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# Class Note for ENGIN 112 at UMass(41)

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Date Created: 02/06/15

College of Engineering University of Massachusetts Amherst ENGIN 112 Introduction to Electrical and Computer Engineering Fall 2008 Discussion A 6 NAND and NOR Implementations Combinational Logic I NAND Circuits As we ve seen NAND circuits are easy to implement in CMOS Also as we ve seen the basic logic operations are AND OR and NOT NAND gates are universal in the sense that they can implement each of these basic operations 0 NOT Note that the operation y X can be implemented as a 2 input NAND gate with X at both inputs X X ii M z X39y X39y NOT X NAND y z W2 L 3H 80 iii OR So Z Xy XY XquotY X NAND Y gt0 gt Twolevel NAND implementations for any function can be derived from the sumofproducts representation for the function using DeMorgan s Rule Example Consider the function FvmxyzWXyZW XZ By using a Karnaugh Map we saw in Discussion 5 that the function simplifies to F xz xy wz Now apply DeMorgan s Rule F XzXywz NH Xyquot WZ X NAND z NAND X NAND y NAND W NAND z So we get the NAND implementation We can also derive a simple graphical method for converting AND OR logic diagrams into NAND implementations by noting that by DeMorgan s Rule X39y X y that is NAND ANDinvert is equivalent to invertOR Do z 13 Z Then to convert a general AND OR logic diagram to NAND form we 1 Replace all AND gates with NAND AN Dinvert Replace all OR gates with invertOR Important note any inputs into invertOR gates that do not come from the outputs of NAND gates must be inverted 3 Replace the invertOR gates with NAND gates Example Consider a function having the following logic gate implementation This is converted first to A B C and then to Q gtF gt need inverter gt H II NOR Circuits NOR circuits are also easy to implement in CMOS Like NAND gates NOR gates are universal in the sense that they can implement each of the basic logic operations AND OR and NOT 0 NOT Note that the operation y X can be implemented as a 2 input NOR gate with X at both inputs X 0 XX X X yzx X Y ii M z X39y X39y X y NOTX NOR NOTy 80 z X DH iii OR Z xy XY X NOR Y 80 Twolevel NOR implementations for any function can be derived from the productofsums representation for the function using DeMorgan s Rule Example i Consider the function FvmxyZWXyzW XZ ln Discussion 5 we saw that F Z567911131415 so F 20 1234810 12 Using a Karnaugh map we see that this simplifies to F w X y z X z WX yz Xz 80 F WX yZ XZ W NOR X NOR y NOR z NOR X NOR 2 Then we have the NOR gate implementation o Do F We can also derive a simple graphical method for converting OR AND logic diagrams into NOR implementations by noting that by DeMorgan s Rule Xy X y that is NOR ORinvert is equivalent to invertAND j 2 2 2 y y Then to convert a general OR AND logic diagram to NOR form we 1 Replace all OR gates with NOR ORinvert Replace all AND gates with invertAND Important note any inputs into invertAND gates that do not come from the outputs of NOR gates must be inverted 3 Replace the invertAND gates with NOR gates Example Consider the function having the logic gate implementation ZZD j F C This is converted first to A B C and then to gt4 need inverter III Combinational Circuits So far we ve looked at logic circuits that convert a number of inputs into a single output but in general circuits can use the same set of inputs to generate a number of different outputs Some circuits have memory that is they store outputs from one time to use as inputs at another time These are called sequential circuits we ll consider them later in the course For now we will consider circuits that have multiple inputs and outputs but no memory these are called combinational circuits combinational n inpUtS circuit 1quot11 l39ll m outputs Suppose first that we are given a combinational circuit diagram how do we determine the output functions The answer is the same as for circuits with one output we trace through gates in the circuit to find the logical functions at each step Since combinational logic can get complicated it is often helpful to use internal variables which are themselves functions of inputs to describe the actions of the gates inside the circuit and at the end make substitutions to put everything in terms of input variables Example Find expressions for the outputs F1 and F2 of the following circuit A B Introduce internal variables T1 AB T2 T C T3 BC A T7 B 39 T2 0 1 B a Then F1 T2T3 T1 CBC ABCBC F2 AT3 ABC Of course Can also describe the inputoutput relations for each of the functions using truth tables Now suppose that we are given functions that we want to implement how do we design a combinational circuit to perform those functions Again the basic approach is the same as for one function we use Karnaugh maps to find simplified expressions for the functions and then connect the logic gates needed for the operations Example Suppose we want to implement a circuit that forms the arithmetic sum of two 2bit binary numbers Let the two input numbers be denoted wx and yz Note that because of a carry the sum has up to three bits denote these AB C For example wx 10 and yz 10 gives AB C 100 Design a combinational circuit that has wxyz as inputs and ABC as outputs First draw the truth table for the necessary functions i5 wxyz wxyz A B C 0000 000 0 0 0 0001 001 0 0 1 0010 010 O 1 O Noyv draw Karnaugh maps to find the function needed 001 1 01 1 0 1 1 to generate each output 0100 001 0 0 1 variable 0101 010 0 1 0 0110 011 0 1 1 0111 100 1 0 0 1000 010 0 1 0 1001 011 0 1 1 1010 100 1 0 0 1011 101 1 0 1 1100 011 0 1 1 1101 100 1 0 0 1110 101 1 0 1 1111 110 1 1 0 K map for A yz W oo 01 11 1o oo 01 1 11 1 1 1 1o 1 1 80 A wy WXZ Xyz K map for B yz W oo 01 11 1o 00 1 1 01 1 1 11 1 1 1o 1 1 80 B wy z WX y W X y W yz W Xy z nyz K map for C yz WX OO 01 11 1O 00 1 11 1 1 1o 1 80 C xz X z Combinational Circuit for generating A and C Note The circuit for generating B is quite complicated We ll soon see a better alternative for implementing an adder in the lecture

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