Matrix Rational Approximation Research Articles

Frequency response and transient analysis of through glass packaging vias using matrix-rational approximation (MRA) technique for three-dimensional ICs

In this paper, a computationally efficient matrix rational approximation (MRA) technique is developed to analyze the electrical behaviour of through packaging vias in glass interposer (TGVs). Using MRA technique, the electrical behaviour of TGVs utilizing differential multi-bit configuration, filled with composite copper-mixed carbon nanotube bundle (Cu-mCNTB), is investigated. A temperature-dependent π-type equivalent circuit of such differential multi-bit (DM) composite TGVs is developed using partial-element equivalent-circuit technique. The effective complex conductivity of Cu-mCNTB is also derived by appropriately considering the effects of temperature-dependent mean free path in CNTs. The frequency dependent differential- and common-mode impedances are obtained up to 100 GHz through PEEC technique. It is analyzed that composite DM-TGVs outperform silicon via counterparts in terms of reduced insertion loss using proposed MRA technique and verified through high-frequency structure simulator (HFSS). The transient analysis including crosstalk-induced propagation delay, peak crosstalk, and peak timing of coupled Cu-mCNTB/composite DM-TGVs is determined through the proposed MRA model, which exhibits error within 1% compared to the SPICE. The robustness of proposed model is examined under a wide variety of test cases including the proposed 5-Cu-mCNTB composite TGV array. For the transient analysis, the CPU runtime using the MRA model is 18.57 × faster than the SPICE. To the best of the authors’ knowledge, it is for the first time that temperature-dependent modeling and crosstalk analysis of coupled TGV structures is reported using MRA technique.

Transient Analysis of Hybrid Cu-CNT On-Chip Interconnects Using MRA Technique

This paper presents the transient analysis of the equivalent single conductor (ESC) model of hybrid Cu-CNT on-chip interconnects for nanopackaging using matrix rational approximation (MRA) modeling technique. The analysis of propagation delay and peak crosstalk noise is carried out for single and coupled Cu-CNT interconnect lines at 14 nm and 22 nm technology nodes. It has been observed that the proposed MRA model provides a speed-up factor of 131 compared to the HSPICE. An error of less than 1% confirms the accuracy of the proposed model compared to the SPICE simulations. It is observed that Cu-CNT lines are more immune to the crosstalk due to lesser coupling effects compared to Cu and CNT interconnects. The efficacy, accuracy, and comprehensive analysis using the proposed model ensures immense application possibility of the proposed model in the VLSI design automation tools at the nanopackaging level.

Exponential Matrix–Rational Approximation (EM-RA) Model for SWCNT Bundle and Hybrid Cu-CNT Interconnects

This article presents a variation-aware exponential matrix–rational approximation (EM-RA) model for the signal integrity analysis in single-walled carbon nanotube (SWCNT) bundle and hybrid copper-CNT (Cu-CNT) on-chip interconnects. The variations include temperature (300–500 K) only, and temperature (300–500 K) and dielectric surface roughness (10–180 pm) both. The transient analysis is carried out for the signal integrity parameters, crosstalk and propagation delay, estimation using the proposed EM-RA model. The results obtained using the proposed model are compared with the industry-level simulation program with an integrated circuit emphasis (SPICE) simulator. The comparison shows excellent agreement for the results with less than 1% error. The proposed model is also compared for the computational time complexity. It is observed that a speed-up factor of 74.24 and 48.35 is attained using the proposed temperature-parameterized model during the analysis of coupled-two and coupled-three interconnect lines, respectively. The speed-up further increases to 485.41 and 378.35 for coupled-two and coupled-three lines, respectively, when the temperature and dielectric surface roughness parametrized EM-RA model is used. This computational CPU time expense efficiency and the accuracy of the proposed model in comparison to the HSPICE simulator ensures immense possibility regarding its implementation in computer-aided design (CAD) tools for carrying out the transient analysis of SWCNT bundle and Cu-CNT interconnect lines. Encouraging results for the eye-diagram analysis reports 95.54% and 99.37% of the eye height and eye width for the SWCNT network, and 99.77% and 99.96% of the eye height and eye width for the Cu-CNT.

Transient analysis of graphene-based on-chip interconnects using closed-form MRA model

As interconnects in very-large-scale-integration technology shrinks into the sub-22 nm regime, carbon nanotubes, graphene nanoribbons (GNRs), and Cu-GNRs are emerging as potential replacements for conventional copper on-chip interconnects. This paper presents an matrix-rational-approximation (MRA) model for the fast transient analysis of graphene-based on-chip interconnects. The proposed model’s key feature is that it allows the mathematical representation of the transfer function matrix of on-chip graphene-based interconnect networks in the frequency-domain using closed-form rational functions. The network’s rational functions coefficients are derived from per-unit-length parameters and predetermined coefficients of the Padé approximation. The graphene-based network’s time-domain model is directly obtained from the rational function frequency-domain model without any numerical integration techniques. This makes the proposed MRA model more numerically efficient than conventional simulation program with integrated circuit emphasis (SPICE) models. The results obtained using the proposed model show less than 1% error when compared with HSPICE simulations, ensuring the proposed model’s excellent accuracy. A wide eye-height of 0.8137 V and eye-width of 219.87 ps are obtained for the eye-diagram analysis of multilayer graphene nanoribbon lines at 4 Gbps.

Edge and Dielectric Surface Roughness-Aware EM-RA Model for MLGNR Interconnects

This paper presents an edge roughness (ER) and dielectric surface roughness (DSR) - aware exponential matrix-rational approximation (EM-RA) model for the transient analysis of multi-layer graphene nanoribbon (MLGNR) interconnects. The proposed model analyzes the signal integrity in terms of the peak crosstalk and the propagation delay. The results obtained using the proposed ER and DSR - aware EM-RA model for the crosstalk and delay analysis are compared with the industry-standard SPICE simulator. The ER and DSR - aware EM-RA model results have excellent accuracy compared to HSPICE with less than 0.5% error. The proposed model provides a speed-up by a factor of 84.36 and 86.05 for carrying out the signal integrity analysis in coupled-two and coupled-three MLGNR interconnect lines, respectively, in comparison to HSPICE. The speed-up further increases with the more refined 2D ER and DSR grid of variations. The CPU memory efficacy of the proposed model is compared with HSPICE. The proposed model's stability is analyzed for a range of local to global interconnects.

Edge-Roughness Aware EM-RA Model for Signal Integrity Analysis in MLGNR Interconnects

This article presents a fast and accurate edge-roughness aware exponential matrix-rational approximation (EM-RA) model to capture the signal integrity effects, namely, crosstalk noise and the propagation delay, in coupled-multilayer graphene nanoribbon (MLGNR) interconnects. The proposed model provides a speed-up factor of 75 for coupled-two MLGNR line and a speed-up factor of 51 for coupled-three MLGNR line in comparison with the industry-level simulation program with integrated circuit emphasis (SPICE) simulator. The proposed model results always lie within a 1% error band compared with the HSPICE simulations, thereby ensuring the proposed model’s excellent accuracy. The robustness of the proposed edge-roughness aware EM-RA is confirmed by its application to a range of test cases in coupled-three MLGNR interconnects network. It is observed that the edge-roughness variations modeling plays a critical role in narrow (≤ 45 nm) MLGNRs. To ensure the signal integrity further, the eye-diagram analysis for the worst case scenario considering fully rough-edged MLGNR lines is carried out that provides the eye-height of 864.53 mV and the eye-width of 220.17 ps. The results obtained using the proposed closed-form parameterized EM-RA model ensures its immense possibility of incorporation in VLSI design automation tools for efficient capturing of the signal integrity effects in MLGNR interconnect networks.

A Temperature and Dielectric Roughness-Aware Matrix Rational Approximation Model for the Reliability Assessment of Copper– Graphene Hybrid On-Chip Interconnects

In this article, a closed-form matrix rational approximation (MRA) model is presented for the reliability assessment of copper–graphene hybrid on-chip interconnect networks. The key feature of this MRA model is its capacity to predict how the different values of temperature and dielectric roughness affect the signal integrity performance of the hybrid interconnect networks. As a result, the proposed MRA model is well suited for very fast parametric sweeps and worst case analysis of the hybrid interconnect networks, which has not been possible using existing closed-form models or even SPICE simulations. Numerical examples show that the proposed model is significantly more efficient than conventional models while exhibiting error less than 5%.

An Efficient SPICE-Compatible Cavity Resonant Model for Microstrip Lines

A SPICE-compatible cavity resonant transmission line (CTL) model for a single-ended microstrip line is first presented by reducing 2-D cavity resonator to 1-D cavity resonator. This model has low efficiency since it consists of infinite higher-order components included inductors, capacitors, resistors, and ideal transformers. A modified CTL model is developed by using a fast algorithm to improve the simulation efficiency, which results in an accurate SPICE model in finite elements. Because the improved cavity resonant model uses higher-order (a little more complicated) elements in finite components, it has high efficiency in the time and frequency domain simulation. The model accuracy and simulation efficiency are validated by comparison with multiconductor transmission line and matrix rational approximation modeled methods.

Closed-Form Delay and Crosstalk Models for $RLC$ On-Chip Interconnects Using a Matrix Rational Approximation

In this paper, a closed-form matrix rational-approximation algorithm is proposed to efficiently model the delay and crosstalk noise of coupled RLC on-chip interconnects. A key feature of the proposed algorithm is that, for any rational order, the approximation is obtained analytically in terms of predetermined coefficients and the per-unit-length parameters. As a result, the proposed method is not limited to fixed number of poles and provides a mechanism to increase the accuracy for cases when inductive effects are significant, the length of the line increases, or when the rise time of the signal becomes sharper. An error criterion is provided to select the order of approximation. The algorithm is tested for various single- and coupled-interconnect scenarios. The 50% delay and overshoot results match that of SPICE with less than 2% average error. The crosstalk results also accurately match those of SPICE with less than 4% average error.

DEPACT: delay extraction-based passive compact transmission-line macromodeling algorithm

With the continually increasing operating frequencies, signal integrity and interconnect analysis in high-speed designs is becoming increasingly important. Recently, several algorithms were proposed for macromodeling and transient analysis of distributed transmission line interconnect networks. The techniques such as method-of-characteristics (MoC) yield fast transient results for long delay lines. However, they do not guarantee the passivity of the macromodel. It has been demonstrated that preserving passivity of the macromodel is essential to guarantee a stable global transient simulation. On the other hand, methods such as matrix rational approximation (MRA) provide efficient macromodels for lossy coupled lines, while preserving the passivity. However, for long lossy delay lines this may require higher order approximations, making the macromodel inefficient. To address the above difficulties, this paper presents a new algorithm for passive and compact macromodeling of distributed transmission lines. The proposed method employs delay extraction prior to approximating the exponential stamp to generate compact macromodels, while ensuring the passivity. Validity and efficiency of the proposed algorithm is demonstrated using several benchmark examples

Efficient sensitivity analysis of lossy multiconductor transmission lines with nonlinear terminations

An efficient approach for sensitivity analysis of lossy multiconductor transmission lines in the presence of nonlinear terminations is described. Sensitivity information is extracted using the recently developed closed-form matrix-rational approximation of the distributed transmission-line model. The method enables sensitivity analysis of interconnect structures with respect to both electrical and physical parameters. An important advantage of the proposed approach is that the derivatives of the modified nodal admittance matrices with respect to per-unit-length parameters are obtained analytically.

A SPICE model for incident field coupling to lossy multiconductor transmission lines

An efficient time-domain macromodel for incident field coupling to lossy multiconductor transmission lines is presented. The model takes the form of ordinary differential equations and can be easily included in SPICE like simulators for transient analysis. The model is based on the closed-form matrix-rational approximation of the exponential matrix describing telegrapher's equations and semi-analytic rational approximation of forcing functions.

A general class of passive macromodels for lossy multiconductor transmission lines

This paper presents a general class of passive macromodeling algorithm for multiport distributed interconnects. A new theorem is described that specifies sufficient conditions for matrix-rational approximation of exponential functions in order to generate a passive macromodel. A proof is given showing that the currently existing passive matrix-rational approximation of exponential functions is a subclass of the generic approach presented in this paper. In addition, a technique to obtain a compact passive macromodel with predetermined coefficients, based on near-optimal approximation, is presented. The proposed model can be easily incorporated with recently developed passive model-reduction techniques.

Critical Points and Error Rank in Best H 2 Matrix Rational Approximation of Fixed McMillan Degree

This paper deals with best rational approximation of prescribed McMillan degree to matrix-valued functions in the real Hardy space of the complement of the unit disk endowed with the Frobenius L2 -norm. We describe the topological structure of the set of approximants in terms of inner-unstable factorizations. This allows us to establish a two-sided tangential interpolation equation for the critical points of the criterion, and to prove that the rank of the error F-H is at most k-n when F is rational of degree k , and H is critical of degree n . In the particular case where k=n , it follows that H=F is the unique critical point, and this entails a local uniqueness result when approximating near-rational functions.