Fundamentals on Testing and Design for Testability Chap

Fundamentals on Testing and Design for Testability Chap 1. Fundamentals. 1

Design Verification, Testing and Diagnosis • Design Verification: Ascertain the design perform its specified behavior • Testing: Exercise the system and analyze the response to ascertain whether it behaves correctly • Diagnosis: To locate the cause of misbehavior after the incorrect behavior is detected Chap 1. Fundamentals. 2

Some Real Defects in Chips • Processing Faults – missing contact windows – parasitic transistors – oxide breakdown • Material Defects – bulk defects (cracks, crystal imperfections) – surface impurities (ion migration) • Time-Dependent Failures – dielectric breakdown – electromigration • Packaging Failures – contact degradation – seal leaks Chap 1. Fundamentals. 3

Faults, Errors and Failures • Fault: A physical defect within a circuit or a system – May or may not cause a system failure • Error: Manifestation of a fault that results in incorrect circuit (system) outputs or states – Caused by faults • Failure: Deviation of a circuit or system from its specified behavior – Fails to do what it should do – Caused by an error • Fault ---> Error ---> Failure Chap 1. Fundamentals. 4

Scenario for Manufacture Test TEST VECTORS MANUFACTURED CIRCUIT RESPONSE CORRECT RESPONSES COMPARATOR PASS/FAIL Chap 1. Fundamentals. 5

Purpose of Manufacture Testing • Verify Manufacture of Circuit – Improve System Reliability – Diminish System Cost • Cost of repair goes up by an order of magnitude each step away from fab line 1000 500 100 Cost per fault (Dollars) 50 10 5 1 0. 5 IC Test Board Test System Test B. Davis, The Economics of Automatic Testing Warranty Repair , Mc. GRAW-HILL, 1982. Chap 1. Fundamentals. 6

Testing and Quality ASIC Fabrication Shipped Parts Yield: Fraction of good parts Testing Quality: Defective parts per million (DPM) Rejects * Quality of shipped part is a function of yield Y and the test (fault) coverage T. Chap 1. Fundamentals. 7

Fault Coverage * Fault coverage T is the measure of the ability of a set of tests to detect a given class of faults that may occur on the device under test. T = # of detected faults # of possible faults Chap 1. Fundamentals. 8

Defect Level * Defect Level, DL is the fraction of the shipped parts that are defective. DL = 1 - Y (1 - T ) Y: yield T: fault coverage Chap 1. Fundamentals. 9

Relating Defect Level to Fault Coverage 1 Y=. 01 . 9 DL = 1 - Y (1 -T) Y = Yield . 8 Y=. 10 . 7. 6 Y=. 25 . 5. 4 Y=. 50 . 3 Y=. 75 . 2 Y=. 90 . 1 0 Y=. 99 0 10 20 30 40 50 60 70 80 90 100 Fault Coverage, T (%) Chap 1. Fundamentals. 10

Defect Level, Yield and Fault Coverage Yield Fault Coverage DPM 50% 75% 90% 95% 99% 90% 90% 90% 67, 000 28, 000 10, 000 5, 000 1, 000 90% 90% 90% 95% 99. 9% 10, 000 5, 000 100 Chap 1. Fundamentals. 11

ASIC • What is ASIC: Application Specific Integrated Circuits – Why we need ASICs • Microelectronic economics – Volume – Time to market – Quality Chap 1. Fundamentals. 12

ASICs' Demand * While ASIC density and complexity have exploded, global market pressures have increased the demand for both and Quick Turnaround Quality . Chap 1. Fundamentals. 13

Test Development Time vs. Testability 40 35 30 25 20 15 10 Measured development times Extrapolated curve 5 0 20 40 60 80 100 Controllability and observability as a percentage of circuit covered Chap 1. Fundamentals. 14

Time-to-Market Model Growth Stagnation Decline Lost revenue due to delay Delay in reaching market * 1/8 Time delay of the product lifetime reduces 1/3 revenue. Chap 1. Fundamentals. 15

Why Testing is Difficult ? • Test application time can be exploded for exhaustive testing of VLSI – For a combinational circuit with 50 inputs, we need 250 = 1. 126 x 1015 test patterns. Assume one test per 10 -7 sec, it takes 1. 125 x 108 sec = 3. 57 yrs. to test such a circuit. – Test generation of sequential circuits are even more difficult. » Lack of Controllability and Observability of Flip-Flops (Latches) • Functional testing may not be able to detect the physical faults Chap 1. Fundamentals. 16

How To Do Test • Fault Modeling – Identify target faults – Limit the scope of test generation – Make analysis possible • Test Generation – Automatical or Manual • Fault Simulation – Assess completeness of tests • Testability Analysis – Analyze a circuit for potential problem on test generation • Design For Testability – Design a circuit for guaranteed test generation – Introduce both area overhead and performance degradation Chap 1. Fundamentals. 17

The New Challenges for VLSI Testing • Chip, Board, Module & System for high – Performance – Density – Integration – Reliability Chap 1. Fundamentals. 18

Reference: • 《Digital Systems testing and testable design》 ISBN： 0－7167－8179－4 Author：Breuer m. ELVIN a. etc • 《VLSI Testing digital and mixed analogure/digital techinques》 ISBN： 085296 901 5 Author： Stanley Lhurst Chap 1. Fundamentals. 19

DEC Alpha Chip (1994) * 64 -bit RISC * 200 MHz * 400 MIPS * 200 Mflops * 16. 8*13. 9 -mm die * 1. 68 million Txs * 431 -pin package * 3. 3 -V * 30 W power consumption. Chap 1. Fundamentals. 20

Multi-Chip Module (MCM) * IBM Enterprise System/9000* Type 9121 Model 320 * Air-Cooled Processor Technology * Integration of bipolar chips, CMOS SRAM chips, and ECL & DCS logic circuitry in a TCM (thermal conduction module) (Ref: IBM J. RES. DEVELOP. , May 1991) Chap 1. Fundamentals. 21

Wafer Scale Integration (WSI) * ELSA (European Large SIMD Array), a wafer-scale two-dimensional array of single-bit processors * MUSE (Matrix Update Systolic Experiment), MIT Lincoln Laboratory Chap 1. Fundamentals. 22

Traditional Design Flow • Conduct testing after design Design Spec. Design Too Large or Too Slow? Yes No Testability Analysis Testability Improvement? Yes Design for Testability No Done Chap 1. Fundamentals. 23

The Infamous Design/Test Wall 30 years of experience proves that test after design does not work! Functionally correct! We're done! Design Engineering Oh no! What does this chip do? ! Test Engineering Chap 1. Fundamentals. 24

New Design Mission • Design circuit to optimally satisfy or trade-off their design constraints in terms of area, performance and testability. TESTABILITY PERFORMANCE AREA Chap 1. Fundamentals. 25

New VLSI Design Flow No Design Spec. Function/ Behavior Test plan Structure Logic Synthesis Testable Design Rules Testability Analysis Satisfied ? Yes Circuit Synthesis Placement/ Routing ATPG TESTS MASK Chap 1. Fundamentals. 26

Why Model Faults ? • I/O function tests inadequate for manufacturing – Functionality vs. component & interconnection testing • Exhaustive testing is Prohibitively expensive Chap 1. Fundamentals. 27

Why Model Faults ? • Fault model identifies target faults – Model faults most likely to occur • Fault model limits the scope of test generation – Creat tests only for the modeled faults • Fault model makes effectiveness measurable by experiments – Fault coverage can be computed for specific test patterns to reflect its effectiveness • Fault model makes analysis possible – Associate specific defects with specific test patterns Chap 1. Fundamentals. 28

Fault Modeling the effects of physical defects on the logic function and timing. • Physical Defects: – Silicon Defects – Photolithographic Defects – Mask Contamination – Process Variations – Defective Oxide Chap 1. Fundamentals. 29

Fault Modeling (cont'd) • Electrical Effects: – Shorts (Bridging Faults) – Opens – Transistor Stuck-On/Open – Resistive Shorts and Opens – Change in Threshold Voltages • Logic Effects: – Logic Stuck-At-0/1 – Slower Transition (Delay Faults) – AND-Bridging, OR-Bridging Chap 1. Fundamentals. 30

Fault Modeling * Stuck-At Faults * Bridging Faults * Transistor Stuck-On/Open Faults * IDDQ Faults * Functional Faults * Memory Faults * PLA Faults * Delay Faults * State Transition Faults Chap 1. Fundamentals. 31

Single Stuck-At Faults Faulty Response True Response Test Vector 0 0 1/0 1 1 1 Assumptions: 1/0 stuck-at-0 • Only one line is faulty. • Faulty line permanently set to 0 or 1. • Fault can be at an input or output of a gate. Chap 1. Fundamentals. 32

Multiple Stuck-At Faults • Several stuck-at faults occur at the same time – Important in high density circuits • For a circuit with k lines – there are 2 k single stuck-at faults – there are 3 k-1 multiple stuck-at faults Chap 1. Fundamentals. 33

Why Single Stuck-At Fault Model? * Complexity is greatly reduced. Many different physical defects may be modeled by the same logical single stuck-at fault. * Single stuck-at fault is technology independent. Can be applied to TTL, ECL, CMOS, etc. * Single stuck-at fault is design style independent. Gate Arrays, Standard Cell, Custom VLSI * Even when single stuck-at fault does not accurately model physical defects, the tests derived for logic faults are still valid for these defects. * Single stuck-at tests cover a large percentage of multiple stuck-at faults. Chap 1. Fundamentals. 34

Multiple Faults • Multiple Stuck-fault coverage by single-fault tests of combinational circuit: – 4 -bit ALU (Hughes & Mc. Cluskey, 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. • No results available for sequential circuits. Chap 1. Fundamentals. 35

Bridging Faults • Two or more normally distinct points (lines) are shorted together – Logic effect depends on technology – Wired-AND for TTL A f B g – Wired-OR for ECL A f B g – CMOS ? Chap 1. Fundamentals. 36

CMOS Transistor Stuck-ON IDDQ ? 0 stuck-on * Transistor stuck-on may cause ambiguous logic level. –depends on the relative impedances of the pull-up & pull-down networks * When input is low, both P and N transistors are conducting causing increased quiescent current, called IDDQ fault. Chap 1. Fundamentals. 37

CMOS Transistor Stuck-OPEN stuck-open 0 ? = previous state * Transistor stuck-open may cause output floating. Chap 1. Fundamentals. 38

CMOS Transistor Stuck-OPEN (cont'd) Initialization vector 0 1 stuck-open 1 0/0 0 memory behaviour • Can turn the circuit into a sequential one • Stuck-open faults require two-vector tests Chap 1. Fundamentals. 39

Fault Coverage in a CMOS Chip 100 80 60 stuck faults only stuck and open faults 40 20 0 1000 2000 3000 Test Vectors Chap 1. Fundamentals. 40

Summary of Stuck-Open Faults * First report: Wadsack, Bell Syst. Tech. J. , 1978 * Recent results: Woodhall, et al, ITC-87 Experiment with 1 -micron CMOS chips: – 4552 chips passed parametric test – 1255 chips (27. 57%) failed tests for stuck-at faults – 44 chips (0. 97%) failed tests for stuck-open faults – 4 chips with stuck-open faults passed tests for stuck-at faults Conclusion – Stuck-at faults are about 29 times more frequent than stuck-open faults – About 91% of chips with stuck-open faults may also have stuck-at faults – Faulty chips escaping tests for stuck-at faults = 0. 121% Chap 1. Fundamentals. 41

Functional Faults * Fault effects modeled at a higher level than logic for function modules, such as Decoders Multiplexers Adders Counters RAMs ROMs Chap 1. Fundamentals. 42

Functional Faults of Decoder f(Li/Lj): Instead of line Li, Line Lj is selected f(Li/Li+Lj ): In addition to Li, Lj is selected f(Li/0): None of the lines are selected A B 2 -bit Decoder AB AB Chap 1. Fundamentals. 43

Memory Faults • Parametric Faults – – Output Levels 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 – Pattern-Sensitive Faults Chap 1. Fundamentals. 44

Memory Faults • Pattern-sensitive faults: the presence of a faulty signal depends on the signal values of the nearby points – Most common in DRAMs 0 0 d b 0 a=b=0 => d=0 a=b=1 => d=1 • Adjacent cell coupling faults – Pattern sensitivity between a pair of cells Chap 1. Fundamentals. 45

Memory Testing * Test sequences can be derived without much difficulty for each specific fault. However, the length of the test sequence can be prohibitive. e. g. A pattern sensitive test is for an n -bit RAM. 5 n 2 long – Testing a 1 -M bit chip at 10 ns per pattern would take 14 hours. – For a 64 -M bit chip it would take 6 years. Chap 1. Fundamentals. 46

PLA Faults * Stuck Faults * Crosspoint Faults Extra/Missing Transistors * Bridging Faults * Break Faults Chap 1. Fundamentals. 47

Stuck Faults in PLA * S-A-0 & S-A-1 on inputs, input invertors, product lines, and outputs * Easy to simulate in gate model Gate-level representation A B C f 1 f 2 P 1 A B C P 1 f 1 P 2 AND-Array OR-Array f 2 P 2 Chap 1. Fundamentals. 48

Missing Crosspoint Faults in PLA * Missing crosspoint in AND-array -- Growth fault * Missing crosspoint in OR-array -- Disappearance fault Equivalent stuck fault representation A B C A f 1 f 2 Growth B C s-a-1 s-a-0 f 1 f 2 Disappearance Chap 1. Fundamentals. 49

Extra Crosspoint Faults in PLA * Extra crosspoint in AND-array -- Shrinkage or disappearance fault * Extra crosspoint in OR-array -- Appearance fault Equivalent stuck fault representation A B C f 1 f 2 Disapp. Shrinkage "1" "0" Appearance Chap 1. Fundamentals. 50

Bridging Faults in PLA * Shorting of adjacent lines (layout dependent) * Faulty value identical on shorted lines * Faulty value AND/OR function of shorted signals * A large number of bridging faults map into stuck or crosspoint faults Chap 1. Fundamentals. 51

Summary of PLA Faults * Crosspoint Faults - 80 ~ 85% covered by stuck-fault tests - Layout-dependence in folded PLA * Bridging Faults - 99% covered by stuck-fault tests - Layout-dependence in all PLA (Ref: Agrawal & Johnson, ICCD-86) Chap 1. Fundamentals. 52

Why Delay Testing? * There are defects on the chip which allows it to pass the DC stuck-fault testing, but causes it to fail when operated at system speed. – A chip may pass testing under 1 MHz operation but not under 10 MHz Chap 1. Fundamentals. 53

Gate-Delay-Fault • Slow to rise, slow to fall – x is slow to rise when channel resistance R 1 is abnormally high VDD R 1 X X L ---> H CL Rp: channel resistance Chap 1. Fundamentals. 54

Gate-Delay-Fault slow * Disadvantage: Delay faults resulting from the sum of several small incremental delay defects may not be detected. Chap 1. Fundamentals. 55

Path-Delay-Fault • Propagation delay of the path exceeds the clock interval. • The number of paths grows exponentially with the number of gates. Chap 1. Fundamentals. 56

State Transition Graph * Each state transition is associated with 4 tuple: (source state, input, output, destination state) S 1 I 2/O 2 I 1/O 1 S 2 S 3 Chap 1. Fundamentals. 57

Single State Transition Fault Model * A fault causes a single state transition to a wrong destination state. S 1 I/O S 2 I/O S 3 Chap 1. Fundamentals. 58

Single State Transition Fault Model (cont'd) * Has M(N-1) faults for a M-transition N-state machine * Is modeled in the state transition level and independent of implementation * Is dominated by most physical faults for all possible implementation Chap 1. Fundamentals. 59

Transition Faults in State Graph * Any irredundant physical faults will cause some changes in the state graph. – Single or multiple transitions will be corrupted. – A transition is corrupted if its: (a) Output label is changed, or (b) Destination state is changed, or (c) Both (a) and (b). – A test to type (b) fault will detect type (a) and (c) faults. Chap 1. Fundamentals. 60

Test for a State Transition Fault * A test sequence for a fault causing a type (b) corruption on a transition consists of three subsequences: (1)Initialization sequence (2) Input label of the faulty transition (3) State-pair differentiating sequence between good and faulty states CS I 1'/O 1' S 1 I 1/O 1 S 2' S 2 S 3' S 3 I'/O' Sn' (1) (2) (3) I/O Sn Chap 1. Fundamentals. 61

Test for a State Transition Fault(cont'd) * Subsequences (1) and (2) already produce faulty output responses if the output label of the target transition is corrupted. * Only need to generate tests for the state transition faults causing wrong destination states. Chap 1. Fundamentals. 62

Multiple-State-Transition (MST) Faults * A test sequence that detects all MST faults detects all irredundant physical fault. * A machine of M transitions and N states has: NM -1 MST faults M(N-1) SST faults Chap 1. Fundamentals. 63

Why Logical Fault Modeling • Fault analysis on logic rather than physical problem – Complexity is reduced • Technology independent – Same fault model is applicable to many technologies – Testing and diagnosis methods remain valid despite changes in technology • Tests derived may be used for physical faults whose effect on circuit behavior is not completely understood or too complex to be analyzed • Stuck-at fault: The most popular logical fault model Chap 1. Fundamentals. 64

Fault Detection • A test(vector) t detects a fault f iff z (t ) * z f (t ) = 1 – t detects f <=> • Example X 1 Z 1 x s-a-1 X 2 Z 2 X 3 Z 1=X 1 X 2 Z 2=X 2 X 3 Z 1 f =X 1 Z 2 f =X 2 X 3 The test 100 detects f because z 1(100)=0 while z 1 f (100)=1 Chap 1. Fundamentals. 65

Sensitization X 1 1 X 2 0 G 1 1 G 3 X 3 1 1 G 2 s-a-1 0/1 G 4 0/1 z 0/1 X 4 1 z (1011)=0 zf (1011)=1 1011 detects the fault f (G 2 stuck-at 1) v/vf : v = signal value in the fault free circuit vf = signal value in the faulty circuit Chap 1. Fundamentals. 66

Sensitization • A test t that detects a fault f – Activates f (or generate a fault effect) by creating different v and vf values at the site of the fault – Propagates the error to a primary output w by making all the lines along at least one path between the fault site and w have different v and vf values • A line whose value in the test changes in the presence of the fault f is said to be sensitized to the fault f by the test • A path composed of sensitized lines is called a sensitized path Chap 1. Fundamentals. 67

Detectability • A fault f is said to be detectable if there exists a test t that detects f ; otherwise, f is an undetectable fault • For an undetectable fault f z f ( x ) = z( x ) – No test can simultaneously activate f and create a sensitized path to a primary output Chap 1. Fundamentals. 68

Undetectable Fault a G 1 s-a-0 b x z c • G 1 output stuck-at-0 fault is undetectable – Undetectable faults do not change the function of the circuit – The related circuit can be deleted to simplify the circuit Chap 1. Fundamentals. 69

Test Set • Complete detection test set: A set of tests that detect any detectable faults in a class of faults • The quality of a test set is measured by fault coverage • Fault coverage: Fraction of faults that are detected by a test set • The fault coverage can be determined by fault simulation – >95% is typically required for single stuck-at fault model – >99. 9% in IBM Chap 1. Fundamentals. 70

Typical Test Generation Flow Select target fault No more faults Generate test for target Done Fault simulate Discard detected faults Chap 1. Fundamentals. 71

Fault Equivalence • A test t distinguishes between faults a and b if • Two 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. – for all t – No test can distinguish between a and b – Any test which detects one of them detects all of them Chap 1. Fundamentals. 72

Fault Equivalence • AND gate: all s-a-0 faults are equivalent • OR gate: all s-a-1 faults are equivalent • NAND gate: all the input s-a-0 faults and the output s-a-1 faults are equivalent • NOR gate: all input s-a-1 faults and the output s-a-0 faults are equivalent • Inverter: input s-a-1 and output s-a-0 are equivalent input s-a-0 and output s-a-1 are equivalent Chap 1. Fundamentals. 73

Equivalence Fault Collapsing • n+2 instead of 2 n+2 faults need to be considered for n-input gates s-a-1 s-a-1 s-a-0 s-a-0 s-a-1 s-a-0 Chap 1. Fundamentals. 74

Equivalence in a Wire A B * Fault equivalence: A sao <---> B sao A sa 1 <---> B sa 1 * B-sa 0 and B-sa 1 need not to be considered. Chap 1. Fundamentals. 75

Fault Equivalence • Two equivalent faults are detected by exactly the same tests s-a-0 x s-a-1 x – three faults shown are equivalent Chap 1. Fundamentals. 76

Fault Dominance • 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. – No need to consider fault b for fault detection Chap 1. Fundamentals. 77

Fault Dominance • • • AND gate: Output s-a-1 dominates any input s-a-1 NAND gate: Output s-a-0 dominates any input s-a-1 OR gate: Output s-a-0 dominates any input s-a-0 NOR gate: Output s-a-1 dominates any input s-a-0 Dominance fault collapsing: The reduction of the set of faults to be analyzed based on dominance relation Chap 1. Fundamentals. 78

Fault Dominance D x A C x E B x • Detect A sa 1: z (t ) Å z f (t ) = (CD Å CE) Å (D Å CE) = D Å CD = 1 Þ (C = 0, D = 1) • Detect C sa 1: z (t ) Å z f (t ) = (CD Å CE) Å (D Å E) = 1 Þ (C = 0, D = 1) or (C = 0, E = 1) C sa 1 --> A sa 1 • Similarly C sa 1 --> B sa 1 C sa 0 --> A sa 0 C sa 0 --> B sa 0 Chap 1. Fundamentals. 79

Equivalence & Dominance in a Single-Gate A B C 1 1 A B C 0 0 0 AB C sa 1 sa 0 00 01 10 11 0 0 0 1 1 * Fault equivalence: A sa 0 <---> B sa 0 <---> C sa 0 * Fault dominance: C sa 1 ---> A sa 1 C sa 1 ---> B sa 1 * A-sa 0 , B-sa 0, and C-sa 1 need not to be considered. Chap 1. Fundamentals. 80

Fault Collapsing • For each n-input gate, we only need to consider n+1 faults Chap 1. Fundamentals. 81

Prime Fault • a is a prime fault if every fault that is dominated by a is also equivalent to a • Representative Set of Prime Fault (RSPF) – A set that consists of exactly one prime fault from each equivalence class of prime faults – True minimal RSPF is difficult to find Chap 1. Fundamentals. 82

Why Fault Collapsing? * Memory & CPU-Time saving ===> To ease the burden for test generation and fault simulation in testing # of total faults equivalent faults prime faults 1 60% 40% Chap 1. Fundamentals. 83

Fault Collapsing for a Combinational Circuit * 30 total faults ==> 12 prime faults Chap 1. Fundamentals. 84

Checkpoint Theorem • Primary-input & Fanout-Branches ==> a sufficient and necessary set of checkpoints in irredundant combinational circuits – In fanout-free combinational circuits, primary inputs are the set of checkpoints • Any test set which detects all signal (multiple) stuck faults on check points will detect all signal (multiple) stuck faults Chap 1. Fundamentals. 85

Fault Collapsing • The set of checkpoint faults can be further collapsed by using equivalence and dominance relation • Example a b d f e h g c – 10 checkpoint faults – a s-a-0 <-> d s-a-0 , c s-a-0 <-> e s-a-0 b s-a-0 -> d -> 0 , b s-a-1 -> d -> 1 – 6 tests are enough Chap 1. Fundamentals. 86

Problem 1. 1 If the yield of good dice is 90%, and we want a defect level not to exceed 0. 1%, what level of testing in terms of fault coverage must be achieved? 1. 2 Given the market entry time verse revenue curves as shown in the figure, fill in the following formula 1 a. Lost Revenue = Total Expected Revenue * [ ]; The answer should be in term of d and w. d is the delay entry, 2 w is the product life. The two market growth rates are the same. 1 b. Given a product with total expected revenue $100 M, product life is 20 months, What is the revenue loss due to the one month late to the market? Chap 1. Fundamentals. 87

1. 3 For state transition fault model, explain why there are M(N-1) faults for a M-transition N-state machine. Similarly explain why there are NM-1 multiple state transition faults. 1. 4 For PLA, there are missing crosspoints and extra crosspoints faults in both AND-plane and OR-plane. Can missing crosspoints be modeled as equivalent stuck-at faults? Why? Can extra crosspoints be modeled as equivalent stuck-at faults? Why? Chap 1. Fundamentals. 88

1. 5 For the PLA design shown in Figure 1, construct the Karnaugh maps for Y 1, Y 2 and Y 3 for each of the following situation. (a) The original PLA; (b) A shrinkage fault caused by an extra connection between bit line b 2 and product line P 4; (c) An appearance fault caused by an extra connection between P 3 and Y 1; (d) A growth fault caused by the missing connection between bit line b 4 and P 3; (e) A disappearance fault caused by the missing connection between P 1 and Y 3; Chap 1. Fundamentals. 89

1. 6 Prove that for combinational circuits faults dominance is a transitive relation, i. e. if f dominates g and g dominates h, then f dominates h. 1. 7 In the following circuit, a. How many single stuck-at faults needed to be considered initially? b. Applying the check point theorem, how many check point faults needed to be considered? c. Using fault dominance and fault equivalence relations to further reduce the number of stuck-at faults? How many remaining faults needed to be considered? a b f j g m c h d i e k Chap 1. Fundamentals. 90

1. 8 In the circuit if any of the following tests detect the fault x 1 s-a-0 ? a. (0, 1, 1, 1) b. (1, 0, 1, 1) c. (1, 1, 0, 1) d. (1, 0, 1, 0) x 1 z x 2 x 3 x 4 Chap 1. Fundamentals. 91