State Equation

A state equation (or transition equation) is a Boolean function expressing the next state of a Sequential circuit in terms of the current state and the inputs.

If a flip-flop holds bit $A$ of the state, its state equation is the Boolean function $A(t+1)=f(\text{current state},\text{inputs})$. With one equation per flip-flop, you have a complete description of the FSM’s transition behavior.

Equivalent to a State table in algebraic rather than tabular form. State equations are typically derived from the state table by reading off each $0/1$ entry of the next-state column and minimizing with a Karnaugh Map.

Example

For a circuit with state bits $A$ and $B$, output $Y$, and input $X$, with D flip-flops (where $D=$ next-state value):

The state equations might look like:

$A(t+1)=AX+BX=X(A+B)$

$B(t+1)=\overline{A}X$

And the output equation:

$Y(t+1)=\overline{X}(A+B)$

The equations describe what each flip-flop’s $D$ input should compute. The first equation says “the next $A$ is $1$ when $X=1$ and either $A$ or $B$ is currently $1$.” The second says “the next $B$ is $1$ when the current $A$ is $0$ and $X=1$.”

Where state equations come from

Two paths:

1. From the state table. For each next-state bit, write down the rows where that bit becomes $1$, then minimize with a K-map to get a compact Boolean expression.

2. Directly from a state diagram. For each transition from state $s$ to state $s^{\prime }$ on input $i$, the bits of $s^{\prime }$ contribute terms to the state equations.

For D flip-flops, the state equation is the input equation — $D=A(t+1)$. For T flip-flops, you’d derive a $T$ input by XORing the current and next states. For JK, use the JK excitation table to find which $J/K$ values produce each desired transition.

Once you have state equations, the implementation is direct: each $D$ input becomes a combinational circuit that computes its equation from the state outputs and the inputs. See Sequence detector (FSM design example) for the full flow worked through.