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Electrical EngineeringTables, Standards, Formulas

EUROPA-TECHNICAL BOOK SERIESfor the electrotechnical, electronic and information technology trades

VERLAG EUROPA-LEHRMITTEL · Nourney, Vollmer GmbH & Co. KGDüsselberger Straße 23 · 42781 Haan-Gruiten · Germany

Europa no. 30337

Prepared and revised by teachers in vocational colleges andengineers from the production industry (see next page).

2nd English edition

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Original title: Tabellenbuch Elektrotechnik, 26nd edition, authors:

Häberle, Gregor Dr.-Ing., department manager Tettnang, Germany

Häberle, Heinz Dipl.-Gwl., VDE Kressbronn, Germany

Jöckel, Hans-Walter Dipl.-Ing. (FH), Oberstudienrat Friedrichshafen, Germany

Krall, Rudolf Dipl.-Päd. Ing., Berufsschuloberlehrer St. Leonhard, Austria

Schiemann, Bernd Dipl.-Ing. Durbach, Germany

Schmitt, Siegfried staatl. gepr. Techniker, Techn. Oberlehrer Bad Bergzabern, Germany

Tkotz, Klaus Dipl.-Ing. (FH) Kronach, Germany

Head of the working group:Dr.-Ing. Häberle, Tettnang, Germany

Graphic design:Design office of the publisher Verlag Europa-Lehrmittel, Ostfildern, Germany

DIN Deutsches Institut für Normung e.V. and VDE Verband der Elektrotechnik Elektronik Informationstechnik e.V. have authorised the citation of excerpts from DIN standards classifi ed as VDE for the registered limited edition of this book, approval no. 302.015. For additional reproductions or editions, a separate authorisation will be required.

Please verify that you are using the most recent version of a standard prior to its application. Standards can be ordered from VDE-VERLAG GmbH, Birmarckstr. 33, 10625 Berlin, www.vde-verlag.de.

2nd English edition 2015Print 5 4 3 2 1All printed versions of this edition may be used concurrently since they are unchanged, except for some corrections to typographical errors.

ISBN 978-3-8085-3270-6

All rights reserved. This publication is protected under copyright law. Any use other than those permitted by the law must be approved in writing by the publisher.

© 2015 by Verlag Europa-Lehrmittel, Nourney, Vollmer GmbH & Co. KG, 42781 Haan-Gruiten, Germanyhttp://www.europa-lehrmittel.de.

Translation: Technische Übersetzungen Eva Schwarz, 76879 Ottersheim, Germanywww.technische-uebersetzungen-eva-schwarz.deMany thanks to Hans Jurgen Porten for his technical review of the translation and his valuable advice.

Typesetting: Satz+Layout Werkstatt Kluth GmbH, 50374 Erftstadt, Germany

Print: M.P. Media-Print Informationstechnologie GmbH, 33100 Paderborn, Germany

MATHEMATICS, PHYSICS, THEORY OF CIRCUITS, COMPONENTS 11 … 66

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TECHNICAL DOCUMENTATION, MEASURING 67 … 120

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ELECTRICAL INSTALLATIONS 121 … 198

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SAFETY, ENERGY SUPPLY 199 … 286

SESE

INFORMATION AND COMMUNICATION TECHNOLOGY SYSTEMS 287 … 334

ICIC

AUTOMATION, DRIVE AND CONTROL SYSTEMS 335 … 424

ACAC

MATERIALS, CONNECTING, JOINING AND BONDING 425 … 458

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THE COMPANY AND ITS ENVIRONMENT, ENVIRONMENTAL TECHNOLOGY, ANNEX 459 … 527

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Preface to the 2nd English edition

This edition is based on the 26th German edition of “Tabellenbuch Elektrotechnik”, a leading compendium in German-speaking countries. The English edition addresses professionals in the various fields of electrical engineering, such as power and building engineering, field engineering, automation systems, machinery, drive systems, components and other electronic systems. This book is intended to • prepare professionals for an activity in an international environment and • help make the world’s leading work processes and standards known outside of the German- speaking region.

Despite the harmonisation of the most important European standards, local regulations may differ slightly from German standards under certain circumstances, which means that where safety matters are con-cerned, the user has to check whether any other local regulations exist.

The book is divided into the following main sections focusing on the specified subjects:

Formula symbols, units and quantities, mathematical symbols, exponents, unit prefi xes, logarithmic unit decibel, force, moment of force, motion rules, work, power, heat, charge, voltage, current, resistance, electric and magnetic fi elds, alternating quantities, switching capacitors and coils, three-phase current, diodes, transistors, IGBTs, thyristors, magnetic fi eld-dependent and photoelectronic components.

Technical drawing, dimensioning, circuit diagrams, circuit symbols, comparison of circuit symbols, preparing documentation, measuring instruments and systems, meas-uring categories, measurement in electrical installations, power meters, bid irectional watt meters, ripple control receivers, oscilloscopes, measuring with sensors.

Working in electrical installations, laying of cables, cable routing, installation circuits, intercom systems, types of dimmers, electrical installation with low-voltage halogen lamps, fi eld-reducing electrical installation, building management and automation, project design based on KNX, house connection, foundation earth electrode, elec-trical installations in residential buildings, calculation of circuit loading, ampacity of cables and wires, lighting engineering, LED lighting.

First aid, workplace health and safety, personal protective equipment, signs for accident prevention, differential current devices, basic protection, fault protection, additional protection, conductors for protective measures, types of power stations, insulator classes, transformers, overhead power cables, buried cables, private power generating systems, smart grids, fuel cells, primary cells, accumulators, UPS systems, electromagnetic compatibility EMC, lighting protection, quality of power supply, harmonics, power factor correction, smoke alarms, AFDD, AFCI, security and monitoring devices, energy conservation directive, energy effi ciency, house-hold appliances, electricity tariffs.

Number systems, codes, Boolean algebra, fl ip-fl ops, D/A converters, A/D con-verters, modulation and demodulation, IT networks, components for data networks, Ethernet, wireless LAN, AS-i bus system, interbus, PROFIBUS, identifi cation systems, connection to the telephone network, internet, aerial systems, satellite systems, safe communication across different fi eld busses.

Rectifi er, switch-mode power supplies, multivibrators, control relays, program-mable logic controllers PLC, word processing in PLCs, control engineering, auxiliary circuits, sequence control with GRAFCET, contactors, motor protection, electrical equipment of machines, automatic control engineering, three-phase motors, single-phase A.C. motors, D.C. motors, effi ciency of drive systems, servomotors, micro-motors, linear drives, design of automation systems, EC Machinery Directive.

Periodic table, specifi c material values, steel standardisation, magnetic materials, insulators, cables and wires, buried cables, connectors, solderless connection technology, ISO threads, screws, bolts and nuts, dowels and plugs.

Organisational structures of companies, teamwork, job planning, cost accounting and key numbers, skills of electrical specialists, realisation of projects, confl ict management, communication with customers, environmental terms, hazardous materials, electronic waste products, standards, glossary, subject index.

We have integrated modifi cations of standards, e.g. in the terms taken from DIN VDE 0100-200. It should be noted in general that standards allow different ways of repre-sentation, e.g. DIN EN 610892 (Documents in Electrical Engineering, Rules) allows the representation of electricity branching with or without a point. As in professional practice, we have taken advantage of this freedom also in the book.

The publisher and authors would be grateful for any suggestions and constructive comments.

Summer 2015 The authors’ working group

• Section M:mathematics, physics, theory of circuits, components

• Section TM:technical documentation,measuring

• Section EI:electrical installations

• Section SE:safety, energy supply

• Section IC:information and communication technologysystems

• Section AC:automation, drive and controlsystems

• Section MC:materials, connec-tion, joining and bonding

• Section CE:the company and its environment, environmental technology, annex

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Contents

Section M:

Mathematics, Physics, Theory of Circuits,

Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Symbols in this Book . . . . . . . . . . . . . . . . . . . . . 12

Subscripts and Signs for Formula Symbols in this Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

International Formula Symbols . . . . . . . . . . . . 14

Quantities and Units . . . . . . . . . . . . . . . . . . . . . 15

Mathematical Symbols . . . . . . . . . . . . . . . . . . . 17

Exponents, Unit Prefi xes, Logarithms, Calculations According to the Rule of Three . . 18

Logarithmic Unit Decibel . . . . . . . . . . . . . . . . . 19

Angles, Trigonometric Functions, Percentage Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Relationships Between Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Lengths and Areas . . . . . . . . . . . . . . . . . . . . . . . 22

Body and Mass . . . . . . . . . . . . . . . . . . . . . . . . . 23

Mass, Force, Pressure, Moment of Force . . . . 24

Motion Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Mechanical Work, Mechanical Power, Energy 26

Transmissions . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Pulleys, Wedges, Winches . . . . . . . . . . . . . . . . 28

Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Charge, Voltage, Electric Current, Resistance . 30

Electric Power, Electric Work . . . . . . . . . . . . . . 31

Electric Field, Capacitor . . . . . . . . . . . . . . . . . . . 32

Alternating Quantities, Wavelength . . . . . . . . . 33

Power of Alternating Sine-wave Current, Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Magnetic Field, Coil . . . . . . . . . . . . . . . . . . . . . . 35

Electric and Magnetic Field Strengths . . . . . . . 36

Current in the Magnetic Field, Induction . . . . . 37

Resistor Circuits . . . . . . . . . . . . . . . . . . . . . . . . . 38

Reference Arrows, Kirchhoff’s Rules, Voltage Dividers . . . . . . . . . . . . . . . . . . . . . . . . . 39

Potentiometer . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Equivalent Voltage Source, Equivalent Current Source, Matching . . . . . . . . . . . . . . . . . . . . . . . . 41

Basic Circuits of Inductances and Capacitances . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Switching Capacitors and Coils . . . . . . . . . . . . 43

Series Connection of R, L, C . . . . . . . . . . . . . . . 44

Equivalent Series Connection and Equivalent Parallel Connection . . . . . . . . . . . . . . . . . . . . . . 46

Simple Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Three-phase Systems (Alternating Current) . . 48

Unbalanced Load, Star-delta Conversion, Bridge Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Resistors and Capacitors . . . . . . . . . . . . . . . . . . 50

Colour Marking of Resistors and Capacitors . . 51

Application Groups and Structures of Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Semiconductor Resistors . . . . . . . . . . . . . . . . . 54

Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Field Effect Transistors, IGBT . . . . . . . . . . . . . . 56

Bipolar Transistors . . . . . . . . . . . . . . . . . . . . . . 57

Thyristor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Thyristor Types and Trigger Diodes . . . . . . . . 59

Rectifi er Terms . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Types of Packages for Diodes, Transistors and ICs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Magnetic Field-dependent Components . . . . . 62

Photoelectronic Components . . . . . . . . . . . . . . 63

Protection Circuits for Diodes and Thyristors . 64

Components for Surge Protection . . . . . . . . . . 65

Cooling of Semiconductor Components . . . . . 66

Section TM:

Technical Documentation, Measuring . . . . . . 67

Graphical Representation of Characteristics . . 68

General Technical Drawing . . . . . . . . . . . . . . . . 69

Graphical Representation of Bodies . . . . . . . 70

Dimension Arrows, Special Representations 71

Dimensioning, Hatching . . . . . . . . . . . . . . . . . . 72

Circuit Diagrams as Functional Documents . . 74

Other Functional Documents . . . . . . . . . . . . . . 75

Location- and Connection-related Documents 76

Marking in Circuit Diagrams . . . . . . . . . . . . . . . 77

Code Letters for Components (Objects) in Circuit Diagrams . . . . . . . . . . . . . . . . . . . . . . 78

Subclasses According to the Task of an Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Contact Marking in Circuit Diagrams . . . . . . . . 80

Circuit Symbols . . . . . . . . . . . . . . . . . . . . . . . . . 81

General Circuit Symbols . . . . . . . . . . . . . . . . . . 82

Additional Circuit Symbols, Switches in Energy Plants . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Measuring Instruments and Devices . . . . . . . . 84

Semiconductor Components . . . . . . . . . . . . . . 85

Binary Elements . . . . . . . . . . . . . . . . . . . . . . . . . 86

Analog Information Processing, Meters and Tariff Switchgears . . . . . . . . . . . . . . . . . . . . . . . 88

Audio Converter, Video Converter and Aerial Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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Circuit Symbols for Installation Circuit Diagrams and Installation Diagrams . . . . . . . . 90

Installation Circuit Diagrams . . . . . . . . . . . . . . 92

Circuit Symbols for Overview Diagrams . . . . . 93

Coils, Transformers, Rotating Generators . . . . 94

Single-phase A.C. Motors and Starters . . . . . . 95

Three-phase Motors and Starters . . . . . . . . . . 96

Converter-fed Motors with A.C./D.C. Drive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Comparison of Circuit Symbols . . . . . . . . . . . . 98

Marks and Symbols on Electrical Equipment (Examples) . . . . . . . . . . . . . . . . . . . 100

Hydraulic and Pneumatic Controls . . . . . . . . . . 101

Symbols in Process Engineering . . . . . . . . . . . 102

Preparing Documentation on Equipment and Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Structure and Contents of Operating Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Electrical Measuring Instruments and Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Pictographs for Measuring . . . . . . . . . . . . . . . . 106

Measuring Circuits for Resistance Calculation 107

Measuring Range Extension . . . . . . . . . . . . . . . 108

Measurements in Electrical Installations . . . . . 109

Low-voltage Power Meters . . . . . . . . . . . . . . . . 112

Electricity Meters, Kilowatt-hour Meters . . . . . 113

Digital Watt Meters (Smart Meters) . . . . . . . . . 114

Oscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Measurement with the Oscilloscope . . . . . . . . 116

Displacement and Angle Measurement with Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Measurement of Force and Pressure with Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Motion Measurement with Sensors . . . . . . . . . 119

Temperature Measurement with Sensors . . . . 120

Section EI:

Electrical Installations . . . . . . . . . . . . . . . . . . . 121

Qualifi cations to Performing Electrotechnical Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Working on Electrical Equipment . . . . . . . . . . . 123

Workshop Equipment . . . . . . . . . . . . . . . . . . . . 124

Cable Installation, Working on Electrical Conductors . . . . . . . . . . . . . . . . . . . . . 125

On-off Circuits, Series Connection . . . . . . . . . . 126

Three- and Four-way Switch Circuits . . . . . . . . 127

Practical Installation of Electrical Circuits . . . . 128

Automatic Switch for Staircase Lighting, Doorbell System with Door Opener . . . . . . . . . 129

Circuits with Latching Relays . . . . . . . . . . . . . . 130

Louvre-control Circuits . . . . . . . . . . . . . . . . . . . 131

Intercom Systems . . . . . . . . . . . . . . . . . . . . . . . 133

Two-wire Door Intercom Systems . . . . . . . . . . 134

Lamp Circuits with Dimmers . . . . . . . . . . . . . . 135

Push-button Dimmer, Types of Dimmers . . . . 136

Automatic Controller with Heat Sensor . . . . . . 137

Automatic Switch with Ultrasonic Motion Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Electrical Installation with Low-voltage Halogen Lamps . . . . . . . . . . . . . . . . . . . . . . . . . 139

Field-reducing Electrical Installation . . . . . . . . 140

Building Management and Automation . . . . . 141

Lines and Areas in a KNX-TP System . . . . . . . 142

Circuit Symbols for KNX . . . . . . . . . . . . . . . . . . 143

Components of KNX-TP Systems . . . . . . . . . . 144

Special Actuators and System Devices for KNX-TP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Sensors for KNX-TP . . . . . . . . . . . . . . . . . . . . . . 146

Actuators for KNX-TP . . . . . . . . . . . . . . . . . . . . 147

Installation Bus with FSK Control KNX-PL . . . 148

Project Design and Commissioning Based on KNX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Local Operating Network LON . . . . . . . . . . . . . 151

LON Components . . . . . . . . . . . . . . . . . . . . . . . 152

Electrical Installations with Wireless Control . 153

Local Control Network LCN . . . . . . . . . . . . . . . 155

House Connection with Protective Equipotentional Bonding . . . . . . . . . . . . . . . . . 156

Foundation Earth Electrode Installed in Concrete or Soil . . . . . . . . . . . . . . . . . . . . . . . . . 157

Main Power Supply Lines in Residential Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Installation of the Meter Cabinet . . . . . . . . . . . 159

Minimum Electrical Equipment in Residential Buildings, Meter Cabinets . . . . . . . . . . . . . . . . . 160

Wiring in Residential Buildings . . . . . . . . . . . . 161

Calculation of Circuit Loading of Lines without Branching . . . . . . . . . . . . . . . . . . . . . . . 162

Calculation of Circuit Loading of Branched Lines . . . . . . . . . . . . . . . . . . . . . . . . . 164

Protection of Conductors against Overload and Short Circuits . . . . . . . . . . . . . . . . . . . . . . . 165

Methods of Installation for Permanent Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Ampacity of Cables and Wires . . . . . . . . . . . . . 167

Ampacity of Flexible or Heat-resistant Cables 169

Ampacity Correction Factor . . . . . . . . . . . . . . . 170

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Minimum Conductor Cross Sections, Ampacity of Power Cables . . . . . . . . . . . . . . . . 171

Overcurrent Protection Devices (Low-voltage Fuses) . . . . . . . . . . . . . . . . . . . . . 172

Overcurrent Protection Devices . . . . . . . . . . . . 173

Bathrooms with Bathtubs or Showers . . . . . . . 174

Special Rooms and Facilities, Working Under Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Saunas, Swimming Pools, Accessible Pools . . 176

Electrical Installations in Hazardous Locations (Risk of Fire) . . . . . . . . . . . . . . . . . . . 177

Electrical Installations in Agricultural Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Electrical Installations in Medical Areas . . . . . 179

Electrical Installations in Teaching Rooms with Experimental Facilities . . . . . . . . . . . . . . . 181

Electrical Installations in Hazardous Locations (Risk of Explosion) . . . . . . . . . . . . . . . . . . . . . . . 182

Power Supply of Workshops and Machine Shops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Lighting Engineering . . . . . . . . . . . . . . . . . . . . . 184

Design of Interior Workshop Lighting . . . . . . . 185

Maintenance Factors of Workplace Lighting . . 186

Calculation of Lighting Systems . . . . . . . . . . . . 187

Lighting and Glare . . . . . . . . . . . . . . . . . . . . . . . 188

Fluorescent Lamps for 230 V . . . . . . . . . . . . . . 189

Incandescent Lamps, Metal-vapour Lamps . . . 190

Energy-saving Lamps, Colour Reproduction . 191

Induction Lamps and Optical Fibres . . . . . . . . 192

Electronic Ballasts for Fluorescent Lamps . . . . 193

Discharge Lamp Circuits . . . . . . . . . . . . . . . . . . 194

LED Lighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

LED-Lamps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Photometric Data of Light Fixtures . . . . . . . . . 197

Fluorescent Tube Systems . . . . . . . . . . . . . . . . 198

Section SE:

Safety, Energy Supply . . . . . . . . . . . . . . . . . . . 199

First Aid at the Workplace . . . . . . . . . . . . . . . . . 200

Personal Protective Equipment PPE . . . . . . . . . 201

Signs for Accident Prevention . . . . . . . . . . . . . 202

Workplace Health and Safety . . . . . . . . . . . . . . 206

Types of Contact, Current Hazards, Types of Faults . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Protective Measures, Protection Classes . . . . . 208

Distribution Systems (Network Layouts) . . . . . 209

Protection against Electric Shock . . . . . . . . . . . 210

Differential Current Devices . . . . . . . . . . . . . . . 211

Fault Protection by Automatic Disconnection from the Power Supply . . . . . . . . . . . . . . . . . . . 212

Other Protection Measures . . . . . . . . . . . . . . . . 214

Additional Fault Protection in Professionally Monitored Systems . . . . . . . . . . . . . . . . . . . . . . 215

Conductors for Protective Measures . . . . . . . . 216

Initial Testing of Protective Measures . . . . . . . 217

Repetitive Testing . . . . . . . . . . . . . . . . . . . . . . . 218

Repair, Modifi cation and Testing of Electrical Equipment . . . . . . . . . . . . . . . . . . . . . 219

Transformers and Chokes, Insulation Testing 221

Calculation of Transformers . . . . . . . . . . . . . . . 222

Additional Operating Parameters of Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Small Transformers . . . . . . . . . . . . . . . . . . . . . . 224

Types of Power Stations . . . . . . . . . . . . . . . . . . 225

Rotating Generators . . . . . . . . . . . . . . . . . . . . . 226

Insulator Classes, Nameplates of Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Transformers for Three-phase Current . . . . . . 228

Transformers in Parallel Operation . . . . . . . . . 229

Power Mains for Energy Supply . . . . . . . . . . . . 230

Overhead Power Lines . . . . . . . . . . . . . . . . . . . 231

Overhead Mains . . . . . . . . . . . . . . . . . . . . . . . . . 232

Sag of Overhead Power Lines . . . . . . . . . . . . . 233

Installation of Buried Cables . . . . . . . . . . . . . . . 234

Private Power Generating Systems . . . . . . . . 235

Reimbursement for Renewable Energies According to EEG . . . . . . . . . . . . . . . . . . . . . . . . 237

Wind Power Stations . . . . . . . . . . . . . . . . . . . . . 238

Photovoltaic Systems . . . . . . . . . . . . . . . . . . . . 239

Photovoltaic Arrays . . . . . . . . . . . . . . . . . . . . . . 240

Smart Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Fuel Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

Protection of Electrical Equipment, IP Codes . 243

Electrochemistry . . . . . . . . . . . . . . . . . . . . . . . . 244

Primary Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Accumulators . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Charging Methods for Rechargeable Batteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

Emergency Power Supply and Emergency Lighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

Stand-by Uninterrupted Power Supply Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

UPS Systems (Uninterrupted Power Supply) . 250

Electromagnetic Compatibility EMC . . . . . . . . 251

Electromagnetic Interferences EMI . . . . . . . . . 252

Measures against EMI . . . . . . . . . . . . . . . . . . . . 253

Internal Lightning Protection . . . . . . . . . . . . . . 254

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External Lightning Protection . . . . . . . . . . . . . . 255

Lightning Arrester Systems . . . . . . . . . . . . . . . 257

Quality of Power Supply . . . . . . . . . . . . . . . . . . 258

Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Compensation, Power Factor Correction . . . . . 260

Compensation of Reactive Power . . . . . . . . . . 261

Monitoring of Final Circuits . . . . . . . . . . . . . . . 263

Alarm and Monitoring Systems . . . . . . . . . . . . 264

Safety and Security Systems in Buildings . . . . 265

Smoke Alarms . . . . . . . . . . . . . . . . . . . . . . . . . . 266

Arc Fault Detection Device AFDD or Arc Fault Circuit Interrupter AFCI . . . . . . . . . . . . 267

Alarm Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 268

Intrusion or Burglar Alarm System . . . . . . . . . 269

Closed-circuit Television CCTV . . . . . . . . . . . . . 270

Temperatures Relevant to Heat Demand Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

Energy Conservation Directive . . . . . . . . . . . . . 272

Heating Energy Consumption and Energy Index of a Single-family House . . . . . . 274

Room Heating . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Underfl oor and Ceiling Panel Heating . . . . . . . 276

Air Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . 277

Electric Cookers . . . . . . . . . . . . . . . . . . . . . . . . . 278

Water Heaters . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Household Appliances . . . . . . . . . . . . . . . . . . . 280

CE Marking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Energy Effi ciency Classes . . . . . . . . . . . . . . . . . 282

Energy-saving Potentials . . . . . . . . . . . . . . . . . 284

Heat Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Electricity Tariffs . . . . . . . . . . . . . . . . . . . . . . . . 286

Section IC:

Information and Communication Technology

Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Binary Numbers and Codes . . . . . . . . . . . . . . . 288

Hexadecimal and Octal Numbers . . . . . . . . . . . 289

ASCII Code in Unicode . . . . . . . . . . . . . . . . . . . 290

Binary Operations . . . . . . . . . . . . . . . . . . . . . . . 291

Boolean Algebra . . . . . . . . . . . . . . . . . . . . . . . . 292

Development of Combinational Circuits . . . . . 293

Code Converters . . . . . . . . . . . . . . . . . . . . . . . . 294

Comparators and Flip-fl ops . . . . . . . . . . . . . . . 295

Digital Counters and Shift Registers . . . . . . . . 296

D/A Converters and A/D Converters . . . . . . . . . 297

Modulation and Demodulation . . . . . . . . . . . . 298

Microcomputers . . . . . . . . . . . . . . . . . . . . . . . . . 299

Visual Display Units, Monitors . . . . . . . . . . . . . 300

PC Ports and Connectors . . . . . . . . . . . . . . . . . 301

Interface Connections, Interface Converters . . 302

Operating System Windows . . . . . . . . . . . . . . . 303

Elements of Windows User Interfaces . . . . . . . 304

IT Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Components of Data Networks . . . . . . . . . . . . 306

Communication via Ethernet . . . . . . . . . . . . . . 308

Installation of an Ethernet . . . . . . . . . . . . . . . . . 309

Industrial Ethernet . . . . . . . . . . . . . . . . . . . . . . . 310

Signal Transmission . . . . . . . . . . . . . . . . . . . . . 311

Wireless Data Transmission . . . . . . . . . . . . . . . 312

Wireless LAN . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Identifi cation Systems . . . . . . . . . . . . . . . . . . . . 314

AS-Interface Bus System . . . . . . . . . . . . . . . . . 315

Safe Communication across Different Field Busses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

PROFIBUS, Process Field Bus . . . . . . . . . . . . . . 317

Remote Control Systems . . . . . . . . . . . . . . . . . 318

Measuring Transducers and Signal Converters for Remote Control Systems . . . . . . . . . . . . . . 319

Programmable Measuring Transducers for Remote Control Systems . . . . . . . . . . . . . . 320

Connection to the Telephone Network . . . . . . 321

Telecommunication via ISDN . . . . . . . . . . . . . . 322

ISDN and Voice over IP (VoIP) . . . . . . . . . . . . . 323

Internet Access . . . . . . . . . . . . . . . . . . . . . . . . . . 324

Internet Applications . . . . . . . . . . . . . . . . . . . . . 325

Backing up and Protecting Data . . . . . . . . . . . . 326

Aerials, Electrical Equipment for Aerial Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Satellite Reception . . . . . . . . . . . . . . . . . . . . . . . 328

Satellite Equipment . . . . . . . . . . . . . . . . . . . . . . 329

Aerial Systems for Satellite Receivers . . . . . . . 330

Broadcasting of Digital Terrestrial Television (DBV-T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Master Aerial Systems . . . . . . . . . . . . . . . . . . . 332

Installation and Safety of Aerial Systems . . . . 333

Broadband Communication Systems . . . . . . . 334

Section AC:

Automation, Drive and Control Systems. . . . 335

Base Circuits of Amplifi ers . . . . . . . . . . . . . . . . 336

Fundamental Principles of the Operational Amplifi er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Circuits with Operational Amplifi ers . . . . . . . . 338

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Tasks of Power Converters . . . . . . . . . . . . . . . . 340

Identifi cation Codes for Converter Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Circuits for Rectifi ers and Power Converters . 342

Bidirectional Connection, Control Characteristics . . . . . . . . . . . . . . . . . . . 343

Operating Quadrants for Drives, Linear Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

Semi-controlled Power Converters . . . . . . . . . 345

Fully Controlled Power Converters . . . . . . . . . 346

Inverters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

D.C. Choppers, Voltage Source Inverters (VSI) . 348

Voltage Source Inverters VSI . . . . . . . . . . . . . . 349

Triggering Circuits for Semiconductors . . . . . 350

Smoothing and Voltage Stabilisation . . . . . . . 351

Fundamentals of Switch-mode Power Supplies . . . . . . . . . . . . . . . . . . . . . . . . . 352

Switch-mode Power Supplies . . . . . . . . . . . . . 353

Switching Transistors and Multivibrators . . . . 354

Solid-state Relays SSR and Safety Relays . . . . 355

Control Engineering . . . . . . . . . . . . . . . . . . . . . 356

Small Controller EASY . . . . . . . . . . . . . . . . . . . 357

Small Controller LOGO! . . . . . . . . . . . . . . . . . . 358

Structograms and Program Flowcharts . . . . . . 359

Programmable Logic Controller PLC, Stored-program Control SPC . . . . . . . . . . . . . . 360

Signal Coupling for PLCs and Microcomputers . . . . . . . . . . . . . . . . . . . . . . . . . 361

Control Statements for PLCs . . . . . . . . . . . . . . 362

Program Examples for PLCs . . . . . . . . . . . . . . . 364

Counters and Timers in PLCs . . . . . . . . . . . . . . 365

Programming Languages SCL (Structured Control Language) and SFC (Sequential Function Chart) . . . . . . . . . . . 366

Program Structure for PLC S7 . . . . . . . . . . . . . 367

Word Processing with PLCs . . . . . . . . . . . . . . . 368

Sequential Control with Sequential Function Charts GRAFCET . . . . . . . . . . . . . . . . . . . . . . . . 369

Alphanumeric Identifi cation of Connections . . 371

Electronic Control of Electrical Consumers . . . 372

Limit Values for Loads Connected to the National Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Auxiliary Circuits . . . . . . . . . . . . . . . . . . . . . . . . 374

Safety-Related Parts of Control Systems . . . . . 375

Architectures of Control Systems . . . . . . . . . . 376

EC Machinery Directive . . . . . . . . . . . . . . . . . . . 377

Electrical Low-voltage Equipment of Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

Contactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

Special Types of Contactors . . . . . . . . . . . . . . . 380

Identifi cation and Actuation of Contactors . . . 381

Utilisation Categories and Test Conditions Applying to Contactors . . . . . . . . . . . . . . . . . . . 382

Contactor Circuits . . . . . . . . . . . . . . . . . . . . . . . 383

Contactor Circuits with Control Devices . . . . . 385

Motor Protection . . . . . . . . . . . . . . . . . . . . . . . . 386

Electronic Motor Protection . . . . . . . . . . . . . . . 387

Control via Motor Switches . . . . . . . . . . . . . . . 388

Optoelectronic Proximity Switches (Light Barriers) . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Proximity Switches (Sensors) . . . . . . . . . . . . . . 390

Ultrasonic Sensors . . . . . . . . . . . . . . . . . . . . . . 391

Automatic Control Engineering . . . . . . . . . . . . 392

Discontinuous Automatic Control Elements . . 393

Digital Control Elements for Continuous Automatic Control . . . . . . . . . . . . . . . . . . . . . . . 394

Analog Control Elements for Continuous Automatic Control . . . . . . . . . . . . . . . . . . . . . . . 395

Automatic Digital Control . . . . . . . . . . . . . . . . . 396

Setting of Control Loops . . . . . . . . . . . . . . . . . . 397

Operating Modes and Temperature Rise Limits of Machines . . . . . . . . . . . . . . . . . . . . . . 399

Effi ciency of Electrical Drives . . . . . . . . . . . . . . 400

Converter-fed Motors for Three-phase A.C. Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

Surface-cooled Squirrel Cage Motors (Standard Motors) . . . . . . . . . . . . . . . . . . . . . . . 402

Operating Data of Squirrel Cage Motors . . . . . 403

Types of Construction of Rotating Electrical Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

Calculation Formulas for Rotating Electrical Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

Conventional D.C. Drives . . . . . . . . . . . . . . . . . 406

Nameplates of Rotating Electrical Machines . . 407

Three-phase Motors . . . . . . . . . . . . . . . . . . . . . 408

Pole-changing Motors . . . . . . . . . . . . . . . . . . . . 409

Troubleshooting on Three-phase Asynchronous Motors . . . . . . . . . . . . . . . . . . . . 410

Single-phase A.C. Motors . . . . . . . . . . . . . . . . . 411

D.C. Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

Servomotors . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

Activation of Servomotors . . . . . . . . . . . . . . . . 414

Stepper Motors . . . . . . . . . . . . . . . . . . . . . . . . . 415

Micromotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

Data of Microdrives, Gears of Micromotors . . 417

Linear Drives . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

Piezo Actuators and Piezo Drives . . . . . . . . . . . 419

Testing on Electrical Machines . . . . . . . . . . . . . 420

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Drive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Selection of the Suitable Drive Motor . . . . . . . 422

Starting of Squirrel Cage Motors . . . . . . . . . . . 423

Design of Automation Systems . . . . . . . . . . . . 424

Section MC:

Materials, Connecting, Joining and Bonding 425

Periodic Table, Chemical Bond . . . . . . . . . . . . . 426

Specifi c Material Values . . . . . . . . . . . . . . . . . . 427

Steel Standardisation . . . . . . . . . . . . . . . . . . . . 428

Conducting Materials in Electrical Engineering (Nonferrous Metals) . . . . . . . . . . . . . . . . . . . . . 429

Magnetisation Characteristics (B-H Curves) . . 430

Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . 431

Solders, Thermal Bimetals, Carbon Brushes . 432

Contact Materials, Overhead Power Lines . . . 433

Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

Synthetic Materials Used as Insulators . . . . . . 436

Other Insulators . . . . . . . . . . . . . . . . . . . . . . . . . 437

Auxiliary Materials . . . . . . . . . . . . . . . . . . . . . . . 438

Cables and Wires . . . . . . . . . . . . . . . . . . . . . . . . 439

Insulated Power Cables and Cords . . . . . . . . . 440

Power Cables and Cords . . . . . . . . . . . . . . . . . . 441

Other Cables for Permanent Installation . . . . . 442

Cables for the Connection of Mobile Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

Cables and Wires for Alarm and Signalling Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

Cables and Wires in Data Networks . . . . . . . . . 445

Wires for ELV Lighting . . . . . . . . . . . . . . . . . . . 446

Multimedia Cabling in Private Homes . . . . . . . 447

Codes for Colour Marking of Power Cables . . 448

Cables for Power Distribution . . . . . . . . . . . . . . 449

Connectors in Power Engineering . . . . . . . . . . 450

Connectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

RJ45 and RJ11 Connectors . . . . . . . . . . . . . . . . 452

Solderless Connection Technology . . . . . . . . . 453

Cable Conduits . . . . . . . . . . . . . . . . . . . . . . . . . . 454

Plugs and Anchors . . . . . . . . . . . . . . . . . . . . . . . 455

Nomenclature and Samples of Screws, Bolts and Nuts . . . . . . . . . . . . . . . . . . . . . . . . . . 456

Metric ISO Threads . . . . . . . . . . . . . . . . . . . . . . 457

ISO System of Limits and Fits . . . . . . . . . . . . . 458

Section CE:

The Company and its Environment,

Environmental Technology, Annex . . . . . . . . 459

Organisational Structures of Companies . . . . 460

Organisation of Work . . . . . . . . . . . . . . . . . . . . 461

Job Planning, Network Planning . . . . . . . . . . . 462

Teamwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

Confl ict Management . . . . . . . . . . . . . . . . . . . . 464

Analysing and Designing Processes . . . . . . . . 465

Preparing a Presentation . . . . . . . . . . . . . . . . . . 466

Presentation of a Project . . . . . . . . . . . . . . . . . . 467

Diagrams and Charts for Presentations . . . . . . 468

Realising Projects . . . . . . . . . . . . . . . . . . . . . . . 469

Systematic Marketing . . . . . . . . . . . . . . . . . . . . 470

Communication with Customers . . . . . . . . . . . 471

Customer Training . . . . . . . . . . . . . . . . . . . . . . . 472

Constituents of a Collective Labour Agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

Legal Transactions of the Company . . . . . . . . 474

Costs and Key Figures . . . . . . . . . . . . . . . . . . . . 475

Cost Accounting . . . . . . . . . . . . . . . . . . . . . . . . . 476

Preparing a Quotation . . . . . . . . . . . . . . . . . . . . 477

Design-of-work and Scope-of-work Specifi cations . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

Computer-aided Planning of Electrical Installations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Certifi cation and Auditing . . . . . . . . . . . . . . . . . 480

Hazardous Substances . . . . . . . . . . . . . . . . . . . 481

Hazard Statements (H-Statements) for Hazardous Substances . . . . . . . . . . . . . . . . . . . 482

Precautionary Statements (P-Statements) for Hazardous Materials . . . . . . . . . . . . . . . . . . 483

Handling of Electronic Waste Products . . . . . . 484

Standards & Codes . . . . . . . . . . . . . . . . . . . . . . 485

Important Standards . . . . . . . . . . . . . . . . . . . . . 486

Important VDE Regulations . . . . . . . . . . . . . . . 489

Parts of DIN VDE 0100 . . . . . . . . . . . . . . . . . . . . 490

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

Overcurrent Protection Devices for Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . 497

Shortcuts of Technical Terms . . . . . . . . . . . . . 498

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . 506

Supporting Companies and Organisations . . . 524

List of Picture Sources . . . . . . . . . . . . . . . . . . . . 527

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Lengths and Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Body and Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Mass, Force, Pressure, Moment of Force . . . . . . . . . . . . 24

Motion Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Mechanical Work, Mechanical Power, Energy . . . . . . . . 26

Transmissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Charge, Voltage, Electric Current, Resistance . . . . . . . . 30

Electric Power, Electric Work . . . . . . . . . . . . . . . . . . . . . . 31

Electric Field, Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Alternating Quantities, Wavelength . . . . . . . . . . . . . . . . 33

Magnetic Field, Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Electric and Magnetic Field Strengths . . . . . . . . . . . . . . . 36

Current in the Magnetic Field, Induction . . . . . . . . . . . . . 37

Symbols in this Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Subscripts and Signs for Formula Symbols in this Book 13

International Formula Symbols . . . . . . . . . . . . . . . . . . . . 14

Quantities and Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Mathematical Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Exponents, Unit Prefi xes, Logarithms, Calculations

According to the Rule of Three . . . . . . . . . . . . . . . . . . . . 18

Logarithmic Unit Decibel . . . . . . . . . . . . . . . . . . . . . . . . . 19

Angles, Trigonometric Functions, Percentage Calculation 20

Relationships Between Trigonometric Functions . . . . . 21

Adjacent b

B

Opp

osite

aCA

Hypotenuse c

90}

å

Circuit theory

Resistor Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Reference Arrows, Kirchhoff’s Rules, Voltage Dividers . 39

Equivalent Voltage Source, Equivalent Current Source,

Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Basic Circuits of Inductances and Capacitances . . . . . . . 42

Switching Capacitors and Coils . . . . . . . . . . . . . . . . . . . . 43

Equivalent Series Connection and Equivalent Parallel

Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Simple Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Three-phase Systems (Alternating Current) . . . . . . . . . . 48

Unbalanced Load, Star-delta Conversion, Bridge Circuit 49

0 0.5 1 1.5 2 2.5

3

2

1

0

G/S

R/Q

F

r

Section M: Mathematics, Physics, Theory of Circuits, Components

Components

Resistors and Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . 50

Semiconductor Resistors . . . . . . . . . . . . . . . . . . . . . . . . . 54

Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Field Effect Transistors, IGBT . . . . . . . . . . . . . . . . . . . . . . 56

Bipolar Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Thyristor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Rectifi er Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Types of Packages for Diodes, Transistors and ICs . . . . 61

Magnetic Field-dependent Components . . . . . . . . . . . . . 62

Photoelectronic Components . . . . . . . . . . . . . . . . . . . . . . 63

Protection Circuits for Diodes and Thyristors . . . . . . . . . 64

Cooling of Semiconductor Components . . . . . . . . . . . . . 66

1 2 4

G

Su

SD Is

P+P+ P

N

3

R2U

¡

R1 ¡1 ¡2G

R2

U

¡ R1

U1 U2G

12

M

####

D(Delta)

Q(Theta)

G(Phi)

U(Psi)

O(Omega)

Meaning

a

c

d

e

f

g

h

i

Œ

m

n

o

p

q

r

s

t

v

w

x

y

z

Symbol

acceleration

1. spec. heat capacity2. electrochemical equivalent3. propagation velocity of waves4. coeffi cient

1. diameter2. distance3. dissipation factor4. duty cycle

elementary charge

1. frequency2. fi lter factor

1. gravitational acceleration, position

height

time-controlled current

1. length2. spacing

1. mass2. number of strands

1. speed, number of revolutions

2. integer 1, 2, 3, ...3. refractive index

overdrive factor

1. number of pole pairs2. pressure3. percentage rate

1. quantity2. shunt current ratio

1. radius2. rate3. differential  resistance

1. section, strength2. normalized slip3. correction

1. time2. transformation ratio

1. velocity2. time-controlled voltage

1. width2. energy density3. command variable

controlled variable

correcting variable

integer, e.g. number of teeth of a gear

Meaning

A

B

C

D

E

F

G

H

I

J

L

M

N

P

Q

R

S

T

U

V

W

X

Y

Z

Symbol

1. area, 2. cross section3. attenuation constant4. apparent power5. amplifi cation factor6. phase of three-phase

system

1. magn. fl ux density2. current gain3. number base4. phase of three-phase

system

1. capacitance2. thermal capacity3. constant 4. coupling factor5. phase of three-phase

system

1. electric fl ux density2. damping factor3. defl ection coeffi cient

1. electric fi eld strength2. illuminance

1. force, 2. factor, 3. fault

1. conductance2. amplifi cation factor3. gravitational force

magnetic fi eld strength

1. electric current2. light intensity

1. current density2. mass moment of inertia

1. level2. inductance

1. moment of force2. memory capacity

number of turns

1. active or effective power2. process value in controller

circuits, often also PV

1. electric charge2. heat3. reactive power4. resonant circuit quality

1. active resistance2. spring rate3. rigidity

1. susceptance2. steepness3. slip (absolute)4. transmission quantity5. set value in controller

circuits, often also SV

1. cycle time2. transmission factor3. temperature in K4. torque

voltage

volume

1. energy2. work

reactance

admittance

1. impedance2. wave impedance3. oscillation impedance

Meaning

a(alpha)

b(beta)

g(gamma)

d(delta)

e0e

(epsilon)

f(zeta)

n(eta)

k(kappa)

h(theta)

l(lambda)

m(mu)

m0

p(pi)

r(rho)

s(sigma)

t(tau)

j(phi)

w(omega)

Symbol

1. angle2. temperature coeffi cient3. triggering angle

1. angle2. short-circuit current amplifi cation factor

1. angle2. conductivity

angle for losses

electric fi eld constant

permittivity

work ratio, utilisation ratio

effi ciency

conductivity (optional symbol)

temperature in °C

wavelength

1. permeability2. friction coeffi cient

magnetic fi eld constant

number 3.1415926…

1. specifi c resistance2. density

1. leakage factor2. stress

time constant

angle, particularly phase-shift angle

1. angular velocity2. angular frequency

Lower-case letters Upper-case letters Lower-case Greek letters

difference

current linkage, (phase-shift angle in NA)

1. magnetic fl ux2. luminous fl ux

electric fl ux

1. solid angle2. resistance

Upper-case Greek letters

Special symbols are created by adding one or more subscripts or other signs to the symbol.

Symbols in this Book

13

M

######

a(alpha)

s(sigma)

j(phi)

D(Delta)

Meaning

0

1

2

3, 4, …

:, e.g. u:;, e.g. u

;

:; , e.g. u:;

‘, e.g. u ’

�Y

a

ab

adm

del

amb

b

c

d

e

eff

f

h

i

in

j

k

—m

Subscript,sign

1. idle2. vacuum3. reference variable

1. input 2. order, sequence

1. output 2. order, sequence

order, sequence

peak value

minimum value

1. peak-to-peak value2. oscillation width

1. related to2. note 3. derivation

delta connection

star connection

1. armature, 2. ambient, 3. actual

absorbed

admissible

delivered

ambient air ...

1. bit2. brake ...

1. cut-off, 2. crest,3. comparison 4. centripetal...

1. referring to DC2. duration, 3. digit4. damping, 5. direction 6. desired7. derivation, derived

1. exterior, 2. effective, 3. error

effective value

1. frequency2. fall ...

high, upper

1. inner, internal 2. induced, 3. current4. ideal, 5. intermediate6. impulse

input

junction

kinetic

1. low, lower, 2. loss

1. magnetic2. mean value3. measuring, measured

Meaning

max

mec

min

n

o

off

out

p

per

q

r

s

sh

st

t

th

tot

v

w

x

y

z

A

B

C

D

E

Subscript

maximum

mechanical

minimum

1. nominal, 2. normal 3. noise

oscillator

switch off, turn off

output

1. parallel, 2. pause3. pulse, 4. potential5. pressure, 6. pre-

permissible

quality

1. reactive quantity2. relative, related to3. rise, 4. resonance5. remanence, retentivity6. reception

1. screen...2. signal...3. specifi c

short circuit

step

1. test, 2. transverse 3. time

thermal, heat ...

total

1. voltage2. visual

1. command variable2. wind...3. wave ...

1. unknown variable2. in x-direction

1. correcting variable2. in y-direction

zigzag connection

1. ammeter2. aerial3. armature, 4. anode5. acceleration, 6. area7. amplifyer, amplifying

1. base 2. building

1. collector, 2. capacitive3. clock pulse, 4. coercitive5. cluster, 6. channel7. maximum (max) contact voltage8. carrier

1. drain, 2. data, 3. discharge

1. emitter2. environment3. earth

Meaning

F

G

H

K

L

N

O

P

R

S

T

V

W

X

Y

Z

Subscript

1. forward, 2. fault 3. (negative) feedback

gate

1. hysteresis2. Hall, 3. height 4. heat sink

cathode

1. inductive 2. load3. left 4. Lorentz 5. loop

nominal, rated

1. operation,2. operational earthing (network)

positive feedback

1. reverse, reward2. active resistance3. right4. regular5. red

1. source, 2. saddle ... 3. smoothing 4. switching5. sluice ...6. sector7. system earthing 8. scanning

1. transformer ...2. transverse3. track 4. test ...

voltage meter

weight

at the x-port

1. at the y-port2. star connection

Zener ...

in direction of the angle a

leakage

phase-shift related

Subscripts may be combined, e.g. VCE for collector-emitter voltage. Subscripts that consist of several letters may be reduced to the fi rst letter.

Lower-case letters

1. referring to a difference

2. confi guration of a three-phase system

Numerals, symbols

Lower-case Greek lettersUpper-case letters

Upper-case Greek letters

Subscripts and Signs for Formula Symbols in this Book

14

M

International Formula Symbols cf. DIN EN 60027-4: 3-08

Quantity Symbol until September 2010

Symbol after September 2010 Unit, unit symbol

Preferred symbol

Reserve symbol

Current and related quantities

Rated current In Irat or Ir IN

Ampere, A

Nominal current IN In or Inom –

Sustained short-circuit current Ikd Ik ISC

Maximum aperiodic short-circuit current Is I:k I:SC

Initial periodic short-circuit current is Ik0 ISCO

Transient current i Ik’ ISC’

Subtransient current is Ik’’ ISC’’

Current load I’ A Not applicable Amperes per metre, A/m

Voltage and related quantities

Rated voltage UN Urat or Ur UN

Volt, VNominal voltage Un Not applicable

Not applicableInduced voltage Ui Ug

Open-loop voltage U0 U0

Power and related quantities

Rated power PN Prat or Pr PN Watt, W

Rated apparent power SN Srat or Sr Volt-ampere, VA

Nominal power Pn Pn or Pnom

Not applicable

Watt, W

Input power P1 or Pi Pin

Output power P2 or Po Pout

Mechanical power P Pmec

Dissipation Pv Pt

Power factor (P.F.) cos j l (lambda)One (no unit)

Active factor – cos j

Moments of force, torques

Torque, moment of force M T M

Newton meter, Nm

Nominal moment/torque Mn Not applicable Not applicable

Rated moment/torque MN Trat or Tr Mrat or Mr

Breakdown torque MK Tb Mb

Holding torque MH TH MH

Pull-up torque MS Tu Mu

Breakaway torque MA Tl Ml

nom = nominal, rat = rated, T = torque, active factor = cosine of fundamental (without overtones), power factor (P.F.) = relation of wattage to apparent power (with overtones)

15

M

######

current linkage

magnetic fi eld strength

magnetic fl ux

magn. fl ux densitymagn. polarisation

inductance

permeability

magn. resistance

SI unit(other unit)

Unit symbol,unit equation

length

area

volume

angle (plane)(see page 20)

solid angle

Quantity

metre(sea mile)(mile)(inch)(foot)

square metre

cubic metre(litre)

radian(degree)

steradian

m1 sm = 1,852 m1 mi = 1,609.344 m1’’ = 25.4 mm1 ft = 12 x 1’’ = 0.3048 m

m2

m3

1 l = 1 dm3 = = 1/1000 m3

rad1° =

π––– 180

rad

sr

Length, area, volume, angle

mass

density

moment of inertia

force

moment of force, torque

impulse

pressure

surface pressure,rigidity, modulus of elasticity

work, energy

power

time

frequency

rotational speed-rotational frequency

angular frequency

velocity

angularvelocity

acceleration

second(minute)(hour)

(day)

hertz

per second(per minute)

per second

metres per second (knot)

radiansper second

–

s1 min = 60 s1 h = 60 min = 3 ,600 s1 d = 24 h

1Hz = 1/s

1/s = 60/min

1/s

m/s1 kn = 1 sm/h = 0.5144 m/s

1 km/h = 1–––3.6

m/s

rad/s

m/s2

Time, frequency, velocity, acceleration

kilogram(carat)(tonne)

–

–

newton

–

newton sec.

pascal(bar)

–

joule (electron volt)

watt

kg1 Kt = 0.2 g1 t = 1,000 kg

kg/m3, kg/dm3

kg · m2

1 N = 1 kg · m/s2

Nm

1 Ns = 1 kg · m/s

1 Pa = 1 N/m2

1 bar = 0.1 MPa= 10 N/cm2

N/mm2

1 J = 1 Nm = 1 Ws1 eV = 0.1602 aJ

1 W = 1 J/s = 1 Nm/s

Mechanics

SI unit(other unit)

Unit symbol,unit equation

electric charge, electric fl ux

electric charge density, electric fl ux density

space charge density

electr. voltage, electr. potential

electr. fi eld strength

electr. capacitance

current loading

permittivity,dielectric constant

electric current

electric current density

electric resistance, active resistance, reactance, impedance

electric conduct-ance, susceptance, admittance

specifi c electric resistance

electric conductivity

power

reactive power

apparent power

inductance

work, energy

Quantity

coulomb

coulombs per square metre

coulombs per cubic metre

volt

volts per metre

farad

amperes per metre

farads per metre

ampere

amperes per m2

ohm

siemens

ohmmetre

siemens per metre

watt

(var)

(VA)

henry

joule (watt-hour)(electron volt)

1 C = 1 A · 1 s = 1 As

C/m2

C/m3

1 V = 1 J/C

1 V/m = 1 N/C

1 F = 1 As/V = 1 C/V

A/m

1 F/m = 1 C/(Vm)

1 A = 1 C/s

A/m2

1 O = 1 V/A

1 S = 1 –––

1 Ω

1 Om = 100 Ωcm1 Omm2/m = 1μOm

1 Sm/mm2

= 1 MS/m

1 W = 1 V · 1 A

1 var = 1 V · 1 A

1 VA = 1 V · 1 A

1 H = 1 Vs/A

1 J = 1 Ws1 Wh = 3.6 kNm1 eV = 0.1602 aJ

Electricity

ampere

amperes per metre

weber

tesla

henry

henrys per metre

–

A

A/m

1 Wb = 1 T · 1 m2

= 1 Vs

1 T = 1 Wb/m2

= 1 Vs/m2

1 H = 1 Vs/A

1 H/m = 1 Vs/(Am)

1/H = A/Vs

Magnetism

Quantities and Units 1

16

M

####

distance in astronomy

velocity of light

light year l.y.

mass in nuclear physics

mass per unit length of textile fi bres and threads

area of plots of land

sound pressure

sound particle velocity

sound velocity (propagation velocity)

volume velocity

sound intensity

specifi c sound impedance

acoustic impedance

mechanical impedance

equivalent absorption area

light intensity

luminance

luminous fl ux

luminous effi cacy

illuminance

optical power of lenses

centigrade temperature

thermodynamic temperature

temperaturedifference

heat, inner energy

heat fl ow

thermal resistance (of components)

thermal conductivity

heat transfer coeffi cient

thermal capacity, entropy

specifi c thermal capacity

quantity of substance

concentration of quantity of sub-stance

molar volume

molality

molar mass

molar thermal capacity

diffusion coeffi cient

SI unit(other unit)

Unit symbol,unit equation

radiant energy

radiant power

radiant intensity

radiant intensity

irradiance

Quantity

joule

watt

watt/sterad.

–

–

1 J = 1 Nm = 1 Ws

1 W = 1 J/s

W/sr

W/(sr · m2)

W/m2

Electromagnetic radiation (except light)

candela

candelas per m2

lumen

lumens per watt

lux

–(dioptre)

cd

cd/m2

lm

lm/W

1 lx = 1 lm/m2

1/m1 dpt = 1/m

Light, optics

degree centigrade

kelvin

kelvin

joule

watt

kelvins per watt

–

–

joules per kelvin

–

°C

K(0 K ‡ – 273.15 °C)

K

1 J = 1 Ws

1 W = 1 J/s

K/W

W/(K · m)

W/(K · m2)

J/K

J/(kg · K)

Heat

mol(e)

–

–

–

–

–

–

mol

mol/m3

m3/mol

mol/kg

kg/mol

J/(mol · K)

m2/s

Chemistry, molecular physics

SI unit(other unit)

Unit symbol,unit equation

activity of a radioactive substance

absorbed dose

absorbed dose rate

dose equivalent

dose equivalent rate

ion dose

ion dose rate

Quantity

bequerel

gray

grays persecond

sievert

sieverts per second

coulombs per kilogram

amperes perkilogram

1 Bq = 1/s

1 Gy = 1 J/kg

Gy/s

1 Sv = 1 J/kg

1 Sv/s = 1 J/(kg · s)

C/kg

1 A/kg= 1 C/(kg · s)

Nuclear reaction, ionising radiation

pascal

metres per second

metres per second

–

–

–

–

–

square metre

1 Pa = 1 N/m2

m/s

m/s

1 m3/s= 1 m2 · 1 m/s

W/m2

Pa · s/m

Pa · s/m3

N · s/m

m2

Acoustics

(astronomical unit) parsec

km/s

km

(nuclear mass unit)

tex

arehectare

1 AU = 149.6 Gm1 pc = 30.857 Pm

c = 3 x 108 m/s ≈ 300,000 km/s

1 l.y. = 9.461 · 1012 km

1 u = 1.66 · 10–27 kg

1 tex = 1 g/kg

1 a = 100 m2

1 ha = 100 a

Other disciplines

Quantities and Units 2

17

M

Symbol

} a, a

_∂_2∂2

infi nite

versus, approaches,

exceeds

function of x

imaginary unit

complex quantity Z

parallel

parallel in the same dir.

parallel in opposite dir.

orthogonal,

perpendicular

triangle

congruent

similar

angle

line segment AB

arc AB

vector A, vector B

absolute value of

vector A

ææ

ΩΩΩºü

™

9

,

@

AB33

AB£

A , B

æ∫Aæ

g1 ææ g2, R1 ææ R2

g ΩΩ h

g1 Ωº g2

g ü h

™ ABC

™ ABC 9 ™ DEF

™ P1P2P3 , ™ ABC

@ ABC = @ (�BA, �BC),

@ (a, b )¯ ¯

P1P233

AB£ = @ g

C = A + B

æ∫Fæ= 50 N

DU = U2 – U1

y * is the fi rst

derivation of y,

fi rst derivative

quotient y * = dy/dx

5 f (x) dx, 5 b

fa (x) dx

D

y *

dy

dx

5

difference

y prime

dy by dx or

dy over dx

integral

a to the power of x

exponential function

general logarithm

logarithm to the basis a

common logarithm

dyadic logarithm

natural logarithm

sine

cosine

tangent

cotangent

arc sine

arc cosine

arc tangent

arc cotangent

sinasin2a+ cos2a =

= (sin a)2+ (cos a)2 = 1

tana = sina/cosacota = 1/tanasina =x π arcsinx = acosa =x π arccosx = atana =x π arctanx = acota =x π arccotx = a

sin

cos

tan

cot

arcsin

arccos

arctan

arccot

ax

exp

log

loga

lg

lb

ln

53, 10x

expx=ex,with e =2.718...

log3 9 = 2

lg 2 = 0.30103...

lb 8 = 3

ln 10 = 2.3025...

n = 1, 2, 3, ..., 6x ∫ a, x approaches

the value a

f (I) = I2 • Ri2 = j2 = – 1

Z\\ = R + jX

6∫

f (x)

i or j

Z\\

◊

÷

‘

π

=

Ï

,

fi

‡

ˆ

˜

‰

›

ˆ

˜

•,®

––, /, :

%

‰

( ), ˛,{ }

æzæ

n!

∑

∏

02n02p

equal to

not equal, unequal

proportional

approximately

corresponds to

less than

greater than

less than or equal to

greater than or equal to

considerably less than

considerably greater than

times, multiplied

divided by

per cent

per thousand, per mil

round, squared,

curly, pointed

brackets

amount of z

n factorial

sum

product

square root of

nth root of

pi

P = U • I

4 Ï 5

u , r

p fi 3.14

1 cm ‡ 20 N

2 ˆ 3

5 ˜ 2

a ‰ 10

n › 7

R ˆ 100 kO

Rx ˜ Rn

a • b = ab, 12 x 3 = 36

7

2 = 7/2 = 7 : 2

1 % = 10–2, 50 % = 0.5

1‰ = 10–3, 8‰ = 0.8%

¸ a (b – c) + d ˛2

æ4æ= 4, æ–7æ= 7

n!=1•2•3•...•n, 3!=6

SI = I1 + I2 + I3 + ...

Wk = k1 • k2 • k3 • ...02216 = 43028 = 2

p = 3.14159...

element of

subset

union of sets

from this follows that

a ◊ M: a is element of M

M1 ÷ M2: M1 is subset of M2

{1, 2} ‘ {3, 4} == {1, 2, 3, 4}

a • b = c π a = c/b

NOT a

AND

OR

NOT AND (NAND)

NOT OR (NOR)

and so on until n

and so on until infi nity

k = 1, 2, 3, ..., n

n = 1, 2, 3,...

022 = 1.41421 ...

SymbolMeaning MeaningExample Example

Boolean algebraGeometry, vectors

Differentiation, integration

Exponents, logarithms

Trigonometry

Set theory

Arithmetic

. . . n

. . .

General symbols

a _ b = } (a _ b)

a _ b or _ (a, b)

a ∂ b or ∂ (a, b)

a _2 b = a _ b

a ∂2 b = a ∂ b

Mathematical Symbols

18

M

####

Exponents

Logarithmic sectioning

Values less than 1 can be expressed by multiples of decimal powers with negative exponents.Values greater than 1 can be expressed by multiples of decimal powers with positive exponents.

Value 0.001 0.01 0.1 1 10 100 1,000 10,000 100,000 1,000,000

Decimal powers 10–3 10–2 10–1 100 101 102 103 104 105 106

Powers of two are used in digital engineering. The base here is 2.

Value 1/128 1/64 1/32 1/16 1/8 1/4 1/2 1 2 4 8 16 32 64 128

Powers of two 2–7 2–6 2–5 2–4 2–3 2–2 2–1 20 21 22 23 24 25 26 27

Metric prefi xes Binary prefi xes

Logarithms

Proportional relation (unit obtained by division)

Inverted proportional relation (unit obtained by multiplication)

Calculation according to the rule of three

Calculation acc. to the rule of three of a proportional relation

The logarithm (log) indicates to which power a base has to be raised in order to obtain the logarithm argu-ment. The following applies

ab = c, logac = b

The common logarithm (lg) has the base 10. The natural logarithm (ln) has the base of the e-function (e=2.718...). The dyadic logarithm (lb) has the base 2.

Extensive number ranges can be represented in a more structured way when using a logarithmic scale.

logac = Inc�lna

=

Inc�lga

Steps of approach Example

1. Statement n elements have a weight of a kg

2. Calculation for 1 object 1 element has a weight of a/n kg

3. Calculation for z objects z elements have a weight of z · a/n kg

1. Statement n workers need a hours

2. Calculation for 1 object 1 worker needs n · a hours

3. Calculation for z objects z workers need n · a/z hours

loga(cm) = m · logac

lgx = lnx / ln10

lnx = lgx / lge

lbx = lgx / lg2

1

2

3

logan��c = 1––n logac

4

5

6

7

loga(cd ) = logac + logad

logac––d

= logac – logad

For large

mass storage

units, often

the meaning

of the

physical

quantities

applies

(decimal

prefixes).

3.5 cm · lg 2

3.5 cm · lg 5

3.5 cm · lg 10

1 2 5 10

3 parts 3 parts4 parts

200

Weight/kg

No

. of

pie

ces

60

40

20

00 400 600

Exponents, Unit Prefi xes, Logarithms, Calculations According to the Rule of Three

Prefix Prefix Meaning Prefix Prefix Meaning Prefix Prefix Meaningsymbol (factor) symbol (factor) symbol (factor)

y Yokto 10–24 da deca 10 – – –z Zepto 10–21 h hecto 102 – – –a atto 10–18 k kilo 103 Ki kibi 210

f femto 10–15 M mega 106 Mi mebi 220 p pico 10–12 G giga 109 Gi gibi 230 n nano 10-9 T tera 1012 Ti tibi 240 µ micro 10-6 P peta 1015 Pi pebi 250 m milli 10-3 E exa 1018 Ei exbi 260 c centi 10-2 Z zetta 1021 Zi zebi 270 d deci 10-1 Y yobi 1024 Yi yobi 280

Prefixes may not be combined. You can assign only one prefix per unit.

19

M

Logarithmic unit decibel

Term, defi nition Formula, note Comments, example

Transmission factor T Gain factor V Attenuation factor D

Increase > 1 and decrease < 1:

S1 S2Transmission

link

S1, S2 quantities referring to

transmission

Power-related measures

Gain ratio G Attenuation ratio A

In order to identify the value as a logarithmic quantity, dB is added instead of a unit, because the value actually has no unit.

Gain ratio

Attenuation ratio

dB refers to decibel (one tenth of a bel,

a unit named after the American scien-

tist Alexander Graham Bell)

Example 1:A fi lter circuit has an input of 500 mW and an output of 250 mW. What are

a) the attenuation factor D and

b) the attenuation ratio A?

a) D = S1/S2

= 500 mW/250 mW = 2

b) A = 10 lg (500 mW/250 mW)= 3.01 dB

Voltage-related measures, pres-sure-related measures

Gain ratio G

Attenuation ratio ASound pressure transmission

ratio Tp

For these quantities, dB is also used instead of a unit.

Gain ratio

Attenuation ratio

Sound pressure transmission ratio

Example 2:An amplifi er has an input of 3 mV and an output of 5 V. What is a) the gain factor, b) the gain ratio?

a) V = U2/U1 = 5 V/3 mV = 1,667

b) G = 20 lg (U1/U2)= 20 lg (5 V/3 mV) = 64.4 dB

Level in dB(*) (* place holder for additional specifi cations)

Sound level, general This quantity expresses the ratio be-tween two values, one of which is an

agreed reference value.

The reference value should be indicated in level specifi cations.

Power level LPIdentifi ed by dB (1 mW) or dBm,

Voltage level LU, identifi ed by dB (1 μV) or dBm = dB (1 mV)

Sound-pressure level Lp

actually identifi ed by

dB (20 μN/m2)

Power level

Voltage level

Sound-pressure level

The agreed reference values are 1 mW for LP, 1mV for LU and 20 μN/m2 for LP.

Example 3:An aerial has an output of 80 mV.LU = ?

LU = 20 lg (U/1 mV) = 38 dBm

Rated sound-pressure levelIdentifi ed by dB(A), dB(B) or dB(C), depending on the correction

The measured quantity is the sound-pressure level. The measuring values are modifi ed with the help of fi lters A, B or C for frequencies other than 1,000 Hz.

The rated sound-pressure level in dB(A) corresponds to a great extent to the human loudness sensation in phon.

A attenuation ratio

D attenuation factor

G gain ratio

LP power level

Lp sound-pressure level

LU voltage level

lg common logarithm

P power

p pressure

T transmission factor

U voltage

V gain factor

Subscripts:

1 input, 2 output of the trans-

mission link

T = V = S2/S11

D = S1/S22

G = –A5

A = –G6

G = 20 lg (U2/U1)7

G = –A8

A = 20 lg (U1/U2)9

A = –G10

Tp = 20 lg (p2/p1)11

LP = 10 lg (P/1 mW)12

LU = 20 lg (U/1 μV)13

LP = 20 lg (p/20 μN/m2)14

G = 10 lg (P2/P1)3

A = 10 lg (P1/P2)

4

Logarithmic Unit Decibel

20

M

####

Figures Explications Notes, formulas

The units referring to angles are de-gree, centesimal degree and radian.

The round angle has

a) 360 ° (degrees)b) 400 gon (centesimal degrees)c) 2p rad (radian)

The unit radian corresponds to the proportion of the circular arc length to the radius in a circle.

1 rad = 360°

2p = 57.296°

Angle dimensions

The longest side (c) of the right-angled triangle is referred to as the hypotenuse. It is the side opposite the right angle. The two other sides (a and b) of the triangle form the right angle. These sides are referred to as the catheti or simply legs of the triangle. The leg (a) opposite the acute angle a is the opposite. The leg contiguous to the angle a is the adjacent (b).

An angle in a right-angled triangle can be defi ned by its angle degrees or as a ratio of two triangle sides. The ratio of the sides depends on the size of the angle. That is why the ratios of two sides in a right-angled triangle are referred to as angle func-tions (function = dependence) or trigonometric functions.

Right-angled triangle

Trigonometric functions

Sine = opposite

hypotenuse

Cosine = adjacent

hypotenuse

Tangent = opposite

adjacent

Cotangent = adjacent

opposite

a, b, c legs of a right-angled triangle

B basic amount

C0 starting capital

Cn capital after n years

I interest per year

n term in years

P percentage amount

p percentage in %, interest rate in %

a, b, g angles in a triangle

a° degrees of an angle

ar radian of an angle

Per cent (pro cent in Latin) means “per hundred”. The total quantity (basic quantity) is always equal to one hundred, the partial quantity (percentage) is expressed in per cent (= hundredths).

23% of 300 € is equal to 69 €

percentage  basic value percentage amount

… … …

Calculation of compound interest

Calculation of interest

Percentagecalculation

percentage = 100 · percentage amount

basic value8

Cn = C0 · ±1 + p

100%≤n

p = P · 100%

B

I =

C0 · p · n

100%

6

7

9

sin a = a

c 2

cos a = bc

3

tan a = a

b 4

cot a = ba

5

ar = a° · p

180°1

Important angles

Round Straight Right angle angle angle

360° 180° 90°

2 · p rad p rad p––2

rad

400 gon 200 gon 100 gon

Still customary in survey engineering: 1 gon = (p/200) rad

23 %

77 %

36

0}90}

1rad

Ro

und

angle

Right angle

Adjacent b

B

Op

po

site

a

CA

Hypotenuse c

90}

å3¢2

¢2

¢ 2¢

sinå coså

¢2

¢ 3¢2

2¢

tanå cotå

å

å

Angles, Trigonometric Functions, Percentage Calculation

Angle functions

Percentage calculation

Angles