Dependent Sources Overview
The four controlled sources that model every transistor and amplifier.
A transistor doesn't magically amplify — it uses a small signal to control a larger one. That controlling relationship is a dependent source.
What Is a Dependent Source?
You've already worked with independent voltage and current sources — they deliver a fixed value regardless of the rest of the circuit. A dependent source (also called a controlled source) is different: its output value depends on a voltage or current at some other location in the circuit. This makes them the natural tool for modeling devices that amplify or convert signals.
Independent sources are drawn as a circle. Dependent sources are drawn as a diamond (rhombus). This visual convention lets you instantly tell whether a source value is fixed or controlled by another circuit variable. The controlling variable is usually labeled elsewhere on the schematic (e.g., Vx or Ix) with a reference indicated by + and − signs for voltages or an arrow for currents.
The Four Types of Dependent Sources
Because the control variable can be a voltage or a current, and the output can be a voltage or a current, there are exactly four combinations:
1. VCVS — Voltage-Controlled Voltage Source. The output voltage is proportional to a controlling voltage elsewhere. The gain k (often written as μ) is dimensionless.
2. CCVS — Current-Controlled Voltage Source. The output voltage is proportional to a controlling current. The gain k (often written as r, or 'transresistance') has units of ohms.
3. VCCS — Voltage-Controlled Current Source. The output current is proportional to a controlling voltage. The gain k (often written as gm, or 'transconductance') has units of siemens (S).
4. CCCS — Current-Controlled Current Source. The output current is proportional to a controlling current. The gain k (often written as β) is dimensionless.
Why Dependent Sources Matter
Dependent sources are not just a textbook abstraction — they are the circuit-level model for every amplifying device. A bipolar junction transistor (BJT) in its small-signal model is a CCCS: a small base current controls a larger collector current. A MOSFET is a VCCS: a gate-source voltage controls a drain current. Operational amplifiers, which you'll meet in the next lesson, are modeled as VCVS devices with very high gain. Whenever you see the word 'amplifier' in circuit analysis, a dependent source is doing the work behind the scenes.
When working with dependent sources, always check the units of the gain constant. A dimensionless gain (VCVS, CCCS) means the output and control variable share the same unit. A gain in ohms (CCVS) or siemens (VCCS) means a unit conversion is happening — current to voltage or voltage to current. Mismatching units is one of the most common mistakes in dependent-source problems.
- The VCCS relationship is Io = gm × Vx. Substituting: Io = 0.1 S × 5 V = 0.5 A.
- The output current (0.5 A) flows through the 100 Ω load resistor.
- By Ohm's law, the voltage across the load is VL = Io × RL = 0.5 A × 100 Ω = 50 V.
- The CCCS relationship is Io = β × Ix.
- Substituting: Io = 4 × 0.2 A = 0.8 A.
Check your understanding
- Dependent sources have an output proportional to a voltage or current elsewhere in the circuit — drawn as a diamond to distinguish them from independent sources (circles).
- The four types are VCVS (dimensionless gain), CCVS (gain in ohms), VCCS (gain in siemens), and CCCS (dimensionless gain).
- Dependent sources are the circuit model for transistors and op-amps — amplification is fundamentally a controlled-source relationship.