1

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ABB Switchgear Manual
ABB Switchgear Manual
More than 50 years after publication of the first edition of the BBC Switchgear Manual
by A. Hoppner, we present to you the current edition of today's ABB Calor Emag
Switchgear Manual in the internet the first time. As always, it is intended for both
experienced switchgear professionals as well as beginners and students.
The ABB Calor Emag Switchgear Manual addresses all relevant aspects of switchgear
technology for power transmission and distribution. Not only the technology of low, medium
and high voltage switchgear and apparatus is considered but also related areas such as
digital control systems, CAD/CAE methods, project planning, network calculation, electromagnetic compatibility (EMC), etc.

Imprint
ABB Pocket Book - Switchgear Manual
10th revised edition
Edited by
ABB Calor Emag Schaltanlagen AG Mannheim and ABB Calor Emag
Mittelspannung GmbH Ratingen
Previous editions:
(published till 1987 by BBC Brown Boveri, since 1988 by ABB)
First edition 1948
Second edition 1951
Second, expanded edition 1951,1955,1956,1957,1958,1960
Third edition 1965
Fourth edition 1973 (in English 1974)
Fifth edition 1975 (also English)
Sixth edition 1977 (in English 1978)
Seventh edition 1979 (in German only)
Eighth edition 1987, 1988 (in English 1988)
Ninth edition 1992,1994 (in English 1993 and 1995)
Tenth edition 1999 (in English 2001)
Published at:
Cornelsen Verlag, Berlin
10th, 10th revised edition, ISBN 3-46448236-7
The tenth edition in English is a translation of the German edition published
towards the end of 1999 by
STAR Deutschland GmbH
Member of STAR Group
However, DIN designation and publication dates of the VDE specifications in
section 17.1 are updated to the start at the end 2000.
ABB does not accept any responsibility whatsoever for potential errors or
possible lack of information in this document. Any reproduction - in whole or
in parts - is forbidden without ABB's prior written consent.

All rights reserved.
Circuit diagrams and data included in this book are pub; lished without
reference to possible industrial property rights (including copyright). The
right for use of industrial property right is not granted.
Extracts from standards are published by permission of "DIN - Deutsches
Institut für Normung e.V." (DIN German Institute for Standardization) and of
"VDE Verband der Elektrotechnik Elektronik Informationstechnik e.V." (VDE
Association for Electrical, Electronic & Information Technologies).
The authoritative standards for the user are the latest editions, which can be
obtained from VDE-VERLAG GMBH, Bismarckstrasse 33, D-10625 Berlin
and from Beuth Verlag GmbH, Burggrafenstrasse 6, D-10787 Berlin.
Copyright © 2004 by ABB Calor Emag Mittelspannung GmbH, Ratingen.
Printed by: Central-Druck Trost GmbH & Co., Heusenstamm
Printed in the Federal Republic of Germany
Provider Information/Impressum
The ABB website is provided by ABB Asea Brown Boveri Ltd,
a company organised under the laws of Switzerland.
ABB Asea Brown Boveri Ltd is registered with the commercial register of
Zurich, Switzerland, under the company number
CH-020.3.900.058-8.
Chairman and CEO: Jürgen Dormann
Address: Affolternstrasse 54, 8050 Zurich, Switzerland
Tel: +41 43 317 7111
Fax: +41 43 317 4420

Table of Contents
1 Fundamental Physical and Technical Terms
1.1 Units of physical quantities
1.1.1 The international system of units (SI)
1.1.2 Other units still in common use; metric, British and US measures
1.1.3 Fundamental physical constants
1.2 Physical, chemical and technical values
1.2.1 Electrochemical series
1.2.2 Faraday's law
1.2.3 Thermoelectric series
1.2.4 pH value
1.2.5 Heat transfer
1.2.6 Acoustics, noise measurement, noise abatement
1.2.7 Technical values of solids, liquids and gases
1.3 Strength of materials
1.3.1 Fundamentals and definitions
1.3.2 Tensile and compressive strength
1.3.3 Bending strength
1.3.4 Loading on beams
1.3.5 Buckling strength
1.3.6 Maximum permissible buckling and tensile stress for tubular rods
1.3.7 Shear strength
1.3.8 Moments of resistance and moments of inertia
1.4 Geometry, calculation of areas and solid bodies
1.4.1 Area of polygons
1.4.2 Areas and centres of gravity
1.4.3 Volumes and surface areas of solid bodies
2 General Electrotechnical Formulae
2.1 Electrotechnical symbols as per DIN 1304 Part 1
2.2 Alternating-current
quantities
2.3 Electrical
resistances
2.3.1 Definitions and specific values
2.3.2 Resistances in different circuit configurations
2.3.3 The influence of temperature on resistance
2.4 Relationships between voltage drop, power loss and conductor cross-section
2.5 Current input of electrical machines and
transformers
2.6 Attenuation constant a of transmission
systems
3 Calculation of Short-Circuit Currents in Three-Phase Systems
3.1 Terms and definitions
3.1.1 Terms as per DIN VDE 0102 / IEC 909
3.1.2 Symmetrical components of asymmetrical three-phase systems
3.2 Fundamentals of calculation according to DIN VDE 0102 / IEC 909

3.3 Impedances of electrical
equipment
3.3.1 System infeed
3.3.2 Electrical machines
3.3.3 Transformers and reactors
3.3.4 Three-phase overhead lines
3.3.5 Three-phase cables
3.3.6 Busbars in switchgear installations
3.4 Examples of calculation
3.5 Effect of neutral point arrangement on fault behaviour in three-phase high-voltage
networks over 1 kV
4 Dimensioning Switchgear Installations
4.1 Insulation rating
4.2 Dimensioning of power installations for mechanical and thermal short-circuit
strength
4.2.1 Dimensioning of bar conductors for mechanical short-circuit strength
4.2.2 Dimensioning of stranded conductors for mechanical short-circuit strength
4.2.3 Horizontal span displacement
4.2.4 Mechanical stress on cables and cable fittings in the event of short circuit
4.2.5 Rating the thermal short-circuit current capability
4.3 Dimensioning of wire and tubular conductors for static loads and electrical surfacefield strength
4.3.1 Calculation of the sag of wire conductors in outdoor installations
4.3.2 Calculation of deflection and stress of tubular busbars
4.3.3 Calculation of electrical surface field strength
4.4 Dimensioning for continuous current rating
4.4.1 Temperature rise in enclosed switchboards
4.4.2 Ventilation of switchgear and transformer rooms
4.4.3 Forced ventilation and air-conditioning of switchgear installations
4.4.4 Temperature rise in enclosed busbars
4.4.5 Temperature rise in insulated conductors
4.4.6 Longitudinal expansion of busbars
4.5 Rating power systems for earthquake safety
4.5.1 General principles
4.5.2 Experimental verification
4.5.3 Verification by calculation
4.6 Minimum clearances, protective barrier clearances and widths of gangways
4.6.1 Minimum clearances and protective barrier clearances in power systems with rated
voltages over 1 kV (DIN VDE 0101)
4.6.2 Walkways and gangways in power installations with rated voltages over 1kV (DIN
VDE0101)
4.6.3 Gangway widths in power installations with rated voltages of up to 1 kV (DIN VDE
0100 Part 729)
4.7 Civil construction requirements
4.7.1 Indoor installations
4.7.2 Outdoor installations
4.7.3 Installations subject to special conditions
4.7.4 Battery compartments
4.7.5 Transformer installation
4.7.6 Fire prevention
4.7.7 Shipping dimensions

5 Protective Measures for Persons and Installations
5.1 Electric shock protection in installations up to 1000V as per DIN VDE 0100
5.1.1 Protection against direct contact (basic protection)
5.1.2 Protection in case of indirect contact (fault protection)
5.1.3 Protection by extra low voltage
5.1.4 Protective conductors, PEN conductors and equipotential bonding conductors
5.2 Protection against contact in installations above 1000V as per DIN VDE 0101
5.2.1 Protection against direct contact
5.2.2 Protection in the case of indirect contact
5.3 Earthing
5.3.1 Fundamentals, definitions and specifications
5.3.2 Earthing material
5.3.3 Dimensioning of earthing systems
5.3.4 Earthing measurements
5.4 Lightning protection
5.4.1 General
5.4.2 Methods of lightning protection
5.4.3 Overhead earth wires
5.4.4 Lightning rods
5.5 Electromagnetic compatibility
5.5.1 Origin and propagation of interference quantities
5.5.2 Effect of interference quantities on interference sinks
5.5.3 EMC measures
5.6 Partial-discharge measurement
5.6.1 Partial-discharge processes
5.6.2 Electrical partial-discharge measurement procedures
5.7 Effects of climate and corrosion protection
5.7.1 Climates
5.7.2 Effects of climate and climatic testing
5.7.3 Reduction of insulation capacity by humidity
5.7.4 Corrosion protection
5.8 Degrees of protection for electrical equipment of up to 72.5 kV (VDE 0470 Part 1, EN
60529)
6 Methods and Aids for Planning Installations
6.1 Planning of switchgear installations
6.1.1 Concept, boundary conditions, pc calculation aids
6.1.2 Planning of high-voltage installations
6.1.3 Project planning of medium-voltage installations
6.1.4 Planning of low-voltage installations
6.1.5 Calculation of short-circuit currents, computer-aided
6.1.6 Calculation of cable cross-sections, computer-aided
6.1.7 Planning of cable routing, computer-aided
6.2 Reference designations and preparation of documents
6.2.1 Item designation of electrical equipment as per DIN 40719 Part 2
6.2.2 Preparation of documents
6.2.3 Classification and designation of documents
6.2.4 Structural principles and reference designation as per IEC 61346
6.3 CAD/CAE methods applied to switchgear engineering
6.3.1 Terminology, standards
6.3.2 Outline of hardware and software for CAD systems
6.3.3 Overview of CAD applications in ABB switchgear engineering
6.4 Drawings
6.4.1 Drawing formats

6.4.2 Standards for representation
6.4.3 Lettering in drawings, line thicknesses
6.4.4 Text panel, identification of drawing
6.4.5 Drawings for switchgear installations
6.4.6 Drawing production, drafting aids
7 Low Voltage Switchgear
7.1 Switchgear apparatus
7.1.1 Low voltage switchgear as per VDE 0660 Part 100 and following parts, EN 60947 ... and IEC 60947
7.1.2 Low voltage fuses as per VDE 0636 Part 10 and following parts, EN 60269-...
IEC602697.1.3 Protective switchgear for household and similar uses
7.1.4 Selectivity
7.1.5 Backup protection
7.2 Low-voltage switchgear installations and distribution boards
7.2.1 Basics
7.2.2 Standardized terms
7.2.3 Classification of switchgear assemblies
7.2.4 Internal subdivision by barriers and partitions
7.2.5 Electrical connections in switchgear assemblies
7.2.6 Verification of identification data of switchgear assemblies
7.2.7 Switchgear assemblies for operation by untrained personnel
7.2.8 Retrofitting, changing and maintaining low-voltage switchgear assemblies
7.2.9 Modular low-voltage switchgear system (MNS system)
7.2.10 Low-voltage distribution boards in cubicle-type assembly
7.2.11 Low-voltage distribution boards in multiple box-type assembly
7.2.12 Systems for reactive power compensation
7.2.13 Control systems for low-voltage switchgear assemblies
7.3 Design aids
7.4 Rated voltage 690
V
7.5 Selected areas of
application
7.5.1 Design of low-voltage substations to withstand induced vibrations
7.5.2 Low voltage substations in internal arc-proof design for offshore applications
7.5.3 Substations for shelter
8 Switchgear and Switchgear Installations for High-Voltage up to and including 52 kV
(Medium Voltage)
8.1 Switchgear apparatus (= 52kV)
8.1.1 Disconnectors
8.1.2 Switch-disconnectors
8.1.3 Earthing switches
8.1.4 Position indication
8.1.5 HV fuse links (DIN EN 60 282-1 (VDE 0670 Part 4))
8.1.6 Is-limiter® - fastest switching device in the world
8.1.7 Circuit-breakers
8.1.8 Vacuum contactors
8.2 Switchgear installations (= 52 kV)
8.2.1 Specifications covering HV switchgear installations
8.2.2 Switchgear as per DIN VDE 0101

8.2.3 Metal-enclosed switchgear as per DIN EN 60298 (VDE 0670 Part 6)
8.2.4 Metal-enclosed air-insulated switchgear as per DIN EN 60298 (VDE 0670 Part 6)
8.2.5 Metal-enclosed gas-insulated switchgear under DIN EN 60298 (VDE 0670 Part 6)
8.2.6 Control systems for medium-voltage substations
8.3 Terminal connections for medium-voltage installations
8.3.1 Fully-insulated transformer link with cables
8.3.2 SF6-insulated busbar connection
8.3.3 Solid-insulated busbar connection
9 High-Current Switchgear
9.1 Generator circuit-breaker
9.1.1 Selection criteria for generator circuit-breakers
9.1.2 Generator circuit-breaker type ranges HG... and HE... (SF6 gas breaker)
9.1.3 Generator circuit-breaker type DR (air-blast breaker)
9.1.4 Generator circuit-breaker type VD 4 G (vacuum breaker)
9.2 High-current bus ducts (generator bus ducts)
9.2.1 General requirements
9.2.2 Types, features, system selection
9.2.3 Design dimensions
9.2.4 Structural design
9.2.5 Earthing system
9.2.6 Air pressure/Cooling system
10 High-Voltage Apparatus
10.1 Definitions and electrical parameters for switchgear
10.2 Disconnectors and earthing
switches
10.2.1 Rotary disconnectors
10.2.2 Single-column (pantograph) disconnector TFB
10.2.3 Two-column vertical break disconnectors
10.2.4 Single-column earthing switches
10.2.5 Operating mechanisms for disconnectors and earthing switches
10.3 Switch-disconnectors
10.4 Circuit-breakers
10.4.1 Function, selection
10.4.2 Design of circuit-breakers for high-voltage (>52kV)
10.4.3 Interrupting principle and important switching cases
10.4.4 Quenching media and operating principle
10.4.5 Operating mechanism and control
10.5 Instrument transformers for switchgear installations
10.5.1 Definitions and electrical quantities
10.5.2 Current transformer
10.5.3 Inductive voltage transformers
10.5.4 Capacitive voltage transformers
10.5.5 Non-conventional transformers
10.6 Surge arresters
10.6.1 Design, operating principle
10.6.2 Application and selection of MO surge arresters
11 High-Voltage Switchgear Installations
11.1 Summary and circuit configuration
11.1.1 Summary
11.1.2 Circuit configurations for high- and medium-voltage switchgear installations
11.2 SF6-gas-insulated switchgear (GIS)

11.2.1 General
11.2.2 SF6 gas as insulating and arc-quenching medium
11.2.3 GIS for 72.5 to 800 kV
11.2.4 SMART-GIS
11.2.5 Station arrangement
11.2.6 Station layouts
11.2.7 SF6-insulated busbar links
11.3 Outdoor switchgear installations
11.3.1 Requirements, clearances
11.3.2 Arrangement and components
11.3.3 Switchyard layouts
11.4 Innovative HV switchgear technology
11.4.1 Concepts for the future
11.4.1.1 Process electronics (sensor technology, PISA)
11.4.1.2 Monitoring in switchgear installations
11.4.1.3 Status-oriented maintenance
11.4.2 Innovative solutions
11.4.2.1 Compact outdoor switchgear installations
11.4.2.2 Hybrid switchgear installations
11.4.3 Modular planning of transformer substations
11.4.3.1 Definition of modules
11.4.3.2 From the customer requirement to the modular system solution
11.5 Installations for high-voltage
direct-current (HDVC) transmission
11.5.1 General
11.5.2 Selection of main data for HDVC transmission
11.5.3 Components of a HDVC station
11.5.4 Station layout
11.6 Static var (reactive power) composition (SVC)
11.6.1 Applications
11.6.2 Types of compensator
11.6.3 Systems in operation
12 Transformers and Other Equipment for Switchgear Installations
12.1 Transformers
12.1.1 Design, types and dimensions
12.1.2 Vector groups and connections
12.1.3 Impedance voltage, voltage variation and short-circuit current withstand
12.1.4 Losses, cooling and overload capacity
12.1.5 Parallel operation
12.1.6 Protective devices for transformers
12.1.7 Noise levels and means of noise abatement
12.2 Current-limiting reactors EN 60289 (VDE 0532 Part 20)
12.2.1 Dimensioning
12.2.2 Reactor connection
12.2.3 Installation of reactors
12.3 Capacitors
12.3.1 Power capacitors
12.3.2 Compensation of reactive power
12.4 Resistor devices
12.5
Rectifiers

13 Conductor Materials and Accessories for Switchgear Installations
13.1 Busbars, stranded-wire conductors and insulators
13.1.1 Properties of conductor materials
13.1.2 Busbars for switchgear installations
13.1.3 Drilled holes and bolted joints for busbar conductors
13.1.4 Technical values for stranded-wire conductors
13.1.5 Post-type insulators and overhead-line insulators
13.2 Cables, wires and flexible cords
13.2.1 Specifications, general
13.2.2 Current-carrying capacity
13.2.3 Selection and protection
13.2.4 Installation of cables and wires
13.2.5 Cables for control, instrument transformers and auxiliary supply in high-voltage
switchgear installations
13.2.6 Telecommunications cables
13.2.7 Data of standard VDE, British and US cables
13.2.8 Power cable accessories for low- and medium- voltage
13.3 Safe working equipment in switchgear installations
14 Protection and Control Systems in Substations and Power Networks
14.1 Introduction
14.2 Protection
14.2.1 Protection relays and protection systems
14.2.2 Advantages of numeric relays
14.2.3 Protection of substations, lines and transformers
14.2.4 Generator unit protection
14.3 Control, measurement and regulation (secondary systems)
14.3.1 D.C. voltage supply
14.3.2 Interlocking
14.3.3 Control
14.3.4 Indication
14.3.5 Measurement
14.3.6 Synchronizing
14.3.7 Metering
14.3.8 Recording and logging
14.3.9 Automatic switching control
14.3.10 Transformer control and voltage regulation
14.3.11 Station control rooms
14.4 Station control with microprocessors
14.4.1 Outline
14.4.2 Microprocessor and conventional secondary systems compared
14.4.3 Structure of computerized control systems
14.4.4 Fibre-optic cables
14.5 Network control and telecontrol
14.5.1 Functions of network control systems
14.5.2 Control centres with process computers for central network management
14.5.3 Control centres, design and equipment
14.5.4 Telecontrol and telecontrol systems
14.5.5 Transmission techniques
14.5.6 Technical conditions for telecontrol systems and interfaces with substations
14.6 Load management , ripple control
14.6.1 Purpose of ripple control and load management

14.6.2 Principle and components for ripple-control systems
14.6.3 Ripple-control command centre
14.6.4 Equipment for ripple control
14.6.5 Ripple control recievers
15 Secondary Installations
15.1 Stand-by power systems
15.1.1 Overview
15.1.2 Stand-by power with generator systems
15.1.3 Uninterruptible power supply with stand-by generating sets (rotating UPS
installations)
15.1.4 Uninterruptible power supply with static rectifiers (static UPS installations)
15.2 High-speed transfer devices
15.2.1 Applications, usage, tasks
15.2.2 Integration into the installation
15.2.3 Design of high-speed transfer devices
15.2.4 Functionality
15.2.5 Types of transfer
15.3 Stationary batteries and battery installations, DIN VDE 0510, Part 2 798
15.3.1 Types and specific properties of batteries
15.3.2 Charging and discharging batteries
15.3.3 Operating modes for batteries
15.3.4 Dimensioning batteries
15.3.5 Installing batteries, types of installation
15.4 Installations and lighting in switchgear installations
15.4.1 Determining internal requirements for electrical power for equipment
15.4.2 Layout and installation systems
15.4.3 Lighting installations
15.4.4 Fire-alarm systems
15.5 Compressed-air systems in switchgear installations
15.5.1 Application, requirements, regulations
15.5.2 Physical basics
15.5.3 Design of compressed-air systems
15.5.4 Rated pressures and pressure ranges
15.5.5 Calculating compressed-air generating and storage systems
15.5.6 Compressed-air distribution systems
16 Materials and Semi-Finished Products
for Switchgear Installations
16.1 Iron and steel
16.1.1 Structural steel, general
16.1.2 Dimensions and weights of steel bars, sections and tubes
16.1.3 Stresses in steel components
16.2 Non-ferrous metals
16.2.1 Copper for electrical engineering
16.2.2 Aluminium for electrical engineering
16.2.3 Brass
16.3 Insulating materials
16.3.1 Solid insulating materials
16.3.2 Liquid insulating materials
16.3.3 Gaseous insulating materials
16.4 Semi-finished products
16.4.1 Dimensions and weights of metal sheets, DIN EN 10130
16.4.2 Slotted steel strip
16.4.3 Screws and accessories

16.4.4 Threads for bolts and screws
16.4.5 Threads for electrical engineering
17 Miscellaneous
17.1 DIN VDE specifications and IEC publications for substation design
17.2 Application of European directives to high-voltage switchgear installations. CE
mark
17.3 Quality in
switchgear
17.4 Notable events and achievements in the history of ABB switchgear
technology

1

1

Fundamental Physical and Technical Terms

1.1

Units of physical quantities

1.1.1 The International System of Units (Sl)
The statutory units of measurement are1)
1. the basic units of the International System of Units (Sl units) for the basic quantities
length, mass, time, electric current, thermodynamic temperature and luminous intensity,
2. the units defined for the atomic quantities of quantity of substance, atomic mass and
energy,
3. the derived units obtained as products of powers of the basic units and atomic units
through multiplication with a defined numerical factor,
4. the decimal multiples and sub-multiples of the units stated under 1-3.

Table 1-1
Basic SI units
Quantity

Units
Symbol

Units
Name

Length
Mass
Time
Electric current
Thermodynamic temperature
Luminous intensity

m
kg
s
A
K
cd

metre
kilogramme
second
ampere
kelvin
candela

mol

mole

Atomic units
Quantity of substance
Table 1-2
Decimals
Multiples and sub-multiples of units
Decimal power

Prefix

Symbol

1012

Tera
Giga
Mega
Kilo
Hekto
Deka
Dezi

T
G
M
k
h
da
d

109
106
103
102
101
10–1
1)DIN

10–2
10–3
10–6
10–9
10–12
10–15
10–18

Zenti
Milli
Mikro
Nano
Piko
Femto
Atto

c
m
µ
n
p
f
a

1301

1

List of units
1

2

No.

Quantity

3

4

Sl unit1)
Name

5

6

7

8

Relationship1)

Remarks

Other units
Symbol

Name

Symbol

1 Length, area, volume
1.1

Length

metre

m

1.2

Area

square metre

m2

1.3

1.4

1.5

1)

Volume

Reciprocal
length

Elongation

cubic metre

see Note to No. 1.1

are
hectare

a
ha

1 a = 102 m2
1 ha = 104 m2

litre

l

1 l = 1 dm3 = 10–3 m3

dioptre

dpt

1 dpt = 1/m

for land measurement
only

m3

reciprocal metre 1/m

metre per
metre

m/m

See also notes to columns 3 and 4 and to column 7 on page 15.

(continued)



2

Table 1-3

only for refractive index of
optical systems
Numerical
value
of
elongation often expressed in
per cent

Table 1-3 (continued)
List of units
1

2

No.

Quantity

3

4

Sl unit1)

5

6

7

8

Relationship1)

Remarks

Other units

Name

Symbol

radian

rad

Name

Symbol

2 Angle
2.1

Plane angle
(angle)

1 rad = 1 m/m
1 full angle = 2 π rad

full angle

2.2
1)

Solid angle

steradian

right angle

v

1v

π
= — rad
2

degree

°

1°

π
= —— rad
180

minute
second

'
"

1'
1"

= 1°/60
= 1’/60

gon

gon

1 gon

π
= —— rad
200

1 sr

= 1m2/m2

sr













see DIN 1315
In calculation the unit rad as
a factor can be replaced by
numerical 1.

see DIN 1315

See also notes to columns 3 and 4 and to column 7 on page 15.

3

(continued)

1

4

Table 1-3 (continued)
List of units
1

2

No.

Quantity

3

4

Sl unit1)

5

6

7

8

Relationship1)

Remarks

Other units

Name

Symbol

kilogramme

kg

Name

Symbol

3 Mass
3.1

3.2

1)

Mass

Mass per unit
length

kilogramme
per metre

Units of weight used
terms
for
mass
expressing quantities
goods are the units
mass, see DIN 1305
gramme
tonne
atomic
mass unit

g
t
u

1g
1t
1u

At the present state of
= 10–3 kg
measuring technology the
= 103 kg
= 1.66053 · 10–27 kg 3-fold standard deviation for
the relationship for u given
in col. 7 is ± 3 · 10–32 kg.

metric carat

Kt

1 Kt

= 0.2 · 10–3 kg

Tex

tex

1 tex = 10–6 kg/m
= 1 g/km

only for gems

kg/m

See also notes to columns 3 and 4 and to column 7 on page 15.

(continued)

as
in
of
of

only for textile fibres and
yarns, see DIN 60905
Sheet 1

Table 1-3 (continued)
List of units
1

2

No.

Quantity

3

4

Sl unit1)

5

6

7

8

Relationship1)

Remarks

Other units

Name

Symbol

Name

Symbol

3.3

Density

kilogramme
per
cubic metre

kg/m3

see DIN 1306

3.4

Specific
volume

cubic metre
per
kilogramme

m3/kg

see DIN 1306

3.5

Moment of
inertia

kilogrammesquare metre

kg m2

see DIN 5497 and
Note to No. 3.5

1)

See also notes to columns 3 and 4 and to column 7 on page 15.

(continued)

5

1

6

Table 1-3 (continued)
List of units
1

2

3
Sl

4
unit1)

5

6

7

8

Relationship1)

Remarks

1 min = 60 s
1 h = 60 min
1 d = 24 h

see DIN 1355

Other units

No. Quantity
Name

Symbol

Name

Symbol

second

s

minute
hour
day
year

min
h
d
a

4 Time
4.1

Time

4.2

Frequency

hertz

Hz

4.3

Revolutions
per second

reciprocal
second

1/s

1)

1 Hz = 1/s

reciprocal
minute

See also notes to columns 3 and 4 and to column 7 on page 15.

(continued)

In the power industry a year
is taken as 8760 hours.
See also Note to No. 4.1.

1/min

1/min = 1/(60 s)

1 hertz is equal to the
frequency of a periodic event
having a duration of 1 s.
If it is defined as the
reciprocal of the time of
revolution, see DIN 1355.

Table 1-3 (continued)
List of units
1

2

No.

Quantity

3

4

Sl unit1)

6

7

8

Relationship1)

Remarks

Other units

Name

Symbol

4.4

Cyclic
frequency

reciprocal
second

1/s

4.5

Velocity

metre per
second

m/s

4.6

Acceleration

metre per
m/s2
second squared

4.7

Angular
velocity

radian per
second

4.8

Angular
acceleration

radian per
rad/s2
second squared

1)

5

Name

Symbol

kilometre
per hour

km/h

1 km/h =

1
—— m/s
3.6

rad/s

See also notes to columns 3 and 4 and to column 7 on page 15.

(continued)

7

1

8

Table 1-3 (continued)
List of units
1

2

No.

Quantity

3

4

Sl unit1)
Name

5

6

7

8

Relationship1)

Remarks

Other units
Symbol

Name

Symbol

5 Force, energy, power
= 1 kg m/s2

5.1

Force

newton

N

1N

5.2

Momentum

newton-second

Ns

1 Ns = 1 kg m/s

5.3

Pressure

pascal

Pa
bar

1)

See also notes to columns 3 and 4 and to column 7 on page 15.

(continued)

bar

1 Pa = 1 N/m2
1 bar = 105 Pa

Units of weight as a quantity
of force are the units of force,
see DIN 1305.

see Note to columns 3 and 4
see DIN 1314

Table 1-3 (continued)
List of units
1

2

No.

Quantity

3

4

Sl unit1)

5

6

7

8

Relationship1)

Remarks

In many technical fields it has
been agreed to express mechanical stress and strength
in N/mm2.
1 N/mm2 = 1 MPa.

Other units

Name

Symbol

Name

Symbol

5.4

Mechanical
stress

newton per
square metre,
pascal

N/m2, Pa

1 Pa = 1 N/m2

5.5

Energy, work,
quantity of heat

joule

J

1J

kilowatt-hour
electron volt

5.6

Torque

newton-metre

5.7

Angular
momentum

newton-second- Nsm
metre

1)

Nm

kWh
eV

= 1 Nm = 1 Ws
see DIN 1345
= 1 kg m2/s2
1 kWh = 3.6 MJ
At the present state of
1 eV = 1.60219 ·10–19 J measuring technology the
3-fold standard deviation for
the relationship given in col. 7
is ± 2 · 10–24 J.
1 Nm = 1 J = 1 Ws
1 Nsm = 1 kg m2/s

See also notes to columns 3 and 4 and to column 7 on page 15.

(continued)

9

1

10

Table 1-3 (continued)
List of units
1

2

No.

Quantity

3

4

Sl unit1)

5.8

Power
energy flow,
heat flow

5

6

7

8

Relationship1)

Remarks

Other units

Name

Symbol

Name

Symbol

watt

W

1 W = 1 J/s
=1 N m/s
= 1 VA

The watt is also termed voltampere (standard symbol
VA) when expressing electrical apparent power, and Var
(standard symbol var) when
expressing electrical reactive
power, see DIN 40110.

1 Pas = 1 Ns/m2
= 1 kg/(sm)

see DIN 1342

6 Viscometric quantities
6.1

Dynamic
viscosity

pascal-second

Pas

6.2

Kinematic
viscosity

square metre
per second

m2/s

1)

See also notes to columns 3 and 4 and to column 7 on page 15.

(continued)

see DIN 1342

Table 1-3 (continued)
List of units
1

2

No.

Quantity

3

4

Sl unit1)
Name

5

6

7

8

Relationship1)

Remarks

The degree Celsius is
the special name for
kelvin when expressing
Celsius temperatures.

Thermodynamic temperature; see Note to No. 7.1 and
DIN 1345.
Kelvin is also the unit for
temperature differences and
intervals.
Expression of Celsius temperatures and Celsius temperature differences, see
Note to No 7.1.

Other units
Symbol

Name

Symbol

7 Temperature and heat
7.1

Temperature

kelvin

K

degree Celsius
(centigrade)

m2/s

see DIN 1341

Entropy, thermal joule
capacity
per kelvin

J/K

see DIN 1345

Thermal
conductivity

W/(K m)

see DIN 1341

7.2

Thermal
diffusivity

7.3

7.4
11

1)

°C

square metre
per second

watt per
kelvin-metre

See also notes to columns 3 and 4 and to column 7 on page 15.

(continued)

1

12

Table 1-3 (continued)
List of units
1

2

No.

Quantity

3

4

Sl unit1)

7.5

Heat transfer
coefficient

5

6

7

8

Relationship1)

Remarks

Other units

Name

Symbol

Name

Symbol

watt per
kelvin-square
metre

W/(Km2)

see DIN 1341

A

see DIN 1324 and

8 Electrical and magnetic quantities
8.1

Electric current,
magnetic
potential
difference

ampere

DIN 1325

8.2

Electric voltage, volt
electric potential
difference

V

1V

=1 W/A

see DIN 1323

83

Electric
conductance

siemens

S

1S

= A/V

see Note to columns 3 and
4 and also DIN 1324

8.4

Electric
resistance

ohm

Ω

1Ω

= 1/S

see DIN 1324

1)

See also notes to columns 3 and 4 and to column 7 on page 15.

(continued)

Table 1-3 (continued)
List of units
1

2

No.

Quantity

3

4

Sl unit1)
Name
8.5

5

6

7

8

Other units
Symbol

Name

Symbol

ampere-hour

Ah

Relationship1)

Remarks

1 C = 1 As
1 Ah = 3600 As

see DIN 1324

1F

see DIN 1357

Quantity of
coulomb
electricity, electric
charge

C

8.6

Electric
capacitance

farad

F

8.7

Electric flux
density

coulomb per
square metre

C/m2

see DIN 1324

8.8

Electric field
strength

volt per metre

V/m

see DIN 1324

8.9

Magnetic flux

weber,
volt-second

Wb, Vs

1 Wb = 1 Vs

8.10 Magnetic flux
tesla
density, (induction)

T

1T

= 1 Wb/m2

see DIN 1325

8.11 Inductance
(permeance)

H

1H

= 1 Wb/A

see DIN 1325

13

1)

henry

See also notes to columns 3 and 4 and to column 7 on page 15.

= 1 C/V

see DIN 1325

(continued)

1

14

Table 1-3 (continued)
List of units
1

2

No.

Quantity

3

4

Sl unit1)

8.12 Magnetic field
intensity

5

6

7

8

Relationship1)

Remarks

Other units

Name

Symbol

Name

Symbol

ampere
per metre

A/m

see DIN 1325

9 Photometric quantities
9.1

Luminous
intensity

candela

cd

see DIN 5031 Part 3. The
word candela is stressed on
the 2nd syllable.

9.2

Luminance

candela per
square metre

cd/m2

see DIN 5031 Part 3

9.3

Luminous flux

lumen

Im

9.4
1)

Illumination

lux

Ix

See also notes to columns 3 and 4 and to column 7 on page 15.

1 Im = 1 cd · sr

see DIN 5031 Part 3

Im/m2

see DIN 5031 Part 3

1 lx = 1

Notes to Table 1-3
To column 7:

Thus,
t = T – T0 = T – 273.15 K.

(1)

A number having the last digit in bold type denotes that this number is defined by
agreement (see DIN 1333).

When expressing Celsius temperatures, the standard symbol °C is to be used.

To No. 1.1:

The difference ∆ t between two Celsius temperatures, e. g. the temperatures
t 1 = T1 – T0 and t 2 = T2 – T0, is

The nautical mile is still used for marine navigation (1 nm = 1852 m). For
conversion from inches to millimetres see DIN 4890, DIN 4892, DIN 4893.
To No. 3.5:
When converting the so-called “flywheel inertia GD2” into a mass moment of
inertia J, note that the numerical value of GD2 in kp m2 is equal to four times the
numerical value of the mass moment of inertia J in kg m2.
To No. 4.1:
Since the year is defined in different ways, the particular year in question should
be specified where appropriate.
3 h always denotes a time span (3 hours), but 3h a moment in time (3 o’clock).
When moments in time are stated in mixed form, e.g. 2h25m3s, the abbreviation
min may be shortened to m (see DIN 1355).
To No. 7.1:
The (thermodynamic) temperature (T), also known as “absolute temperature”, is
the physical quantity on which the laws of thermodynamics are based. For this
reason, only this temperature should be used in physical equations. The unit
kelvin can also be used to express temperature differences.

∆ t = t 1 –t 2 = T1 – T2 = ∆ T

(2)

A temperature difference of this nature is no longer referred to the thermodynamic temperature T0, and hence is not a Celsius temperature according to the
definition of Eq. (1).
However, the difference between two Celsius temperatures may be expressed
either in kelvin or in degrees Celsius, in particular when stating a range of
temperatures, e. g. (20 ± 2) °C
Thermodynamic temperatures are often expressed as the sum of T0 and a
Celsius temperature t, i. e. following Eq. (1)
T = T0 + t

(3)

and so the relevant Celsius temperatures can be put in the equation straight
away. In this case the kelvin unit should also be used for the Celsius temperature
(i. e. for the “special thermodynamic temperature difference”). For a Celsius
temperature of 20 °C, therefore, one should write the sum temperature as
T = T0 + t = 273.15 K + 20 K = 293.15 K

(4)

Celsius (centigrade) temperature (t) is the special difference between a given
thermodynamic temperature T and a temperature of T0 = 273.15 K.

15

1

16

1.1.2 Other units still in common use; metric, British and US measures
Some of the units listed below may be used for a limited transition period and in certain exceptional cases. The statutory requirements vary
from country to country.
ångström
atmosphere physical
atmosphere technical
British thermal unit
calorie
centigon
degree
degree fahrenheit
dyn
erg
foot
gallon (UK)
gallon (US)
gauss
gilbert
gon
horsepower
hundredweight (long)
inch (inches)
international ampere
international farad
international henry
international ohm
international volt
international watt
kilogramme-force, kilopond

Å
atm
at, ata
Btu
cal
c
deg, grd
°F
dyn
erg
ft
gal (UK)
gal (US)
G.Gs
Gb
g
hp
cwt
in, "
Aint
Fint
Hint
Ωint
Vint
Wint
kp, kgf

length
pressure
pressure
quantity of heat
quantity of heat
plane angle
temperature difference
temperature
force
energy
length
volume
liquid volume
magnetic flux density
magnetic potential difference
plane angle
power
mass
length
electric current
electrical capacitance
inductance
electrical resistance
electrical potential
power
force

1 Å = 0.1 nm = 10–10m
1 atm = 101 325 Pa
1 at = 98 066.5 Pa
1 Btu ≈ 1055.056 J
1 cal = 4.1868 J
1 c = 1 cgon = 5 π · 10–5 rad
1 deg = 1 K
TK = 273.15 + (5/9) · (tF – 32)
1 dyn = 10–5 N
1 erg = 10–7 J
1 ft = 0.3048 m
1 gal (UK) ≈ 4.54609 · 10–3 m3
1 gal (US) ≈ 3.78541 · 10–3 m3
1 G = 10–4T
1 Gb = (10/4 π) A
1 g = 1 gon = 5 π · 10–3 rad
1 hp ≈ 745.700 W
1 cwt ≈ 50.8023 kg
1 in = 25.4 mm = 254 · 10–4 m
1 Aint ≈ 0.99985 A
1 Fint = (1/1.00049) F
1 Hint = 1.00049 H
1 Ωint = 1.00049 Ω
1 Vint = 1.00034 V
1 Wint ≈ 1.00019 W
1 kp = 9.80665 N ≈ 10 N

Unit of mass
maxwell
metre water column
micron
millimetres of mercury
milligon
oersted
Pferdestärke, cheval-vapeur
Pfund
pieze
poise
pond, gram
-force
pound1)
poundal
poundforce
sea mile, international
short hundredweight
stokes
torr
typographical point
yard
Zentner
1)

ME
M, Mx
mWS
µ
mm Hg
cc
Oe
PS, CV
Pfd
pz
P

mass
magnetic flux
pressure
length
pressure
plane angle
magnetic field strength
power
mass
pressure
dynamic viscosity

1 ME = 9.80665 kg
1 M = 10 nWb = 10–8 Wb
1 mWS = 9806.65 PA ≈ 0,1 bar
1 µ = 1 µm = 10–6 m
1 mm Hg ≈ 133.322 Pa
1 cc = 0.1 mgon = 5 π · 10–7 rad
10e = (250/π) A/m
1 PS = 735.49875 W
1 Pfd = 0.5 kg
1 pz = 1 mPa = 10–3 Pa
1 P = 0.1 Pa · s

p, gf
Ib
pdl
Ibf
n mile
sh cwt
St
Torr
p
yd
z

force
mass
force
force
length (marine)
mass
kinematic viscosity
pressure
length (printing)
length
mass

1 p = 9.80665 · 10–3 N ≈ 10 mN
1 Ib ≈ 0.453592 kg
1 pdl ≈ 0.138255 N
1 Ibf ≈ 4.44822 N
1 n mile = 1852 m
1 sh cwt ≈ 45.3592 kg
1 St = 1 cm2/s = 10–4 m2/s
1 Torr ≈ 133.322 Pa
1 p = (1.00333/2660) m ≈ 0.4 mm
1 yd = 0.9144 m
1 z = 50 kg

UK and US pounds avoirdupois differ only after the sixth decimal place.

17

1

18

Table 1-4
Metric, British and US linear measure
Metric units of length

British and US units of length

Kilometre

Metre

Decimetre

Centimetre

Millimetre

Mile

Yard

Foot

Inch

Mil

km

m

dm

cm

mm

mile

yd

ft

in or "

mil

1
0.001
0.0001
0.00001
0.000001
1.60953
0.000914
0.305 · 10–3
0.254 · 10–4
0.254 · 10–7

1 000
1
0.1
0.01
0.001
1 609.53
0.9143
0.30479
0.02539
0.254 · 10–4

10 000
10
1
0.1
0.01
16 095.3
9.1432
3.0479
0.25399
0.254 · 10–3

100 000
100
10
1
0.1
160 953
91.432
30.479
2.53997
0.00254

1 000 000
1 000
100
10
1
1 609 528
914.32
304.79
25.3997
0.02539

0.6213
0.6213 · 10–3
0.6213 · 10–4
0.6213 · 10–5
0.6213 · 10–6
1
0.5682 · 10–3
0.1894 · 10–3
0.158 · 10–4
0.158 · 10–7

1 093.7
1.0937
0.1094
0.01094
0.001094
1 760
1
0.3333
0.02777
0.0277 · 10–3

3 281
3.281
0.3281
0.03281
0.003281
5 280
3
1
0.0833
0.0833 · 10–3

39 370
39.370
3.937
0.3937
0.03937
63 360
36
12
1
0.001

3 937 · 104
39 370
3 937.0
393.70
39.37
6 336 · 104
36 000
12 000
1 000
1

Special measures: 1 metric nautical mile = 1852 m
1 metric land mile = 7500 m

1 Brit. or US nautical mile = 1855 m
1 micron (µ) = 1/1000 mm = 10 000 Å

Table 1-5
Metric, British and US square measure
Metric units of area

British and US units of area

Square
kilometres

Square
metre

Square
decim.

Square
centim.

Square
millim.

Square
mile

Square
yard

km2

m2

dm2

cm2

mm2

sq.mile

sq.yd

1
1 · 10–6
1 · 10–8
1 · 10–10
1 · 10–12
2.58999
0.8361 · 10–6
9.290 · 10–8
6.452 · 10–10
506.7 · 10–18

106

1·
1
1 · 10–2
1 · 10–4
1 · 10–6
2 589 999
0.836130
9.290 · 10–2
6.452 · 10–4
506.7 · 10–12

106

100 ·
100
1
1 · 10–2
1 · 10–4
259 · 106
83.6130
9.29034
6.452 · 10–2
506.7 · 10–10

108

100 ·
10 000
100
1
1 · 10–2
259 · 108
8 361.307
929.034
6.45162
506.7 · 10–8

1010

100 ·
1 000 000
10 000
100
1
259 · 1010
836 130.7
92 903.4
645.162
506.7 · 10–6

Special measures: 1 hectare (ha)
= 100 are (a)
1 are (a)
= 100 m2
1 Bad. morgen = 56 a = 1.38 acre
1 Prussian morgen = 25.53 a = 0.63 acre
1 Württemberg morgen = 31.52 a = 0.78 acre
1 Hesse morgen = 25.0 a = 0.62 acre
1 Tagwerk (Bavaria) = 34.07 a = 0.84 acre
1 sheet of paper = 86 x 61 cm
gives 8 pieces size A4 or 16 pieces A5
or 32 pieces A6

0.386013
0.386 · 10–6
0.386 · 10–8
0.386 · 10–10
0.386 · 10–12
1
0.3228 · 10–6
0.0358 · 10–6
0.2396 · 10–9
0.196 · 10–15

103

1 196 ·
1.1959
0.01196
0.1196 · 10–3
0.1196 · 10–5
30 976 · 10 2
1
0.11111
0.7716 ·10–3
0.607 · 10–9

Square
foot

Square
inch

sq.ft

sq.in
104

1076 ·
10.764
0.10764
0.1076 · 10–2
0.1076 · 10–4
27 878 · 103
9
1
0.006940
0.00547 · 10–6

Circular
mils
cir.mils
106

1 550 ·
1 550
15.50
0.1550
0.00155
40 145 · 105
1296
144
1
0.785 · 10–6

1 section (sq.mile) = 64 acres = 2,589 km2
1 acre = 4840 sq.yds = 40.468 a
1 sq. pole = 30.25 sq.yds = 25.29 m2
1 acre = 160 sq.poles = 4840 sq.yds = 40.468 a
1 yard of land = 30 acres = 1214.05 a
1 mile of land = 640 acres = 2.589 km2








197.3 · 1013
197.3 · 107
197.3 · 105
197.3 · 103
1 973
5 098 · 1012
1 646 · 106
183 · 106
1.27 · 106
1

USA

Brit.

19

1

20

Table 1-6
Metric, British and US cubic measures
Metric units of volume

British and US units of volume

US liquid measure

Cubic
metre

Cubic
decimetre

Cubic
centimetre

Cubic
millimetre

Cubic
yard

Cubic
foot

Cubic
inch

Gallon

Quart

Pint

m3

dm3

cm3

mm3

cu.yd

cu.ft

cu.in

gal

quart

pint

1
1 · 10–3
1 · 10–6
1 · 10–9
0.764573
0.0283170
0.1638 · 10–4
3.785 · 10–3
0.9463 · 10–3
0.4732 · 10–3

1 000
1
1 · 10–3
1 · 10–6
764.573
28.31701
0.0163871
3.785442
0.9463605
0.4731802

1 000 · 103
1 000
1
1 · 10–3
764 573
28 317.01
16.38716
3 785.442
946.3605
473.1802

1 000 · 106 1.3079
1 000 · 103 1.3079 · 10–3
1 000
1.3079 · 10–6
1
1.3079 · 10–9
764 573 · 10 3 1
28 317 013 0.037037
16387.16
0.2143 · 10–4
3 785 442
0.0049457
946 360.5
0.0012364
473 180.2
0.0006182

35.32
0.03532
0.3532 · 10–4
0.3532 · 10–7
27
1
0.5787 · 10–3
0.1336797
0.0334199
0.0167099

61 · 103
61.023
0.061023
0.610 · 10–4
46 656
1 728
1
231
57.75
28.875

264.2
0.2642
0.2642 · 10–3
0.2642 · 10–6
202
7.48224
0.00433
1
0.250
0.125

1 056.8
1.0568
1.0568 · 10–3
1.0568 · 10–6
808
29.92896
0.01732
4
1
0.500

2 113.6
2.1136
2.1136 · 10–3
2.1136 · 10–6
1 616
59.85792
0.03464
8
2
1

1
Table 1-7
Conversion tables
Millimetres to inches, formula: mm x 0.03937 = inch
mm 0

1

2

3

4

5

6

7

8

9

10
10
20
30
40
50

0.03937
0.43307
0.82677
1.22047
1.61417
2.00787

0.07874
0.47244
0.86614
1.25984
1.65354
2.04724

0.11811
0.51181
0.90551
1.29921
1.69291
2.08661

0.15748
0.55118
0.94488
1.33858
1.73228
2.12598

0.19685
0.59055
0.98425
1.37795
1.77165
2.16535

0.23622
0.62992
1.02362
1.41732
1.81102
2.20472

0.27559
0.66929
1.06299
1.45669
1.85039
2.24409

0.31496
0.70866
1.10236
1.49606
1.88976
2.28346

0.35433
0.74803
1.14173
1.53543
1.92913
2.32283

5

6

7

8

9

0.39370
0.78740
1.18110
1.57480
1.96850

Inches to millimetres, formula: inches x 25.4 = mm
inch 0
10
10
20
30
40
50

1

2

3

4

25.4
50.8
76.2
101.6
254.0
279.4
304.8
330.2
355.6
508.0
533.4
558.8
584.2
609.6
762.0
787.4
812.8
838.2
863.6
1 016.0 1 041.4 1 066.8 1 092.2 1 117.6
1 270.0 1 295.4 1 320.8 1 246.2 1 371.6

127.0 152.4
177.8
381.0 406.4
431.8
635.0 660.4
685.8
889.0 914.4
939.8
1 143.0 1 168.4 1 193.8
1 397.0 1 422.4 1 447.8

203.2
457.2
711.2
965.2
1 219.2
1 473.2

228.6
482.6
736.6
990.8
1 244.6
1 498.6

Fractions of inch to millimetres
inch
¹⁄₆₄
¹⁄₃₂
³⁄₆₄
¹⁄₁₆
⁵⁄₆₄
³⁄₃₂
⁷⁄₆₄
¹⁄₈
⁹⁄₆₄
⁵⁄₃₂
¹¹⁄₆₄
³⁄₁₆
¹³⁄₆₄

mm

inch

mm

inch

mm

inch

mm

inch

mm

0.397
0.794
1.191
1.587
1.984
2.381
2.778
3.175
3.572
3.969
4.366
4.762
5.159

⁷⁄₃₂
¹⁵⁄₆₄
¹⁄₄
¹⁷⁄₆₄
⁹⁄₃₂
¹⁹⁄₆₄
⁵⁄₆
²¹⁄₆₄
¹¹⁄₃₂
²³⁄₆₄
³⁄₈
²⁵⁄₆₄
¹³⁄₃₂

5.556
5.953
6.350
6.747
7.144
7.541
7.937
8.334
8.731
9.128
9.525
9.922
10.319

²⁷⁄₆₄
⁷⁄₁₆
²⁹⁄₆₄
¹⁵⁄₃₂
³¹⁄₆₄
¹⁄₂
³³⁄₆₄
¹⁷⁄₃₂
³⁵⁄₆₄
⁹⁄₁₆
³⁷⁄₆₄
¹⁹⁄₃₂
³⁹⁄₆₄

10.716
11.112
11.509
11.906
12.303
12.700
13.097
13.494
13.891
14.287
14.684
15.081
15.478

⁵⁄₈
⁴¹⁄₆₄
²¹⁄₃₂
⁴³⁄₆₄
¹¹⁄₁₆
⁴⁵⁄₆₄
²³⁄₃₂
⁴⁷⁄₆₄
³⁄₄
⁴⁹⁄₆₄
²⁵⁄₃₂
⁵¹⁄₆₄
¹³⁄₁₆

15.875
16.272
16.669
17.066
17.462
17.859
18.256
18.653
19.050
19.447
19.844
20.241
20.637

⁵³⁄₆₄
²⁷⁄₃₂
⁵⁵⁄₆₄
⁷⁄₈
⁵⁷⁄₆₄
²⁹⁄₃₂
⁵⁹⁄₆₄
¹⁵⁄₁₆
⁶¹⁄₆₄
³¹⁄₃₂
⁶³⁄₆₄
1
2

21.034
21.431
21.828
22.225
22.622
23.019
23.416
23.812
24.209
24.606
25.003
25.400
50.800

1.1.3 Fundamental physical constants
General gas constant: R = 8.3166 J K–1 mol–1
is the work done by one mole of an ideal gas under constant pressure (1013 hPa) when
its temperature rises from 0 °C to 1 °C.
Avogadro’s constant: NA (Loschmidt’s number NL): NA = 6.0225 · 1023 mol–1
number of molecules of an ideal gas in one mole.
When Vm = 2.2414 · 104 cm3 · mol-1: NA/Vm = 2.686 1019 cm –3.
Atomic weight of the carbon atom: 12C = 12.0000
is the reference quantity for the relative atomic weights of fundamental substances.
21

Base of natural logarithms: e = 2.718282
Bohr’s radius: r1 = 0.529 · 10–8 cm
radius of the innermost electron orbit in Bohr’s atomic model
R = 1.38 · 10–23 J · K–1
Boltzmann’s constant: k = —–
NA
is the mean energy gain of a molecule or atom when heated by 1 K.
Elementary charge: eo = F/NA = 1.602 · 10–19 As
is the smallest possible charge a charge carrier (e.g. electron or proton) can have.
Electron-volt: eV = 1.602 · 10–19 J
Energy mass equivalent: 8.987 · 1013 J · g–1 = 1.78 · 10–27 g (MeV)–1
according to Einstein, following E = m · c2, the mathematical basis for all observed
transformation processes in sub-atomic ranges.
Faraday’s constant: F = 96 480 As · mol–1
is the quantity of current transported by one mole of univalent ions.
Field constant, electrical: εo = 0.885419 · 10–11 F · m–1.
a proportionality factor relating charge density to electric field strength.
Field constant, magnetic: µ0 = 4 · π · 10–7 H · m–1
a proportionality factor relating magnetic flux density to magnetic field strength.
Gravitational constant: γ = 6.670 · 10–11 m4 · N–1 · s–4
is the attractive force in N acting between two masses each of 1 kg weight separated
by a distance of 1 m.
Velocity of light in vacuo: c = 2.99792 · 108 m · s–1
maximum possible velocity. Speed of propagation of electro-magnetic waves.
Mole volume: Vm = 22 414 cm3 · mol–1
the volume occupied by one mole of an ideal gas at 0 °C and 1013 mbar. A mole is that
quantity (mass) of a substance which is numerically equal in grammes to the molecular
weight (1 mol H2 = 2 g H2)
Planck’s constant: h = 6.625 · 10–34 J · s
a proportionality factor relating energy and frequency of a light quantum (photon).
Stefan Boltzmann’s radiation constant: δ = 5.6697 · 10–8 W · m–2 K–4 relates radiant
energy to the temperature of a radiant body. Radiation coefficient of a black body.
Temperature of absolute zero: T0 = – 273.16 °C = 0 K.
Wave impedance of space: Γ0 = 376.73 Ω
coefficient for the H/E distribution with electromagnetic wave propagation.

Γ0 = 
µ0 / ε0 = µ0 · c = 1/ (ε0 · c)
Weston standard cadmium cell: E0 = 1.0186 V at 20 °C.
Wien’s displacement constant: A = 0.28978 cm · K
enables the temperature of a light source to be calculated from its spectrum.
22

1
1.2 Physical, chemical and technical values
1.2.1 Electrochemical series
If different metals are joined together in a manner permitting conduction, and both are
wetted by a liquid such as water, acids, etc., an electrolytic cell is formed which gives
rise to corrosion. The amount of corrosion increases with the differences in potential. If
such conducting joints cannot be avoided, the two metals must be insulated from each
other by protective coatings or by constructional means. In outdoor installations,
therefore, aluminium/copper connectors or washers of copper-plated aluminium sheet
are used to join aluminium and copper, while in dry indoor installations aluminium and
copper may be joined without the need for special protective measures.
Table 1-8
Electrochemical series, normal potentials against hydrogen, in volts.
1. Lithium
2. Potassium
3. Barium
4. Sodium
5. Strontium
6. Calcium
7. Magnesium
8. Aluminium
9. Manganese

approx.
approx.
approx.
approx.
approx.
approx.
approx.
approx.
approx.

– 3.02
– 2.95
– 2.8
– 2.72
– 2.7
– 2.5
– 1.8
– 1.45
– 1.1

10. Zinc
11. Chromium
12. Iron
13. Cadmium
14. Thallium
15. Cobalt
16. Nickel
17. Tin
18. Lead

approx.
approx.
approx.
approx.
approx.
approx.
approx.
approx.
approx.

– 0.77
– 0.56
– 0.43
– 0.42
– 0.34
– 0.26
– 0.20
– 0.146
– 0.132

19. Hydrogen
20. Antimony
21. Bismuth
22. Arsenic
23. Copper
24. Silver
25. Mercury
26. Platinum
27. Gold

approx.
approx.
approx.
approx.
approx.
approx.
approx.
approx.
approx.

0.0
+ 0.2
+ 0.2
+ 0.3
+ 0.35
+ 0.80
+ 0.86
+ 0.87
+ 1.5

If two metals included in this table come into contact, the metal mentioned first will
corrode.
The less noble metal becomes the anode and the more noble acts as the cathode. As
a result, the less noble metal corrodes and the more noble metal is protected.
Metallic oxides are always less strongly electronegative, i. e. nobler in the electrolytic
sense, than the pure metals. Electrolytic potential differences can therefore also occur
between metal surfaces which to the engineer appear very little different. Even though
the potential differences for cast iron and steel, for example, with clean and rusty surfaces
are small, as shown in Table 1-9, under suitable circumstances these small differences
can nevertheless give rise to significant direct currents, and hence corrosive attack.
Table 1-9
Standard potentials of different types of iron against hydrogen, in volts
SM steel, clean surface
cast iron, clean surface

approx. – 0.40
approx. – 0.38

cast iron, rusty
SM steel, rusty

approx. – 0.30
approx. – 0.25

1.2.2 Faraday’s law
1. The amount m (mass) of the substances deposited or converted at an electrode is
proportional to the quantity of electricity Q = l · t.
m ~ l·t
23

2. The amounts m (masses) of the substances converted from different electrolytes by
equal quantities of electricity Q = l · t behave as their electrochemical equivalent
masses M*. The equivalent mass M* is the molar mass M divided by the
electrochemical valency n (a number). The quantities M and M* can be stated in
g/mol.
M*
m = —l·t
F
If during electroysis the current I is not constant, the product
t2

l · t must be represented by the integralt  l dt.
1

The quantity of electricity per mole necessary to deposit or convert the equivalent mass
of 1 g/mol of a substance (both by oxidation at the anode and by reduction at the
cathode) is equal in magnitude to Faraday's constant (F = 96480 As/mol).
Table 1-10
Electrochemical equivalents1)
Valency
Equivalent
n
mass2)
g/mol
Aluminium
Cadmium
Caustic potash
Caustic soda
Chlorine
Chromium
Chromium
Copper
Copper
Gold
Hydrogen
Iron
Iron
Lead
Magnesium
Nickel
Nickel
Oxygen
Silver
Tin
Tin
Zinc
1)
2)

3
2
1
1
1
3
6
1
2
3
1
2
3
2
2
2
3
2
1
2
4
2

8.9935
56.20
56.10937
30.09717
35.453
17.332
8.666
63.54
31.77
65.6376
1.00797
27.9235
18.6156
103.595
12.156
29.355
19.57
7.9997
107.870
59.345
29.6725
32.685

Quantity
precipitated,
theoretical
g/Ah

Approximate
optimum current
efficiency
%

0.33558
2.0970
2.0036
1.49243
1.32287
0.64672
0.32336
2.37090
1.18545
2.44884
0.037610
1.04190
0.69461
3.80543
0.45358
1.09534
0.73022
0.29850
4.02500
2.21437
1.10718
1.21959

85
95
95
95
95
—
10
65
97
—
100
95
—
95
—
95
—
100
98
70
70
85

… 98
… 95

… 18
… 98
… 100

… 100
… 100
… 98

…
…
…
…

100
95
95
93

Relative to the carbon -12 isotope = 12.000.
Chemical equivalent mass is molar mass/valency in g/mol.

Example:
Copper and iron earthing electrodes connected to each other by way of the neutral
conductor form a galvanic cell with a potential difference of about 0.7 V (see Table 1-8).
These cells are short-circuited via the neutral conductor. Their internal resistance is de24

1
termined by the earth resistance of the two earth electrodes. Let us say the sum of all
these resistances is 10 Ω. Thus, if the drop in “short-circuit emf” relative to the “opencircuit emf” is estimated to be 50 % approximately, a continuous corrosion current of
35 mA will flow, causing the iron electrode to decompose. In a year this will give an
electrolytically active quantity of electricity of
h
Ah
35 mA · 8760 —
a = 306 —
a– .
Since the equivalent mass of bivalent iron is 27.93 g/mol, the annual loss of weight from
the iron electrode will be
27.93 g/mol
3600 s
m = ————————— · 306 Ah/a · ————— = 320 g/a.
96480 As/mol
h
1.2.3 Thermoelectric series
If two wires of two different metals or semiconductors are joined together at their ends
and the two junctions are exposed to different temperatures, a thermoelectric current
flows in the wire loop (Seebeck effect, thermocouple). Conversely, a temperature
difference between the two junctions occurs if an electric current is passed through the
wire loop (Peltier effect).
The thermoelectric voltage is the difference between the values, in millivolts, stated in
Table 1-11. These relate to a reference wire of platinum and a temperature difference of
100 K.
Table 1-11
Thermoelectric series, values in mV, for platinum as reference and temperature
difference of 100 K
Bismut ll axis
– 7.7
Bismut ⊥ axis
– 5.2
Constantan
– 3.37 … – 3.4
Cobalt
– 1.99 … –1.52
Nickel
– 1.94 … – 1.2
Mercury
– 0.07 … + 0.04
Platinum
±0
Graphite
0.22
Carbon
0.25 … 0.30
Tantalum
0.34 … 0.51
Tin
0.4 … 0.44
Lead
0.41 … 0.46
Magnesium
0.4 … 0.43
Aluminium
0.37 … 0.41
Tungsten
0.65 … 0.9
Common thermocouples
Copper/constantan
(Cu/const)
up to 500 °C
Iron/constantan
(Fe/const)
up to 700 °C
Nickel chromium/
constantan
up to 800 °C

Rhodium
Silver
Copper
Steel (V2A)
Zinc
Manganin
Irdium
Gold
Cadmium
Molybdenum
Iron
Chrome nickel
Antimony
Silicon
Tellurium

0.65
0.67 … 0.79
0.72 … 0.77
0.77
0.6 … 0.79
0.57 … 0.82
0.65 … 0.68
0.56 … 0.8
0.85 … 0.92
1.16 … 1.31
1.87 … 1.89
2.2
4.7 … 4.86
44.8
50

Nickel chromium/nickel
(NiCr/Ni)
up to 1 000 °C
Platinum rhodium/
platinum
up to 1 600 °C
Platinum rhodium/
platinum rhodium
up to 1 800 °C

25

1.2.4 pH value
The pH value is a measure of the “acidity” of aqueous solutions. It is defined as the
logarithm to base 10 of the reciprocal of the hydrogen ion concentration CH3O1).
pH ≡ –log CH3O.

1 m = 1 mol/ l hydrochloric acid (3.6 % HCl

pH scale
—– 0

0.1 m hydrochloric acid (0.36 % HCl)—–—–—–—–—–—–—– 
 —– 1
gastric acid—–—–—–—–—–—–—– 
—– 2
vinegar ( ≈ 5 % CH3 COOH)—–—–—–—–—–—–—–
—– 3
acid
marsh water—–—–—–—–—–—–—–

—– 4
—– 5
—– 6

river water—–—–—–—–—–—–—–  —– 7



tap water 20 Ωm—–—–—–—–—–—–—– 

neutral

see water 0.15 Ωm (4 % NaCl)—–—–—–—–—–—–—–  —– 8

 —– 9
—– 10
0.1 m ammonia water (0.17 % NH3)—–—–—–—–—–—–—–
saturated lime-water (0.17 % CaOH2)—–—–—–—–—–—–—–
0.1 m caustic soda solution (0.4 % NaOH)—–—–—–—–—–—–—–
Fig. 1-1
pH value of some solutions
1)

CH3O = Hydrogen ion concentration in mol/l.

1.2.5 Heat transfer
Heat content (enthalpy) of a body: Q = V · ρ · c · ∆ϑ
V volume, ρ density, c specific heat, ∆ϑ temperature difference
Heat flow is equal to enthalpy per unit time:
Φ = Q/t
Heat flow is therefore measured in watts (1 W = 1 J/s).
26

—– 11
alkaline
—– 12
—– 13

1
Specific heat (specific thermal capacity) of a substance is the quantity of heat required
to raise the temperature of 1 kg of this substance by 1 °C. Mean specific heat relates to
a temperature range, which must be stated. For values of c and λ, see Section 1.2.7.
Thermal conductivity is the quantity of heat flowing per unit time through a wall 1 m2 in
area and 1 m thick when the temperatures of the two surfaces differ by 1 °C. With many
materials it increases with rising temperature, with magnetic materials (iron, nickel) it
first falls to the Curie point, and only then rises (Curie point = temperature at which a
ferro-magnetic material becomes non-magnetic, e. g. about 800 °C for Alnico). With
solids, thermal conductivity generally does not vary much (invariable only with pure
metals); in the case of liquids and gases, on the other hand, it is often strongly
influenced by temperature.
Heat can be transferred from a place of higher temperature to a place of lower
temperature by
– conduction (heat transmission between touching particles in solid, liquid or gaseous
bodies).
– convection (circulation of warm and cool liquid or gas particles).
– radiation (heat transmission by electromagnetic waves, even if there is no matter
between the bodies).
The three forms of heat transfer usually occur together.

Heat flow with conduction through a wall:

λ · A · ∆ϑ
Φ = —
s

A transfer area, λ thermal conductivity, s wall thickness, ∆ϑ temperature difference.
Heat flow in the case of transfer by convection between a solid wall and a flowing
medium:

Φ = α · A · ∆ϑ
α heat transfer coefficient, A transfer area, ∆ϑ temperature difference.
Heat flow between two flowing media of constant temperature separated by a solid
wall:

Φ = k · A · ∆ϑ
k thermal conductance, A transfer area, ∆ϑ temperature difference.
In the case of plane layered walls perpendicular to the heat flow, the thermal conductance coefficient k is obtained from the equation
1
1
— = —
—+
k
α1

∑

1
s
—
—n + –—

λn

α2

Here, α1 and α2 are the heat transfer coefficients at either side of a wall consisting of n
layers of thicknesses sn and thermal conductivities λn.
27

Thermal radiation
For two parallel black surfaces of equal size the heat flow exchanged by radiation is

Φ12 = σ · A(T14 – T24)
With grey radiating surfaces having emissivities of ε1 and ε2, it is

Φ12 = C12 · A (T14 – T24)
σ = 5.6697 · 10–8 W · m–2 · K–4 radiation coefficient of a black body (Stefan Boltzmann’s
constant), A radiating area, T absolute temperature.
Index 1 refers to the radiating surface, Index 2 to the radiated surface.
C12 is the effective radiation transfer coefficient. It is determined by the geometry and
emissivity ε of the surface.
C12 = σ · ε1

Special cases: A1  A2

σ

A1 ≈ A2

C12 = —————
1
1
– + – –1

ε1

A2 includes A1

ε2

σ
C12 = ————————

( )

1
A
1
– + —1 · — – 1
ε1 A2 ε2

Table 1-12
Emissivity ε (average values ϑ < 200 °C)
Black body
Aluminium, bright
Aluminium, oxidized
Copper, bright
Copper, oxidized
Brass, bright
Brass, dull
Steel, dull, oxidized
Steel, polished

28

1
0.04
0.5
0.05
0.6
0.05
0.22
0.8
0.06

Oil
Paper
Porcelain, glazed
Ice
Wood (beech)
Roofing felt
Paints
Red lead oxide
Soot

0.82
0.85
0.92
0.96
0.92
0.93
0.8-0.95
0.9
0.94

1
Table 1-13
Heat transfer coefficients α in W/(m2 · K) (average values)
Natural air movement in a closed space
Wall surfaces
Floors, ceilings: in upward direction
in downward direction
Force-circulated air
Mean air velocity w = 2 m/s
Mean air velocity w > 5 m/s

10
7
5
20
6.4 · w0.75

1.2.6 Acoustics, noise measurement, noise abatement
Perceived sound comprises the mechanical oscillations and waves of an elastic medium
in the frequency range of the human ear of between 16 Hz and 20 000 Hz. Oscillations
below 16 Hz are termed infrasound and above 20 000 Hz ultrasound. Sound waves can
occur not only in air but also in liquids (water-borne sound) and in solid bodies (solidborne sound). Solid-borne sound is partly converted into audible air-borne sound at the
bounding surfaces of the oscillating body. The frequency of oscillation determines the
pitch of the sound. The sound generally propagates spherically from the sound source,
as longitudinal waves in gases and liquids and as longitudinal and transverse waves in
solids.
Sound propagation gives rise to an alternating pressure, the root-mean-square value of
which is termed the sound pressure p. It decreases approximately as the square of the
distance from the sound source. The sound power P is the sound energy flowing
through an area in unit time. Its unit of measurement is the watt.
Since the sensitivity of the human ear is proportional to the logarithm of the sound
pressure, a logarithmic scale is used to represent the sound pressure level as
loudness.
The sound pressure level L is measured with a sound level metre as the logarithm of
the ratio of sound pressure to the reference pressure po, see DIN 35 632
p
L = 20 lg — in dB.
po
Here: po reference pressure, roughly the audible threshold at 1000 Hz.
po = 2 · 10–5 N/m2 = 2 · 10–4 µbar
p = the root-mean-square sound pressure
Example:
p = 2 · 10–3 N/m2 measured with a sound level metre, then
2 · 10–3
sound level L = 20 lg ————
= 40 dB.
2 · 10–5
The loudness of a sound can be measured as DIN loudness (DIN 5045) or as the
weighted sound pressure level. DIN loudness (λ DIN) is expressed in units of DIN phon.
29

The weighted sound pressure levels LA, LB, LC, which are obtained by switching in
defined weighting networks A, B, C in the sound level metre, are stated in the unit dB
(decibel). The letters A, B and C must be added to the units in order to distinguish the
different values, e. g. dB (A). According to an ISO proposal, the weighted sound
pressure LA in dB (A) is recommended for expressing the loudness of machinery noise.
DIN loudness and the weighted sound pressure level, e.g. as recommended in IEC
publication 123, are related as follows: for all numerical values above 60 the DIN
loudness in DIN phon corresponds to the sound pressure level LB in dB (B), for all
numerical values between 30 and 60 to the sound pressure level LA in dB (A). All noise
level values are referred to a sound pressure of 2 · 10–5 N/m2.
According to VDI guideline 2058, the acceptable loudness of noises must on average
not exceed the following values at the point of origin:

Area

Industrial
Commercial
Composite
Generally residential
Purely residential
Therapy (hospitals, etc.)

Daytime
(6–22 hrs)
dB (A)

Night-time
(22–6 hrs)
dB (A)

70
65
60
55
50
45

70
50
45
40
35
35

Short-lived, isolated noise peaks can be disregarded.
Disturbing noise is propagated as air- and solid-borne sound. When these sound waves
strike a wall, some is thrown back by reflection and some is absorbed by the wall. Airborne noise striking a wall causes it to vibrate and so the sound is transmitted into the
adjacent space. Solid-borne sound is converted into audible air-borne sound by
radiation from the bounding surfaces. Ducts, air-shafts, piping systems and the like can
transmit sound waves to other rooms. Special attention must therefore be paid to this
at the design stage.
There is a logarithmic relationship between the sound pressure of several sound
sources and their total loudness.
Total loudness of several sound sources:
A doubling of equally loud sound sources raises the sound level by 3 dB (example:
3 sound sources of 85 dB produce 88 dB together). Several sound sources of different
loudness produce together roughly the loudness of the loudest sound source.
(Example: 2 sound sources of 80 and 86 dB have a total loudness of 87 dB). In
consequence: with 2 equally loud sound sources attenuate both of them, with sound
sources of different loudness attentuate only the louder.
An increase in leveI of 10 dB signifies a doubling, a reduction of 10 dB a halving of the
perceived loudness.

30

1
In general, noises must be kept as low as possible at their point of origin. This can often
be achieved by enclosing the noise sources.
Sound can be reduced by natural means. The most commonly used sound-absorbent
materials are porous substances, plastics, cork, glass fibre and mineral wool, etc. The
main aim should be to reduce the higher-frequency noise components. This is also
generally easier to achieve than eliminating the lower-frequency noise.
When testing walls and ceilings for their behaviour regarding air-borne sound, one
determines the difference “D” in sound level “L” for the frequency range from 100 Hz to
3200 Hz.
p
D = L1 – L2 in dB where L = 20 lg — dB
po
L1 = sound level in room containing sound source
L2 = sound level in room receiving the sound
Table 1-14
Attenuation figures for some building materials in the range 100 to 3200 Hz
Structural
component

Attenuation
dB

Brickwork rendered,
12 cm thick
Brickwork rendered,
25 cm thick
Concrete wall, 10 cm thick
Concrete wall, 20 cm thick

45
50
42
48

Wood wool mat, 8 cm thick

50

Straw mat, 5 cm thick

38

Structural
component

Attenuation
dB

Single door without
extra sealing
Single door with
good seal
Double door without seal
Double door with extra
sealing
Single window without
sealing
Spaced double window
with seal

to 20
30
30
40
15
30

The reduction in level ∆L obtainable in a room by means of sound-absorbing materials
or structures is:
A
A1

T
T2

∆L = 10 lg —2 = 10 lg —1 dB
In the formula:
V
A = 0.163 – in m2
T
V = volume of room in m3
T = reverberation time in s in which the sound level L falls by 60 dB after sound
emission ceases.
Index 1 relates to the state of the untreated room, Index 2 to a room treated with noisereduction measures.
31

32

1.2.7 Technical values of solids, liquids and gases
Table 1-15
Technical values of solids
Material

E-aluminium F9
Alu alloy AlMgSi 1 F20
Lead
Bronze CuSnPb
Cadmium
Chromium

Density

Melting
or
freezing
point

Boiling
point

kg/dm3

°C

°C

ρ

2.70
2.70
11.34
8.6 . . 9
8.64
6.92

658
≈ 645
327
≈ 900
321
1 800

2270

767
2 400

≈ 17.5
31.6
8.5

42
92

360
234
452

12.3
≈ 11.5
≈ 11

71
46
46

464
485
540

0.10
0.0058
0.25 . . 0.10 ≈ 0.005
0.6 . . 1
0.0045

14.2
16.8
1.3
7.86

309
22

0.022
0 0038
0.48 . . 0.50 ≈ 0.00005

5

130
410
502
711

16.5
16.5
25.0

385
385
167

393
393
1034

Gold
Constantan Cu + Ni
Carbon diamond
Carbon graphite

19.29
1 063
8 . . 8.9
1 600
3.51
≈ 3 600
2.25

2 700

between 0 °C and 100 °C

Ω mm2/m

1 730

2 500

1)

J/(kg · K)

4 200
2 330
2 330
1110

0.02874
0.0407
0.21

Temperature
coefficient α
of electrical
resistance
at 20 °C
1/K

920
920
130

1 530
≈ 1 350
≈ 1 200

1 083
1 083
650

Specific
electrical
resistance ρ
at 20 °C

220
190
34

7.88
≈ 7.8
≈ 7.25

8.92
8.92
1.74

Mean
spec.
heat c at
0 . . 100 °C

23.8
23
28

Iron, pure
Iron, steel
Iron, cast

E-copper F30
E-copper F20
Magnesium

Linear
Thermal
thermal
conductiexpansion vity λ at
α
20 °C
mm/K
x 10–6 1)
W/(m · K)

≈ 0.027
0.762
0.028

0.01786
0.01754
0.0455

0.0042
0.0036
0.0043
0.004
0.0042

0.00392
0.00392
0.004
(continued)

Table 1-15 (continued)
Technical values of solids
Material

Density

Melting
or
freezing
point

Boiling
point

kg/dm3

°C

°C

ρ

Brass (Ms 58)
Nickel
Platinum

8.5
8.9
21.45

912
1 455
1 773

3 000
3 800

Mercury
Sulphur (rhombic)
Selenium (metallic)

13.546
2.07
4.26

38.83
113
220

357
445
688

Silver
Tungsten
Zinc
Tin

10.50
19.3
7.23
7.28

960
3 380
419
232

1 950
6 000
907
2 300

1)

Linear
Thermal
thermal
conductiexpansion vity λ at
α
20 °C
mm/K
–6
1)
x 10
W/(m · K)
17
13
8.99
61
90
66
19.5
4.50
16.50
26.7

110
83
71
8.3
0.2
421
167
121
67

Mean
spec.
heat c at
0 . . 100 °C

Specific
electrical
resistance ρ
at 20 °C

J/(kg · K)

Ω mm2/m

Temperature
coefficient α
of electrical
resistance
at 20 °C
1/K

397
452
134

≈ 0.0555
≈ 0.12
≈ 0.11

0.0024
0.0046
0.0039

139
720
351

0.698

0.0008

233
134
387
230

0.0165
0.06
0.0645
0.119

0.0036
0.0046
0.0037
0.004

between 0 °C and 100 °C

33

1

Table 1-16
34

Technical values of liquids
Material

Chemical
formula

Density

kg/dm3

Melting
or
freezing
point
°C

ρ

Boiling
point at
760 Torr

Expansion
coefficient
x 10–3

Thermal
conductivity
λ at 20 °C

Specific
heat c p
at 0 °C

Relative
dielectric
constant εr
at 180 °C

°C

at 18 °C

W/(m · K)

J/(kg · K)

56.3
78.0
35.0

1.43
1.10
1.62

0.2
0.14

2 160
2 554
2 328

21.5
25.8
4.3

— 33.5
184.4
80.1

0.84
1.16

4 187
2 064
1 758

14.9
7.0
2.24

2 030
2 428

6.29
56.2
2.2

Acetone
Ethyl alcohol
Ethyl ether

C3H6O
C2H6O
C4H10O

0.791
0.789
0.713

— 95
— 114
— 124

Ammonia
Aniline
Benzole

NH3
C6H7N
C6H6

0.771
1.022
0.879

— 77.8
— 6.2
+ 5.5

Acetic acid
Glycerine
Linseed oil

C2H4O2
C3H8O3

1.049
1.26
0.94

+ 16.65
— 20
— 20

117.8
290
316

1.07
0.50

Methyl alcohol
Petroleum
Castor oil

CH4O

0.793
0.80
0.97

— 97.1

64.7

1.19
0.99
0.69

0.21
0.16

2 595
2 093
1 926

31.2
2.1
4.6

Sulphuric acid
Turpentine
Water

H2S O4
C10H16
H2O

1.834
0.855
1.001)

— 10.5
— 10
0

0.57
9.7
0.18

0.46
0.1
0.58

1 385
1 800
4 187

> 84
2.3
88

1)

at 4 °C

338
161
106

0.022
0.14
0.29
0.15

Table 1- 17
Technical values of gases
Material

Chemical
formula

ρ1)

Density

Melting
point

Boiling
point

Thermal
conductivity λ

Specific
heat cp at 0 °C

kg/m3

°C

°C

10–2 W/(m · K)

J/(kg · K)

Relative1)
dielectric
constant εr

Ammonia
Ethylene
Argon

NH3
C2H4
Ar

0.771
1.260
1.784

— 77.7
— 169.4
— 189.3

— 33.4
— 103.5
— 185.9

2.17
1.67
1.75

2 060
1 611
523

Acetylene
Butane
Chlorine

C2H2
C4H10
Cl2

1.171
2.703
3.220

— 81
— 135
— 109

— 83.6
— 0.5
— 35.0

1.84
0.15
0.08

1 511

Helium
Carbon monoxide
Carbon dioxide

He
CO
CO2

0.178
1.250
1.977

— 272
— 205
— 56

— 268.9
— 191.5
— 78.5

1.51
0.22
1.42

5 233
1 042
819

1.000074
1.0007
1.00095

Krypton
Air
Methane

Kr
CO2 free
CH4

3.743
1.293
0.717

— 157.2
— 182.5

— 153.2
— 194.0
— 161.7

0.88
2.41
3.3

1 004
2 160

1.000576
1.000953

Neon
Ozone
Propane

Ne
O3
C2H8

0.8999
2.22
2.019

— 248.6
— 252
— 189.9

— 246.1
— 112
— 42.6

4.6

Oxygen
Sulphur hexafluoride
Nitrogen
Hydrogen

O2
SF6
N2
H2

1.429
6.072)
1.250
0.0898

—
—
—
—

—
—
—
—

1 038
670
1042
14 235

1.000547
1.00212)
1.000606
1.000264

1)
2)

35

3)

218.83
50.83)
210
259.2

192.97
63
195.81
252.78

2.46
1.282)
2.38
17.54

502

1.0072
1.001456
1.00056

1.97

at 0 °C and 1013 mbar
at 20 °C and 1013 mbar
at 2.26 bar

1

1.3 Strength of materials
1.3.1 Fundamentals and definitions
External forces F acting on a cross-section A of a structural element can give rise to
tensile stresses (σz), compressive stresses (σd), bending stresses (σb), shear stresses
(τs) or torsional stresses (τt). If a number of stresses are applied simultaneously to a
component, i. e. compound stresses, this component must be designed according to
the formulae for compound strength. In this case the following rule must be observed:
Normal stresses σz. σd. σb,
Tangential stresses (shear and torsional stresses) τs, τt.
are to be added arithmetically;
Normal stresses σb with shear stresses τs,
Normal stresses σb with torsional stresses τt,
are to be added geometrically.

a)

b)

E

Fig. 1-2
Stress-strain diagram, a) Tensile test with pronounced yield point, material = structural
steel; b) Tensile test without pronounced yield point, material = Cu/Al, ε Elongation,
σ Tensile stress, σs Stress at yield point, σE Stress at proportionality limit, Rp02 Stress
with permanent elongation less than 0.2 %, σB Breaking stress.
Elongation ε = ∆ l/l0 (or compression in the case of the compression test) is found
from the measured length l0 of a bar test specimen and its change in length ∆ l = l – l0
in relation to the tensile stress σz, applied by an external force F. With stresses below
the proportionality limit σE elongation increases in direct proportion to the stress σ
(Hooke’s law).
Stress σ
σ
The ratio ———————— = —–E = E is termed the elasticity modulus.
εE
Elongation ε
E is an imagined stress serving as a measure of the resistance of a material to deformation due to tensile or compressive stresses; it is valid only for the elastic region.
According to DIN 1602/2 and DIN 50143, E is determined in terms of the load σ0.01, i.e.
the stress at which the permanent elongation is 0.01 % of the measured length of the
test specimen.
36

1
If the stresses exceed the yield point σs, materials such as steel undergo permanent
elongation. The ultimate strength, or breaking stress, is denoted by σB, although a bar
does not break until the stress is again being reduced. Breaking stress σB is related to
the elongation on fracture δ of a test bar. Materials having no marked proportional limit
or elastic limit, such as copper and aluminium, are defined in terms of the so-called
Rp0.2-limit, which is that stress at which the permanent elongation is 0.2 % after the
external force has been withdrawn, cf. DIN 50144.
For reasons of safety, the maximum permissible stresses, σmax or τmax in the material
must be below the proportional limit so that no permanent deformation, such as
elongation or deflection, persists in the structural component after the external force
ceases to be applied.
Table 1-18
Material

Elasticity
modulus E
N/mm2 1)

Structural steel in general, spring steel (unhardened), cast steel
Grey cast iron
Electro copper, Al bronze with 5 % Al, rolled
Red brass
E-AlMgSi 0.5
E-AI
Magnesium alloy
Wood

210 000
100 000
110 000
90 000
75 000
65 000
45 000
10 000

1) Typical

values.

Fatigue strength (endurance limit) is present when the maximum variation of a stress
oscillating about a mean stress is applied “infinitely often” to a loaded material (at least
107 load reversals in the case of steel) without giving rise to excessive deformation or
fracture.
Cyclic stresses can occur in the form of a stress varying between positive and negative
values of equal amplitude, or as a stress varying between zero and a certain maximum
value. Cyclic loading of the latter kind can occur only in compression or only in tension.
Depending on the manner of loading, fatigue strength can be considered as bending
fatigue strength, tension-compression fatigue strength or torsional fatigue strength.
Structural elements which have to withstand only a limited number of load reversals can
be subjected to correspondingly higher loads. The resulting stress is termed the fatigue
limit.
One speaks of creep strength when a steady load with uniform stress is applied, usually
at elevated temperatures.
1.3.2 Tensile and compressive strength
If the line of application of a force F coincides with the centroidal axis of a prismatic bar
of cross section A (Fig.1- 3), the normal stress uniformly distributed over the cross37

section area and acting perpendicular to it is
F
A

σ = — .
With the maximum permissible stress σmax for a given material and a given loading, the
required cross section or the maximum permissible force, is therefore:
F
A = ——— or F = σmax · A.

σmax

Example:
A drawbar is to be stressed with a steady
load of F = 180 000 N.
The chosen material is structural steel
St 37 with σmax = 120 N/mm2.
Required cross section of bar:
E
180 000 N
A = ——— = ————————2 = 1500 mm2.
σmax 120 N/mm
Fig. 1- 3

Round bar of d = 45 mm chosen.

1.3.3 Bending strength
The greatest bending action of an external force, or its greatest bending moment M,
occurs at the point of fixing a in the case of a simple cantilever, and at point c in the
case of a centrally loaded beam on two supports.
l

l/2

F

l/2

l

Fig. 1-4
Maximum bending moment at a: M = Fl; at c: M = Fl/4
In position a and c, assuming the beams to be of constant cross section, the bending
stresses σb are greatest in the filaments furthermost from the neutral axis. M may be
greater, the greater is σmax and the “more resistant” is the cross-section. The following
cross sections have moments of resistance W in cm, if a, b, h and d are stated in cm.
The maximum permissible bending moment is M = W · σmax and the required moment
of resistance
M
W =σ
—–— .
max

38

1
Example:
A mild-steel stud

(

σmax

= 70 N/mm2

)

with an unsupported length of

l = 60 mm is to be loaded in the middle with a force F = 30 000 N. Required moment of
resistance is:
M
F·l
30 000 N · 60 mm
W = —
—
—
—
— = ————— = ——————————
— = 6.4 · 103 mm3.
σmax
4 · σmax
4 · 70 N/mm2
According to Table 1- 22, the moment of resistance W with bending is W ≈ 0.1 · d 3.
3

The diameter of the stud will be: d = 
10 W,

3

3

d = 
64
000 = 
64 · 10 = 40 mm.

1.3.4 Loadings on beams
Table 1-19
Bending load
Case

Reaction force
Bending
moment

A

= F

l
Mmax = F l

A

= Q

l
Ql
Mmax = ——
2

l

A

F
= B = —
2

Fl
Mmax = ——
4

l

A

Q
= B = —
2

Ql
Mmax = ——
8

Required
moment of
resistance, max.
permissible load

Deflection

Fl
W = ———

F l3
f = ———
3EJ

σmax

σmax W
F =—
———
—
l

Ql
W = ————
2 σmax

Q l3
f = ———
8EJ

2 σmax W
Q = ———
———
l

Fl
W = ———
4 σmax

F l3
f = ————
48 E J

4 σmax W
F = ———
———
l

Ql
W = ————
8 σmax

5 Q l3
f = —— · ——
384 E J

8 σmax W
Q = ———
———
l

(continued)

39

Table 1-19 (continued)
Bending load
Case

Reaction force
Bending
moment

l

Required
moment of
resistance, max.
permissible load

Deflection

F a2 b2
f = —————
3EJl

A

Fb
= ——

Fab
W = ————

B

Fa
= ——

σ Wl
F = —max
————

l σmax

l

ab

l

Mmax = A a = B b

for F1 = F2 = F 1)

l

A

Fa
f = ————
24 E J
[3 (l + 2 a) 2 – 4 a2]

σmax W
F = ————

Mmax = F a

l

Fa
W = ——
—

σmax

= B =F

a

A

F1 e + F2 c
= ——
—————

Aa
W1 = ———

B

F a + F2 d
= —1—————
—

Bc
W2 = ———

l

l

σmax

F1 a2 e2 + F2l2 d 2
f =—
————————
3EJl

σmax

Determine beam for
greatest “W”

l

A

Q
= B = —

l

Ql
Mmax = —
—

12

Ql
W = ————

12 σzul

12 σzul W
Q = ——————

l

A and B = Section at risk.
F = Single point load, Q = Uniformly distributed load.
1) If

40

F1 und F2 are not equal, calculate with the third diagram.

Q
l3
f = —
—
— · ——
EJ
384

1
1.3.5 Buckling strength
Thin bars loaded in compression are liable to buckle. Such bars must be checked both
for compression and for buckling strength, cf. DIN 4114.
Buckling strength is calculated with Euler's formula, a distinction being drawn between
four cases.
Table 1-20
Buckling

10 E J
F = ————
4 s l2

l

Case I
One end fixed, other end free

4 s F l2
J = ————
10 E

10 E J
F = ———
—
s l2

l

Case II
Both ends free to move along bar axis

s F l2
J = ————
10 E

20 E J
F = ———
—
s l2

l

Case III
One end fixed, other end free to move
along bar axis

l

Case IV

E
J
F
I

Both ends fixed, movement along
bar axis

= Elasticity modulus of material
= Minimum axial moment of inertia
= Maximum permissible force
= Length of bar

s F l2
J = —
—
—
—
20 E

40 E J
F = ———
—
s l2
s F l2
J = —
—
—
—
40 E

s = Factor of safety:
for cast iron
= 8,
for mild carbon steel = 5,
for wood
= 10.
41

1.3.6 Maximum permissible buckling and tensile stress for tubular rods
Threaded steel tube (gas pipe) DIN 2440, Table 11)
or seamless steel tube
DIN 24482).
D4 – d 4
10 E
10 E
D4 – d4
Fbuck = ———
· J = ———
· ————— where J ≈ ———— from Table 1- 22
s l2
20
20
s l2
Ften = A · σmax
in which F
E
J
s

σmax
A
D
d

l

Force
Elasticity modulus = 210 000 N/mm2
Moment of inertia in cm4
Factor of safety = 5
Max. permissible stress
Cross-section area
Outside diameter
Inside diameter
Length

Fig. 1- 5

Table 1-21
Moment
of
inertia
J
cm4

Weight Fbuck for tube length l ≈
of
tube
0.5 m 1 m
1.5 m 2 m
kg/m
N
N
N
N

17.2 2.35 109.6
21.3 2.65 155.3
26.9 2.65 201.9

0.32
0.70
1.53

0.85
1.22
1.58

5400 1350 600 340 220
11800 2950 1310 740 470
25700 6420 2850 1610 1030

1
33.7 3.25 310.9
0.8
25
2
144.5
0.104 31.8 2.6 238.5

3.71
0.98
2.61

2.44
1.13
1.88

62300 15600 6920 3900 2490 1730 18650
16500 4100 1830 1030 660 460 17350
43900 11000 4880 2740 1760 1220 28600

Nomi- Dimensions
nal diameter
D
D
a
inch mm mm
10
15
20
25

1)
2)

³⁄₈
¹⁄₂
³⁄₄

Crosssections
A
mm2

Ften

2.5 m 3 m
N
N
N
150 6600
330 9300
710 12100

No test values specified for steel ST 00.
σmax = 350 N/mm2 for steel ST 35 DIN 1629 seamless steel tube, cf. max. permissible buckling stress
for structural steel, DIN 1050 Table 3.

42

1
1.3.7 Shear strength1)
Two equal and opposite forces F acting perpendicular to the axis of a bar stress this
section of the bar in shear. The stress is
F
A

τs = – or for given values of F and τs max, the required cross section is
F
A = –——–

τs max

F = 15 000 kp ≈ 1.5 · 105 N

Fig. 1- 6
Pull-rod coupling

Stresses in shear are always combined with a bending stress, and therefore the
bending stress σb has to be calculated subsequently in accordance with the following
example.
Rivets, short bolts and the like need only be calculated for shear stress.
Example:
Calculate the cross section of a shackle pin of structural steel ST 50-12), with
Rp 0.2 min = 300 N/mm2 and τs max = 0.8 Rp 0.2 min, for the pull-rod coupling shown in
Fig. 1- 6.
1. Calculation for shear force:
F
150 000 N
A = ———— = —————————————
= 312 mm2
2 τs max
2 · (0.8 · 300) N/mm2
yields a pin diameter of d ≈ 20 mm, with W = 0.8 · 103 mm3 (from W ≈ 0.1 · d 3, see
Table 1- 22).

1)

2)

For maximum permissible stresses on steel structural components of transmission towers and
structures for outdoor switchgear installations, see VDE 0210.
Yield point of steel ST 50 -1 σ0.2 min = 300 N/mm2, DIN 17 100 Table 1 (Fe 50-1).

43

2. Verification of bending stress:
The bending moment for the pin if F l/ 4 with a singlepoint load, and F l/ 8 for a
uniformly distributed load. The average value is
Fl
Fl
—— + ——
4
8
3
—= — Fl
M b = ———————
2
16
F = 1.5 · 105 N, l = 75 mm becomes:

when

3
— · 1.5 · 105 N · 75 mm ≈ 21 · 105 N · mm;
Mb = —
16
M
W

21 · 105 N · mm
0.8 · 10 mm

N
mm

N
mm

σB = –—b = ——————3———3— ≈ 262 · 103 ———
= 2.6 · 105 ———
2
2
i. e. a pin calculated in terms of shear with d = 20 mm will be too weak. The required
pin diameter d calculated in terms of bending is
21 · 105 N · mm
Mb
W= —
—
—= —————————2— = 7 · 103 mm2 = 0.7 cm3
σmax
300 N/mm
d

≈


10 · 
W
 = 
10 · 7 · 103 mm3 = 
70 = 41.4 mm ≈ 42 mm.
3

3

3

i. e. in view of the bending stress, the pin must have a diameter of 42 mm instead of
20 mm.

44

1
1.3.8 Moments of resistance and moments of inertia
Table 1-22
Crosssection

Moment of resistance
torsion
bending1)
W 4)
W 4)
cm3
cm3

Moment of inertia
polar1)
axial2)
Jp
J
cm4
cm4

0.196 d 3
≈ 0.2 d 3

0.098 d 4
≈ 0.1 d 4

0.098 d 3
≈ 0.1 d 3

0.049 d 4
≈ 0.05 d 4

D4 – d4
D4 – d4
0.196 ———— 0.098 ————
D
D

0.098 (D 4 – d 4) 0.049 (D 4 – d 4)
D4 – d4
≈ ————
20

0.208 a3

0.167 a 4

0.083 a 4

bh
—— (b 2 + h 2)
12

b h3
—— = 0.083 b h 3
12

0.018 a3

0.208 k b 2 h 3) b h 2
—— = 0.167 b h 2
6

B H 3 – b h3
———————
6H

B H 3 – b h3
———————
12

B H 3 – b h3
———————
6H

B H 3 – b h3
———————
12

B H 3 – b h3
———————
6H

B H 3 – b h3
———————
12

b h 3 + bo h o3
——————
—
6h

b h 3 + bo h o3
———————
12

1)

Referred to CG of area.
Referred to plotted axis.
Values for k: if h : b = 1 1.5
2
3
4
——————————————————————————
then k = 1 1.11 1.18 1.27 1.36
4) Symbol Z is also applicable, see DIN VDE 0103
2)
3)

45

1.4 Geometry, calculation of areas and solid bodies
1.4.1 Area of polygons

Regular polygons (n angles)
The area A, length of sides S and radii of the outer
and inner circles can be taken from Table 1- 23 below.

Table 1-23
Number of
sides
n
3
4
5
6
8
10
12

Area A
S2 ×
0.4330
1.0000
1.7205
2.5981
4.8284
7.6942
11.196

Side S

Outer radius

Inner radius

R2 ×

r2 ×

R×

r×

R
S×

r×

r
R×

S×

1.2990
2.0000
2.3776
2.5981
2.8284
2.9389
3.0000

5.1962
4.0000
3.6327
3.4641
3.3137
3.2492
3.2154

1.7321
1.4142
1.1756
1.0000
0.7654
0.6180
0.5176

3.4641
2.0000
1.4531
1.1547
0.8284
0.6498
0.5359

0.5774
0.7071
0.8507
1.0000
1.3066
1.6180
1.9319

2.0000
1.4142
1.2361
1.1547
1.0824
1.0515
1.0353

0.5000
0.7071
0.8090
0.8660
0.9239
0.9511
0.9659

0.2887
0.5000
0.6882
0.8660
1.2071
1.5388
1.8660

Irregular polygons
g1 h1
g2 h2
A= —
—— + —
—— + …
2
2
1
= – (g 1 h 1 + g 2 h 2 + …)
2

Pythagoras theorem
c 2 = a 2 + b 2;
a 2 = c 2 – b 2;
b 2 = c 2 – a 2;

46

c = 
a2 + b2
a = 
c2 – b2
b = 
c2 – a2

1
1.4.2 Areas and centres of gravity
Table 1- 24
Shape of surface

A = area

Triangle

1
A=–ah
2

Trapezium

a+b
A = ——— · h
2

Rectangle

A=ab

U = 2 (a + b)

Circle
segment

α0
br
A = —— = —— r π
2
180

U =2r+b

α0

b = r π ——
180
Semicircle

1
A = – π r2
2

Circle

d2
A = r2 π = π —
4

U = perimeter
S = centre of gravity (cg)
e = distance of cg

U =a+b+c
1
e =–h
3

U =a+b+c+d
h a+2b
e = – · ————
3 a+b

2 sin α 180
e = – r ——0— · ———
π
3 α
U = r (2 + π) = 5.14 r
1 r
e = – · – = 0.425 r
3 π
U =2πr = πd

π

Annular
segment

A = —— α0 (R 2 – r 2) U = 2 (R – r) + B + b
180
2 R 2 – r 2 sin α 180
e = – · ——
—— · ——0— · ——
α
π
3 R2 – r 2

Semiannulus

A = —— α0 (R 2 – r 2)
2

Annulus

A = π (R 2 – r 2)

Circular
segment

α0
sh
π r α0
A = —— r 2 π – —— U = 2 
r 2 – h 2 + ————
180
2
90

π

r 2 – h2
s = 2 

Ellipse

a—
b π
A=—
4

if b < 0.2 R, then
e ≈ 0.32 (R + r)
U = 2 π (R + r)

s2
e = ————
12 · A
U = –π 1.5 (a + b) – 
ab
2

[

]
47

1.4.3 Volumes and surface areas of solid bodies
Table 1-25
Shape
of body

V = volume

O = Surface
A = Area

Solid
rectangle

V=abc

O = 2 (a b + a c + b c)

Cube

3
V = a3 = —d——
2.828

O = 6 a2 = 3 d 2

Prism

V=Ah

Pyramid

1
V=–Ah
3

O=Uh+2A
A = base surface

O = A + Nappe

Cone

1
V=–Ah
3

O = π r s + π r2
2 + r2
s = 
h

Truncated
cone

πh
V = (R 2 + r 2 + R r) · —–
3

O =(R + r) π s + π (R2 + r2)
h2 + (R 
– r)2
s = 

Truncated
pyramid

1
AA1)
V = – h (A + A1 + 
3

O = A + A1 + Nappe

Sphere

4
V = – π r3
3

O = 4 π r2

Hemisphere

2
V = – π r3
3

O = 3 π r2

Spherical
segment

Spherical
sector

48

V = π h2

(r — 31– h)

2
V = – π r2 h
3

O = 2 π r h + π (2 r h – h 2) =
π h (4 r – h)

πr
O = —— (4 h + s)
2
(continued)

1
Table 1-25 (continued)
Shape
of body

V = Volume

O = Surface
A = Area

Zone
of sphere

πh
V = —— (3a 2 + 3b 2 + h 2)
3

O = π (2 r h + a 2 + b 2)

Obliquely
cut cylinder

h + h1
V = π r 2 ————
2

O = π r (h + h1) + A + A1

Cylindrical
wedge

2
V = – r2 h
3

0 = 2rh + – r 2 + A
2

Cylinder

V = π r 2h

O = 2 π r h + 2 π r2

Hollow
cylinder

V = π h (R 2 – r 2)

O = 2 π h (R + r) + 2 π (R 2 – r2)

Barrel

V=—l·
15
(2 D 2 + Dd + 0.75 d 2)

D+d
π
O = ———— π d + – d 2
2
2
(approximate)

Frustum

A – A1
V = ————
+ A1
2

O = A + A1 + areas of sides

Body of rotation (ring)

V=2πA
A = cross-section

O = circumference of crosssection x 2 π 

Pappus’
theorem for
bodies of
revolution

Volume of turned
surface (hatched) x
path of its
centre of gravity
V=A2π

Length of turned line x path
of its centre of gravity
O = L 2 π 1

π

π

(

)h

49

50

2 General Electrotechnical Formulae
2

2.1 Electrotechnical symbols as per DIN 1304 Part 1

Table 2-1
Mathematical symbols for electrical quantities (general)
Symbol

Quantity

Sl unit

Q
E
D
U

quantity of electricity, electric charge
electric field strength
electric flux density, electric displacement
electric potential difference
electric potential
permittivity, dielectric constant
electric field constant, εo = 0.885419 · 10 –11 F/m
relative permittivity
electric capacitance
electric current
electric current density
specific electric conductivity
specific electric resistance
electric conductance
electric resistance
electromotive force

C
V/m
C/m2
V
V
F/m
F/m
1
F
A
A/m2
S/m

ϕ
ε
εo
εr
C
I
J
x, γ, σ

ρ

G
R

θ

Ωm

S

Ω
A

Table 2-2
Mathematical symbols for magnetic quantities (general)
Symbol

Quantity -

Sl unit

Φ

magnetic flux
magnetic induction
magnetic field strength
magnetomotive force
magnetic potential
permeability
absolute permeability, µo = 4 π · 10–7 · H/m
relative permeability
inductance
mutual inductance

Wb
T
A/m
A
A
H/m
H/m
1
H
H

B
H
V

ϕ
µ
µo
µr

L
Lmn

51

Table 2-3
Mathematical symbols for alternating-current quantities and network quantities
Symbol

Quantity

Sl unit

S
P
Q
D

apparent power
active power
reactive power
distortion power
phase displacement
load angle
power factor, λ = P/S, λ cos ϕ 1)
loss angle
loss factor, d = tan δ
impedance
admittance
resistance
conductance
reactance
susceptance
impedance angle, γ = arctan X/R

W, VA
W
W, Var
W
rad
rad
1
rad
1

ϕ

ϑ

λ
δ

d
Z
Y
R
G
X
B

γ

Ω
S

Ω
S

Ω

S
rad

Table 2-4
Numerical and proportional relationships
Symbol

Quantity

Sl unit

η

efficiency
slip
number of pole-pairs
number of turns
transformation ratio
number of phases and conductors
amplitude factor
overvoltage factor
ordinal number of a periodic component
wave content
fundamental wave content
harmonic content, distortion factor
increase in resistance due to skin effect,
ζ = R~ / R —

1
1
1
1
1
1
1
1
1
1
1
1
1

s
p
w, N
ü
m

γ

k

ν

s
g
k

ζ
1)

Valid only for sinusoidal voltage and current.

2.2 Alternating-current quantities
With an alternating current, the instantaneous value of the current changes its direction
as a function of time i = f (t). If this process takes place periodically with a period of
duration T, this is a periodic alternating current. If the variation of the current with
respect to time is then sinusoidal, one speaks of a sinusoidal alternating current.
52

The frequency f and the angular frequency ω are calculated from the periodic time T
with

2

1
2π
f = – and ω = 2 π f = — .
T
T
The equivalent d. c. value of an alternating current is the average, taken over one
period, of the value:

i =

1T
1 2π
i dt =
i dω t .
∫ 
∫ 
T0
2π 0

This occurs in rectifier circuits and is indicated by a moving-coil instrument, for example.
The root-mean-square value (rms value) of an alternating current is the square root of
the average of the square of the value of the function with respect to time.
I =

1 T 2
1 2π 2
⋅ ∫ i dt =
⋅ ∫ i dω t .
T 0
2π 0

As regards the generation of heat, the root-mean-square value of the current in a
resistance achieves the same effect as a direct current of the same magnitude.
The root-mean-square value can be measured not only with moving-coil instruments,
but also with hot-wire instruments, thermal converters and electrostatic voltmeters.
A non-sinusoidal current can be resolved into the fundamental oscillation with the
fundamental frequency f and into harmonics having whole-numbered multiples of the
fundamental frequency. If I1 is the rms value of the fundamental oscillation of an
alternating current, and I2, I3 etc. are the rms values of the harmonics having
frequencies 2 f, 3 f, etc., the rms value of the alternating current is
I =

I +I +I +…

2
1

2
2

2
3

If the alternating current also includes a direct-current component i – , this is termed an
undulatory current. The rms value of the undulatory current is
I =

I +I +I +I +…

2
–

2
1

2
2

2
3

The fundamental oscillation content g is the ratio of the rms value of the fundamental
oscillation to the rms value of the alternating current
I
g = ––1 .
I
The harmonic content k (distortion factor) is the ratio of the rms value of the harmonics
to the rms value of the alternating current.



I 22 + I 23 + …
k = —————— =
I


1 –
g

2

The fundamental oscillation content and the harmonic content cannot exceed 1.
In the case of a sinusoidal oscillation
the fundamental oscillation content
the harmonic content

g = 1,
k = 0.
53

Forms of power in an alternating-current circuit
The following terms and definitions are in accordance with DIN 40110 for the sinusoidal
wave-forms of voltage and current in an alternating-current circuit.
apparent power

S = UI = 
P 2 + Q 2,

active power

P = UI · cos ϕ = S · cos ϕ,

reactive power

Q = UI · sin ϕ = S · sin ϕ,

power factor

P
cos ϕ = – ,
S

reactive factor

Q
sin ϕ = – .
S

When a three-phase system is loaded symmetrically, the apparent power is
S = 3 U1I1 = 
3 · U · I1,
where I1 is the rms phase current, U1 the rms value of the phase to neutral voltage and
U the rms value of the phase to phase voltage. Also
active power

3 · U · I1 · cos ϕ,
P = 3 U1I1 cos ϕ = 

reactive power

Q = 3 U1I1 sin ϕ = 
3 · U · I1 · sin ϕ.

The unit for all forms of power is the watt (W). The unit watt is also termed volt-ampere
(symbol VA) when stating electric apparent power, and Var (symbol var) when stating
electric reactive power.
Resistances and conductances in an alternating-current circuit
impedance

U
S
R 2 + X2
Z = – = –2 = 
I
I

resistance

U cos ϕ
P
Z 2– X2
R = ——–— = —2 = Z cos ϕ = 
I
I

reactance
inductive reactance

U sin ϕ
Q
Z2– R2
X = ——–— = —
= Z sin ϕ = 
I
I2
Xi = ω L

capacitive reactance

1
Xc = —–
ωC

admittance

I
S
1
G2 + B2 = –
Y = – = —2 = 
Z
U
U

conductance

P
R
I cos ϕ
Y 2 – B2 = —2
G = ——— = —2 = Y cos ϕ = 
Z
U
U

conductance

I sin ϕ
Q
X
Y 2 – G 2 = —2
B = ——— = —2 = Y sin ϕ = 
Z
U
U

inductive susceptance

1
B i = —–
ωL

capacitive susceptance

Bc = ω C

54

Complex presentation of sinusoidal time-dependent a. c. quantities
Expressed in terms of the load vector system:
U = I · Z, I = U · Y
The symbols are underlined to denote
that they are complex quantities
(DIN 1304).
Fig. 2- 1
Equivalent circuit diagram

Fig. 2-2
Vector diagram of resistances

Fig. 2-3
Vector diagram of conductances

If the voltage vector U is laid on the real reference axis of the plane of complex
numbers, for the equivalent circuit in Fig. 2-1 with Z = R + j X i: we have
U = U,
I = Iw –j Ib = I (cos ϕ – j sin ϕ),
P
Q
Iw = – ; Ib = – ;
U
U
S 1) = U I* = U I (cos ϕ + j sin ϕ) = P + j Q,
S = S = U I = P 2 + Q 2,
U
U
U
Z = R + j Xi = — = ——————— = — (cos ϕ + j sin ϕ ),
I
I I (cos ϕ – j sin ϕ)
U
U
where R = — cos ϕ and Xi = — sin ϕ,
I
I
I
I
Y = G–jB = — = — (cos ϕ – j sin ϕ)
U U
I
I
where G = — cos ϕ and B i = — sin ϕ.
U
U
1)

S : See DIN 40110
I* = conjugated complex current vector

55

2

ω = 2 π f is the angular frequency and ϕ the phase displacement angle of the voltage
with respect to the current. U, I and Z are the numerical values of the alternating-current
quantities U, I and Z.

Table 2-5
Alternating-current quantities of basic circuits
Z

Z 

1.

R

R

2.

jωL

ωL

3.

– j / (ω C )

1/ω C

4.

R + j ω L1)

R 2 + (ω L)2

5.

R – j / (ω C )

6.

j (ω L – 1/(ω C ))

7.

R + j(ωL–1/(ω C)) 2)

R2 + (ωL–1/(ω C))2

8.

RωL
———––
ωL–jR

RωL
——————–
R 2 + (ω L)2

9.

R – j ω C R2
———––—–
—
1 + (ω C )2 R 2

10.

j
—–—————
1/ (ω L) – ω C

1
——————–——–
(1/ω L)2 – (ω C)2

1
—————–——–—–—
1/R + j (ω C – 1/(ω L))

1
————————–————–
1/R 2 + (ω C – 1/ (ω L))2

Circuit

11.

4)

R 2 + 1/(ω C)2
2)

3)

[Y = 1/R 2 + j (ω C – 1/ (ω L))]

12.

1)
2)

5)

R + j (L (1– ω 2 LC) – R 2 C)
———–——–——–—–—–—
(1 – ω 2 L C)2 + (R ω C)2

(ω L – 1/(ω C))2

R
————–——
1 + (ω C)2 R 2

R 2 + [L (1– ω 2 LC) – R 2 C ]2
————————————––
(1 – ω2 L C)2 + (R ω C)2

With small loss angle δ (= 1/ϕ ) ≈ tan δ (error at 4° about 1 ‰): Z ≈ ω L (δ + j).
Series resonance (voltage resonance) for ω L = 1 / (ω C ):
1
L /C ƒres = ———— Zres = R.
Xres = XL = Xc= 
2 π 
LC

Close to resonance (∆ ƒ< 0.1 ƒres) is Z ≈ R + j Xres · 2 ∆ ƒ / ƒres with ∆ ƒ = ƒ – ƒres

3)

With small loss angle δ (= 1/ϕ ) ≈ tan δ = –1/(ω C R ):

δ+j
Z = —–—
ωC
4)

5)

56

1
Bres = 
C/L: ƒres = ————
2 π LC
Close to resonance (∆ ƒ  < 0.1 ƒres ):
Y = G + j B res · 2 ∆ ƒ with ∆ ƒ = ƒ – ƒres
e. g. coil with winding capacitance.

Y res = G.

Table 2-6
Current / voltage relationships
Ohmic
resistance
R

Capacitance
(capacitor)
C

u =

iR

1
–
C

i =

u
–
R

du
C·—
dt

1
–
L

Time law

u =

û sin ω t

û sin ω t

û sin ω t

hence

u =

î R sin ω t = û sin ω t

1
– —— î cos ω t = – û cos ω t
ωC

ω L î cos ω t = û cos ω t

i =

û
— sin ω t = î sin ω t
R

ω C û cos ω t = î cos ω t

1
– —– û cos ω t = – î cos ω t
ωL

General law

Elements of calculation

∫

i dt

Inductance
(choke coil)
L
di
L·—
dt

∫

u dt

î =

û/R

ωCû

û / (ω L)

û =

îR

î /(ω C)

îωL

ϕ =

0

π
1
arctan ——— = – –
ωC·0
2
i leads u by 90 °

ωL π
arctan —– = –
0
2
i lags u by 90 °

u and i in phase
f =

ω

57

—–
2π

ω

—–
2π

ω

—–
2π

(continued)

2

58

Table 2-6 (continued)

Alternating current
impedance

Diagrams

Ohmic
resistance
R

Capacitance
(capacitor)
C

Inductance
(choke coil)
L

Z =

R

– j
——
ωC

jωL

|Z| =

R

1
—–

ωL

ωC

2.3

Electrical resistances

2

2.3.1 Definitions and specific values
An ohmic resistance is present if the instantaneous values of the voltage are
proportional to the instantaneous values of the current, even in the event of
time-dependent variation of the voltage or current. Any conductor exhibiting this
proportionality within a defined range (e. g. of temperature, frequency or current)
behaves within this range as an ohmic resistance. Active power is converted in an
ohmic resistance. For a resistance of this kind is
P
R = –2 .
I
The resistance measured with direct current is termed the d. c. resistance R – . If the
resistance of a conductor differs from the d. c. resistance only as a result of skin effect,
we then speak of the a. c. resistance R ∼ of the conductor. The ratio expressing the
increase in resistance is
R∼
R–

a. c. resistance
d. c. resistance

ζ = —– = ———–———– .
Specific values for major materials are shown in Table 2-7.
Table 2-7
Numerical values for major materials
Conductor

Aluminium, 99.5 % Al, soft
Al-Mg-Si
Al-Mg
Al bronze, 90 % Cu, 10 % Al
Bismuth
Brass
Bronze, 88 % Cu, 12 % Sn
Cast iron
Conductor copper, soft
Conductor copper, hard
Constantan
CrAI 20 5
CrAI 30 5
Dynamo sheet
Dynamo sheet alloy (1 to 5 % Si)
Graphite and retort carbon
Lead
Magnesium
Manganin
Mercury
Molybdenum
Monel metal
Nickel silver

Specific
electric
resistance ρ
(mm2 Ω/m)
0.0278
0.03…0.033
0.06…0.07
0.13
1.2
0.07
0.18
0.60…1.60
0.01754
0.01786
0.49…0.51
1.37
1.44
0.13
0.27…0.67
13…100
0.208
0.046
0.43
0.958
0.054
0.42
0.33

Electric
conductivity
x = 1/ ρ
(m/mm2 Ω)

Temperature
coefficient α

Density

(K–1)

(kg/dm3)

20–3

36
4·
33…30
3.6 · 10–3
17…14
2.0 · 10–3
7.7
3.2 · 10–3
0.83
4.5 · 10–3
14.3
1.3…1.9 · 10–3
5.56
0.5 · 10–3
1.67…0.625 1.9 · 10–3
57
4.0 · 10–3
56
3.92 · 10–3
2.04…1.96 –0.05 · 10–3
0.73
0.05 · 10–3
0.69
0.01 · 10–3
7.7
4.5 · 10–3
3.7…1.5
—
0.077…0.01 –0.8…–0.2 · 10–3
4.8
4.0 · 10–3
21.6
3.8 · 10–3
2.33
0.01 · 10–3
1.04
0.90 · 10–3
18.5
4.3 · 10–3
2.8
0.19 · 10–3
3.03
0.4 · 10–3

2.7
2.7
2.7
8.5
9.8
8.5
8.6…9
7.86…7.2
8.92
8.92
8.8
—
—
7.8
7.8
2.5…1.5
11.35
1.74
8.4
13.55
10.2
—
8.5

(continued)

59

Table 2-7 (continued)
Numerical values for major materials
Conductor

Ni Cr 30 20
Ni Cr 6015
Ni Cr 80 20
Nickel
Nickeline
Platinum
Red brass
Silver
Steel, 0.1% C, 0.5 % Mn
Steel, 0.25 % C, 0.3 % Si
Steel, spring, 0.8 % C
Tantalum
Tin
Tungsten
Zinc

Specific
electric
resistance ρ
(mm2 Ω/m)

Electric
conductivity
x = 1/ ρ
(m/mm2 Ω)

Temperature
coefficient α

Density

(K–1)

(kg/dm3)

1.04
1.11
1.09
0.09
0.4
0.1
0.05
0.0165
0.13…0.15
0.18
0.20
0.16
0.12
0.055
0.063

0.96
0.90
0.92
11.1
2.5
10
20
60.5
7.7…6.7
5.5
5
6.25
8.33
18.2
15.9

0.24 · 10–3
0.13 · 10–3
0.04 · 10–3
6.0 · 10–3
0.18…0.21 · 10–3
3.8…3.9 · 10–3
—
41 · 10–3
4…5 · 10–3
4…5 · 10–3
4…5 · 10–3
3.5…10–3
4.4 · 10–3
4.6 · 10–3
3.7 · 10–3

8.3
8.3
8.3
8.9
8.3
21.45
8.65
10.5
7.86
7.86
7.86
16.6
7.14
19.3
7.23

Resistance varies with temperature, cf. Section 2.3.3

2.3.2 Resistances in different circuit configurations
Connected in series (Fig. 2-4)

Fig. 2-4
Total resistance = Sum of individual resistances
R = R1 + R2 + R3 + …
The component voltages behave in accordance with the resistances U1 = I R1 etc.
U
The current at all resistances is of equal magnitude I = – .
R
Connected in parallel (Fig. 2-5)

Fig. 2-5
Total conductance = Sum of the individual conductances
1
– = G = G1 + G2 + G3 + …
R
60

1
R = –.
G

2

In the case of n equal resistances the total resistance is the n th part of the individual
resistances. The voltage at all the resistances is the same. Total current
U
U
I = – = Sum of components I1 = — etc.
R
R1
The currents behave inversely to the resistances
R
R
R
I1 = I — ; I2 = I — ; I3 = I — .
R2
R3
R1
Transformation delta-star and star-delta (Fig. 2-6)
Fig. 2-6
Conversion from delta to star connection with the same total resistance:
R d2 R d3
RS1 = ——————
—–
R d1 + R d2 + R d3
R d3 R d1
RS2 = ——————
—–
R d1 + R d2 + R d3
R d1 R d2
RS3 = ——————
—–
R d1 + R d2 + R d3
Conversion from star to delta connection with the same total resistance:
R S1 R S2 + R S2 R S3 + R S3 R S1
R d1 = ——————–—————
——–
R S1
R S1 R S2 + R S2 R S3 + R S3 R S1
R d2 = ——————–——————
——
R S2
R S1 R S2 + R S2 R S3 + R S3 R S1
R d3 = ——————–————
———–
R S3
Calculation of a bridge between points A and B (Fig. 2-7)
To be found:
1. the total resistance R tot between points A and B,
2. the total current I tot between points A and B,
3. the component currents in R 1 to R 5.
Given:
voltage
U =
resistance R1 =
R2 =
R3 =
R4 =
R5 =

220 V.
10 Ω,
20 Ω,
30 Ω,
40 Ω,
50 Ω.

Fig. 2-7
61

First delta connection CDB is converted to star connection CSDB (Fig. 2-8):
R2 R5
20 · 50
= ———–—— = 10 Ω,
R25 = ———–——
20 + 30 + 50
R2 + R3 + R5
30 · 50
R3 R5
= ———–—— = 15 Ω,
R35 = ———–——
20 + 30 + 50
R2 + R3 + R5
20 · 30
R2 R3
= ———–—— = 6 Ω,
R23 = ———–——
20 + 30 + 50
R2 + R3 + R5
(R1 + R25) (R4 + R35)
+ R23 =
Rtot = ———–——————
R1 + R25 + R4 + R35
(10 +10) (40 + 15)
= ———–—————— + 6 = 20.67 Ω.
10 + 10 + 40 + 15

Fig. 2-8

U
220
Itot = —– = ——– = 10.65 A.
20.67
Rtot
20.67 – 6
Rtot – R23
= 10.65 · ————– = 7.82 A,
IR1 = Itot ——–—–
10 + 10
R1 + R25
20.67 – 6
Rtot – R23
—–—– = 10.65 · ————– = 2.83 A,
IR4 = Itot —–
40 + 15
R4 + R35
By converting the delta connection CDA to star connection CSDA, we obtain the following values (Fig. 2-9):
R15 = 5 Ω; R45 = 20 Ω; R14 = 4 Ω; IR2 = 7.1 A; IR3 = 3.55 A.

Fig. 2-9

With alternating current the calculations are somewhat more complicated and are
carried out with the aid of resistance operators. Using the symbolic method of
calculation, however, it is basically the same as above.
2.3.3 The influence of temperature on resistance
The resistance of a conductor is
l
l·ρ
R = ––— = ——
A
x·A
where
l = Total length of conductor
A = Cross-sectional area of conductor
ρ = Specific resistance (at 20 °C)
x

1
= – Conductance

ρ

α = Temperature coefficient.
Values for ρ, x and α are given in Table 2-7 for a temperature of 20 °C.
For other temperatures ϑ1) (ϑ in °C)
ρϑ = ρ20 [1 + α (ϑ – 20)]
1)

Valid for temperatures from – 50 to + 200 °C.

62

and hence for the conductor resistance

l

2

Rϑ = – · ρ20 [1 + α (ϑ – 20)].
A
Similarly for the conductivity
xϑ = x20 [1 + α (ϑ – 20)]–1
The temperature rise of a conductor or a resistance is calculated as
R / R –1

w
k
∆ ϑ = ————–
·
α

The values R k and R w are found by measuring the resistance of the conductor or
resistance in the cold and hot conditions, respectively.
Example:
The resistance of a copper conductor of l = 100 m and A = 10 mm2 at 20 °C is
100 · 0.0175
R20 = —————– = 0.175 Ω.
10
If the temperature of the conductor rises to ϑ = 50 °C, the resistance becomes
100
R50 = —– · 0.0175 [1 + 0.004 (50 – 20)] ≈ 0.196 Ω.
10

2.4

Relationships between voltage drop, power loss and conductor
cross section

Especially in low-voltage networks is it necessary to check that the conductor crosssection, chosen with respect to the current-carrying capacity, is adequate as regards
the voltage drop. It is also advisable to carry out this check in the case of very long
connections in medium-voltage networks. (See also Sections 6.1.6 and 13.2.3).
Direct current
voltage drop
percentage
voltage drop

2·l·P
2·l·I
∆ U = R'L · 2 · l · I = ——— = ————
x·A

x·A·U

∆U

R ’L · 2 · l · I
∆ u = —— 100 % = ——
——— 100 %
Un
Un

power loss

2 · l · P2
∆ P = I 2 R'L 2 · l = ———— 2

percentage
power loss

∆P
I 2 R'L · 2 · l
∆ p = —— 100 % = —————
100 %

conductor
cross section

A

x·A·U

Pn

Pn

2·l·I
2·l·I
2·l·P
= ———— = ————– 100 % = —————
100 %
∆ p · U2 · x
x·∆u·U
x·∆U
63

Single-phase alternating current
voltage drop2)
percentage
voltage drop2)

∆ U = I · 2 · l (R 'L · cos ϕ + X'L · sin ϕ)
∆U
I · 2 · l (R'L · cos ϕ + X 'L · sin ϕ)
∆ u = —– 100 % = —————–————————
Un

Un

power loss

2 · l · P2
∆ P = I 2R 'L · 2 · l = —————––——
x · A · U 2 · cos2 ϕ

percentage
power loss

∆P
I 2 · R'L · 2 · l
∆ p = —— 100 % = —————–
100 %

conductor
cross-section1)

Pn

Pn

2 · l cos ϕ
A = ———————————–—
∆U
x —— — X'L· 2 · l · sin ϕ
I
2 · l cos ϕ
= ——————————————–
∆ u · Un
x ———— — X'L· 2 · l · sin ϕ
I · 100 %

(

)

(

)

Three-phase current
voltage drop2)
percentage
voltage drop2)

∆ U = 
3 · I · l (R'L · cos ϕ + X'L · sin ϕ)
∆U

3 · I · l (R ’L · cos ϕ + X'L · sin ϕ)
∆ u = —– 100 % = —————–—————————
100 %
Un

Un

power loss

l · P2
∆ P = 3 · I 2 R'L · l = —————
–——–
x · A · U 2 · cos2 ϕ

percentage
power loss

∆P
3 I 2 · R'L · l
∆ p = —— 100 % = —————
100 %

conductor
cross-section1)

Pn

Pn

l · cos ϕ
A = ———————————–
∆U
x ——– – X'L · l · sin ϕ

3·I
l · cos ϕ
= ————————————–———
∆u·U
x ——————– — X'L · l · sin ϕ

3 · I · 100 %

(
(

l = one-way length of

)

)

conductor

R 'L = Resistance
per km

P = Active power to be
transmitted (P = Pn)

U = phase-to-phase
voltage

X 'L = Reactance
per km

I = phase-to-phase
current

In single-phase and three-phase a.c. systems with cables and lines of less than 16 mm2
the inductive reactance can usually be disregarded. It is sufficient in such cases to
calculate only with the d.c. resistance.
1)
2)

Reactance is slightly dependent on conductor cross section.
Longitudinal voltage drop becomes effectively apparent.

64

Effective resistances per unit length of PVC-insulated cables with copper conductors as
per DIN VDE 0271 for 0.6/1 kV
Number
of conductors and
crosssection

D. C.
resistance
at 70 °C

Ohmic
resistance at
70 °C

Inductive
reactance

Effective resistance per unit length
R 'L · cos ϕ + X 'L · sin ϕ
at cos ϕ
0.95
0.9
0.8
0.7

0.6

mm2

R 'L–
Ω / km

R 'L~
Ω / km

X 'L
Ω / km

Ω / km

Ω / km

Ω / km

Ω / km

Ω / km

4 × 1.5
4 × 2.5
4×4
4×6
4 × 10
4 × 16
4 × 25
4 × 35
4 × 50
4 × 70
4 × 95
4 × 120
4 × 150
4 × 185
4 × 240
4 × 300

14.47
14.47
8.71
8.71
5.45
5.45
3.62
3.62
2.16
2.16
1.36
1.36
0.863
0.863
0.627
0.627
0.463
0.463
0.321
0.321
0.231
0.232
0.183
0.184
0.149
0.150
0.118
0.1202
0.0901 0.0922
0.0718 0.0745

0.115
0.110
0.107
0.100
0.094
0.090
0.086
0.083
0.083
0.082
0.082
0.080
0.080
0.080