1. Chapter 4 Combinational Logic

2. 4.1 Introduction Logic circuits for digital systems may be  combinational or sequential. A combinational circuit consists of logic gates  whose outputs at any time are determined from only the present combination of inputs. 2

3. 4.2 Combinational Circuits Logic circuits for digital system  Sequential circuits  contain memory elements  the outputs are a function of the current inputs and the  state of the memory elements the outputs also depend on past inputs  3

4. A combinational circuits  n 2 possible combinations of input values  Combinational circuits n input m output Combinatixnal variables variables xoxic Circuit Specific functions  Adders, subtractors, comparators, decoders, encoders,  and multiplexers 4

5. 4-3 Analysis Procedure A combinational circuit  make sure that it is combinational not sequential  No feedback path  derive its Boolean functions (truth table)  design verification 

6. A straight-forward procedure F = AB+AC+BC 2 T = A+B+C 1 1 T = ABC 2 T = F2'T1 3 F = T3+T2 1 6

7. F = T +T = F 'T +ABC  1 3 2 2 1 = (AB+AC+BC)'(A+B+C)+ABC = (A'+B')(A'+C‘)(B'+C')(A+B+C)+ABC = (A'+B'C')(AB'+AC'+BC'+B'C)+ABC = A'BC'+A'B‘C+AB‘C'+ABC 7

8. The truth table  8

9. 4-4 Design Procedure The design procedure of combinational circuits  State the problem (system spec.)  determine the inputs and outputs  the input and output variables are assigned symbols  derive the truth table  ed b x derive the simplifi erive the simplified Bool oolean functions xan functionx  draw the logic diagram and verify the correctness  9

10. code conversion example BCD to excess-3 code  The truth table  11

11. The maps 12

12. The simplified functions  z = D'  y = CD +C'D‘ x = B'C + B‘D+BC'D' w = A+BC+BD Another i mplementation   z = D'  y = CD +C'D' = CD + (C+D)' x = B'C + B'D+BC'D‘ = B'(C+D) +B(C+D)' w = A+BC+BD 13

13. The logic diagram  14

14. 4-5 Binary Adder-Subtractor Half adder  0 + 0 = 0 ; 0 + 1 = 1 ; 1 + 0 = 1 ; 1 + 1 = 10  two input variables: x, y  two output variables: C (carry), S (sum)  truth table  15

15. S = x'y+xy'  C = xy the flexibility for implementation  S=xÅy  S = (x+y)(x'+y')  S‘= xy+x'y'   S = (C+x'y')' C = xy = (x'+y')x  16

16. 17

17. Z 0 0 0 0 X 0 0 1 1 + Y + 0 + 1 + 0 + 1 C S 0 0 0 1 0 1 1 0 Z 1 1 1 1 X 0 0 1 1 + Y + 0 + 1 + 0 + 1 C S 0 1 1 0 1 0 1 1 Functional Block: Full-Adder • A full adder is similar to a half adder, but includes a carry-in bit from lower stages. Like the half-adder, it computes a sum bit, S and a carry bit, C. • For a carry-in (Z) of 0, it is the same as the half-adder: • For a carry- in(Z) of 1:

18. Full-Adder  the arithmetic sum of three input bits  three input bits  x, y: two significant bits  z: the carry bit from the previous lower significant bit  Two output bits: C, S  18

19. 19

20. S = x'y'z+x'yz'+ xy'z'+xyz  C = xy + xz + yz S = zÅ (xÅy)  = z'(xy'+x‘y)+z(xy'+x'y)' = z‘xy'+z'x'y+z(xy+x‘y') = xy'z'+x'yz'+xyz+x'y'z C = z(xy'+x'y)+xy = xy'z+x'yz+ xy 20

21. Binary adder 21

22. Binary subtractor A-B = A+(2’s complement of B)  4-bit Adder-subtractor  M=0, A+B; M=1, A+B’+1  26

23. Overflow  The storage is limited  Add two positive numbers and obtain a negative  number Add two negative numbers and obtain a positive  number V = 0, no overflow; V = 1, overflow  Example: 27

24. 4-6 Decimal Adder Add two BCD's  9 inputs: two BCD's and one carry-in  5 outputs: one BCD and one carry-out  Design approaches  use uinary full Adders  A truth table with 2^9 entries  the sum <= 9 + 9 + 1 = 19  binary to BCD 

25. BCD Adder: The truth Table 

26. Modifications are needed if the sum > 9  C = 1  K = 1  Z Z = 1  8 4 Z Z = 1  8 2 mo mo d x ification: ification: - - (10) (10) or +6 or +6   d d C = K +Z Z + Z Z 8 4 8 2

27. Block diagram 

28. Binary Multiplier Partial products  – AND operations fig. 4.15 Two-bit by two-bit binary multiplier.

29. 4-bit by 3-bit binary multiplier  Fig. 4.16 Four-bit by three-bit binary multiplier. Digital Circuits

30. 4-9 Decoder A n-to-m decoder  n a binary code of n bits = 2 distinct information n n input variables; up to 2 output lines only one output can be active (high) at any time

31. An implementation  Fig. 4.18 Three-to-eight-line decoder. 38 Digital Circuits

32. Combinational logic implementation  each output = a minterm  use a decoder and an external OR gate to  implement any Boolean function of n input variables

33. Demultiplexers  a decoder with an enable input  receive information in a single line and transmits  n it in one of 2 possible output lines Fig. 4.19 Two-to-four-line decoder with enable input

34. Decoder Examples • 3-to-8-Line Decoder: example: Binary-to-octal conversion. D0 = m0 = A2’A1’A0’ D1= m1 = A2’A1’A0 …etc

35. Expansion  two 3-to-8 decoder: a 4-to-16 deocder  Fig. 4.20 4 16 decoder constructed with two 3 x 8 decoders a 5-to-32 decoder? 

36. A0 A1 A2 3-8-line Decoder D0 – D7 E 3-8-line Decoder D8 – D15 A3 A4 E 2-4-line Decoder 3-8-line Decoder D16 – D23 E 3-8-line Decoder D24 – D31 E Decoder Expansion - Example 2 • Construct a 5-to-32-line decoder using four 3-8-line decoders with enable inputs and a 2-to-4-line decoder.

37. Combination Logic Implementation each output = a minterm  use a decoder and an external OR gate to  implement any Boolean function of n input variables A full-adder  S(x,y,z)=S(1,2,4,7)  C(x,y,z)= C(x,y,z)= S S (3,5,6,7) (3,5,6,7) Fig. 4.21 Implementation of a full adder with 1 decoder

38. two possible approaches using decoder  OR(minterms of F): k inputs  NOR(minterms of F'): 2 - k inputs n  In general, it is not a practical implementation 

39. 4-10 Encoders The inverse function of decoder  a decoder z = D + D + D + D 1 3 5 7 The encoder can be implemented y = D + D + D + D 2 3 6 7 with three OR gates. x = D + D + D + D 4 5 6 7

40. An implementation  limitations  illegal input: e.g. D =D x1  3 6 The output = 111 (¹3 and ¹6) 

41. Priority Encoder resolve the ambiguity of illegal inputs  only one of the input is encoded  D has the highest priority  3 has the lowest priority D  0 X: don't-care conditions  V: valid output indicator 

42. ■ The maps for simplifying outputs x and y fig. 4.22 Maps for a priority encoder

43. ■ Implementation of priority x = D + D Fig. 4.23 2 3 Four-input priority encoder y= ¢ D + D D 3 1 2 V = D + D + D + D 0 1 2 3

44. 4-11 Multiplexers select binary information from one of many input  lines and direct it to a single output line 2 input lines, n selection lines and one output line n  e.g.: 2-to-1-line multiplexer  Fig. 4.24 Two-to-one-line multiplexer

45. 4-to-1-line multiplexer  Fig. 4.25 Four-to-one-line multiplexer

46. Note n n-to- 2 decoder  add the 2 input lines to each AND gate n  OR(all AND gates)  an enable input (an option) 

47. Fig. 4.26 Quadruple two-to-one-line multiplexer

48. Boolean function implementation MUX: a decoders an OR gate  n 2 -to-1 MUX can implement any Boolean function  of n input variable a better solution: implement any Boolean function of n+1 input variable n of these variables: the selection lines  the remaining variable: the inputs

49.  an example: F(A,B,C) = S(1,2,6,7) Fig. 4.27 Implementing a Boolean function with a multiplexer

50. procedure:  assign an ordering sequence of the input variable  the rightmost variable (D) will be used for the input  lines assign the remaining n-1 variables to the selection  Lines with construct the truth table lines w.r.t. their corresponding sequ  consider a pair of consecutive minterms starting  from m 0 determine the input lines 