Binary Parallel Adder, Decimal Adder

1. Digital Logic Design
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Combinational Logic
Prof. Shehzad Ali

2. Introduction
 Logic circuits for digital systems may be combinational or
sequential.
 Consists of logic gates whose outputs at any time are
determined from only the present combination of inputs.
 Performs an operation that can be specified logically by a
set of Boolean functions.
Combinational Circuit
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3. Digital Circuits
Combinational Circuits
Logic circuits for digital system
◦ Combinational circuits
◦ the outputs are a function of the current inputs
◦ Sequential circuits
◦ contain memory elements
◦ the outputs are a function of the current inputs and the state of
the memory elements
◦ the outputs also depend on past inputs
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4. Digital Circuits
Combinational circuits
A combinational circuits
◦ 2
n
possible combinations of input values
◦ Specific functions
◦ MSI (Medium-Scale Integration) circuits or standard cells
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Combinational
Logic Circuit
n input
variables
m output
variables
 Adders, subtractors, comparators, decoders, encoders and
multiplexers

5. Sequential Circuit
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 Employs storage elements in addition to logic gates.
 Their outputs are a function of the inputs and the state of
the storage elements.
 Because the state of the storage elements is a function of
previous inputs, the outputs of a sequential circuit depend
not only on present value of inputs, but also on past
inputs.
Sequential Circuit

6. Digital Circuits
Design Procedure
The design procedure of combinational circuits
◦ State the problem (system spec.)
◦ determine the inputs and outputs
◦ the input and output variables are assigned symbols
◦ derive the truth table
◦ derive the simplified Boolean functions
◦ draw the logic diagram and verify the correctness
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7. 7
Half adder
 Is a combinational circuit that performs the addition of two bits.
0 + 0 = 0 ; 0 + 1 = 1 ; 1 + 0 = 1 ; 1+ 1 = 10
Elementary Operations
Truth Table
 two input variables
 x, y.
 two output variables.
 C (output carry), S (least
significant bit of the sum).
Binary Adder-Subtractor

8. 8
Half adder
 S = x'y+xy'
 C = xy
Simplified Boolean Function
(Sum of Products)
Logic Diagram (Sum of
Products)
Binary Adder-Subtractor

9. 9
Half adder
S=xy
C = xy
Simplified Boolean Function (XOR
and AND gates)
Logic Diagram
(XOR and AND gates)
Binary Adder

10. Digital Circuits
Binary Adder
◦ S = x'y+xy'
◦ C = xy
For the SUM bit:
SUM = A XOR B = A ⊕ B
For the CARRY bit:
CARRY = A AND B = A.B
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11. Digital Circuits 11
Functional Block: Full-Adder
 It is a combinational circuit that performs the arithmetic sum of three
bits (two significant bits and previous carry).
 It is similar to a half adder, but includes a carry-in bit from lower
stages.
 Two half adders can be employed to implement a full adder.
Inputs & Outputs
 Three input bits:
 x, y : two significant bits
 Z : the carry bit from the previous lower significant bit.
 Two output variables:
 C (output carry), S (least significant bit in sum).
Binary Adder-Subtractor

12. Digital Circuits
 For a carry-in (Z) of 0,
it is the same as
half-adder:
 For a carry- in
(Z) of 1:
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Z 0 0 0 0
X 0 0 1 1
+ Y + 0 + 1 + 0 + 1
C S 00 0 1 0 1 1 0
Z 1 1 1 1
X 0 0 1 1
+ Y + 0 + 1 + 0 + 1
C S 0 1 1 0 1 0 11
Functional Block: Full-Adder
Operations
Binary Adder

13. Digital Circuits
Full-Adder
Full-Adder
◦ The arithmetic sum of three input
bits
◦ three input bits
◦ x, y: two significant bits
◦ z: the carry bit from the previous lower
significant bit
◦ Two output bits: C, S
x y z C S
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 1

14. Digital Circuits 14
Full-Adder
Then the Boolean expression for a full adder is as follows.
For the SUM (S) bit:
SUM = (A XOR B) XOR Cin = (A ⊕ B) ⊕ Cin
For the CARRY-OUT (Cout) bit:
CARRY-OUT = A AND B OR Cin(A XOR B) = A.B + Cin(A ⊕ B)