D and T flip-flops; T to JK conversion

D flip-flop

The output copies the D input at each active clock edge: $Q_{n+1} = D$.

 D --+------------>+------+  S*  +------+
     |             | NAND |----->| NAND |---> Q
     |   CLK --+-->|  G1  |  Q'->|  G3  |
     |         |   +------+      +------+
     |         |   +------+  R*  +------+
     |         +-->| NAND |----->| NAND |---> Q'
     +--[NOT]----->|  G2  |  Q-->|  G4  |
                   +------+      +------+
 S = D and R = D' : the forbidden S=R=1 never occurs

T flip-flop

The T (toggle) flip-flop holds its state for T = 0 and complements it for T = 1: $Q_{n+1} = T \oplus Q_n$. It is made from a J-K flip-flop with J and K tied together.

          +---------+
 T --+----| J     Q |--- Q
     |    |         |
     | CLK|>  JK    |
     |    |         |
     +----| K    Q' |--- Q'
          +---------+

T flip-flops are used in counters and as divide-by-2 circuits (T = 1).

Converting T flip-flop into J-K flip-flop

Step 1: excitation table of T flip-flop (T = 1 whenever the state must change)

Step 2: conversion table. For each J, K and Q, take Q+ from the J-K characteristic table and write the T needed:

Step 3: K-map for T

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

• Group Q = 0, JK = 11 and 10: $J\overline{Q}$.
• Group Q = 1, JK = 01 and 11: $KQ$.

$$T = J\,\overline{Q} + K\,Q$$

Step 4: logic circuit

          +-----+
 J ------>|     |
 Q' ----->| AND |--+
          +-----+  |  +----+     +---------+
                   +->|    |     |         |
                      | OR |---->| T     Q |---> Q
                   +->|    |CLK->|>   T    |
          +-----+  |  +----+     |      Q' |---> Q'
 K ------>|     |  |             +---------+
 Q  ----->| AND |--+
          +-----+
 (Q and Q' are fed back from the outputs)

Check: J = K = 1 gives $T = \overline{Q} + Q = 1$ (toggle); J = K = 0 gives T = 0 (hold); J = 1, K = 0 gives $T = \overline{Q}$, so a 0 toggles to 1 and a 1 holds (set); J = 0, K = 1 gives T = Q (reset).