6.07 Internal resistance.

In the electrical circuits we have looked at so far, we have regarded the voltage source supplying the circuit as being a perfect voltage source. i.e.:

We have assumed that the voltage supplied to the circuit remains constant, regardless of how much current is being supplied.
We have also assumed that energy is only converted into heat, by the resistance in the external circuit.

In practice this is not the case as:

The terminal voltage of a battery decreases, as the current it supplies to a circuit increases.
This is the same for all real voltage sources. (However power supply designers do produce stabilised power supplies, where feedback circuits are used to maintain a relatively constant output voltage).
All power supplies get warm during use, which shows that some of the energy that they provide, is actually being converted to heat inside the power supply itself!

This is illustrated in the animation below.

This variation of supply voltage is potentially a difficult thing to explain, especially as the actual reasons for the variation would depend on the type of power supply being used.
For example:

For a battery: the voltage drops because the rate of chemical reactions transferring charge to the battery terminals, cannot match the rate at which the charge is leaving the terminals, to flow around the circuit.
For a generator: the current produces stronger magnetic fields inside the generator, which slow the generator and reduce the supply voltage.

Fortunately we can avoid these details and regardless of the actual nature of the power supply, we can represent it with a power supply model which consists entirely of simple electrical components. One way of doing this is by representing the power supply as a perfect voltage source, (an e.m.f.) in series with an internal resistance. When this power supply model is applied to an external circuit, then the circuit current also flows through the internal resistance. This produces an internal voltage drop inside the power supply, which therefore reduces the voltage across the power supply terminals. The power dissipated by the internal resistance, represents the heat generated in the power supply. This is illustrated in the animation below.

internal resistance

The terminal voltage (V) is equal to the e.m.f. voltage (E), minus the internal voltage drop (Ir).
(using ohms law: internal voltage drop = current (I) x internal resistance (r)).

V = E - Ir.

To model any real power supply, we just have to determine the correct values of E and r to use.

When the power supply is not connected to a circuit, there will be no current flowing, therefore:

V = E - 0 x r .
V = E .
i.e. the e.m.f voltage, is equal to the open circuit terminal voltage of the power supply.

The internal resistance can be determined, by connecting a circuit of known resistance and measuring the current that flows.

I = E/(R + r).
therefore r = (E/I) - R .

The power supplied by the e.m.f., is given by P = EI and the power dissipated in the power supply, is given by P = I²r.
The energy provided by the e.m.f., is given by W = EIt and the energy dissipated in the power supply, is given by W =I²rt.