EE201: Digital Circuits and Systems
EE201:
4 Sequential Circuits
page 1 of 11
Digital Circuits and Systems
Section 4 – Sequential Circuits
4.1 Overview of Sequential Circuits:
Definition
• The circuit whose outputs and next state
depend on both the input signals and the
present state of the circuit
Principle [spot the error!]
Memory Elements
Present
State
Clock
Combinational Logic
Input
Signals
O
S
EE201: Digital Circuits and Systems
4 Sequential Circuits
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4.2 Flip-Flops
SR Flip-Flop
S
R
SET
CLR
Q
Q
S
0
0
1
1
R
0
1
0
1
Q’
Q
0
1
X
Q Q’
0 0
0 1
1 0
1 1
S
0
1
0
X
R
X
0
1
0
J
0
0
1
1
K Q’
0 Q
1 0
0 1
1 !Q
Q Q’
0 0
0 1
1 0
1 1
J
0
1
X
X
K
X
X
1
0
D
0
1
Q’
0
1
Q Q’
X 0
X 1
D
0
1
T
0
1
Q’
Q
!Q
Q Q’
0 0
0 1
1 0
1 1
T
0
1
1
0
JK Flip-Flop
J
K
SET
CLR
Q
Q
D Flip-Flop
D
SET
CLR
Q
Q
T Flip-Flop
D
T
SET
CLR
Q
Q
EE201: Digital Circuits and Systems
4 Sequential Circuits
page 3 of 11
4.3 Design of Sequential Circuits
Algorithm:
• Obtain the description of circuit and create the
State Diagram
• Determine the State Table
• Minimize the number of states
• Assign binary codes to each state
• Determine the number of flip-flops needed and
give a letter symbol to each of them
• Choose the type of flip-flops
• Starting from State Table, derive the Excitation
Table and the Output Table
• Derive the minimized circuit output functions
and flip-flop input functions
• Draw the Logic Diagram
EE201: Digital Circuits and Systems
4 Sequential Circuits
page 4 of 11
4.4 Example of Design
Design a sequential logic circuit whose output Z is 1 except
when the input X = 1 for at least four clock periods. Then the
output Z is 0. Use J-K flip-flops.
• State Diagram [X/Z]
1/1
1/1
A
B
0/1
C
0/1
D
1/0
0/1
0/1
• State Coding
Present
State
A
B
C
D
1/1
Code
0
0
1
1
0
1
0
1
• Flip Flops
We require two JK flip-flops.
Let’s name them JKA and JKB
EE201: Digital Circuits and Systems
4 Sequential Circuits
page 5 of 11
• State Table
Present
State
A
B
C
D
Next State
X=0
X=1
A
B
A
C
A
D
A
D
Output Z
X=0
X=1
1
1
1
1
1
1
1
0
• Excitation Table
QA
0
0
0
0
1
1
1
1
QB
0
0
1
1
0
0
1
1
X
0
1
0
1
0
1
0
1
Q’A
0
0
0
1
0
1
0
1
Q’B
0
1
0
0
0
1
0
1
JA
0
0
0
1
x
x
x
x
KA
x
x
x
x
1
0
1
0
• Minimisations and Equations
JA
X\ QA QB
0
1
00
0
0
01
0
1
11
x
x
10
x
x
J A = XQ B
JB
0
1
x
x
0
1
x
x
KB
x
x
1
1
x
x
1
0
Z
1
1
1
1
1
1
1
0
EE201: Digital Circuits and Systems
4 Sequential Circuits
KA
X\ QA QB
0
1
00
X
X
01
x
x
11
1
0
10
1
0
KA = X
JB
X\ QA QB
0
1
00
0
1
01
x
x
11
x
x
10
0
1
JB = X
KB
X\ QA QB
0
1
00
x
x
01
1
1
11
1
0
10
x
x
K B = X + Q A = XQ A
Z
X\ QA QB
0
1
00
0
0
01
0
0
11
0
1
10
0
0
Z = XQ AQ B
page 6 of 11
EE201: Digital Circuits and Systems
4 Sequential Circuits
page 7 of 11
4.5 State Reduction
Definition of Equivalent States
• Two or more states of a sequential circuit are equivalent
if for the same values for the inputs, have exactly the same
output and determine the sequential circuit transition to the
same next state or to equivalent states.
Algorithm for State Reduction
• If more 2 or more states are equivalent, one of them can be
substituted with the other one
• The other states that have transitions to one of the removed
states have to have their next states changed into the
remaining equivalent state
Advantages of State Reduction
• By reducing the number of states, it is possible that the
number of flip-flops and/or amount of combinational
circuitry needed to implement the sequential circuit will
decrease, reducing the cost of the circuit
EE201: Digital Circuits and Systems
4 Sequential Circuits
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Example of State Reduction
• Let’s assume that there is the following state table:
Present
State
Next State
Output
X=0
X=1
X=0
X=1
A
A
B
0
0
B
C
D
0
0
C
A
D
0
0
D
E
F
0
1
E
A
F
0
1
F
G
F
0
1
G
A
F
0
1
• States G and E are equivalent (same next states for the
same inputs and same outputs for the same inputs)
• State reduction => state G will be replaced by E
Present
State
Next State
Output
X=0
X=1
X=0
X=1
A
A
B
0
0
B
C
D
0
0
C
A
D
0
0
D
E
F
0
1
E
A
F
0
1
F
E
F
0
1
EE201: Digital Circuits and Systems
4 Sequential Circuits
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• States D and F are equivalent
• State reduction => state F will be replaced by D
Present
State
Next State
Output
X=0
X=1
X=0
X=1
A
A
B
0
0
B
C
D
0
0
C
A
D
0
0
D
E
D
0
1
E
A
D
0
1
• By reducing the number of states, the number of flip-flops
and amount of combinational circuitry needed to implement
this sequential circuit could have decreased, reducing the
cost of the circuit.
• However in this example the number of flip-flops remains
the same.
EE201: Digital Circuits and Systems
4 Sequential Circuits
page 10 of 11
4.6 State Assignment
Definition of State Assignment
• The process of assigning a binary code to each state
Possibilities for State Assignment
• There are many possibilities to assign binary codes to
states
• The number of possible options increases exponentially
with the number of states
• The cost of the combinational circuit strongly depends
on the state assignment chosen
• Although various State Assignment methods have been
proposed, there is no assignment procedure that
guarantees a minimal cost for the resulting
combinational circuit
Examples of State Assignment
States
A
B
C
D
E
Assign.
1
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
Assign.
2
0 0 0
0 1 0
0 1 1
1 0 1
1 1 1
Assign.
3
0 0 0
1 0 0
0 1 0
1 0 1
0 1 1
EE201: Digital Circuits and Systems
4 Sequential Circuits
page 11 of 11
4.7 Homework
• Having the following state table, design the sequential
circuit following state assignment 1, 2 and 3,
respectively.
Present
State
Next State
Output
X=0
X=1
X=0
X=1
A
A
B
0
0
B
C
D
0
0
C
A
D
0
0
D
E
D
0
1
E
A
D
0
1