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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: …
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 …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
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
22
For the following circuit diagram:• Find the Boolean expression• Find the truth table
X
X =
XOR
NOT
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!
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!
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
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: 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
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
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 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?
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
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.
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