QUIZ
What do these bits represent?
1001 0110
1 Solution It depends on what is being represented! Unsigned integer: …
Signed integer (2’s complement): … 1001 0110 Fraction: …
IBM 437 character: …
Latin-1 character: …
Huffman-compressed text: … R/G/B level of pixel in image: … 2 Audio sample: … Video sample: … QUIZ What do these bits represent as Huffman- compressed text?
1001 0110
3 Solution What do these bits represent as Huffman- compressed text?
1001 0110 … L D ? Bits from the next byte
4 All hail the HARDWARE!
Chapter 4
Gates and Circuits (with some transistors thrown in for good measure) Abstractions and more abstractions …
We are here 6 Abstractions and more abstractions … Computers Made of lots of different circuits (CPU, memory, controllers, etc.)
Circuits Made from gates combined to perform more complicated tasks
Gates Devices that perform basic logical operations on electrical signals. They’re built out of transistors
Transistors Very small electronic switches
7 There are 3 layers of computer abstraction that we examine in this chapter:
Circuits
Gates
Transistors
8 4.2 Gates There 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 expressions Uses Boolean algebra, a mathematical notation for expressing two-valued logic
2. Logic diagrams A graphical representation of a circuit; each gate has its own symbol 3. Truth tables A table showing all possible input value and the associated output 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
Figure 4.1 Various representations of a NOT gate
11 AND Gate
An AND gate accepts two input signals If both are 1, the output is 1; otherwise, the output is 0
Figure 4.2 Various representations of an AND gate
12 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
A AND 0 = 0, irrespective of B A AND 1 = A
16 OR Gate An OR gate accepts two input signals If both are 0, the output is 0; otherwise, the output is 1
Figure 4.3 Various representations of a OR gate
17 XOR Gate If both inputs are the same, the output is 0; otherwise, the output is 1
XOR is called the exclusive OR Pronunciations: zor, ex-or
18 QUIZ: recognize the gate and draw its symbol!
19 Source: AP CS Principles – Course and Exam Descriptions QUIZ: elementary properties
A AND 0 = ? A AND 1 = ? 1 AND B = ? A OR 1 = ? A XOR 0 = ? A XOR 1 = ?
21 XOR Quiz
For the following circuit diagram: • Find the Boolean expression NOT • Find the truth table
X
X =
22 NAND Gate If 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!
(A⋅B)’ (A⋅B)’’ = A⋅B
Rule: Two “bubbles” next to each other cancel out!
24 QUIZ
Can we build an inverter (NOT) from a NAND gate?
25 Solution
Can we build an inverter (NOT) from a NAND gate?
26 NOR Gate If both inputs are 0, the output is 1; otherwise the output is 0
Compare to OR:
27 Quiz
Can we make an OR gate out of a NOR and an INVERTER?
28 Not in text Whatever happened with the inverted XOR?
It’s called XNOR, pronounced [ex-nor]
29 Review of Gates
A NOT gate inverts its single input An AND gate produces 1 iff both input values are 1 An OR gate produces 0 iff both input values are 0 A XOR gate produces 0 iff input values are the same If and only if All inverted gates have the opposite outputs
It’s OK to put the gate representations on your memory-sheet, but you should be able to remember the word descriptions above!
30 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:
Circuits Gates √ Transistors
32 4.3 Constructing Gates Transistor = 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
High voltage, a.k.a. +
Base or gate
Low voltage, a.k.a. -
When 1 is applied When 0 is applied on the base/gate, on the base/gate, the switch closes the switch opens
34 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 Figure 4.8 The connections of a transistor
35 The names of transistor terminals -setting the record straight FYI-
Bipolar Junction Transistor (BJT) Field-Effect Transistor (FET) Collector, Base, Emmiter Drain, Gate, Source
“Mongrel” transistor (our text)
36 The easiest gates to create are NOT, NAND, and NOR
We can explain their operation for any combination of inputs! We do this by replacing the transistors with switches!
37 If there is no path from output to Ground, Vout = 1
If 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
EOL 1/2 42 QUIZ: Review of gates’ operation A NOT gate … An AND gate … An OR gate … A XOR gate … All inverted gates …
43 Quiz For the following circuit diagram: • Find the Boolean expression • Find the truth table
X
X =
44 Quiz For the following circuit diagram: • Find the Boolean expression • Find the truth table
X
X =
45 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.
Remember: connection to Ground means logical zero!
47 4.4 Intro: From gates to circuits We can go either way between gates and circuits:
Analysis Find the truth table for the circuit:
X
48 From gates to circuits Design Find the logic diagram of the circuit described by the following truth table:
1
0 ?
Hint: The table is similar to which of the fundamental gates?
49 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’ + B Draw the circuit diagram:
51 SOLUTION
Write the Boolean expression: X = A’ + B Draw the diagram:
52 4.4 Circuits Combinational circuit The input values explicitly determine the output
Remember: We describe the circuit operations using Boolean expressions Logic diagrams Truth tables
53 Combinational Circuit
AB+AC
Three inputs require 23 = 8 rows to describe all possible input combinations Boolean expression is:
54 Another Combinational Circuit
Boolean expression A(B + C)
Does this truth table look familiar?
55 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!
Null elements A · 0 = 0 A + 1 = 1 Idempotency A · A = A A + A = A Double complement (A’)’ = A
57 De Morgan’s Laws
Null elements A · 0 = 0 A + 1 = 1 Idempotency A · A = A A + A = A Double complement (A’)’ = A
58 DeMorgan’s law applied directly to gates
59 DeMorgan’s law QUIZ
Apply DeMorgan’s Law directly on the gate diagram below to obtain equivalent circuits:
60 DeMorgan’s law QUIZ
Apply DeMorgan’s Law directly on the gate diagram below to obtain equivalent circuits:
61 Individual work (in notebook): Write the circuit forms for all Boolean properties in this table! (left and right)
Null elements A · 0 = 0 A + 1 = 1 Idempotency A · A = A A + A = A Double complement (A’)’ = A
62 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 adder A circuit that computes the sum of two bits and produces the correct Truth table carry bit Do you recognize these outputs?
64 Half Adder
Circuit diagram
Boolean expressions Sum = A ⊕ B Carry = A·B How many transistors are in this circuit?
65 Full Adder This adder takes the Carry-in value into account!
Do you recognize these circuits? 66 Adding multiple bits - “Ripple Carry” Adder -
67 SKIP Multiplexers
68 There are 2 basic types of circuits
Combinational circuit The input values explicitly determine the output
Sequential circuit The output is a function of the input values and the existing state of the circuit
69 4.5 Circuits for Memory a.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.
How many Figure 4.12 An S-R latch transistors are here?
71 The state of an S – R latch The value of X is considered to be the state of the circuit.
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.
Figure 4.12 An S-R latch
72 QUIZ
If X = 1, S = 1, R = 1, what is Y? If X = 0, S = 1, R = 1, what is Y?
Figure 4.12 An S-R latch
73 How the inputs control the outputs
To cause X = 1, make S = 0 (while keeping R = 1).
To cause X = 0, make R = 0 (while keeping S = 1).
Digital lingo: Set means “make 1”, Reset means “make 0”
74 QUIZ
What happens if both S and R are made 0 at the same time?
75 Solution 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 forbidden combination of inputs!
76 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
3rd gen.
4th gen. 5th gen. VVLSI (?) more than 1B
78 Integrated Circuits
Figure 4.13 An SSI chip containing NAND gates
How many VLSI chip: AMD Phenom II CPU transistors are here? contains 768 million transistors 79 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 AMD CEO Lisa Su, May 2017
Xilinx Virtex Ultrascale FPGAs >20 billion
Source: http://en.wikipedia.org/wiki/Transistor_count 80 The most important integrated circuit in any computer is the Central Processing Unit, or CPU
motherboard CPU
Each CPU chip has a large number of pins through which communication takes place in a computer system 81 Each CPU chip has a large number of pins (Pin Grid Array = PGA) through which communication takes place
82 Image source: http://www.digitale-fotografien.com/freie-galerie/sonstiges/prozessor/ Image source: http://www.etech-web.com/bga-reballing.htm Image source: http://www.electronicsweekly.com/news/design/test-and- measurement/boundary-scan-testing-grows-as-bgas-proliferate-2006-01/
Latest packaging technology: Ball Grid Array = BGA
83 The first midterm for this class is next week, during the lab
Homework - see next slide Review packets Bring: 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 QUIZ: Draw the equivalent circuits corresponding to the LHS and RHS of these Boolean algebra properties. Name each property.
86 QUIZ: Why is this circuit called Half Adder?
Boolean expressions Sum = A ⊕ B Carry = 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