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1、Chapter 3 Digital Electronics3.1 IntroductionA circuit that employs a numerical signal in its operation is classified as a digital circuit.Computers,pocket calculators, digital instruments, and numerical control (NC) equipment are common applications of digital circuits. Practically unlimited quanti
2、ties of digital information can be processed in short periods of time electronically. With operational speed of prime importance in electronics today,digital circuits are used more frequently. In this chapter, digital circuit applications are discussed.There are many types of digital circuits that h
3、ave applications in electronics, including logic circuits, flip-flop circuits, counting circuits, and many others. The first sections of this unit discuss the number systems that are basic to digital circuit understanding. The remainder of the chapter introduces some of the types of digital circuits
4、 and explains Boolean algebra as it is applied to logic circuits.3.2 Digital Number SystemsThe most common number system used today is the decimal system,in which 10 digits are used for counting. The number of digits in the system is called its base (or radix).The decimal system，therefore，has a base
5、 of 10.Numbering systems have a place value，which refers to the placement of a digit with respect to others in the counting process. The largest digit that can be used in a specific place or location is determined by the base of the system. In the decimal system the first position to the left of the
6、 decimal point is called the units place. Any digit from 0 to 9 can be used in this place.When number values greater than 9 are used,they must be expressed with two or more places.The next position to the left of the units place in a decimal system is the tens place.The number 99 is the largest digi
7、tal value that can be expressed by two places in the decimal system.Each place added to the left extends the number system by a power of 10.Any number can be expressed as a sum of weighted place values.The decimal number 2583,for example, is expressed as (21000)+(5100)+(810)+(31).The decimal number
8、system is commonly used in our daily lives. Electronically, however, it is rather difficult to use. Each digit of a base 10 system would require a specific value associated with it, so it would not be practical.3.2.1 Binary Number SystemElectronic digital systems are ordinarily the binary type，which
9、 has 2 as its base. Only the numbers 0 or 1 are used in the binary system.Electronically,the value of 0 can be associated with a low-voltage value or no voltage. The number 1 can then be associated with a voltage value larger than 0. Binary systems that use these voltage values are said to have posi
10、tive logic. Negative logic,by comparison,has a voltage assigned to 0 and no voltage value assigned to 1 .Positive logic is used in this chapter.The two operational states of a binary system,1 and 0,are natural circuit conditions. When a circuit is turned off or has no voltage applied，it is in the of
11、f, or 0,state. An electrical circuit that has voltage applied is in the on，or 1,state. By using transistor or ICs，it is electronically possible to change states in less than a microsecond. Electronic devices make it possible to manipulate millions of 0s and is in a second and thus to process informa
12、tion quickly.The basic principles of numbering used in decimal numbers apply in general to binary numbers.The base of the binary system is 2,meaning that only the digits 0 and 1 are used to express place value. The first place to the left of the binary point,or starting point，represents the units，or
13、 is,location. Places to the left of the binary point are the powers of 2.Some of the place values in base 2 are 2=1,2=2,2=4,2=8,2=16,25=32,and 26=64.When bases other than 10 are used,the numbers should have a subscript to identify the base used.The number 100is an example.The number 100(read“one,zer
14、o,zero, base 2”)is equivalent to 4 in base 10,or 410.Starting with the first digit to the left of the binary point,this number has value (020)+(021)+(122).In this method of conversion a binary number to an equivalent decimal number，write down the binary number first. Starting at the binary point，ind
15、icate the decimal equivalent for each binary place location where a 1 is indicated. For each 0 in the binary number leave a blank space or indicate a 0 Add the place values and then record the decimal equivalent.The conversion of a decimal number to a binary equivalent is achieved by repetitive step
16、s of division by the number 2.When the quotient is even with no remainder,a 0 is recorded.When the quotient has a remainder. as 1 is recorded.The division process continues until the quotient is 0.The binary equivalent consists of the remainder values in the order last to first.3.2.2 Binary-coded De
17、cimal (BCD) Number SystemWhen large numbers are indicated by binary numbers，they are difficult to use. For this reason，the Binary-Coded Decimal(BCD) method of counting was devised. In this system four binary digits are used to represent each decimal digit.To illustrate this procedure，the number 105，
18、is converted to a BCD number.In binary numbers，10510=10001012.To apply the BCD conversion process，the base 10 number is first divided into digits according to place values.The number 10510 gives the digits 1-0-5.Converting each digit to binary gives 0001-0000-0101BCD.Decimal numbers up to 99910 may
19、be displayed by this process with only 12 binary numbers. The hyphen between each group of digits is important when displaying BCD numbers.The largest digit to be displayed by any group of BCD numbers is 9.Six digits of a number-coding group are not used at all in this system.Because of this, the oc
20、tal (base 8) and the hexadecimal (base 16) systems were devised. Digital circuits process numbers in binary form but usually display them in BCD,octal,or hexadecimal form.3.2.3 Octal Number SystemThe octal (base 8) number system is used to process large numbers by digital circuits.The octal system o
21、f numbers uses the same basic principles as the decimal and binary systems.The octal number system has a base of 8. The largest number used in a base 8 system is 7. The place values starting at the left of the octal point are the powers of eight: 80=1,81=8,82=64,83=512,84=4096,and so on. The process
22、 of converting an octal number to a decimal number is the same as that used in the binary-to-decimal conversion process. In this method,however,the powers of 8 are used instead of the powers of 2. The number for changing 3828 to an equivalent decimal is 25810.Converting an octal number to an equival
23、ent binary number is similar to the BCD conversion process. The octal number is first divided into digits according to place value. Each octal digit is then converted into an equivalent binary number using only three digits.Converting a decimal number to an octal number is a process of repetitive di
24、vision by the number 8.After the quotient has been determined，the remainder is brought down as the place value.When the quotient is even with no remainder，a 0 is transferred to the place position.The number for converting 409810 to base 8 is 100028.Converting a binary number to an octal number is an
25、 important conversion process of digital circuits. Binary numbers are first processed at a very high speed. An output circuit then accepts this signal and converts it to an octal signal displayed on a readout device.Assume that the number 1101001002 is to he changed to an equivalent octal number. Th
26、e digits must first be divided into groups of three，starting at the octal point.Each binary group is then converted into an equivalent octal number.These numbers are then combined，while remaining in their same respective places，to represent the equivalent octal number.3.2.4 Hexadecimal Number System
27、The hexadecimal number system is used in digital systems to process large number values.The base of this system is 16，which means that the largest number used in a place is 15.Digits used by this system are the numbers 0-9 and the letters A-F. The letters A-P are used to denote the digits 10-15，resp
28、ectively. The place values to the left of the hexadecimal point are the powers of 16:160=1，161=16，162=256, l63=4096，164=65536， and so on.The process of changing a hexadecimal number to a decimal number is similar to that outlined for other conversions. Initially，a hexadecimal number is recorded in p
29、roper digital order.The place values，or powers of the base，are then positioned under the respective digits in step 2.In step 3，the value of each digit is recorded. The values in steps 2 and 3 are then multiplied together and added. The sum gives the decimal equivalent value of a hexadecimal number.T
30、he process of changing a hexadecimal number to a binary equivalent is a simple grouping operation. Initially，the hexadecimal number is separated into digits. Each digit is then converted to a binary number using four digits per group. The binary group is combined to form the equivalent binary number
31、.The conversion of a decimal number to a hexadecimal number is achieved by repetitive division，as with other number systems. In this procedure the division is by 16 and remainders can be as large as 15.Converting a binary number to a hexadecimal equivalent is the reverse of the hexadecimal to binary
32、 process. Initially，the binary number is divided in groups of four digits，starting at the hexadecimal point. Each number group is then converted to a hexadecimal value and combined to form the hexadecimal equivalent number.3.3 Binary Logic CircuitsIn digital circuit-design applications binary signal
33、s are far superior to those of the octal，decimal，or hexadecimal systems. Binary signals can be processed very easily through electronic circuitry，since they can be represented by two stable states of operation. These states can be easily defined as on or off, 1 or 0，up or down，voltage or no voltage，
34、right or left，or any other two-condition states. There must be no in-between state.The symbols used to define the operational state of a binary system are very important.In positive binary logic，the state of voltage，on，true，or a letter designation (such as A ) is used to denote the operational state
35、 1 .No voltage，off，false，and the letter A are commonly used to denote the 0 condition. A circuit can be set to either state and will remain in that state until it is caused to change conditions.Any electronic device that can be set in one of two operational states or conditions by an outside signal
36、is said to be bistable. Relays，lamps，switches，transistors, diodes and ICs may be used for this purpose. A bistable device has the capability of storing one binary digit or bit of information.By using many of these devices，it is possible to build an electronic circuit that will make decisions based u
37、pon the applied input signals. The output of this circuit is a decision based upon the operational conditions of the input. Since the application of bistable devices in digital circuits makes logical decisions，they are commonly called binary logic circuits.If we were to draw a circuit diagram for su
38、ch a system，including all the resistors，diodes，transistors and interconnections，we would face an overwhelming task, and an unnecessary one.Anyone who read the circuit diagram would in their mind group the components into standard circuits and think in terms of the system functions of the individual
39、gates. For this reason，we design and draw digital circuit with standard logic symbols. Three basic circuits of this type are used to make simple logic decisions.These are the AND circuit, OR circuit, and the NOT circuit.Electronic circuits designed to perform logic functions are called gates.This te
40、rm refers to the capability of a circuit to pass or block specific digital signals.The logic-gate symbols are shown in Fig.3-1.The small circle at the output of NOT gate indicates the inversion of the signal. Mathematically，this action is described as A=.Thus without the small circle，the rectangle w
41、ould represent an amplifier (or buffer) with a gain of unity.An AND gate has two or more inputs and one output. If all inputs are in the 1 state simultaneously，then there will be a 1 at the output.The AND gate in Fig. 3-1 produces only a 1 out-put when A and B are both 1. Mathematically，this action
42、is described as AB=C. This expression shows the multiplication operation. An OR gate has also two or more inputs and one output. Like the AND gate，each input to the OR gate has two possible states:1 or 0.The output of OR gate in Fig.3-1 produces a when either or both inputs are l.Mathematically，this
43、 action is described as A+B=C. This expression shows OR addition. This gate is used to make logic decisions of whether or not a 1 appears at either input.An IF-THEN type of sentence is often used to describe the basic operation of a logic state.For example，if the inputs applied to an AND gate are al
44、l 1，then the output will be 1 .If a 1 is applied to any input of an OR gate，then the output will be 1 .If an input is applied to a NOT gate，then the output will be the opposite or inverse.The logic gate symbols in Fig. 3-1 show only the input and output connections. The actual gates，when wired into
45、a digital circuit, would have supply and grounding connections as well.Fig. 3-2 shows the inner connections of 74LS08，i.e. a quadruple，two-input AND gate chip.Notice that the power supply is applied between pin 14 and 7.3.4 Combination Logic GatesWhen a NOT gate is combined with an AND gate or an OR
46、 gate，it is called a combination logic gate. A NOT-AND gate is called a NAND gate，which is an inverted AND gate. Mathematically the operation of a NAND gate is AB=. A combination NOT-OR ，or NOR，gate produces a negation of the OR function.Mathematically the operation of a NOR gate is A+B=.A 1 appears
47、 at the output only when A is 0 and B is 0.The logic symbols are shown in Fig. 3-3.The bar over C denotes the inversion，or negative function，of the gate.The logic gates discussed here illustrate basic gate operation.In actual digital electronic applications，solid-state components are ordinarily used
48、 to accomplish gate functions.Boolean algebra is a special form of algebra that was designed to show the relationships of logic operations.Thin form of algebra is ideally suited for analysis and design of binary logic systems.Through the use of Boolean algebra，it is possible to write mathematical ex
49、pressions that describe specific logic functions.Boolean expressions are more meaningful than complex word statements or or elaborate truth tables.The laws that apply to Boolean algebra are used to simplify complex expressions. Through this type of operation it may be possible to reduce the number of logic gates needed to achieve a specific function before the circuits are designed.In Boolean algebra the variables of an equation are assigned by letters of the
