JK characteristic equation and toggle mode

Characteristic equation of J-K flip-flop

The characteristic table lists the next state $Q_{n+1}$ for every combination of J, K and present state $Q_n$:

(J K = 00 hold, 01 reset, 10 set, 11 toggle.)

Plotting $Q_{n+1}$ on a 3-variable K-map:

K-map for Q+ (rows Q, columns JK)
Q\JK   00  01  11  10
 0      0   0   1   1
 1      1   0   0   1
 Groups: J.Q' (row 0, cells 11,10)
         K'.Q (row 1, cells 00,10)

• Row Q = 0, columns JK = 11 and 10 give $J\,\overline{Q_n}$.
• Row Q = 1, columns JK = 00 and 10 give $\overline{K}\,Q_n$.

$$Q_{n+1} = J\,\overline{Q_n} + \overline{K}\,Q_n$$

Making a toggle (T) flip-flop

Connect J and K together and call the common input T (or tie J = K = 1 for permanent toggling).

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

With J = K = T, the equation becomes

$$Q_{n+1} = T\,\overline{Q_n} + \overline{T}\,Q_n = T \oplus Q_n$$

so T = 0 holds the state and T = 1 toggles it at every clock edge (a divide-by-2 circuit when T = 1).

Input and output waveforms (positive edge triggered, Q initially 0)

       1   2   3   4   5   6   7
CLK   _|‾|_|‾|_|‾|_|‾|_|‾|_|‾|_|‾|
J     ‾‾‾|___|‾‾‾‾‾‾‾|___|‾‾‾‾‾‾‾|
K     _______|‾‾‾‾‾‾‾‾‾‾‾|___|‾‾‾|
Q     _|‾‾‾‾‾‾‾|___|‾‾‾|___|‾‾‾|__