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But it will be g Older Posts. Subscribe to: Posts Atom. Popular Video Category. Popular Posts. Friends, today i'm going to share with you a very essential book for the students who are studying in electronics related fields. The two numbers to be added are represented by the proper logic levels at A and B, and the SUM of these two numbers will appear at output F. Here's how we might do it with decimal numbers. The digital levels illustrated in Fig.
The equivalent decimal numbers are shown in parentheses. The desired function appears at the F bus. There are tour AND operations. Comparison Comparing the magnitude of two numbers is an important logical operation. The circuit in Fig. The other two outputs will be low. A in the TTL family is a 4-bit comparator similar to Fig. Also, the ALU can be used with the same results. A comparator. In the case of a computer, information is frequently entered by typing on a keyboard or perhaps by using a magnetic floppy disk.
Useful information can be obtained from the computer by examining the visual displays on a cathode-ray tube CRT or by reading material produced on a printer. Clearly there is a requirement to connect multiple input devices, one at a time, to the system.
The digital circuit used for this operation is a multiplexer. Likewise, there is a need to connect the system output to a number of different destinations, one at a time. The digital circuit used for this purpose is a demultiplexer. The term multiplex means "many into one. There are n input lines. Each line is used to shift digital data serially. There is a single output line which is connected to the computer system input port.
Operation of the circuit can be explained by using the "switch" as a model. Each setting of the digital control levels on the C bus will connect the switch to one of the input lines.
Data from that particular input is then entered into the computer. Changing the C bus levels will connect a different input. Thus, data from multiple sources can be connected to a single input port, one at a time.
It has 16 input lines and a single output line. The opposite of multiplex is demultiplex, which means "one into many. This digital circuit simply connects the single data input line to one of the n output lines, one at a time, according to the levels on the C bus.
Thus serial data from the computer output port can be directed to different destinations, one at a time. Any information entered into a digital system must be in the form of a digital number. A circuit that changes data into the required digital form is called an encoder. The encoder shown in Fig. It may be used with a keyboard. For instance, depressing the number 4 key on a keyboard will cause input line 4 to this encoder to be high the other inputs are all low.
The result will be decimal 4, binary , at the encoder output as shown. Taking digital information from the output of a computer and changing it into another form is accomplished with a decoder, for example, changing the digital number O decimal 6 into its decimal fonn. The decoder in Fig. As shown, the binary input OI IO will cause output line 6 to be high, while all other output lines remain low.
There are many different types of encoders and decoders. A number of them will be discussed in detail in Chapter 4. What binary number will be stored in the counter in Fig. How many flip-flops are required to construct a digital counter capable of counting events? State whether or not the ALU in Fig.
What are the digital output levels of the encoder in Fig. A computer intended to perform a very specific task, constructed with a minimum number of components, might be referred to as a microcomputer. Small portable, or desktop, computers are usually in the microcomputer class. Computers with greater capacities, often used in business, are called minicomputers. A large mainframe computer system capable of storing and manipulating massive quantities of data, for example, a digital computer system used by a bank or an insurance company, might then be called a maxicomputer.
Uses What can a digital computer be used for? Numerical computation is surely one possible use. The inclusion of an ALU with additional logic circuits provides arithmetic capabilities addition, subtraction, multiplication, division. The logic portion of the ALU means the computer can be used to make logical decisions.
Beyond these basic functions, a digital computer can be used to process data balance bank accounts , to rapidly perform otherwise time-consuming tasks determine payroll amounts and print out paychecks , to precisely monitor and control intricate processes life support systems in a hospital operating room , to use speech for communication with humans automatic telephone systems and voice recognition -the list is almost end- less, and is limited only by the ingenuity and resourcefulness of individual users!
Basic Configurations A microcomputer designed to control a given machine, process, or system might be represented as in Fig. The control signals produced by the computer appear as the output bus and are sent to an output device. Here, the signals are properly conditioned and sent to the mechanism being controlled. The controlled entity must then send signals indicating its present condition back to the computer via an input device and via the input bus. The computer analyzes these present condition signals, determines any necessary action, and sends required correction signals out to the system.
A microcomputer system might be designed to irrigate the lawn area of a park. Watering is to be done only at night, when the soil moisture falls below a given value. The system "sensors" in this case would be 1 a probe to detect soil moisture and 2 a light detector to distinguish between daylight Controlled IA and darkness. The input bus is serviced with A digital computer based system aMUX.
Audio output. This allows the connection of a number of different input devices: A keyboard for typewritten entry of alphanumeric information A disk drive or tape drive for entering data stored in magnetic form A microphone for voice input The DEMUX on the output bus allows numerous possibilities for receiving information from the computer: The familiar CRT for a visual display A printer to provide printed material called hard copy A disk or tape drive to record data in magnetic form Perhaps a speaker for audio information A minicomputer such as this can be used for many different tasks.
It can be used as a word processor, for data processing, for communication via telephone both voice and fax , for training in an educational setting, for computer games, and so on!
The block diagram in Fig. For instance, a maxicomputer will likely have more than one printer, and perhaps even different types of printers. It will generally have a large number of users, all of whom desire access to the system at the same time. One workstation must then be provided for each user. A keyboard and a CRT are the minimum components required at each workstation. The digital circuits used to construct maxicomputer systems are necessarily more complicated than minicomputer systems, and they may operate at a much faster rate.
Let's take a look inside a typical digital computer. It is constructed using an ALU along with a number of registers and counters. All the operations within the CPU, and indeed within the computer itself, must be Clock carefully coordinated. A digital signal refened to as the system clock is used as a reference to time when specific operations take place. The clock signal is usually a periodic, rectangular waveform as illustrated CPU in Fig.
Using a crystal in the clock circuit allows the accuracy and stability of the clock frequency, to The "heart" and "brain" of be controlled with great precision. The clock provides a digital computer a "heartbeat" for the computer. A block diagram of a digital computer is started by drawing the CPU and clock as shown in Fig. The CPU is capable of computation and decision, but it must have specific instructions telling it exactly what to do and when to do it.
This set of instructions is called a program. A program is a detailed list of CPU operations written by a human programmer. The pro- grammer decides what the computer is to do and when it should be done, and then writes a list of instructions to be carried out in the proper order. The program is Program Data entered into the computer, using perhaps a keyboard, memory memory and stored in the computer memory. It will execute the instruction and then fetch the next instruction.
With this repeated fetch- and-execute cycle, the CPU will accomplish the desired task. A memory block used. A portion of the memory block in Fig. This is the area where the information being processed by the computer is stored. Since the CPU takes "reads" data from memory, as well as returns "writes" data into memory, the memory data bus is bidirectional. By contrast, the program data bus is not bidirectional, since information on this bus is always from memory to CPU.
The CPU communicates with the "outside world" by means of the input encoders and the output decoders. This configuration is sometimes quite inefficient, since all information entering or exiting the computer must pass through the CPU. The CPU operates at a much faster rate than most external devices, and it must wait while data are being entered or exited: A direct memory access DMA block is generally included to alleviate this problem.
As seen in Fig. While information is being transfened via the DMA, the CPU is free to cany on its computational or logical operations. This greatly improves system efficiency as well as speed of operation. Before data can be entered into the computer, a signal on the input requestline asks the computer for "per- mission" to input information. For instance, depressing the enter key on a keyboard will generate an input request signal. Program Data memory memory. This request-acknowledge sequence is often called handshaking.
A similar handshaking must occur when the CPU is ready to deliver data to an external device. However, in this case, the CPU makes an output request, and the external device gives permission. All of these blocks are operated in synchronism with the clock, but additional direction must be provided. The controller is the unit that decides which block "goes first" establishes priorities , decides the order.
As such, the controller communicates individually with each block as illustrated in Fig. This block diagram is representative of the architecture of many digital computers. Amicroprocessoris often used as the basic IC around which amicrocomputerorminicomputer is constructed. Numerous computers have been designed, beginning with the microprocessor. Improvements to this basic IC have led to the development of a family of microprocessor units including the and the Referring to the block diagram in Fig.
The purpose of this section is to provide a general understanding of a digital computer. In the remainder of this text, the blocks used to build any digital system, including a digital computer, will be studied in detail. Why is the system clock considered the heart of a digital computer system? What is a computer program? What functions are ca1Tied out in an ALU? What is the purpose of the DMA in Fig. A digital IC is constructed by an interconnection of resistors, transistors, and perhaps small capacitors, all of whic.
The entire circuit resides on a tiny piece of semiconductor material called a chip. The semiconductor wafer is typically a slice of monocrystalline silicon about 0. The wafer is divided checkerboard fashion into or so rectangular areas.
Each area will become a single chip. In this fashion, identical digital circuits are manufactured simultaneously on the same silicon wafer. After the processing steps are completed, the wafer is separated into individual chips as seen in Fig. Each chip is a digital circuit, for example, an inverter or an AND gate. An individual digital circuit may have only a few components, but some circuits have a few hundred components!
Each chip is then mounted in a suitable package, as shown in Fig. The package illustrated here is a pin dual-inline package DIP. Some additional packages for ICs are shown in Fig. IC Families ICs are categorized by size according to the number of gates contained on each chip.
For instance, a has six inverters in a pin DIP. ICs having more than 12 but fewer than gates are called medium-scale integration MS! ICs are further categorized according to the type of transistors used. The two basic transistor types are bipolar and metal-oxide-semiconductor MOS. MOS is slower, but requires much less power and also occupies a much smaller chip area for a given function.
MOS is therefore preferred for LSI and is widely used in applications such as pocket calculators, wristwatches, hearing aids, and so on. For the moment, let's consider the overall characteristics of each digital IC family. These two families are now widely available from a number of different manufacturers. Otherwise, the logical operations of these two families are the same. In each case, the XX por- tion of the part number refers to a specific device.
For instance, "04" stands for inverter, and a is a TTL inverter. When there is no danger of confusion, it is common practice to shorten the description by omitting the first two digits. Thus, a inverter is referred to as a 04, and the AND gate is designated as Table 1. Note that a is logically the same as a ; it is simply guaranteed to operate over a wider temperature range. In the interest of higher operating speed, the 74XX family was improved with the introduction of the 74HXX where the H stands for high speed family of devices.
The price paid for increased speed was an increase in power required to operate each gate. This led to another family of devices designed to minimize power requirements-the 74LXX where the L stands for low power series. A major improvement in the TTL series came with the development of a special transistor arrangement called a Schottky transistor.
These devices greatly improved operating speed, but again at the cost of increased power consumption. The 74LSXX family offers high-speed operation with minimal power consumption and today is preferred in most designs. The original also remains popular. One might anticipate the development of other families with characteristics to match specific needs.
The lOOK , series is even faster, with a delay time of only 1 ns. CMOS is preferred where individual logic circuits are used and where very low power consumption is required. It was slow, was not compatible with TTL, and is rarely seen in modern designs. These devices are pin-for-pin replacements for similarly numbered TTL devices. For instance, a is an IC that contains six inverters, and the 74C04 also contains six inverters. The 74CXX design will require considerably less operating power but will be restricted to lower operating speeds.
Complete digital logic systems can be constructed using entirely 74XX devices or 74HCXX devices, and the two types of devices can even be used together, provided certain precautions are observed. The necessary precautions involve voltage levels, current requirements, and switching times; this comes under the subject of interfacing, which will be addressed in Chapter These new symbols are now being used on manufacturers' data sheets along with traditional symbols.
Symbols of the first type, called distinctive-shape symbols, are exactly as have been shown throughout this chapter. The second system, which is called the rectangular- shape system, uses a rectangular box with a special symbol for each type of gate. The IEEE standard does not express a preference for either shape. Most people presently involved in digital electronics seem to prefer. More detailed information is a b c available m Appendix 7. The standard logic symbol for an inverter is a Standard symbol, b New shown in Fig.
The new IEEE symbol is shown in part b of this figure. A rectangular box is used for the gate, the input is labeled A, and the output is labeled Y. The 1 inside the box signifies that the input must be active in order to have an active output.
The triangle on the output line signifies that the output is active when low. Thus, when the input is active high , the output will be active low. The truth table is shown in Fig.
The is a hex inverter; that is, it is an IC that contains six inverters. The DIP package for this device is shown in Fig. The DIP package and pinout for the are shown in Fig.
The pinout and symbols for the quad 2-input OR gate are shown in Fig. The term quad means "four," and there are four gates in this DIP. This symbol means "at least one input must be high in order for the output to be high. Vee lo-r 2 B 3. Which requires more power to operate? What is the significance of the triangle on the output line of the inverter in Fig.
The voltages in Fig. Voltages within the forbidden region are not allowed. This illustration is often called Volts a profile, and it can be used to define the operation of any digital logic circuit. Thus, there must be an input profile and an output profile for each family.
An understanding of the correct voltage levels for each digital family is absolutely essential. Measuring logic levels in the laboratory, interconnecting different logic families, and connecting digital logic circuits with other digital circuits require a detailed knowledge of voltage levels. Profiles for other circuits are Logic level profile easily obtained from manufacturers' data books.
Output Input Look carefully at Fig. Similarly, any output voltage within the low range is within the input range recognized as a low.
Clearly an output voltage from any 74XX circuit can be used as the input signal to any other 74XX circuit! This family of circuits is thus said to be compatible. This shouldn't come as a surprise, since any circuit within a family should be able to "drive" any other circuit within the same family. There are, however, differences in the number of circuits that can be connected to the output in each fam- ily.
This consideration, calledfanout, is discussed in Chapter By comparing the profiles in Figs. The voltage levels are not compatible.
Interconnecting different families like this is called interfacing. Both interfacing and fanout are considered in Chapter Noise Margin We shall end the discussion on digital IC signal levels by introducing the concept of noise margin. Consider, output of a digital device is connected to input of another digital device see Fig.
Now, refer to Fig. Device 1 Device 2 An addition of any random noise voltage greater than 2. Thus, there exists a noise margin, the amount of noise that can get added without Noise affecting output of any possibility of logic value misinterpretation. If it is positive, there is no such issue. However, for NML the corrupting noise voltage should be positive when misinterpretation may occur.
L,max - VOL. This introductory chapter in digital principles is intended to provide a dear concept of a digital signal in an electronic circuit or system. Both ideal and realistic digital voltage levels are presented. How these levels vary with time waveforms is illustrated, and the concepts of rise time, fall time, and duty cycles are defined. Simple ideal switch models are used to illustrate digital Circuit operation.
The flip-flop is introduced as a basic memory element, and both serial and parallel shift registers are discussed. The basic conceptual operation of a number of common MSI and LSI digital circuits is covered: encoders, decoders, multiplexers, demultiplexers, ALUs, counters, and comparators.
These basic elements are then discussed in the contextoftheir use. DMA Direct memory access. CMOS Complementary metal-oxide silicon. CRT Cathode-ray tube. LSI Large-scale integration. The MSI Medium-scale integration. VLSI Very large-scale integration. This is similar to the popular ECL family of 1. Is this positive or negative decimal number 3. What is the width of each 1. Does this resemble a series of negative pulses?
Directly under that VOH. V08 ,min. What has are given as O Vdc?. To reduce the number of connections, data are sometimes shifted in groups of 2. That is, 1. If this is done, the number of connections is cut in half. Hint: Use the binary numbers in Table 1. How many connections are needed here if this is done? Compare the times required to shift a 1. What must the requires ns. What if the buses were only three bits each? Use only a. With the switch b. Moving c. Draw a diagram to illustrate this.
How much time is required to 1. ALU b. CPU cessor and shifts data in parallel between its c. ECL microprocessor and the input-output port reg- 1. How many connections wires must be the block diagram in Fig. What would you 1. Describe any wants to know the difference between a sensors needed, and give a general discussion and a 74HC What is the total power required? T three inputs must be low. The possibilities.
Positive logic The term V0 H. This is a 4-bit number. It would take only I s for the papdlel The term V0 L. Aport is usually a register that serves region only during the short time while as a place to eitp. When not switching static , f'. Ten 7. Yes c. Yes 8. The clock provides a periodic digital 9. It is a specific list of instructions, is open. To produce f'. Arithmetic computations and logical three inputs mustbehigh.
ECL is faster but requires more power. Quad means "four. The output is active when low. Digital Logic. A digital circuit having one or more input signals but only one output signal is called a gate.
Connecting the basic gates in different ways makes it possible to produce circuits that perform arithmetic and other functions associated with the human brain an ALU. Because they simulate mental processes, gates are often called logic circuits. Hardware description languages HDL are an alternative way of describing logic circuits.
This uses a set of textual codes that is machine computer readable. The concept is relatively new and is useful for design, testing and fabrication of complex digital circuits. We'll have a soft introduction ofHDL towards the end of this chapter. We'll learn it in detail in later part of this book introducing features relevant to each chapter.
Is an action right or wrong? A motive good or bad? A conclusion true or false? Much of our thinking involves trying to find the answer to two-valued questions like these.
Logic next attracted mathematicians, who intuitively sensed some kind of algebraic process running through all thought. Augustus De Morgan came close to finding the link between logic and mathematics.
But it was George Boole who put it all together. He invented a new kind of algebra that replaced Aristotle's verbal methods. Boolean algebra did not have an impact on technology, however, until almost a century later. In Shannon applied the new algebra to telephone switching circuits.
Because of Shannon's work, engineers soon realized that Boolean algebra could be used to analyze and design computer circuits. Three logic circuits, the inverter, the OR gate, and the AND gate, can be used to produce any digital system. The function of each of these gates was introduced in Chapter 1. Let's look more closely at the operation of each circuit and-also at their Boolean expressions. In one truth table, the symbols Hand L are used, while the binary numbers O and 1 are used.
You will find both symbols tables used in other texts, as well as in manufacturers' data sheets. The important idea is that there are only two possible voltage levels low and high associated with a digital circuit.
This fits nicely with the binary number system, since it has only two values 0 and 1. This is often referred to as two-state operation. Later in this chapter, we will consider negative logic, where the higher voltage level is assigned binary zero.
Figure 2. This IC contains six inverters. For instance, if you only need one inverter, you can connect an input signal to pin 1 and take the output signal from pin 2; the other inverters can be left unconnected. In Boolean algebra a variable can be either O or 1. The output Yofnot gate is always complement of input A. However, almost all digital signals do in fact change with time, as illustrated by the waveforms in Chapter 1 Sec. Here are two examples that illustrate how to use the truth table information with signals that vary with time.
A I-kHz square wave drives pin 1 of a see Fig. What does the voltage waveform at pin 2 look like? Solution Figure 2. Assuming you have set the sweep timing to get the upper waveform pin I , then you would see an inverted square wave on pin 2. If a Hz square wave drives pin 3 of a , what is the waveform on pin 4? Solution Pins 3 and 4 are the input and output pins of an inverter see Fig. A giance at Fig. Again, the output waveform is the complement of the input waveform, Because of two-state operation, rectangular wavefonns like this are the normal shape of digital signals.
Incidentally, a timing diagram is a picture of the input and output waveforms of a digital circuit. Examples of timing diagrams are shown in Figs. Cfl F. For instance, the output of a 2-in- 1 0 I 1 l l put OR gate is high if either or both inputs are high. Binary 0 0 0 0 addition is discussed in detail in Chapter 6. If A or B or C is high, Y will be high. The truth 0 0 table summarizes all input possibilities.
For example, the first ABC entry is , the next is , then , and so on, up to the final entry of Since all binary numbers are present, all input possibilities are included.
Incidentally, the number of rows in a truth table equals 2", where n is the number of inputs. For a 2-input OR gate, the truth table has 22, or 4 rows. A 3-input OR gate has a truth table with 23, or 8 rows, while a 4-input OR gate results in 24, or 16 rows, and so on. An OR gate can have as many inputs as desired.
No matter how many inputs, the action of any OR gate is summarized like this: One or more high inputs produce a high output. Whenever you see this symbol, remember the output is high if either input is high. Shown in Fig. For these gates, the output is high when any input is high. The only way to get a low output is by having all inputs low. When there are many input signals, it's common drafting practice to extend the input side as needed to allow sufficient space between the input lines.
For instance, Fig. The same idea applies to any type of gate; extend the input side when necessary to accommodate a large number of input signals. Timing Diagram Figure 2. The input voltages drive pins 1 and 2 of a Notice that the output pin 3 is low only when both inputs are low. The output is high the rest of the time because one or more input pins are high.
Solution With two input signals A and B , four input cases are possible: low-low, low-high, high-low, and high- high. For convenience, let L stand for low and Hfor high. Here is what happens for each input possibility.
CASE l A is! With these inputs the upper inverter has a high output, while the lower inverter has a low output. Since the OR gate still has a high input, the output Y is high. Now, the upper. With both inputs high, each inverter has a low output. This time, the ORgate has all inputs in the low state, so that Y is low, as shown by the final entry of Fig. Logic circuit and truth table of Example 2. Incidentally, the circuit of Fig.
The other gates in these digital ICs are not connected, which is all right because you don't have to use all of the available gates. The truth table Fig. In other words, the a b AND gate is an all-or-nothing gate; a high output occurs only when all inputs are high. Three Inputs Figure 2. The inputs are A, B, and C.
When all inputs are 0 0 0 0 low, Y is low. If even one input is low, Y is in the low 0 0 I 0 state. Remember: For any of these gates, the output is high only if all inputs are high. As before, it's common drafting practice to extend the input sides when there are many input signals. See Appendix 3 for pinout diagrams. Notice that the output pin 3 is high only when both inputs are high between C and D, G and H, etc.
The output is low the rest of the time. Solution We get the final truth table here in slightly different way. Consider, one logic gate at as shown in Fig. Finally, the 4'" column shows OR operation on column 3and 4 to give the final output Y.
With both input voltages in the low state, each inverter has a A B y high output This means the AND gate has a high output, the first entry of Fig. With these inputs the upper inverter has a high output, while L H L the lower inverter has a low output Since the AND gate produces a low output, Y is low.
With both inputs high, each inve1ter has a low output Again, the AND gate has a low as shown bythe final. Note that input-output relations described i11 Fig. AND-OR networks always produce sum-of- products equations. Solution Thislogic circuit u,ic. As shown in this equation, parentheses may be used to indicate a logical product ANDing , Also notice that the final answer is a product of sums.
OR-AND networks always produce product-of-sums equations. A gate whose output is H if any input is His an gate. Alternatively, inverters on the A and B input lines may produce the complemented variables, as shown in Fig. This example illustrates one method oflogic design. Here, we address an interesting question. Is it possible to use only one type of gate for this purpose? If possible, one needs to procure only one type of gate for his design. And more importantly, fabrication of Integrated Circuit that perfonns a logic operation becomes easier when gate of only one kind is used.
Gates, which can perform this task, are called universal logic gates. The bubble small circle on the output is a reminder of the inversion that takes place after the ORing. So from now, we will call the circuit a NOR gate and will use the symbol of Fig.
With a NOR gate, all inputs must be low to get a high output. If any input is high, the output is low. The small triangle on the output is equivalent to the bubble used on the standard symbol. The new rectangular symbol for the is l 0 0 shown in Fig. This logic circuit is often drawn in the abbreviated form shown in Fig. The bubbles on the inputs are a reminder of the a. We will refer to the abbreviated drawing of Fig. We have already b. Therefore, these two circuits are equivalent and thus interchangeable.
In words, it says the complement of a sum equals the product of the complements. This can also be proved by comparing the truth tables shown in Fig. Note that this equivalence can be extended to gates or circuits for larger number of inputs, too.
If input is 0, then both the inputs to NOR gate are 0. Following NOR gate truth table Table 2. Similarly, if input is 1, both the inputs to NOR gate are 1 that gives output 0. Therefore output of circuit, shown in Fig. Thus output of circuit in Fig. The above equivalences can be proved simply, by applying Boolean theorems and we'll discuss those theorems in next chapter.
Since, we can perform all the Boolean operations using only NOR gates it is termed as universal logic gate. Eye of the Beholder Which brings us to a principle. Truth tables, logic circuits, and Boolean equations are different ways of looking at the same thing. Whatever we learn from one viewpoint applies to the other two. If we prove that truth tables are identical, this immediately tells us the co1Tesponding logic circuits are interchangeable, and their Boolean equations are equivalent.
When designing, we often startwith a truth table, generate a Boolean equation, and arrive at a logic circuit. A is a quad 2-input NOR gate. What is the Boolean equation for the output of Fig. Appendix 3 shows the pinout diagram. If you invert a signal twice, you get the original signal back again. Put another way, double inversion has no effect on the logic state; double invert a low and you still have a low; double-invert a high and. Therefore, each double inversion in Fig.
Therefore, Fig. Why would anyone want to replace Fig, 2. In general, this idea applies to any circuit that you can rean-ange with De Morgan's theorem. You can build whichever equivalent circuit is convenient. Equivalence among logic circuits: Example 2. Ith tllbJf,. Qutput Yis low up to transition5, high between 5 and. With a NAND gate, all inputs must be high to get a 0 1 low output. If any input is low, the output is high.
The new rectangular symbol for the is shown in Fig. Bubbled OR Gate Figure 2. The circuit is often drawn in the abbreviated form shown in Fig. We have already analyzed this circuit in Example 2. It says the complement of a product equals the sum of the complements. A similar exercise that compares truth tables of three input NAND gate and three input bubbled OR gate show they are identical and we can write, A. Note that this equivalence can be extended to gates or circuits with any number of inputs.
If input is 0, then both the inputs to NAND gate are. Similarly, if input is 1, both the inputs to NAND gate are 1 that gives output 0. Furthermore, the NAND gate is available in more configurations than other gates, as shown in Table 2. Notice that the NAND gate is available as a 2-, 3-, 4-, or 8-input gate. The other gates have fewer configurations, with the OR gate available only in 2-input fonn. Prove that Fig. Solution Ile. Each double inversion in Fig. Therefore, Figs.
Anyone who sees Fig. Furthermore, when troubleshooting the circuit, they can ignore the bubbles and visualize the easy-to-analyze AND-OR circuit of Fig. Solution Let us analyze the equivalent circuit of Fig. Table 2. By analyzing each input possibility, we can determine the resulting output Fo. This is the first entry of Table 2. Proceeding like this, we can arrive at the output for the remaining possibilities ofTable2.
Solution All you have to do is convert the low-high states of Table 2. To agree with the truth table, output Yis low up to transition 3,. NANI gate? A variety of circuits like this are available as TTL chips. Because of the inversion, the output has the equation shown below.
What do we do when we need a 6- or 8-wide circuit? There are two additional inputs, labeled bubble and arrow. Expanders What do we connect to the arrow and bubble inputs of an expandable gate? We connect the output of an ex- pander as in Fig. Connect bubble to bubble and arrow to arrow.
Visualize the outputs of Fig. In other words, Fig. We can connect more expanders. The is a dual 4-input expander. This means that we can add two more expanders in Fig. Up to now, we have used a binary O for low voltage and a binary 1 for high voltage. This is called positive logic. People are comfortable with positive logic because it feels right.
But there is another code known as negative logic where binary O stands for high voltage and binary 1 for low voltage. Even though it seems unnatural, negative logic has many uses. The following discussion introduces some of the terminology and concepts for both types of logic. Look at the gate of Fig. We have been calling it an OR gate. This is correct, provided we are using positive logic. That is, if either input is high in Fig. In a positive logic system, binary O stands for low and binary 1 for high.
So, we can convert Table 2. Note that Y is a 1 if either A or B is 1. This sounds like an OR gate. And it is, because we are using positive logic.
To avoid ambiguity, we can call Fig. In a negative logic system, binary 1 stands for low and binary O for high. With this code, we can convert Table 2. Now, watch what happens. The output Y is a 1 only when both A and B are 1. This sounds like an AND gate! And it is, because we are now using negative logic. In other words, gates are defined by the way they process the binary Os and 1s. If you use binary 1 for low voltage and binary O for high voltage, then you liave to refer to Fig.
As you see, the gate of Fig. But what you call it depends on whether you see positive or negative logic. Use whichever name applies. With positive logic, call it a positive OR gate.
With negative logic, call it a negative AND gate. In a similar way, we can show the truth table of other gates with positive or negative logic. These definitions are always valid. If you get confused from time to time, refer to Table 2. Voltage Definitions of Basic Gates. Vn,egatjveAND Output is high if any input is high.
Assertion-level logic Why do we even bother with negative logic? The reason is related to the concept of active-low signals. For instance, the multiplexer has an active-low input strobe; this input turns on the chip only when it is low negative true.
This is an active-low signal; it causes something to happen when it is low, rather than high. As another example, the decoder has 16 output lines; the decoded output signal is low negative true. In other words, all output lines have a high voltage, except the decoded output line. Besides TTL devices, microprocessor chips like the have a lot of active-low input and output signals.
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