Download Chapter 6 – Combinational and Sequential Circuit

Chapter 4 – Combinational and Sequential
Circuit
4.1
4.2
4.3
Combinational Circuit and Sequential
Circuit Criterions
Some Examples of Combinational
Circuit: Parallel Adder, Decoder, etc
Some Examples of Sequential Circuits:
Flip-flop, Register, Serial Adder, etc.
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4.1) Combinational Circuit and Sequential Circuit Criterions
Logic Circuits can be divided into :
1. Combinational Logic Circuit
2. Sequential Logic Circuit
Combinational Logic Circuit
Combinational Logic circuit contains logic gates where its output is determined by the
combination of the current input, regardless of the output or the prior combination of
input. Basically, combinational circuit can be depicted by Diagram 1 below:
n input
combinational
.
m output
circuit
Diagram 1
Examples of Combinational circuits in the computer system are decoder, parallel adder,
and multiplexer.
(Note: Students are encouraged to obtain examples of combinational circuits stated
above)
Sequential Logic Circuit
Sequential Logic Circuit contains logic gates arranged in parallel and its output is not
only determined by the combination of the current input, but also the prior output. The
circuit also contains memory elements that enable it to store the information of the prior
output. Generally, sequential circuits can be depicted by Diagram 2 below:
n input
sequential
logic
circuit
m output
memory
elements
Diagram 2
2
Examples of sequential circuits in the computer system are like registers, counters and
serial adders
4.2) Some Examples of Combinational Circuit: Parallel Adder,
Decoder, etc
The circuits learnt in chapter 3 are combinational circuits. The steps to design
combinational circuits are as the following:
1. Understand the problem
2. Determine the number of input and output variables that are needed
3. Give symbols for the stated input and output
4. Construct a truth table that defines the relationship between the input and
output
5. Obtain the Boolean function or the logical expression from the truth table in
(4) using Karnaugh Map or other known methods.
6. Draw a logic circuit based on the expression obtained from (5) above.
(Note : Chapter 3.8 is an example of designing a combinational circuit – Alarm
System)
Below are examples of designing combinational circuits that are in the computer system
that is the adder. Because computers use binary system for its data, its adder is based on
the addition of the binary system. There are 2 kinds of addition, which are identified to be
half addition and full addition.
Half addition is the addition of 2 bits data (doesn’t involve carry) that produces 2 bits
output, that is the result and the carrier. Full addition is the addition of 3 bits data (2 bits
data and 1 bit carry) that produces 2 bits output (sum and carry). Logic circuit for half
addition is known as Half Adder while the logic circuit for full addition is known as Full
Adder
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Designing a Circuit for Half Adder
The steps are as below:
1. Problem: to build a logic circuit for the addition of 2 bits data
2. Number of input : 2
Number of output : 2
3. Variables for input: x and y
Variables for output : s (sum) and c (carry)
4. The Truth Table for the problem :
INPUT
x
0
0
1
1
y
0
1
0
1
OUTPUT
s
c
0
0
1
0
1
0
0
1
5. The expressions for r and c using Karnaugh Map
For s
y
x
0
1
1
1
s=xy+xy
=x + y
For c
x
y
1
c = xy
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6. A logic circuits for Half Adder (HA)
x
s = xy + xy
y
c = xy
OR
x
x + y=s
y
xy = c
A Block Diagram for HA is as below:
x
input
s
HA
y
output
c
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Designing a Circuit for Full Adder (FA)
The same method used to design HA.
1. Problem: Build logic circuit for the addition of 3 bit data
2. Number of input : 3
Number of output : 2
3. Variables for input: x , y and ci
Variables for output : s (sum) and co (carry)
4. The truth table for the problem :
INPUT
OUTPUT
x
y
ci
s
co
0
0
0
0
0
0
0
1
1
0
0
1
0
1
0
0
1
1
0
1
1
0
0
1
0
1
0
1
0
1
1
1
0
0
1
1
1
1
1
1
5. Obtain the expression for r and co using Karnaugh Map (Students are required
to try this out themselves):
will obtain
s = x y pi + x y ci + x y ci + x y ci
= x + y + ci
and
co = x y + y ci + x ci
6. Draw the circuit for FA (Students are required to try this out themselves):
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Generally, the block diagram for FA is shown as below :
x
r
y
FA
co
ci
To construct a 4-bit parallel adder, 3 FA and 1 HA are required like the diagram below
with the input as X = x3x2x1x0 and Y = y3y2y1y0 (X and Y are binary numbers 4-bit) and
the output (addition result) is r3r2r1r0.
INPUT
x3 y3
x2 y2
x1 y1
x0 y0
FA
FA
FA
HA
c2
OUTPUT
c3
c1
s3
c0
s2
s1
s0
OR
INPUT
x3 y3
x2 y2
x1 y1
FA
FA
FA
c2
OUTPUT
c3
s3
x0 y0
0
FA
c1
c0
s2
s1
7
s0
4.3) Some Examples of Sequential Circuits: Flip-flop, Register,
Serial Adder, etc.
Sequential circuits are a kind of logic circuit where the current output not only depends
on the current input but also on the past history of inputs. Another and generally more
useful way to view it is that the current output of a sequential circuit depends on the
current input and the current state of that circuit. The simplest form of sequential circuit
is the flip-flop. Flip-flop is a kind of logic circuit that is capable of exhibiting 2 stable
conditions. It is also known as 1-bit memory element and is mostly used to make
important computer components such as registers, counters, memory etc.
There are a variety of flip-flops, all of which share two properties:
1. The flip-flop is a bistable device either 0 or 1. It exists in one of two states and, in the
absence of input, remains in that state. Thus, the flip-flop can function as a 1-bit
memory.
2. The flip-flop has two outputs, which are always the complements of each other.
These are generally labeled Q and Q.
Table 1 shows symbolic graphic and feature table for three types of flip-flop that are S-R,
J-K and D flip-flops. Flip-flop is a form of memory element used to construct sequential
circuits that are more complex, such as registers etc. Sequential circuits can be divided
into
1. Synchronous
2. Asynchronous
In synchronous sequential circuit, all flip-flops are moved by the same clock pulse so that
all flip-flops involved change simultaneously.
In asynchronous circuit, the change of flip-flop condition depends on the change that
occurs on the input and the late time that is in the circuit.
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Name
Graphical Symbol
S
Feature Table
Q
Clock
–
R
Q
S-R
J
Q
S
R
Qn+1
0
0
Qn
0
1
0
1
0
1
1
1
-
J
K
Qn+1
0
0
Qn
0
1
0
1
0
1
1
1
Change
condition
D
Qn+1
0
0
1
1
Clock
–
K
Q
J-K
D
Q
Clock
D
–
Q
Table 1: A few basic Flip-flops
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S-R Flip-flop
S-R flip-flop has 2 inputs, S (set) and R (reset) like Diagram 3 below. In the diagram
below, (also for JK and D flip-flops), there use another input called clock. It is to control
the movement of input that is input will only occur when given a clock pulse
(synchronous circuit)
The features of S-R flip-flop can be depicted in Table 2 below. It can be summarized
that:
1. If the value of both S and R are 0, the flip-flop will remain in its present condition
(either 0 or 1).
2. If S = 0 and R = 1 (reset), then the flip-flop condition will change to 0 (its output, Q =
0).
3. If S = 1 (set) and R = 0, then the flip-flop condition will change to 1 (output, Q = 1).
4. This circuit does not allow combinational input of input S = 1 and R = 1.
S
Q
clock
–
Q
R
Diagram 3 : S-R Flip-flop
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S
R
Qn
Qn+1
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
0
1
0
0
1
1
0
1
1
1
1
0
-
1
1
1
-
Table 2 : Feature table of S-R Flip-flop
J-K Flip-flop
J-K flip-flop also has 2 inputs, J and K. The function of clock is same as S-R flip-flop.
Unlike S-R flip-flop, J-K flip-flop allows all combination of inputs. The logic circuit for
J-K flip-flop is shown in Diagram 4 below. Table 3 shows the features of J-K flip-flop.
From the table, it can be summarized that:
1. If J = 0 and K = 0, it will maintain the flip-flop condition like before
2. If J = 0 and K = 1, it will cause flip-flop to change to condition 0 (reset).
3. If J = 1 and K = 0, it will cause flip-flop to change to condition 1 (set).
4. If J = 1 and K = 1, it will change the flip-flop condition, that is it will become
complementary to the initial or prior condition
It can be observed that J-K flip-flop is built to address the input problem of S = R = 1 in
S-R flip-flop. Features 1 till 3 are same as S-R flip-flop.
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J
Q
Clock
–
Q
K
Diagram 4: J-K Flip-flop
J
K
Qn
Qn+1
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
0
1
0
0
1
1
0
1
1
1
1
0
1
1
1
1
0
Table 3: Features table of J-K flip-flop
D Flip-flop
Logic circuit for D flip-flop is shown in Diagram 5 below. This flip-flop only has one
input that is D. The clock function is same as S-R and J-K flip-flops. The features of D
flip-flop can be illustrated by Table 4. From the table, it can be seen that this flip-flop
produces the same output as its input regardless of the condition of the stated flip-flop.
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This feature is very suitable to be used as memory element and this flip-flop is mostly
used to make registers and computer memory (RAM)
Q
clock
–
Q
D
Diagram 5 : D Flip-flop
D
Qn
Qn+1
0
0
0
0
1
0
1
0
1
1
1
1
Table 4 : Feature table of D Flip-flop
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A few examples of Flip-flop (Sequential Circuit) usage
As priory stated, flip-flop is an example of the simplest form of sequential circuit. It is
also a form of memory element where a flip-flop can store 1 bit of data. In this section,
examples of sequential circuits that use flip-flop will be given:
1. Register
2. Adder
Register
Register is an important component in the computer. Generally, it can be categorized
into:
1. Storage Register (or Parallel Register)
2. Shift Register (or Serial Register)
Parallel register is made up of a set of 1-bit (flip-flop) that can be written on and read
simultaneously. This register is used to store data (output=input). The amount of flip-flop
used depends on the size of the register that is to be built. If a parallel register that can
store 8 bits of data is to be built, then 8 flip-flops are needed. Diagram 6 below is a 4 bit
parallel register that uses flip-flop D. (Note: all kinds of flip-flop can be used to build
storage register, but its circuit will differ because ever flip-flop has its own features)
I1
I2
D
Q
Clock
Q
–
I3
D
Q
Clock
Q
–
I4
D
Q
Clock
Q
–
D
Q
Clock
Q
–
Clock
Pulse
Q1
Q2
Q3
Diagram 6: A 4-bit parallel register that uses D Flip-flop
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Q4
In the above diagram, 4 bits of input is admitted simultaneously, that is I1, I2, I3 and I4,
whereas its output is also is simultaneous or parallel, that is Q1, Q2, Q3 and Q4.
In shift register, only one output is produced at a time. There are 2 types of shift register
that is shift to right and shift to left. Shift to right register means the rightmost bit of the
stated will be taken out first followed by the following bits after a given clock beat. It’s
vice versa for move to shift to left register. Diagram 7 below is an example of 4-bit shift
to right register that utilizes J-K flip-flop.
Input
J
Q
Clock
K
J
Q
J
Clock
–
Q
K
Q
Clock
–
K
Q
J
Q
Output
Clock
–
Q
K
–
Q
Clock
Pulse
Diagram 7: Shift to Right Register Using J-K Flip-flop
Parallel Adder
In the computer environment, there are 2 types of adders:
1. Parallel Adder
2. Serial Adder
Parallel adder is an adder that performs addition concurrently for each bit involved.
Adder in section 4.2 is called a serial adder. Serial Adder performs addition bit by bit
starting with the rightmost bit, followed by the following bits. Diagram 8 below is an
example of a serial 4-bit adder. This adder uses two Shift to Right Registers, X and Y to
hold operand 1 (A = A3A2AIA0) and operand 2 (B = B3B2B1B0), a full adder (see section
4.2) and a flip-flop (usually D flip-flop) to hold the carrier value.
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The addition process in the adder are as below :
X=X+Y
that is the X and Y registers will hold operand 1 and operand 2 and the addition result
will be kept in the X register. Hence, in the addition, the value in the Y (Operand 2)
register cannot change while the X register holds the addition result (the value of operand
1 will be lost)
Note: observe and understand the data movement in the stated circuit after every clock
pulse is given.
A
33
X Register
A
A
A
2
1
Ai
Si
0
Full
Adder
B
3
B
B
B
2
1
Y Register 0
Bi
Ci
Ci+1
D flipflop
Carry
Clock
Pulse
Diagram 8 : 4-bit Serial Adder
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