1. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 1 Fundamentals of Logic Design Counters and Similar Sequential Networks Objectives: To design counter using basic flip-flops To describe other simple sequential networks To derive input equations for a design.

2. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 2 Counters and Similar Sequential Networks Introduction Counter Code Converter Shift Register Simple Sequential Network Application

3. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 3 Counters and Similar Sequential Networks Binary Counter Counter Decoder Basic Counting: Counting Up Counting Down

4. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 4 Counters and Similar Sequential Networks Binary Counter Synchronous Counters Ripple Counters Two types:

5. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 5 Counters and Similar Sequential Networks Binary Counter Synchronous Counter(Synchronous) FF Pulse Flip-flop operations is synchronized by pulse i.e. Flip-flops change states simultaneously.

6. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 6 Counters and Similar Sequential Networks Binary Counter Ripple Counter (Asynchronous) FF Pulse Flip-flop state change of one flip-flop triggers the next flip-flop in line

7. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 7 Counters and Similar Sequential Networks Synchronous Binary Counter Design: 3-bits counter Requirements:- Synchronous Count Up Use T flip-flop/ J-K flip-flop Reset to 0

8. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 8 Counters and Similar Sequential Networks Synchronous Binary Counter 3-bits counter 0123456701234567 000 001 010 011 100 101 110 111 Reset 000 001 010 011 100 101 110 111 Reset State Graph

9. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 9 Counters and Similar Sequential Networks Synchronous Binary Counter 3-bits counter (cont’d) ABC 000 001 010 011 100 101 110 111 Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 State Table

10. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 10 Counters and Similar Sequential Networks Synchronous Binary Counter 3-bits counter: Using T Flip-flop T Q Q + 00110011 01010101 01100110 Q Q’ T CK

11. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 11 Counters and Similar Sequential Networks Synchronous Binary Counter Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 T-FF Inputs T A T B T C 0 0 1 0 1 1 0 0 1 1 1 1 0 0 1 0 1 1 0 0 1 1 1 1 A B C 0 0 0 A + B + C + 0 0 1 T B =0 T A =0 T C =1 3-bits counter: Using T Flip-flop (cont’d)

12. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 12 Counters and Similar Sequential Networks Synchronous Binary Counter Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 T-FF Inputs T A T B T C 0 0 1 0 1 1 0 0 1 1 1 1 0 0 1 0 1 1 0 0 1 1 1 1 A B C 0 0 1 A + B + C + 0 1 0 T B =1 T A =0 T C =1 3-bits counter: Using T Flip-flop (cont’d)

13. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 13 Counters and Similar Sequential Networks Synchronous Binary Counter Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 T-FF Inputs T A T B T C 0 0 1 0 1 1 0 0 1 1 1 1 0 0 1 0 1 1 0 0 1 1 1 1 A B C 0 1 0 A + B + C + 0 1 1 T B =0 T A =0 T C =1 3-bits counter: Using T Flip-flop (cont’d)

14. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 14 Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 T-FF Inputs T A T B T C 0 0 1 0 1 1 0 0 1 1 1 1 0 0 1 0 1 1 0 0 1 1 1 1 Counters and Similar Sequential Networks Synchronous Binary Counter 00 01 11 10 0 1 BC A TATA 00 01 11 10 0 1 BC A TBTB 1 1 1 1 1 1 T A = BCT B = C and T C = 1 3-bits counter: Using T Flip-flop (cont’d)

15. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 15 Counters and Similar Sequential Networks Synchronous Binary Counter TATA QAQA Q’ A TCTC QCQC Q’ C TBTB QBQB Q’ B Pulse 3-bits counter: Using T Flip-flop (cont’d) OR

16. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 16 Counters and Similar Sequential Networks Synchronous Binary Counter TATA QAQA Q’ A TCTC QCQC Q’ C TBTB QBQB Q’ B Pulse 3-bits counter: Using T Flip-flop (cont’d) 1

17. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 17 Q Q’ J K CK Counters and Similar Sequential Networks Synchronous Binary Counter 3-bits counter: Using J-K Flip-flop J K Q Q + 01XX01XX XX10XX10 00110011 01010101

18. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 18 3-bits counter: Using J-K Flip-flop (cont’d) Counters and Similar Sequential Networks Synchronous Binary Counter Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 J-K flip-flop Inputs J A K A J B K B J C K C 0 X 0 X 1 X 0 X 1 X X 1 0 X X 0 1 X 1 X X 1 X 1 X 0 0 X 1 X X 0 1 X X 1 X 0 X 0 1 X X 1 X 1 X 1 A B C 0 0 0 A + B + C + 0 0 1 J A =0 K A =X J B =0 K B =X J C =1 K C =X

19. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 19 3-bits counter: Using J-K Flip-flop (cont’d) Counters and Similar Sequential Networks Synchronous Binary Counter Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 J-K flip-flop Inputs J A K A J B K B J C K C 0 X 0 X 1 X 0 X 1 X X 1 0 X X 0 1 X 1 X X 1 X 1 X 0 0 X 1 X X 0 1 X X 1 X 0 X 0 1 X X 1 X 1 X 1 A B C 0 0 0 A + B + C + 0 0 1 J A =0 K A =X J B =1 K B =X J C =X K C =1

20. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 20 3-bits counter: Using J-K Flip-flop (cont’d) Counters and Similar Sequential Networks Synchronous Binary Counter Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 J-K flip-flop Inputs J A K A J B K B J C K C 0 X 0 X 1 X 0 X 1 X X 1 0 X X 0 1 X 1 X X 1 X 1 X 0 0 X 1 X X 0 1 X X 1 X 0 X 0 1 X X 1 X 1 X 1 A B C 0 0 0 A + B + C + 0 0 1 J A =0 K A =X J B =X K B =0 J C =1 K C =X

21. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 21 3-bits counter: Using J-K Flip-flop (cont’d) Counters and Similar Sequential Networks Synchronous Binary Counter 00 01 11 10 0 1 BC A JAJA 00 01 11 10 0 1 BC A KAKA 1 X X X X X X X X 1 J A = BCK A = BC

22. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 22 3-bits counter: Using J-K Flip-flop (cont’d) Counters and Similar Sequential Networks Synchronous Binary Counter 00 01 11 10 0 1 BC A JBJB 00 01 11 10 0 1 BC A KBKB X 1 X X 1 J B = CK B = C X11 X X X X

23. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 23 3-bits counter: Using J-K Flip-flop (cont’d) Counters and Similar Sequential Networks Synchronous Binary Counter 00 01 11 10 0 1 BC A JCJC 00 01 11 10 0 1 BC A KCKC 1 XX XX J C = 1K C = 1 1 1 1 X 11 11 X X X

24. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 24 Pulse 3-bits counter: Using J-K Flip-flop (cont’d) Counters and Similar Sequential Networks Synchronous Binary Counter KAKA QAQA Q’ A JAJA KBKB QBQB Q’ B JBJB KCKC QCQC Q’ C JCJC

25. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 25 Pulse 1 3-bits counter: Using J-K Flip-flop (cont’d) Counters and Similar Sequential Networks Synchronous Binary Counter KAKA QAQA Q’ A JAJA KBKB QBQB Q’ B JBJB KCKC QCQC Q’ C JCJC

26. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 26 Counters and Similar Sequential Networks Next State Maps Introducing Next State Map Deriving input equation for J-K flip-flop using short cut method

27. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 27 Introducing Next State Maps Counters and Similar Sequential Networks Next State Maps vs Input Equation Maps Present-Next State Mapping Derive from State Table An intermediate step Flip-Flop specific Can be derived from State Table Next State Maps

28. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 28 Introducing Next State Maps Counters and Similar Sequential Networks State Table Next State Maps Input Equation Maps Circuit Realization Next State Maps

29. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 29 Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 J-K flip-flop Inputs J A K A J B K B J C K C 0 X 0 X 1 X 0 X 1 X X 1 0 X X 0 1 X 1 X X 1 X 1 X 0 0 X 1 X X 0 1 X X 1 X 0 X 0 1 X X 1 X 1 X 1 Counters and Similar Sequential Networks Use to derive FF input Equation via K-Maps Next State Maps Introducing Next State Maps

30. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 30 Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 J-K flip-flop Inputs J A K A J B K B J C K C 0 X 0 X 1 X 0 X 1 X X 1 0 X X 0 1 X 1 X X 1 X 1 X 0 0 X 1 X X 0 1 X X 1 X 0 X 0 1 X X 1 X 1 X 1 Counters and Similar Sequential Networks Use to derive Next State Maps Introducing Next State Maps Next State Maps

31. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 31 Using J-K Flip-flop 3-bits Prime Number Counter with Next State Maps Counters and Similar Sequential Networks Next State Maps Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 1 - - - 1 0 1 - - - 1 1 1 - - - 0 0 0 001 011 101 111

32. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 32 00 01 11 10 0 1 BC A A+A+ 1 X X X 00 01 11 10 0 1 BC A B+B+ 11 X X X 00 01 11 10 0 1 BC A C+C+ 1 11 1 X X X Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 1 - - - 1 0 1 - - - 1 1 1 - - - 0 0 0 Using J-K Flip-flop 3-bits Prime Number Counter with Next State Maps Counters and Similar Sequential Networks Next State Maps 1

33. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 33 Using J-K Flip-flop 3-bits Prime Number Counter with Next State Maps Counters and Similar Sequential Networks Next State Maps 00 01 11 10 0 1 BC A JAJA 1 X X X X J K Q Q + 01XX01XX XX10XX10 00110011 01010101 00 01 11 10 0 1 BC A A+A+ 1 X X X 1 00 01 11 10 0 1 BC A KAKA X X X X X J A = B X X 1 K A = B

34. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 34 Using J-K Flip-flop 3-bits Prime Number Counter with Next State Maps Counters and Similar Sequential Networks Next State Maps J K Q Q + 01XX01XX XX10XX10 00110011 01010101 00 01 11 10 0 1 BC A JBJB X X X 1 00 01 11 10 0 1 BC A KBKB 1 X X X X J B = C X X 1 K B = 1 00 01 11 10 0 1 BC A B+B+ 11 X X X 1 X X

35. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 35 Using J-K Flip-flop 3-bits Prime Number Counter with Next State Maps Counters and Similar Sequential Networks Next State Maps 00 01 11 10 0 1 BC A JCJC X X X X X J K Q Q + 01XX01XX XX10XX10 00110011 01010101 00 01 11 10 0 1 BC A KCKC X X X X J C = 1 X 1 K C = AB 00 01 11 10 0 1 BC A C+C+ 1 11 1 X X X 1 X

36. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 36 Using J-K Flip-flop 3-bits Prime Number Counter with Next State Maps Counters and Similar Sequential Networks Next State Maps Pulse 1 KAKA QAQA Q’ A JAJA KBKB QBQB Q’ B JBJB KCKC QCQC Q’ C JCJC

37. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 37 Counters and Similar Sequential Networks Next State Maps SHORTCUT to derive input equations of J-K flip-flop from Next State Maps. 00 01 11 10 0 1 BC A B+B+ 11 X X X 00 01 11 10 0 1 BC A C+C+ 1 11 1 X X X 00 01 11 10 0 1 BC A A+A+ 1 X X X 1 Next State Maps

38. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 38 00 01 11 10 0 1 BC A A+A+ 1 X X X 1 Counters and Similar Sequential Networks Next State Maps SHORTCUT to derive input equations of J-K flip-flop from Next State Maps. J A = BK A = B 00 01 11 10 0 1 BC A B+B+ 11 X X X J B = C K B = 1 00 01 11 10 0 1 BC A C+C+ 1 11 1 X X X K B = AB J C = 1

39. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 39 Counters and Similar Sequential Networks Initial State: Binary Counter Design 000 001 011 101 111 Only 5 states used 3-bits binary would have 8 state! How does missing states relates to initial state specification?

40. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 40 Counters and Similar Sequential Networks Initial State: Binary Counter Design 000 001 011 101 111 Missing states: 010 100 110 What if the Binary counter designed has initial/present states of those three? Not a Self-Starting Design Assume that initial state will always be 000.

41. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 41 Counters and Similar Sequential Networks Initial State: Binary Counter Design 000 001 011 101 111 A more complete and robust design incorporates all possibilities, i.e. for 3-bits binary: all 8 combinations need to be taken into account.

42. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 42 Counters and Similar Sequential Networks Initial State: Binary Counter Design 000 001 011 101 111 Better solution 010 Note: It is not always the case that initial condition of a design has to be 000.

43. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 43 Counters and Similar Sequential Networks Initial State: Binary Counter Design 000 001 011 101 111 Better solution (3-bits Prime Number Counter) 010 Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 0 0 0

44. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 44 Counters and Similar Sequential Networks Initial State: Binary Counter Design Present State Next State A B C 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A + B + C + 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 0 0 0 00 01 11 10 0 1 BC A B+B+ 11 00 01 11 10 0 1 BC A C+C+ 1 11 1 00 01 11 10 0 1 BC A A+A+ 1 1 Next State Maps Better solution (3-bits Prime Number Counter)

45. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 45 Counters and Similar Sequential Networks Initial State: Binary Counter Design 00 01 11 10 0 1 BC A B+B+ 11 00 01 11 10 0 1 BC A C+C+ 1 11 1 00 01 11 10 0 1 BC A A+A+ 1 1 J A = BCK A = B+C’ J B = C K B = 1 K C = AB J C = A’B’ Better solution (3-bits Prime Number Counter)

46. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 46 Pulse 1 KAKA QAQA Q’ A JAJA KBKB QBQB Q’ B JBJB KCKC QCQC Q’ C JCJC Counters and Similar Sequential Networks Initial State: Binary Counter Design Better solution (3-bits Prime Number Counter) B C B C’ C A’ B’ A B

47. Digital 1 1010101010101010101010101010101010101010101010101010101010101 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010 Hisham 47 Counters and Similar Sequential Networks Summary Similar techniques used in counters can be applied for other type of sequential network design which has single pulse input to initiate state changes with other no other external inputs. Approach for simple sequential network design: 1. Derive State (Transition) Table 2. Derive Next State Maps 3. Derive Input Equations from K-Maps 4. Realize the circuits.