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Electric generators are devices that transform other forms of energy into electrical energy.

An example of an electric generator is a car battery, which converts chemical energy into electrical.
On electrical circuts schemes, batteries are represented with parallel bars of different sizes, where the shortest bar is the lowest potential. The first figure represents a single battery, the second figure is a series association of batteries.

Types of Generators

On the generator terminals is maintained a potential difference, which is due to the process of converting different types of energies in electric potential. The main types are:

Electrochemical Generator:
It transforms chemical energy into electrical energy. They are made of plates of different metals, suitably chosen, partially immersed in acidic, basic or saline solutions.
It is an association of electrochemical generators that increase the electric potential and/or the maximum current, depending on the type of association.
Electromechanical Generator:
It transforms mechanical energy into electricity. The main types are the dynamos and alternators, while the former are able to establish direct current, the latter generate alternating current.

Electromotive Force \((\mathscr{E})\)

Despite the name, this quantity is just the work done per unit charge on a unit load transported within the generator. \(\mathscr{E}\) is the increment in the electric potential gained by the charges of the eletrical current that passes through the generator. (By convention, from the negative terminal to the positive one.) The unit of \(\mathscr{E}\) is the same as the potential difference; in the \(IS\) is the Volt, \([\mathscr{E}]= V = \frac{J}{C}\) , $$\mathscr{E} = \frac{w}{q}$$

Features of a Generator

The characteristic of a generator is its \(\mathscr{E}\) and its internal resistance \(r\). The internal resistance is inevitable, for a generator and any electrical appliance, since any material has some electrical resistance. Therefore, the \(\mathbb{V}\) between the terminals \(A\) and \(B\) of a generator is: $$\mathbb{V} = V_B - V_A = \mathbb{V}_{BA} = \mathscr{E} - ri,$$ where \(i\) is the current that the generator provides to the circuit. Each generator has a characteristic curve, which is the Cartesian graph of \(\mathbb{V}\) (at the generator terminals) as a function of the current \(i\) that goes through it.

Power and Efficiency

Power Loss \((P_l)\):
In an electric generator, the electrical power is lost in the internal resistance (internal dissipation). The power loss is the amount of Joules per second that are lost uselessly into the generator. That is, for a generator with internal resistance \(r \), the power lost is $$ P_d = ri ^ 2, $$ where \(i \) is the current passing through the generator.
Net Power \((P_n)\) :
It is the net power that the generator provides to the system that it feeds, that is, the amount of electric power (Joules per second) the generator effectively provides, i.e., $$P_n = \mathbb{V} i = \mathscr{E} i - ri^2$$ where \(r\) is the internal resistance of the generator, \(i\) is the current passing through it and \(\mathbb{V}\) is the potential difference at its terminals.
Total Power \((P_T)\):
It is the total electric power produced by the generator. Thus, it is the sum of net power and the dissipated one. It means the amount of Joules of some kind of energy (in the case of batteries, chemical energy) are transformed into electrical energy, each second. $$P_T = P_n + P_d = \mathscr{E} i$$
Efficiency \((\eta)\):
It is the ratio between the power output and the power supplied. The performance of a generator is greater for a smaller internal resistance and a lower intensity of the current running in the system. Therefore, we have $$ \eta = \frac{P_n}{P_T} = \frac{\mathscr{E} i - ri^2}{\mathscr{E}i} = 1 - \frac{r i}{\mathscr{E}},$$ where \(r\) is the internal resistance of the generator, \(i \) is the current passing through it and \(\mathscr{E}\) is its electromotive force.
Ideal Generator:
It is one that is able to supply electric power without dissipating any energy. We can think of the ideal generator as a generator with zero internal resistance. Of course, it is a hypothetical model.
Short Circuit:
A generator is short-circuited if its terminals are directly connected or connected by a negligible resistance wire. The \(U\) between the terminals of a generator in a short circuit is zero and the current intensity is the maximum possible. In this condition, the power output of a generator is zero and the power provided by it is completely dissipated in the internal resistance, which ultimately heats too much and deteriorate. The intensity of the short-circuit current is $$ i_{cc} = \frac{\mathscr{E}}{r},$$ where \(r\) is the internal resistance of the generator and \(\mathscr{E}\) is its electromotive force.
Maximum Power:
It is obtained when the current flowing in the generator is half the short-circuit current, mathematically: $$ i_{m} = \frac{i_{cc}}{2} = \frac{\mathscr{E}}{2r},$$ where \(r\) is the internal resistance of the generator and \(\mathscr{E}\) is its electromotive force. The maximum power $$ P_m = \frac{\mathscr{E}^2}{4r}$$ is produced on a generator when the potential difference is half its electromotive force: $$\mathbb{V} = \frac{\mathscr{E}}{2}.$$ In a condition where maximum power is transferred, the efficiency is \(\frac{1}{2}\). The circuit resistance must be equal to the internal resistance of the generator so that it generates maximum power.

Generators Associations

Equivalent Generator \((\mathscr{E}_e)\) :
In a combination of generators, the equivalent generator is the one that drives the same current of the association when it is connected to the same circuit.
Series Association:
When multiple generators are associated in series, the \(\mathscr{E}_e\) of the equivalent generator is equal to the sum of \(\mathscr{E}_i\) of each associated generator. And the equivalent internal resistance \(r_e\) is the sum of \(r_i\), i.e., $$\mathscr{E}_e = \mathscr{E}_1 +\mathscr{E}_2 + ... +\mathscr{E}_n,$$ $$r_e = r_1 + r_2 + ... + r_n.$$ In this case, one obtains a generator with a higher \(\mathscr{E}_e\).
Parallel Association:
In a parallel association, to avoid that some of the associated generators might act as a receiver, one must only use generators with the same electromotive force and the same internal resistance. The combinations of equivalent generators have an equivalent electromotive force that is equal to the one of a single generator of the association, and the equivalent internal resistance \(r_e\) is equal to the value of the internal resistance of one of the associated generators \(r\) divided by the number \(n\) of associated generators. Therefore, the \(U\) of the equivalent generator is given by: $$\mathbb{V} = \mathscr{E} - \frac{r}{n}i,$$ where it is clear that this association has the same electromotive force of a single generator, but there is a decrease on its internal resistance. Thus, despite the association not having a larger \(\mathbb{V}\), batteries can power the circuit for a longer period of time, since each battery contributes the same fraction to the total power.


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