Page 746 - Engineering Digital Design

712 CHAPTER 14/ASYNCHRONOUS STATE MACHINE DESIGN AND ANALYSIS

•| st 2 nd
level level
1st 2nd 9 ates Sates
level level
gates gates Direct path
Direct path X -f—' At E j—>/T 1 }—i/RoV—• y a
st
Initiator VJ/^^ar^.y 1 Invariant
x —pJ At E j-x RG w-w ) y a
st
Initiator ^^''^^-^ ^-^ 1 Invariant
s^t \ 2|
I
^^^^^^\
nd
2 Invariant t i Irtdireet ""^-4 T 3H v b
nd
i path Kjy 2 Invariant
_ <*, *~ *•
l^Wnyrt, —*
(a) (b)
FIGURE 14.26
Illustrations of the path delay requirements for E-hazard formation in two-level logic showing causal
delays AtE, initiator input X, first and second invariants, gate delays r,, race gates (RG), and correction
delays to eliminate the E-hazard. (a) First-level race gate, (b) Second-level race gate.

In both Eqs. (14.16) and (14.17) the quantity A? £ is the asymmetric path delay, shown in
Fig. 14.26, that is required to cause the E-hazard to form (yb wins the race); T, are the path
delays associated with the gates (including any inverters) and leads. In these equations the
correction delay At correct is assumed to be zero. If a counteracting delay A.t correct is added
in the indicated feedback path of the 2nd invariant, then the requirements for eliminating
the E-hazard are given by

(TI + T 2 + &t correct), (14.18)

for the first-level race gate, and
(r 2 + T 3 + r 4 + At correct\ (14.19)

for the second-level race gate. Thus, if At E is of sufficient magnitude to cause an E-
hazard to become active according to the requirements of Eqs. (14.16) and (14.17), the
second invariant yb wins the race and the FSM is guaranteed to malfunction. However,
if a counteracting (correcting) delay is added in the feedback path of the 2nd y-variable
invariant, the inequality is reversed, as in Eqs. (14.18) and (14.19). Under this condition,
the initiator X wins the race and the E-hazard is eliminated.
The minimum requirements for E-hazard formation are summarized in Fig. 14.27. The
state diagram segment, shown in Fig. 14.27a, specifies the first- and second-level race gate
SOP terms that must be contained in the first invariant function 7, before an E-hazard is
possible. Notice that the first invariant is the second y-variable to change while the second
invariant is the first to change. The minimum requirements for E-hazard formation are
continued in Fig. 14.27b, where now another type of E-hazard is identified, the d-trio. The