QUIZ

What do these bits represent?

1001 0110

1

Fraction: …

SolutionIt depends on what is being represented!

1001 0110

2

Unsigned integer: …

Signed integer (2’s complement): …

IBM 437 character: …

Latin-1 character: …

Huffman-compressed text: …

R/G/B level of pixel in image: …Audio sample: …Video sample: …

QUIZWhat do these bits represent as Huffman-compressed text?

1001 0110

3

Solution

What do these bits represent as Huffman-compressed text?

1001 0110 …

4

L D ? Bits from the next byte

Chapter 4

Gates and Circuits (with some transistors thrown in for

good measure)

All hail the HARDWARE!

Abstractions and more abstractions …

6We are here

Abstractions and more abstractions …ComputersMade of lots of different circuits (CPU, memory, controllers, etc.)

CircuitsMade from gates combined to perform more complicated tasks

GatesDevices that perform basic logical operations on electrical

signals. They’re built out of transistors

TransistorsVery small electronic switches

7

There are 3 layers of computer abstraction that we examine in this

chapter:

8

Circuits

Gates

Transistors

4.2 GatesThere are six basic gates:

– NOT– AND– OR– XOR– NAND– NOR

Real-life logic diagrams are black and white with gates distinguished only by their shape

We use color for emphasis (and fun) in this text

9

How do we describe the behavior of gates and circuits?

1. Boolean expressionsUses Boolean algebra, a mathematical notation for expressing two-valued logic

2. Logic diagramsA graphical representation of a circuit; each gate has itsown symbol

3. Truth tablesA table showing all possible input value and the associatedoutput values

10

NOT Gate (a.k.a. inverter)

A NOT gate accepts one input signal (0 or 1) and returns the opposite signal as output

11

Figure 4.1 Various representations of a NOT gate

AND Gate

An AND gate accepts two input signalsIf both are 1, the output is 1; otherwise, the output is 0

12

Figure 4.2 Various representations of an AND gate

AND gate quiz #1

The inputs to an AND gate have the values A = 1 and B = 0. What is the output?

13

AND gate quiz #2

The input B to an AND gate has the value B = 0. What is the output?

14

AND gate quiz #3

The input B to an AND gate has the value B = 1. What is the output?

15

Conclusion: the AND gate has the following interesting properties

16

A AND 0 = 0, irrespective of BA AND 1 = A

OR GateAn OR gate accepts two input signalsIf both are 0, the output is 0; otherwise,the output is 1

17

Figure 4.3 Various representations of a OR gate

XOR Gate

18

XOR is called the exclusive ORPronunciations: zor, ex-or

If both inputs are the same, the output is 0;

otherwise, the output is 1

QUIZ: recognize the gate and draw its symbol!

19

Source: AP CS Principles – Course and Exam Descriptions

QUIZ: elementary properties

21

A AND 0 = ?A AND 1 = ?1 AND B = ?A OR 1 = ?A XOR 0 = ?A XOR 1 = ?

Quiz

22

For the following circuit diagram:• Find the Boolean expression• Find the truth table

X

X =

XOR

NOT

NAND GateIf both inputs are 1, the output is 0;otherwise, the output is 1

Compare to AND:

We can build an AND gate out of a NAND and an INVERTER!

24

(A⋅B)’ (A⋅B)’’ = A⋅B

Rule: Two “bubbles” next to each other cancel out!

QUIZ

Can we build an inverter (NOT) from a NANDgate?

25

Solution

Can we build an inverter (NOT) from a NANDgate?

26

NOR Gate

27

If both inputs are 0, the output is 1;otherwise the output is 0

Compare to OR:

Quiz

Can we make an OR gate out of a NOR and an INVERTER?

28

Whatever happened withthe inverted XOR?

It’s called XNOR, pronounced [ex-nor]

29

Not in text

Review of Gates

A NOT gate inverts its single input An AND gate produces 1 iff both input values are 1An OR gate produces 0 iff both input values are 0A XOR gate produces 0 iff input values are the same

All inverted gates have the opposite outputs

30

If and only if

It’s OK to put the gate representations on your memory-sheet, but you should be able to remember the word descriptions above!

Gates with More Inputs

A three-input AND gate produces an output of 1 iff all input values are 1

31

Remember: There are 3 layers of computer abstraction that we

examine in this chapter:

32

Circuits

Gates

Transistors√

4.3 Constructing GatesTransistor = device that acts either as a wire that

conducts electricity or as a resistor that blocks the flow of electricity, depending on the voltage level of an input signal

Made of a semiconductor material• Neither good conductor of electricity nor a good insulator …• …but with a little help can be either!

Acts like a switch, but w/o moving parts:• Switch open• Switch closed

33

How transistors operate as switches

34

High voltage, a.k.a. +

Low voltage, a.k.a. -

When 1 is applied on the base/gate, the switch closes

When 0 is applied on the base/gate, the switch opens

Base or gate

Transistors

A transistor has three terminals:– A collector/source, typically

connected to the positive terminal of a power source (5 volts, 3.5 volts, etc.)

– An emitter/drain, typically connected to the “ground” (0 volts)

– A base/gate, which controls the flow of current between source and emitter

35

Figure 4.8 The connections of a transistor

The names of transistor terminals-setting the record straight FYI-

36

Bipolar Junction Transistor (BJT)Collector, Base, Emmiter

Field-Effect Transistor (FET)Drain, Gate, Source

“Mongrel” transistor (our text)

The easiest gates to create are NOT, NAND, and NOR

37

We can explain their operation for any combination of inputs!We do this by replacing the transistors with switches!

If there is no path from output to Ground, Vout = 1If there is a path from output to Ground, Vout = 0

38

Non-inverted gates:The AND gate is obtained as a NAND followed by an

inverter. Draw its transistor diagram!

39

QUIZ What gate do we have below?

40

QUIZ What gate do we have below?

41

Quick work for next time:

Read pp.93-104 of text

Answer in notebook the end-of-chapter questions: 1 – 11, 18 – 29, 59

42EOL 1/2

QUIZ: Review of gates’ operationA NOT gate …

An AND gate …

An OR gate …

A XOR gate …

All inverted gates …

43

Quiz

44

For the following circuit diagram:• Find the Boolean expression• Find the truth table

X

X =

Quiz

45

For the following circuit diagram:• Find the Boolean expression• Find the truth table

X

X =

QUIZ: What gate is implemented below?

46

QUIZ: Based on the switching behavior of transistors, verify the behavior of the OR gate when A=V1=1 and B=V2=0.

47

Remember: connection to Ground means

logical zero!

4.4 Intro: From gates to circuits

48

We can go either way between gates and circuits:

AnalysisFind the truth table for the circuit:

X

From gates to circuits

49

DesignFind the logic diagram of the circuit described by the following truth table:

Hint: The table is similar to which of the fundamental gates?

1

0?

SOLUTION

Having only one 0 in the output column, the circuit most resembles the OR gate!It is different from the OR gate only in this respect: …

Write the Boolean expression:

50

SOLUTION

Write the Boolean expression: X = A’ + BDraw the circuit diagram:

51

SOLUTION

Write the Boolean expression: X = A’ + BDraw the diagram:

52

4.4 CircuitsCombinational circuitThe input values explicitly determine the output

Remember: We describe the circuit operations using Boolean expressionsLogic diagramsTruth tables

53

Combinational Circuit

Three inputs require 23 = 8 rows to describe all possible input combinations

Boolean expression is:

54

AB+AC

Another Combinational Circuit

Boolean expression A(B + C)

55

Does this truth table look familiar?

Circuit equivalence: Two circuits that produce the same output for identical input are called equivalent

Boolean algebra allows us to apply provable mathematical principles to help design circuits:

A(B + C) = AB + AC (distributive law) so circuits must be equivalent

56

Each property (law) of Boolean Algebra translates directly into a property of equivalent circuits!

57

Null elements A · 0 = 0 A + 1 = 1Idempotency A · A = A A + A = ADouble complement (A’)’ = A

De Morgan’s Laws

58

Null elements A · 0 = 0 A + 1 = 1Idempotency A · A = A A + A = ADouble complement (A’)’ = A

DeMorgan’s law applied directly to gates

59

DeMorgan’s law QUIZ

60

Apply DeMorgan’s Law directly on the gate diagram below to obtain equivalent circuits:

DeMorgan’s law QUIZ

61

Apply DeMorgan’s Law directly on the gate diagram below to obtain equivalent circuits:

Individual work (in notebook): Write the circuit forms for all Boolean properties in this table! (left and right)

62

Null elements A · 0 = 0 A + 1 = 1Idempotency A · A = A A + A = ADouble complement (A’)’ = A

Very Useful Combinational Circuit:the Adder

At the digital logic level, addition is performed in binary

Addition operations are carried out by special circuits called adders

63

Half-Adder truth table

The result of adding two binary digits could produce a carry value

Recall that 1 + 1 = 10 in base two

Half adderA circuit that computes the

sum of two bits and produces the correct carry bit

64

Truth table

Do you recognize these outputs?

Half Adder

Circuit diagram

Boolean expressions

Sum = A ⊕ BCarry = A·B

65

How many transistors are in

this circuit?

Full AdderThis adder takes the Carry-in value into account!

66

Do you recognize these circuits?

Adding multiple bits- “Ripple Carry” Adder -

67

SKIP Multiplexers

68

There are 2 basic types of circuits

Combinational circuitThe input values explicitly determine the output

Sequential circuitThe output is a function of the input values and the

existing state of the circuit

69

4.5 Circuits for Memorya.k.a. Sequential Circuits

A sequential circuit is one whose output depends not only on the current values of its inputs, but also on the past sequence of those inputs (history).

It can be used to store information, i.e. as memory.

70

The S – R latch

There are several ways to build S – R latches using various kinds of gates, but there’s always feedback.

71

Figure 4.12 An S-R latchHow many

transistors are here?

The state of an S – R latch

If X is 1, we say that the circuit is storing a 1; if X is 0, the circuit is storing a 0

The design guarantees that the two outputs X and Y are (almost always) complements of each other.

72

Figure 4.12 An S-R latch

The value of X is considered to be the state of the circuit.

QUIZ

73

Figure 4.12 An S-R latch

If X = 1, S = 1, R = 1, what is Y?

If X = 0, S = 1, R = 1, what is Y?

Digital lingo: Set means “make 1”, Reset means “make 0”

74

To cause X = 1, make S = 0 (while keeping R = 1).

To cause X = 0, make R = 0 (while keeping S = 1).

How the inputs control the outputs

QUIZ

What happens if both S and R are made 0 at the same time?

75

If both S and R are made 0 at the same time, the outputs X and Y are both 1.This goes against the interpretation of X and Y being opposites!For this reason, it is considered a forbiddencombination of inputs!

76

Solution

Integrated Circuit (a.k.a. IC or chip) = A piece of silicon on which multiple gates have been embedded

Silicon pieces are mounted on a plastic or ceramic package with pins along the edges that can be soldered onto circuit boards or inserted into appropriate sockets

77

Integrated Circuits

Integrated circuits (IC) are classified by the number of gates contained in them

78

VVLSI (?) more than 1B

3rd gen.

4th gen.5th gen.

Integrated Circuits

79

Figure 4.13 An SSI chip containing NAND gates

VLSI chip: AMD Phenom II CPU contains 768 million transistors

How many transistors are here?

80

As of 2017, the highest transistor counts in commercially available CPUs are:• 7.2 billion in Intel's Xeon E5 (22 cores).• >10 billion in Oracle’s Spark M7 and M8 (32 cores)• 19.2 billion in AMD’s Epyc (32 cores)

Nvidia GV100 GPU: 21.1 billion

Xilinx Virtex Ultrascale FPGAs >20 billion

Source: http://en.wikipedia.org/wiki/Transistor_count

AMD CEO Lisa Su, May 2017

The most important integrated circuit in any computer is the Central Processing Unit, or CPU

Each CPU chip has a large number of pins through which communication takes place in a computer system

81

motherboard CPU

Each CPU chip has a large number of pins (Pin Grid Array = PGA) through which communication takes place

82Image source: http://www.digitale-fotografien.com/freie-galerie/sonstiges/prozessor/

Latest packaging technology: Ball Grid Array = BGA

83

Image source: http://www.electronicsweekly.com/news/design/test-and-measurement/boundary-scan-testing-grows-as-bgas-proliferate-2006-01/

Image source: http://www.etech-web.com/bga-reballing.htm

The first midterm for this class is next week, during the lab

Homework - see next slideReview packetsBring: Notebook, Memory Sheet

84

Homework for ch.4 is due Tue, Oct.9(after exam)

End-of-chapter 37, 48, 49, 50, 62, 66, 70, 72, 73

Thought question #4

Notes: • At #48, “grounded” means “connected directly to the

ground”.• In order to solve #50(e), you need to solve #73 first!

85

86

QUIZ: Draw the equivalent circuits corresponding to the LHS and RHS of these Boolean algebra properties.

Name each property.

QUIZ: Why is this circuit called Half Adder?

Boolean expressions

Sum = A ⊕ BCarry = A·B

87

88

Chapter review questions

• Identify the basic gates and describe the behavior of each– Includes the inverted gates!

• Describe how gates are implemented using transistors

• Combine basic gates into circuits– Analysis vs. design

• Describe the behavior of a gate or circuit using Boolean expressions, truth tables, and logic diagrams

89

Chapter review questions

• Write deMorgan’s laws in Boolean form and in circuit form

• Compare and contrast a half adder and a full adder

• Explain how an S-R latch operates• Describe the characteristics of the four (five?)

generations of integrated circuits• What is the CPU?

90