Gives more insight into how Boolean difference is used to test circuits

1. Boolean Differences
Examples
Fault Diagnosis and Failure Tolerance
Alexander Kwasi Amoah
Kwame Nkrumah University of Science and Technology
Department of Computer Engineering, Student
4th May 2017

2. Outline
Boolean Difference in Simple Terms Illustration
Example A
 Dealing with faulty primary input
Example B
 Dealing with faulty internal node
Example C
 When faults cannot be detected by any input vector
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3. Boolean Difference in Simple Terms Illustration3

4. Example A - Dealing with faulty primary input
Using Boolean Difference, create a test for the circuit shown below with a fault y SA0
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5. Example A - Solution
 Objective: Find test vectors that can test for SA0 at node y using Boolean
Difference method
 Output of fault-free circuit, f, is given by 𝑓 = 𝑥. 𝑦 + 𝑦. 𝑧
 Output of the faulted circuit (y SA0. i.e. put y = 0) is given as
𝑔 = 𝑥. 𝑦 + 𝑦. 𝑧 = 𝑧
 Next, we compute T as 𝑇 = 𝑓⨁𝑔 = 𝑓𝑔 + 𝑓 𝑔, where T = Test
 : 𝑇 = 𝑥. 𝑦 + 𝑦. 𝑧 𝑧 + 𝑥. 𝑦 + 𝑦. 𝑧 𝑧
 : 𝑇 = 𝑥. 𝑦 𝑦. 𝑧 𝑧 + 𝑥. 𝑦. 𝑧
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6. Example A – Solution : cont.
 : 𝑇 = 𝑥 + 𝑦 𝑦 + 𝑧 𝑧 + 𝑥. 𝑦. 𝑧
 : 𝑇 = 𝑥 + 𝑦 𝑦 + 𝑧 𝑧 + 𝑥. 𝑦. 𝑧
 : 𝑇 = 𝑥 + 𝑦 𝑦𝑧 + 𝑥. 𝑦. 𝑧
 : 𝑇 = 𝑥𝑦𝑧 + 𝑥𝑦 𝑧
 But T must be a 1 for the required test vectors distinguishing between the
two circuits:  𝑇 = 𝑥𝑦𝑧 + 𝑥𝑦 𝑧 = 1
 For y SA0 we let y = 1
 : 𝑇 = 𝑥𝑧 + 𝑥 𝑧 = 1
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7. Example A – Solution : Truth Table for T7
Truth Table for 𝑻 = 𝒙𝒛 + 𝒙 𝒛 T = 1 Note that y = 1
x z 𝒙 𝒛 𝒙𝒛 𝒙 𝒛 𝑻 = 𝒙𝒛 + 𝒙 𝒛 Condition Satisfied Required Test Vector (xyz)
0 0 1 1 0 0 0 No -
0 1 1 0 1 0 1 Yes 011
1 0 0 1 0 1 1 Yes 110
1 1 0 0 0 0 0 No -
Thus a test has been successfully created to test the circuit at SA0 at node y. Meaning
that applying the test vectors identified, (011 and 110), different outputs will be
realized for the two circuits as shown in the figure on next slide

8. Example A – Solution : 0s and 1s Application8

9. Example A – Solution : 0s and 1s Application, cont.9

10. Example B - Dealing with faulty internal node
Using Boolean Difference, create a test for the circuit shown below with a fault w SA0
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11. Example B - Solution
 Objective: Find test vectors that can test for SA0 at node w using Boolean
Difference method
 Output of fault-free circuit, f, is given by 𝑓 = 𝑥. 𝑦 + 𝑦. 𝑧
 Output of the faulted circuit (w SA0. i.e. put 𝑤 = 𝑦. 𝑧 = 0) is given as
𝑔 = 𝑥. 𝑦
 Next, we compute T as 𝑇 = 𝑓⨁𝑔 = 𝑓𝑔 + 𝑓 𝑔, where T = Test
 : 𝑇 = 𝑓⨁𝑔 = 𝑥. 𝑦 + 𝑦. 𝑧 𝑥. 𝑦 + (𝑥. 𝑦 + 𝑦. 𝑧)𝑥. 𝑦
 : 𝑇 = (𝑥. 𝑦)( 𝑦. 𝑧)𝑥. 𝑦 + (𝑥. 𝑦 + 𝑦. 𝑧)𝑥. 𝑦
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12. Example B – Solution : cont.
 : 𝑇 = ( 𝑥 + 𝑦)(𝑦 + 𝑧)𝑥. 𝑦 + (𝑥. 𝑦 + 𝑦. 𝑧)( 𝑥 + 𝑦)
 : 𝑇 = 𝑥. 𝑦. 𝑧 + 𝑦. 𝑧
 : 𝑇 = 𝑦. 𝑧( 𝑥 + 1)
 : 𝑇 = 𝑦. 𝑧
 But T must be a 1 for the required test vectors distinguishing between the
two circuits:  𝑇 = 𝑦. 𝑧 = 1
 For w SA0 we let 𝑤 = 𝑦. 𝑧 = 1
 : 𝑇 = 𝑤 = 𝑦. 𝑧 = 1
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13. Example B – Solution : Truth Table for T13
Truth Table for 𝑻 = 𝒚. 𝒛 T = 1 Note that x = X
y z 𝒚 𝑻 = 𝑦. 𝑧 Condition Satisfied Required Test Vector (xyz)
0 0 1 0 No -
0 1 1 1 Yes X01
1 0 0 0 No -
1 1 0 0 No -
Thus a test has been successfully (Test Vectors = 001 and 101) created to test the
circuit at SA0 at node w. Meaning that applying the test vectors identified, different
outputs will be realized for the two circuits as shown in the figure on next slide

14. Example B – Solution : 0s and 1s Application14

15. Example B – Solution : 0s and 1s Application, cont.15

16. Example C - When faults cannot be detected by any input vector
Using Boolean Difference, create a test for the circuit shown below with a fault z SA0
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17. Example C - Solution
 Objective: Find test vectors that can test for SA0 at node z using Boolean
Difference method
 Output of fault-free circuit, f, is given by
𝑓 = 𝑥. 𝑦 + 𝑥. 𝑦. 𝑧 = 𝑥. 𝑦(1 + 𝑧) = 𝑥. 𝑦
 Output of the faulted circuit (z SA0. i.e. put 𝑧 = 0) is given as 𝑔 = 𝑥. 𝑦
 Next, we compute T as 𝑇 = 𝑓⨁𝑔 = 𝑓𝑔 + 𝑓 𝑔, where T = Test
 : 𝑇 = 𝑓⨁𝑔 = 𝑥. 𝑦 𝑥. 𝑦 + 𝑥. 𝑦 𝑥. 𝑦 = 0
 The condition for testability (𝑓⨁𝑔 = 1) cannot be satisfied, hence, the
fault is undetectable
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18. Example C – Solution : Confirmation18

19. 19 Differential Fault
Fault ListIntersection
Deductive Simulation
Thank you
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Presentation by
Alexander K. Amoah