It Logic Gates

Logic Gates

A logic gate is a circuit which uses digital signals as its inputs and outputs. What makes a circuit a gate is that each output depends entirely on the signals applied at the inputs. If these input signals change, then the output signal may also change. Digital circuits which use logic gates are usually arranged so that a logic 1 appears at an output only for some definite combination of input signals - for this reason these circuits are sometimes called combinational logic circuits.

Objectives  

 

Perform the three basic logic operations. Describe the operation of and construct the truth tables for the AND, NAND, OR, and NOR gates, and the NOT (INVERTER) circuit. Write the Boolean expression for the logic gates and combinations of logic gates. Implement logic circuits using basic AND, OR, and NOT gates.

Objectives (cont’d)  

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Use DeMorgan's theorems to simplify logic expressions. Use either of the universal gates (NAND or NOR) to implement a circuit represented by a Boolean expression. Explain the advantages of constructing a logiccircuit diagram using the alternate gate symbols versus the standard logic-gate symbols. Describe the concept of active-LOW and activeHIGH logic symbols.

Boolean Constants and Variables  Boolean

0 and 1 do not represent actual numbers but instead represent the state, or logic level. Logic 0 False Off Low No Open switch

Logic 1 True On High Yes Closed switch

Three Basic Logic Operations  OR  AND  NOT

Truth Tables 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. Inputs A 0 0 1 1

B 0 1 0 1

Output x 1 0 1 0

A ? B

x

OR Operation  Boolean

expression for the OR operation: x =A + B  The above expression is read as “x equals A OR B”

OR Gate  An

OR gate is a gate that has two or more inputs and whose output is equal to the OR combination of the inputs.

AND Operation  Boolean

expression for the AND operation: x =A B  The above expression is read as “x equals A AND B”

AND Gate  An

AND gate is a gate that has two or more inputs and whose output is equal to the AND product of the inputs.

NOT Operation     

The NOT operation is an unary operation, taking only one input variable. Boolean expression for the NOT operation: x= A The above expression is read as “x equals the inverse of A” Also known as inversion or complementation. Can also be expressed as: A’

A

NOT Circuit  Also

known as inverter.  Always take a single input  Application:

Describing Logic Circuits Algebraically  Any

logic circuits can be built from the three basic building blocks: OR, AND, NOT  Example 1: x = A B + C  Example 2: x = (A+B)C  Example 3: x = (A+B)  Example 4: x = ABC(A+D)

Examples 1,2

Examples 3

Example 4

Evaluating Logic-Circuit Outputs x

= ABC(A+D)

 Determine

the output x given A=0, B=1, C=1,

D=1.  Can also determine output level from a diagram

Solution

NOR Gate • Boolean expression for the NOR operation: • x=A+B

NAND Gate  Boolean

x=AB

expression for the NAND operation:

Boolean Theorems (SingleVariable)        

x* 0 =0 x* 1 =x x*x=x x*x’=0 x+0=x x+1=1 x+x=x x+x’=1

Boolean Theorems (Multivariable)         

x+y = y+x x*y = y*x x+(y+z) = (x+y)+z=x+y+z x(yz)=(xy)z=xyz x(y+z)=xy+xz (w+x)(y+z)=wy+xy+wz+xz x+xy=x x+x’y=x+y x’+xy=x’+y

DeMorgan’s Theorems  (x+y)’=x’y’

 (xy)’=x’+y’

 Extension

to N variables

Implications and alternative symbol for NOR function

Implications and alternative symbol for NAND function

Universality of NAND Gates

Universality of NOR Gates

Available ICs

Alternate Logic-Gate Representation