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14.10 DETECTION AND ELIMINATION OF TIMING DEFECTS 713

I — > I . in state a
a ab
~ ab
ab
st
1 Level
Race Gate x x
if 'bc=- i r
x. = Initiator input
Only a single change in the initiator x. is
allowed in f gb and f bcwith x and all other
y i = First invariant
Path to inputs held constant.
nd
y. = Second invariant 2 Level
Race Gate For E-hazard formation

For D-trio formation
S
'ab *cb
(b)
FIGURE 14.27
Minimum requirements for first-order E-hazard and d-trio formation in two-level SOP logic, (a) State
diagram segment showing first- and second-level race gate requirements, only one of which will be
met in the first-invariant function F,. (b) Minimum requirements for E-hazard and d-trio formation
indicating assumed input conditions for l ab and l bc-

d-trio (delay-trio) is a special E-hazard that returns the FSM to the intended state but only
following a second (error) transition to another state. Thus, the transition path for a d-trio
is a -» b ->• c -> b, while that for a E-hazard isa^b^-cora^b-*c^-x, where
state x lies beyond state c in Fig. 14.27a. The latter E-hazard path is possible if the input
conditions are such that I af, c f cx in addition to those indicated in Fig. 14.27b. Clearly,
the minimum requirements are the same for the E-hazard and d-trio formation, except the
E-hazard does not return the FSM to the intended next state. Another important minimum
requirement for E-hazard and d-trio formation is that the initiator jc, is permitted to have
only one change in f ab and f\, c while holding Xj and all other inputs constant.
To summarize, an E-hazard or d-trio can form iff an unintended asymmetric delay A/£
of sufficient magnitude is explicitly located as shown in Fig. 14.26, and if the minimum
requirements indicated in Fig. 14.27 are met. A cursory check of the state diagram is all that
is necessary to show whether or not the minimum requirements for E-hazard (and d-trio)
formation are met. If they are not met, these potential defects cannot form and no further
analysis is necessary. If the minimum requirements are met, the second stage of the analysis
is to determine the requirements for the indirect path — that is, the requirements to allow
the second j-variable invariant to win the race at the race gate.
Only first-order E-hazards are considered in this text. The reason is that second and higher
order E-hazards are far less likely to be activated than first-order E-hazards. A second-order
E-hazard, for example, requires two successive invariants in the indirect path (IP), which
greatly increases the minimum path delay requirement for activation of the E-hazard.