NH 67, Karur Trichy Highways, Puliyur C.F, Karur District UNITII COMBINATIONAL CIRCUITS


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1 NH 67, Karur Trichy Highways, Puliyur C.F, Karur District DEPARTMENT OF ELETRONICS AND COMMUNICATION ENGINEERING COURSE NOTES SUBJECT: DIGITAL ELECTRONICS CLASS: II YEAR ECE SUBJECT CODE: EC2203 UNITII COMBINATIONAL CIRCUITS Design procedure Half adder Full Adder Half subtractor Full subtractor Parallel binary adder, parallel binary Subtractor Fast Adder  Carry Look Ahead adder Serial Adder/Subtractor  BCD adder Binary Multiplier Binary Divider  Multiplexer/ Demultiplexer decoder  encoder parity checker parity generators code converters  Magnitude Comparator. INTRODUCTION In digital circuit theory, combinational logic (also called combinatorial logic) is a type of logic circuit whose output is a pure function of the present input only. This is in contrast to sequential logic, in which the output depends not only on the present input but also on the history of the input. In other words, sequential logic has memory while combinational logic does not. APPLICATIONS Combinational logic is used in computer circuits to do Boolean algebra on input signals and on stored data. Practical computer circuits normally contain a mixture of combinational and sequential logic. For example, the part of an arithmetic logic unit, or ALU, that does mathematical calculations is constructed in accord with combinational logic, although the ALU is controlled by a sequencer that is constructed in accord with sequential logic. For n input variables, there are 2 n possible input combinations. For each input there is only one output.
2 A combinational circuit is described by m Boolean functions, one for each output variable. Each output function is expressed in terms of n input variables. ANALYSIS PROCEDURE Procedure starts with getting Boolean expression and truth table from the logic circuit or by computer simulation program. Check first whether the given circuit is combinational or not. To get Boolean expression from circuit, i) Label all gates outputs which are the direct function of input variables with arbitrary symbols and determine Boolean function for each gate output ii) Label all gates outputs which are the functions of input variables and previously labeled gates and determine Boolean function for these gates iii) iv) Example: Repeat step ii until the output of the circuits are obtained. By repeated Substitution of previously defined symbols, outputs in terms of input variables is obtained T1, T2 and F2 are defined first. T1 = A+B+C T2 = ABC F2= AB+AC+BC T3 by next step. T3 = F2 T1 = (AB+AC+BC) A+B+C Logic diagram for analysis example F1 is the finalized expression for one output F1 = T3+T2 = A BC +A B C+AB C +ABC
3 Truth table for the above example Truth table is constructed directly from the Boolean expression. From the circuit, start constructing from the calculation for each logic gate until the output of the circuit is reached. DESIGN PROCEDURE Design starts from the specification of the problem and culminates in a logic circuit diagram or set of Boolean functions from which the circuit is obtained. Steps: i) Determine the number of inputs and outputs from the specification of the problem. ii) Derive the truth table. iii) Obtain simplified Boolean functions for each output as function of inputs iv) Draw the logic diagram and verify the correctness of the design Verbal specification of the problem should be interpreted correctly in truth table. Output binary functions in truth table are simplified by algebraic manipulation, map method or by computer simulation depending upon the application. Example: BCD to Excess3 code. Here the BCD is considered under 4bit case. Excess3 also have the same number of bits. Conversion table is shown below. Truth table for BCD to Excess3 code conversion KMap simplification for the outputs (w,x,y,z) is done as shown under to arrive at the simplified Boolean function.
4 The two level logic diagram for the derived Boolean expression is drawn directly. But the function may be simplified further as
5 Logic diagram for BCD to Excess3 conversion ARITHMETIC CIRCUITS Various arithmetic operations are encountered in computing tasks. Most basic operation is addition of two binary digits. Starting with the addition of 2 LSb and adding the next higher order bit with the generated carry and proceeds through the number Half adder It is a Combinational circuit doing the addition of two bits only. Output is Sum and Carry. Figure 2.3 Half adder The output Boolean expressions are, So = Ao Bo +AoBo Co = AoBo The logic diagram for the above expression can be expressed as follows: logic circuit of Half adder Simplified circuit Now this is only a half adder, although we are generating a carry, we cannot add in the carry from the previous addition. FULL ADDER For all other bits (except the lsb) a half adder will not suffice because there may be a carry input from a previous stage. ie) It is a Combinational circuit doing the addition of three bits only. (Two information bit+one previous carry).
6 Figure 2.6 Full adder and Truth table Full adder can be built by using two half adders as shown below. Two bits are added and the sum is given as one of the input to the next half adder along with the previous carry. Carry in cascaded blocks are added separately to produce the final carry. Remember the Boolean operation For half adder: S K = A K B K Figure 2.7 Full adder in terms of half adder C K = A K B K For full adder: S K = A K B K C K1 C K = A K B K + C K1 (A K B K )
7 Proof: Figure 2.8 Simplified Full adder THE PARALLEL (BINARY) ADDER Also called ripple carry adder used to add two nbit numbers. it consists of n full adders where the carry output of each stage is the carry in of the next stage. Simplified Full adder
8 Initial carry, C 1, is zero. But to get Co, propagation delay, 2 gate delay, inside Full adder is encountered. Totally 8x2 gate delays are there to get C3 finally. This is avoided by using Carry look ahead generator which is generating Carries simultaneously. Full adder Circuit From the above full adder circuit, Where G i is Carry generate as it produces the output carry regardless of the input carry and P i is Carry propagate as it is associated with the carry propagation from input to the output. We can have the following Boolean expression from the figure Above Boolean functions is implemented by the logic diagram shown in 2.11.
9 Logic diagram of a Carry look ahead generator A 4bit Binary adder is built with the Carry look ahead generator as shown in figure bit full adders with look ahead carry In Connection with the above diagram, all the output carries are generated simultaneously by having G i, P i and Co as the inputs. DECIMAL ADDER Decimal numbers are presented in coded form in all case of digital devices for manipulation. Accepting numbers in coded form and presenting the result in same form. In these cases 9 inputs (8 data+1 Previous carry) and 5 outputs(4 data+1 output carry) are needed. BCD ADDER One example of the decimal adder is BCD adder. It is useful because many digital devices process + display numbers in tens in BCD each number is defined by a binary code of 4 bits. We will now look at into the effort of using a binary adder to add 2 single digit BCD numbers. The maximum possible BCD number is 9. So for adding two BCD number, 9+9+1(for Carry) = 19 is the maximum value the BCD adder should represent at its output.
10 Following table represents the BCD, binary values for the corresponding decimal number till 19. We can use this table to detect the binary combinations which are greater than the BCD number ie) 9. From the table, (Look at the Binary Sum column) For the first group of numbers, partitioned till 9, BCD and binary are equal. In the next 6 combinations, Z 8 is 1 always also with 1s in either Z 4 or Z 2 places. This condition is stated as Z 8 Z 4 + Z 8 Z 2 In the last 4 combinations, K is 1. As a consequence we can deduce the condition of getting carry once the result of the summation of a binary adder exceeds the number 9. C = K+ Z 8 Z 4 + Z 8 Z 2
11 Block diagram of a BCD adder So When C =1, 0110 is added to the result to convert the binary number to BCD number. The entire process is implemented as a block diagram in figure 2.13 BINARY MULTIPLIER Binary multiplier performs the operation similar to the decimal multiplication using partial products method. Consider two 2bit number B1 B0 and A1 A0 B1 B0 A1 A0 A0B1 A0B0 A1B1 A1B0 C3 C2 C1 C0
12 Block diagram of 2bit multiplier For J multiplier bits and K multiplicand bits, we need (J*K) AND gates (J1) Kbit adders to produce (J+K) bits as result. MAGNITUDE COMPARATOR It compares two numbers and gives relative magnitudes. 3variables used to specify the result for three conditions (>, =, <) Thus for an nbit comparator 2 2n entries are needed in the truth table. So design is cumbersome unless we use algorithmic procedure. Consider for 4bit comparison of A A3 A2 A1 A0 and B B3 B2 B1 B0 A=B If all the bits are equal. ExNOR gate is used to check this status as X i = A i B i + A i B I for i = 0,1,2,3 Finally the condition is, (A=B)= X 3 X 2 X 1 X 0 A>B if MSB of A= 1 and that of B = 0. This can be stated as, (A>B) = A3B3 + X3A2B2 + X3X2A1B1 + X3X2X1A0B0 B>A if MSB of A= 1 and that of B = 0. This can be stated as, (A>B) = A3 B3 + X3A2 B2 + X3X2A1 B1 + X3X2X1A0 B0
13 The stated condition is implemented simply by the logic diagram as shown in figure below. A0 A1 B0 B1 C0 C1 C2 C3 4bit magnitude comparator The same circuit can be used to compare the magnitudes of two BCD digits. CODE CONVERSION Code conversion is necessary to achieve the compatibility between two different systems following different coding schemes. The code that has to be converted is applied as the inputs and the output gives the transferred code. BCD to Excess3 conversion was given already. EXCESS3 TO BCD CODE CONVERSION The truth table for excess3 to BCD is constructed and from the table the equation for conversion of BCD to excess3 is obtained. It is then implemented with the help of logic gates.
14 TRUTH TABLE EXCESS3 BCD A B C D W X Y Z By applying Kmap method the simplified circuit is shown below A B C D W 330k LED X 330k LED Y 330k LED 7486 Z 330k LED Excess3 to BCD converter BINARY TO GRAY CODE CONVERTER The first bit of the gray code will be the same as the first bit of the binary code. Second bit of gray code is obtained by the XOR operation of first two bits in the
15 binary code.similarly third bit is obtained by the XOR combination of second and third bit in binary and so on. TRUTH TABLE DECIMAL BINARY CODE GRAY CODE D C B A G3 G2 G1 G D C B A IC G0 330k LED G1 330k LED G2 330k LED G3 330k LED Binary to Gray converter
16 GRAY TO BINARY CODE CONVERTER TRUTH TABLE GRAY CODE BINARY CODE G3 G2 G1 G0 D C B A G1G2 G3 G4 D C 330k 330k LED LED IC B 330k LED IC A 330k LED IC 7486 Gray to Binary converter
17 PARITY CHECKING AND GENERATION It is the simple error detecting code by appending or prepending one (parity) bit in each data word at the transmitter. The bit added will make the total number of 1s in the word (including parity bit) as even or odd. The receiver detects the error by counting the number of 1s in each word. IC is used to check/generate even/odd parity of the 9bit data. INPUT PIN configuration The data (X0 X7) along with the Even or odd input (9bits in total) is checked by the IC. It will act as the parity generator if PE = 1, PO = 0 (shown in first two entry in truth table). These inputs also taken into account while counting for the 1s. even output will glow to generate a 1 to make the total number of 1s even. The last two entries show the IC operation as parity checker. It checks the number of 1s in those 9bits and gives output accordingly. Both the outputs are produced simultaneously. Output is derived from any of these two pins depending up on the parity we follow. TRUTH TABLE OUTPUT No.of.high PE PO EVEN ODD Data input (X0 X7) EVEN ODD EVEN ODD IC is cascaded to increase the word length capability from 9bit to 16bit. It is shown below. MODE = 1 for checking parity and for MODE = 0, for generating parity.
18 X8 > E EVEN X0 > X9 > X1 > X10 > X2 > X11 > E ODD X3 > X12 > X4 > X13 > X5 X14 > IC X6 > X15 > X7 > EVEN O/P > 330k LED IC ODD O/P 330k LED MODE EVEN > EVEN 1 2 ODD > ODD 7404 bit parity checker/generator DECODERS Often, digital information represented in some binary form must be converted into some alternative digital form. This is achieved by a multipleinput, multiple output network referred to as a decoder. The most commonly used decoder is the nto2 n line decoder: The structure of a such decoder is straightforward. Consider the truth table of a 3to 8line decoder:
19 This corresponds to the logic diagram shown below: A particular application for this decoder is binary tooctal conversion. The input variables represent a binary number, and the outputs represent the eight digits in the octal number system. DECODERS WITH AN ENABLE INPUT Some decoders include one or more enable inputs to control the circuit operation. The logic diagram and truth table of a 2to4line decoder are shown below:
20 A decoder with enable input can function as a demultiplexer. The above decoder can function as a 4to1line demultiplexer when E is taken as a data input line and A and B are taken as the selection inputs. Decoders with enable inputs can be connected together to form a larger decoder circuit. A 4to16line decoder realized using two 3to8line decoders is shown below: When w = 0, the top decoder is enabled and the other is disabled. The bottom decoder outputs are all 0 s, and the top eight outputs generate minterms 0000 to When w = 1, the enabled conditions are reversed; the bottom decoder generates minterms 1000 to 1111, while the outputs of the top decoder are all 0 s. COMBINATIONAL LOGIC IMPLEMENTATION An nto2nline decoder is a minterm generator. Recall that any Boolean function is describable by a sumofminterms. Thus, by using ORgates in conjunction with an n to2nline decoder realizations of Boolean functions are possible. However, these realizations do not correspond to minimal sumofproducts. Consider the pair of expressions: (1,2,4,5) F1(X2,X1,X0) = (1,5,7) F2(X2,X1,X0) = Using a single 3to8line decoder and two ORgates, the following realization is obtained:
21 When more than ½ the total number of minterms must be ORed, it is usually more economical to use NORgates rather than ORgates to do the summing. Consider the pair of expressions: (0,1,3,4,5,6) F1(X2,X1,X0) = (1,2,3,4,6) F2(X2,X1,X0) = These may be realized with a 3to8line decoder and two ORgates having a total of 11 terminals between them. However, a more efficient realization is to rewrite the expressions as: F1 (X2,X1,X0) = F1 (X2,X1,X0) = F2 (X2,X1,X0) = F2 (X2,X1,X0) = This corresponds to the realization shown below: (2,7) (0,5,7) A total of five gateinput terminals are needed. ENCODER Perform the inverse operation of decoders. An encoder has 2n (or fewer) input lines and n output lines. The output lines generate the binary code corresponding to the input value. An example of an encoder is the octaltobinary encoder whose truth table is as follows:
22 The equations for the three outputs are: Z = D1 + D3 + D5 + D7 y = D2 + D3 + D6 + D7 x = D4 + D5 + D6 + D7 The encoder can be realized with three ORgates. PRIORITY ENCODER The encoder defined before has the limitation that only one input can be active at any given time. If two inputs are active simultaneously, the output produces an undefined combination. This is resolved by establishing an input priority function. The truth table of a fourinput priority encoder is: In addition to the two outputs, x and y, the circuit has a third output V; this is a valid bit indicator and is set to 1 when one or more inputs are equal to 1. X s in the output represent don tcare conditions. X s in the input columns are for representing the truth table in condensed form. Instead of listing all 16 minterms of four variables, the truth table uses an X to represent either 1 or 0. According to the table, D3 has the highest priority followed by D2 and D1. The maps for simplifying outputs x and y are shown below:
23 The condition for output V is an OR function of all the input variables: V = D0+D1+D2+D3 The priority encoder is implemented as follows: MULTIPLEXERS A multiplexer is a circuit that selects binary information from one of many input lines and directs it to a single output. Normally, there are 2 n input lines and n selection lines whose bit combination determine which input is selected. The logic and block diagrams of a 2to1line multiplexer are shown below:
24 The circuit has two data input lines, I1 and I2, one output line Y, and one selection line S. When S = 1, the lower AND gate is enabled and I1 has path to the output. This multiplexer acts like a switch that selects one of the two sources. A 4to1line multiplexer is shown below: A multiplexer is also called a data selector, since it selects one of many inputs and steers the binary information to the output line. In general, a 2 n to1line multiplexer is constructed from an nto2 n decoder by adding to it 2 n input lines, one to each AND gate. The outputs of the AND gates are applied to a single OR gate. As in decoders, multiplexers may have an enable input to control the operation of the unit. By interconnecting several multiplexers in a treelike structure, it is possible to produce a larger multiplexer. For example, a 16to1 line multiplexer may be constructed using five 4to1line multiplexers as follows:
25 MUX/DEMUX TRANSMISSION SYSTEM One of the primary applications of multiplexers is to provide for the transmission of information from several sources over a single path. This process is known as multiplexing. Eg. the multiplexing of conversations on the telephone system. When a multiplexer is used in conjunction with a demultiplexer, an effective means is provided for connecting information from several source locations to several destination locations. This basic application is illustrated below:
26 By using n of the structures shown above in parallel, an nbit word from any of four source locations is transferred to the four destination locations. SUMMARY Thus, this unit provides a brief knowledge over adder, subtractor circuits. It also provides a sound knowledge on multiplexers, demultiplexers, decoder, encoder, parity generator and magnitude comparator circuits.
NH 67, Karur Trichy Highways, Puliyur C.F, Karur District DEPARTMENT OF INFORMATION TECHNOLOGY CS 2202 DIGITAL PRINCIPLES AND SYSTEM DESIGN
NH 67, Karur Trichy Highways, Puliyur C.F, 639 114 Karur District DEPARTMENT OF INFORMATION TECHNOLOGY CS 2202 DIGITAL PRINCIPLES AND SYSTEM DESIGN UNIT 2 COMBINATIONAL LOGIC Combinational circuits Analysis
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