IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 32,NO. 1,FEBRUARY 2009 175 Fast Methodology for Determining Eye Diagram Characteristics of Lossy Transmission Lines Wei-Da Guo, Jeng-Hau Lin, Chien-Min Lin, Member, IEEE, Tian-Wei Huang, Senior Member, IEEE, and Ruey-Beei Wu, Senior Member, IEEE Abstract—As the speed of signal through an interconnection increases toward the multigigabit ranges, the effects of lossy transmission lines on the signal quality of printed circuit boards becomes a critical issue. To evaluate the eye diagram and thus the signal integrity in the modern digital systems, this paper proposes a fast methodology that employs only two anti-polarity one-bit data patterns instead of the pseudo-random bit sequence as input sources to simulate the worst-case eye diagram. Analytic expressions are derived for the impulse response of the lossy trans- mission lines due to the skin-effect loss, while the Kramers–Kronig relations are employed to deal with the noncausal problem related to the dielectric loss. Two design graphs that can be used to rapidly predict the eye diagram characteristics versus the conductive and dielectric losses are then constructed and based on which, the maximally usable length of transmission lines under a certain signal specification can be easily acquired. At last, the time-do- main simulations and experiments are implemented to verify the exactitude of proposed concept. Index Terms—Eye diagram, impulse response, Kramers–Kronig (K–K) relations, lossy transmission line, pseudo-random bit se- quence, signal integrity. I. INTRODUCTION E YE diagram is a very helpful metric of intuitively and quickly assessing the performance quality of digital sig- nals through various interconnection structures, such as chip carrier [1], [2], connector [3], [4], through silicon via [5], op- tical backplane [6], delay line [7], and so on. Since the switching time and feature size of circuit devices keep on decreasing, many nonideal effects previously regarded to be negligible now be- come the critical design challenges for meeting the requirements of signal/power integrity and electromagnetic interference [8]. Several works have been devoted to the prediction of the eye Manuscript received March 12, 2008; revised June 02, 2008. Current version published February 13, 2009. This work was supported in part by the National Science Council, Republic of China under Grant NSC 96-2221-E-002-083, in part by the National Taiwan University (NTU) Excellence Research Program under 95R0062-AE00-08, and in part by the Taiwan Semiconductor Manufac- turing Company (TSMC) under Grant 96-FS-B01. This work was recommended for publication by Association Editor J. Tan upon evaluation of the reviewers comments. W.-D. Guo, J.-H. Lin, T.-W. Huang, and R.-B. Wu are with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, 10617 Taipei, Taiwan (e-mail: f92942062@ntu. edu.tw; rbwu@ew.ee.ntu.edu.tw). C.-M. Lin is with the Backend Technology Development Division, Taiwan Semiconductor Manufacturing Company, 30077 Hsin- Chu, Taiwan. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TADVP.2008.2006276 diagram performance for these effects in some specific appli- cations [9], [10]. Among them, the prominent one is the fre- quency-dependent loss of transmission lines that mainly comes from the finite conductivity of imperfect conductor and the nat- urally electric polarization of dielectric material. The dispersion accompanied by the frequency dependent loss of the transmission lines will exhibit a relative long-tail response on the transmitted signal. Hence, the lossy lines may induce se- rious intersymbol interference (ISI) problem, resulting in the occurrence of poor eye diagram performance or even the incor- rect functionality of logic gates, especially for the systems with the long-distance data transmission inside. Even if a few com- pensation schemes were presented to alleviate the lossy effects [11]–[13], it is imperious to evaluate the level of their influences on the eye diagram performance. To simulate the eye diagram performance of a high-speed digital system, a pseudo-random bit sequence (PRBS) is often adopted as the input excitation. Then, the successive output waveforms can be overlapped on a specific time window to produce the eye diagram for signal-integrity analyses [14]. The relationship between the eye diagram characteristics and the lossy effects has been empirically investigated in previous study [15]. However, it still lacked the complete discussion for the eye-opening determination and the required large number of bits in PRBS would be very time consuming in constructing the sufficient responses, which are both the major motives for this research. This paper is organized as follows. In Section II, a much faster but still accurate methodology which employs only two anti-po- larity one-bit data patterns as the input excitation is proposed to gauge the worst-case eye diagram performance at the receiving end of a well-matched transmission-line system. With the ap- propriate simplification on the transfer function of lossy trans- mission lines, the corresponding impulse response is thus de- rived in Section III and can be divided into three parts, which are associated with the effects of transmission line length, conduc- tive loss, and dielectric loss, respectively. The Kramers–Kronig (K–K) relations are, therefore, introduced to resolve the non- causality of impulse responses related to the dielectric loss [16]. In Section IV, the conductive and dielectric losses are further quantified to construct two design graphs for the rapid predic- tion of resultant eye diagram characteristics and based on which, one can easily acquire the maximally usable length of lines under a certain signal specification. The comparisons of eye di- agram characteristics between the simulated and measured re- sults are shown in Section V for verification and the conclusions are addressed in Section VI. 1521-3323/$25.00 © 2009 IEEE