Introduction to Electrical Engineering  Basic vocational knowledge (Institut für Berufliche Entwicklung, 213 p.) 
2. Fundamental Quantities of Electrical Engineering 

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 crosssectional 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
_{}
where
R 
resistance 
r 
specific resistance 
l 
length of the conductor 
A 
crosssectional 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 
crosssectional 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 Englishspeaking 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 subunits of ohm (W) and siemens (S) are
1 MW 
= 
1 megaohm 
= 
10^{6} W 
= 
1,000,000 W 
1 kW 
= 
1 kiloohm 
= 
10^{3} 
= 
1,000 W 
1 mW 
= 
1 milliohm 
= 
10^{3} 
= 
0.001W 
 
  
 
 
  
 
1 kS 
= 
1 kilosiemens 
= 
10^{3} 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 subunit 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 crosssectional areas of 2,5 mm^{2}.
Given:
l = 175 mm
A = 2,5 mm^{2}
r_{Cu} = 0.0178 (W · mm^{2})/m
(k_{Cu }= 1/r_{Cu }» 56 · 10^{6 }S/m)
To be found:
R
G
Solution:
R= r · l/A
G=1/R
R=0.0178 · (W · mm^{2})/m
175 m/2.5 mm^{2 }= 1.246 W
G = 1/1.246 W = 0.804 S
Example 2.2.
A copper conductor having a crosssectional area of 6 mm is to be replaced by an aluminium conductor of the same resistance. What is the size of the crosssectional area of the aluminium conductor?
Given:
A_{Cu} = 6 mm^{2}
r_{Cu} = 0.0178 (W · mm^{2})/m
r_{Al} = 0.0286 (W · mm^{2})/m
To be found:
A_{Al}
Solution:
R_{Cu} = R_{Al}
R_{Cu }= r_{Cu }·_{ }1/A_{cu}
R_{Al }= r_{Cu }·_{ }1/A_{Al}
r_{Cu }·_{ }1/A_{al} = r_{Al }·_{ }1/A_{Al}
A_{Al} = r_{Al}/r_{Cu} · A_{cu}
A_{Al} = 0.0286/0.0178 · 6 mm^{2} = 9.64 mm^{2}
For the aluminium conductor, the standardised crosssectional o area of 10 mm^{2} 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 a^{1)}
^{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
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):
· Nonferromagnetic 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 · mm^{2}/m 
S · m/mm^{2} 
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:
R_{20} = 18 W
D u = 85 °C  20 °C = 65 K
a » +0.004 1/K
To be found:
R_{85}
Solution: From equation (2.5) we obtain by transposing a value for the change of resistance
DR = a R_{20} Du
This amount must be added to the resistance R_{20} in order to determine the final resistance R_{85}.
R_{85} = R_{20} + DR
R_{85} = R_{20} + aR_{20}Du
R_{85} = R_{20} (1 + aDu)
R_{85} = 18W (1 + 0.004 1/K · 65K)
R_{85} = 18W (1 + 0.26)
R_{85} = 18W · 1.26
R_{85} = 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 nonferromagnetic 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 subunits are MW, KW, mW.
The unit of conductance is siemens = S = 1/W; the most frequently used subunits 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 2core copper line with a crosssectional area of 2.5 mm^{2} 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.