Analysis of Sequential Circuits
The behaviour of a sequential circuit is determined from the inputs, the outputs and the states of its flipflops. Both the output and the next state are a function of the inputs and the present state.
The suggested analysis procedure of a sequential circuit is set out in Figure 6 below.
Figure 6. Analysis procedure of sequential circuits.
We start with the logic schematic from which we can derive excitation equations for each flipflop input. Then, to obtain nextstate equations, we insert the excitation equations into the characteristic equations. The output equations can be derived from the schematic, and once we have our output and nextstate equations, we can generate the nextstate and output tables as well as state diagrams. When we reach this stage, we use either the table or the state diagram to develop a timing diagram which can be verified through simulation.
This example is taken from D. D. Gajski, Principles of Digital Design, Prentice Hall, 1997, p.230.
Example 1.1. Modulo4 counter
Derive the state table and state diagram for the sequential circuit shown in Figure 7.
Figure 7. Logic schematic of a sequential circuit. 
SOLUTION:
STEP 1: First we derive the Boolean expressions for the inputs of each flipflops in the schematic, in terms of external input Cnt and the flipflop outputs Q1 and Q0. Since there are two D flipflops in this example, we derive two expressions for D1 and D0:
D1 = Cnt'*Q1 + Cnt*Q1'*Q0 + Cnt*Q1*Q0' 
These Boolean expressions are called excitation equations since they represent the inputs to the flipflops of the sequential circuit in the next clock cycle.
STEP 2: Derive the nextstate equations by converting these excitation equations into flipflop characteristic equations. In the case of D flipflops, Q(next) = D. Therefore the next state equal the excitation equations.
Q0(next) = D0 = Cnt'*Q0 + Cnt*Q0' 
Q1(next) = D1 = Cnt'*Q1 + Cnt*Q1'*Q0 + Cnt*Q1*Q0' 
STEP 3: Now convert these nextstate equations into tabular form called the nextstate table.
Present State
 Next State
 


Each row is corresponding to a state of the sequential circuit and each column represents one set of input values. Since we have two flipflops, the number of possible states is four  that is, Q1Q0 can be equal to 00, 01, 10, or 11. These are present states as shown in the table.
For the next state part of the table, each entry defines the value of the sequential circuit in the next clock cycle after the rising edge of the Clk. Since this value depends on the present state and the value of the input signals, the next state table will contain one column for each assignment of binary values to the input signals. In this example, since there is only one input signal, Cnt, the nextstate table shown has only two columns, corresponding to Cnt = 0 and Cnt = 1.
Note that each entry in the nextstate table indicates the values of the flipflops in the next state if their value in the present state is in the row header and the input values in the column header.
Each of these nextstate values has been computed from the nextstate equations in STEP 2.
STEP 4: The state diagram is generated directly from the nextstate table, shown in Figure 8.
Figure 8. State diagram
Each arc is labelled with the values of the input signals that cause the transition from the present state (the source of the arc) to the next state (the destination of the arc).
In general, the number of states in a nextstate table or a state diagram will equal 2^{m }, where m is the number of flipflops. Similarly, the number of arcs will equal 2^{m }x 2^{k} , where k is the number of binary input signals. Therefore, in the state diagram, there must be four states and eight transitions. Following these transition arcs, we can see that as long as Cnt = 1, the sequential circuit goes through the states in the following sequence: 0, 1, 2, 3, 0, 1, 2,.... On the other hand, when Cnt = 0, the circuit stays in its present state until Cnt changes to 1, at which the counting continues.
Since this sequence is characteristic of modulo4 counting, we can conclude that the sequential circuit in Figure 7 is a modulo4 counter with one control signal, Cnt, which enables counting when Cnt = 1 and disables it when Cnt = 0.
Below, we show a timing diagram, representing four clock cycles, which enables us to observe the behaviour of the counter in greater detail.
Figure 9. Timing Diagram 
In this timing diagram we have assumed that Cnt is asserted in clock cycle 0 at t_{0} and is disasserted in clock cycle 3 at time t_{4}. We have also assumed that the counter is in state Q1Q0 = 00 in the clock cycle 0. Note that on the clock's rising edge, at t_{1}, the counter will go to state Q1Q0 = 01 with a slight propagation delay; in cycle 2, after t_{2}, to Q1Q0 = 10; and in cycle 3, after t_{3} to Q1Q0 = 11. Since Cnt becomes 0 at t_{4}, we know that the counter will stay in state Q1Q0 = 11 in the next clock cycle.
In Example 1.1 we demonstrated the analysis of a sequential circuit that has no outputs by developing a nextstate table and state diagram which describes only the states and the transitions from one state to the next. In the next example we complicate our analysis by adding output signals, which means that we have to upgrade the nextstate table and the state diagram to identify the value of output signals in each state.
This example is taken from D. D. Gajski, Principles of Digital Design, Prentice Hall, 1997, p.234.
Example 1.2
Derive the next state, the output table and the state diagram for the sequential circuit shown in Figure 10.
Figure 10. Logic schematic of a sequential circuit.
SOLUTION:
The input combinational logic in Figure 10 is the same as in Example 1.1, so the excitation and the nextstate equations will be the same as in Example 1.1.
Excitation equations:
D0 = Cnt'*Q1 + Cnt*Q1'*Q0 + Cnt*Q1*Q0' 
Nextstate equations:
Q0(next) = D0 = Cnt'*Q0 + Cnt*Q0' 
Q1(next) = D0 = Cnt'*Q1 + Cnt*Q1'*Q0 + Cnt*Q1*Q0' 
In addition, however, we have computed the output equation.
Output equation: Y = Q1Q0
As this equation shows, the output Y will equal to 1 when the counter is in state Q1Q0 = 11, and it will stay 1 as long as the counter stays in that state.
Nextstate and output table:
Present State
 Next State
 Output
 



State diagram:
Figure 11. State diagram of sequential circuit in Figure 10.
Timing diagram:
Figure 12. Timing diagram of sequential circuit in Figure 10.
Note that the counter will reach the state Q1Q0 = 11 only in the third clock cycle, so the output Y will equal 1 after Q0 changes to 1. Since counting is disabled in the third clock cycle, the counter will stay in the state Q1Q0 = 11 and Y will stay asserted in all succeeding clock cycles until counting is enabled again.
Design of Sequential Circuits
The design of a synchronous sequential circuit starts from a set of specifications and culminates in a logic diagram or a list of Boolean functions from which a logic diagram can be obtained. In contrast to a combinational logic, which is fully specified by a truth table, a sequential circuit requires a state table for its specification. The first step in the design of sequential circuits is to obtain a state table or an equivalence representation, such as a state diagram.
A synchronous sequential circuit is made up of flipflops and combinational gates. The design of the circuit consists of choosing the flipflops and then finding the combinational structure which, together with the flipflops, produces a circuit that fulfils the required specifications. The number of flipflops is determined from the number of states needed in the circuit.
The recommended steps for the design of sequential circuits are set out below.
.State Reduction
Any design process must consider the problem of minimising the cost of the final circuit. The two most obvious cost reductions are reductions in the number of flipflops and the number of gates.
The number of states in a sequential circuit is closely related to the complexity of the resulting circuit. It is therefore desirable to know when two or more states are equivalent in all aspects. The process of eliminating the equivalent or redundant states from a state table/diagram is known as state reduction.
Example: Let us consider the state table of a sequential circuit shown in Table 6.
Table 6. State table
Present State  Next State
 Output
 



It can be seen from the table that the present state A and F both have the same next states, B (when x=0) and C (when x=1). They also produce the same output 1 (when x=0) and 0 (when x=1). Therefore states A and F are equivalent. Thus one of the states, A or F can be removed from the state table. For example, if we remove row F from the table and replace all F's by A's in the columns, the state table is modified as shown in Table 7.
Table 7. State F removed
Present State  Next State
 Output
 



It is apparent that states B and E are equivalent. Removing E and replacing E's by B's results in the reduce table shown in Table 8.
Table 8. Reduced state table
Present State  Next State
 Output
 



The removal of equivalent states has reduced the number of states in the circuit from six to four. Two states are considered to be equivalent if and only if for every input sequence the circuit produces the same output sequence irrespective of which one of the two states is the starting state.
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