In electrical circuits, sources play a crucial role in providing power for the operation of the circuit. These sources can be broadly categorized into two types: independent and dependent.

Independent voltage or current sources supply a fixed amount of voltage or current, respectively, which is unaffected by other elements within the circuit. These are represented using specific symbols. Independent voltage sources are symbolized with polarities (+ and -), indicating the direction of the potential difference, while independent current sources use arrows to denote the direction of the current flow.

An interesting aspect of independent sources is that a short circuit can be considered as a zero-voltage independent voltage source because it allows current to flow freely with no voltage drop. Conversely, an open circuit can be thought of as a zero-current independent current source, given that it prevents any current flow regardless of the applied voltage.

On the other hand, dependent sources are voltage or current generators whose values are controlled by other elements within the same circuit. They are depicted by a diamond shape in circuit diagrams.

These dependent sources are further divided into four categories based on their controlling and controlled parts: voltage-controlled voltage sources (VCVS), current-controlled voltage sources (CCVS), voltage-controlled current sources (VCCS), and current-controlled current sources (CCCS).

For instance, in a circuit where the voltage output of a dependent source is controlled by the current flowing through a resistor, if a short circuit is added in series with this resistor, the controlling current becomes equivalent to the current in the short circuit. The ratio of the source voltage to the controlling current represents the gain of the dependent source.

Similarly, the controlling voltage can be considered as open-circuit voltage. This means that the voltage or current of the dependent sources is proportional to the controlling voltage or current.