1. Logic Circuits Basic Building Blocks Dr John Cowell
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CSCI1412 Lecture 10
Logic Circuits
Basic Building Blocks
Dr John Cowell

2. Overview Logic gates & Truth tables Gate Operations Logic functions
Digital Logic
Combinational Circuits
Logic gates & Truth tables
Gate Operations
Logic functions
Adder circuits
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3. Digital Logic

4. Digital Logic Circuits
The function of a Digital Logic Circuit determines the values of the outputs for specified value of inputs
The function is normally implemented using electronic circuitry
Inputs and outputs are each in either of two states: TRUE HIGH ON 1 5V Or FALSE LOW OFF 0 0V
Digital Logic
Circuit
Outputs
Inputs

5. Digital Logic …
The Digital Logic Circuit may be as simple as a single logic gate or as complex as a computer’s processor
Binary logic deals with binary variables (0 or 1) and with operations that assume a logical meaning
It is used to describe, in algebraic or tabular form, the manipulation and processing of binary information
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6. Boolean Variables (algebra)
Boolean algebra differs in a major way from ordinary algebra in that boolean constants and variables are allowed to have only two possible values, 0 or 1
Boolean 0 and 1 are represented by the state of a voltage variable, or what is called the logic level
In boolean algebra, there are three basic logic operations: OR AND and NOT
These are implemented by logic gates
constructed from diodes, transistors, and resistors
connected in such a way that the circuit output is the result of a basic logic operation (OR, AND, NOT) performed upon the inputs
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7. Combinational Logic Circuits
There are Combinational logic circuits and Sequential logic circuits, but most realistic logic circuits are a mixture of both
Combinational logic circuits
Output values depend only on the current values of the inputs
there is no feedback from previous outputs (more on this later)
They are called combinational as they combine the operations of more than one logic gate
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8. Logic Gates and Truth Tables

9. Logic Gates
The manipulation of binary information is done by logic circuits galled gates
Integrated circuits (ICs) contain a number of logic gates, ranging from three or four in simple chips to many thousands in more complex parts
Each gate is represented with a distinct graphic symbol and its operation is described by means of an algebraic expression
The input-output relationship of the binary variables for each gate is represented in tabular form by a truth table
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10. Truth Table
A truth table is a means for describing how a logic circuit's output depends on the logic levels present at the circuit's inputs
In the following two-input logic circuit, the table lists all possible combinations of logic levels present at inputs A and B along with the corresponding output level X
When either input A OR B is 1, the output X is 1. Therefore the "?" in the box is an OR gate (OR gate is explained later)
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11. Gate Operations

12. AND Operation truth table
The expression X = A * B reads as "X equals A AND B“
The multiplication sign stands for the AND operation
same for ordinary multiplication of 1s and 0s
The AND operation produces a result of 1 occurs only for the single case when all of the input variables are 1
The output is 0 for any case where one or more inputs are 0
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13. OR Operation The expression X = A + B reads as "X equals A OR B“
truth table
The expression X = A + B reads as "X equals A OR B“
The + sign stands for the OR operation, not for ordinary addition
The OR operation produces a result of 1 when any of the input variable is 1
The OR operation produces a result of 0 only when all the input variables are 0
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14. NOT Operation
truth table
The NOT operation is unlike the OR and AND operations in that it can be performed on a single input variable
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15. NOT…..
For example, if the variable A is subjected to the NOT operation, the result x can be expressed as
x = A'
where the prime (') represents the NOT operation
This expression is read as:
x equals NOT A
x equals the inverse of A
x equals the complement of A
Each of these is in common usage and all indicate that the logic value of x = A' is opposite to the logic value of A
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16. XOR Operation Exclusive OR Returns: True only if either input is True
Means: this or that, but not both
Often used for:
cryptography
generating parity bits for error checking and fault tolerance
ie network messages

17. De Morgans Theorem

18. DeMorgan's Theorem (x+y)' = x' * y' (x*y)' = x' + y'
DeMorgan's theorems are extremely useful in simplifying expressions in which a product or sum of variables is inverted
The two theorems are:
(x+y)' = x' * y'
(x*y)' = x' + y'
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19. Universality of NAND Gate
It is possible to implement any logic expression using only NAND gates
ie no other type of gate is required
This is because NAND gates, in the proper combination, can be used to perform each of the Boolean operations OR, AND, and INVERT
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20. NAND Gate Examples
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21. NOT Gate fron NAND Gate A
A*A
(A*A)' 1

22. AND Gate from NAND GatesB
C=(A*B)'
X=(C*C)' 1

23. OR Gate From NAND Gates A B A' B'
X=(A' * B')' 1

24. Logic Functions

25. Logic Functions A = 1010 0101 B = 0000 1111 A*B = 0000 0101
Whilst the microcomputer can perform the normal arithmetical operations on binary numbers, it can also perform logical operations:
AND
For clearing bits OR
For setting bits
A =
B =
A*B =
A =
B =
A+B =
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26. NOT & XOR functions A = 1010 0101 NOT A = 0101 1010 A = 1010 0101
For inverting bits
XOR
For checking bits
A =
NOT A =
A =
B =
A xor B =
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27. Sequential Logic Circuits
Sequential logic circuits differ from combinatorial logic circuits in two main respects
The output of the system depends not only on the present external input(s) but also on the previous inputs
The same external input(s) can give a different output response
An important feature of sequential logical circuits, not present in combinational logic circuits, is the presence of feedback where the output from one or more logic gates is fed back into the input(s) of logic gates further back in the circuit
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28. Adder Circuits

29. Addition, using logic gates
Consider:
The circuit is performing binary addition of B to A (recalling that in binary 1+1 = 0 carry 1)
Such a circuit is called a half-adder 1
  CARRY  
  SUM  
  B  
  A  
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30. Full-Adder
A full adder is made by combining two half-adders and an additional OR-gate
A full adder adds two binary numbers AND the carry bit.
By extending the full adder long bibary n umbers can be added.
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31. Truth Table for a Full Adder
Input Carry
Output Carry
Sum 1

32. Find out more! What are ? READ: Sequential logic circuits
Programmable logic
READ:
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33. Lecture Summary Logic gates & Truth tables Gate Operations
Digital Logic
Combinational Circuits
Logic gates & Truth tables
Gate Operations
Logic functions
Adder circuits
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