LOGIC CIRCUITS II - LOGIC EQUATIONS (SS2 FIRST TERM)

LOGIC EQUATIONS

Aside representing the functioning of a logic gate with truth table and grammatical definition, the use of logic equations can be used not only to represent logic gates and circuits, but also with the usage of some theorems and equivalences, to reduce the number of terms involved, simplifying the equation.

Symbolic logic uses values, variables and operations;

TRUE is represented as 1 while FALSE as 0.

Variables are represented by letters and can have one or two values, either 0 or 1. Operations are functions of one or more variables.

AND gate equation

The AND gate operation can also be expressed by a Boolean algebraic equation. For 2 – input AND gate, the equation is;

X = A.B

The symbol for AND gate operation is a center dot. It does not mean multiplication. The equation read X equals to A and B, which simply mean that the output of the gate is a logic 1 when A and B inputs are in their 1 states.

OR gate equation

The Boolean algebraic equation expression is given as;

X = A + B

The equation read X equals to A or B, which simply mean that the output of the gate is a logic 1 when A or B inputs are in their 1 states.

NOT gate equation

The NOT gate operation can be expressed by a Boolean algebraic equation as;

X = A

A complement bar is placed over the assigned input letter. The expression is read as X is equal A which simply means that the output state is opposite of the logic state applied to the input.

ALTERNATIVE LOGIC CIRCUIT

These are gates that are formed from combination of two logic gates. There are two types of alternative logic gate:

NAND GATE

A NAND gate is the combination of an AND gate and NOT gate. It operates the same as an AND gate but the output will be opposite. Remember, the NOT gate does not always have to be the output leg; it could be used to invert an input signal also.

LOGIC SYMBOL FOR THE “NAND” GATE

Notice the circle on output C.

TRUTH TABLE FOR THE “NAND” GATE

INPUT	INPUT	OUTPUT
A	B	C
0	0	1
0	1	1
1	0	1
1	1	0

 

 

 

 

NAND GATE EQUATION

The NAND gate operation can also be expressed by a Boolean algebra equation. For a 2 – input NAND gate, the equation is:

X = A.B

This equation read X equal to A and B NOT, which simply means that the output of the gate is not a logic 1 when A and B inputs are their 1 states.

NOR GATE

A NOR gate is the combination of both an OR gate and NOT gate. It operates the same as an OR gate, but the output will be the opposite.

 

TRUTH TABLE FOR THE “NOR” GATE

INPUT	INPUT	OUTPUT
A	B	C
0	0	1
0	1	0
1	0	0
1	1	0

 

 

 

  

NOR GATE EQUATION

The NOR gate operation can also be expressed by a Boolean algebra equation. For a 2 – input NOR gate, the equation is:

X = A + B

The expression is the same as the OR gate with an over bar above the entire portion of the equation representing the input. This equation read X equal to A or B NOT, which simply means that the output of the gate is not a logic 1 when A or B are in their 1 states.

 USES OF LOGIC GATES

Logic gates are in fact the building block of digital electronics, they are formed by the combination of transistors (either BJT or MOSFET) to realise some digital operations like logical OR, NOT, AND etc. Every digital product like computers, mobile phones, calculators, even digital watches contains logical gates.

 XOR GATE

The XOR (exclusive – OR) gate acts in the same way as the logical “either or”. The output is “True” if either but not both, of the inputs are “true”. The output is “false” or if both inputs are “true”.

 LOGIC SYMBOL FOR “XOR” GATE

TRUTH TABLE FOR THE “XOR” GATE

INPUT	INPUT	OUTPUT
A	B	Y
0	0	0
0	1	1
1	0	1
1	1	0

           

 

XOR COMPARATOR

Comparator is a combinational logic circuit that compares the magnitudes of two binary quantities to determine which one has the greater magnitude.

In other word, comparator determines the relationship of two binary quantities. A XOR can be used as basic comparator.

 

One of the most common uses for XOR gates is to add two binary numbers. For this operation to work, the XOR gate must be used in combination with an AND gate.