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close this bookIntroduction to Electrical Engineering - Basic vocational knowledge (Institut für Berufliche Entwicklung, 213 p.)
close this folder2. Fundamental Quantities of Electrical Engineering
View the document2.1. Current
View the document2.2. Voltage
View the document2.3. Resistance and Conductance

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


(2.2)

where

R

resistance

r

specific resistance

l

length of the conductor

A

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
(2.3)

where

G

conductance

R

resistance

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

where

G

conductance

k

specific conductance; k=1/r

A

cross-sectional area of the conductor

l

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

=

103

=

1,000 W

1 mW

=

1 milliohm

=

10-3

=

0.001W















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.

Given:

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

To be found:

R
G

Solution:

R= r · l/A
G=1/R
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?

Given:

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

To be found:

AAl

Solution:

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

(2.5)

where

a

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


r

k

a

Conductor Material

W · mm2/m

S · m/mm2

1/K

silver

0.016

62.5

» + 0.004

copper

0.0178

56

» + 0.004

aluminium

0.0286

55

» + 0.004

zinc

0.063

16

» + 0.004

lead

0.21

4.8

» + 0.004

nickel

0.10

10

» + 0.006

iron, pure

0.10

10

» + 0.006

Novokonstand 1)

0.45

2.3

» 0

constantan 2)

0.5

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.

Given:

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

To be found:

R85

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.