2. Logic Gates and Logic Circuits
Digital Systems
Section 4
Logic Gates and Logic Circuits

3. NOR Gates Lecture Digital Systems credential: bring a computer
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4. NAND Gates Lecture Digital Systems credential: bring a computer
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wafer :
This can be an hidden slide. I just want to use this to do my own planning.
I have rearranged Culler’s lecture slides slightly and add more slides. This covers everything he covers in his first lecture (and more) but may
We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20.
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs.

5. Exercise: NOR and NAND Gates
Lecture
Digital Systems
Exercise: NOR and NAND Gates
Find the expression for the following circuit and create the corresponding truth table. D E G H
F = (A+BC) + AB A B C D E G H F 1 AB 1
Proof that
(A+BC) + AB = AB ?
credential:
bring a computer
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wafer :
This can be an hidden slide. I just want to use this to do my own planning.
I have rearranged Culler’s lecture slides slightly and add more slides. This covers everything he covers in his first lecture (and more) but may
We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20.
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs.

6. IC Logic Gates 74XX Series
Lecture
Digital Systems
IC Logic Gates 74XX Series
74LS00
74LS32
74LS08

7. Universality of NOR Gates
Lecture
Digital Systems
Universality of NOR Gates
It is possible to implement any logic expression using only NOR gates.

8. Universality of NAND Gates
Lecture
Digital Systems
Universality of NAND Gates
It is possible to implement any logic expression using only NAND gates.

9. Truth Table of Inverter
Lecture
Digital Systems
Summary: Logic Gates
Basic logic functions are realized in logic gates. Each logic gate has a symbol.
The functionality of a basic logic function can be represented through a truth table.
A truth table completely specifies the outputs of the basic logic function for all possible input combinations.
Inverter
Truth Table of Inverter A F
Symbol of Inverter
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We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20.
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs.
The logic operation of an inverter is: F = A’.
As implied by its name, an input of 0 to this gate is inverted to 1, and an input of 1 to 0.

10. Summary: Logic Gates A F B AND Gate
Lecture
Digital Systems
Summary: Logic Gates
AND Gate
Truth Table of AND Gate A B F
Symbol of AND Gate
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We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20·
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs·
The logic operation of an AND Gate is: F = A · B.
If the two input signals are asserted (high), then the output will also be asserted (high).
Otherwise, the output will be de-asserted (low).

11. Summary: Logic Gates A F B OR Gate
Lecture
Digital Systems
Summary: Logic Gates
OR Gate
Truth Table of OR Gate A B F
Symbol of OR Gate
The logic operation of an OR Gate is: F = A + B.
If either of the two input signals are asserted (high), or both of them are, then the output will also be asserted (high).
Otherwise, the output will be de-asserted (low).
Note that if there are n inputs to a gate, then the truth table of the gate will have 2n rows.
credential:
bring a computer
die photo
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This can be an hidden slide· I just want to use this to do my own planning·
I have rearranged Culler’s lecture slides slightly and add more slides· This covers everything he covers in his first lecture (and more) but may
We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20·
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs·

12. Truth Table of NAND Gate
Lecture
Digital Systems
Summary: Logic Gates
NAND Gate
Truth Table of NAND Gate A B F
Symbol of NAND Gate
The logic operation of an NAND Gate is: F = (A · B)’.
It is a combination of an AND Gate and an Inverter.
credential:
bring a computer
die photo
wafer :
This can be an hidden slide· I just want to use this to do my own planning·
I have rearranged Culler’s lecture slides slightly and add more slides· This covers everything he covers in his first lecture (and more) but may
We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20·
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs·

13. Summary: Logic Gates A F B NOR Gate
Lecture
Digital Systems
Summary: Logic Gates
NOR Gate
Truth Table of NOR Gate A B F
Symbol of NOR Gate
The logic operation of an NOR Gate is: F = (A + B)’.
It is a combination of an OR Gate and an Inverter.
credential:
bring a computer
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wafer :
This can be an hidden slide· I just want to use this to do my own planning·
I have rearranged Culler’s lecture slides slightly and add more slides· This covers everything he covers in his first lecture (and more) but may
We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20·
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs·

14. Summary: Logic Gates A F B XOR Gate (Exclusive OR)
Lecture
Digital Systems
Summary: Logic Gates
XOR Gate (Exclusive OR)
Truth Table of XOR Gate A B F
Symbol of XOR Gate
The logic operation of an XOR Gate is: F = A  B = A·B’ + A’·B
A XOR Gate asserts its output only when both inputs differ (one is true, the other is false).
credential:
bring a computer
die photo
wafer :
This can be an hidden slide· I just want to use this to do my own planning·
I have rearranged Culler’s lecture slides slightly and add more slides· This covers everything he covers in his first lecture (and more) but may
We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20·
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs·

15. Summary: Logic Gates F XNOR Gate (Exclusive NOR)
Lecture
Digital Systems
Summary: Logic Gates
XNOR Gate (Exclusive NOR)
Truth Table of XOR Gate F
Symbol of XOR Gate
The logic operation of an XOR Gate is: F = (A  B)’ = A·B + A’·B’
A XNOR Gate asserts its output only when both inputs are identical (whether both are true, or both are false).
XNOR is the negation of XOR.
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This can be an hidden slide· I just want to use this to do my own planning·
I have rearranged Culler’s lecture slides slightly and add more slides· This covers everything he covers in his first lecture (and more) but may
We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20·
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs·

16. Digital Systems
Section 5
Combinatorial Logic

17. Describing Circuit Functionality in Waveforms
Lecture
Digital Systems
Describing Circuit Functionality in Waveforms
Waveforms provide another approach in representing the functionality of a digital circuit.
The value of each input signal is either high (1) or low (0), and for each combination, the output signal is defined.
Waveforms for AND Gate, OR Gate, and Inverter

18. Logic Gates in Waveforms
Lecture
Digital Systems
Logic Gates in Waveforms 1
time F x 1 x y F
time
NOT Gate
OR Gate
AND Gate

19. Lecture
Digital Systems
Example: Waveforms
Given the following waveform, can you find the gate that connects input A, B, and C to the output OUT?
OR Gate A
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We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20·
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs·

20. Example: Toll Booth Controller
Lecture
Digital Systems
Example: Toll Booth Controller
Consider the design of an automatic toll gate controller. The inputs are: payment card sensor (P) and car sensor (S). The outputs are: gate lift signal (L) and gate close signal (C).
If a driver put the valid card near the sensor, the gate should raise. Afterwards, when the car has cleared the gate, it will close again.
How to connect the inputs to the output?
Logic Circuit
Payment card?
Car present?
Lift gate Close gate
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We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20.
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs.
L = P · S
C = S’ A

21. Example: Automatic Lamp Switch
Lecture
Digital Systems
Example: Automatic Lamp Switch
An automatic lamp switch should turn on a lamp with a signal F if someone comes close to a door (motion sensor A is high) and the day is already dark (light sensor B is low).
Design an expression to connect the inputs to the output.
Draw the logic circuit.
F = A · B’ A

22. Example: Seat Belt Warning
Lecture
Digital Systems
Example: Seat Belt Warning
For the design of circuit of seat belt warning light, sensor S = 1 if seat belt fastened, sensor K = 1 if car key is inserted, and sensor P = 1 if person is in seat.
Warning light W should turn on if a person is in seat and the car key is inserted, but the seat belt is not yet fastened.
Capture the Boolean equation and convert it to logic circuit.
W = S’ · K · P A K P S W
Seat Belt Warning

23. Example: Automatic Door Opener
Lecture
Digital Systems
Example: Automatic Door Opener
A door should be automatically open (F = 1) if a person is detected (P = 1), or a hold switch is pressed (H = 1) while there is no people detected (P = 0) . However, the door must be unlocked (C = 0) first using a force key before the door can operate.
Design the logic equation and draw the logic circuit.
F = H’·P·C’ + H·C’ = automatic + manual A F H C P
Automatic Door Opener

24. Lecture
Digital Systems
Logic Design Process
Step 1: Capture the function Create a truth table to describe the desired behavior of the system.
Step 2: Create the logic expression For each output, an equation is assigned in the form of sum of minterms. If possible, the equations may be simplified.
Step 3: Implement as a gate-based circuit For each output, a corresponding circuit is created. Sharing gates among multiple outputs is allowed.

25. Example: Boolean Function
Lecture
Digital Systems
Example: Boolean Function
Find the logic function of the following circuit.
Write down its truth table.
F = A · (B + C’) A B C C’
B+C’
F = A · (B+C’) 1

26. Circuit Representation
Lecture
Digital Systems
Circuit Representation A F
F = A’
Logic Circuit
Boolean
Function
Truth Table
Transition can be made among logic circuit, Boolean function / expression, and truth table.

27. Venn Diagram Representation
Lecture
Digital Systems
Venn Diagram Representation
Venn Diagram is a diagram representing mathematical or logical sets pictorially as circles or closed curves within an enclosing rectangle (the universal set).
Common elements of the sets are represented by the areas of overlap among the circles.

28. Venn Diagram Representation
Lecture
Digital Systems
Venn Diagram Representation

29. Venn Diagram Representation
Lecture
Digital Systems
Venn Diagram Representation
credential:
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This can be an hidden slide. I just want to use this to do my own planning.
I have rearranged Culler’s lecture slides slightly and add more slides. This covers everything he covers in his first lecture (and more) but may
We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20.
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs.

30. Venn Diagram Representation
Lecture
Digital Systems
Venn Diagram Representation
credential:
bring a computer
die photo
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This can be an hidden slide. I just want to use this to do my own planning.
I have rearranged Culler’s lecture slides slightly and add more slides. This covers everything he covers in his first lecture (and more) but may
We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20.
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs.

31. Venn Diagram Representation
Lecture
Digital Systems
Venn Diagram Representation
Constant 1
Constant 0 x x'
Variable x
Variable x’

32. Venn Diagram Representation
Lecture
Digital Systems
Venn Diagram Representation x y
x · y
x + y x y z
x · y’
x · y + z

33. Exercise with Venn Diagram
Lecture
Digital Systems
Exercise with Venn Diagram
Draw and give shade to the Venn diagram of:
a) x·(y+z). b) x·y + x’y’z. x y z x y z

34. Exercise with Venn Diagram
Lecture
Digital Systems
Exercise with Venn Diagram
Express the shaded areas in terms of the variables A, B, and C. A B C A B C A B C
B + A·C
AB’C’ + A’BC
(A+B)·C’

35. Exercise with Venn Diagram
Lecture
Digital Systems
Exercise with Venn Diagram
Verify xy + x’z + yz = xy + x’z by using Venn diagram. x y z
x·y x y z
x·y x y z
x’·z y x z
y·z x y z
x’·z y z x
x·y + x’·z + y·z y z x
x·y + x’·z

36. Exercise with Venn Diagram
Lecture
Digital Systems
Exercise with Venn Diagram
Using Venn diagram, prove that (A + B)(A’ + B’) = AB’ + A’B.
Using De Morgan’s theorem, find also the complement of (A + B)(A’ + B’).

37. Lecture
Digital Systems
Homework 3 (1/2)
A house alarm (F) should sound if the alarm (A) is enabled and either the window is shaken (S) or the door is opened (D). Determine the Boolean equation F of this system.
An expansion project (P) will be conducted if it is approved by the manager (G). If the project is not approved by the manager, the project can still be conducted if it is approved by either construction supervisor (C) only or maintenance supervisor (M) only. Determine the Boolean equation P of this system.
See next slide.

38. Lecture
Digital Systems
Homework 3 (2/2)
Determine the expression that defines the shaded area of the following Venn diagram. U W V
Draw a 4-set Venn diagram and shade the area corresponding to the expression AB + ACD’.
credential:
bring a computer
die photo
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This can be an hidden slide. I just want to use this to do my own planning.
I have rearranged Culler’s lecture slides slightly and add more slides. This covers everything he covers in his first lecture (and more) but may
We will save the fun part, “ Levels of Organization,” at the end (so student can stay awake): I will show the internal stricture of the SS10/20.
Notes to Patterson: You may want to edit the slides in your section or add extra slides to taylor your needs.
Please write your Class number after your Student ID.
Deadline: 1 day before class.
Monday, 25 September 2017 (Class 2).
Tuesday, 26 September 2017 (Class 1).