Made Easy Test Series:Flip-Flop

Question

A Finite State Machine(FSM) is implemented using the D-FFs A and B with logic gates as shown below. The four possible states of FSM are $Q_{A}Q_{B}=00,01,10,11$.

Assume that $X_{in}$ is held at constant logic level throughout the operation of FSM. Where the FSM is initialized to the $Q_{A}Q_{B}=00$ and clocked, after a few clock cycle, it starts cycling through.

$A)$ All of the four possible states if $X_{in}=1$

$B)$ Only two of the four possible states if $X_{in}=0$

$C)$ Only two of the four possible states if $X_{in}=1$

$D)$ All of the four possible states if $X_{in}=0$

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How do we check $X_{in}$ here? Can we check it arbitrarily, or checked with prev states??

Comments

Ans:-> B

We will approach this problem by trial and error.

See that D(a) = Q(a) XOR Q(b)

and D(b) = complement( Q(a) AND X(in) )

Now put Xin = 0 and test starting from  0 0.

We will have only 2 states --> 0 1 and 1 1

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@Hirak

Why only 2 states?? explain more

Answer 1

There are two cases, first if $X_{in}=0$ and second if $X_{in}=1$ And its given that state of $X_{in}$ does not change in the middle of its working.

 

Case 1 : $X_{in}=0$

If one input of NAND gate is 0, then its output will be always 1. Hence in this case, $D_B$ would always receive $1$ as input. We should make truth table accordingly as below :

Clock	$Q_B$	$Q_A$	$D_B=1$	$D_A=Q_A \bigoplus Q_B$ (calculated from previous states)
0 (initial)	0	0	1	0
1	1	0	1	1
2	1	1	1	0
3	1	0	1	1
4	1	1

We can see that the output $Q_BA_A$ "after a few clock cycles" is like 10-11-10-11 and so on. Only two states $10$ and $11$ are in cycle.

Now see the options B and D. Option B matches here. But still we will take case 2 also to check what happens :

Case 2 : $X_{in}=1$

If one input of NAND gate is 1, then its output will be compliment of other input. Hence in this case, $D_B$ would always receive $\bar{Q_A}$ as input. We should make truth table accordingly as below :

Clock	$Q_B$	$Q_A$	$D_B=\bar{Q_A}$ (calculated from previous states)	$D_A=Q_A \bigoplus Q_B$ (calculated from previous states)
0 (initial)	0	0	1	0
1	1	0	1	1
2	1	1	0	0
3	0	0	1	0
4	1	0	1	1
5	1	1	0	0
6	0	0

We can see that the output $Q_BA_A$ 00-10-11-00-10-11 and so on. Three states $00$, $10$ and $11$ are in cycle.

Check the options A and C, Both are incorrect.

 

Hence considering both Cases, Correct answer is OPTION B.