ECE- 1551 DIGITAL LOGIC

LECTURE 5: BINARY LOGIC AND DIGITAL LOGIC GATES

Assistant Prof. Fareena SaqibFlorida Institute of Technology

Fall 2016, 01/28/2016

Recap

Unsigned Numbers and Signed Numbers. Signed binary system

signed‐magnitude Signed- Complement

Arithmetic Operations Addition Subtraction

Binary Codes

Agenda

Binary logic Digital logic gates

Binary Logic

Binary logic consists of binary variables and a set of logical operations. Variables can take on two discrete values 0 or 1. Operations assume logical meaning. There are three basic logical operations: AND, OR,

and NOT.

Variables: The two values the variables assume may be called by different names (true and false, yes

and no, etc. Consider it in terms of bits and assign the values 1 and 0.

The variables are designated by letters of the alphabet, such as A, B, C, x, y, z, etc.,

Operations There are three basic logical operations: AND, OR, and NOT.

Each operation produces a binary result, denoted by z.

Binary Logic-Basic Logical Operations

1. AND: This operation is represented by a dot or by the absence of an operator.

For example, x . y = z or xy = z is read “x AND y is equal to z.” The logical operation AND is interpreted to mean that z = 1 if and only if x = 1 and y = 1;

otherwise z = 0. (Remember that x, y, and z are binary variables and can be equal either to 1 or 0, and nothing else.) The result of the operation x . y is z.

2. OR: This operation is represented by a plus sign. For example, x + y = z is read “x OR y is equal to z,” meaning that z = 1 if x = 1 or if y = 1

or if both x = 1 and y = 1. If both x = 0 and y = 0, then z = 0.

3. NOT: This operation is represented by a prime (sometimes by an overbar). For example, x’ = z is read “not x is equal to z,” meaning that z is what x is not. In other words, if x = 1, then z = 0, but if x = 0, then z = 1. The NOT operation is also

referred to as the complement operation, since it changes a 1 to 0 and a 0 to 1, i.e., the result of complementing 1 is 0, and vice versa.

Binary Logic

Definitions of logical operations may be listed in a compact form called truth tables.

A truth table is used to illustrate how the output of a gate responds to all possible combinations on the inputs to the gate.

Logic Gates

In electronics, a logic gate is an idealized or physical device implementing a Boolean function

A Boolean function describes how to determine a Boolean value output based on some logical calculation from Boolean inputs. 

(From Textbook) Logic gates are electronic circuits that operate on one or more input signals to produce an output signal.

xy

Z= x.y xy

Z= x+y x X’

Two input AND gate Two input OR gate Not gate or Inverter

AND Gate

The output of an AND gate is true (logic 1) if and only if all of the inputs to the gate are true (logic 1).

The AND gate can be illustrated with a series connection of manual switches or transistor switches.

AND Gate

The number of combinations of a truth table is equal to 2N where N is the number of inputs.

So a 2 input gate would have 22 outputs or 4

INPUT OUTPUTA B X0 0 00 1 01 0 01 1 1

OR Gate

The output of an OR gate is true (logic 1) if any or all of the inputs are true (logic 1). 

The OR gate can be illustrated with a parallel connection of manual switches or transistor switches.

OR Gate

INPUT OUTPUTA B A OR B0 0 00 1 11 0 11 1 1

Not Gate - Inverter

The output of an inverter is the complement (opposite) of the input. When the input to an inverter is high (1) the output is low (0); When the input is low, the output is high.

In Boolean Algebra the inverter operation is shown by placing a bar over the variable.

INPUT OUTPUT

A NOT A0 11 0

Input-output signals for gates- Summary

NAND, NOR, XOR, and XNOR Gates

NAND is compliment of AND

NOR is compliment of OR

Interesting Fact:  NOR gates alone (or alternatively NAND gates alone) can be used to reproduce the functions of all the other logic gates. They are also called universal logic gates

INPUT OUTPUTA B X0 0 10 1 11 0 11 1 0

INPUT OUTPUTA B X0 0 10 1 01 0 01 1 0

AB (A.B)’

A

B(A+B)’

XOR Gate

XOR is exclusive OR is a digital logic gate that implements an exclusive or; that is, a true output (1/HIGH) results if one, and only one, of the inputs to the gate is true. If both inputs are false (0/LOW) or both are true, a false output results.

INPUT OUTPUT

A B X0 0 00 1 11 0 11 1 0

A

BA xor B

XNOR Gate

XNOR is compliment of XOR

INPUT OUTPUTA B X0 0 10 1 01 0 01 1 1

Using Integrated Circuit (IC) Logic Gates

All logic gates are available in both TTL and CMOS logic families.

All the gates are available in configurations of from two inputs per gate up to eight inputs per gate.

A TTL or CMOS manual should be consulted for proper circuit configuration and pin assignment.

Data sheets include  limits and conditions set by the manufacturer as well as DC and AC characteristics.

Example: Enable and Disable Functions Using And gate

Example: Enable and Disable Functions Using And IC

Logic Circuits (Discussed in future)

Logic circuits include such devices as multiplexers, registers, arithmetic logic units (ALUs), and computer memory, all the way up through complete microprocessors, which may contain

In modern practice, most gates are made from field-effect transistors (FETs), particularly MOSFETs (metal–oxide–semiconductor field-effect transistors).more than 100 million gates.

Compound logic gates AND-OR-Invert (AOI) and OR-AND-Invert (OAI) are often employed in circuit design because their construction using MOSFETs is simpler and more efficient than the sum of the individual gates. (will cover in later lectures)

XOR Implantation using BASIC gates

Example

[(AB)’ + (CD’ + C’D)’]’

Next Class

Boolean Algebra – Chapter 2