Fundamental logic circuits

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FUNDAMENTAL LOGIC CIRCUITS

LEARNING OBJECTIVES

Upon completing this chapter, you should be able to do the following:

Identify general logic conditions, logic states, logic levels, and positive and negative logic as these terms and characteristics apply to the inputs and outputs of fundamental logic circuits.

Indentify the following logic circuit gates and interpret and solve the associated Truth Tables:

• AND
• OR
• Inverters (NOT circuits)
• NAND
• NOR
• Identify variations of the fundamental logic gates and interpret the associated Truth Tables.
• Determine the output expressions of logic gates in combination.
• Recognize the laws, theorems, and purposes of Boolean algebra.

INTRODUCTION

In chapter 1 you learned that the two digits of the binary number system can be represented by the state or condition of electrical or electronic devices. A binary 1 can be represented by a switch that is closed, a lamp that is lit, or a transistor that is conducting. Conversely, a binary 0 would be represented by the same devices in the opposite state: the switch open, the lamp off, or the transistor in cut-off.

In this chapter you will study the four basic logic gates that make up the foundation for digital equipment. You will see the types of logic that are used in equipment to accomplish the desired results. This chapter includes an introduction to Boolean algebra, the logic mathematics system used with digital equipment. Certain Boolean expressions are used in explanation of the basic logic gates, and their expressions will be used as each logic gate is introduced.

COMPUTER LOGIC

Logic is defined as the science of reasoning. In other words, it is the development of a reasonable or logical conclusion based on known information.

GENERAL LOGIC

Consider the following example: If it is true that all Navy ships are gray and the USS Lincoln is a Navy ship, then you would reach the logical conclusion that the USS Lincoln is gray.

To reach a logical conclusion, you must assume the qualifying statement is a condition of truth. For each statement there is also a corresponding false condition. The statement "USS Lincoln is a Navy ship" is true; therefore, the statement "USS Lincoln is not a Navy ship" is false. There are no in-between conditions.

Computers operate on the principle of logic and use the TRUE and FALSE logic conditions of a logical statement to make a programmed decision.

The conditions of a statement can be represented by symbols (variables); for instance, the statement "Today is payday" might be represented by the symbol P. If today actually is payday, then P is TRUE. If today is not payday, then P is FALSE. As you can see, a statement has two conditions. In computers, these two conditions are represented by electronic circuits operating in two LOGIC STATES. These logic states are 0 (zero) and 1 (one). Respectively, 0 and 1 represent the FALSE and TRUE conditions of a statement.

When the TRUE and FALSE conditions are converted to electrical signals, they are referred to as LOGIC LEVELS called HIGH and LOW. The 1 state might be represented by the presence of an electrical signal (HIGH), while the 0 state might be represented by the absence of an electrical signal (LOW).

If the statement "Today is payday" is FALSE, then the statement "Today is NOT payday" must be TRUE. This is called the COMPLEMENT of the original statement. In the case of computer math, complement is defined as the opposite or negative form of the original statement or variable. If today were payday, then the statement "Today is not payday" would be FALSE. The complement is shown by placing a bar, or VINCULUM, over the statement symbol (in this case, P). This variable is spoken as NOT P. Table 2-1 shows this concept and the relationship with logic states and logic levels.

Table 2-1. - Relationship of Digital Logic Concepts and Terms

Example 1: Assume today is payday

 STATEMENT SYMBOL CONDITION LOGIC STATE LOGIC LEVEL Original: TODAY IS PAYDAY P TRUE 1 HIGH Complement: TODAY IS NOT PAYDAY P FALSE 0 LOW Example 2: Assume today is not payday Complement: TODAY IS NOT PAYDAY P FALSE 0 LOW Complement: TODAY IS NOT PAYDAY P TRUE 1 HIGH

In some cases, more than one variable is used in a single expression. For example, the expression ABCD is spoken "A AND B AND NOT C AND D."

POSITIVE AND NEGATIVE LOGIC

To this point, we have been dealing with one type of LOGIC POLARITY, positive. Let's further define logic polarity and expand to cover in more detail the differences between positive and negative logic.

Logic polarity is the type of voltage used to represent the logic 1 state of a statement. We have determined that the two logic states can be represented by electrical signals. Any two distinct voltages may be used. For instance, a positive voltage can represent the 1 state, and a negative voltage can represent the 0 state. The opposite is also true.

Logic circuits are generally divided into two broad classes according to their polarity - positive logic and negative logic. The voltage levels used and a statement indicating the use of positive or negative logic will usually be specified on logic diagrams supplied by manufacturers.

In practice, many variations of logic polarity are used; for example, from a high-positive to a low-positive voltage, or from positive to ground; or from a high-negative to a low-negative voltage, or from negative to ground. A brief discussion of the two general classes of logic polarity is presented in the following paragraphs.

Positive Logic

Positive logic is defined as follows: If the signal that activates the circuit (the 1 state) has a voltage level that is more POSITIVE than the 0 state, then the logic polarity is considered to be POSITIVE. Table 2-2 shows the manner in which positive logic may be used.

Table 2-2. - Examples of Positive Logic

As you can see, in positive logic the 1 state is at a more positive voltage level than the 0 state.

Negative Logic

As you might suspect, negative logic is the opposite of positive logic and is defined as follows: If the signal that activates the circuit (the 1 state) has a voltage level that is more NEGATIVE than the 0 state, then the logic polarity is considered to be NEGATIVE. Table 2-3 shows the manner in which negative logic may be used.

Table 2-3. - Examples of Negative Logic

NOTE: The logic level LOW now represents the 1 state. This is because the 1 state voltage is more negative than the 0 state.

In the examples shown for negative logic, you notice that the voltage for the logic 1 state is more negative with respect to the logic 0 state voltage. This holds true in example 1 where both voltages are positive. In this case, it may be easier for you to think of the TRUE condition as being less positive than the FALSE condition. Either way, the end result is negative logic.

The use of positive or negative logic for digital equipment is a choice to be made by design engineers. The difficulty for the technician in this area is limited to understanding the type of logic being used and keeping it in mind when troubleshooting.

NOTE: UNLESS OTHERWISE NOTED, THE REMAINDER OF THIS BOOK WILL DEAL ONLY WITH POSITIVE LOGIC.

LOGIC INPUTS AND OUTPUTS

As you study logic circuits, you will see a variety of symbols (variables) used to represent the inputs and outputs. The purpose of these symbols is to let you know what inputs are required for the desired output.

If the symbol A is shown as an input to a logic device, then the logic level that represents A must be HIGH to activate the logic device. That is, it must satisfy the input requirements of the logic device before the logic device will issue the TRUE output.

Look at view A of figure 2-1. The symbol X represents the input. As long as the switch is open, the lamp is not lit. The open switch represents the logic 0 state of variable X.

Figure 2-1. - Logic switch: A. Logic 0 state; B. Logic 1 state.

Closing the switch (view B), represents the logic 1 state of X. Closing the switch completes the circuit causing the lamp to light. The 1 state of X satisfied the input requirement and the circuit therefore produced the desired output (logic HIGH); current was applied to the lamp causing it to light.

If you consider the lamp as the output of a logic device, then the same conditions exist. The TRUE (1 state) output of the logic device is to have the lamp lit. If the lamp is not lit, then the output of the logic device is FALSE (0 state).

As you study logic circuits, it is important that you remember the state (1 or 0) of the inputs and outputs.

So far in this chapter, we have discussed the two conditions of logical statements, the logic states representing these two conditions, logic levels and associated electrical signals and positive and negative logic. We are now ready to proceed with individual logic device operations. These make up the majority of computer circuitry.

As each of the logic devices are presented, a chart called a TRUTH TABLE will be used to illustrate all possible input and corresponding output combinations. Truth Tables are particularly helpful in understanding a logic device and for showing the differences between devices.

The logic operations you will study in this chapter are the AND, OR, NOT, NAND, and NOR. The devices that accomplish these operations are called logic gates, or more informally, <emphasis type="i.GIF">gates. These gates are the foundation for all digital equipment. They are the "decision-making" circuits of computers and other types of digital equipment. By making decisions, we mean that certain conditions must exist to produce the desired output.

In studying each gate, we will introduce various mathematical SYMBOLS known as BOOLEAN ALGEBRA expressions. These expressions are nothing more than descriptions of the input requirements necessary to activate the circuit and the resultant circuit output.

THE AND GATE

The AND gate is a logic circuit that requires all inputs to be TRUE at the same time in order for the output to be TRUE.