Combinational Circuits Design in Digital System Design.pptx

Department of ECE
22EC401 DIGITAL SYSTEM DESIGN
Class: II ECE
Academic Year: 2023-2024 (Even Sem)
MODULE I - LOGIC SIMPLIFICATION AND LOGIC FAMILIES
Course handling faculty: Mr.M.Navin Kumar, Dr.R.R.Thirrunavukkarasu, Dr.S.Prema
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Syllabus:
Module 1: LOGIC SIMPLIFICATION AND LOGIC FAMILIES 15 Hrs
Logic Simplification: Review of Boolean Algebra and DeMorgan’s Theorem, SOP & POS
forms, Canonical forms, Karnaugh maps upto 4 variables, Quine McCluskey method,
Binary codes, Code Conversion. Logic families: Logic levels, propagation delay, power
dissipation, fan-out and fan-in, noise margin, logic families and their characteristics-RTL,
TTL, ECL, CMOS.
22EC401 DIGITAL SYSTEM DESIGN 2
Module II
Module 2: MSI COMBINATIONAL LOGIC CIRCUITS AND SEQUENTIAL LOGIC
DESIGN 15 Hrs
MSI devices like Comparators, Multiplexers, Encoder, Decoder, Half and Full Adders,
Subtractors, Serial and Parallel Adders, BCD Adder. Flip flops — SR, JK, T, D,
Master/Slave FF — operation and excitation tables, Triggering of FF, Analysis and design of
clocked sequential circuits — Design — Moore/Mealy models, state minimization, state
assignment, circuit implementation — Design of Counters- Ripple Counters, Ring Counters,
Shift registers, Universal Shift Register.

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Module contents
MSI COMBINATIONAL LOGIC CIRCUITS AND SEQUENTIAL
LOGIC DESIGN: MSI devices like Comparators, Multiplexers,
Encoder, Decoder, Half and Full Adders, Subtractors, Serial and
Parallel Adders, BCD Adder. Flip flops — SR, JK, T, D,
Master/Slave FF — operation and excitation tables, Triggering of
FF, Analysis and design of clocked sequential circuits — Design
— Moore/Mealy models, state minimization, state assignment,
circuit implementation — Design of Counters- Ripple Counters,
Ring Counters, Shift registers, Universal Shift Register.
Module II

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2.1 Introduction
 Traditional methods of combinational circuit design involve
simpliﬁcation and realisation using gates.
 Using these methods, complex functions have been integrated
(MSI) and are easily available in IC form.
 Few examples are multiplexers, demultiplexers, adders, parity
generators/checkers, priority encoders, decoders,
comparators, etc.
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2.1 Introduction
 Design is greatly simplified because the laborious and time
consuming simplification methods are generally not required.
 Also improves the reliability of the system by reducing the
number of external wired connections.
 To be familiar with the functions performed, the options
available, and the limitations of these devices in order to make
an effective and optimum use of such devices.
 Usage of these devices significantly reduce IC package count
thereby reducing the system cost.
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2.2 Magnitude Comparator
2.2.1 Introduction
 A magnitude digital Comparator is a combinational circuit that
compares two digital or binary numbers in order to find out
whether one binary number is equal, less than or greater than the
other binary number.
 We logically design a circuit for which we will have two inputs one for
A and other for B and have three output terminals, one for A > B
condition, one for A = B condition and one for A < B condition.
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https://drive.google.com/file/d/1NsqPt1svqlKAK3qT5ynx6i-2AWDsCm6a/view?usp=sharing

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2.2.2 1-Bit Magnitude Comparator
 A comparator used to compare two bits is called a single bit comparator.
 It consists of two inputs each for two single bit numbers and three outputs to
generate less than, equal to and greater than between two binary numbers.
 The truth table for a 1-bit comparator is given below:
 From the above truth table logical expressions for each output can be
expressed as follows:
A>B: AB'
A<B: A'B
A=B: A'B' + AB
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 A comparator used to compare
two binary numbers each of two
bits is called a 2-bit Magnitude
comparator. It consists of four
inputs and three outputs to
generate less than, equal to and
greater than between two binary
numbers.
 The truth table for a 2-bit
comparator is given below:
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From the above truth table K-map for each output can be drawn as follows:
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A>B: A1B1’ + A0B1’B0’ + A1A0B0’
A=B: A1’A0’B1’B0’ + A1’A0B1’B0 + A1A0B1B0 + A1A0’B1B0’
: A1’B1’ (A0’B0’ + A0B0) + A1B1 (A0B0 + A0’B0’)
: (A0B0 + A0’B0’) (A1B1 + A1’B1’)
: (A0 Ex-Nor B0) (A1 Ex-Nor B1)
A<B: A1’B1 + A0’B1B0 + A1’A0’B0
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 A comparator used to compare two binary numbers each of four bits is called a
4-bit magnitude comparator.
 It consists of eight inputs each for two four bit numbers and three outputs to
generate less than, equal to and greater than between two binary numbers.
 In a 4-bit comparator the condition of A>B can be possible in the
following four cases:
If A3 = 1 and B3 = 0
If A3 = B3 and A2 = 1 and B2 = 0
If A3 = B3, A2 = B2 and A1 = 1 and B1 = 0
If A3 = B3, A2 = B2, A1 = B1 and A0 = 1 and B0 = 0
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Similarly the condition for A<B can be possible in the following four cases:
If A3 = 0 and B3 = 1
If A3 = B3 and A2 = 0 and B2 = 1
If A3 = B3, A2 = B2 and A1 = 0 and B1 = 1
If A3 = B3, A2 = B2, A1 = B1 and A0 = 0 and B0 = 1
The condition of A=B is possible only when all the individual bits of one number
exactly coincide with corresponding bits of another number.
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 From the above statements logical expressions for each output can be
expressed as follows:
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Magnitude Comparator
A3A2A1A0 B3B2 B1 B0
A<B A=B A>B
3
3
3
3
3 B
A
B
A
x 

2
2
2
2
2 B
A
B
A
x 

1
1
1
1
1 B
A
B
A
x 

0
0
0
0
0 B
A
B
A
x 

0
1
2
3
)
( x
x
x
x
B
A 

0
0
1
2
3
1
1
2
3
2
2
3
3
3
)
( B
A
x
x
x
B
A
x
x
B
A
x
B
A
B
A 




0
0
1
2
3
1
1
2
3
2
2
3
3
3
)
( B
A
x
x
x
B
A
x
x
B
A
x
B
A
B
A 





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2.2.5 Cascading Comparator
A comparator performing the comparison operation to more than four bits by
cascading two or more 4-bit comparators is called cascading comparator. When
two comparators are to be cascaded, the outputs of the lower-order comparator
are connected to corresponding inputs of the higher-order comparator.
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2.2.6 Applications
1. Comparators are used in central processing units (CPUs) and
microcontrollers (MCUs).
2. These are used in control applications in which the binary numbers
representing physical variables such as temperature, position, etc.
are compared with a reference value.
3. Comparators are also used as process controllers and for Servo
motor control.
4. Used in password verification and biometric applications.
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2.3 Multiplexers
2.3.1 Introduction
 The multiplexer (or data selector) is a logic circuit that allows one of the
data inputs at the output.
 Figure shows a 4:1 Multiplexer with 4 input lines, 2 selection lines and 1
output line
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2.3 Multiplexers
2.3.1 4:1 Multiplexer
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https://drive.google.com/file/d/1bssbmNAP0OFe6clx4w_NFPp6ptYBvTN8/view?usp=sharing

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2.3 Multiplexers
2.3.2 Multiplexers and their use in Combinational Logic Design
 One of the most widely used standard logic circuits in digital design.
 Because of its widespread use, it has been fabricated as MSI IC and is
commercially available in various sizes, such as 2:1, 4:1, 8:1, and 16:1
multiplexers.
https://drive.google.com/file/d/1zuDJmE3m9edehjoFDxTMntuGLeWN3kDv/view?usp=
sharing
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2.3 Multiplexers
2.3.2 Multiplexers and their use in Combinational Logic Design
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2.3 Multiplexers
2.3.3 A 4:1 Multiplexer with Strobe Input Using NAND Gates
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2.3 Multiplexers
 32:1 Multiplexer Using
Two 16:1 Multiplexers
and One OR Gate
https://drive.google.com/file/d/13nMAGW1V2AS5xN1wGePBg80bX-A_fTeP/view?usp=sharing
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2.3 Multiplexers
 32.1
Multiplexer
Using Two 16:1
Multiplexers
and One 2:1
Multiplexer

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2.3 Multiplexers
2.3.4 Combinational Logic Design Using Multiplexers
 For using the multiplexer as a logic element, either the truth table or
one of the canonical forms of logic expression must be available. The
design procedure is given below:
1. Identify the decimal number corresponding to each minterm in the
expression. The input lines corresponding to these numbers are to
be connected to logic 1 level.
2. All other input lines are to be connected to logic 0 level.
3. The inputs are to be applied to select inputs.
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2.3 Multiplexers
2.3.4 Example: Implement the expression using a multiplexer
f (A, B, C, D) = Σm(0, 2, 3, 6, 8, 9, 12, 14)
Solution
 Since there are four variables, therefore, a multiplexer with four select inputs is
required. The circuit of 16:1 multiplexer connected to implement the above
expression is shown in Fig.
 This implementation requires only one IC package.
 In case the output of the multiplexer is active-low, the logic 0 and logic 1 inputs
of Fig. are to be interchanged.
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2.3 Multiplexers
2.3.4 Example: Implement the expression using a multiplexer
f (A, B, C, D) = Σm(0, 2, 3, 6, 8, 9, 12, 14)
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2.3 Multiplexers
2.3.4 Realise the logic function of the truth table given in Table
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2.3 Multiplexers
Solution
i. First Method: This can be realised using the method used in Example. Here,
the input lines 2, 4, 6, 7, 9, 10, 11, 12, and 15 are to be connected to logic 1
and the input lines 0, 1, 3, 5, 8, 13, and 14 are to be connected to logic 0.
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2.3 Multiplexers
Solution
Second Method:
 A four variable truth table or logic expression can be realised by using an 8:1
multiplexer instead of a 16:1 multiplexer. For this, partition the truth table as
shown by dotted lines. Here the inputs A, B, and C are to be connected to S2S
and S select inputs respectively. Now, we observe the relationship between
input D and output Y for each group of two rows.
 There are four possible values of Y and these are 0, 1, D, and D’. These are
given in Table. From this table, we note the output Y for each of the
combinations of A, B, and C, and then make the connections accordingly. The
implementation of this function using an 8:1 multiplexer is shown in Fig.
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2.3 Multiplexers
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2.3 Multiplexers
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2.3 Multiplexers
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2.3 Multiplexers
2.3.5 Demultiplexer
 The Demultiplexer performs the reverse operation of a multiplexer. It
accepts a single input and distributes it over several outputs. Figure
gives the block diagram of a Demultiplexer. The select input code
determines to which output the data input will be transmitted.
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2.3 Multiplexers
2.3.5 Demultiplexer
• The number of output lines is n and the number of select lines is m,
where n = 2m.
• The data input D will appear on the output line selected by the select
input.
• For example, if the decimal equivalent of the select input is 4, then the
data will appear on D4 output line.
• This circuit can also be used as binary-to-decimal decoder
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2.3 Multiplexers
2.3.5 Demultiplexer/Decoder
 This circuit can also be used as binary-to-decimal decoder with binary
inputs applied at the select input lines and the output will be obtained
on the corresponding line. The data input line is to be connected to
logic 1 level.
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2.3 Multiplexers
2.3.6 Example:
Implement the following multi-output combinational logic circuit using a 4-
to-16-line decoder.
F1 = Σm (1, 2, 4, 7, 8, 11, 12, 13)
F2 = Σm (2, 3, 9, 11)
F3 = Σm (10, 12, 13, 14)
F4 = Σm (2, 4, 8)
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2.3 Multiplexers
2.3.6 Example:
Solution
• The four-bit input ABCD is applied at the Select input terminals S3, S2, S1, and
S0. The output F1 is required to be 1 for minterms 1, 2, 4, 7, 8, 11, 12, and 13.
• Therefore, a NAND gate is connected as shown.
• Similarly NAND gates are used for the outputs F2, F3, and F4.
• Here, the decoder’s outputs are active-low, therefore a NAND gate is required
for every output of the combinational circuit.
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2.3 Multiplexers
2.3.6 Example:
Realization
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2.4 Decoder, Encoder & Priority Encoder
 Extract “Information” from the code
 Binary Decoder
 Example: 2-bit Binary Number
Binary
Decoder
x1
x0
Only one
lamp will
turn on
0
0
1
0
0
0
2.4.1 Decoder
https://drive.google.com/file/d/1Hjeh3GabRyMgsYVAKwhO0DI3V4SKEi0S/view?usp=sharing
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2-to-4 Line Decoder
I1 I0 Y3 Y2 Y1 Y0
0 0 0 0 0 1
0 1 0 0 1 0
1 0 0 1 0 0
1 1 1 0 0 0
Binary
Decoder
I1
I0
y3
y2
y1
y0
I1
I0
Y3
Y2
Y1
Y0
0
1
3 I
I
Y  0
1
2 I
I
Y 
0
1
1 I
I
Y  0
1
0 I
I
Y 
2.4.1 Decoder

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3-to-8 Line Decoder
Binary
Decoder
I2
I1
I0
Y7
Y6
Y5
Y4
Y3
Y2
Y1
Y0
I2
I0
Y7
Y6
Y5
Y4
Y3
Y2
Y1
Y0
I1
0
1
2 I
I
I

0
1
2 I
I
I

0
1
2 I
I
I

0
1
2 I
I
I

0
1
2 I
I
I

0
1
2 I
I
I

0
1
2 I
I
I

0
1
2 I
I
I

2.4.1 Decoder

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“Enable” Control
Binary
Decoder
I1
I0
E
Y3
Y2
Y1
Y0
E I1 I0 Y3 Y2 Y1 Y0
0 x x 0 0 0 0
1 0 0 0 0 0 1
1 0 1 0 0 1 0
1 1 0 0 1 0 0
1 1 1 1 0 0 0
E
I0
Y3
Y2
Y1
Y0
I1
2.4.1 Decoder

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Expansion
I2 I1 I0 Y7 Y6 Y5 Y4 Y3 Y2 Y1 Y0
0 0 0 0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0 1 0 0
0 1 1 0 0 0 0 1 0 0 0
1 0 0 0 0 0 1 0 0 0 0
1 0 1 0 0 1 0 0 0 0 0
1 1 0 0 1 0 0 0 0 0 0
1 1 1 1 0 0 0 0 0 0 0
I2 I1 I0
Binary
Decoder
I0
I1
E
Y3
Y2
Y1
Y0
Y7
Y6
Y5
Y4
Y3
Y2
Y1
Y0
Binary
Decoder
I0
I1
E
Y3
Y2
Y1
Y0
2.4.1 Decoder

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Active-High / Active-Low
I1 I0 Y3 Y2 Y1 Y0
0 0 0 0 0 1
0 1 0 0 1 0
1 0 0 1 0 0
1 1 1 0 0 0
I1 I0 Y3 Y2 Y1 Y0
0 0 1 1 1 0
0 1 1 1 0 1
1 0 1 0 1 1
1 1 0 1 1 1
Binary
Decoder
I1
I0
Y3
Y2
Y1
Y0
I1
I0
Y3
Y2
Y1
Y0
Binary
Decoder
I1
I0
Y3
Y2
Y1
Y0
2.4.1 Decoder

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 Each output is a minterm
 All minterms are produced
 Sum the required minterms
Example: Full Adder
S(x, y, z) = ∑(1, 2, 4, 7)
C(x, y, z) = ∑(3, 5, 6, 7)
I2
I1
I0
Y7
Y6
Y5
Y4
Y3
Y2
Y1
Y0
Binary Decoder
x
y
z
S C
2.4.1 Decoder
https://drive.google.com/file/d/1m6lTCZD0S5fkIEQpMWPV8UVaAcA0Nsba/view?usp=sharing
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2.4.1 Implementation Using Decoders
I2
I1
I0
Y7
Y6
Y5
Y4
Y3
Y2
Y1
Y0
Binary
Decoder
x
y
z
S C
I2
I1
I0
Y7
Y6
Y5
Y4
Y3
Y2
Y1
Y0
Binary
Decoder
x
y
z
S C
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Implementation Using Decoders
Implement the following multi-output combinational logic circuit using a 4-to-16-line
decoder.
F1 = Σm (1, 2, 4, 7, 8, 11, 12, 13)
F2 = Σm (2, 3, 9, 11)
F3 = Σm (10, 12, 13, 14)
F4 = Σm (2, 4, 8)
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Implementation Using Decoders
Module 2
 The four-bit input ABCD is applied at the Select input terminals S
3
, S
2
, S
1
, and S
0
.
The output F
1
is required to be 1 for minterms 1, 2, 4, 7, 8, 11, 12, and 13. Therefore,
a NAND gate is connected as shown. Similarly NAND gates are used for the outputs
F
2
, F
3
and F
4
. Here, the decoder’s outputs are active-low, therefore a NAND gate is
required for every output of the combinational circuit.

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2.4.2 Encoders
 Put “Information” into code
 Binary Encoder
 Example: 4-to-2 Binary Encoder
x3 x2 x1 y1 y0
0 0 0 0 0
0 0 1 0 1
0 1 0 1 0
1 0 0 1 1
Binary
Encoder
y1
y0
x1
x2
x3
Only one
switch
should be
activated at a
time
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2.4.2 Encoders
Octal-to-Binary Encoder (8-to-3)
I7 I6 I5 I4 I3 I2 I1 I0 Y2 Y1 Y0
0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 1 0 0 0 1
0 0 0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 0 0 0 1 1
0 0 0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 1 0 1
0 1 0 0 0 0 0 0 1 1 0
1 0 0 0 0 0 0 0 1 1 1
Binary
Encoder
Y2
Y1
Y0
I7
I6
I5
I4
I3
I2
I1
I0
1
3
5
7
0
2
3
6
7
1
4
5
6
7
2
I
I
I
I
Y
I
I
I
I
Y
I
I
I
I
Y












I7
I6
I5
I4
I3
I2
I1
I0
Y2
Y1
Y0
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2.4.3 Priority Encoders
 4-Input Priority Encoder
I3 I2 I1 I0 Y1 Y0 V
0 0 0 0 0 0 0
0 0 0 1 0 0 1
0 0 1 x 0 1 1
0 1 x x 1 0 1
1 x x x 1 1 1
Priority
Encoder
V
Y1
Y0
I3
I2
I1
I0
I0
I1
I2
I3
Y1
Y0
V
0
1
2
3
1
2
3
0
2
3
1
I
I
I
I
V
I
I
I
Y
I
I
Y








Y1 I1
1 1 1 1
I2
I3
1 1 1 1
1 1 1 1
I0
https://drive.google.com/file/d/1T59H03KvUe87PgP3Ug1MOokTPLHu1z1j/view?usp=sharing
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2.4.4 Encoder / Decoder Pairs
Y2
Y1
Y0
I7
I6
I5
I4
I3
I2
I1
I0
I2
I1
I0
Y7
Y6
Y5
Y4
Y3
Y2
Y1
Y0
Binary
Encoder
Binary
Decoder
https://drive.google.com/file/d/1Hjeh3GabRyMgsYVAKwhO0DI3V4SKEi0S/view?usp=s
haring
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2.5 Half and FullAdders & Subtractors
2.5.1 Introduction
 An adder is a digital logic circuit in electronics that implements
addition of numbers.
 In many computers and other types of processors, adders are used to
calculate addresses, similar operations and table indices in the ALU
and also in other parts of the processors.
 These can be built for many numerical representations like excess-3
or binary coded decimal.
 Adders are classified into two types: Half adder and Full adder.
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2.5.1 Half Adder
 The half adder circuit has two inputs: A and B, which add two input
digits and generate a Carry (C) and Sum (S)
 The half adder adds two binary digits called as augend and addend
and produces two outputs as sum and carry; XOR is applied to both
inputs to produce sum and AND gate is applied to both inputs to
produce carry.
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2.5.1 Design of Half adder
 By using half adder, you can design simple addition with the help of
logic gates.
 Let’s see an addition of single bits.
0+0 = 0
0+1 = 1
1+0 = 1
1+1 = 10
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 These are the least possible single-bit combinations. But the result for
1+1 is 10, the sum result must be re-written as a 2-bit output. Thus, the
equations can be written as
0+0 = 00
0+1 = 01
1+0 = 01
1+1 = 10
 The output ‘1’of ‘10’ is carry-out. ‘SUM’ is the normal output and
‘CARRY’ is the carry-out.
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 Truth table
 Now it has been cleared that 1-bit adder can be easily implemented
with the help of the XOR Gate for the output ‘SUM’ and an AND Gate
for the ‘Carry’.
 The half-adder is useful when you want to add one binary digit
quantities.
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Design
• The simplest expression uses the exclusive OR function: Sum=A B.
• An equivalent expression in terms of the basic AND, OR, and NOT is:
SUM=A’.B+A.B’
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Design
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2.5.2 Full Adder
 This type of adder is a little more difficult to implement than a half-
adder.
 The main difference between a half-adder and a full-adder is that the
full-adder has three inputs and two outputs.
 The first two inputs are A and B and the third input is an input carry
designated as Cin.
 When a full adder logic is designed we will be able to string eight of
them together to create a byte-wide adder and cascade the carry bit
from one adder to the next.
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https://drive.google.com/file/d/1vj-0WyD0LqhE89VFQjls19J17xXoLXoC/view?usp=sharing

Department of ECE
2.5.2 Full Adder
 The full adder circuit has three inputs: A, B and Cin, which add the
three input numbers and generate output carry (Cout) and sum (S).
 The full adder adds 3 one bit numbers, where two can be referred to
as operands (A & B) and one can be referred to as bit carried in (Cin)
and produces 2-bit output, and these can be referred to as output
carry (Cout) and sum (S).
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https://drive.google.com/file/d/1fkq0hXMG_YQfTryRxnGMpq-Cbop-NGuo/view?usp=sharing

Department of ECE
2.5.3 Half Subtractor
 Half Subtractor is used for the purpose of subtracting two single bit numbers.
 Half subtractors have no scope of taking into account “Borrow-in” from the
previous circuit.
 To overcome this drawback, full subtractor comes into play.
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Department of ECE
2.5.3 Half Subtractor Design
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Inputs Outputs
A B D (Difference) b (Borrow)
0 0 0 0
0 1 1 1
1 0 1 0
1 1 0 0

Department of ECE
2.5.4 Full Subtractor
 Full Subtractor is a combinational logic circuit.
 It is used for the purpose of subtracting two single bit numbers.
 It also takes into consideration borrow of the lower significant stage.
 Thus, full subtractor has the ability to perform the subtraction of three bits.
 Full subtractor contains 3 inputs and 2 outputs (Difference and Borrow) as
shown-
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https://drive.google.com/file/d/1Unow3pvAvygc_kk4c5rAE_Rks1PeridA/view?usp=sharing

Department of ECE
2.5.4 Full Subtractor Design
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Inputs
Outputs
A B Bin
Bout
(Borrow)
D
(Difference)
0 0 0 0 0
0 0 1 1 1
0 1 0 1 1
0 1 1 1 0
1 0 0 0 1
1 0 1 0 0
1 1 0 0 0
1 1 1 1 1

Department of ECE
2.6 Serial and ParallelAdders
2.6.1 Introduction
 As, we already know, a full adder was used to add two 1-bit binary
numbers and the additional carry bit (Cin).
 But, to add two n-bit binary numbers, we will require n-number of full
adders.
 The carry-out of each full-adder is also connected to the carry-in of the
next full-adder in the higher-order.
 A Binary Parallel adder is used to add two numbers in parallel form
and to produce the sum bits as parallel outputs.
 In two numbers, one is addend and the other is augend and both are
added parallelly to get the sum.
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Department of ECE
2.6.1 Introduction
 As, we already know, a full adder was used to add two 1-bit binary
numbers and the additional carry bit (Cin).
 But, to add two n-bit binary numbers, we will require n-number of full
adders.
 The carry-out of each full-adder is also connected to the carry-in of the
next full-adder in the higher-order.
 A Binary Parallel adder is used to add two numbers in parallel form
and to produce the sum bits as parallel outputs.
 In two numbers, one is addend and the other is augend and both are
added parallelly to get the sum.
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https://drive.google.com/file/d/1zr59hARmII5xIEjOtus_3xobI-SJmGHD/view?usp=sharing

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2.4.1 Binary Parallel Adder
 Two n+1-bit binary number A and B of the form
A: An An-1 An-2 ... A3 A2 A1 A0 (Augend)
B: Bn Bn-1 Bn-2 ... B3 B2 B1 B0 (Addend)
4-bit Parallel adder
 In the block diagram, A0 and B0 represent the LSB of the four-bit
words A and B
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Department of ECE
c3 c2 c1 .
+ x3 x2 x1 x0
+ y3 y2 y1 y0
────────
Cy S3 S2 S1 S0
FA
x3 x2 x1 x0
FA
FA
FA
y3 y2 y1 y0
S3 S2 S1 S0
C4 C3 C2 C1
0
Binary Adder
x3x2x1x0 y3y2y1y0
S3S2S1S0
C0
Cy
Carry
Propagate
Addition
2.4.1 Binary Parallel Adder
Module 2

Department of ECE
CPA
A3 A2 A1 A0 B3 B2 B1 B0
S3 S2 S1 S0
C0
Cy
CPA
A3 A2 A1 A0 B3 B2 B1 B0
S3 S2 S1 S0
C0
Cy
x3 x2 x1 x0
y3 y2 y1 y0
x7 x6 x5 x4
y7 y6 y5 y4
S3 S2 S1 S0
S7 S6 S5 S4
0
2.4.1 Binary Parallel Adder (Carry Propagate Adder)

Department of ECE
2.4.2 CLA adder
Motivation behind Carry Look-Ahead Adder :
 In ripple carry adders, for each adder block, the two bits that are to be
added are available instantly.
 However, each adder block waits for the carry to arrive from its
previous block.
 So, it is not possible to generate the sum and carry of any block until
the input carry is known.
 The ith block waits for the i-1th block to produce its carry. So there will
be a considerable time delay which is carry propagation delay.
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https://drive.google.com/file/d/1N7H_GgZU3JN6bzUimKHi45FlqmvKJXHJ/view?usp=sharing

Department of ECE
2.4.2 CLA adder
 Consider the above 4-bit ripple carry adder. The sum S4 is produced by the
corresponding full adder as soon as the input signals are applied to it.
 But the carry input C4 is not available on its final steady state value until carry
C3 is available at its steady state value.
 Similarly C3 depends on C2 and C2 on C1.
 Therefore, though the carry must propagate to all the stages in order that
output S3 and carry C4 settle their final steady-state value.
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Department of ECE
2.4.2 CLA adder
 Consider the above 4-bit ripple carry adder. The sum S4 is produced by the
corresponding full adder as soon as the input signals are applied to it.
 But the carry input C4 is not available on its final steady state value until carry
C3 is available at its steady state value.
 Similarly C3 depends on C2 and C2 on C1.
 Therefore, though the carry must propagate to all the stages in order that
output S3 and carry C4 settle their final steady-state value.
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2.4.2 CLA adder
 The propagation time is equal to the propagation delay of each adder block,
multiplied by the number of adder blocks in the circuit.
 For example, if each full adder stage has a propagation delay of 20
nanoseconds, then S3 will reach its final correct value after 60 (20 × 3)
nanoseconds.
 The situation gets worse, if we extend the number of stages for adding more
number of bits.
Carry Look-ahead Adder :
A carry look-ahead adder reduces the propagation delay by introducing more
complex hardware. In this design, the ripple carry design is suitably transformed
such that the carry logic over fixed groups of bits of the adder is reduced to two-
level logic.
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2.4.1 CLA adder
Carry Look-ahead Adder :
Consider the full adder circuit shown with corresponding truth table. We define
two variables as ‘Carry Generate’ (Gi) and ‘Carry Propagate’ (Pi) then,
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2.4.1 CLA adder

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 4-bits plus 4-bits
 Operands and Result: 0 to 9
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+ x3 x2 x1 x0
+ y3 y2 y1 y0
────────
Cy S3 S2 S1 S0
X +Y x3 x2 x1 x0 y3 y2 y1 y0 Sum Cy S3 S2 S1 S0
0 + 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0
0 + 1 0 0 0 0 0 0 0 1 = 1 0 0 0 0 1
0 + 2 0 0 0 0 0 0 1 0 = 2 0 0 0 1 0
0 + 9 0 0 0 0 1 0 0 1 = 9 0 1 0 0 1
1 + 0 0 0 0 1 0 0 0 0 = 1 0 0 0 0 1
1 + 1 0 0 0 1 0 0 0 1 = 2 0 0 0 1 0
1 + 8 0 0 0 1 1 0 0 0 = 9 0 1 0 0 1
1 + 9 0 0 0 1 1 0 0 1 = A 0 1 0 1 0
2 + 0 0 0 1 0 0 0 0 0 = 2 0 0 0 1 0
9 + 9 1 0 0 1 1 0 0 1 = 12 1 0 0 1 0
Invalid Code
Wrong BCD Value
0001 1000
2.7 BCD Adder

Department of ECE
Truth table for corrective Design
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X +Y x3 x2 x1 x0 y3 y2 y1 y0 Sum Cy S3 S2 S1 S0 Required BCD Output Value
9 + 0 1 0 0 1 0 0 0 0 = 9 0 1 0 0 1 0 0 0 0 1 0 0 1 = 9
9 + 1 1 0 0 1 0 0 0 1 = 10 0 1 0 1 0 0 0 0 1 0 0 0 0 = 16
9 + 2 1 0 0 1 0 0 1 0 = 11 0 1 0 1 1 0 0 0 1 0 0 0 1 = 17
9 + 3 1 0 0 1 0 0 1 1 = 12 0 1 1 0 0 0 0 0 1 0 0 1 0 = 18
9 + 4 1 0 0 1 0 1 0 0 = 13 0 1 1 0 1 0 0 0 1 0 0 1 1 = 19
9 + 5 1 0 0 1 0 1 0 1 = 14 0 1 1 1 0 0 0 0 1 0 1 0 0 = 20
9 + 6 1 0 0 1 0 1 1 0 = 15 0 1 1 1 1 0 0 0 1 0 1 0 1 = 21
9 + 7 1 0 0 1 0 1 1 1 = 16 1 0 0 0 0 0 0 0 1 0 1 1 0 = 22
9 + 8 1 0 0 1 1 0 0 0 = 17 1 0 0 0 1 0 0 0 1 0 1 1 1 = 23
9 + 9 1 0 0 1 1 0 0 1 = 18 1 0 0 1 0 0 0 0 1 1 0 0 0 = 24
2.7 BCD Adder

Department of ECE
 Correct Binary Adder’s Output (+6)
 If the result is between ‘A’ and ‘F’
 If Cy = 1
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S3 S2 S1 S0 Err
0 0 0 0 0
1 0 0 0 0
1 0 0 1 0
1 0 1 0 1
1 0 1 1 1
1 1 0 0 1
1 1 0 1 1
1 1 1 0 1
1 1 1 1 1
S1
S2
S3
1 1 1 1
1 1
S0
Err = S3 S2 + S3 S1
2.7 BCD Adder

Department of ECE
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Binary Adder
A3 A2 A1 A0 B3 B2 B1 B0
S3 S2 S1 S0
Ci
Cy
Binary Adder
A3 A2 A1 A0 B3 B2 B1 B0
S3 S2 S1 S0
Ci
Cy
0 0
0
0
S3 S2 S1 S0
Cy
x3 x2 x1 x0 y3 y2 y1 y0
Err
2.7 BCD Adder
https://drive.google.com/file/d/1FxgQuWkDL6mHqYThoYRo0uLpkJ7zzl_-/view?usp=sharing

Combinational Circuits Design in Digital System Design.pptx

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