frequencies of periodic puse waveform. One of them is significant frequency [6]. ignificant frequency is expressed by signa transition time t r. The significant frequency is defined such that the signa energy from to becomes 75% of a signa energy. In the range 7 T w /t r 3, is given by.34/t r [6]. On the other hand, is often used for extraction. Ref. [7] concudes that the extraction at is accurate enough to estimate signa deay and overshoot/undershoot. extraction is enough when frequency-dependence is weak, e.g. narrow interconnects or ow frequency. But as shown in Figure 3, RLC adder extracted at or the significant frequency causes considerabe amount of errors in transient anaysis. B. Proposed Method Conventiona methods based on input puse shape focus on the frequency components at the near-end of interconnects. However the far-end waveform is more important for circuit designer because the waveform directy affects signaing deay. The far-end waveform becomes totay different because of attenuation and refection. We propose an extraction frequency that aims to express accurate far-end waveforms. Figure 4 shows step responses obtained with a mode and a adder extracted at significant frequency. The experimenta setup is the same as Figure 3. As shown in Figure 4, the adder extracted at modes the incident wave of interconnects we, but a remarkabe error occurs at the far-end. This error is mainy caused by overestimation of attenuation. On transmission-ines, characteristic impedance and attenuation constant are important factors which decide the waveform at 69

3 Magnitude of Incident wave Near-end Near-end R d Z =34GHz Fig. 4. Waveform at near-end and far-end. (interconnect structure is shown in Figure, Z = 55Ω, R d = Ω) the far-end. Approximatey, characteristic impedance is expressed as Z = L/C and is proportiona to square root of inductance L. The attenuation constant α is expressed as α = R/Z. The attenuation constant is roughy proportiona to resistance R and square root of inductance L. From the above observation, variation of resistance strongy affects waveform propagation. Moreover, as shown in Figure, the variation of resistance is arger than that of inductance. At 34GHz of Figure, inductance decreases by about 3% from and resistance increases by about 3% from. The inductance decreases because of proximity effect and the interna-inductance decreasing. Therefore the inductance vaue saturates at high frequency. On the other hand, resistance increases exponentiay as frequency become higher. Therefore the estimation of resistance is crucia to anayze far-end waveform. The attenuation strongy depends on interconnect structure such as interconnect ength. From above discussion, we have to consider interconnect structure when determining an extraction frequency. To determine an extraction frequency from the viewpoint of the waveform at the far-end, we have to specify the dominant frequency component at the far-end. From the theory of open-ended transmission-ine resonators, when the quarter waveength λ/4 is equa to interconnect ength, transmissionines are equivaent to a series resonator shown in Figure 5. When quarter waveength λ/4 is equa to interconnect ength, the frequency f res is expressed by f res = c/λ = c/4, () where c is the veocity of eectromagnetic wave. When the frequency is f res, the impedance of series resonator become minimum and the attenuation of frequency component f res is minimum. Figure 6 shows a transfer characteristic of a transmission-ine. The interconnect structure is the same as Figure and interconnect ength is 5mm. The reative permittivity of io is 4., so the veocity of eectromagnetic wave is.5 8 m/s. In this case, resonance frequency f res is 7.5GHz. The votage gain becomes maximum at the resonance frequency f res. Therefore the frequency component f res strongy affects the waveform at the far-end. The frequency spectrum at the far-end is as shown in Figure 7 when a transmission-ine is driven by a votage source and a resistor. The frequency f res is the first peak of frequency components regardess of various transition times. We hence consider the frequency f res = c/4 as a representative frequency. In LIs, the phase veocity of eectromagnetic wave c is constant because it is determined by the permittivity and permiabiity of insuator. Frequency f res is determined ony by interconnect λ/4 Open-end Open-ended transmission-ine eries resonator Fig. 5. Open-ended transmission-ine and equivaent series resonator. Votage gain [db] f res =7.5GHz 5mm Vout -4 Frequency [GHz] Fig. 6. Transfer characteristics of a transmission-ine shown in Figure, interconnect ength is 5mm. ength. We propose this f res as an extraction frequency and rewrite f res to in foowing sections. C. Limitations of the Proposed Method We here examine the imitation of the proposed method. The proposed method assumes that the inductance effect of interconnects is significant and interconnects behave as transmission-ines. This assumption at first seems to make a imitation. However when the inductance effects are negigibe, RC ump mode is enough to mode interconnects. The second assumption is that the resonance frequency is uniquey decidabe. For exampe, the resonance frequency cannot be determined on branched interconnect because of mutipe-refection. But in high-performance interconnects, impedance matching is appied at the branch to avoid mutiperefection. Additionay on amost goba interconnects, repeaters are inserted and the fan-out of driver is. The proposed method is based on open-ended transmissionine resonator. In most CMO circuits, transmission-ines are terminated by input capacitance of receivers, which is sma enough to assume open-ended. However on transmission-ines terminated by resistance or so, the resonance frequency f res is not equa to c/4. In such case, we have to decide resonance frequency by other way. Therefore these assumptions does not reduce the appication area of the proposed method so much. The proposed method is vaid for the most of high-performance interconnects. IV. EXPERIMENTAL REULT This section shows some experimenta resuts. We verify the modeing accuracy of each representative frequency by circuit Ampitude [db] mm f =7.5GHz res tr =5ps -. Frequency [GHz] Vin t r =ps tr =3ps Fig. 7. Frequency spectrum of waveform at the far-end. 693

4 TABLE I RANGE OF PARAMETER AND REPREENTATIVE FREQUENCIE. Parameter range ps t r ps.5mm mm W s G Wg W s Micro-strip um um um um Fig. 8. Cross-sections of interconnects. Corresponding freq. range 3.4GHz 34GHz 3.75GHz 75GHz W g G g W s Co-panar W s g simuation. We first expain experimenta conditions and some metrics of accuracy. We then verify the accuracy under various experimenta conditions. A. Experimenta Conditions and the Metrics of Accuracy In this section, we expain experimenta conditions and metrics of accuracy. To verify the accuracy of the proposed method comprehensivey, we examine under various frequency-dependence and various waveforms. Frequency-dependence of interconnects is determined by the interconnect structures. Waveform variation is expressed by puse transition time. We therefore vary the foowing parameters and evauate the proposed and the conventiona representative frequencies. puse transition time ( changes). interconnect ength ( changes). interconnect structure and driver strength. First, the effect of puse transition time is examined. Transition time decides significant frequency, so varies and is fixed in this experiment. We then verify the cases that interconnect ength changes. Frequency varies as changing interconnect ength, and is fixed. The ranges of each parameter and the range of corresponding representative frequencies are isted in Tabe I. We experiment the above conditions in various interconnect structures and driver output impedance. As the interconnect structure, two popuar interconnect structures; micro-strip and co-panar are used. To evauate crosstak noise, we ocate two signa interconnects. The cross-sections of two interconnect structures are shown in Figure 8. W s is the width ona interconnect, W g is the width of ground ine, is the spacing between signa interconnects and g is the spacing between the signa interconnect and the ground ine. The frequencydependence of interconnect characteristics is significant on the thick, wide and ong interconnects such as cock ines, bus and goba interconnects. For such interconnects, wide interconnects are used to reduce interconnect oss, and the spacing between interconnects are adjusted considering the inductance and capacitive couping. Therefore we verify interconnect structures in µm W s 8µm, 8µm W g 4µm, µm 8µm and µm g 8µm. In transient anaysis, we evauate the votage waveform of the experimenta circuit as shown in Figure 9. One of two ines is stimuated by the input puse, and the other is kept quiet. We ca the stimuated ine as Aggressor, and the quiet ine as Victim. The near-end of each ine are hed by a resistance, W g G Victim Aggressor Fig. 9. Experimenta circuit for transient anaysis Deay V / dd Agg. V noise Vpp Vict. Vict. Agg Fig.. Definition of deay time, peak-to-peak votage and crosstak. which represents the output impedance of the driver. The characteristic impedance of verified interconnects are within Ω Ω. The driver output impedance is varied from Ω to Ω. The far-end of each ine is connected to the capacitor oad that corresponds to the input capacitance of a receiver. The vaue of capacitor oads is fixed to 5fF. To verify modeing accuracies, evauation metrics are necessary. We use V dd / propagation deay time (Deay), ampitude of overshoot/undershoot (V pp ) and ampitude of farend crosstak noise (V noise ) as evauation metrics. Figure shows the definition of deay time, peak-to-peak votage and crosstak. We evauate these metrics of the adder extracted at each representative frequencies and frequency-dependent mode. We consider the resut of the frequency-dependent mode as reference data. This means that the evauation resuts that are cose to those of frequency dependent mode are accurate. B. Transition time vs. Accuracy We here show the resuts when transition time is changed. ignificant frequency is decided by transition time. When transition time t r is ps, is 34GHz and when t r ps, becomes 3.4GHz. Figure and Figure show the simuated peak-to-peak votage and deay time. We use a co-panar interconnect structure with 8µm signa wire width, µm ground wire width, 4µm spacing between each interconnects and 5mm ength. The output impedance of the drivers is 5Ω. The simuated crosstak noise votage is aso shown in Figure 3. Tabe II shows the maximum errors when the transition time varied. From Figure, extraction at causes about 9% error constanty in the peak-to-peak votage. The extraction at causes over % error when the transition time is sma. ignificant frequency becomes extremey high when transition time is sma. Therefore attenuation on interconnect is overestimated. From Figure, the adder extracted at causes about 9% error in the deay time. extraction overestimates the inductance vaue, so the veocity ona is underestimated. Therefore deay time is overestimated especiay when transition time is sma. The extraction at achieves ess than 3% errors in the peak-to-peak votage and the deay time. From Figure 3, there is the same trend as the peak-topeak votage in the ampitude of crosstak noise. extrac- 694

5 Peak-to-peak votage normaized by suppy votage Transition time [ps] Fig.. Votage peak-to-peak when the transition time is changed. igna deay time [ps] Transition time [ps] t r t r Fig.. Deay time when the transition time is changed. tion causes error constanty and causes remarkabe error when the transition time is sma. As seen in Tabe II, extraction causes about % overestimation in V pp, deay and V noise. Resistance and inductance extraction at causes over % underestimation in V pp and V noise. The adder extracted at steadiy provides the most accurate estimation among the three, and the maximum error is about 8%. We here show one exampe of typica waveforms. Figure 4 shows the waveforms at the far-end of the aggressor and the victim interconnects. From Figure 4, the overshoot and crosstak are overestimated on the adder extracted at, and are underestimated on the adder extracted at. From viewpoint of the signa deay, we can see that overestimates the deay time. From the observation of waveforms, the equivaent circuit extracted at is the most accurate. C. Interconnect ength vs. Accuracy Next, the accuracy versus the interconnect ength is discussed. Frequency depends on the interconnect ength and the wave veocity. The wave veocity is determined by reative permittivity. Therefore we can assume that the veocity is constant in the same technoogy. Figure 5 shows the peak-topeak votage, and Figure 6 shows the deay time normaized by the deay time of mode. Figure 7 shows the ampitude of the crosstak noise. The simuation condition is the same as ection B. As seen in Figure 5, the adder extracted at TABLE II MAXIMUM ERROR WHEN THE TRANITION TIME CHANGED. Extraction Freq. Error in V pp +9.% 3.%.5% Error in Deay +9.% +.9%.% Error in V noise +.8% 7.9%.4% Aggressor Victim 5mm Aggressor Victim Fig. 4. The waveforms at the far-end of the aggressor and victim. achieves the minimum error in peak-to-peak votage. extraction aways overestimates the V pp, and extraction causes underestimation when the interconnect ength becomes ong. As shown in Figure 6, extraction causes about % error when the interconnect ength becomes ong. The errors of and extraction are amost same and beow 4%. From Figure 7, crosstak noise becomes arger as the interconnect ength becomes ong in the region where the interconnect ength is sma. The noise ampitude is amost constant when the ength is more than mm. Figure 7 shows that extraction causes overestimation and causes underestimation of the crosstak noise. The maximum errors are isted in Tabe III. As you see, and may cause over % errors but the maximum error of is about 3%. These resuts indicates the adder extracted at is robust against the change of the interconnect ength. D. Resuts of Overa Experiments In the above sections, we show that the frequency cacuated from interconnect ength achieves the most accurate anaysis. Tabe IV shows the maximum errors in a of the resuts we evauate. We carefuy choose the experimenta conditions so that we can cover most part of the reaistic cases. The tota number of experiments is about 4,. The adder extracted or causes errors beyond %. When a wide micro-strip interconnect is driven by a strong driver, and tend to cause arge error. The proposed frequency achieves the error beow 8%. The above discussions prove that Crosstak noise normaized by suppy votage Transition time [ps] Fig. 3. Crosstak noise peak-to-peak when the transition time changed. t r Peak-to-peak votage normaized by suppy votage Interconnect ength [mm] Fig. 5. Votage peak-to-peak when the interconnect ength changed. 695

6 igna deay time normaized by deay of mode Interconnect ength [mm] Fig. 6. Normaized deay time when the interconnect ength changed. Crosstak noise normaized by suppy votage Interconnect ength [mm] Fig. 7. Crosstak noise peak-to-peak when the interconnect ength changed. the adder extracted at the proposed frequency provides the most accurate modeing of frequency-dependent interconnects among the three frequencies. E. Toerance to Extraction Frequency Variation We here discuss the effect of estimation error on modeing accuracy. As mentioned in ection C, the proposed method is based on open-ended transmission-ine theory. However in rea chips, interconnects are terminated by input capacitor of the receiver and, rigidy speaking, the sink is not idea open-end. The resonance frequency is not equa to exacty, but the difference is usuay quite sma because input capacitor of CMO receiver is sma. Figure 8 shows the extraction frequency versus errors. X- axis is the extraction frequency and Y-axis is the error from frequency-dependent mode. The experimenta setup is the same as that of Figure 4, 5mm wire ength and ps transition time. The proposed frequency is 7.5GHz. As shown in Figure 8, the errors in V pp and in V noise become minimum at the proposed frequency. The error in deay becomes minimum at about GHz, but the error is amost constant above GHz. From Figure 8, the errors are beow % in the region of ± 3%. This resut indicates that the proposed method is accurate enough even if the proposed frequency has a certain error in comparison with the exact resonance frequency. We can aso see that extraction at and significant frequency = 34GHz is far from the frequency with the minimum error around. The errors at and significant frequency are above % whereas that of the proposed method is beow %. TABLE III MAXIMUM ERROR WHEN THE INTERCONNECT LENGTH CHANGED. Extraction Freq. Error in V pp +%.4% 5.7% Error in Deay +9.% +3.% +.5% Error in V noise +8.7%.8%.3% TABLE IV MAXIMUM ERROR IN OVERALL EXPERIMENT. Extraction Freq. Error in V pp +.5% 4.6% 8.% Error in Deay +7.% +4.8% +3.% Error in V noise +37.4% +7.9% 8.% Error from frequency-dependent mode [%] =7.5GHz Error in V noise Error in V pp Error in deay =34GHz Extraction frequency [GHz] Fig. 8. Extraction frequency vs. errors. V. CONCLUION The frequency that shoud be used to extract RLC vaues is discussed. When we use frequency-independent equivaent circuits for circuit design, the extraction frequency must be carefuy determined to maximize the fideity in interconnect characteristics. We propose an RLC extraction scheme that uses the frequency determined by interconnect ength. We experimentay verify that the proposed frequency achieves the most accurate estimation in deay time and ampitude of overshoot or undershoot. The maximum error is within 5% in peakto-peak votage and deay, and the maximum error in crosstak is within 8% in our experiments. With the proposed representative frequency, RLC extraction at a singe frequency becomes accurate enough to mode interconnect characteristics, and hence we can expoit many effective design and anaysis techniques deveoped ignoring frequency-dependence. REFERENCE [] H. A. Wheeer, Formuas for the kin-effect, Proceedings of Institute of Radio Engineers, vo.3, pp.4 44, ept 94. [] B. Krauter and. Mehrotra, Layout Based Frequency Dependent Inductance and Resistance Extraction for On-Chip Interconnect Timing Anaysis, Proc. DAC, pp.33 38, 998. [3] D. B. Kuznetsov and J. E. chutt-ainé, Optima Transient imuation of Transmission Lines, IEEE Trans. CA, vo.43, no., pp., Feb 996. [4] A. Deutsch, P. W. Coteus, G. V. Kopcsay, H. H. mith, C. W. urovic, B. L. Krauter, D. C. Edestein, and Phiip J. Reste, On-Chip Wiring Design Chaenges for Gigahertz Operation, Proceedings of IEEE, vo.89, no.4, pp , Apr. [5] H. B. Bakogu, Circuits, Interconnections, and Packaging for VLI, Addison-Wesey Pubishing Company, Inc., 99. [6] C.-K. Cheng, J. Liis,. Lin, and N. H. Chang, Interconnect Anaysis and ynthesis, A Wiey-Interscience Pubication.,. [7] Y. Cao, X. Huang, D. yvester, T.-J. King, and C. Hu, Impact of On-Chip Interconnect Frequency-Dependent R(f)L(f) on Digita Design, Proc. of Internationa AIC/OC Conference, pp , ep. [8] Avant! Corp., Raphae Reference Manua, May 998. [9] Avant! Corp., tar-hspice Manua, Dec. 696