22 Electric circuit elementsInternal resistance
It was stated earlier in the chapter that an ideal voltage source is independent of
the current flowing through it. Practical voltage sources have internal resistance
which means that the voltage at its terminals varies as the current through it
changes. The equivalent circuit of a practical voltage source then takes the form
shown in Fig. 2.15 where r represents the internal resistance of the source and
A and B are its terminals. The terminal voltage is thus VAB.
, r,,, B
E V~
Figure 2.15Example 2.11
A battery has an internal resistance of 0.5 f~ and a terminal voltage of 15 V
when it supplies no current. Determine the terminal voltage when the current
through it is 5 A.Solution
The diagram is as shown in Fig. 2.15. Let the battery terminal voltage when it
supplies no current be E (this is called the open circuit voltage). Then, when a
current I flows, the terminal voltage VAB = E- Ir where r is the internal
resistance. When I = 5 A, VAB = 15 -- 5 • 0.5 = 15 -- 2.5 = 12.5 V.Effect of temperature
The resistance of metals increases with temperature while for insulators it
decreases with temperature. There are some materials for which there is
virtually no change in resistance over a wide range of temperatures.
For a given material it is found that
R = Rs[1 + a~(T- T~)] (2.16)
where R is the resistance at a temperature T, Rs is the resistance at temperature
T~, and as is the temperature coefficient of resistance corresponding to T~ and is
defined as the change in resistance per degree change of temperature divided
by the resistance at some temperature Ts. It is measured in (~ -1) which is read
as 'per degree Celsius'. For a standard temperature Ts = 0 ~ as for copper is
0.0043 per ~ for manganin (an alloy of copper, magnesium and nickel) it is
0.000 003 per ~
If a certain material has a resistance of R0 at a standard temperature of 0 ~
and a resistance temperature coefficient of a0, then at temperatures T1 and T2,