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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
TMTM
ELECTRICAL INSTALLATIONS 121 … 198
EIEI
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
MCMC
THE COMPANY AND ITS ENVIRONMENT, ENVIRONMENTAL TECHNOLOGY, ANNEX 459 … 527
CECE
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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
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