# Independent Versus Dependent Controlled Sources

In the previous chapters, each source appearing in the analysis of dc or ac networks was an independent source, such as $E$ and $I$ (or $E$ and $I$) in Fig. 1.
Fig. 1: Independent sources.

### What is an independent source?

The term independent specifies that the magnitude of the source is independent of the network to which it is applied and that the source displays its terminal characteristics even if completely isolated.

### What is a dependent source?

A dependent or controlled source is one whose magnitude is determined (or controlled) by a current or voltage of the system in which it appears.
Currently two symbols are used for controlled sources. One simply uses the independent symbol with an indication of the controlling element, as shown in Fig .2. In Fig .2(a), the magnitude and phase of the voltage are controlled by a voltage $V$ elsewhere in the system, with the magnitude further controlled by the constant $k_1$. In Fig. 2(b),
Fig. 2: Controlled or dependent sources.
the magnitude and phase of the current source are controlled by a current I elsewhere in the system, with the magnitude further controlled by the constant $k_2$. To distinguish between the dependent and independent sources, the notation of Fig .3 was introduced. In recent years many respected publications on circuit analysis have accepted the notation of Fig. 3, although a number of excellent publications in the area of electronics continue to use the symbol of Fig. 2, especially in the circuit modeling for a variety of electronic devices such as the transistor and FET. This text will employ the symbols of Fig. 3.
Fig. 3: Special notation for controlled or dependent sources.
Possible combinations for controlled sources are indicated in Fig .4. Note that the magnitude of current sources or voltage sources can be controlled by a voltage and a current, respectively. Unlike with the independent source, isolation such that $V$ or $I = 0$ in Fig. 4(a) will result in the short-circuit or open-circuit equivalent as indicated in Fig. 4(b). Note that the type of representation under these conditions is controlled by whether it is a current source or a voltage source, not by the controlling agent ($V$ or $I$).
Fig. 4: Conditions of $V = 0 V$ and $I =0 A$ for a controlled source.