Efficient Metastability Characterization for Schmitt-Triggers

Despite their attractiveness as metastability filters, Schmitt-Triggers can suffer from metastability themselves. Therefore, in the selection or construction of a suitable Schmitt-Trigger implementation, it is a necessity to accurately determine the metastable behavior. Only then one is able to compare different designs and thus guide proper optimizations, and only then one can assess the potential for residual metastable upsets. However, while the state of the art provides a lot of research and practical characterization approaches for flip-flops, comparatively little is known about Schmitt-Trigger characterization. Unlike the flip-flop with its single metastable point, the Schmitt-Trigger exhibits a whole range of metastable points depending on the input voltage. Thus the task of characterization gets much more challenging. In this paper we present different approaches to determine the metastable behavior of Schmitt-Triggers using novel methods and mechanisms. We compare their accuracy and runtime by applying them to three common circuit implementations. The achieved results are then used to reason about the metastable behavior of the chosen designs which turns out to be problematic in some cases. Overall the approaches proposed in this paper are generic and can be extended beyond the Schmitt-Trigger, i.e., to efficiently characterize metastable states in other circuits as well.

I. Introduction

To use digital abstraction in electronic circuits we have to “digitize” an essentially analog input, i.e. either assign logic HI or LO depending on whether it is above or below a threshold. In order to prevent oscillation of the output due to noise in case of an input voltage close to the threshold - like in case of a comparator circuit -, the Schmitt-Trigger (S/T) uses a higher threshold for rising transitions than for falling ones, leading to a hysteresis (blue lines in Fig. 1). This, however, directly translates into a dependence of the threshold on the current output state, which, in turn, implies a positive feedback from the output to the input. As a consequence, the S/T must be susceptible to metastability. This intuitive argument has been more formally supported by Marino [1] already, and more recently Steininger et al. [2] have detailed several practically relevant scenarios where metastability may occur and where it may not. While there exist analytic solutions to calculate certain properties such as the threshold voltages [3], none have been presented so far regarding metastability. Fig. 1:

Stable and metastable states of a latch (dots) and an s/t (lines) forming the characteristic z-shaped curve.