In the same way that combinatorial circuits are generalizations of
gates, sequential circuits are generalizations of
flipflops.
In general, we define a synchronous sequential circuit,
or just sequential circuit as a circuit with m
inputs, n outputs, and a distinguished clock input.
The description of the circuit is made with the help of a state
table.
We have intentionally simplified our definition of sequential
circuit. With our definition, the number of different states of the
circuit is completely determined by the number of outputs. A more
general definition separates the concept of output and the concept
of state. Four our purposes, the additional generality is
not needed, and we stall stick with our simple definition.
The same "goodness" criteria apply to the design of sequential
circuits as to combinatorial circuits, i.e., number of transistors,
speed, power consumption etc. Thus, as with combinatorial circuits,
we are not going to discuss methods for obtaining optimal circuits,
but only a very general method that in the worst case may waste a
large number of tansistors. However, the purpose is not to turn the
students into circuit designers, but to give them an idea of how
sequential circuits work.
For a sequential circuit with m inputs and n
outputs, our method uses n Dflipflops (one for
each output), and a combinatorial circuit with m + n inputs
and n outputs. Here is the general structure of the
resulting circuit:
Since we are using Dflipflops, the outputs of the
circuit after the next clock pulse is exactly the same as the output
of the combinatorial circuit. We can therefore use our general
method for building combinatorial circuits, this time applied to the
state table of the sequential circuit.
As an example, let us construct a 2bit counter with an input
indicating whether to count up or down (0 means down
and 1 means up). The counter should stop at 00
when counting down and at 11 when counting up. Here is
the state table for such a counter:
u/d y1 y0  y1' y0'

0 0 0  0 0
0 0 1  0 0
0 1 0  0 1
0 1 1  1 0
1 0 0  0 1
1 0 1  1 0
1 1 0  1 1
1 1 1  1 1
With our general method, all we need to do is to build a
sequential circuit from the truth table corresponding exactly to
this state table. Here is the resulting circuit:
