Synchronous vs asynchronous circuits; SR to JK

Synchronous vs asynchronous sequential circuits

Converting S-R flip-flop into J-K flip-flop

Step 1: excitation table of the available S-R flip-flop

Step 2: conversion table. For each J, K and present state Q, find Q+ from the J-K characteristic table, then write the S and R needed for that change:

Step 3: K-maps

K-map for S
Q\JK   00  01  11  10
 0      0   0   1   1
 1      x   0   0   x

K-map for R
Q\JK   00  01  11  10
 0      x   x   0   0
 1      0   1   1   0

• S: the 1s are at Q = 0, JK = 11 and 10; the don't-cares at Q = 1 cannot enlarge this group (Q = 1, JK = 11 is 0), so $S = J\overline{Q}$.
• R: the 1s are at Q = 1, JK = 01 and 11, so $R = KQ$.

$$S = J\,\overline{Q}, \qquad R = K\,Q$$

Step 4: circuit

            +-----+      +---------+
 J -------->|     |      |         |
 Q' ------->| AND |----->| S     Q |---> Q
            +-----+      |         |
            +-----+ CLK->|>  SR    |
 K -------->|     |      |         |
 Q  ------->| AND |----->| R    Q' |---> Q'
            +-----+      +---------+
 (Q and Q' are fed back from the outputs)

Check: J = K = 1 with Q = 0 gives S = 1, R = 0 (Q becomes 1); with Q = 1 gives S = 0, R = 1 (Q becomes 0). So the circuit toggles, and S = R = 1 can never occur because S needs Q = 0 while R needs Q = 1.