DIGITAL LOGIC GATES

Digital Logic | Logic Gates :

In Boolean Algebra, there are three basic operations,  which are analogous to dis-junction, conjunction, and negation in propositional logic. Each of these operations has a corresponding logic gate. Apart from these there are a few other logic gates as well.
Digital logic gates may have more than one input, (A, B, C, etc.) but generally only have one digital output, (Q). Individual logic gates can be connected together to form combinational or sequential circuits, or larger logic gate functions.
Integrated Circuits or IC’s as they are more commonly called, can be grouped together into families according to the number of transistors or “gates” that they contain. For example, a simple AND gate my contain only a few individual transistors, were as a more complex microprocessor may contain many thousands of individual transistor gates. Integrated circuits are categorised according to the number of logic gates or the complexity of the circuits within a single chip with the general classification for the number of individual gates given as:

Logic Gates –

• AND gate(.) – The AND gate gives an output of 1 if both the two inputs are 1, it gives 0 otherwise.
• OR gate(+) – The OR gate gives an output of 1 if either of the two inputs are 1, it gives 0 otherwise.

NOT gate(‘) – The NOT gate gives an output of 1 input is 0 and vice-versa.

• XOR gate(⊕) – The XOR gate gives an output of 1 if either both inputs are different, it gives 0 if they are same.

Three more logic gates are obtained if the output of above-mentioned gates is negated.

• NAND gate(↑)- The NAND gate (negated AND) gives an output of 1 if both inputs are 0, it gives 1 otherwise.
• NOR gate(↓)- The NOR gate (negated OR) gives an output of 1 if both inputs are 0, it gives 1 otherwise.

XNOR gate(⊙)- The XNOR gate (negated XOR) gives an output of 1 both inputs are same and 0 if both are different.

Boolean Algebra :

Boolean Algebra is used to analyze and simplify the digital (logic) circuits. It uses only the binary numbers i.e. 0 and 1. It is also called as Binary Algebra or logical Algebra

Following are the important rules used in Boolean algebra.

1.   Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW.2.   Complement of a variable is represented by an overbar (-).

3.  ORing of the variables is represented by a plus (+) sign between them. For example ORing of A, B, C is represented as A + B + C.

4.  Logical ANDing of the two or more variable is represented by writing a dot between them such as A.B.C. Sometime the dot may be omitted like ABC.

Digital Logic States :

The Digital Logic Gate is the basic building block from which all digital electronic circuits and microprocessor based systems are constructed from. Basic digital logic gates perform logical operations of AND, OR and NOT on binary numbers.

In digital logic design only two voltage levels or states are allowed and these states are generally referred to as Logic “1” and Logic “0”, High and Low, or True and False. These two states are represented in Boolean Algebra and standard truth tables by the binary digits of “1” and “0” respectively

Most digital logic gates and digital logic systems use “Positive logic”, in which a logic level “0” or “LOW” is represented by a zero voltage, 0v or ground and a logic level “1” or “HIGH” is represented by a higher voltage such as +5 volts, with the switching from one voltage level to the other, from either a logic level “0” to a “1” or a “1” to a “0” being made as quickly as possible to prevent any faulty operation of the logic circuit.

Every Logic gate has a graphical representation or symbol associated with it. Below is an image which shows the graphical symbols and truth tables associated with each logic gate.

Logic AND Gate :

A Logic AND Gate is a type of digital logic gate whose output goes HIGH to a logic level 1 when all of its inputs are HIGH.

The output state of a “Logic AND Gate” only returns “LOW” again when ANY of its inputs are at a logic level “0”. In other words for a logic AND gate, any LOW input will give a LOW output.

The logic or Boolean expression given for a digital logic AND gate is that for Logical Multiplication which is denoted by a single dot or full stop symbol, ( . ) giving us the Boolean expression of:  A.B = Q.

Logic OR Gate :

A Logic OR Gate is a type of digital logic gate whose output goes HIGH to a logic level 1 when one or more of its inputs are HIGH.

The output, Q of a “Logic OR Gate” only returns “LOW” again when ALL of its inputs are at a logic level “0”. In other words for a logic OR gate, any “HIGH” input will give a “HIGH”, logic level “1” output.

The logic or Boolean expression given for a digital logic OR gate is that for Logical Addition which is denoted by a plus sign, ( + ) giving us the Boolean expression of:  A+B = Q.

Exclusive-OR Gate :

The Exclusive-OR logic function is a very useful circuit that can be used in many different types of computational circuits.

In the previous tutorials, we saw that by using the three principal gates, AND Gate, the OR Gate and the NOT Gate, we can build many other types of logic gate functions, such as a NAND Gate and a NOR Gate or any other type of digital logic function we can imagine.

But there are two other types of digital logic gates which although they are not a basic gate in their own right as they are constructed by combining together other logic gates, their output Boolean function is important enough to be considered as complete logic gates. These two “hybrid” logic gates are called the Exclusive-OR (Ex-OR) Gate and its complement the Exclusive-NOR (Ex-NOR) Gate.

Previously, we saw that for a 2-input OR gate, if A = “1”, OR B = “1”, OR BOTH A + B = “1” then the output from the digital gate must also be at a logic level “1” and because of this, this type of logic gate is known as an Inclusive-OR function. The gate gets its name from the fact that it includes the case of Q = “1” when both A and B = “1”.

If however, an logic output “1” is obtained when ONLY A = “1” or when ONLY B = “1” but NOT both together at the same time, giving the binary inputs of “01” or “10”, then the output will be “1”. This type of gate is known as an Exclusive-OR function or more commonly an Ex-Or function for short. This is because its boolean expression excludes the “OR BOTH” case of Q = “1” when both A and B = “1”.

In other words the output of an Exclusive-OR gate ONLY goes “HIGH” when its two input terminals are at “DIFFERENT” logic levels with respect to each other.

An odd number of logic “1’s” on its inputs gives a logic “1” at the output. These two inputs can be at logic level “1” or at logic level “0” giving us the Boolean expression of:  Q = (A ⊕ B) = A.B + A.B

The Exclusive-OR Gate function, or Ex-OR for short, is achieved by combining standard logic gates together to form more complex gate functions that are used extensively in building arithmetic logic circuits, computational logic comparators and error detection circuits.

Logic NAND GATE :

The Logic NAND Gate is a combination of a digital logic AND gate and a NOT gate connected together in series.

The logic or Boolean expression given for a logic NAND gate is that for Logical Addition, which is the opposite to the AND gate, and which it performs on the complements of the inputs. The Boolean expression for a logic NAND gate is denoted by a single dot or full stop symbol, ( . ) with a line or Overline, ( ‾‾ ) over the expression to signify the NOT or logical negation of the NAND gate giving us the Boolean expression of:  A.B = Q.

Logic NOR Gate :

The Logic NOR Gate gate is a combination of the digital logic OR gate and an inverter or NOT gate connected together in series

The logic or Boolean expression given for a logic NOR gate is that for Logical Multiplicationwhich it performs on the complements of the inputs. The Boolean expression for a logic NOR gate is denoted by a plus sign, ( + ) with a line or Overline, ( ‾‾ ) over the expression to signify the NOT or logical negation of the NOR gate giving us the Boolean expression of:  A+B = Q.

Exclusive-NOR Gate :

The Exclusive-NOR Gate function is a digital logic gate that is the reverse or complementary form of the Exclusive-OR function.

Basically the “Exclusive-NOR Gate” is a combination of the Exclusive-OR gate and the NOT gate but has a truth table similar to the standard NOR gate in that it has an output that is normally at logic level “1” and goes “LOW” to logic level “0” when ANY of its inputs are at logic level “1”.

However, an output “1” is only obtained if BOTH of its inputs are at the same logic level, either binary “1” or “0”. For example, “00” or “11”. This input combination would then give us the Boolean expression of:  Q = (A ⊕ B) = A.B + A.B