Dependent Sources

Dependent sources, also called controlled sources, are two-port elements whose output voltage or current is controlled by a voltage or current elsewhere in the circuit. They are used to model amplifiers, transistors, and operational amplifiers in linear circuit analysis. GATE regularly tests the four types and their application in small-signal equivalent circuits.

Core Concept of Dependent Sources

Unlike independent sources that maintain a fixed voltage or current, a dependent source has its output controlled by another circuit variable. The controlling variable can be a voltage or a current at some other branch in the same circuit. This coupling between two parts of the circuit is what makes dependent sources useful for modeling electronic devices.

Dependent sources are represented by diamond shapes in circuit diagrams (as opposed to circles for independent sources). They are active elements since they can supply power to the circuit. However, they do not supply power independently. They require an independent source elsewhere in the circuit to establish the controlling variable.

The Four Types

The VCVS (Voltage-Controlled Voltage Source) produces an output voltage proportional to a controlling voltage: Vout = mu x Vcontrol. The gain mu is dimensionless (V/V). This models ideal amplifiers and the output stage of op-amps.

The VCCS (Voltage-Controlled Current Source) produces an output current proportional to a controlling voltage: Iout = gm x Vcontrol. The transconductance gm has units of Siemens (A/V). This is the standard model for MOSFETs in saturation region: Id = gm x Vgs.

The CCVS (Current-Controlled Voltage Source) produces an output voltage proportional to a controlling current: Vout = rm x Icontrol. The transresistance rm has units of Ohms (V/A). This appears in transresistance amplifier models.

The CCCS (Current-Controlled Current Source) produces an output current proportional to a controlling current: Iout = beta x Icontrol. The gain beta is dimensionless (A/A). This is used in the small-signal model of BJTs where beta = hfe represents the common-emitter current gain.

Mathematical Expression

In node voltage or mesh current analysis, dependent sources introduce an additional constraint equation linking two circuit variables. The controlling variable must first be expressed in terms of the chosen unknowns (node voltages or mesh currents) and then substituted into the KVL or KCL equations. This is the standard procedure for circuits with dependent sources in GATE problems.

Given:
Circuit with VCCS: transconductance gm = 0.05 A/V
Controlling voltage Vgs across a 1 kohm resistor
Input current source Ii = 2 mA drives the input port
Output port has load RL = 2 kohm

Why this formula applies:
Vgs = Ii x Rin (controlling voltage developed across input resistor)
Output current Iout = gm x Vgs (VCCS relationship)
Output voltage Vout = Iout x RL

Formula:
Vgs = Ii x Rin
Iout = gm x Vgs
Vout = Iout x RL

Substitution:
Vgs = 2 x 10^-3 x 1 x 10^3 = 2 V
Iout = 0.05 x 2 = 0.1 A
Vout = 0.1 x 2000

Calculation:
Vout = 200 V
Voltage gain = Vout / Vgs = 200 / 2 = 100

Final Answer:
Output voltage = 200 V, Voltage gain Av = gm x RL = 0.05 x 2000 = 100 V/V

Exam Tip: In GATE node/mesh analysis questions with dependent sources, always write the controlling variable in terms of node voltages or mesh currents first, then substitute. Never leave the dependent source gain as an unknown. A common trap is forgetting that the controlling branch and the controlled source can be in different parts of the circuit.

Mechanism: Dependent Sources in Analysis

• Dependent sources do not have a fixed output value. Their output is always proportional to the controlling variable in another branch.
• VCVS and VCCS are controlled by voltage. CCVS and CCCS are controlled by current. The subscripts indicate: first letter = controlling type, second letter = output type.
• In node voltage analysis: identify the controlling variable, express it in terms of node voltages, substitute into the source expression, then apply KCL normally.
• In mesh analysis: identify the controlling variable, express it in terms of mesh currents, substitute, then apply KVL.
• Thevenin and Norton equivalents can still be found for circuits with dependent sources, but the test source method must be used to find Rth since dependent sources prevent using simple resistance combination.
• Power check: sum of all power delivered by independent sources equals power absorbed by passive elements plus power delivered or absorbed by all dependent sources.

Quick Revision

• VCVS: Vout = mu x Vcontrol. Gain mu is dimensionless (V/V). Diamond shape symbol.
• VCCS: Iout = gm x Vcontrol. Transconductance gm in Siemens (A/V). Used in MOSFET model.
• CCVS: Vout = rm x Icontrol. Transresistance rm in Ohms (V/A).
• CCCS: Iout = beta x Icontrol. Current gain beta is dimensionless (A/A). Used in BJT model.
• All dependent sources are active elements and must have an independent source to provide the controlling signal.
• Exam trap: Dependent sources cannot be deactivated (set to zero) when finding Rth. Use the test source (apply Vtest or Itest and find the ratio Vtest/Itest) method for Thevenin resistance.
• Superposition applies to circuits with dependent sources, but the controlling variable must be included in every superposition sub-problem.