Lesson Objectives:
In addition to AND, OR, and NOT gates, other logic gates like NAND and NOR are also used in the design of digital circuits.
The NOT circuit inverts the logic sense of a binary signal.
The small circle (bubble) at the output of the graphic symbol of a NOT gate is formally called a negation indicator and designates the logical complement.
The objectives of this lesson are to learn about:
1. Universal gates - NAND and NOR.
2. How to implement NOT, AND, and OR gate using NAND gates only.
3. How to implement NOT, AND, and OR gate using NOR gates only.
4. Equivalent gates.
5. Two-level digital circuit implementations using universal gates only.
6. Two-level digital circuit implementations using other gates.
NAND Gate:
The NAND gate represents the complement of the AND operation. Its name is an abbreviation of NOT AND.
The graphic symbol for the NAND gate consists of an AND symbol with a bubble on the output, denoting that a complement operation is performed on the output of the AND gate.
The truth table and the graphic symbol of NAND gate is shown in the figure.
The truth table clearly shows that the NOR operation is the complement of the OR.
Universal Gates:
A universal gate is a gate which can implement any Boolean function without need to use any other gate type.
The NAND and NOR gates are universal gates.
In practice, this is advantageous since NAND and NOR gates are economical and easier to fabricate and are the basic gates used in all IC digital logic families.
In fact, an AND gate is typically implemented as a NAND gate followed by an inverter not the other way around!!
Likewise, an OR gate is typically implemented as a NOR gate followed by an inverter not the other way around!!
NAND Gate is a Universal Gate:
To prove that any Boolean function can be implemented using only NAND gates, we will show that the AND, OR, and NOT operations can be performed using only these gates.
Implementing an Inverter Using only NAND Gate
The figure shows two ways in which a NAND gate can be used as an inverter (NOT gate).
1. All NAND input pins connect to the input signal A gives an output A’.
2. One NAND input pin is connected to the input signal A while all other input pins are connected to logic 1. The output will be A’.
Implementing AND Using only NAND Gates
An AND gate can be replaced by NAND gates as shown in the figure (The AND is replaced by a NAND gate with its output complemented by a NAND gate inverter).
Implementing OR Using only NAND Gates
An OR gate can be replaced by NAND gates as shown in the figure (The OR gate is replaced by a NAND gate with all its inputs complemented by NAND gate inverters).
Thus, the NAND gate is a universal gate since it can implement the AND, OR and NOT functions.
NAND Gate is a Universal Gate: To prove that any Boolean function can be implemented using only NOR gates, we will show that the AND, OR, and NOT operations can be performed using only these gates.
Implementing an Inverter Using only NOR Gate
The figure shows two ways in which a NOR gate can be used as an inverter (NOT gate).
1. All NOR input pins connect to the input signal A gives an output A’.
2. One NOR input pin is connected to the input signal A while all other input pins are connected to logic 0. The output will be A’.
Implementing OR Using only NOR Gates
An OR gate can be replaced by NOR gates as shown in the figure (The OR is replaced by a NOR gate with its output complemented by a NOR gate inverter)
Implementing AND Using only NOR Gates
An AND gate can be replaced by NOR gates as shown in the figure (The AND gate is replaced by a NOR gate with all its inputs complemented by NOR gate inverters
Equivalent Gates:
The shown figure summarizes important cases of gate equivalence. Note that bubbles indicate a complement operation (inverter).
A NAND gate is equivalent to an inverted-input OR gate.
An AND gate is equivalent to an inverted-input NOR gate.
A NOR gate is equivalent to an inverted-input AND gate.
An OR gate is equivalent to an inverted-input NAND gate.
This is how to do the universal gates.
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