CS/EE 3700 : Fundamentals of Digital System Design Chris J. Myers Lecture 9: Asynchronous Sequential...

CS/EE 3700 : Fundamentals of Digital System Design

Chris J. Myers

Lecture 9: Asynchronous Sequential Circuits

Chapter 9

Asynchronous Circuits

• Asynchronous sequential circuits do not use a clock or flip-flops for state variables.

• Changes in state occur in response to changes on the inputs.

• This chapter describes single input change (SIC) fundamental-mode async. circuits.

• Only one input allowed to change at a time.• Time between input changes must be sufficient for

the circuit to stabilize.

Figure 9.1 Analysis of the SR latch

R

S Q Y y

Present Next state

state SR = 00 01 10 11

y Y Y Y Y

0 0 0 1 0

1 1 0 1 0

(b) State-assigned table

(b) Circuit with modeled gate delay

Figure 9.2 FSM model for the SR latch

Present Next state Outputstate SR = 00 01 10 11 Q

A A A B A 0

B B A B A 1

(a) State table

(b) State diagram

1000

110100

10

A 0 B 1

1101

SR

Figure 9.3 Mealy representation of the SR latch

Present Nextstate Output, Q

state SR = 00 01 10 11 00 01 10 11

A A A B A 0 0 0

B B A B A 1 1

(a) State table

(b) State diagram

10/100/1

11/001/000/0

10/ –

A B

01 –11 –

SR/Q

–

–

–

Terminology

• state table = flow table

• state-assigned table = transition table or excitation table.

Figure 8.90 The general model for a sequential circuit

Combinational circuit

Y k

Y 1

y k

y 1

w 1

w n

z 1

z m

Outputs

Next-statevariables

Present-state variables

Inputs

Figure 9.4 The gated D latch

D

C

Q Y y

(a) Circuit

Present Nextstate

state CD = 00 01 10 11y Y Y Y Y Q

0 0 0 0 1 0

1 1 1 0 1 1

(b) Excitationtable

Figure 9.4 The gated D latch

Present Nextstate state CD = 00 01 10 11 Q

A A A A B 0

B B B A B 1

(c) Flowtable

(d) State diagram

x10x

x00x

11

A 0 B 1

10

CD

Figure 9.5 Circuit for the master-slave D flip-flop

D

Clk

Q

Q

D

C

Q y s y m

Master Slave

Q

D

Clk

Q

Q

Figure 9.6 Excitation and flow tables for Example 9.2

Present Next state

state CD = 00 01 10 11 Output

y m y s Y m Y s Q

00 0 0 0 0 0 0 10 0

01 00 00 0 1 11 1

10 11 11 00 1 0 0

11 1 1 1 1 01 1 1 1

(a) Excitationtable

Present Next state Outputstate CD = 00 01 10 11 Q

S1 S 1 S 1 S 1 S3 0

S2 S1 S1 S 2 S4 1

S3 S4 S4 S1 S 3 0

S4 S 4 S 4 S2 S 4 1

(b) Flow table

Figure 9.6 Excitation and flow tables for Example 9.2

Present Next state Outputstate CD = 00 01 10 11 Q

S1 S 1 S 1 S 1 S3 0

S2 S1 S1 S 2 S4 1

S3 S4 S4 S1 S 3 0

S4 S 4 S 4 S2 S 4 1

(b) Flow table

Present Next state Outputstate CD = 00 01 10 11 Q

S1 S 1 S 1 S 1 S3 0

S2 S1 S 2 S4 1

S3 S4 S1 S 3 0

S4 S 4 S 4 S2 S 4 1

(c) FlowTable with unspecifiedentries

–

–

Figure 9.7 State diagram for the master-slave D flip-flopC

x10x10

11

S2 1 S4 1 10

11x00x

11

S1 0 S3 0 10

0x0x

CD

Figure 9.8 Circuit for Example 9.3

z y 1

y 2

Y 1

Y 2

w 1

w 2

Figure 9.9 Excitation and flow tables

Present Next state

state w 2 w 1 = 00 01 10 11 Output

y 2 y 1 Y 2 Y 1 Y 2 Y 1 Y 2 Y 1 Y 2 Y 1 z

00 0 0 01 10 11 0

01 11 0 1 11 11 0

10 00 1 0 1 0 1 0 1

11 1 1 10 10 10 0

(a) Excitation table

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A B C D 0

B D B D D 0

C A C C C 1

D D C C C 0

(b) Flow table

Figure 9.10 Modified flow table for Example 9.3

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A B C 0

B D B D 0

C A C C C 1

D D C C C 0

––

Figure 9.11 State table for Example 9.3

01

x11x

10

w2w1

A/0

C/1

B/0

D/0 00

00

00 0011

x11x

01

Figure 9.12 Flow table for a simple vending machine

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A B C 0

B D B 0

C A C C 1

D D C C 0

w 2 dime w 1 nickel

–––––

Steps in the Analysis Process

• Cut feedback paths and insert delay element.– Input to delay element is next-state, and output is the

present-state.– Cut set may not be unique.

• Derive next-state and output expressions from the circuit.

• Derive the excitation table.• Derive a flow table.• Derive a state diagram, if desired.

Synthesis of Asynchronous Circuits

• Devise a state diagram.

• Derive the flow table.

• Minimize number of states.

• Perform race-free state assignment.

• Derive excitation table.

• Obtain next-state and output expressions.

• Construct a hazard-free circuit.

Figure 9.13 Parity-generating asynchronous FSM

(a) State diagram

Present Next state OutputState w = 0 w = 1 z

A A B 0

B C B 1

C C D 1

D A D 0

(b) Flow table

1

1

0

A/0

D/0

B/1

C/1 0

0

0

1

1

Figure 9.14 State assignment

Present Next state

state w = 0 w = 1 Output

y 2 y 1 Y 2 Y 1 z

00 0 0 01 0

01 10 0 1 1

10 1 0 11 1

11 00 1 1 0

(a) Poor state assignment

Present Next state

state w = 0 w = 1 Output

y 2 y 1 Y 2 Y 1 z

00 0 0 01 0

01 11 0 1 1

11 1 1 10 1

10 00 1 0 0

(b) Good state assignment

Figure 9.14 State assignment

Figure 9.15 Circuit that implements an FSM

y 1

y 2

w

z

Figure 9.16 Synchronous solution for Example 9.4

w

z D Q

Q

Figure 9.17 State diagram for a modulo-4 counter

1

1

0

A 0

H 0

B 1

G 3

0

0

1

1 C 1

F 3

D 2

E 2

1

0 1

0

0

1

1

0

0

Figure 9.18 Flow and excitation tables for a modulo-4 counter

Present Nextstate Outputstate w = 0 w = 1 z

A A B 0

B C B 1

C C D 1

D E D 2

E E F 2

F G F 3

G G H 3

H A H 0

(a) Flowtable

Figure 9.18 Flow and excitation tables for a modulo-4 counter

Present Next state

state w = 0 w = 1 Output Mod-8 output

y 3 y 2 y 1 Y 3 Y 2 Y 1 z 2 z 1 z 3 z 2 z 1

000 0 00 001 00 000

001 011 0 01 01 001

011 0 11 010 01 010

010 110 0 10 10 011

110 1 10 111 10 100

111 101 1 11 11 101

101 1 01 100 11 110

100 000 1 00 00 111

(b) Excitation table (c) Output for countingthe edges

Figure 9.19 Arbitration example

Arbiter

Request1

Grant1

Request2

Grant2

Device 1

Device 2

Sharedresource

(a) Arbitration structure

Request (r)

Grant (g)

(b) Handshake signaling

Figure 9.20 State diagram for the arbiter

01A 0 0 B 01

C 10

0011

00

00

10

01

10

0111

1011

r 2 r 1

Figure 9.21 Implementation of the arbiter

Present Nextstate Outputstate r 2 r 1 = 00 01 10 11 g

2 g 1

A A B C B 00

B A B C B 01

C A B C C 10

(a) Flowtable

Present Nextstate

state r 2 r 1 = 00 01 10 11 Output

y 2 y 1 Y 2 Y 1 g 2 g 1

A 00 0 0 01 10 01 00

B 01 00 0 1 10 0 1 01

C 10 00 01 1 0 1 0 10

D 11 01 10 dd

(b) Excitationtable

Figure 9.22 The arbiter circuit

y 2

g 1

g 2

r 1

r 2

THIS CIRCUIT IS WRONG!

Figure 9.23 An alternative for avoiding a critical race

Present Next state Outputstate r 2 r 1 = 00 01 10 11

g 2 g 1

A A B C B 00

B A B A B 01

C A A C C 10

(a) Modified flow table

Present Next state

state r 2 r 1 = 00 01 10 11Output

y 2 y 1 Y 2 Y 1 g 2 g 1

00 0 0 01 10 01 00

01 00 0 1 00 0 1 01

10 00 00 1 0 1 0 10

(b) Modified excitation table

Figure 9.24 Mealy model for the arbiter FSM

1x 1000 00

x1 0100 00

10 0 –

B C

01 0–

Present Next state Output g 2 g 1

state r 2 r 1 = 00 01 10 11 00 01 10 11

B B B C B 00 01 –0 01

C C B C C 00 0– 10 10

(a) Flow diagram

Figure 9.25 Mealy model implementation of the arbiter FSM

Present Next state Output

state r 2 r 1 = 00 01 10 11 00 01 10 11

y Y g 2 g 1

0 0 0 1 0 00 01 d 0 01

1 1 0 1 1 00 0 d 10 10

(b) Excitation table

State Reduction

• Usually start with primitive flow table (i.e., only a single stable state per row).

• Asynchronous FSMs likely have many unspecified (don’t care) entries.

• Two-step state reduction process:– Apply partitioning procedure in which don’t

care entries must match.– Merge rows exploiting don’t cares.

Figure 9.26 Derivation of an FSM for the simple vending machine

A 0

B 0 C 1

D 0

E 1 F 1

0 0

0

0 0

0

D

N

N

N

N

D

D

0

D

(a) Initial statediagram

Figure 9.26 Derivation of an FSM for the simple vending machine

Present Nextstate OutputState DN = 00 01 10 11 z

A A B C 0

B D B 0

C A C 1

D D E F 0

E A E 1

F A F 1

(b) Initial flowtable

–

–

–

–

–

–

–

–

–

–

Figure 9.27 First-step reduction of the vending machine FSM

Present Next state Outputstate DN = 00 01 10 11 z

A A B C 0

B D B 0

C A C 1

D D E C 0

E A E 1

–––––––

–

Figure 9.12 Flow table for a simple vending machine

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A B C 0

B D B 0

C A C C 1

D D C C 0

w 2 dime w 1 nickel

–––––

Merging Procedure

• May be many possibilities for row mergers.• Two states Si and Sj are compatible if there

are no state conflicts for any input.– both Si and Sj have same successor, or– both Si and Sj are stable, or– the successor of Si or Sj or both is unspecified.

• Si and Sj must also have same output when specified.

Figure 9.28 A primitive flow table

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A H B 0

B F B C 0

C H C 1

D A D E 1

E D G E 1

F F D 0

G F G 0

H H E 0

––

––

––

–

–

–

–

– –

Figure 9.29 Merger diagram (which preserves the Moore model)

E G

B

F

C

H

A D

Figure 9.30 Reduced Moore-type flow table

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A A B D 0

B B D B C 0

C A C 1

D A D B D 1

– –

Figure 9.31 Complete merger diagram

G

A

F

C

H

B

D

E

State Reduction Procedure

1. Use partitioning procedure to eliminate equivalent states in primitive flow table.

2. Construct merger diagram.

3. Choose subsets of equivalent states including each state in only one subset.

4. Derive reduced flow table.

5. Repeat 2 to 4 until no reduction.

Figure 9.32 Flow table for Example 9.8

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A F C 0

B A B H 1

C G C D 0

D F D 1

E G E D 1

F F K 0

G G B J 0

H L E H 1

J G J 0

K B E K 1

L A L K 1

–

–

–

–

–

–

–

–

–

–

–

––

–

Figure 9.33 Reduction obtained by using the partitioning procedure

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A F C 0

B A B H 1

C G C D 0

D F D 1

E G E D 1

F F H 0

G G B J 0

H B E H 1

J G J 0

–

–

–

–

–

–

–

–

–

––

–

Figure 9.34 Merger diagram

G F

A

J

C

E

D

H

B

Figure 9.35 Reduction obtained from the merger diagram

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A A C B 0

B A B D B 1

C G C D 0

D G A D D 1

G G B G 0

–

–

Figure 9.36 Merger diagram

C B D G A

Figure 9.37 Reduced flow table for Example 9.8

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A A C B 0

B A B D B 1

C C B C D 0

D C A D D 1

Figure 9.38 Flow table for Example 9.9

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A G E 0

B K B D 0

C F C H 1

D C E D 0

E A E D 1

F F C J 0

G K G D 1

H E H 1

J F J D 0

K K C B 0

––

––

––

–––

–

–

Figure 9.39 Reduction resulting from the partitioning procedure

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A G E 0

B F B D 0

C F C H 1

D C E D 0

E A E D 1

F F C B 0

G F G D 1

H E H 1

––

––

––

–––

Figure 9.40 Merger diagrams

D B C A

E F H G

(a) Preserving the Moore model

G H C A

B D F E

(b) Complete merger diagram

Figure 9.41 An FSM specified in the form of a Mealy model

Present Next state Output z

state w 2 w 1 = 00 01 10 11 00 01 10 11

A A G E 0

B F B D 0 0 0

C F C H 1 1

D C E D 0

E A E D 1

F F C B 0 0

G F G D 1

H E H 1 1

––

––

––

––– ––

––

––

––

–

––––

–

–––

– –

Figure 9.42 Reduced flow table for Example 9.9

Present Next state Output z

state w 2 w 1 = 00 01 10 11 00 01 10 11

A A B D A 0 1 1

B C B B D 0 1 0 0

C C C B A 0 1 0 1

D A C D D 1 0 – –

–

Figure 9.43 Flow table for Example 9.9

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A B C 0

B F B H 0

C F C H 0

D D G C 1

E A E H 0

F F E C 0

G D G H 0

H G C H 1

––

––

––

––

Figure 9.44 Reduction after the partitioning procedure

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z

A A B C 0

B A B H 0

C A C H 0

D D G C 1

G D G H 0

H G C H 1

––

––

––

Figure 9.45 Merger diagram

H

B

C

G

D

A

Figure 9.46 Reduced flow table for Example 9.10

Present Next state Output z

state w 2 w 1 = 00 01 10 11 00 01 10 11

A A A A D 0 0 0

D D D A D 1 0 1

––

State Assignment

• Impossible to ensure that a change of two or more state variables occur at the same time.

• To achieve reliable operation, should make state variables change one at a time.

• Hamming distance is number of bits different in two bit strings.

• Ideal state assignment has Hamming distance of 1 for all state transitions.

Figure 9.47 Transitions

(a) Corresponding to Figure 9.14a

(b) Corresponding to Figure 9.14 b

C 10= D 11=

A 00= B 01=

w 0= w 0=

w 1=

w 1=

D 10= C 11=

A 00= B 01=

w 0= w 0=

w 1=

w 1=

Present Nextstate

state w = 0 w = 1 Output

y 2 y 1 Y 2 Y 1 z

00 0 0 01 0

01 11 0 1 1

11 1 1 10 1

10 00 1 0 0

Present Nextstate

state w = 0 w = 1 Output

y 2 y 1 Y 2 Y 1 z

00 0 0 01 0

01 10 0 1 1

10 1 0 11 1

11 00 1 1 0

Figure 9.48 Transitions for the arbiter

Present Nextstate Outputstate r 2 r 1 = 00 01 10 11 g 2 g 1 A A B C B 00

B A B C B 01

C A B C C 10

Present Nextstate

state r 2 r 1 = 00 01 10 11 Output

y 2 y 1 Y 2 Y 1 g 2 g 1

A 00 0 0 01 10 01 00

B 01 00 0 1 10 0 1 01

C 10 00 01 1 0 1 0 10

D 11 01 10 dd

(a) Transitions in Figure 9.21 a

(b) Using the extra state D

C 10=

A 00= B 01=

0010

01

01

C 10= D 11=

A 00= B 01=

00 10

10

01

11

01

01

11

10

10

00

00

Figure 9.49 Modified flow table

Present Next state Outputstate r 2 r 1 = 00 01 10 11 g 2 g 1

A A B C B 00

B A B D B 01

C A D C C 10

D B C 10– –

Transition Diagram

• Transition diagram illustrates all transitions in a flow table.

• Good state assignment means no diagonals in the transition diagram.

• Must embed the transition diagram onto ak-dimensional cube.

• n state variables can be embedded onto an n-dimensional cube.

Figure 9.50 Relabeled flow table

Present Next state Outputstate r 2 r 1 = 00 01 10 11 g 2 g 1

A 1 2 4 3 00

B 1 2 4 3 01

C 1 2 4 5 10

Figure 9.51 Transition diagrams

(c) Selected transition diagram

(a) Transitions in Figure 9.50 (b) Complete transition diagram

C 10=

A 00= B 01=1 2 3

1 4 2 4

C 10=

A 00= B 01=1 2 34

1 4 2 2 4 1

C 10=

A 00= B 01=1 2 34

1 4 2

Deriving Transition Diagrams

• Derive relabeled flow table.– Transitions through unstable states that lead to

stable state are given the same number.

• Represent each row of flow table by vertex.• Join Vi and Vj by edge if they have same

number in any column.• For any column in which Vi and Vj have

same number, label edge with that number.

Figure 9.52 Flow tables

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z 2 z 1

A A B C A 00

B B B D C 01

C A C D C 10

D B D A 11

(a) Flow table

Present Next state Outputstate w 2 w 1 = 00 01 10 11 z 2 z 1

A 1 4 7 2 00

B 3 4 7 6 01

C 1 5 7 6 10

D 3 7 2 11

(b) Relabeled flow table

–

–

Figure 9.53 Transition diagrams

B

(a) First transition diagram (b) Second transition diagram

D 10=

A 00= 4 7 ,

2 7 , 6 7 ,

(c) Augmented transition diagram

C 11=

1 7 ,

3 7 ,

7 D 10=

A 00= C 01= 1 7 ,

2 7 , 6 7 ,

B 11=

4 7 ,

7

3 7 ,

D 10=

A 00= C 01= 1 7 ,

2 7 4 , , 6 7 ,

B 11= 3 7 4 , ,

B 01=

Figure 9.54 Realization of an FSM

Present Next state Output z 2 z 1

state w 2 w 1 = 00 01 10 11 00 01 10 11

A A D D A 00 00 11 00

B B B D C 01 01 11 01

C A C B C –0 10 1– 10

D B B D A –1 0– 11 00

(a) Modified flow table

Present Next state Output

state w 2 w 1 = 00 01 10 11 00 01 10 11y 2 y 1 Y 2 Y 1 z 2 z 1

A 00 0 0 10 10 0 0 00 00 11 00

B 11 1 1 1 1 10 01 01 01 11 01

C 01 00 0 1 11 0 1 10 10

D 10 11 11 1 0 00 11 00

(b) Excitation table

–0 1–

–1 0–

Figure 9.55 FSM for Example 9.14

Present Nextstate Outputstate w 2 w 1 = 00 01 10 11 z 2 z 1

A A A C B 00

B A B D B 01

C C B C D 10

D C A D D 11

(a) Flow table

Present Nextstate Outputstate w 2 w 1 = 00 01 10 11 z 2 z 1

A 1 2 6 4 00

B 1 3 7 4 01

C 5 3 6 8 10

D 5 2 7 8 11

(b) Relabeled flow table

Figure 9.56 Transition diagrams

(a) Transition diagram (b) Augmented transition diagram

A

D C 5 8 ,

2 3

(c) Embedded transition diagram

B

7 6

1 4 , A

D C 5 8 ,

2

3

B

7 6

1 4 ,

5 8 ,

3

7

G

E

D

A B

E

F C

G

7

5 8 ,

1 4 , 3

3 2

7

y 2 y 3

y 1

F

6

5 8 ,

Figure 9.57 Modified tables for Example 9.14

Present Nextstate Outputstate w 2 w 1 = 00 01 10 11 z 2 z 1

A A A C B 00

B A B E B 01

C C F C G 10

D G A D D 11

E D –1

F B 01

G C D 1–

(a) Modified flow table

–– –

– –

– –

–

Figure 9.57 Modified tables for Example 9.14

Present Next state

state w 2 w 1 = 00 01 10 11 Output

y 3 y 2 y 1 Y 3 Y 2 Y 1 z 2 z 1

A 000 0

00 0

00 100 001 00

B 001 000 0

01 011 0

01 01

C 100 1

00 101 1

00 110 10

D 010 110 000 0

10 0

10 11

E 011 010

F 101 001 01

G 110 100 010

(b) Excitation table

–– –

– –

– –

–

–1

1–

Figure 9.58 Embedded transition diagram if two nodes per row are used

B1

A1 A2

B2

D1C1

C2

1 4

5 8

2

5 8 1 4

7

y 2

y 3

y 1

D2

6

3

Figure 9.59 Modified flow and excitation tables for Example 9.15

Present Nextstate Outputstate w 2 w 1 = 00 01 10 11 z 2 z 1

A1 A

1 A

1 C1 B1 00

A2 A

2 A

2 A1 B2 00

B1 A1 B

1 B2 B

1 01

B2 A2 B

2 D2 B

2 01

C1 C

1 C2 C

1 D1 10

C2 C

2 B1 C

2 D2 11

D1 C1 A2 D

1 D

1 11

D2 C2 D1 D

2 D

2 11

(a) Modified flow table

Figure 9.59 Modified flow and excitation tables for Example 9.15

Present Next state

state w 2 w 1 = 00 01 10 11 Output

y 3 y 2 y 1 Y 3 Y 2 Y 1 z 2 z 1

A1 000 0

00 0

00 100 010 00

A2 001 0

01 0

01 000 011 00

B1 010 000 0

10 011 0

10 01

B2 011 001 0

11 111 0

11 01

C1 100 1

00 110 1

00 101 10

C2 110 1

10 010 1

10 111 10

D1 101 100 001 1

01 1

01 11

D2 111 110 101 1

11 1

11 11

(b) Excitation table

Figure 9.60 State assignment with one-hot encoding

State Present Next state Outputassignment State w 2 w 1 = 00 01 10 11 z 2 z 1

0001 A A A E F 00

0010 B F B G B 01

0100 C C H C I 10

1000 D I J D D 11

0101 E C –0

0011 F A B 0–

1010 G D –1

0110 H B 01

1100 I C D 1–

1001 J A 00

– –

– –

– –– –

– ––

–

–

–

– –

Hazards

• In asynchronous circuits, undesirable glitches must not occur.

• Glitches caused by structure of circuit and propagation delays are called hazards.

• Designer must eliminate all hazards from an asynchronous circuit.

Figure 9.61 Definition of hazards

1 1 0 0

1 0 0 1

(a) Static hazard

(b) Dynamic hazard

1

0

1

0

Figure 9.62 An example of a static hazard

x 1 x 2 x 3 00 01 11 10

1

0

1

(b) Karnaugh map

1

f

x 3

(a) Circuit with a hazard

1

1

x 1

x 2 p

q

x 3

x 1

x 2

f

(c) Hazard-free circuit

Remove Static 1-Hazards

• Hazard exists whenever 2 adjacent 1s in a K-map are not covered by a single product.

• To remove all static hazards, find a cover that includes each pair of adjacent 1s.

Figure 9.63 Two-level implementation of master-slave D flip-flop

00 01 11 10

1

1 1

00

01

11

10

(b) Karnaugh maps for Y m and Y s in Figure 9.6a

1 1

1

y m y s

CD

1

1

00 01 11 10

1

1 1

00

01

11

10

1 1 1

y m y s

CD

1 1

D

C

Y m

Y s

y m

y s

p

q

r

(a) Minimum-cost circuit

Q

Figure 9.63 Two-level implementation of master-slave D flip-flop

00 01 11 10

1

1 1

00

01

11

10

(b) Karnaugh maps for Y m and Y s in Figure 9.6a

1 1

1

y m y s

CD

1

1

00 01 11 10

1

1 1

00

01

11

10

1 1 1

y m y s

CD

1 1

D

C

Y m

Y s

y m

y s

(c) Hazard-free circuit

Q

Figure 9.64 Function for Example 9.17

x 1 x 2 x 3 x 4 00 01 11 10

1 1 1

00

01

11

10

1 1

1 1

d d

d

1

Figure 9.65 Static hazard in a POS circuit

x 1 x 2 x 3 00 01 11 10

1

0

1

(b) Karnaugh map

1

f

x 3

(a) Circuit with a hazard

0

0

x 2

x 1 p

q

0 0

1

1

x 3

x 2

x 1

f

(c) Hazard-free circuit

Figure 9.66 Circuit with a dynamic hazard

(a) Circuit

x 2

x 1

x 3

x 4

b

a

c d

f

x 2 x 3 x 4

x 1

b

a

c

d

f

One gate delay

(b) Timing diagram

Significance of Hazards

• A glitch in an asynchronous circuit can cause the circuit to enter an incorrect state and possibly become stable in that state.

• Next-state logic must be hazard-free.

• Synchronous circuits can have hazards as long as they are stable by the setup time of the flip-flops.

Vending Machine Controller

• It accepts nickels and dimes.

• A total of 15 cents to release candy.

• No change given if 20 cents is deposited.

• Coins deposited one at a time.

Figure 9.67 Initial state diagram for the vending-machine controller

A 0

B 0

0

D

N

N

N

D D

C 0

J 0

K 1 L 1 N

N

D

D 0

E 0 F 1

G 0

H 1 I 1

0

N

N

N

0

0

0 0 0 D

D

0

D

D

0

0

0

0

0

Figure 9.68 Initial flow table for the vending-machine controller

Present Next state Outputstate DN = 00 01 10 11 z

A A B C 0

B D B 0

C J C 0

D D E F 0

E G E 0

F A F 1

G G H I 0

H A H 1

I A I 1

J J K L 0

K A K 1

L A L 1

–

–

–

–

–

–

–

–

–

–

–

–

–

–

–

–

–

–

–

–

Figure 9.69 First step in state minimization

Present Next state Outputstate DN = 00 01 10 11 z

A A B C 0

B D B 0

C G C 0

D D E F 0

E G E 0

F A F 1

G G H F 0

H A H 1

–

–

–

–

–

–

–

–

–

–

–

–

–

Figure 9.70 Merger diagram

B

H

A

F

D

G

C

E

Figure 9.71 Reduced flow tables

Present Next state Outputstate DN = 00 01 10 11 z

A A B C 0

B D B 0

C G C C 0

D D C F 0

F A F F 1

G G F F 0

(a) Minimized flow table

Present Next state Outputstate DN = 00 01 10 11 z

A 1 2 4 0

B 5 2 0

C 8 3 4 0

D 5 3 7 0

F 1 6 7 1

G 8 6 7 0

(b) Relabeled flow table

–

–

–

–

–

–

–

–

–

–

–

–

–

–

Figure 9.72 State diagram for the vending-machine controller

A 0

F 1

B 0

G 0

D 0

C 0

00 01 00

00 01

01 00

00

00 0110

0110

0110

DN

10 10

Figure 9.73 Determination of the state assignment

B

A F

D

G 100

110

7

4

1

6 7

111

2

3

C

4

001 000

101

5

011 010 8

B

A F

D

G

C

7

1

2

5

4

6 7

3

7 8

(a) Transition diagram

(b) Embedded on the cube

4

Figure 9.74 Excitation table

Present Next state

state DN = 00 01 10 11 Output

y 3 y 2 y 1 Y 3 Y 2 Y 1 z

A 000 0

00 010 100 0

B 010 011 0

10 0

C 111 101 1

11 1

11 0

D 011 0

11 111 001 0

F 001 000 0

01 0

01 1

G 101 1

01 001 001 0

100 110 0

110 111 0

– –

–

–

–

–

–

–

–

–

–

–

–

Figure 9.75 Karnaugh maps for the functions in Figure 9.74

00 01 11 10

d

1

00

01

11

10

1 1

d 1

1

y 1 y 2

DN

d

d

1

d

00 01 11 10

d

1 1

00

01

11

10

1 1

d 1

d

d

y 1 y 2

DN

d

d

d

d

1

1

y 3 1 = y 3 0 =

(a) Map for Y 1

Figure 9.75 Karnaugh maps for the functions in Figure 9.74

00 01 11 10

d

00

01

11

10

1 1

d

1

y 1 y 2

DN

d

1

1

d d

00 01 11 10

d

00

01

11

10

1

d

1 d

d

y 1 y 2

DN

d

d

d

d

1

1

y 3 0 = y 3 1 =

(b) Map for Y 2

Figure 9.75 Karnaugh maps for the functions in Figure 9.74

00 01 11 10

d

00

01

11

10

1

d

1

y 1 y 2

DN

d

d d

00 01 11 10

d

1

00

01

11

10

1 1

d

1 d

d

y 1 y 2

DN

d

d

d

d

1

1

y 3 1 = y 3 0 =

(c) Map for Y 3

Summary

• Analysis of asynchronous circuits.

• Synthesis of asynchronous circuits.– State reduction– State assignment– Hazard-free logic design