Dependent and Independent Sources

Definition

A source is an active circuit element that can deliver electrical energy to a circuit.

• An independent source provides a specified voltage or current that does not depend on any other circuit variable.
• A dependent source provides a voltage or current whose magnitude is controlled by another voltage or current in the same or another part of the circuit.

Dependent sources are also called controlled sources because their output depends on a controlling electrical quantity.

Main Content

1. Independent Sources

Meaning and characteristics

An independent source is a source whose voltage or current remains fixed according to its specification, regardless of other variables in the circuit, ideally. For example, a 10 V voltage source always tries to maintain 10 V across its terminals, and a 2 A current source always tries to supply 2 A through the connected branch. These sources are called “independent” because their values are not determined by any other element in the network.

Types and examples

Independent sources are mainly of two types:

• Independent voltage source: Maintains a constant voltage. Example: a battery, DC supply, or ideal generator.
• Independent current source: Maintains a constant current. Example: ideal current supply used in testing circuits.
  In practical systems, real sources are not perfectly ideal because they have internal resistance and limited output range, but in circuit theory they are often treated as ideal for analysis.

2. Dependent Sources

Meaning and characteristics

A dependent source is one whose output depends on another voltage or current in the circuit. The control variable may be a voltage or a current, and the source may itself be a voltage source or a current source. These sources are extremely important in modeling active devices such as transistors, op-amps, and amplifiers.

Four common types of dependent sources

1. Voltage-Controlled Voltage Source (VCVS): Output voltage depends on another voltage.
   Example: $V_o = \mu V_x$, where $\mu$ is the voltage gain.

2. Current-Controlled Voltage Source (CCVS): Output voltage depends on a current.
   Example: $V_o = r I_x$, where $r$ is transresistance.

3. Voltage-Controlled Current Source (VCCS): Output current depends on a voltage.
   Example: $I_o = g V_x$, where $g$ is transconductance.

4. Current-Controlled Current Source (CCCS): Output current depends on another current.
   Example: $I_o = \beta I_x$, where $\beta$ is current gain.

3. Practical Importance and Circuit Representation

Graphical symbols and identification

Independent sources are usually shown by a circle symbol, while dependent sources are shown by a diamond-shaped symbol. The symbol helps identify whether the source is controlled or not. The polarity marks on voltage sources and arrows on current sources indicate the reference direction or polarity.

Role in analysis and modeling

Dependent sources are widely used to represent the behavior of real electronic components. For example, a transistor can be represented as a dependent source because its output current is controlled by an input voltage or current. Similarly, op-amp circuits often use dependent source relationships in their equivalent models. They are also used in network theorems such as Thevenin’s and Norton’s methods when analyzing active circuits.

Working / Process

1. Identify the source type

Determine whether the element is independent or dependent. Check if its value is fixed by the source itself or controlled by another circuit variable.

2. Find the controlling variable

For dependent sources, identify the quantity that controls the source output, such as a voltage $V_x$ or current $I_x$. Also note the proportionality constant, such as gain, transconductance, or resistance.

3. Apply circuit laws and solve

Use Kirchhoff’s Voltage Law (KVL), Kirchhoff’s Current Law (KCL), Ohm’s law, and source equations together to analyze the circuit. For dependent sources, write the control relation and combine it with the rest of the network equations to find unknown voltages and currents.

Advantages / Applications

Useful for modeling real devices

Dependent sources help model transistors, operational amplifiers, and other active components whose behavior depends on input signals.

Simplifies circuit analysis

Independent sources provide standard excitation to circuits, making it easier to study voltage, current, and power relationships under different conditions.

Essential in network theorems and design

Both types of sources are used in Thevenin/Norton equivalent circuits, amplifier design, feedback systems, and control-based electronic circuit analysis.

Summary

• Independent sources provide fixed voltage or current values.
• Dependent sources produce output based on another circuit variable.
• These sources are basic building blocks in circuit analysis and electronic modeling.
• Important terms to remember: independent source, dependent source, voltage source, current source, controlled source.