Are Your PLDs Metastable?
4
metastable for an unusually long period of time is zero, how-
ever, due to factors such as noise, temperature imbalance
within the chip, transistor differences, and variance in input
timing. During the second phase of metastability, for very
small deviations around the metastable voltage, V
m
, the
flip-flop behaves like two cross-coupled linear amplifier stag-
es that gain V
d
= V
q
– V
q
. When the gain of the cross-coupled
loop exceeds unity, the differential voltage increases expo-
nentially with time.
The length of time the flip-flop takes to resolve cannot be ex-
actly determined. The probability that the flip-flop will resolve
within a specific length of time, however, can be predicted.
This probability depends on the electrical parameters of the
flip-flop acting as a linear amplifier around the metastability
voltage. The solution (Reference 11) to the differential voltage
V
d
(t) driving the resolving phase is given by
Eq. 2
where t depends directly on the amplifier gain and capaci-
tance, and where V
d
(t
0
) represents the differential voltage at
some time t
0
. You can use this equation to determine the
length of time that the output voltage will take to drift from the
metastable voltage V
m
to a specified voltage difference V
d
.
Horstmann (Reference 5) states that a flip-flop, like any other
system with two stable states, can be described by an energy
function with two local energy minima where P(x) = 0 (Figure
9). Any bistable system has at least one metastable state,
which is an unstable energy level within the system and rep-
resents the local maximum of the energy function. The sys-
tem’s gradient can be represented by a force, F(x), that is zero
at stable and metastable states (inflection points of the energy
function).
Figure 10 shows a simplified first-order model of an RS
flip-flop used to predict and visualize metastability. A flip-flop
energy transfer curve (Figure 11) shows the relationship be-
tween the two outputs. The two stable states are local energy
minima of the system. The metastable state, M, is a local
energy maximum and represents an unstable state with loop
gain near M that is greater than one.
Figure 12 shows the trigger line for the first-order approxima-
tion of the flip-flop. The dashed line RS represents the de-
vice’s normal trigger line, which does not follow the transfer
curve because, during triggering, the feedback loop has not
been established. If at varying points along the trigger line the
feedback loop is re-established, the nodes of the device follow
the curves that lead to the line S
0
– S
1
. Once on this line, the
circuit exponentially drifts toward stability at either S
0
or S
1
,
depending on which side of the line Q = Q
the feedback loop
Figure 8. Two Phases of Metastability
V
d
t() V
d
t
0
()e
tt
0
–
()
τ
----------------
=
Figure 9. Energy/Force Function of a Bistable System
Figure 10. First-Order Flip-Flop
Figure 11. Energy Transfer Diagram of Simple RS Flip-Flop
S
Vin1
V
o
Q
V
o
Q
Vin2
R
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