# Tag: Free Book on Digital

• ## 7.9 Converting Truth Tables into Boolean Expressions

In designing digital circuits, the designer often begins with a truth table describing what the circuit should do. The design task is largely to determine what type of circuit will perform the function described in the truth table. While some people seem to have a natural ability to look at a truth table and immediately…

• ## 7.8 DeMorgan’s Theorems

A mathematician named DeMorgan developed a pair of important rules regarding group complementation in Boolean algebra. By group complementation, I’m referring to the complement of a group of terms, represented by a long bar over more than one variable. You should recall from the chapter on logic gates that inverting all inputs to a gate…

• ## 7.7 The Exclusive-OR Function: The XOR Gate

What Is a XOR Gate? One element conspicuously missing from the set of Boolean operations is that of Exclusive-OR, often represented as XOR. Whereas the OR function is equivalent to Boolean addition, the AND function to Boolean multiplication, and the NOT function (inverter) to Boolean complementation, there is no direct Boolean equivalent for Exclusive-OR. This…

• ## 7.6 Circuit Simplification Examples

Let’s begin with a semiconductor gate circuit in need of simplification. The “A,” “B,” and “C” input signals are assumed to be provided from switches, sensors, or perhaps other gate circuits. Where these signals originate is of no concern in the task of gate reduction. How to Write a Boolean Expression to Simplify Circuits Our…

• ## 7.5 Boolean Rules for Simplification

Boolean algebra finds its most practical use in the simplification of logic circuits. If we translate a logic circuit’s function into symbolic (Boolean) form, and apply certain algebraic rules to the resulting equation to reduce the number of terms and/or arithmetic operations, the simplified equation may be translated back into circuit form for a logic…

• ## 7.4 Boolean Algebraic Properties

The Commutative Property Another type of mathematical identity, called a “property” or a “law,” describes how differing variables relate to each other in a system of numbers. One of these properties is known as the commutative property, and it applies equally to addition and multiplication. In essence, the commutative property tells us we can reverse…

• ## 7.3 Boolean Algebraic Identities

In mathematics, an identity is a statement true for all possible values of its variable or variables. The algebraic identity of x + 0 = x tells us that anything (x) added to zero equals the original “anything,” no matter what value that “anything” (x) may be. Like ordinary algebra, Boolean algebra has its own…

• ## 7.2 Boolean Arithmetic

Let us begin our exploration of Boolean algebra by adding numbers together: The first three sums make perfect sense to anyone familiar with elementary addition. The last sum, though, is quite possibly responsible for more confusion than any other single statement in digital electronics, because it seems to run contrary to the basic principles of…

• ## 7.1 Introduction to Boolean Algebra

Mathematical rules are based on the defining limits we place on the particular numerical quantities dealt with. When we say that 1 + 1 = 2 or 3 + 4 = 7, we are implying the use of integer quantities: the same types of numbers we all learned to count in elementary education. What most…

• ## 6.6 Programmable Logic Controllers (PLC)

Before the advent of solid-state logic circuits, logical control systems were designed and built exclusively around electromechanical relays. Relays are far from obsolete in modern design, but have been replaced in many of their former roles as logic-level control devices, relegated most often to those applications demanding high current and/or high voltage switching. Systems and…

• ## 6.5 Fail-safe Design

Logic circuits, whether comprised of electromechanical relays or solid-state gates, can be built in many different ways to perform the same functions. There is usually no one “correct” way to design a complex logic circuit, but there are usually ways that are better than others. In control systems, safety is (or at least should be)…

• ## 6.4 Motor Control Circuits

The interlock contacts installed in the previous section’s motor control circuit work fine, but the motor will run only as long as each push button switch is held down. If we wanted to keep the motor running even after the operator takes his or her hand off the control switch(es), we could change the circuit…

• ## 6.3 Permissive and Interlock Circuits

A practical application of switch and relay logic is in control systems where several process conditions have to be met before a piece of equipment is allowed to start. A good example of this is burner control for large combustion furnaces. In order for the burners in a large furnace to be started safely, the…

• ## 6.2 Digital Logic Functions

We can construct simple logic functions for our hypothetical lamp circuit, using multiple contacts, and document these circuits quite easily and understandably with additional rungs to our original “ladder.” If we use standard binary notation for the status of the switches and lamp (0 for unactuated or de-energized; 1 for actuated or energized), a truth…