Chapter 3 Fault Modeling

Outlines

Introduction Fault Models Properties of Stuck-at Faults Stuck-at Fault Collapsing

Fault Model and Structural Tests

Fault model is the foundation of structural testing methods Structural tests

Use the information of interconnected components (e.g., gates) to derived test regardless of the functions Define faults
fault coverage (quality evaluation) ATPG to generate tests for faults DfT for enhancing fault detection.

Characteristics of Fault Models

Model the effects of physical defects on the logic function and timing Identifies target faults

Model faults (defects) most likely to occur Depends on the process, design platform, design style, design level, etc.
Create tests only for the modeled faults

Limits the scope of test generation

Levels of Fault Models

Physical Defects

Silicon Defects Photolithographic Defects Mask Contamination Process Variation Defective Oxides

Electrical Effects

Shorts (Bridging Faults) Opens Transistor Stuck-On/Open Resistive Shorts/Opens Change in Threshold Voltages
Logical Stuck-at 0/1 Slower Transition (Delay Faults) AND-bridging, OR-bridging
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Logical Effects

Defect, Fault, and Error

Defect

Physical imperfection
The unintended difference between the implemented hardware and its intended design

Error

A wrong output signal produced by a defective system

Fault

A representation of a defect at the abstracted function level

One of the gate input terminal was mistakenly connected to ground Defect: short to ground Error: f = 0 when a = b = 1 Fault: b stuck at 0
6

a b

Common Fault Models

Stuck-at faults Interconnect short and opens Transistor stuck-on/open faults Memory faults Delay faults

Single Stuck-At Fault Model

Assumptions:

Only One line is faulty Faulty line permanently set to 0 or 1 Fault can be at an input or output of a gate
One of the gate input terminal was mistakenly connected to ground Fault: b stuck at 0 signal b will always be 0
8

a b

The Popularity of Single StuckAt Faults

Complexity is greatly reduced Technology independent

Can be applied to TTL, ECL, CMOS, BiCMOS etc.

Design style independent

Gate array, standard cell, custom VLSI

Empirically many defects accidentally detected by test derived based on single stuck-at fault

Detection capability of un-modeled defects

Multiple Stuck-At Faults

Several stuck-at faults occur at the same time For a circuit with k lines

there are 2k single stuck-at faults there are 3k-1 multiple stuck-at faults A line could be stuck-at-0, stuck-at-1, or fault-free One out of 3k resulting circuits is fault-free

Most Multiple faults are covered by single-fault tests of combinational circuit:

4-bit ALU (Hughes & McCluskey, ITC-84) All double and most triple-faults covered. Large circuits (Jacob & Biswas, ITC-87) Almost 100% multiple faults covered for circuits with 3 or more outputs.

Bridging Faults For CMOS Logic

Two or more normally distinct points (lines) are shorted together

Could be AND-bridging or OR-bridging

depends on the inputs

VDD

(A=B=0) and (C=1, D=0) (f and g) is OR-bridging fault

pull to VDD

VDD

f
B A GND
bridging

g
C D

pull to zero GND

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(A=B=0) and (C=1, D=1) (f and g) is AND-bridging fault

Interconnet Opens

Open defects are due to a defect which splits up a node into two or more distinct nodes.

Large open (break): No current can go between the two ends of the open when a voltage is applied across it. Narrow open: The opening is usually < 100 nm. In this case a small leakage current (by tunnel effect) go across the open.
VDD VDD

C A Open defects as a large resistor

f B D C

A
GND GND
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CMOS Transistor Stuck-On

VDD

IDDQ current

Example: N transistor is always ON

0
stuck-on GND

Output level?

Transistor Stuck-On

May cause ambiguous logic level

Depends on the relative impedances of the pull-up and pull-down networks Both P and N transistors are conducting, causing increased quiescent current, called IDDQ fault

When Input Is Low in the example

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CMOS Transistor Stuck-Open

Transistor stuck-open

May cause the output to be floating The faulty cell has sequential behavior
VDD

Stuck-open might require two vector tests

stuck-open

Initialization vector

(0/0) ->(1/0)
Faulty value

1 -> 0 GND

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Pseudo-Stuck-At Fault Model

It is similar to single stuck-at fault model, except that every cell output is considered observable by IDDQ testing.

The fault site at a gate input requires sensitization and propagation to an output of the same gate (but not to an output of the device) in order to be given credit for IDDQ fault detection.
1/0

a=1 b=1 c=0

f=0

Pseudo-Stuck-At Fault Model

The exhaustive pseudo-stuck-at patterns for IDDQ testing cover all leakage faults in fully complementary combinational CMOS circuits.
ab 01 10 cd 01 11 Detected Pseudo Stuck-At Faults a-sa1 d-sa0 b-sa1 d-sa0 Detected leakage Faults (total 6C2 = 15 leakage faults) <b, 0> <c, 1> <d, 0> <a, 1> <a, b> <a, d> <b, c> <c, d> <0, 1> <a, 0> <c, 0> <d, 0> <b, 1> <a, b> <b, c> <b, d> <0, 1>

11

00

a-sa0 b-sa0 d-sa1

<a, 0> <c, 0> <c, 1> <d, 1> <a, c> <a, d> <b, c> <b, d> <0, 1>

2-input NAND2

S.T., Zachariah, A comparative study of pseudo stuck -at and leakage fault model, IEEE Intl Conf. on VLSI Design 1999

Memory Faults

Parametric Faults

Speed Power Consumption Noise Margin Data Retention Time

Functional Faults

Stuck Faults in Address Register, Data Register, and Address Decoder Cell Stuck Faults Adjacent Cell Coupling Faults 0 0 0 the presence of a faulty signal depends on 0 d b the signal values of the neighboring cells Pattern-Sensitive Faults 0 a 0 Pattern sensitivity between cells a=b=0 => d=0

a=b=1 => d=1

Delay Fault Models

Delay faults change the timing of circuit.

Testable when circuit operates at higher speed

Insufficient stuck-at tests

Chip with timing defects may pass the low speed stuck-fault testing, but fail at the system speed Transition Fault, Gate Delay Fault, Line Delay Fault, Path Delay Fault, Segment Delay Fault

Types

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Fault Model Summary

Current practice

Single stuck-at faults Transition faults

Other models to enhance defect coverage

N-detect faults
Interconnect open faults and bridging faults Stuck-open (CMOS)

Path delay faults (high performance circuits)

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Properties of Stuck-at Faults

Definition Of Fault Detection

A test (vector) t detects a fault f iff

t detects f z(t) zf(t) Note that there can be multiple outputs for z(t) A fault f is said to be detectable If there exists a test t that detects f; otherwise, f is undetectable. Example
X1

x
s-a-1
X2

Z1

Z1=X1X2 Z1f =X1

Z2

Z2=X2X3 Z2f =X2X3

X3

The test (x1,x2,x3) = (100) detects f because z1(100)=0 while z1f (100)=1

Redundancy of Faults

For combinational circuits, an undetectable fault is corresponding to a redundant wire.

Undetectable faults do not change the function of the circuit The wire can be connected to a constant value without changing circuits function.

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Examples of Simplifying Circuits with Redundancy

If l s.a.1 is undetectable, the gate can be simplified as

s. a. 1

l x m n

l m n

m n

If l s.a.0 is undetectable, the entire gate can be removed & replaced by a constant 0 wire
s. a. 0

l m n

l
m n

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Cause of Circuit Redundancy

Design to satisfy certain physical characteristics

Speed: carry look-ahead adders Fault tolerant circuits: triple module redundant (TMR)

Un-intentional redundancy by design partitioning.

Under specified requirements.

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Example: Redundancy Removal

Redundant faults: e s-a-1, d s-a-0 and d s-a-1 The NAND gate with d is redundant
a

e e sa1
c

d f

b a c b
simplify the redundant signals and gates

Fault Masking

The existence of an undetectable fault may invalidates the test for another fault.

Test pattern masking by undetectable faults

e a e sa1 c

d f

Fault a is undetectable, i.e., f(t) = f(t) for all t.

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 Fault b can be detected by test vector 1101. (X10X) 1 a 0 0 1 d 1 1 c 0 1 0/1 1/0 1 g b b g sa0 1/0

0/1

In the presence of the undetectabe fault a, the test vector 1101 becomes invalidate for fault b.

1 1 a c 0 d 1 1

1
1

0 1/0 0/1

g b g sa0 1/0

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 But fault b can also be detected by test vector 010X with fault a.

0 1 a d X 1 0

c
b

1
1/0 0/1

0/1

1 g b g sa0 1/0

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Stuck-at Fault Collapsing

Fault Equivalence Fault Dominance Checkpoint Theorem

Fault Equivalence

Distinguishing test
A

test t distinguishes faults a and b if

Za t Zb t 1

Equivalent Faults
faults, a & b are said to be equivalent in a circuit , iff the function under a is equal to the function under b for any input combination (sequence) of the circuit. No test can distinguish between a and b In other words, test-set(a) = test-set(b)
Two

Equivalence Analysis of a Single Gate

AB
A C

C 0 0

A B C A B C sa1 sa1 sa1 sa0 sa0 sa0 1 1 1

00 01

10
11

0
1

1
0 0 0

Fault Equivalence Class

(A s-a-0, B s-a-0, C s-a-0)

Faults that can be ignored:

A s-a-0, B s-a-0, or C s-a-0

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Fault Equivalence of Faults on a Gate

AND gate:

all s-a-0 faults are equivalent all s-a-1 faults are equivalent all the input s-a-0 faults and the output s-a-1 faults are equivalent all input s-a-1 faults and the output s-a-0 faults are equivalent

OR gate:

x s-a-0

x s-a-0

NAND gate:

same effect

NOR gate:

Inverter:

input s-a-1 and output s-a-0 are equivalent input s-a-0 and output s-a-1 are equivalent

Equivalence Fault Collapsing of a Single Gate

n+2 instead of 2(n+1) faults need to be considered for n-input gates

s-a-1
s-a-1 s-a-1 s-a-0

s-a-0 s-a-0

s-a-1 s-a-0

s-a-1 s-a-1

s-a-1 s-a-0

s-a-0 s-a-0

s-a-1 s-a-0

Equivalent Fault Group

In a combinational circuit, many faults may form an equivalent group

These equivalent faults can be found by sweeping the circuit from the primary outputs to the primary inputs Transitive Rule: When a == b and b == g, then a == g

s-a-0
x

s-a-1
x

s-a-1
x
Three faults shown are equivalent !

Finding Equivalent Fault Groups

Construct a Graph

A node is a fault When there is a link between two node, the two faults are equivalent
a s-a-0

a s-a-0
b s-a-0
x x

c s-a-1
x b s-a-0 c s-a-1

Example of Finding Equivalent Fault Groups

Construct a bigger fault equivalent graph by sweeping the netlist from POs to PIs
a b d e

c
a s-a-0 c s-a-1

b s-a-0

d s-a-1

e s-a-1

Limitation of Structural Analysis

c

a
b s-a-0 f s-a-1

There is no rule to guarantee equivalence between faults of branches. It is a special case here b s.a.0 == f s.a.1

In general, they cannot be identified by simple structure analysis.

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Fault Dominance

Dominance Relation

A fault b is said to dominate another fault a in an irredundant circuit, iff every test (sequence) for a is also a test (sequence) for b . I.e., test-set(b) > test-set(a) No need to consider fault b for fault detection once a is detected

Test(a)

Test(b)

a is dominated by b

Fault Dominance

AND gate:

Output s-a-1 dominates any input s-a-1 Output s-a-0 dominates any input s-a-1

Easier-to-test x s-a-1

NAND gate:

x s-a-1

OR gate:

Output s-a-0 dominates any input s-a-0

Output s-a-1 dominates any input s-a-0

harder-to-test

NOR gate:

Dominance fault collapsing:

The reduction of the set of faults to be analyzed based on dominance relation

Dominance Analysis of a Single Gate

AB
A C

C 0 0

A B C A B C sa1 sa1 sa1 sa0 sa0 sa0 1 1 1

00 01

10
11

0
1

1
0 0 0

Fault Dominance Relations

(C s-a-1 > A s-a-1) and (C s-a-1 > B s-a-1)

Faults that can be ignored for test generation:

C s-a-1

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Complete Fault Collapsing of a Single Gate

Equivalence + Dominance

For each n-input gate, we only need to consider n+1 faults during test generation
s-a-0
s-a-0 s-a-0

s-a-1
s-a-1

s-a-1

s-a-1

s-a-1

s-a-0

s-a-1

s-a-0

s-a-0

Dominance Analysis for a Group of Faults

Construct a dominance graph

A node is a fault When fault a dominates fault b, then an arrow is pointing from a to b

a s-a-1

a s-a-1

b s-a-1

x x

c s-a-0
x

b s-a-1

c s-a-0
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Example of Finding Dominant Fault Groups

Construct a bigger fault equivalent graph by sweeping the netlist from POs to PIs
a

d e

a s-a-1

c s-a-0

b s-a-1

d s-a-0

e s-a-0

Fault Collapsing Flow

Start Sweeping the netlist from PO to PI To find the equivalent fault groups Equivalence analysis

Sweeping the netlist To construct the dominance graph

Dominance analysis

Discard the dominating faults

Select a representative fault from each remaining equivalence group

Generate collapsed fault list

Done
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Checkpoint Theorem

Checkpoints Theorem

Checkpoints = primary inputs + fanout branches In a combinational circuit, any test which detects all single stuck faults on all checkpoints detects all single faults in the circuit.

The faults marked by X are checkpoint faults.

Why Inputs + Branches Are Enough ?

Sweeping the circuit from PO to PI to examine every gate and replace output faults by input faults until a PI or branch is met.

Based on an reversed levelized order: EDABC If a branch is met, start the above process from the stem again.

In this example, checkpoints are marked in solid blue.

A
D

PO branch faults are ignored, since they are represented by gate output faults, e.g., Ds output

E
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An Example of Fault Collapsing + Checkpoint

10 checkpoint faults

a s-a-0 == d s-a-0, c s-a-0 == e s-a-0 b s-a-0 > d s-a-0, b s-a-1 > d s-a-1

Note that the branch dominance is not obvious.

6 out of 10 faults are enough.
a
d

f
g

b
e c