Understanding metastability in Technion Paper

Question

I am trying to understand metastability as introduced in the Technion IEEE paper (link). But I am struggling a little with some of the concepts, and wanted to ask about that.

This is the flip-flop circuit they are using:

Question: They mention that when the input D to the FF changes right as CLK2 is turning high, it can introduce metastability in the master latch. They present the following graph by playing around with the relative timing about D and CLK2 to show how metastability is introduced in the circuit.

Caption under the image: Empirical circuit simulations of entering metastability in the master latch of Figure 2 (left). Charts show multiple inputs D, internal clock (CLK2) and multiple corresponding outputs Q (voltage vs. time). The input edge is moved in steps of 100ps, 1ps and 0.1fs in the top, middle and bottom charts, respectively.

I do not understand why moving the input edge by different time steps produces such varied plots. The relative input - clock timings are the same, only the time steps are varied. Why does it take the circuit so long to stabilize in the 3rd case compared to the first?

I looked at the literature references they mentioned, and also in my notes, Wiki, university lectures on metastability but to no avail.

Thank you!

Answer 1

Why does it take the circuit so long to stabilize in the 3rd case compared to the first?

The 3rd case is much closer to the balance point of this particular gate. The search time step determines how close the closest curve will be.

As an analogy:

You are trying to balance a ball on a hill, but don't know where the hill is. You try different points at 1 meter interval. Some balls roll left, some balls roll right. Then you try points every 1 cm in the interval where the direction changed. Finally, you try every 1 mm and find a spot where the ball stays balanced.

Theoretically there exists a point where the flip-flop will stay metastable for infinite time. But in practice there is always noise and fluctuations that will eventually push it to one side or the other.

Answer 2

They are trying to show you that meta stability is essentially a chaotic process, i.e. a butterfly flaps its wings here and a tornado happens 1000km away.

They are showing you that as long as the timing requirements of the FF are violated, i.e. setup-time violation, that is input edge is too near the clock edge, then even a slight variation in the timing can provoke big changes in the output.

In the first graph they show that a difference of 100ps between steps causes differences in the output, as expected, but as the steps become smaller, there is a part of the output that remains the same, but still the output final state cannot be predicted, even if the steps are extremely close together. No matter how small a variation in input timing, the output end state cannot be predicted and can be either 0 or 1, and the time to settle to that state increases.