We may classify circuit element into two broad categories, passive elements, and active elements, by considering the energy delivered to or by them.

A circuit element is said to be passive if the total energy delivered to it from the rest of the circuit is always non-negative. That is, for all t we have

\[w(t)=\int_{-\infty }^{t}{p(t)\text{ }dt=}\int_{-\infty }^{t}{vi\text{ }dt\ge 0}\text{ }\cdots \text{ (1)}\]

The polarities of v and I are as shown in the following figure.

Fig.1: Typical element with voltage and current

Examples of passive elements are resistors, inductors, and capacitors.

An active element is one that is not passive, of course. That is, (1) does not hold for all the time. Examples of active elements are generators, batteries, and electronic devices that require power supplies.

In this section, we will talk about two very important active elements, the independent voltage source, and the independent current source.

An independent voltage source is a two-terminal element, such as a battery or a generator that maintains a specified voltage between its terminals. The voltage is completely independent of the current through the element. The symbol for a voltage source having v volts across its terminals is shown in figure 2. The polarity is as shown, indicating that terminal a is v volts above terminal b. thus if v>0, then terminal a is at a higher potential than terminal b. The opposite is true, of course, if v<0.

Fig.2: independent voltage source

In figure 2, the voltage v may be time varying, or it may be constant, in which case we would probably label it V. another symbol that is often used for a constant voltage source, such as a battery with V volts across its terminals, is shown in figure 3. In the case of constant sources, we can use figure 2 and 3 interchangeably.

Fig.3: Constant Voltage Source

An independent current source is a two-terminal element through which a specified current flows. The current is completely independent of the voltage across the element. The symbol for an independent current source is shown in figure 4, where I is the specified current. The direction of the current is indicated by the arrow.

Fig.4: Independent Current Source

Independent sources are usually meant to deliver power to the external circuit and not to absorb it. Thus if v is the voltage across the source and its current I is directed out of the positive terminal, the source is delivering power, given by p=vi, to the external circuit. Otherwise, it is absorbing power. For example, in figure 5 (a), the battery is delivering 24W to the external circuit. In figure 5 (b), the battery is absorbing 24W, as would be the case when it is being charged.

Fig.5: (a) Source Delivering and (b) Source Absorbing power

The sources we have discussed in here are ideal elements. That is, they are mathematical models that approximate the actual or physical elements only under certain conditions. For example, an ideal automobile battery supplies a constant 12V, no matter what external circuit is connected to it. Since its current is completely arbitrary, it could theoretically deliver an infinite amount of power. This, of course, is not possible in the case of an actual device. A real 12V battery supplies approximately constant voltage only as long as the current it delivers is low. When the current exceeds a few hundred amperes, the voltage drops appreciably from 12V.