Asst. Prof. Thavatchai Tayjasanant, PhD. Power System Research Lab 12 th Floor, Building 4 Tel: (02)

Transcription

1 Aircraft Electricity and Electronics Asst. Prof. Thavatchai Tayjasanant, PhD a co Power System Research Lab 12 th Floor, Building 4 Tel: (02)

2 Chapter 6: Logic Circuits 6.1 Basic Logic Circuit Concepts 6.2 Representation of Numerical Data in Binary Form 6.3 Combinatorial Logic Circuits 6.4 Synthesis of Logic Circuits 6.5 Minimization of Logic Circuits 2

3 6.1 Basic Logic Circuit Concepts Analog Signal Amplitudes are in a continuous range Each amplitude has a unique significance. Digital Signal Only a few restricted ranges of amplitude (discrete amplitudes) Each amplitude in a given range has the same significance. Most common are binary signals represented by the logic values 1 or 0 3

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5 Advantages of the Digital Approach When the noise is added d to: An analog signal impossible to determine the precise amplitude A digital signal can still determine the logic values (noise amplitude is not too large) It is possible to manufacture exceedingly complex digital circuits economically, while this is impossible for analog circuits. 5

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7 Positive versus Negative Logic Positive logic*: higher amplitude in a binary system represents 1 and the lower- amplitude range represents 0. Negative logic: 1 for lower amplitude and 0 for higher amplitude 1 = high, true or on, 0 = low, false or off * = Positive logic will be used throughout this course. 7

8 Digital Words A single binary digit = bit small information To represent more information groups of logic variables called digital words A byte consists of 8 bits A nibble consists of 4 bits 8

9 ADC = Analog-to-digital converter Digital computer uses digitized input signal to compute a sequence of values for the output t signal Digital it Signal Processing (DSP) DAC = Digital-to-analog converter 9

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11 6.2 Representation of Numerical Data in Binary Form Binary Numbers Digital words can represent numerical data. Base 10: Base 2: = = = With N bits, we can form 2 N distinct words, e.g. with three bits, we can form 2 3 (= 8) distinct words. 11

12 Conversion of Decimal Numbers to Binary Form Decimal Integer to Binary Repeatedly divide by two until the quotient is zero then stop. The remainders read in reverse order give the binary form 12

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14 Conversion of Decimal Numbers to Binary Form (Cont.) Decimal Fraction to Binary The fraction part of the number is repeatedly multiply by two. Stop when the desired degree of precision has been reached, e.g. closest six-bit. 14

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16 Conversion of Decimal Numbers to Binary Form (Cont.) Decimal Number (having both integer and decimal fraction) to Binary Convert each part separately and then combine both parts, e.g

17 Addition Binary Arithmetic Rule of addition is given below. 17

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19 Hexadecimal and Octal Numbers Binary numbers are inconvenient because it takes many bits to represent large numbers. Hexadecimal (base 16) and octal (base 8) numbers are easily converted to and form binary numbers. Note that Octal numbers use three-bit word and dhexadecimal numbers use four-bit word. 19

20 Conversion Octal and Hexadecimal numbers to Binary Replace each digit by its binary equivalent using Table 7.1 (next page) = Binary to Octal and Hexadecimal numbers For converting binary numbers to octal, arrange the bits in groups of three starting from the binary ypoint and work outward = =

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22 Binary-Coded Decimal (BCD) Format Sometimes, binary numbers are represented in binary form by writing the four-bit equivalents for each digit = BCD Calculators frequently represent numbers internally in BCD format. Calculations are often carried out in true binary fashion in computers. 22

23 Complement Arithmetic One s complement Replace 1s by 0s, and vice versa: Two s complement Adding 1 to the one s complement, neglecting the carry (if any) out of the most significant bit Copy the number (from right to left) until after the first 1 is copied. Then the remaining bits are inverted. 23

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25 Complement Arithmetic (Cont.) Subtraction using Complement Complements are useful for representing negative numbers and performing subtraction in computers. Signed two s complement representation, the first bit is taken as the sign bit (0 for positive and 1 for negative). Subtraction is done by first finding the two s complement of the subtrahend and then adding. 25

26 Perform using eight-bit signed two s complement arithmetic 26

27 There is a range that eight-bit words can represent, i.e. from -128 to

28 Overflow and Underflow In performing arithmetic using two s s- complement arithmetic, we must be aware of the possibility of Overflow result exceeds the maximum value that can be represented by the word length in use. Underflow result is less than the minimum value that can be represented by the word length in use. 28

29 6.3 Combinatorial Logic Circuits Logic Gates Circuits that combine several logic- variable inputs to produce a logic-variable output. AND Gate AND operation = Logical Multiplication AB is 1 if and only if A and B are both 1. Truth table Table listing all the possible combinations of the input variables and the corresponding output values. 29

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31 AA = A 1 = A A A0 0 = 0 AB = BA ( BC) ( AB) C ABC A = = 31

32 Logic Inverter It is the NOT operation on a logic variable. It is represented by placing a bar over the symbol. The bubble at the output is used to indicate inversion. OR Gate OR operation = logical addition A+B is 1 if A or B (or both) are 1. 32

33 A A = 0 A = A 33

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35 ( A + B) + C = A + ( B + C) = A + B + C ( B C) AB AC A + = + A + 0 = A A + 1 = 1 A + A =1 1 A + A = A 35

36 Boolean Algebra Boolean algebra = Mathematical theory of logic variables Steps: 1) Produce a truth table 2) List all possible combinations 3) Show that both sides of the expression yield the same results. 36

37 Implementation of Boolean Expressions Implementation can be implemented by interconnection of AND gates, OR gates and inverters. Manipulate logic expression to find a simpler equivalent expression. F = ABC+ ABC+ CD+ CE+ DD+ DE = CA ( + D+ E) + DE 37

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39 De Morgan s Laws ABC = A + B + C A+ B+ C= ABC If the variables in a logic expression are replaced by their inverses, 1) AND operation is replaced by OR 2) OR operation is replaced by AND, The entire expression is inverted, the resulting expression yields the same values. 39

40 De Morgan s Laws (Cont.) De Morgan s Laws provide an alternative way to write logic expressions. Any logic function can be implemented by using AND gates and inverters. Any logic function can be implemented by using OR gates and inverters. 40

41 Additional Logic Gates NAND = AND gate followed by an inverter NOR = OR gate followed by an inverter XOR (exclusive-or) Modulo-two addition (XOR yields 1 if A is 1 or if B is 1) Buffer Single input and produce the same output Equivalence gate = high output only if both inputs have the same value XOR followed by an inverter 41

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43 Logical Sufficiency of NAND gates or of NOR gates Inverter = Inputs to a NAND gate tied together AND gate = NAND gate followed by an inverter Therefore, basic logic functions can be realized by using only NAND gates. 43

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45 Logical Sufficiency of NAND gates or of NOR gates (Cont.) Similarly, basic logic functions can be realized by using only NOR gates. Are NAND gates or NOR gates more common? 45

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47 6.4 Synthesis of Logic Circuits Methods to implement logic circuits given the specification for the output in terms of the inputs. Sum-of Products (SOP) Implementation Product terms that include all of the input variables (or their inverses) are called minterms. Finally, write an expression for the output variable as a logical sum of minterms. 47

48 Steps for SOP Implementation Form a product of all the input variables (or their inverses) for each row of the truth table for which the result is logic 1. The output is the sum of these products. 48

49 D= m( 0267,,, ) 49

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51 Product of Sums (POS) Implementation Sum terms that include all of the input variables (or their inverses) are called maxterms. Finally, write an expression for the output variable as a logical product of maxterms. Steps for POS Implementation Form a sum of all the input variables (or their inverses) for each row of the truth th table for which the result is logic 0. The output is the product of these sums. 51

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54 Decoders, Encoders, and Translators Many useful combinatorial circuits such as decoders, encoders and translators are available as integrated circuits. Examples: Calculator or Watch (BCD form) Calculator display needs a decoder to translate the four-bit binary-coded decimal words into seven-bit words (seven-segmentsegment display) 54

55 All segments except G are displayed for zero number so seven-bit word = four-bit binary-coded decimal word of 0000 (= number 0) 55

56 6.5 Minimization of Logic Circuits Logic functions can be expressed either as SOP or POS. Direct implementation may not yield the best circuit in terms of minimizing the number of gates required. Karnaugh Maps Graphical approach used to simplify logic expressions. Rectangular array of squares. Each square is one of minterms of logic variables 56

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58 Karnaugh Maps (Cont.) The left and right (as well as top and bottom) edges are considered to be adjacent. Two squares with a common edge = 2-cube Four = 4-cube Also four squares in the map corners form a 4-cube 58

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60 Spot the set of largest cube that cover the 1s in a Karnaugh map. Minimum SOP: 60

61 Conclusions Logic gates are the basic building blocks for complex digital systems. Complex combinatorial logic functions can be achieved simply pyby interconnecting NAND gates (or NOR gates). Logic gates can be interconnected to form flip-flops. A complex digital system consists of many gates, flip-flops, and registers. 61

62 Summary Digital it signals are more immune to the effect of noise than analog signals. Numerical data can be represented in decimal, binary, octal, hexadecimal, a or BCD. Logic variables take two values: 1 and 0. They can be combined by the AND, OR, and inverters. A truth table lists all combinations of input variables and the corresponding output. NAND (or NOR) gates are sufficient to realize any combinatorial logic functions. Any logic functions can be written as SOP or POS. Karnaugh maps is used to minimize the number of gates needed to implement a given logic function. 62