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Reconvergent Fanout Analysis of Bounded Gate

Delay Faults

Dept. of ECE, Auburn University

Auburn, AL 36849

Master’s DefenseHillary Grimes

Thesis Advisor: Dr. Vishwani D. AgrawalThesis Committee: Dr. Victor P. Nelson and Dr. Charles E. Stroud

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Outline

• Background

• Problem Statement

• Ambiguity Lists

Fault-Free Circuit Simulation

Detection Threshold Evaluation

• Experimental Setup

• Results and Discussion

• Conclusions

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Delay Testing

• Delay testing ensures a manufactured

design meets it’s timing specifications

• Gate Delay Fault Model

Assume that a delay fault is lumped at a faulty

gate

All other gates have their delays within the

specified (min, max) range.

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A Gate Delay Test

• A delay test consists of a vector pair

First vector (V1) initializes the circuit

Second vector (V2) produces the required

transitions

• For a slow-to-rise gate delay fault:

V1 – places a logic 0 at the fault site

V2 – stuck-at-0 test for the same fault site

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Fault-Free Circuit Simulation

• IV – initial value for V1-V2 transition – stable

logic value at gate after V1 is applied

• FV – final value for V1-V2 transition – stable

logic value at gate after V2 is applied

• EA – earliest arrival time for gate output

after V2 is applied

• LS – latest stabilization time for gate output

after V2 is applied

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Fault-Free Circuit Simulation

1,31,31,3

1,31,31,3

1,2 1,2

1,2

3,4

1 3

2 5

3 5

5 9

4 11

0

1 1

EA = 5 LS = 9

IV = 1

FV = 0

EA = ∞LS = -∞

EA = 0LS = 0

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Faulty Waveforms

• Faulty Propagating Value - FPV

Signal’s logic value in the presence of a stuck-

at-IV fault at the fault site

• Propagates the fault’s logic effect through

the circuit

• Used to determine whether or not a delay

fault of any size (transition fault) is

detected

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Fault Propagating Values

1,31,31,3

1,31,31,3

1,2 1,2

1,2

3,4

2 5

3 5

5 9

4 11

0

1 1

Slow-ToFall

1 3

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Fault Propagating Values

1,31,31,3

1,31,31,3

1,2 1,2

1,2

3,4

1 3

2 5

3 5

5 9

4 11

0

1 1

FPV = 0

FPV = 1

FPV = 1

FPV = 1

FPV = 1

FPV = 0

FPV ≠ FV

Assuming the delayfault size is large

enough, it is detected

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Detection Threshold

• For gate delay faults, we also need to

know the size (δ) of faults detected

• Detection Threshold – minimum size delay

fault detectable by the test

• Requires timing information about faulty

waveforms to be propagated along with

FPVs

RTa & RTb – signal is at FPV between the times

RTa to RTb+δ

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Detection Threshold Evaluation

1,31,31,3

1,31,31,3

1,2 1,2

1,2

3,4

1 3

2 5

3 5

5 9

4 11

0

1 1

FPV = 1

FPV = 1

FPV = 0

FPV = 1

FPV = 1

FPV = 0

Ts = 12

RTa = -∞RTb = 1

RTa = -∞RTb = ∞

RTa = -∞RTb = 2

RTa = -∞RTb = 3

RTa = -∞RTb = 5

RTa = -∞RTb = 4

Ts and RTb at the output determine

detection threshold

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Detection Threshold Evaluation

FPV = 0RTa = -∞RTb = 4

4 11

Ts = 12

Detection Threshold = 8

• The output signal is at FPV between times

RTa (RTb+δ): -∞ (4+δ)

• Detection Threshold is Ts – RTb: (12-4)=8

Fault detected if it’s size is greater than 8:

δ > 8

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Detection Gap

• Calculating the “detection gap” provides a

way to relate the detection threshold of a

detected gate delay fault to the slack at

the fault site

• Detection gap is: DT(G) – slack(G)

DT(G) detection threshold at fault site G

slack(G) sum of all minimum gate delays

along the longest delay path through G

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Detection Gap

• The smaller the detection gaps are for a

set of vectors, the better quality that set

provides for detecting gate delay faults

• If a test detects a fault with gap = 0:

The smallest possible gate delay fault has been

detected

• If detection gap > 0:

There is a possibility a better test exists to

detect the fault with a smaller threshold

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An Illustration of Detection Gap

• A test that detects the fault through path

p1 would be better than a test that detects

the fault through path p2

PIPO

p1 - longest delaypath through gate

p2

Gate

Ts

p1delay

p2delay

gap

DT(p2)

slack

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Problem Statement

• When signals produced by a common

fanout point reconverge, the inputs to the

reconvergent gate are correlated

• Conventional simulation ignores this

correlation when bounded gate delays are

used

Produces pessimistic results in both bounded

delay simulation and gate delay fault simulation

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Reconvergent Fanout Analysis

1,31,31,3

1,31,31,3

1,2 1,2

1,2

3,4

1 x 33 5

5 9

4 6 11

Fall occurs at time ‘x’

x+1 5

Output rises at least 1 unit after ‘x’

Hazard cannot occur

0

1 1

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Correct Detection Threshold

FPV = 0RTa = - ∞RTb = 6

Ts = 12

Detection Threshold = 6

• The output signal is at FPV between times

RTa to RTb+δ

• In an accurate analysis, RTb=6, not 4

Fault detected if it’s size is greater than 6:

δ > 6

4 6 11

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Ambiguity Lists

• Ambiguity Lists generated at fanout points

contain

originating fanout name

ambiguity interval – min and max delays from

fanout to gate

• Ambiguity lists at the inputs of a

reconvergent gate help determine its output

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Ambiguity Lists

• List propagation is similar to fault list

propagation in concurrent fault simulation

• For accurate detection threshold evaluation,

ambiguity lists are propagated during both

fault-free and faulty waveform calculations

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Ambiguity Lists – Fault-Free Circuit

• Ambiguity lists propagated through all

gates during fault-free circuit simulation

• If signal correlations are such that no

hazard can occur, the hazard is

suppressed: (EA = ∞) & (LS = -∞)

• Otherwise, the ambiguity lists are

propagated to the gate’s output, and

ambiguity intervals are updated

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Fault-Free Circuit Simulation

1,31,31,3

1,31,31,3

1,2 1,2

1,2

3,4

1 3

2 5

5 9

0

1 1

EA = 1LS = 3

EA = ∞LS = -∞

EA = 2LS = 5

EA = 5LS = 9

EA = 6LS = 11

EA = 0LS = 0

EA = ∞LS = -∞

3 5

4 6 11

EA = ∞LS = -∞

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Ambiguity Lists – Detection Threshold Evaluation

• Ambiguity lists propagated through

downcone of the fault site

• If signal correlations are such that no

hazard can occur, the hazard is

suppressed: (RTa = -∞) & (RTb = ∞)

• Otherwise, the hazard lists are propagated

to the gate’s output, and ambiguity

intervals are updated

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Detection Threshold Evaluation

1,31,31,3

1,31,31,3

1,2 1,2

1,2

3,4

1 3

2 5

3 5

5 9

0

1 1

FPV = 1RTa = -∞RTb = 1

FPV = 1RTa = -∞RTb = ∞

FPV = 0RTa = - ∞RTb = 2 FPV = 1

RTa = - ∞RTb = 5

FPV = 1RTa = -∞RTb = ∞

FPV = 0RTa = - ∞RTb = 6

4 6 11

Tc = 12

Now, RTb at the output is correctly

evaluated as 6

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Experimental Setup• C program implemented to perform gate

delay fault simulation on combinational

circuits

• Simple wireload model used for gate

delays

Bounded delays set to (3.5n ± 14%), where n is

the number of fanouts

Program can accept any available gate delay

data, which may be normally available from

process technology characterization

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Experimental Setup

• Program inputs

Netlist in bench format

Vector file

• Program outputs for vector set

Average detection gap of detected gate delay

faults

Fault coverage of faults detected with gap ≤ 3.5

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Experiment A

• Gate delay fault simulation 1,000

vectors

• Ambiguity lists propagated during faulty

waveform calculations only

• Average detection gap and fault coverage

of faults detected with gap ≤ 3.5 (nominal

gate delay) recorded

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Experiment A

• For fault coverage, faults are counted as

detected if they are detected:

though the longest path through the gate

through a path which is less than the longest

path by only one gate delay

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Results: Experiment A

Without Reconvergent

Fanout Analysis

With Reconvergent

Fanout Analysis

Circuit

Average

Detection

Gap

Faults

Detected with

Gap ≤ 3.5

Average

Detection

Gap

Faults

Detected with

Gap ≤ 3.5

c432 99.7 8.83% 98.0 8.83%

c499 36.5 5.69% 35.4 5.69%

c880 19.3 43.69% 17.0 43.92%

c1355 51.3 3.80% 47.8 6.31%

c1908 57.0 15.85% 51.9 17.66%

c2670 40.5 28.78% 29.9 31.70%

c3540 54.1 20.07% 49.1 17.49%

c5315 24.8 42.59% 8.2 46.79%

c7552 41.0 11.46% 24.8 20.26%

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Experiment B

• Bounded delay simulation of the fault-free

circuit 10,000 vectors

• Ambiguity lists propagated through every

gate

• Largest EA and LS values at circuit

outputs for all vectors recorded to

illustrate difference seen at outputs when

ambiguity lists are used

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Results: Experiment B

Without Reconvergent

Fanout Analysis

With Reconvergent

Fanout Analysis

Circuit Largest EA Largest LS Largest EA Largest LS

c3540 96.0 204.0 121.6 196.8

C5315 76.8 204.0 91.2 194.4

C6288 158.4 576.0 236.8 504.0

C7552 91.2 204.0 104.0 201.6

• Using reconvergent fanout analysis generally

results in larger EA and smaller LS values at

outputs

• More apparent for circuits that contain a large

number of reconvergent fanouts, such as in

multiplier circuit c6288

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Experiment C

• Gate delay fault simulation 10,000

vectors

• Ambiguity lists propagated during both

fault-free circuit simulation and detection

threshold evaluation

• Average detection gap and fault coverage

of faults detected with gap ≤ 3.5 recorded

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Results: Experiment C

Without Reconvergent

Fanout Analysis

With Reconvergent

Fanout Analysis

Circuit

Average

Detection

Gap

Faults

Detected with

Gap ≤ 3.5

Average

Detection

Gap

Faults

Detected with

Gap ≤ 3.5

c432 110.4 7.35% 108.9 7.08%

c499 51.7 4.91% 44.0 12.85%

c880 16.4 48.41% 12.9 48.86%

c1355 50.8 4.80% 42.2 13.62%

c1908 55.2 21.70% 47.1 25.10%

c2670 41.8 31.25% 36.0 36.54%

c3540 50.4 32.60% 44.0 33.19%

c5315 21.7 55.72% 6.1 57.31%

c7552 39.4 13.43% 22.5 22.83%

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Discussion

• When reconvergent fanout analysis is

used, the average detection gap is smaller

and more faults are detected with smaller

gaps

• To accurately evaluate detection

thresholds, signal correlations must be

considered in both fault-free waveform

and faulty waveform calculations

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Conclusion

• Propagating ambiguity lists during simulation

provides useful information about signal

correlations due to reconvergent fanouts

• The use of this information during both fault-

free and faulty waveform calculations

produces more accurate results for gate delay

fault simulation

• This min-max delay simulator has found

application in hazard-free delay test generation

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Future Work

• During simulation, ambiguity lists can grow

quite large

Efficiency in list propagation needs to be

improved

• Can information provided by propagating

ambiguity lists help reduce pessimism in

static timing analysis?

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Publications related to this work

• S. Bose, H. Grimes and V. D. Agrawal, “Delay Fault

Simulation with Bounded Gate Delay Model”, in Proc.

IEEE International Test Conference, paper 26.3, 2007.

• H. Grimes and V. D. Agrawal, “Analyzing Reconvergent

Fanouts in Gate Delay Fault Simulation”, in Proc. 17th

IEEE North Atlantic Test Workshop, May 2008.

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Thank You