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Digital logic: Gates, Truth tables, logic equations

Yipeng Huang

Rutgers University

April 20, 2021

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Table of contents

Announcements

Transistors: The building block of computers

Combinational logicBasic gatesMore-than-2-input gates

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Looking ahead

Class plan

1. PA5 due Monday, 4/26.2. Digital logic. Reading assignment: CS:APP Chapter 4.2. Recommended

reading: Patterson & Hennessy, Computer organization and design, appendixon "The Basics of Logic Design." Available online via Rutgers Libraries.

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Table of contents

Announcements

Transistors: The building block of computers

Combinational logicBasic gatesMore-than-2-input gates

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Computer organizationLayer cake

I SocietyI Human beingsI ApplicationsI AlgorithmsI High-level programming languagesI InterpretersI Low-level programming languagesI CompilersI ArchitecturesI MicroarchitecturesI Sequential/combinational logicI TransistorsI SemiconductorsI Materials scienceI PhysicsI Mathematics

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Why binary

Figure: Source: CS:APP

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To build logic, we need switches

Vacuum tubes a.k.a. valves

Figure: Source: By Stefan Riepl (Quark48) -Self-photographed, CC BY-SA 2.0https://commons.wikimedia.org/w/index.php?curid=14682022

Transistors

Figure: Source: Wikimedia

I The first transistor developed at BellLabs, Murray Hill, New Jeresy

I https://www.bell-labs.com/about/locations/murray-hill-new-jersey-usa/

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MOSFETs

MOS: Metal-oxide-semiconductorI A sandwich of conductor-insulator-semiconductor.

FET: Field-effect transistorI Gate exerts electric field that changes conductivity of semiconductor.

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NMOS, PMOS, CMOS

PMOS: P-type MOS

I positive gate voltage, acts as opencircuit (insulator)

I negative gate voltage, acts as shortcircuit (conductor)

NMOS: N-type MOS

I positive gate voltage, acts as shortcircuit (conductor)

I negative gate voltage, acts as opencircuit (insulator)

CMOS: Complementary MOS

I A combination of NMOS and PMOS to build logical gates such as NOT,AND, OR.

I We’ll go to slides posted in supplementary material to see how they work.

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Combinational vs. sequential logic

Combinational logic

I No internal state nor memoryI Output depends entirely on inputI Examples: NOT, AND, NAND, OR,

NOR, XOR, XNOR gates, decoders,multiplexers.

Sequential logic

I Has internal state (memory)I Output depends on the inputs and

also internal stateI Examples: latches, flip-flops, Mealy

and Moore machines, registers,pipelines, SRAMs.

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Table of contents

Announcements

Transistors: The building block of computers

Combinational logicBasic gatesMore-than-2-input gates

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NOT gate

A A

A A0 11 0

Table: Truth table for NOT gate

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AND gate, NAND gate

AB

AB

A B AB0 0 00 1 01 0 01 1 1

Table: Truth table for AND gate

AB

AB

A B AB0 0 10 1 11 0 11 1 0

Table: Truth table for NAND gate

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OR gate, NOR gate

AB

A + B

A B A + B0 0 00 1 11 0 11 1 1

Table: Truth table for OR gate

AB

A + B

A B A + B0 0 10 1 01 0 01 1 0

Table: Truth table for NOR gate

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XOR gate, XNOR gate

AB

A ⊕ B

A B A ⊕ B0 0 00 1 11 0 11 1 0

Table: Truth table for XOR gate

AB

A ⊕ B

A B A ⊕ B0 0 10 1 01 0 01 1 1

Table: Truth table for XNOR gate

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More-than-2-input AND gate

AB

AB

CABC

A B C ABC0 0 0 00 0 1 00 1 0 00 1 1 01 0 0 01 0 1 01 1 0 01 1 1 1

Table: Truth table for three-input AND gate

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More-than-2-input OR gate

AB

A + B

CA + B + C

A B C A + B + C0 0 0 00 0 1 10 1 0 10 1 1 11 0 0 11 0 1 11 1 0 11 1 1 1

Table: Truth table for three-input AND gate