Home-immediately access 800+ free online publications. Download CD3WD (680 Megabytes) and distribute it to the 3rd World. CD3WD is a 3rd World Development private-sector initiative, mastered by Software Developer Alex Weir and hosted by GNUveau_Networks (From globally distributed organizations, to supercomputers, to a small home server, if it's Linux, we know it.)ar.cn.de.en.es.fr.id.it.ph.po.ru.sw

CLOSE THIS BOOKIntroduction to Electrical Engineering - Basic vocational knowledge (Institut fr Berufliche Entwicklung, 213 p.)
2. Fundamental Quantities of Electrical Engineering
VIEW THE DOCUMENT2.3. Resistance and Conductance

Introduction to Electrical Engineering - Basic vocational knowledge (Institut fr Berufliche Entwicklung, 213 p.)

2. Fundamental Quantities of Electrical Engineering

2.1. Current

Flowing quantities of electricity cause effects which are utilised in practice, i.e. in electrical engineering. Since flowing quantities of air are called an air current, flowing quantities of water a water current, the phenomenon of flowing quantities of electricity is called electrical current.

The carriers of the quantities of electricity are called charge carriers. Mostly the latter are electrons, in rare cases ions. An electron has the smallest imaginable charge which, therefore, is called elementary charge. In electro-technology, electrons are considered as practically massless charge carriers because of their small volume and extremely small mass.

Electrons are constituents of atoms, the basic units of which the material world is constructed. An atom consists of a nucleus and the electrons surrounding it. Atoms or groups of atoms which have lost or gained one or more electrons are called ions. Ions are charge carriers having mass. When an ion - as compared with the chargelss neutral atom - has more electrons, then it is called negatively charged, when it has less electrons, it is called positively charged. The electron itself has a negative charge.

As a current consists of flowing quantities of electricity (charge quantities), it can only flow in such substances which possess freely mobile, non-stationary charge carriers. Substances with many mobile charge carriers are called conductors. They include all metals (especially silver, copper, aluminium and iron) and electrolytes (salt solutions). As the current in metals is carried by electrons, it is called electron current, whereas the current flowing through electrolytes is called ion current because the flowing charge carriers are ions.

Fig. 2.1. Electron current; the free electrons move through the atomic lattice of the conductor

1 - Atomic union
2 - Conductor electrons

Fig. 2.2. Ion current

1 - Metallic feed lines
2 - Electrolyte
3 - Electrons
4 - Negative ions
5 - Positive ions
6 - Neutral molecules

Substances in which the charge carriers are fixed or stationary, that is to say, they are not freely mobile, are called non-conductors or insulators. Current cannot flow through them. The most important non-conductors are procelain, glass, plastics.

There are substances whose electrical conductivity is such that they are between conductor and non-conductor. They conduct current so badly that they cannot be termed as conductor but they allow a small current to flow so that they cannot be used as a non-conductor. These substances are called semiconductor. The most important semiconductors are silicon, germanium and selenium. Semiconductors are of particular practical importance to electrical engineering.

We cannot perceive electrical currents directly but only indirectly we become aware of three characteristic effects of current.

These are

1. the generation of heat in conductors through which current flows
2. the magnetic field associated with the current
3. transport of substance by ion currents

Re 1. - Every electric current generates electric heat in conductors. It is utilised in electric heating engineering, for example, cooking plate, flat iron. The generation of electric heat can be imagined in such a way that the flowing charge carriers collide with the stationary particles forming the skeleton of the material or substance. As a consequence, the energy of the braked charge carriers is converted into irregular oscillatory energy, namely thermal energy, of the stationary particles.

Fig. 2.3. Development of heat in the current carrying conductor

Re 2. - Every electric current is accompanied by a magnetic field. It surrounds the current spatially like an eddying fluid its axis of eddy. There is no current without a magnetic eddy and no magnetic field eddy without current. Proof of this can easily be given by means of a magnetic needle which with initial direction parallel to the current will be turned so that it is across to the current. The mutual coupling of current and magnetic field is of eminent practical importance, for example, for an economic production of electrical energy (see Section 5).

Fig. 2.4. Magnetic field associated with the current

1 - North pole
2 - South pole

Re 3. - When a fluid conductor with ions is interposed in a metallic current path, material changes will take place at the two feed wires when current flows. These material changes are the result of the material particles flowing with the ions, in other words, a consequence of the transport of substance associated with the current. From the ions, the electrons can migrate into the current supply leads or out of them; this cannot be effected by the material particles which, consequently, are deposited at these leads. If, for example, the fluid conductor is a copper sulphate solution, the copper particles are separated at one electrode in the form of a metallic coat. This process is called electrolysis. It is used for the winning of metals, especially metals in a pure state, for the deposition of metallic coats and protective coverings (galvanisation).

Fig. 2.5. Transport of matter in case of conduction by ions

In order to define the intensity of a current, the term current intensity (formula sign I) has been introduced.

Obviously, it is independent of the place of the line, the line material and the line cross-sectional area but it is only determined by the number of charge carriers (quantity of charge Q) flowing through the line in a certain time t. When, in a certain time, many charge carriers flow through the conductor, then the current intensity is high, vice versa it is low.

The following holds:



I = Q/t


current intensity


charge quantity



The sign of the current intensity indicates the current direction. It is an arbitrarily established mathematical direction of counting and should not be confused with the aktual flowing direction of the moved charge carriers. One has defined:

The current intensity is positive when the current direction is equal to the direction of flow of the positive charge carriers or when it is opposite to the direction of flow of negative carriers (e.g. electrons).

The unit of current intensity is called ampere = A in honour of the French physicist Marie Andr Ampre (1775 - 1836).

[I] = A

Other usually used units of the ampere are

1 kA = 1 kiloampere = 103 A = 1.000 A
1 mA = 1 milliampere = 10-3 A = 0.001 A
1 µA = 1 microampere = 10-6 A = 0.000001 A

In electrical engineering, current intensities may occur in largely different magnitudes. Table 2.1. shows a few values.

Table 2.1. Current Intensities for a Few Applications

Melting furnace

100,000 A


100 kA

Aluminium production

10,000 A


10 kA


1,000 A


1 kA

Starter for motor-car

100 A

Household appliances up to

6 A


0,5 A


500 mA

Torch lamp

0,2 A


200 mA

After the establishment of the basic unit for the current intensity, units for the quantity of electricity can be derived from equation (2.1.), namely

Q = I · t
[Q] = [I] · [t]
[Q] = A · s and from 1A · 1s = 1C = 1 coulomb follows
[Q] = C

The product of A · s is called coulomb in honour of the French physicist Charles Auguste de Coulomb (1736 - 1806).

A larger unit of the quantity of electricity is the ampere-hour (a.h). As 1 hour has 3,600 seconds, the following relation holds for the conversion of A.s into A.h:

1 A · h = 1 A · 3600 s = 3600 A · s = 3600 C

The electrical current is the phenomenon of flowing quantities of electricity. The carriers of the quantities of electricity are called charge carriers; these are electrons and ions. As to their conductivity for electrical current, the various substances are divided into conductors, non-conductors (insulators) and semiconductors. The three characteristic effects of current are

- generation of heat in conductors through which current passes
- the magnetic field associated with the current
- transport of substance by ion currents

The current intensity is determinded by the quantity of charge flowing through the conductor during a certain time. It results from the relation I = Q/t. The unit of current intensity is the ampere = A; the most frequently used sub-units are kA, mA and µA.

From the definition equation of the current intensity, the basic unit for the quantity of electricity is derived; it is the ampere-second (A.s) = coulomb (C). A frequently used sub-unit is the ampere-hour (A.h).

Questions and problems:

1. How many A are 27 mA; 5,1 kA; 80 µA; 1,000 mA; 6,500 µA; 0,04 kA

2. How many C are

0,5 A.h; 84 A.h; 0,000278 A.h?

3. A quantity of electricity of 108,000 C is flowing through a line within 5 hours. Find the current intensity.

4. An electric current having the intensity of 2 A flows through a line for a period of 2 hours. Calculate the transported quantity of electricity in the units C and A.h.

2.2. Voltage

In order that a current flows through a conductor, an electrical “pressure” must be exerted on the freely mobile charge carriers. This “pressure” is the electrical drive phenomenon on the charge carriers which is called voltage. There is no current without an electrical voltage.

The original drive phenomenon for current is called primary electromotive force. It is generated in a voltage source. It imparts energy to the charge carriers which thus are driven through the conductor.

Since every conductor offers resistance more or less to the passage of current, the charge carriers lose energy when passing through. This loss can be characterised as voltage drop.

A current can only flow through a conductor; therefore, the current path formed by the conductor must be closed.

When a charge carrier has received drive energy from a voltage source, it passes through the conductor, completely transferring the energy taken up to this conductor. After exactly one circulation, the charge carrier differs by nothing from its state bevor it started the circulation, that is to say, it cannot have stored energy.

The primary electromotive force is designated by the formula sign E, the voltage drop by U. In practice, no difference is made between these two terms and they are called voltage in short. Primary electromotive force and voltage drop have the same unit which is called volt - V in honour of the Italian physicist Alessandro Volta (1745 - 1827).

[E] = V
[U] = V

Frequently used sub-units of volt are

1 MV = 1 megavolt = 106 V = 1,000,000 V
1 kV = 1 kilovolt = 103 V = 1,000 V
1 mV = 1 millivolt = 10-3 V = 0.001 V
1 µV = 1 microvolt = 10-6 V = 0.000001 V

In electrical engineering, voltages may occur in quite different magnitudes. Table 2.2. shows some values.

Table 2.2. Voltage Values for a Few Applications

Lightning up to

10.000,000 V


10 MV

Extra-high voltage lines

600,000 V


600 kV

High-voltage lines

60,000 V


60 kV

Sparking-plug in an internal combustion engine

15,000 V


15 kV

Lighting network

220 V

Motor-car battery

12 V

Receiving voltage of a wireless set

0.000,01 V


10 µV

The primary electromotive force is a prerequisite for an electrical current. Table 2.5. shows the various possibilities of producing a primary electromotive force, the designations of the respective voltage sources and their main applications.

For the winning of electrical energy, the generation of the primary electromotive force by chemical and magnetic-field actions is of particular importance. On principle, these voltage sources operate as follows

· Primary electromotive force by chemical action

When immersing two conductors of different kinds into an electrolyte, then one will find an excess of electrons at one conductor (negative pole) and an electron deficit at the other conductor (positive pole). This charge carrier difference externally acts as electrical primary electromotive force. Diluted sulphuric acid H2SO4 is suitable as electrolyte; as conductor rods (electrodes), copper Cu and zinc Zn are particulary suitable (Fig. 2.6.).

Table 2.3. Ways of Producing Primary Electromotive Forces

Causes of the production of the electromotive force

Designation of the voltage source

Examples of use

chemical action

galvanic cell; battery; accumulator

voltage supply to portable devices; starting battery in motor-cars

thermal action

thermoelectric element (thermocouple)

measuring the temperature at points which are not readily accessible; remote temperature measurement

action of magnetic field (induction)


economical generation of electrical energy in power stations

action of light

photovoltaic cell; solar cell

measuring the intensity of illumination

charge separation by

- influence

influence machine

generation of high and extra-high voltages by means of which, for example, the properties of insulating materials are tested

- mechanical charge movement

belt-type generator

displacement of charge (polarisation) on a non-conductor by means of pressure

piezoelectric element

measurement of pressure; sound pick-up for records; microphone

Fig. 2.6. Galvanic element also known as galvanic cell

Other substances are also suitable (especially coal and zinc in a thickened ammonium chloride solution).

In accordance with the general tendency to balance differences in concentration, the basic units of construction of the solid conductors are eager to migrate as ions in the electrolyte. On the other hand, the electrolyte tries to press its ions into the solid conductor. This impetus of motion is different in the different conductor materials so that, as a result, a primary electromotive force acts externally.

When current flows, these voltage sources disintegrate due to the transport of substance and become useless; this is also occurring when stored too long. Rechargeable voltages sources do not show this disadvantage, therefore, they are called accumulators (storage batteries). Lead accumulators and nickel-iron or nickel-cadmium accumulators are of particular importance.

· Primary electromotive force by magnetic-field action (induction)

This production of voltage is of greatest technical importance and it is used in all cases when primary electromotive force is to be generated by mechanical motion. According to a law of nature (law of induction) the following happens:

When the magnetic flux enclosed by a conductor loop is changed, the charge carriers in the conductor are subjected to an impetus to move. Then, the entire conductor loop is a primary electromotive force source.

The change of the magnetic flux may, for example, be due to the fact that the conductor loop is turned inside the magnetic field or the magnet is approached to are moved away from this loop.

Fig. 2.7. Primary electromotive force 2 generated by induction

1 - Direction of motion
2 - Direction of the primary electromotive force
3 - North pole
4 - South pole

As symbol of a voltage source, the graphical symbol shown in Fig. 2.8. is used. The electrode with an excess of electrons is called negative pole (-); the electrode with an electron deficit is called positive pole (+).

Fig. 2.8. Graphical symbol of a (direct) voltage source; the arrow indicating the direction may be omitted

The direction of voltage corresponds to the direction of current defined in Section 2.1.; thus, the primary electromotive force E is directed from - to + whereas the voltage drop U runs from + to -.

The voltage direction is indicated by an arrow.

The electrical drive exerted on the charge carriers is called voltage. The drive phenomenon originally generated in a voltage source is called primary electromotive force E; the loss in voltage caused when current flows through a conductor is called voltage drop U. As unit of the voltage, the volt - V - has been laid down; the most frequently used sub-units are MV, kV and µV. For the winning of electrical energy, the generation of the primary electromotive force by chemical action and by the action of the magnetic field is of particular importance.

Questions and problems:

1. How many V are

500 mV; 2,5 kV; 350 µV; 0,6 MV?

2. Give reasons for the fact why in a current passage the sum of all voltage drops must be equal to the entire primary electromotive force!

2.3. Resistance and Conductance

Every conductor and every electrical device (electric bulb, heater, electromotor, wireless reciever, etc.) has the property of resisting any current passage. This property is called electrical resistance (formula sign B), Depending on the material used and the design of the conductor or the device, it has a different magnitude.

For a conductor, the geometrical dimensions and the conductor material are decisive for the value of the resistance. The formula for calculating the resistance is called resistance rating formula. It is easily understood and can be checked by experiment that a long thin wire will offer a higher resistance to the current passage than a short thick one. When designating the line length by 1 and the line cross-sectional area by A, then the resistance R is proportional to 1/A, hence,

R ~ 1/A

Finally, the resistance is dependent on the conductor material; for example, iron as a conductor is inferior to copper (iron has a higher resistance). This dependence on material is covered by a material constant which is termed as specific resistance or resistivity (formula sign r 1)). Hence,

1) r Greek letter rho






specific resistance


length of the conductor


cross-sectional area of the conductor

The higher the resistance, the poorer the conduction of the current. The permeability to current of a conductor is called conductance (formula sign G) and, hence, is inversely proportional to the resistance.

G = 1/R






Similar relations apply to the material constant. In the place of the specific resistance, the specific conductance (formula k 1)) can be stated as reciprocal value; k=1/r. From the equations (2.2.) and (2.3), the rating equation for the electrical conductance is obtained as follows

1) k Greek letter kappa





specific conductance; k=1/r


cross-sectional area of the conductor


length of the conductor

The unit of the resistance is called ohm in honour of the German physicist Georg Simon Ohm (1789 - 1854) and abbreviated by the Greek letter W 2)

2) W Greek letter omega

[R] = W

A conductor has a resistance of 1W if a voltage of 1 V drops when a current of 1 A passes this conductor.

The unit of the conductance is called siemens = S in honour of the German physicist Werner von Siemens (1816 - 1892). (In English-speaking countries, the unit siemens has not been generally adopted.) The correlation between the units siemens and ohm is given by equation (2.3).

[G] = S = 1/W

Frequently used sub-units of ohm (W) and siemens (S) are

1 MW


1 megaohm


106 W


1,000,000 W

1 kW


1 kiloohm




1,000 W

1 mW


1 milliohm





1 kS


1 kilosiemens


103 S


1,000 S

1 mS


1 millisiemens


10-3 S


0.001 S

1 µS


1 microsiemens


10-6 S


0.000001 S

Now, units can be given also for the specific resistance and the specific conductance by rearranging the equations (2.2) and (2.4).

For r from equation (2.2.) we have

r = R · A/I

[r] = W · m²/m = W · m

A frequently used sub-unit is W · mm²/m = 10-6 W·m

From equation (2.4), for k we have

k = G l/A

[k] = S · m/m² = S/m = 1/(W · m)

Table 2.4. shows for a few substances the values of r and k.

Example 2.1.

Calculate the resistance and conductance of a copper wire having a length of 175 m and a cross-sectional areas of 2,5 mm2.


l = 175 mm
A = 2,5 mm2
rCu = 0.0178 (W · mm2)/m
(kCu = 1/rCu » 56 · 106 S/m)

To be found:



R= r · l/A
R=0.0178 · (W · mm2)/m
175 m/2.5 mm2 = 1.246 W
G = 1/1.246 W = 0.804 S

Example 2.2.

A copper conductor having a cross-sectional area of 6 mm is to be replaced by an aluminium conductor of the same resistance. What is the size of the cross-sectional area of the aluminium conductor?


ACu = 6 mm2
rCu = 0.0178 (W · mm2)/m
rAl = 0.0286 (W · mm2)/m

To be found:



RCu = RAl
RCu = rCu · 1/Acu
RAl = rCu · 1/AAl
rCu · 1/Aal = rAl · 1/AAl
AAl = rAl/rCu · Acu
AAl = 0.0286/0.0178 · 6 mm2 = 9.64 mm2

For the aluminium conductor, the standardised cross-sectional o area of 10 mm2 is selected.

The most striking influence on the resistance of a conductor or device is exerted by the temperature.

The temperature dependence of the electrical resistance can be quantitatively expressed by the temperature coefficient a1)

1) a Greek letter alpha

The temperature coefficient states the fraction by which the resistance changes with a change in temperature of 1 K:

a = (DR/R) · 1/Du




temperature coefficient

D 2)

R/R change in resistance related to the initial resistance

Du 3)

temperature change

2) D Greek letter delta
3) u Greek letter theta

The unit of the temperature coefficient is

[a] = 1/K (K = Kelvin)

In metallic conductors, the resistance increases with increasing temperature. This is due to the fact that the more intensively oscillating crystal lattices offer a higher resistance to the electron current; hence, a is positive.

In electrolytes and semiconductors, the resistance diminishes with increasing temperature. This is due to the fact that with rise in temperature more charge carriers are released which then are available as free charge carriers for the transport of electricity; hence, a is negative.

For practice, the following approximate values of the temperature coefficient will suffice (see also Table 2.4):

· Non-ferromagnetic pure metals (no metal alloys)

a » + 0.004 1/K

The resistance of a copper conductor of 100W, for example, will increase by 0.4W to 100.4W in the event of an increase in temperature of 1 K; in case of a rise in temperature of 80 K (e.g. from 20 °C to 100 °C) it will increase by 32 W to 132 W.

· Ferromagnetic metals (iron, nickel)

a » + 0.006 1/K

· Metal alloys of a special composition (novoconstant, constantan)

a » 0

These special metal alloys are of particular importance to measuring techniques if resistors independent of temperature are required.

· Electrolytes

a » - 0.02 1/K

· Semiconductors

a is negative and largely dependent on temperature; a numerical value cannot be stated; it should be drawn from special Tables for the temperatures involved.

Table 2.4. Specific Resistance r, Conductance k and Temperature Coefficient a of a Few Conductor Materials




Conductor Material

W · mm2/m

S · m/mm2





» + 0.004




» + 0.004




» + 0.004




» + 0.004




» + 0.004




» + 0.006

iron, pure



» + 0.006

Novokonstand 1)



» 0

constantan 2)



» 0

1) Novokonstant: 82.5 % Cu; 12 % Mn; 4 % Al; 1.5 % Fe
2) constantan: 54 % Cu; 45 % Ni; 1 % Mn

Example 2.3.

A coil of copper wire has a resistance of 18 W at room temperature (20 °C). During operation, the temperature rises to 85 °C. Find the resistance of the coil at this temperature.


R20 = 18 W
D u = 85 °C - 20 °C = 65 K
a » +0.004 1/K

To be found:


Solution: From equation (2.5) we obtain by transposing a value for the change of resistance

DR = a R20 Du

This amount must be added to the resistance R20 in order to determine the final resistance R85.

R85 = R20 + DR
R85 = R20 + aR20Du
R85 = R20 (1 + aDu)
R85 = 18W (1 + 0.004 1/K · 65K)
R85 = 18W (1 + 0.26)
R85 = 18W · 1.26
R85 = 22.68W

Components which are used to limit the current by means of certain resistance values and which are constructed specifically for this purpose are called resistors. Resistor is a component for the realisation of a certain resistance value.

The general graphical symbol of a resistor is shown in Fig. 2.9.

Fig. 2.9. Graphical symbol of a resistor

Resistance and conductance are properties of electrical conductors and devices. The resistance characterises the resistance offered to the passage of current; the conductance indicates how well the conductor or device in question allows the current to pass. The correlation between resistance and conductance results from the relation

R = 1/G

The rating equation of the resistance and of the conductance is

R = r · l/A and G = k · A/l

The material constant r is called specific resistance, k is called specific conductance.

The resistance (and the conductance, too) is primarily depending on temperature. The magnitude of the temperature dependence is covered by the temperature coefficient a which indicates the relative change in resistance per degree of change in temperature. For non-ferromagnetic metals, a = +0.004 1/K; this means that the resistance of these materials increases with increasing temperature. As unit of the resistance, the ohm = W is specified; the most frequently used sub-units are MW, KW, mW.

The unit of conductance is siemens = S = 1/W; the most frequently used sub-units are kS, mS, µS.

A component which is specially built to realise a certain resistance value is called resistor.

Questions and problems:

1. How many W are

2 MW 15 kW; 350 mW; 0.5 µS; 4 S; 2 mS?

2. For the supply of energy to a consumer situated at a distance of 150 m, a 2-core copper line with a cross-sectional area of 2.5 mm2 per conductor is used. Calculate the resistance and the conductance of the line (take into consideration the outgoing and the return conductors).

3. Calculate the temperature (related to a reference temperature of 20 °C) at which the resistance of a copper wire will double.