Asynchronous Sequential Logic
For most figures:
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We know… don’t we?
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We have learned how to analyze and design circuits with memory, don’t we?
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We have designed really complicated circuits and can do so for any problem, right?
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How about taking a look at our lovely latch circuit?
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Asynchronous sequential circuits • • • • •
No clock pulses Change of an internal state occurs when there is a change in the inputs. More difficult to design/analyze than synchronous sequential circuits. Useful in cases where speed is important. More economical to use.
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Combinational circuits with feedback loops (connected through delayed lines).
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Secondary variables: input end of the feedback loops., e.g. Y1, y2,… yk.
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Example
• • • •
One input: x Two feedback loops Two excitation variables: – Y1 and Y2 Two secondary variables: – y1 and y2
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Transition table • • • •
Y1 = xy1 + x’y2 Y2 = xy1’ + x’y2 The input (x) is also part of the state Stable states are the circled ones where – Y1 = y1 and Y2 = y2
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Understanding the transition table • •
Four stable states: – y1y2x = {000,011,110,101} If y1y2x = 000 and x: 0 1 – Then Y1Y2x = 011
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Procedure for analysis
• • • • • •
Determine all feedback loops Assign Yi's (excitation variables), yi's (the secondary variables) Derive the Boolean functions of all Yi's Plot each Y function in a map Construct the state table Circle the stable states
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Flow table •
A flow table is a state transition table with its internal state being symbolized with letters
output
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(a) is called a primitive flow table since it has only one stable state in each row 8
Designing an asynch. seq. ckt.
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Assign – a=0 – b=1
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Race conditions •
•
•
Race conditions occur when two or more state variables are supposed to change simultaneously due to a change in the input. For instance: – From y1y2=00 to y1y2 = 11 – Three possible transitions: • 00 11 • 00 10 11 • 00 01 11 Critical vs non-critical race conditions
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Non-critical races •
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Three possible transitions: – 00 11 – 00 10 11 – 00 01 11 If all possible transitions lead us to the same final state, then it is a noncritical race.
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Critical races •
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Three possible transitions: – 00 11 – 00 10 11 – 00 01 11 If different transitions lead to different final states, then it is a critical race.
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Cycles •
Cycle: a unique sequence of unstable states that the circuit goes.
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Unstability •
Y = (x1 y)’x2 = x1’x2 + x2 y’
• •
If x1x2y = 111 Y = 0 If x1x2Y = 110 Y = 1
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Oscillation between 1 and 0. Y will be a square wave.
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SR latch - revisited
Forbidden input: 11
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Asynchronous sequential circuits • • • • •
No clock pulses Change of an internal state occurs when there is a change in the inputs. More difficult to design/analyze than synchronous sequential circuits. Useful in cases where speed is important. More economical to use.
•
Combinational circuits with feedback loops (connected through delayed lines).
•
Secondary variables: input end of the feedback loops., e.g. Y1, y2,… yk.
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Transition table • • • •
Y1 = xy1 + x’y2 Y2 = xy1’ + x’y2 The input (x) is also part of the state Stable states are the circled ones where – Y1 = y1 and Y2 = y2
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Procedure for analysis
• • • • • •
Determine all feedback loops Assign Yi's (excitation variables), yi's (the secondary variables) Derive the Boolean functions of all Yi's Plot each Y function in a map Construct the state table Circle the stable states
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Flow table •
A flow table is a state transition table with its internal state being symbolized with letters
output
•
(a) is called a primitive flow table since it has only one stable state in each row 20
Designing an asynch. seq. ckt.
•
Assign – a=0 – b=1
21
Race conditions •
•
•
Race conditions occur when two or more state variables are supposed to change simultaneously due to a change in the input. For instance: – From y1y2=00 to y1y2 = 11 – Three possible transitions: • 00 11 • 00 10 11 • 00 01 11 Critical vs non-critical race conditions
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Critical races •
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Three possible transitions: – 00 11 – 00 10 11 – 00 01 11 If different transitions lead to different final states, then it is a critical race.
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Cycles •
Cycle: a unique sequence of unstable states that the circuit goes.
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Analysis example
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Design with a latch
WRONG!
How were we designing in Sequential Logic?
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Design Procedure •
Gated latch with two inputs G and D. If G=1, then Q (latch output) should follow D. When G becomes 0, then the value of D at the time of the transition from G=1 to G=0 is retained at Q; i.e., once G becomes 0, the value of D does not change Q.
Let us start by giving a state to each row.
Inputs
Output
State
D
G
Q
Comments
a
0
1
0
D=Q because G=1
b
1
1
1
D=Q because G=1
c
0
0
0
After state a or d
d
1
0
0
After state c
e
1
0
1
After state b or f
f
0
0
1
After state e
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Primitive flow table formation •
Step 1: Stable states, and outputs are inserted. Since we allow only one of DG to change, put ‘-’ to where they both change.
Inputs
Output
State
D
G
Q
Comments
a
0
1
0
D=Q because G=1
b
1
1
1
D=Q because G=1
c
0
0
0
After state a or d
d
1
0
0
After state c
e
1
0
1
After state b or f
f
0
0
1
After state e 30
Primitive flow table formation-2 • •
Step 1: Stable states, and outputs are inserted. Step 2: Unstable states are determined. Outputs at these unstable states don’t matter. Use ‘-’ to indicate this.
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Reduction of the flow table
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Transition table Set a=0 and b=1
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Gated latch logic
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Gated latch with SR latch
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Assigning output to unstable states •
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•
Set the output of the transient state to 0 if the start and finish states have the output 0. Set the output of the transient state to 1 if the start and finish states have the output 1. Set the output of the transient state to ‘-’ if the start and finish states have different outputs.
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Design Procedure •
Gated latch with two inputs G and D. If G=1, then Q (latch output) should follow D. When G becomes 0, then the value of D at the time of the transition from G=1 to G=0 is retained at Q; i.e., once G becomes 0, the value of D does not change Q.
Let us start by giving a state to each row.
Inputs
Output
State
D
G
Q
Comments
a
0
1
0
D=Q because G=1
b
1
1
1
D=Q because G=1
c
0
0
0
After state a or d
d
1
0
0
After state c
e
1
0
1
After state b or f
f
0
0
1
After state e
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Primitive flow table formation •
Step 1: Stable states, and outputs are inserted. Since we allow only one of DG to change, put ‘-’ to where they both change.
Inputs
Output
State
D
G
Q
Comments
a
0
1
0
D=Q because G=1
b
1
1
1
D=Q because G=1
c
0
0
0
After state a or d
d
1
0
0
After state c
e
1
0
1
After state b or f
f
0
0
1
After state e 38
Primitive flow table formation-2 • •
Step 1: Stable states, and outputs are inserted. Step 2: Unstable states are determined. Outputs at these unstable states don’t matter. Use ‘-’ to indicate this.
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Reduction of the flow table
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Transition table Set a=0 and b=1
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Gated latch logic
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Gated latch with SR latch
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Assigning output to unstable states •
•
•
Set the output of the transient state to 0 if the start and finish states have the output 0. Set the output of the transient state to 1 if the start and finish states have the output 1. Set the output of the transient state to ‘-’ if the start and finish states have different outputs.
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A formal procedure for reducing flow table
Any Equivalent States?
• In (a,b), • x=0 -> next states: (c,d) • x=1 -> next states: (a,b) • In (c,d), • x=0 -> next states: (a,b) • x=1 -> next state: d
If c=d, then we can say a=b. In other words, (a,b) implies (c,d).
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A formal procedure for reducing flow table
Place a X for all pairs of states whose outputs differ. Fill in the remaining cells with values of implied states. For each confirmed implication, put a tick.
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Implication table from the flow table • What happens if the circuit is not completely specified? • Compatible states: two incompletely specified states that can be combined (i.e., there is no conflict! )
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Merger Diagram for Finding Maximal Compatibles • Links are drawn between compatible states.
• An n-state compatible is represented by an n-sided polygon with all its diagonals connected.
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How to Merge using the Merger Diagram?
The resulting set of compatibles need to be: • Covering: • Contains all initial states.
• Closed: • Contains all the implied states. The closed & covering set of compatibles determine the reduced flow table.
So, what is the subset of (a,b), (a,c,d), (b,e,f) that satisfy coverage and closedness? For coverage, (a,c,d) and (b,e,f) are sufficient. Do (a,c,d) and (b,e,f) satisfy closedness? • Are the implied states of (a,c), (a,d), (c,d), (b,e), (b,e), (b,f) included in the selected set?
Another Example for Merger Diagram
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Quizzzz… 1. 2. 3.
Derive the implication table for the following transition table. Draw the merger diagram and determine the set of maximal compatibles. Determine the minimum set of maximal compatibles and draw the reduced transition table.
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Hazards A hazard is a condition in which a change in a single variable produces a momentary change in output when no change should occur. • All inputs are 1 at the beginning. • When x2 becomes 0, due to delay caused by the NOT gate, Y may temporarily become 0.
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Types of Hazards
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Removing Hazards • Hazars are produced when changing from one term to another! • This causes a problem because during the transition, neither of the terms might be 1. • So, make redundant terms/groups so that the new terms make sure that the transition is covered.
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The Circuit After Removing the Hazard
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Another way to avoid hazards Implement the circuit with SR latches!!! Why would this remove hazards?
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Implementation with SR latches to avoid hazards
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Essential Hazards
• Due to delays, feedback loops might produce hazards. • These hazards cannot be corrected by adding gates/terms. • They can be avoided only by adjusting the delays in the feedback loops according to the delays in the input signals.
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Quiz
the last of the Asyncronous Logic part
• Check whether the following circuit has a hazard. • If it does, remove the hazard by both a) b)
adding redundant terms to the Y function, implementing the circuit with SR-latches.
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