# Simulation with Bounds

Chapter

## Abstract

There is a well known law of nature, attributed to someone named Murphy, that is often cited in jest to explain unfortunate events. By this law, the most pessimistic predictions of the future always come to pass. Although this law may appear to have some validity at times, due to human psychology, fortunately our world is not exclusively governed by it^{2}.

## Keywords

Circuit Model Logic Gate Digital Circuit Ring Oscillator Coupling Capacitor
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## References

- 2.The completion of this book serves as a counterexample to Murphy’s law [38].Google Scholar
- 3.The quotes emphasize the fact that, while the time derivative plays its normal role in the circuit, it need not actually be the derivative of the corresponding signal in the excitation.Google Scholar
- 4.These logic gates must have d.c. output voltages that are roughly monotonic functions of their input voltage vectors, e.g., NAND gates but not EXCLUSIVE-OR (XOR) gates.Google Scholar
- 5.Recall that a signal waveform has a unique derivative, and the two arc therefore strongly correlated.Google Scholar
- 6.Again the quotes indicate that corresponding time derivatives and signals need not match in a circuit behavior, even though the time derivative plays the role of a time derivative in the circuit.Google Scholar
- 7.If the original bound is a continuous function, the optimal monotonic bounds as defiined here are as well.Google Scholar
- 8.In the context of bounding algorithms, a known input is one that is restricted to a known interval.Google Scholar
- 9.The simple relationships listed here require all voltages to fall within the range of [O,VI)D].Google Scholar
- 10.Relaxation is actually a class of algorithms. If the latest guesses for each variable are always used in a sequential algorithm, as in the example here, it is called Gauss-Seidel relaxation.Google Scholar
- 11.In general, as long as a bound is self-consistent through all feedback paths among blocks, it is an “optimal” bound given the information provided by the bounds used for the behavior of each block.Google Scholar
- 12.Note that, oddly enough, it is positive feedback that is desirable in the bounding context.Google Scholar
- 13.By slight changes in the bounds on each block that are pictured in figure 3–11 for
**a= -0.5**one can form a number of distinct “rectangles” that are self-consistent. Each one is still a valid bound on the entire system, though.Google Scholar - 14.The effect of ignoring correlations among device characteristics is classified as
*conservative bound specifiication.*Only the effect of ignoring correlations between two instances of the same device, used for bound generation, is considered in this section.Google Scholar - 15.As explained further in chapter four, an upper bound on v
_{B}is produced, roughly speaking, from an upper bound on the current entering the load capacitor through i_{L}=C_{L}v_{B}. An upper bound on i_{L}can be calculated by assuming v_{B}is at its lower bound.Google Scholar - 16.A precharged logic gate is a fairly common subcircuit that contains a dynamic node, but since it is restored once each clock cycle, it is not overly sensitive to uncertainty in leakage current.Google Scholar
- 17.These time windows should have a duration that is roughly equal to the time that it takes a signal to pass once around the feedback loop.Google Scholar
- 18.Even if A and B were not monotonic in x, they could be bounded by trying many values for x, as they represent a device model that need only be evaluated once for each new fabrication process. More generally, correlations among variables inside a device model can be easily incorporated because computation is not a limiting resource in bounding device models, only in bounding circuit behaviors.Google Scholar
- 19.Waveforms that first’rise and then fall are possible in combinational logic circuits but are not considered here for simplicity. They can be incorporated with the decomposition pictured in figure 3–5.Google Scholar

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© Springer Science+Business Media New York 1986