Logic Gates: Boolean Expressions and Digital Circuits

Logic Gates and Related Devices

• Logic gates are electronic circuits used to implement elementary logic (Boolean) expressions.
• Logic gates are the basic building blocks of combinational logic.
• Three basic logic gates include OR, AND, and NOT gates.
• Other logic gates are derived from the basic gates: NAND, NOR, EXCLUSIVE-OR, and EXCLUSIVE-NOR.
• Buffers and drivers are related devices.
• Truth tables and Boolean expressions are used to understand logic gate functions.
• Logic gates perform operations on two or more logic variables

Positive and Negative Logic

• Binary variables have logic '0' or logic '1' states.
• Digital systems represent logic states with different voltage or current levels.
• Positive logic system: Higher potential represents logic '1', lower represents logic '0'.
• Negative logic system: Higher potential represents logic '0', lower represents logic '1'.
• A positive OR behaves like a negative AND, and vice versa.
• A positive NOR behaves like a negative NAND, and vice versa.

Truth Table

• A truth table lists all possible input combinations and corresponding outputs of a logic system
• Boolean expressions relate the output with the inputs of a certain logic system.
• A logic circuit with n binary inputs has 2^n possible input combinations.

Logic Gates

• Digital systems including computers use logic gates
• Each basic logic gate is a piece of hardware or electronic circuit used to implement basic logic expressions.
• Boolean algebra is used to do manipulation with binary variables and simplify logic expressions.
• Electronic circuits called logic gates implement in a digital system.
• OR, AND, and NOT are the three basic logic gates.

OR Gate

• The OR gate performs an ORing operation on two or more logic variables
• The OR operation on two independent logic variables A and B is written as Y = A + B
• For a positive logic system, the output of an OR gate gives a logic '0' only when all inputs are logic '0', otherwise output is logic '1'.

AND Gate

• The AND gate is a logic circuit having two or more inputs and one output.
• Output of the AND gate translates to HIGH only when all inputs are in the HIGH state, otherwise the output is LOW.
• The AND operation on two independent logic variables A and B is written as Y = A.B.

NOT Gate

• The NOT gate is a one-input, one-output logic circuit whose output is always the complement of the input.
• It is also known as a 'complementing circuit' or an 'inverting circuit'
• Notation on a logic variable X is denoted as X

EXCLUSIVE-OR Gate

• The EX-OR gate is a two-input, one-output gate.
• EX-OR gate output is a logic '1' when the inputs are unlike and a logic '0' when the inputs are like.
• Multiple-input EX-OR logic functions can be implemented using more than one two-input gates.
• The output of a multiple-input EX-OR logic function is a logic ‘1' when the number of 1s in the input sequence is odd, and a logic '0' when the number of 1s in the input sequence is even (including zero)
• An all Os input sequence produces a logic '0' at the output.
• The output of a two-input EX-OR gate is expressed by Y = (A + B) = AB+ AB

NAND Gate

• NAND stands for NOT AND.
• An AND gate followed by a NOT circuit makes a NAND gate
• The output of a NAND gate is a logic '0' when all its inputs are a logic '1
• NAND gate operation is logically expressed as Y = A.B

NOR Gate

• NOR stands for NOT OR.
• An OR gate followed by a NOT circuit makes a NOR gate
• The output of a NOR gate is a logic '1' when all its inputs are logic '0
• The output of a two-input NOR gate is logically expressed as Y = (A+B)

EXCLUSIVE-NOR Gate

• EXCLUSIVE-NOR (EX-NOR) means NOT of EX-OR, i.e. the logic gate that is achieved by complementing the output of an EX-OR gate.
• The output of a two-input EX-NOR gate is a logic '1' when the inputs are like and a logic '0' when they are unlike.

INHIBIT Gate

• Produces a certain fixed logic level at the output irrespective of changes in the input logic level.

Universal Gates

• OR, AND, and NOT gates can construct the logic circuit for any given Boolean expression.
• NOR and NAND gates individually implement logic circuits corresponding to a given Boolean expression.
• Combinations of NAND or NOR gates can perform functions of basic logic gates

Gates with Open Collector/Drain Outputs

• Gates where an external resistor (pull-up resistor) connects between the output and the DC power supply.
• Used to make the logic gate perform the intended logic function.
• Referred to as gates with open collector output (TTL logic family) or open drain output (MOS logic family).
• Advantages include the capability of providing an ANDing operation when outputs of several gates are tied together through a common pull-up resistor,

Tristate Logic Gates

• Tristate logic gates have three possible output states (logic '1,' logic '0,' high-impedance).
• The high-impedance state is controlled by the ENABLE input
• Advantage: inputs and outputs can be connected in parallel to a common bus line
• When tristate devices are paralleled, only one of them is enabled at a time.

AND-OR-INVERT Gates

• AND-OR and OR-AND gates implement sum-of-products and product-of-sums Boolean expressions.
• Logic gates can be designated them as two-wide, two-input AND-OR-INVERT or OR-AND-INVERT gates.
• The gate is two-wide as there are two gates at the input, and two-input as each of the gates has two inputs.

Schmitt Gates

• Logic gates have a single-input threshold voltage level.
• Schmitt gates interpret varying input voltages according to two threshold voltages, one for LOW-to-HIGH and the other for HIGH-to-LOW transition.
• Schmitt gates distinguish from conventional gates by the small 'hysteresis' symbol reminiscent of the B-H loop for a ferromagnetic material.
• Hysteresis increases noise immunity and is used in applications where noise is expected on input signal lines.

Special Output Gates

• Some applications need both noninverted and inverted outputs from the same gate.
• Examples are a single-input gate with both an inverter and a noninverting buffer, or a two-input logic gate that is both an AND and a NAND.
• Such gates are called complementary output gates are particularly useful in circuits where PCB space is at a premium

Fan-Out of Logic Gates

• Not practical to drive the inputs of an unlimited number of logic gates from the output of a single logic gate.
• Limited by the current-sourcing capability for a HIGH output, the current-sinking capability for a LOW output, and the input requirements of the driven gates in both states.
• The number of logic gate inputs driven without causing a false output is called fan-out, a characteristic of the logic family

Buffers and Transceivers

• Logic gates have a limited load-driving capability, so a buffer can be used.
• Various buffers exist for a single input inverting or noninverting, NAND, NOR, OR, or AND.
• Drivers are circuits with even larger drive capability than buffers.
• Buffers are usually tristate devices for use in bus-oriented systems.
• A transceiver is a bidirectional buffer with direction control and ENABLE inputs for data flow and compatibility with bus-oriented systems.

IEEE/ANSI Standard Symbols

• Introduced in 1984 under IEEE/ANSI Standard 91-1984.
• This standard uses a rectangular symbol for all devices instead of a different symbol shape for each device.
• Used for representing different types of gates are the ones that are better known to all of us and have been in use for many years.
• A right triangle indicates inversion of a logic level/active LOW input or output. Absence of a triangle indicates active HIGH.

Some Common Applications of Logic Gates

• Used to provide specific functions in larger digital circuits and build digital building blocks.
• The OR gate can be used in all those situations where the occurrence of any one or more than one event needs to be detected or acted upon
• The AND gate is commonly used as an ENABLE or INHIBIT gate to allow or disallow passage of data