Closed Form Expressions for Delay to Ramp Inputs for On-Chip VLSI RC Interconnect

In high speed digital integrated circuits, interconnects delay can be significant and should be included for accurate analysis. Delay analysis for interconnect has been done widely by using moments of the impulse response, from the explicit Elmore delay (the first moment of the impulse response) expression, to moment matching methods which creates reduced order trans impedance and transfer function approximations. However, the Elmore delay is fast becoming ineffective for deep submicron technologies, and reduced order transfer function delays are impractical for use as early-phase design metrics or as design optimization cost functions. This paper describes an approach for fitting moments of the impulse response to probability density functions so that delay can be estimated accurately at an early physical design stage. For RC trees it is demonstrated that the inverse gamma function provides a provably stable approximation. We used the PERI [13] (Probability distribution function Extension for Ramp Inputs) technique that extends delay metrics for ramp inputs to the more general and realistic non-step inputs. The accuracy of our model is justified with the results compared with that of SPICE simulations. Keywords¾ Moment Matching, On-Chip Interconnect, Probability Distribution function, Cumulative Distribution function, Delay calculation, Slew Calculation, Beta Distribution, VLSI

Fast high-order variation-aware IC interconnect analysis

Interconnects constitute a dominant source of circuit delay for modern chip designs. The variations of critical dimensions in modern VLSI technologies lead to variability in interconnect performance that must be fully accounted for in timing verification. However, handling a multitude of inter-die/intra-die variations and assessing their impacts on circuit performance can dramatically complicate the timing analysis. In this thesis, three practical interconnect delay and slew analysis methods are presented to facilitate efficient evaluation of wire performance variability. The first method is described in detail in Chapter III. It harnesses a collection of computationally efficient procedures and closed-form formulas. By doing so, process variations are directly mapped into the variability of the output delay and slew. This method can provide the closed-form formulas of the output delay and slew at any sink node of the interconnect nets fully parameterized, in-process variations. The second method is based on adjoint sensitivity analysis and driving point model. It constructs the driving point model of the driver which drives the interconnect net by using the adjoint sensitivity analysis method. Then the driving point model can be propagated through the interconnect network by using the first method to obtain the closedform formulas of the output delay and slew. The third method is the generalized second-order adjoint sensitivity analysis. We give the mathematical derivation of this method in Chapter V. The theoretical value of this method is it can not only handle this particular variational interconnect delay and slew analysis, but it also provides an avenue for automatical linear network analysis and optimization. The proposed methods not only provide statistical performance evaluations of the interconnect network under analysis but also produce delay and slew expressions parameterized in the underlying process variations in a quadratic parametric form. Experimental results show that superior accuracy can be achieved by our proposed methods

A fast and retargetable framework for logic-IP-internal electromigration assessment comprehending advanced waveform effects

A new methodology for system-on-chip-level logic-IP-internal electromigration verification is presented in this paper, which significantly improves accuracy by comprehending the impact of the parasitic RC loading and voltage-dependent pin capacitance in the library model. It additionally provides an on-the-fly retargeting capability for reliability constraints by allowing arbitrary specifications of lifetimes, temperatures, voltages, and failure rates, as well as interoperability of the IPs across foundries. The characterization part of the methodology is expedited through the intelligent IP-response modeling. The ultimate benefit of the proposed approach is demonstrated on a 28-nm design by providing an on-the-fly specification of retargeted reliability constraints. The results show a high correlation with SPICE and were obtained with an order of magnitude reduction in the verification runtime.Peer ReviewedPostprint (author's final draft

Assessment of 50%-Propagation-Delay for Cascaded PCB Non-Linear Interconnect Lines for the High-Rate Signal Integrity Analysis

This paper presents an enlarged study about the 50-% propagation-time assessment of cascaded transmission lines (TLs). First and foremost, the accurate modeling and measurement technique of signal integrity (SI) for high-rate microelectronic interconnection is recalled. This model is based on the reduced transfer function extracted from the electromagnetic (EM) behavior of the interconnect line RLCG-parameters. So, the transfer function established takes into account both the frequency dispersion effects and the different propagation modes. In addition, the transfer function includes also the load and source impedance effects. Then, the SI analysis is proposed for high-speed digital signals through the developed model. To validate the model understudy, a prototype of microstrip interconnection with w = 500 µm and length d = 33 mm was designed, simulated, fabricated and tested. Then, comparisons between the frequency and time domain results from the model and from measurements are performed. As expected, good agreement between the S-parameters form measurements and the model proposed is obtained from DC to 8 GHz. Furthermore, a de-embedding method enabling to cancel out the connectors and the probe effects are also presented. In addition, an innovative time-domain characterization is proposed in order to validate the concept with a 2.38 Gbit/s-input data signal. Afterwards, the 50-% propagation-time assessment problem is clearly exposed. Consequently an extracting theory of this propagation-time with first order RC-circuits is presented. Finally, to show the relevance of this calculation, propagation-time simulations and an application to signal integrity issues are offered

A Fast Symbolic Computation Approach to Statistical Analysis of Mesh Networks with Multiple Sources *

Abstract-Mesh circuits typically consist of many resistive links and many sources. Accurate analysis of massive mesh networks is demanding in the current integrated circuit design practice, yet their computation confronts numerous challenges. When variation is considered, mesh analysis becomes a much harder task. This paper proposes a symbolic computation technique that can be applied to the moment-based analysis of mesh networks with multiple sources. The variation issues are easily taken care of by a structured computation mechanism, which can naturally facilitate sensitivity based analysis. Applications are addressed by applying the computation technique to a set of mesh circuits with varying sizes

Minimizing structural vibrations with Input Shaping (TM)

A new method for commanding machines to move with increased dynamic performance was developed. This method is an enhanced version of input shaping, a patented vibration suppression algorithm. This technique intercepts a command input to a system command that moves the mechanical system with increased performance and reduced residual vibration. This document describes many advanced methods for generating highly optimized shaping sequences which are tuned to particular systems. The shaping sequence is important because it determines the trade off between move/settle time of the system and the insensitivity of the input shaping algorithm to variations or uncertainties in the machine which can be controlled. For example, a system with a 5 Hz resonance that takes 1 second to settle can be improved to settle instantaneously using a 0.2 shaping sequence (thus improving settle time by a factor of 5). This system could vary by plus or minus 15% in its natural frequency and still have no apparent vibration. However, the same system shaped with a 0.3 second shaping sequence could tolerate plus or minus 40% or more variation in natural frequency. This document describes how to generate sequences that maximize performance, sequences that maximize insensitivity, and sequences that trade off between the two. Several software tools are documented and included

Lagrangian relaxation-based multi-threaded discrete gate sizer

In integrated circuit design gate sizing is one of the key optimization techniques which is repeatedly invoked to trade-off delays for area and/or power of the gates during logic design and physical design stages. With increasing design sizes of a million gates and larger, discrete gate sizes and non-convex delay models the gate sizing algorithms that were designed for continuous sizes and convex delay models are slow and timing inaccurate. Of the several published discrete gate sizing algorithms, recent works have shown that Lagrangian relaxation based gate sizers have produced designs with the lowest power on average with high timing accuracy. But they are also very slow due to a large number of expensive timing updates spread across hundreds of iterations of solving the Lagrangian sub-problem. In this thesis we present a Lagrangian relaxation based multi-threaded discrete gate sizer for fast timing and power reduction by swapping the gate sizes and the threshold voltages. We developed two parallelization enabling techniques to reduce the runtime of Lagrangian sub-problem solver, namely, mutual exclusion edge (MEE) assignment and directed acyclic graph (DAG) based netlist traversal. MEEs are dummy edges assigned to reduce computational dependencies among gates sharing one or more common fan-ins. DAG based netlist traversal facilitates simultaneous resizing of gates belonging to different topological levels. We designed a Lagrange multiplier update framework that enables rapid convergence of the timing recovery and power recovery algorithms. To reduce the runtime of timing updates, we proposed a simple and fast-to-compute effective capacitance model and several mechanisms to calibrate the timing models to improve their accuracy. Compared to the state-of-the-art gate sizer, our proposed gate sizer is on average 15x faster and the optimized designs have only 1.7\% higher power. In digital synchronous designs simultaneous gate sizing and clock skew scheduling provides significantly more power saving. We extend the gate sizer to simultaneously schedule the clock skew. It can achieve an average of 18.8\% more reduction in power with only 20\% increase in the runtime