# Extra Problem 2 Solution

See the following figure:

First obtain the output functions for the S-R latch (top figure). Break the feedback loop at Q1; the output functions are thus

```Q1+ = (S Q2)'
= S' + Q2'
= S' + Q1 R
Q2  = (Q1 R)'
= Q1' + R'
```

or you can examine the truth table:

``` S  R  |  Q1+ Q2
-------+---------
0  0  |   1   1
0  1  |   1   0
1  0  |   0   1
1  1  |  Q1  Q1'
```

Note that we must be careful about the output Q2; it isn't necessarily the complement of Q1 !

Now look at the bottom figure; this is the given circuit with the S-R latches replacing the cross-coupled NANDs. Substituting using the above characteristic equations for the S-R latches, we get:

```Y1+ = S1' + R1 Y1
= X1 + Y4 Y1
= X1 + (Y2' + R2') Y1
= X1 + (Y2' + X2) Y1
= X1 + Y2' Y1 + X2 Y1
Y2+ = S2' + R2 Y2
= (X1' Y3)' + X2' Y2
= X1 + Y3' + X2' Y2
= X1 + (Y1' + R1')' + X2' Y2
= X1 + Y1 Y4 + X2' Y2
= X1 + Y1 (Y2' + R2') + X2' Y2
= X1 + Y1 Y2' + Y1 X2 + X2' Y2
```

The flow table looks like: (asterisks show stable positions)

```(Y1+ Y2+)

(X1 X2)
00  01  11  10

(Y1 Y2) 00   00* 00* 11  11
01   01* 00  11  11
11   01  11* 11* 11*
10   11  11  11  11
```

Note that there is a race condition (as (Y1 Y2) goes from (00) to (11)), but it is not critical.