Babylon University
The college of science for girls
Logic Design
Lecture 5
Combinational Logic
Logic circuit for digital systems may be combinational or sequential. A combinational circuit consist of logic gates whose outputs at any time are determined directly from the present combination of inputs without regard to previous inputs.
A combinational circuit consist of input variables, logic gates, and output variables.
The logic gates accept signals from the inputs and generate signals to the outputs.
Combinational
Logic circuit
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n input m output
variables variables
for n input variables, there are 2 possible combinations of binary input values .for each possible input combination, there is one and one possible output combination, A combinational circuit can be described by m Boolean functions, one for each output variables. Each output function is expressed in terms of the n input variables.
Design procedure
1-the problem is stated
2-the number of available input variables and required output variables is determined
3-the input and output variables are assigned letter symbols.
4-the truth table that defines the required relationships between inputs and outputs is
Derived
5-the simplified Boolean function for each output is obtained
6-the logic diagram is drawn.
ADDER
Digital computers perform a variety of information processing takes, among the basic functions encountered are the various arithmetic operation. Is the addition of two binary digits
Half-Adder
This circuit needs two binary-inputs and two binary outputs
Input Output
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X Y
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(S) sum (C) carry
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0 0
0 1
1 0
1 1
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0 0
1 0
1 0
0 1
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The simplified Boolean functions for the two outputs can be obtained directly from the truth table the simplified sum of products expressions are:-
S = x`y + x y
C = x y
Ahmed M. Shhaab
Another implementation of
A half-adder is:
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Full-Adder
A full-adder is a combinational circuit that forms the arithmetic sum of three input bits. It consists of three inputs (x,y,z) and two outputs; S (sum) and C (carry)
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X Y Z
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S C
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0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
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0 0
1 0
1 0
0 1
1 0
0 1
0 1
1 1
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The simplified Boolean functions for outputs C & S are obtained using K-map:-
00 01 11 10
C = xy + xz + yz
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S = x`y`z +x`y z` + xy`z` + xyz
Ahmed M. Shhaab
Another implementation for the full-adder is by using two half-adders and one OR gate
S = z ( x y )
=z` ( xy` + x`y) + z(xy+ x`y )
=xy`z` + x`yz` + x`y`z +xyz
And the carry output is :-
C = z (xy`+ x`y ) + xy
=x y`z + x`yz + xy
Ahmed M. Shhaab
SUBTRACTION
Just as there are half-and full adders, there are half and full subtractions.
Half-Subtraction
A half-subtraction is a combinational circuit that subtracts two bits and produces their difference it also it has an output to specify if a 1 has been borrowed.
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X Y
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D B
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0 0
0 1
1 0
1 1
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0 0
1 1
1 0
0 0
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It is a obvious that the logic for D exactly the same as the logic for outputs in the half-adder
Full-Subtraction
A full subtraction is a combinational circuit that that performs a subtraction between two bits taking into account that a 1 may have been borrowed by a lower significant stage.
This circuit has three inputs and two outputs.
D = x`y`z` + x`yz + xy`z` + xyz
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X Y Z
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D B
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0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
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0 0
1 1
1 1
0 1
1 0
0 0
0 0
1 1
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Again we note the logic function for output D is the full subtraction is exactly the same as outputs in the full-adder. More over the output B resembles the function for C in the full-adder, except that the input variable X is complemented, therefore, it is possible to convert a full-adder into a full-sub merely complementing input X prior, to its application to the gates that from the carry output.
Ahmed M. Shhaab
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .