## Chapter 18 FUNDAMENTAL MODE SEQUENTIAL CIRCUITS

Chapter 18 FUNDAMENTAL MODE SEQUENTIAL CIRCUITS Lesson 2 ANALYSIS OF ASYNCHRONOUS SEQUENTIAL CIRCUIT Ch18L2- "Digital Principles and Design", Raj...
Author: Jacob Owen
Chapter 18

FUNDAMENTAL MODE SEQUENTIAL CIRCUITS

Lesson 2

ANALYSIS OF ASYNCHRONOUS SEQUENTIAL CIRCUIT

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Outline • • • •

Analysis Procedure Excitation cum Transition table State table and state Diagram Flow table and diagram

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Analysis Procedure — Fundamental Mode circuit • Step 1: Draw logic circuit diagram to be

analyzed • Step 2: Draw circuit such that feedback-lop is clear, feedback loop minimized to have minimum feedback inputs and outputs. Find circuit x’q1, x’q2 , ... and inputs yq0, yq1 and yq2 inputs to memory section are there. Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Analysis Procedure • Step 3: Perform state variables xQ assignments and excitation variables yQ (means latches and other delaycircuit elements inputs) assignments. [For example state variable assignments xq1 and xq2 and excitation variables yq0, yq1 and yq2.] Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Analysis Procedure • Step 4 (i) Find the expressions for the excitations from the flip-flop characteristics equations as per the excitations. In other words, find Y = Fo (Xi, xQ) and thus Yj and yQ. (ii) Make an excitation table.

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Analysis Procedure • Step 5 Make excitation-cum-transition table (or simply transition table) from the expressions for xQ’ = FQ (Xi, xQ, yQ) .

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Analysis Procedure • Step 6 Show the stable states and transitions by horizontal and vertical directed arcs at excitation-cum-transition table and make flow diagram by making the state, flow and primitive flow tables • Step 7: Find reduced flow table for the restricted inputs. Flow diagram is then drawn for final result of analysis. [Refer next lesson] Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Example: Draw Logic Circuit-i X

D Q1 Q1

J

Q2

K Q2

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Example: Draw Feedback loop and combination circuit distinctly yq0, yq1, yq2

m- latches xq0 xq xq1 Memory Section xq2

xq0, xq1, xq2 Present Combinational Y0 state yq0 Circuit yq1 X yq2

Next state Feedback yq0, yq1, yq2

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Example: Find Expressions for Delay Section Inputs • D = yq0 • J = yq1 • K = yq2

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Example: Find expressions for Combinational Circuit Inputs • X = 0 or 1 • xq1 = D = yq0 after a delay • xq2 = J. xq2 + K. xq2 after a delay = yq1. xq2 + yq2. xq2 after a delay

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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• • • •

Example: Find Expressions for Combinational Circuit Outputs Y0= X . xq2.+ xq1 yq0 = X . (xq2 + xq1 ) = D yq1 = xq1 = J yq2 = K = X

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Outline • • • •

Analysis Procedure Excitation cum Transition table State table and State Diagram Flow table and diagram

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Excitation cum Transition table • Tabular representation of present state xq input of the latches, X input, next state yq output, which will be feedback to the latches from the next state Y [YO and yq at the output stages]. • Gives the memory-section outputs that follow the excitations from yq output feedback Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Excitation Table for YO = X. Q2 +Q1; Q1’ =D and Q2’ = Qn+1 = J. Q n + K. Q n

State

Excitation Inputs

YO

(Q1,Q2) [D, (J, K)]X=0

[D, (J, K)]X=1

X=0 X =1

(0, 0)

0, (0, 0)

1, (0,1)

0

0

(0, 1) (1, 0)

0, (0, 0) 0, (1, 0)

1, (0,1) 1, (1,1)

1 1

0 1

(1, 1)

0, (1, 0)

1, (1,1)

1

1

Y is present output state after the X inputs but before transition Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Excitation-cum-Transition Table for Y = X. xq2 + xq1; xq1’ =D and xq2’ = xqn+1 = J. xq n + K. xq State

n

Transition Outputs

Y

(Q, Q2) [Q1’, Q2’]X=0 [Q1’, Qq2’]X=1 X=0 X =1 (0, 0)

0, 0

1, 0

0

0

(0, 1) (1, 0)

0, 1 0, 1

1, 0 1, 1

1 1

0 1

(1, 1)

0, 1

0, 0

1

1

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Outline • Analysis Procedure • Excitation cum Transition table for drawing flow table and diagram • State table and State Diagram • Flow table and diagram Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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State Table • State table can made easily from excitationcum-transition table • A set of present Q0, Q1,.. denotes a state • Each set assigned a state-name Si as follows: • (Q1, Q2) = (0,0) → S0 • (Q1, Q2) = (0,1) → S1 • (Q1, Q2) = (0,0) → S2 • (Q1, Q2) = (0,0) → S3 Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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State table showing stable next states • A state table gives the stable next state and output before this state. This state is that which finally exists after one or more feedback-cycles of next states from the memory section after change of an input Xk. That is fine when the finally what state(s) is achieved that matters Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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State Table for Y = X. xq2 + xq1; xq1’ =D and xq2’ = xqn+1 = J. xq n + K. xq n State (yq)

Next State-Transition Outputs YO [xq]X=0

[xq]X=1

S0

S0

S2

S1

S1

S2

X=0 X =1

0 1 1 1

0 0 1 1

S2 S1 S3 S3 S1 S0 YO is present output state after the xq outputs but before transition yq next state

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State Diagram for transitions to stable states 1/1

0/0

S0

S3 1/1

1/0

S2 0/1

1/0 0/1

0/1

S1

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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State Minimization • A set of present (xq 0, xq1, … , xqm – 1) is denoted by a state. There are z (=2m) maximum possible stable states S0, S1, S2, … , Sz–1 in asynchronous circuit with zmemory or delay section circuits. [The 2n stable states can be reduced by state minimization method of finding the equivalent states. In that case state table will have less number of rows and state diagram less number of nodes Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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State table showing intermediate states • A state table also gives the unstable next states shown by vertical arcs in excitation cum transition table and output changes before this state. The final state is that which finally exists after one or more feedback-cycles of next states from the memory section after change of an input Xk. That is fine when the finally what state(s) is achieved that matters Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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State Table for Y = X. xq2 + xq1; xq1’ =D and xq2’ = xqn+1 = J. xq n + K. xq n State (yq)

Next State-Transition Outputs YO [xq]X=0

[xq]X=1

S0

S0

S2

S1

S1

S2

X=0 X =1

0 1 1 1

S2 S1 S3 S3 S1 S0 YO is present output state after the xq outputs but before transition yq next state

0 0 1 1 25

State Diagram for transitions to stable states 1/1

0/0

S0

S3 1/1

1/0

S2 0/1

1/0 0/1

0/1

S1

Note that for X = 1, S0 first changes to intermediate state S2 and then to S3

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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State table giving all states, whether certain set of inputs X, Y exist or not • A state table gives all the next state(s) whether a certain set of inputs is specific as nonexistent after a certain sequence of set on inputs or existent. For example, when two simultaneous input bit changes not permitted (in fundamental mode circuit), at S2, the 01 will not follow 11 when X changes first Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Example, when 10 changes to 01 • S0 →S2 →S3 → S1 if Y changes first • S0 →S2 →S1 if X changes first

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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State Diagram for transitions to stable states through intermediate states in two paths 1/1

0/0

S0

S3 1/1

1/0

S2 0/1

1/0 0/1

0/1

S1 S0 →S2 →S3 → S1

S0 →S2 →S1

When 1/0 becomes 0/1, it does not happen simultaneously. State table is showing both paths

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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State table does not give • The information about state that will not exist due to certain specified input constraints as it gives entries for all cases

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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State table gives too much • The information about state that will not exist due to certain specified input constraints as it gives entries for all cases

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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State table does not give too much • It does not provide the information about the flow (next sequence) of the stable states and outputs for the final stable existence after one or more feedback-cycles of next states from the memory section after change of an input Xk For a condensed view of the asynchronous circuit behavior, the flow part is required Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Outline • • • •

Analysis Procedure Excitation cum Transition table State table and state Diagram Flow table and diagram

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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State table does not give • It does not provide the information about the flow (next sequence) of the stable states and outputs for the final stable existence after one or more feedback-cycles of next states from the memory section after change of an input Xk For a condensed view of the asynchronous circuit behavior, the flow part is required Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Flow table • Gives the present state, for the different input combinations next states (stable) and present outputs, which corresponds to ones leading to stable state. These states exist after one or more feedback-cycles to next stable states at the memory section after change of an input Xk Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Flow Table for Y = X. xq2 + xq1; xq1’ =D and xq2’ = xqn+1 = J. xq n + K. xq n State (yq) S0

Next State-Transition Outputs YO [xq]X=0 S0

[xq]X=1 S3

X=0 X =1

0 -

S2 S1 S3 0 S3 S1 S0 1 YO is present output state after the xq outputs but before transition yq next state

1 1 36

Flow Table in Example • Does not show row for S1, because S2 has Y= 1 for both X= 0 and 1, therefore, transition for S1 to S2 for X/Y = 1/0 is unstable and flow table shows only stable states, • Shows dash for Y as for S0 the excitations X/Y = 1/0 does not exist in stable case Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Flow Diagram for transitions to stable states 0/0

S2

1/1

S0

1/1 1

S3 0/0

1 0/1

0/1

S1

When 1/0 becomes 0/1, it does not happen simultaneously. Flow table is showing one path and stable state only

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Flow diagram • States S0, S1, S2 and S3 are labeled at the centers of each circle representing a node • Each arc or circular arc is labeled with present input and the present output at the transition. If there is dash, then slash sign and Y for that are not shown Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Flow diagram • Each arc or circular arc can have more then one set of (pre-transition input/ post transition output) labeled on it if there are more than one sets of (pretransition input/ post transition output) that are having the same transition from a node to another. Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Flow diagram • If there are only inwards arcs at a node from other nodes , it shows that from that state, there are no transitions that are stable and eventually state will flow back to the same. • In flow table, it shows missing row(s) for that node • In flow table, dash at Y shows, that output will undergo transition and that X/Y non existent in stable states after intermediate cycles Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Flow Table • Unstable states, which lead to other unstable states, are not shown in flow table and at a table next state entry, only the eventually occurring states are shown. Flow table gives the condensed view of the circuit Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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Summary

Analysis of a asynchronous fundamental mode circuit is done by drawing circuit, from which memory section and combinational circuits, from which excitation-cum-transition tables, state table and finally flow table is drawn

• •

Excitation-cum-transition table shows the excitation input change by horizontal directed arc, stable states by marking over it. Table shows the excitation input change by horizontal directed arc, stable states by marking over it Then state table is drawn by assigning a state to each set xq. Flow table and flow diagram are then drawn for final result of analysis

End of Lesson 2 on

ANALYSIS OF ASYNCHRONOUS SEQUENTIAL CIRCUIT

THANK YOU

Ch18L2- "Digital Principles and Design", Raj Kamal, Pearson Education, 2006

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