Number System. Lesson: Number System. Lesson Developer: Dr. Nirmmi Singh. College/ Department: S.G.T.B Khalsa College, University of.

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1 Lesson: Number System Lesson Developer: Dr. Nirmmi Singh College/ Department: S.G.T.B Khalsa College, University of Delhi 1

2 NUMBER SYSTEM Table of contents Chapter 1: Number System 1.1 Learning outcomes 1.2 Introduction 1.3 Analogue and digital circuits 1.4 Number System Binary Number System Binary to decimal and decimal to binary conversion Octal System Octal to decimal and decimal to octal conversion Octal to binary and binary to octal conversion Hexadecimal System Decimal to hexadecimal and hexadecimal to decimal conversion Hexadecimal to binary and binary to hexadecimal conversion 1.5 Logic gates NOT gate OR gate AND gate 1.6 Universal Logic gates NAND gate Bubbled OR gate NAND gate as Universal gate NOR gate Bubbled AND gate NOR gate as Universal gate 1.7 Exclusive OR gate 1.8 Exclusive NOR gate 1.9 Realizing logic expressions using logic gates 1.10 Summary 1.11 Some interesting facts 1.12 Exercise 1.13 Glossary 1.14 References 2

3 1.1 Learning outcomes: Binary number system Octal number system Hexadecimal system Conversion from one number system to other Logic gates: basic and universal gates 1.2 Introduction: We usually use decimal system (0,1,2,3, ) although there are other many other number systems like binary, octal, hexadecimal etc.. These numbers are expressed with a base. For example, the base is 2 in the binary system while for octal it is 8 and for hexadecimal it is 16. But, in the digital circuits, we use the binary system only that is just two numbers 0 or 1. In other words, the output of the digital circuits is either high, 1 or low, 0. Algebra that deals with the binary system is Boolean algebra. It is used in simplifying logic circuits which are formed by interconnecting the logic gates like OR, AND gates etc. In other words, a logic gate is an electronic circuit which makes logical decisions. 1.3 Analogue and digital circuits: Electronic circuits and systems are majorly divided into two categories: analogue and digital circuits. Analogue circuits are: Designed for small signals Used in linear fashion Example: an operational amplifier with a given gain is an example of analogue circuit as the output is an amplified version of the input itself which is a linear operation. Continuous and all possible values for a given physical quantity are recorded. Digital circuits are: Used with large signals Considered to be non linear Discontinuous and represent only a finite number of discrete values. 3

4 Digital circuits involves circuits that work in two states that is the binary system having two set of values: 0 and 1 The operation of digital circuits is described in terms of two voltage levels: high level and low level. High level means H = 1 and low level means L=0. Choosing H=1 and L=0 is positive logic And L=1 and H=0 is negative logic. 1=+5 Volts dc 0=0 Volts dc 1.4 Number systems: Apart from the decimal system, we have binary system, octal system and hexadecimal systems out of which it is the binary system that holds importance in the digital world Binary Number system: Binary means two so a system having two states is what we call as binary. This system has exact two values: 0 and 1. This is the simplest number system as it uses only two digit. This has base 2. The abbreviation for binary digit is bit. If a binary digit has four bits, it is called a nibble and if a binary digit has eight bits, it is called a byte which is now the Basic unit of data in computers. Positions and weights: In a decimal system, a number is represented in units, hundreds, thousands and so on Which are the weights or value assigned to each digit. As one moves towards the left the weight of each digit is an increasing power of 10. For example: 239= which in powers of 10 is represented as 239=2x10^2 + 3x10^1 + 9x10^0 hundreds tens ones Similarly, we can assign weights to binary numbers also. The position of 0 or 1 in the binary number represents its weight within the number. Just like in the case of decimal number, the weight of each digit on the left is an increasing power of 2. For example: (180) 10 = ( ) 2 = 1x2^7 + 0x2^6 +1x2^5 +1x2^4 + 0x2^3 + 1x2^2 + 0x2^1 +0x2^0 Assigning weights makes it easier to convert a binary number into its decimal equivalent Binary to decimal and decimal to binary conversion: Binary to decimal conversion: In order to convert a binary number into its decimal equivalent, we rewrite the binary number in terms of weights. The least significant digit on the extreme right will have weight 1 that is 2^0. As one moves towards left the weights are in ascending powers of 2. For example consider the binary number This is equivalent to 1x2^3 + 1x2^2 + 1x2^1 + 0x2^0 4

5 = =14 In order to convert binary fractions into decimal equivalent, the weights towards the right of the binary fractions are given by ½,1/4,1/8,1/16 etc which in powers of 2 are as the following : 2^-1,2^-2, 2^-3 and so on. For example consider the binary fraction its decimal equivalent will be 0.111=1x2^-1 + 1x2^-2 +1x2^-3 = = For mixed numbers, the rules to be followed are the same. That is proper weights are to be assigned to 0s and 1s depending on whether the part is integer part or fractional part. For example ( )=1x2^2+1x2^1+1x2^0+0x2^-1 +0x2^-2 +1x2^-3 = = Decimal to binary conversion: The easiest method to convert a decimal number to a binary number is simply by dividing the decimal number by 2 progressively. Then the binary number is obtained by taking into account the remainder after every step or division in the reverse order that is from bottom to top (double dabble method). For example 62 Division by 2 remainder obtained After step 2) 61 2) ) ) 7 1 2) So the binary equivalent after reading the remainder in the reverse order will be For fractions, they are multiplied by 2 and a carry is recorded in integer position. Every time it is the resultant fraction multiplied by 2 again giving an integral carry that is either 0 or 1. These carries when read from top to bottom that is downwards gives the binary fraction. For example 0.65 x 2=1.3, it is written as 0.3 with a carry of x 2=0.6, 0.6 with a carry of x 2= with a carry of x 2= with a carry of x 2= with a carry of x 2= with a carry of 1 And so on. This multiplication by 2 continues until one gets the desired number of digits. So in this case, the six digit binary fraction for the above decimal fraction is this is just an approximate value as this multiplication can continue further. 5

6 For mixed decimal number, both the rules are to be followed that is for integer and for fractions. For example let us convert into its binary equivalent. Conversion of integer part: 2)65 2)32 1 2)16 0 2) 8 0 2) 4 0 2) The binary number for integral part is Conversion of fractional part: 0.725x2= with a carry of x2= with a carry of x2= with a carry of x2= with a carry of x2= with a carry of x2= with a carry of 0 and so on. So we take an approximate result up to six binary digits As a result, the binary number for (65.725) 10 is ( ) Octal system: The octal system uses the first seven digits that is from 0 to 7. The base of the system is eight and so is the name octal given to the system.8 does not come in octal system so 10 comes after 7. These digits from 0 to 7 have the same meaning as the decimal symbols. Octal Decimal (tibasicdev.wikidot.com/binadhex) Here also each digit is assigned a weight. the least significant digit or position has weight 1 or 8^0. the higher positions as we move left have weights in the increasing powers of 8 respectively. The rules for conversion from octal to decimal or binary system and vice versa are the same as that of binary to decimal conversion and vice versa with a difference that here powers of 8 are involved instead of powers of 2 and Octal to decimal and decimal to octal conversion: Octal to decimal conversion: 6

7 To convert the integral part from octal to decimal, each octal digit is multiplied by it weight and then added while for the fractional part, each octal digit is multiplied by 8^-1, 8^-2 towards the right and so on and then added. For example let us convert ( ) 8 to decimals 3x8^2+3x8^1+7x8^0+1x8^-1 +2x8^-2 +8x8^-3 = =( ) 10 Decimal to octal conversion: Here the division in the case of integral part is with 8 and also the multiplication in the fractional part is by 8. For example, let us convert ( ) 10 to an octal number Integral part : division remainder 8)555 8)69 3 8) so (555) 10 = (153) 8 Fractional part : multiplication carry or integer generated 0.212x8= x8= x8= Etc. So (0.212) 10 =(154) 8 So ( ) 10 = ( ) Octal to binary and binary to octal conversion: Octal to binary conversion: Octal to binary conversion is possible as 8 is just the third power of 2. This conversion is very simple and easy. Each octal digit is replaced by its 3 bit binary digit and the complete binary number is thus obtained. For example consider (234) 8 Now 3 bit binary equivalent for 2 is 010, for 3 is 011 and for 4 is 100. So the binary equivalent of the given number will be ( ) 2. Similar is the case for fractions also. Binary to octal conversion: In this conversion, the binary digits are divided into group of three bits. Then each group is converted into its decimal equivalent. For example, consider ( ) 2 Dividing into group of three bits by adding 0 if required we get Now 011 is 3, 101 is 5, 100 is 4 and 010 is 2. So the octal number will be (35.42) Hexadecimal system: 7

8 Hexadecimal system has base 16 and thus uses 16 symbols. They are much shorter in comparison to the binary numbers and are thus easy to write. They find use in microprocessor work. The 16 symbols are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E and F. Here, A,B,C,D,E and F represent the decimal numbers 10,11,12,13,14 and 15 respectively. In this system also each significant digit has been assigned a weight or a value. The least significant digit is assigned the value 1 that is 16^0 while as we move left the weight assigned to each digit is in the increasing powers of 16 that is 16^1, 16^2, 16^3 etc respectively Decimal to hexadecimal and hexadecimal to decimal conversion: Decimal to hexadecimal conversion: Decimal number is converted into its hexadecimal equivalent by dividing successively by 16 and considering the remainder in the reverse order. For example: consider (334) 10 its hexadecimal equivalent will be Division by 16 remainder 16)334 16)20 14 or E 1 4 So (334) 10 =(14E) 16 Hexadecimal to decimal conversion: In this case each digit is multiplied by its weight in terms of powers of 16 For example: (E9A5.23) 16 converts into it decimal equivalent as: E9A5.23 = 14x16^3+9x16^2+10x16^1+5x16^0+2x16^-1 +3x16^-2 = =( ) Hexadecimal to binary and binary to hexadecimal conversion: Its easy and simple to convert a binary number to its hexadecimal equivalent and vice versa. To convert hexadecimal number into its binary equivalent, each significant digit is replaced by its 4-bit binary equivalent and we get the resultant number For example consider (4C2) 16 4-bit binary digit for 4 is 0100, for C is 1100 and for 2 is So the binary equivalent for(4c2) 16 will be ( ) 2 To convert a binary number into hexadecimal number, the given binary number is divided into groups of four and then each group is replaced by its decimal equivalent but in terms of hexadecimal symbols. For example, consider ( ) corresponds to 15 or F, 0010 to 2, 0101 to 5 and 1100 to 12 or C. So the hexadecimal equivalent of the given binary number will be (F25C) 16 8

9 The following is a table for 4 bit binary numbers, equivalent to decimal 0 to 15 and their hexadecimal symbols. Binary Decimal Hexadecimal A B C D E F 1.5 Logic gates: A gate is just a digital circuit having one or more input signals but the output is one. These gates find importance in making combinational logic circuits that is the output signal will appear only for certain combinations of the input signal. In other words, these are the building blocks for today s integrated circuits. OR, NOT, AND, NOR and NAND gates are some commonly used gates. OR,AND and NOT gates are the basic gates which makes it possible to build the logic circuits that can perform arithmetic functions. NOR and NAND gates are the Universal gates. While gates like Exclusive OR and Exclusive NOR gates can be obtained from the basic circuits. Logic gates find applications in : (i) Building some complex devices like binary counters (ii) Calculators and computers (iii) Digital communication (iv) Digital music instruments, home appliances Positive and negative logic: In the logic circuits, 0 and 1 the two binary digits are represented by voltage levels. If the more positive voltage level (+5 Volts) represents the logic 1; that is High=1 while the more negative level (0 volts) represents the logic 0; that is LOW=0 then the logic is positive logic. But if the more positive voltage level represents the logic 0 ; that is HIGH=0, and the more negative voltage level represents the logic 1 ; that is LOW=1 then the logic is negative logic. We will make use of only the positive logic that is H=1, L=0 9

10 1.5.1 NOT GATE (inverter gate) : The NOR gate is the only gate which has got one input. It performs the function called inversion that is the output is the compliment of the input. So when a HIGH input is given then the output is LOW and vice versa. The truth table and the symbol for the NOT gate are shown in the figure 1.5.1(a) Input A Output Y=A' Figure (a) Symbol of NOT Gate A NOT gate can be realized using a transistor which is shown in figure 1.5.1(b). When the input is HIGH the transistor will be in ON state drawing maximum collector current so the output at Y becomes LOW as the whole voltage drop will be across the resistance R. But when the input is LOW, the transistor will be in the OFF state and the output Y will, therefore be HIGH that is Y = V cc as there will be no voltage drop across R now. 10

11 Figure (b) OR Gate: An OR gate has got two or more inputs while one output. The gate is called OR gate as the output voltage is HIGH if any one of the inputs or all the inputs are HIGH. While the output is LOW if both the inputs are LOW. The truth table and the symbol for a two input OR gate is shown in the fig.1.5.2(a) A B Y Fig 1.5.2(a) In Boolean Form, if A and B are the inputs while Y is the output, then the OR logic is Y=A+B The + sign only represents the logic operation OR and not the basic arithmetic addition operation. Fig 1.5.2(b) An OR gate can be realized using diodes which is shown in the fig 1.5.2(b). 11

12 Here A and B are the two inputs to the two diodes D 1 and D 2 respectively while Y is the output which is noted down across R, the load resistance. If A=0, B=0 both the diodes will be non conducting and so there will be no voltage drop across R and the output Y=0 If A=0, B=1 then the diode D 1 will not conduct but D 2 will conduct then voltage across R will be 5 volts and Y=1 If A=1, B=0 then the diode D 2 will not conduct but D 1 will conduct so Y=1 If A=1,B=1 then both the diodes will conduct and hence Y=1 The fig 1.5.2(c) shows a three input OR gate, its truth table and its realization using diodes. A B C Y Fig (c) Here A,B and C are the three inputs and Y is the single output. If all the three inputs are LOW then Y will be LOW but if any one of the inputs is high then the output will be HIGH. In Boolean form the OR logic will be Y=A+B+C While realizing the OR logic using diodes, here instead of two diodes, now three diodes are used. If all the inputs are LOW, none of the diodes will conduct and the output will also be LOW, but if one of the inputs is HIGH, then the diode corresponding to that input will conduct and hence, the output will be HIGH. Point to be remembered: 12

13 An OR gate can have more than three inputs also but the logic will remain the same. There may be any number of inputs, but if even one of the inputs is HIGH, the output will be HIGH AND Gate : The AND gate also has two or more than two inputs and one output. The gate is so called as in this gate the output will be high only if all the inputs are HIGH otherwise the output will be LOW. The truth table and the symbol for a two input AND gate is shown in the fig1.5.3(a) A B Y Fig 1.5.3(a) In Boolean form, if A and B are the inputs of AND gate and Y is its output then the AND logic will be Y=A.B Where. represents the AND logic and not the basic arithmetic multiplication. The fig 1.5.3(b) shows the symbol of a three input AND gate and its truth table A B C Y

14 Fig 1.5.3(b) Here A,B and C are the three inputs and Y is the output. Here, the output is HIGH only if all the three inputs are HIGH otherwise the output will be LOW. For this reason, AND gate is also called ALL gate. In Boolean form, three input AND logic is Y=A.B.C AND gate can also be realized using diodes which is shown in the fig (c): Fig (c) i A and B are the inputs to the two diodes D 1 and D 2 respectively and Y is the output. If A=0 and B=0 then both the diodes will be forward biased and will conduct and the whole voltage drop will be across R and the output Y=0 If A=0 and B=1 then D 1 will conduct while D 2 will not conduct and again Y=0 If A=1 and B=0 then D 2 will conduct while D 1 will not conduct and again Y=0 If A=1 and B=1 then both the diodes will be reversed biased and not conduct so Y=1 Points to be remembered: There may be any number of inputs for an AND gate, but only if all the inputs will be high, then the output will be HIGH. Even if one inputs will be high, the output will remain LOW. 1.6 Universal logic gates: NOR, NAND gates OR,AND and NOT gates that we have studied are the basic gates with the help of which we can realize any logic function. But these gates cannot be converted among themselves as for example one cannot get AND operation by any combination of wether NOT or OR gate.so 14

15 we consider such gates which we call as universal gates because each such gate individually is capable of performing a logic operation. In other words, only one such type of gate in itself can be used to make a complete logic circuit. Two such universal gates are the NAND and NOR gates NAND Gate: NAND gate is the combination of AND and NOT gates. It has also got two or three inputs with one output and it works on the logic which is compliment of AND gate that is if A and B are the inputs then the output Y=(A.B). When all the inputs are LOW, the output is HIGH; when one of the inputs is HIGH, then also the output is HIGH but when both the inputs are HIGH, the output will be LOW just opposite to that of AND gate. The symbol and truth table for a two input NAND gate is shown in the fig.1.6.1, the small circle or bubble represents the inversion operation. A B Y Fig Bubbled OR gate: A bubbled OR gate is nothing but inverters on the input of OR gate. The circuit is shown in the fig On seeing both NAND and bubbled OR gates, we notice that the output Y and the inputs A and B are identical. Thus, the two circuits are equivalent and interchangeable. In terms of Boolean algebra, Y=(A.B) for NAND gate Y=A +B for bubbled OR gate Since both the gates are equivalent so (A.B) = A + B (De Morgan s Theorem) 15

16 Fig NAND gate as universal gate: NAND gate is one of the Universal gates as it can be used to implement any of the three gates OR,NOT and AND gate or any of their combinations. NOT, AND, OR gates from NAND gate: The logic circuits for obtaining NOT, AND and OR gates from a NAND gate are shown in the figures (a), 1.6.3(b),1.6.3(c) respectively. In the case of obtaining NOT gate, both the inputs are tied so that same and single input goes to the NAND gate. If the input is 0, then both the inputs are 0 and according to NAND logic, the output will be 1. If the input is 1, then both the inputs will be 1 and the output, thus will be 0. Thus, the output is the compliment of the input and thus we get the NOT gate. Fig 1.6.3(a) In the case of AND gate, two NAND gates are used. The second NAND gate is used as a NOT gate which inverts the logic of the first NAND gate to attain the AND logic Fig 1.6.3(b) In the case of OR gate, three NAND gates are used. The inputs in each of the first two NAND gates are tied to get NOT gates and then the output from those two gates goes to the input of the third NAND gate to get OR logic. 16

17 Fig 1.6.3(c) NOR Gate: It is the combination of OR and NOT gate and works on the logic which is compliment of OR. That is if A and B are the inputs, then the output is Y=(A+B). When both the inputs are LOW only then the output is HIGH, but if one of the inputs or both the inputs is HIGH, then the output will be LOW. The truth table and the symbol for two input NOR gate is shown in the fig A B Y Fig Bubbled AND gate A bubbled AND gate is just that the inputs going to the AND gate are inverted. On looking at both the NOR and Bubbled AND gate, we notice that the inputs A and B and the output Y are identical for both the gates and hence the two gates are equivalent and can be used in place of each other. 17

18 In term of Boolean expression Y=(A+B) for NOR gate Y=A.B for the bubbled AND gate Since both the gates are equivalent so (A+B) = A.B (De Morgan s theorem) Fig NOR gate as universal gate: NOR gate is the other universal gate which can be used to obtain the basic gates that is OR, NOT and AND gates and their combinations. NOT, AND, OR gate from NOR gate The logic circuits for obtaining NOT, OR, AND gates from NOR gate are shown in the figures 1.6.6(a), 1.6.6(b) and 1.6.6(c) respectively. In order to obtain NOT gate, both the inputs to the NOR gate are tied to get one input. If the input is 0, then both the inputs to the NOR gate will be 0 and as per the logic of NOR gate, the output will be 1. If the input is 1, then both the inputs to the NOR gate will be one and the output will be0. So the output is just the compliment of the input which is the logic for NOT gate. Fig 1.6.6(a) In order to obtain OR gate, we use two NOR gates. The second NOR gate performs the function of NOT gate and inverts the logic from the first NOR gate to attain the OR gate. 18

19 Fig 1.6.6(b) In order to obtain AND gate, we use three NOR gates. The first two NOR gates act as NOT gates and their output goes as the input to the third NOR gate whose output give the logic for the AND gate. Fig 1.6.6(c) 1.7 Exclusive OR Gate: Exclusive OR gate or Ex-OR gate also has two or more than two inputs with one output. In a two input Ex-OR gate, the output will be HIGH only if exclusively either of the two inputs will be HIGH and the output is LOW if both the inputs are either HIGH or LOW simultaneously. That is why the name is given as Exclusive gate. The truth table and the logic symbol for an Ex-OR gate are given in the figure 1.7 (a): A B Y

20 Fig 1.7(a) The truth table shows that the output is HIGH, when any one inputs is HIGH. This feature makes this gate exclusive and different from OR gate. If there are more than two inputs, then the output of the Ex-OR gate will be HIGH only if the odd number of inputs is HIGH. But when the number of even inputs is HIGH then the output will be LOW. In term of Boolean form, if A and B are the inputs and Y is the output then for this gate Y=A+B=AB + A B The Ex-OR gate can be realized using AND,NOT,OR, shown in fig 1.7(b) Fig 1.7(b) 1.8 Exclusive NOR GATE: The Exclusive NOR gate is nothing but Ex-OR gate followed by an inverter that is a NOT gate. An Ex-NOR gate has got the output HIGH only when both the inputs are in same state that is either both are 0 or 1 while the output is LOW if the inputs are in different states. If the number of input is more than two, then the output of Ex-NOR gate is HIGH when the even number of inputs is 1 and the output will be LOW if the number of odd inputs will be 1 which makes it different from NOR gate. The truth table and the logic symbol for Ex-NOR gate is shown in the fig1.8 A B Y

21 1 1 1 Fig 1.8 In terms of Boolean Algebra, if A and B are the inputs while Y is the output then Y=AB+A B Y=(A+B) =(A B+A B) =(AB ).(A B) (De Morgan s Theorem) =(A +B)(A+B ) =(AB+A B ) 1.9 Realizing logic expressions using gates: We are now going to do some examples of how certain logic expressions can be realized using the logic gates. Some complicated logical expressions can be also be simplified using Boolean algebra and then realized using logic gates but that will be taken care of in the next module: Example 1: Realize the logic expression Y=B C +A C +A B using basic gates. Solution: There are three product terms which are realized using AND gates but they are complimented using NOT gates first. Then these additive expressions are realized using OR gates as following: 21

22 Example 2: Design the logic for the following Boolean expression using logic gates. Y=(A+B C) Solution: The second term is the product term to be realized with AND gate with one term complimented with the help of NOT gate and then they are ORed using OR gate 1.10 Summary: To summarize, we have studied the various numbers systems that exist. For example, binary system, octal and hexadecimal system etc. Binary systems can be converted to octal or decimal or hexadecimal etc We have also studied the different logic gates OR, NOT,AND. A gate is just a digital circuit having one or more input signals but the output is one. 22

23 These gates find importance in making combinational logic circuits that is the output signal will appear only for certain combinations of the input signal. In other words, these are the building blocks for today s integrated circuits. NAND and NOR gates are the universal gates with which we can make OR,NOT and NAND gates We also have Exclusive OR and Exclusive NOR gates used in digital circuits Some interesting facts: About the number systems: Number systems are used to describe basically the quantity of something or represent certain information. Most important is the Binary System which is useful in computer and electrical engineering. Transistors, the basic component in making circuits operate on the Binary System. 0 means no current while 1 means current is allowed to flow Computers and electronics work with bytes which are the eight digit binary numbers. Octal system uses less symbols than our conventional number system. Eight symbols are used to represent all the quantities. 8 doesnot exist so 10 comes after 7 ( About the logic gates: Logic gates can be implemented not only using diodes or transistors but also using vacuum tubes, relay logic, fluidic logic, optics etc. In modern practice most of the gates are made from Field Effect Transistors, particularly MOSFET (Metal Oxide Semiconductor Field Effect Transistor) Charles Sanders Peirce ( ) showed that NOR gates alone or NAND gates alone can produce other logic gates. But this work remained unpublished till The fist published proof was given by Henry M. Sheffer in 1913, so the NAND logical operation is sometimes called Sheffer Stroke while the logical NOR is sometimes called Peirce s arrow. 23

24 Charles s Sanders Peirce ( IC7400-IC containing NAND gates ( Three-state logic gates A tristate buffer can be thought of as a switch. If B is on, the switch is closed. If B is off, the switch is open. A three-state logic gate is a type of logic gate that can have three different outputs: high (H), low (L) and high-impedance (Z). The high-impedance state plays no role in the logic, which is strictly binary. These devices are used on buses of the CPU to allow multiple chips to send data. A group of three-states driving a line with a suitable control circuit is basically equivalent to a multiplexer, which may be physically distributed over separate devices or plug-in cards. In electronics, a high output would mean the output is sourcing current from the positive power terminal (positive voltage). A low output would mean the output is 24

25 sinking current to the negative power terminal (zero voltage). High impedance would mean that the output is effectively disconnected from the circuit. ( Exercise: Muiltiple choice questions: Q1. The digital system operates on system (i)binary, (ii) decimal, (iii) hexadecimal, (iv) octal Ans: (i) Q2. The octal system uses powers of for positional values (i) 2 (ii) 8 (iii) 10 (iv) 16 Ans: (ii) Q3. The decimal equivalent for the number (1001) 2 is (i) 2, (ii) 4 (iii) 9 (iv) 12 Ans: (iii) Q4. In positive logic, the logic state 1 corresponds to (i) positive voltage (ii) higher voltage level (iii) zero voltage (iv) lower voltage level Ans: (ii) Q4. The output of a two-input OR gate is 0 only if (i) both the inputs are 0 (ii) one of the inputs is 0 (iii) both the inputs are 1 Ans: (i) Q5. The function of a NOT gate is to (i) Stop a signal (ii) Invert a signal (iii) Recompliment a signal Ans: (ii) Q6. What is the base of a decimal system (i) 2 (ii) 8 (iii) 10 (iv) 16 Ans: (iii) Q7. A NAND gate is OFF when (i) All the inputs are high (ii) When one of the inputs is high 25

26 (iii) When both the inputs are low Ans: (iii) Q8. Is a three input NOT gate possible (i) Yes (ii) No Ans: (ii) : Subjective Type questions: Q1. Write the truth table and the logic gate for a three input XOR gate Q2. explain how NOT, OR and AND gates can be realized using diodes and transistors? Q3. Name the gates that are used as Universal gates. Why are they called so? Q4. Convert the following numbers into their octal equivalents: (i) (ii) Q5. Convert the following binary numbers to hexadecimal equivalents (i) (ii) Q6. Convert the following decimal numbers into their binary and octal equivalents: (i) 64.2 (ii) Q7. Find the Boolean equation for the output of the given logic circuit. What would be the output if A=1,B=0,C=1,D=1 Q8. What would be the expression for the output of the given logic circuit. Also find the output when A=1,B=0,C=1,D=0 26

27 Q9. Give the output equation for the following logic circuit: Q10. Write the logic functions for the following circuit and compute the result for A=0,B=1,C=0,D=1 Q11. Realize Y=(A+B)(A +C)(B+D) using basic gates Q12. Realize Y=(AB) +A+(B+C) using NAND gates only 27

28 Q13. Realize Y=A+BCD using NAND gates only Q14. Realize Y=(A+C)(A+D )(A+B+C ) using NOR gates only Q15. Draw the logic diagram of X-OR gate and discuss its operation Q16. What is an X-NOR gate. Give its truth table Q17. Construct a two input X-OR gate using NAND gates only Glossary: Number systems: Number systems are used to describe basically the quantity of something or represent certain information. For example the conventional decimal system, binary system used in digital circuits, octal and hexadecimal systems. Decimal system: It is the conventional system that we use everywhere. I has base 10 and is represented by 10 digits from 0 to 9. Binary system: It is the most simplest system used in digital logic circuits. It has base 2 and takes two values only, 0 or 1. Octal system: An octal system has base 8 and uses 8 symbols from 0 to 7. 8 does not come in this system. 10 comes after 7 here. Hexadecimal system: This system has got base 16 and uses 16 symbols. Analogue circuits: Analogue circuits are designed for small signals and are used in linear fashion. For example OP Amp. Digital circuits: Digital circuits are used with large signals, are considered to be non linear, are discontinuous and represent a finite set of discrete values. Logic gates: A gate is just a digital circuit having one or more input signals but the output is one. NOT,OR,AND,NAND and NOR are some commonly used logic gates. Positive logic: In a logic circuit,if the more positive voltage level (+5 Volts) represents the logic 1; that is High=1 while the more negative level (0 volts) represents the logic 0; that is LOW=0 then the logic is positive logic. 28

29 Negative logic: In a logic circuit, if the more positive voltage level represents the logic 0 ; that is HIGH=0, and the more negative voltage level represents the logic 1 ; that is LOW=1 then the logic is negative logic References: Digital Principles and Applications, Donald P.Leach, Albert Paul Malvino & Goutam Saha, 7 th Edition, 2011, Tata Mc Graw Hill companies Digital Electronics, S.Salivahanan & S.Arivazhagan, 2010, Vikas Publishing House Pvt. Ltd. Digital Electronics, An Introduction to Theory and Practice, W.H. Gothmann, 1982, PHI 29