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716 CHAPTER 14/ASYNCHRONOUS STATE MACHINE DESIGN AND ANALYSIS

Fig. 14.27:

AB^AB, AB = I cb ^f b = A, AB = I ba c f c = AB, I cb£f ab.

Only a single change in the initiator is indicated in c —> b and b —>• a with B held constant.
Here, state 11 (state c) is the origin state, A is the initiator input, the intended path is
11 -> 01 (c —> b), the E-hazard (error) path is 11 -> 01 ->• 00, y 0 is the first invariant,
and y\ is the second invariant.
The remaining requirements for E-hazard formation in the FSM of Fig. 14.28a are
obtained from Eqs. (14.20) and Fig. 14.29 in accordance with Fig. 14.27 and the indirect
path (IP) requirements given previously.
1. A delay At£ placed on the initiator A input to the first invariant jo causes a critical
race to the race gate RG E between the initiator A and the second invariant y\. If Ar £
exceeds the minimum path delay requirements, y\ wins the race and the E-hazard is
formed. If A?E is not of sufficient magnitude, the initiator input A wins the race and
no E-hazard will form.
2. The path to the ORing (2nd level) race gate (RG E) is indicated by the term yoA in YQ,
as shown in Eqs. (14.20). No ANDing RG is possible according to Figs. 14.27a and
14.28a.
3. The indirect path (!PE) must not be inconsistent with B, y\, yo in Y\ and must contain
y\ or y \ and A or A in YQ. Therefore, the IP is via the term y\ A in Y\, and either y\ A
or y\B in YQ.
4. Based on the foregoing and with reference to Fig. 14.29, the theoretical minimum
path delay requirements for E-hazard formation is given by the inequalities

(Af £ + T 7) > (r 2 + T 5 + TIO + T 8) = (TINV + 3r p)

or

A? £ > (r, NV + 2r p), (14.22)
where r p is the path delay through a gate (e.g., a two- or three-input NAND gate),
and T/NV = ?2 is the path delay through an inverter. Thus, Eq. (14.22) does not take
into account the gate path delay dependence on fan-in.

D-trio Analysis With reference to the FSM in Fig. 14.28a and Eqs. (14.21), the following
constitute the minimum requirements for d-trio formation as set forth in Fig. 14.27:
AB^AB, AB = I ad ^f d = AB, AB = I dc c f a = B, I ad^f cd = AB,

and only a single change of the initiator is indicted in a —> b and b —>• c with A constant.
In this case state 00 (state a) is the origin state, B is the initiator, the intended path is
00 —> 10 (a —> b), the d-trio (error) path is 00 —> 10 —>• 11 -»• 10, jo is the first invariant, and
ji is the second invariant.
The remaining requirements for d-trio formation in the FSM of Fig. 14.28a are obtained
from Eqs. (14.21), from Figs. 14.27 and 14.29, and from the IP path requirements given
previously.