Characteristic equation

Also called: next-state equation, flip-flop equation, Q(t+1) equation, characteristic expression

A Boolean equation giving a latch's or flip-flop's next state Q⁺ in terms of its inputs and present state Q, e.g. Q⁺ = JQ' + K'Q.

A characteristic equation is the characteristic table written as a formula. It gives Q⁺, the next value of Q, as a Boolean function of the inputs and the present Q.

The equations to know:

D flip-flop: Q⁺ = D.
T flip-flop: Q⁺ = .
JK flip-flop: Q⁺ = .
SR flip-flop: Q⁺ = , valid only when S and R are never both 1.
D latch with enable E: Q⁺ = .

How to get one: write the full characteristic table with Q as an input column, then simplify the Q⁺ column like any other function, for example with a K-map.

How to use one: plug in the present inputs and Q to get the next state. It is also the key to flip flop conversion: to make a D flip-flop act like any other type, wire its D input to that type's characteristic equation.

Read an equation in halves by the value of Q. In the JK equation, when Q = 0 only can be 1, so Q⁺ = J. When Q = 1 only can be 1, so Q⁺ = . That is the whole JK table in two lines.

J\KQ	00	01	11	10
0	0m0	1m1	0m3	0m2
1	1m4	1m5	0m7	1m6

Worked examples

Example

Deriving the JK equation from its K-map

Put Q⁺ for each (J, K, Q) on a K-map. Q⁺ = 1 at minterms 1, 4, 5 and 6 (the map above).

1.
Minterm 1 (001): hold with Q = 1. Minterm 5 (101): set with Q = 1. Both stay at 1.
2.
Minterm 4 (100): set from 0. Minterm 6 (110): toggle from 0.
3.
Group 4 and 6 (J = 1, Q = 0, K varies): .
4.
Group 1 and 5 (K = 0, Q = 1, J varies): .
5.
Q⁺ = .

Example

Using an equation to predict Q

A D latch has E = 0, D = 1 and Q = 0. Use Q⁺ = .

1.
ED = 0·1 = 0.
2.
E'Q = 1·0 = 0.
3.
Q⁺ = 0 + 0 = 0. The latch is closed, so it holds its 0 even though D = 1.

Common mistakes

Leaving Q out of the equation for types that hold or toggle. Only the D flip-flop's equation is free of Q.
Using Q⁺ = for S = R = 1. The SR equation is only valid when that input is never applied.
Writing the D latch as Q⁺ = D. Without the enable terms it would be a flip-flop's equation.

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Characteristic equation is taught in Latches and Flip-Flops, Counters and Finite State Machines.

Latches and Flip-Flops →Counters →Finite State Machines →

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