Clocked Synchronous State Machine Design

•Design a clocked synchronous state machine with two inputs, A and B and a single output Z that is equal to 1 if:

–A has the same value at each of the two previous clock ticks, or–B has been 1 since the last time that the first condition was true.

•Otherwise Z should be 0.

Clocked Synchronous State Machine Design

• A has the same value at each of the two previous clock ticks, or

• B has been 1 since the last time that the first condition was true.

Output00 01 11 10 Z

Initial State INIT 0

S*

MeaningInput - AB

S

Output00 01 11 10 Z

Initial State INIT A0 A0 A1 A1 0Got a 0 on A A0 0Got a 1 on A A1 0

S*

MeaningInput - AB

S

Clocked Synchronous State Machine Design

• A has the same value at each of the two previous clock ticks, or

• B has been 1 since the last time that the first condition was true.

• Try:– (A,B) = (0,0),(0,0),(0,1),(0,0)– (A,B) = (0,0),(0,0),(0,1),(1,0)

Output00 01 11 10 Z

Initial State INIT A0 A0 A1 A1 0Got a 0 on A A0 OK OK A1 A1 0Got a 1 on A A1 A0 A0 OK OK 0

Got two equal A OK

S*

MeaningInput - AB

S

Output00 01 11 10 Z

Initial State INIT A0 A0 A1 A1 0Got a 0 on A A0 OK OK A1 A1 0Got a 1 on A A1 A0 A0 OK OK 0

Got two equal A OK ? OK OK ? 1

S*

MeaningInput - AB

S

Clocked Synchronous State Machine Design

• A has the same value at each of the two previous clock ticks, or

• B has been 1 since the last time that the first condition was true.

• Try the sequence (A,B) = (1,0), (1,0), (1,1), (0,0).

Output00 01 11 10 Z

Initial State INIT A0 A0 A1 A1 0Got a 0 on A A0 OK OK A1 A1 0Got a 1 on A A1 A0 A0 OK OK 0

Two equal, A=0 last OK0 OK0 OK0 OK1 A1 1Two equal, A=1 last OK1 A0 OK0 OK1 OK1 1

S*

MeaningInput - AB

S

State Minimization

• Equivalence:– For a given input, two states are

equal if their outputs are the same, and their next state is the same or an equivalent one.

Output0 1 Z

A A B 0B C D 0C A D 0D E F 1E A F 1F G F 1G A F 1

S*

Input - XS

State Assignment• Procedures concerned with

methods for assigning binary values to states in such a way as to reduce the cost of the combinational circuit that drives the flip-flops.

• Methods:– Choose an initial coded state

which the machine can easily be forced to at reset.

– Minimize the number of state variables that change on each transition.

– Maximize the number of state variables that do not change

State Assignment• Methods:

– Exploit symmetries by assigning state variables differing only in one bit to both states or group of states.

– If there are unused states, then choose the best of the available state-variable combinations to achieve the foregoing goal.

– Decompose the set of state variables into individual bits or fields, with well defined meaning.

– Consider using more than the minimum number of state variables to make decomposed assignments possible.

State Assignment

• Unused states:– Minimal risk – assumes that it is

possible for the state machine to get into one unused or illegal state.

– Minimal cost – assumes that the state machine will never enter an unused state.

Designing State Machines Using State Diagrams

• State-diagram design is simpler but it is more prone to errors.– State table is an exhaustive listing

of the next states for each state/input combination. No ambiguity is possible.

– When constructing a state diagram there is no guarantee that the transition expressions written on the arcs leaving a particular state, cover all input combination exactly once.

Designing State Machines Using State Diagrams

• Design a state machine to control the tail lights of a 1965 Ford Thunderbird. The tail lights are composed of three light on each side which operate for the turns in the manner shown in the picture below.

Designing State Machines Using State Diagrams

• The state machine has:– Three inputs: left, and right turns,

and hazard.

– Six outputs: LA, LB, LC, RA, RB, and RC.

– Free running clock with frequency equal to the flashing rate.

Designing State Machines Using State Diagrams

• LA = L1+L2+L3+LR3• LB = L2+L3+LR3• LC = L3+LR3• RA = R1+R2+R3+LR3• RB = R2+R3+LR3• RC = R3+LR3

Designing State Machines Using State Diagrams

• Mutual exclusion: For each state the logical product of each pair of transition expression on arcs leaving that state is zero.

• All inclusion: For each state the logical sum of the transition expressions on all arcs leaving the state is one.

Designing State Machines Using State Diagrams

Designing State Machines Using State Diagrams

• IDLE state of 000.

• Q1 and Q0 are used to count in gray code sequence (IDLE→L1→L2→L3→IDLE) (IDLE→R1→R2→R3→IDLE).

• Q2 identifies LEFT or RIGHT turn.

• HAZ state of 100.

Designing State Machines Using State Diagrams

Feedback Sequential Circuits

• Fundamental mode circuits:– Most common example of

feedback sequential circuits.– Inputs are not normally allowed

to change simultaneously.

• Feedback sequential circuits may be Mealy or Moore circuits.

• A circuit with n feedback loops has n binary state variables and 2n states.

Feedback Sequential Circuits

• Analysis:– Assumes that input changes occur one

at a time to allow enough time between successive changes for the circuit to settle into a internal stable state.

– Break the feedback loops so that the next value stored in each loop can be predicted as a function of the circuit inputs and the current value stored in all loops.

Feedback Sequential Circuits

• Y* = (C D ) + (C D’ + Y’)’• Y* = C D + C’ Y + D Y

• Q = Y* = C D + C’ Y + D Y• QN = C D’ + Y’

Feedback Sequential Circuits

• Total state: combination of internal state (value of feedback loop) and input state (current input value) .

• Stable total state: Total state whose next state predicted by the state table is the same as the current internal state.

• Unstable total state: Total state whose next state predicted by the state table is the same as the current internal state.

• Note that Q and QN are outputs, not state variables.

Feedback Sequential Circuits

• Simultaneous input changes don’t always cause unpredictable behavior.

Races• A race is said to occur when multiple

internal variables change state as a result of a single input changing state.

• Noncritical race: the final state does not depend on the order in which the state variables change.

• Starting at state 011/00 change CLK to 1.

Races• Critical race: the final state depends on the

order in which the state variables change.

• State 010/10 has been changed from 000 to 110.

• Try starting at state 011/00 change CLK to 1 and see what state you end up at.

State Tables and Flow Tables

• Once it has been determined that a transition table does not have any critical races, the state-variable combinations can be named and outputs can be determined to obtain a state/output table.

State Tables and Flow Tables

• Flow table eliminates:– Rows for unused internal states (states

that are stable for no input combination).– Next state entries for total states that

cannot be reached from a stable total state as the result of a single input change.

• It eliminates multiple hops and show only the ultimate destination of each transition.

State Tables and Flow Tables

• Assume internal state S0/10.• Change D to 1, then 0.• Change clock to 0.• Change D to 1, then 0.• What happens when clock changes

to 1.

SSI Latches andFlip-Flops

• PLDs and FPGAs have to a large extent eliminated their use.

Debouncer• Bouncing – Behavior of mechanical

of switches which causes their contacts to close, and open several times before finally reaching a resting or stable closed state.

• Typically switches bounce for 10 – 20 ms, which is a very long time compared to the switching speeds of logic gates.

Debouncer• Debouncing – providing a single

signal change or pulse for each switch transition.

Registers• Collection of two or more D flip-

flops with a common clock input.• Often used to store a collection of

related bits, such as a byte of data.• May also be used to store unrelated

bits of data.

Registers

Registers

PLD

PLD

PLD

PLD

PLD