State Graphs FSMs. Page 1

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1 State Graphs FSMs Page 1

2 Binary Counter State Graph 00 Q1 Q0 N1 N State graphs are graphical representations of TT s They contain the same information: no more, no less 10 State Transition Page 2

3 Design Procedure Using State Graphs 1. Draw the state graph 2. Create an equivalent transition table 3. If transition table contains input don t cares, - unfold it to a full transition table 4. Complete the design using KMaps, gates, FF s Page 3

4 State Graphs With Moore Outputs 00 Z Q1 Q0 N1 N0 Z Z Write the output next to the states it is asserted in Underline it to make it more clear Page 4

5 Another SG With A Moore Output Q1 Q0 N1 N0 Z Z Z 01 Moore output shows up in multiple TT rows Page 5

6 State Graphs and Mealy Outputs Q1Q0N1N0 Y / Y Mealy output is associated with an arc in SG Page 6

7 Properly Formed State Graphs A properly formed state graph is both: Complete and Conflict-free Page 7

8 An Incomplete State Graph This SG is incomplete. Can you see why? What happens in state 10 when =0? Page 8

9 An Incomplete State Graph This SG is incomplete. Can you see why? Q1Q0N1N There is a missing row in the TT as well Page 9

10 Complete State Graphs For a SG to be complete: Paths leaving each state must cover all cases To check: OR together conditions on all arcs leaving a given state If result = 1, state is complete If result 1, state is incomplete Page 10

11 Checking for Completeness + = 1 State 11 is OK + = 1 State 00 is OK State 10 is not OK 10 + = 1 State 01 is OK Page 11

12 Alternate Check for Completeness Full transition table (no input don t cares) should have 2 n rows where: n = (#inputs + #state variables) This only has 7 rows, something is missing Q1Q0N1N Page 12

13 Additional Completeness Considerations If some input combination will never occur Don t need to enforce completeness CLR CLR 01 What about the case of CLR? If =CLR= 1 will never occur, this incomplete state is OK Page 13

14 Conflicts in State Graphs This SG has a conflict, can you find it? CLR CLR 00 CLR CLR CLR 11 CLR CLR 01 CLR CLR 10 CLR CLR Page 14

15 Conflicts in State Graphs This SG has a conflict, can you find it? CLR CLR 00 CLR CLR CLR 11 CLR CLR 01 CLR 10 CLR CLR CLR What happens in state 10 when CLR== 1? Page 15

16 Conflicts in State Graphs This SG has a conflict, can you find it? CLR CLR 00 CLR CLR CLR 11 CLR CLR 01 CLR 10 CLR CLR One arc says to go to state 00, another says to go to state 11 CLR This is a conflict Page 16

17 Conflicts in State Graphs The corresponding transition table has a problem as well CLR Q1 Q0 N1 N CLR CLR 11 CLR CLR CLR CLR CLR CLR CLR 01 CLR CLR Page 17

18 CLR Q1 Q0 N1 N Conflicts in State Graphs The corresponding transition table has a problem as well CLR CLR 11 Do you want 2 TT rows active at the same time? CLR CLR CLR CLR CLR CLR CLR 01 CLR CLR Page 18

19 Conflict-Free State Graphs For a State Graph to not have conflicts: Paths leaving each state must not conflict with one another To check: For each pair of arcs leaving a given state, AND together their conditions If result = 0, arcs have no conflict If result 0, arcs have a conflict Page 19

20 Checking for Conflicts CLR CLR (CLR ) = 0 These arcs OK CLR CLR 11 CLR 00 CLR CLR CLR 01 (CLR ) (CLR ) = 0 These arcs OK CLR CLR CLR 0 These arcs not OK 10 CLR CLR Remember you must do all pairs of arcs leaving each state Page 20

21 Alternate Check for Conflicts Full transition table (no input don t cares) should have 2n rows where: n = (#inputs + #state variables) This has 18 rows, something is wrong CLR Q1 Q0 N1 N Page 21

22 Additional Conflict Considerations If some input combination will never occur Don t need to worry about conflicts CLR CLR 01 If =CLR= 1 will never occur, the conflict shown here is OK. But, you will have to make a decision on how to write the TT Page 22

23 Summary - Properly Formed State Graphs A properly formed state graph is both Complete Conflict-free Perform tests to ensure you have covered all the cases once and only once Page 23

24 Q1Q0N1N Summary SG s vs. TT s A state graph is simply a graphical way of writing a Transition Table No additional information You should be adept at converting between them Design is always done from TT to KMaps to gates/ff s Page 24

25 Finite State Machines Page 25

26 State Machine Concepts State, current state, next state, state registers IFL, OFL, Moore outputs, Mealy outputs Transition tables With output don t cares (X s) With input don t cares (- s) State graphs And their correspondence to TT s Page 26

27 Counters as State Machines A counter is a state machine Where the state encodings are significant Q0 Q1 Q2 Q3 7 Segment Decoder Page 27

28 State Machines A state machine is a sequential circuit which progresses through a series of states in reponse to inputs The output values are usually significant The state encodings are usually not significant Unlike with counters Page 28

29 State Encodings In this machine, the state encodings don t matter The output values do Event 1 Output1 Event 1 Output2 Event 2 Output3 Event 2 Event 3 Page 29

30 Buttons A State Machine Controller for a Photocopier Lamps Timers Inputs Finite State Machine Outputs Motors Switches Display 1) FSM receives inputs from copier 2) FSM generates control outputs in response 3) States help it remember where it is in the copy process Page 30

31 A Sequence Detector FSM S0 Because the encodings don t matter, we will use symbolic state names S3 Z S1 S2 What does this machine do? Page 31

32 A Sequence Detector FSM It is a sequence detector. S0 It has 1 input: and one output: Z S3 Z S1 S2 It detects the sequence on the input. When detected, output Z is asserted. Page 32

33 A Sequence Detector FSM As long as =1, we stay in state S0 S S3 Z S1 S2 Page 33

34 A Sequence Detector FSM When =0, we go to state S1 S S3 Z S1 S2 Page 34

35 A Sequence Detector FSM As long as =0, we stay in state S1 S S3 Z S1 S2 Page 35

36 A Sequence Detector FSM When =1, we go to state S2 S S3 Z S1 S2 Page 36

37 A Sequence Detector FSM If =1 again, we go to state S3 S S3 Z S1 S2 SUCCESS! Raise the Z output Page 37

38 A Sequence Detector FSM Once we enter state S3, we never leave S0 S3 Z S1 S2 Page 38

39 A Sequence Detector FSM What if we don t see the second =1? S S3 Z S1 S2 Page 39

40 Implementing the Sequence Detector FSM 1. Create symbolic Transition Table 2. Assign state encoding 3. Create conventional Transition Table 4. Do standard implementation steps CS NS Z 0 S0 S1 0 1 S0 S0 0 0 S1 S1 0 1 S1 S2 0 0 S2 S1 0 1 S2 S3 0 - S3 S3 1 Symbolic TT S0 = 00 S1 = 01 S2 = 10 S3 = 11 State Assignment Q1 Q0 N1 N0 Z Conventional TT Page 40

41 Sequence Detector Implementation N1 = Q1 Q0 + Q1 + Q0 N0 = + Q1 Z = Q1 Q0 Q1 Q0 Q1 Q0 N1 D Q CLK Q1 Z Q1 N0 D Q Q0 CLK ECE 238L STATEGRAPHS/FSM Page 41

42 A Problem With the Sequence Detector This FSM only detects first occurrence of 011 on input Here is a timing diagram actual screenshot from a simulation Page 42

43 Improved Sequence Detector This starts over each time 011 is detected S0 Z S3 S1 S2 Page 43

44 Improved Detector Timing Diagram Z=1 any time state is S3 Looking at, does it look like Z is a cycle late? Page 44

45 Mealy Version of Sequence Detector S0 Output is asserted during the second 1 of the 011 sequence S3 S1 / Y S2 Page 45

46 Mealy Version Timing Diagram Y Note how Mealy output follows input during state S2 Output is a cycle earlier than in Moore machine it appears in S2 rather than in S3 Page 46

47 Simplified Mealy Sequence Detector A characteristic ofmealy machines is they often require fewer states than Moore machines Here is simplified but equivalent FSM / Y S0 S1 S2 Page 47

48 Example FSM s Two Car Wash Controllers Page 48

49 Basic Car Wash FSM Operation 1. Wait for a token to be inserted 2. Reset timer 3. Turn on water pump until timer expires 4. Start over This assumes existence of: Token acceptance mechanism Timer Digitally-controlled water pump Page 49

50 Basic Car Wash FSM SG TOKEN Wait for token S_IDLE TOKEN TDONE S_TOKEN CLRT Clear timer SPRAY S_SPRAY Spray car while waiting for timer to expire TDONE Page 50

51 TOKEN TDONE CS NS CLRT SPRAY 0 - S_IDLE S_IDLE S_IDLE S_TOKEN S_TOKEN S_SPRAY S_SPRAY S_SPRAY S_SPRAY S_IDLE 0 1 TOKEN TDONE Q1 Q0 N1 N0 CLRT SPRAY TOKEN TDONE Q1 Q0 N1 N0 CLRT SPRAY X X X X X X X X X X X X X X X X Page 51

52 Basic Car Wash Implementation SPRAY Q0 N1 D Q Q1 Q1 TDONE CLK CLRT TOKEN Q1 Q0 N0 D Q Q0 CLK This is what you get from a KMap solution Page 52

53 TOKEN Simplified State Graph Clear timer while waiting for a token and eliminate a state S_IDLE CLRT TDONE SPRAY S_SPRAY TOKEN TOKEN TDONE CS NS CLRT SPRAY 0 - S_IDLE S_IDLE S_IDLE S_SPRAY S_SPRAY S_SPRAY S_SPRAY S_IDLE 0 1 TDONE Page 53

54 Simplified Car Wash Implementation State Encoding: S_IDLE = 0, S_SPRAY = 1 NS = CS TOKEN + CS TDONE CLRT = CS SPRAY = CS CS CS TOKEN CS TDONE NS D Q SPRAY CLRT CLK Page 54

55 A Fancy Car Wash Controller Two types of washes: Regular wash (spray only) = 1 token Deluxe wash (spray, soap, spray) = 2 tokens Customer: Inserts 1 token and pushes START for Regular Inserts 2 tokens for Deluxe Page 55

56 TOKEN S0 Straightforward process to write transition table and reduce to gates T1DONE TOKEN CLRT1 S1 TOKEN START START TOKEN START T1DONE S4 SPRAY SPRAY, CLRT2 S2 T1DONE T2DONE S3 T2DONE T1DONE SOAP, CLRT1 Creativity in FSM design is designing the initial state graph Page 56

57 Enhancements to Fancy Controller Page 57

58 T1DONE TOKEN S0 ACCEPTCOIN TOKEN There is no way to cause coinbox to refuse tokens. Coins inserted after wash commences are presumably kept by the machine. CLRT1, ACCEPTCOIN S1 TOKEN START START TOKEN START T1DONE S4 SPRAY SPRAY, CLRT2 S2 T1DONE T2DONE S3 T2DONE T1DONE SOAP, CLRT1 Let s add an ACCEPTCOIN output to control when coinbox will accept tokens Page 58

59 T1DONE TOKEN S0 ACCEPTCOIN TOKEN If customer pushes START at precisely the same time as he inserts another TOKEN, the FSM will take the token, but deliver a Regular Wash. ACCEPTCOIN CLRT1 S1 TOKEN START START TOKEN START T1DONE S4 SPRAY SPRAY, CLRT2 S2 T1DONE T2DONE S3 T1DONE SOAP, CLRT1 That is, START has precedence over TOKEN T2DONE Page 59

60 T1DONE TOKEN S0 ACCEPTCOIN TOKEN This changes the precedence so the machine won t steal the second token, but will give a deluxe wash in this case. ACCEPTCOIN CLRT1 S1 TOKEN START START TOKEN TOKEN T1DONE S4 SPRAY SPRAY, CLRT2 S2 T1DONE T2DONE S3 T1DONE SOAP, CLRT1 T2DONE Page 60

61 Completeness and Conflict Revisited State S1 is a typical problem spot Carefully analyze the state graph to ensure no conflicts exist and that all cases covered. Page 61

62 Resetting State Machines Ability to reset the FSM is essential for testing most systems Always include a reset capability Add CLR signal to state graph Use flip flops with clear inputs Either method will work Page 62

63 Another Example An Electronic Key Lock Page 63

64 Example: Electronic Key Lock 1) There are 10 keypads 0-9 2) The unlock sequence is ) When a pad is pushed the signal for that number is asserted 4) When any of the keypads are pushed a PUSHED signal is asserted 5) If you push a number out of sequence, you get an error indicator and you get to start over 6) After three wrong tries, you must wait ½ hr. before trying again 7) If you entered the correct sequence, the lock unlocks. 8) Once the lock has opened, nothing happens until it is manually locked again. Inputs: Keypads signals 0-9 PUSHED signal ECNT3 (error count = 3) WAITDONE LOCKED Outputs: CLRTIMER CLRCNTR ERROR UNLOCK Page 64

65 LOCKED LOCKED/ CLRCNTR D PUSHED PUSHED 9/ UNLOCK PUSHED PUSHED A Start PUSHED 7 B PUSHED 8 C WAITDONE/ CLRCNTR ECNT3 PUSHED 7 / PUSHED 8 / PUSHED 9 / F ECNT3/ CLRTIMER E ERROR WAITDONE Page 65

66 Let s create the transition table PUSHED A Start LOCKED LOCKED/ CLRCNTR PUSHED 7 D PUSHED B PUSHED 9/ UNLOCK PUSHED PUSHED 8 C WAITDONE WAITDONE/ CLRCNTR F PUSHED 7 / ECNT3 ECNT3/ CLRTIMER E PUSHED 8 / PUSHED 9 / ERROR PUSHED ECNT3 WAITDONE LOCKED STATE NEXTSTATE ERROR CLRCNTR CLRTIMER UNLOCK A A A E A B B B B E B C C C C E C D D D D A E A E F F F F A Page 66

67 Now let s expand State A PUSHED ECNT3 WAITDONE LOCKED STATE NEXTSTATE ERROR CLRCNTR CLRTIMER UNLOCK A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A Page 67

68 This is going to get ugly fast 7 inputs + 3 state bits = 1024 rows in TT Can you do a 10-variable KMap? There is a technique called One-Hot state machines we can use instead Page 68