Universal Gates: NAND and NOR Gates and their Conversions
Introduction to Universal Gates
In digital logic circuits, gates are fundamental building blocks that perform logical operations on binary inputs to produce binary outputs. Universal gates, such as the NAND gate and the NOR gate, play a crucial role in digital logic design by allowing the implementation of various logical operations using a single gate type. Conversions from NAND to other basic gates (NOT, AND, OR) and from NOR to other basic gates demonstrate the versatility and universal property of these gates. Understanding these conversions and properties is essential for any digital logic designer to create efficient and functional digital circuits. These gates are essential in designing complex digital systems, as they offer flexibility and reduce the number of unique gate types required in a circuit, which reducing the complexity of circuitry and improving efficiency.
Two commonly used universal gates are the NAND (NOT-AND) gate and the NOR (NOT-OR) gate.
NAND Gate
Overview and Truth Table
The NAND gate is a digital logic gate that performs the logical NOT-AND operation. It takes two or more inputs and produces an output that is the inverse of the AND operation's output. The NAND gate's truth table is as follows:
| Input A | Input B | Output |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
The symbol for the NAND gate is a standard logic gate symbol with a small circle at the output end, indicating the inversion operation.
Universal Property of NAND Gate
The NAND gate is considered a universal gate because it can be used to implement all other basic logic gates, including AND, OR, and NOT gates. This universality can be demonstrated through various conversions.
Conversion from NAND to NOT Gate
To convert a NAND gate into a NOT gate, both inputs are connected together, and the output is taken from this common connection point. In this configuration, when both inputs are the same, the output will be the inverse of that input, effectively emulating the behavior of a NOT gate.
Conversion from NAND to AND Gate
To convert a NAND gate into an AND gate, the output of the NAND gate is connected back to one of its inputs, while the other input remains independent. The input that is connected to the output acts as the controlling input. When both inputs of the NAND gate are high (1), the output will be low (0), effectively implementing the AND gate's behavior.
Conversion from NAND to OR Gate
To convert a NAND gate into an OR gate, two NAND gates are used in combination. The inputs of both NAND gates are connected together, and the outputs of both gates are connected to a third NAND gate. The resulting output of this third NAND gate is the desired OR gate behavior. This conversion is achieved by using De Morgan's theorem, which states that the complement of an AND operation is equivalent to the OR operation of the complements.
NOR Gate
Overview and Truth Table
The NOR gate is another fundamental digital logic gate that performs the logical NOT-OR operation. It takes two or more inputs and produces an output that is the inverse of the OR operation's output. The truth table for the NOR gate is as follows:
| Input A | Input B | Output |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
The symbol for the NOR gate is a standard logic gate symbol with a small circle at the output end, indicating the inversion operation.
Universal Property of NOR Gate
Similar to the NAND gate, the NOR gate is also a universal gate and can be used to implement all other basic logic gates through appropriate configurations.
Conversion from NOR to NOT Gate
To convert a NOR gate into a NOT gate, both inputs are connected together, and the output is taken from this common connection point. In this setup, when both inputs are the same, the output will be the inverse of that input, effectively implementing the NOT gate behavior.
Conversion from NOR to OR Gate
To convert a NOR gate into an OR gate, the output of the NOR gate is connected back to one of its inputs, while the other input remains independent. The input that is connected to the output acts as the controlling input. When both inputs of the NOR gate are low (0), the output will be high (1), effectively implementing the OR gate's behavior.
Conversion from NOR to AND Gate
To convert a NOR gate into an AND gate, two NOR gates are used in combination. The inputs of both NOR gates are connected together, and the outputs of both gates are connected to a third NOR gate. The resulting output of this third NOR gate is the desired AND gate behavior. This conversion is again achieved using De Morgan's theorem.