Voltage Dependent Source Explained: VCVS vs VCCS Theory and Real-World OTA Picks

A voltage dependent source is an active circuit element whose output voltage or current is mathematically proportional to a controlling voltage measured across two other nodes in the same network. Unlike a standard battery or bench power supply (an independent source) that pushes a fixed value regardless of the rest of the circuit, a dependent source acts as a bridge, linking two separate parts of a schematic. This concept fundamentally changes how we analyze real circuits: it allows engineers to take inherently non-linear physical components—like transistors, vacuum tubes, and operational amplifiers—and model them as predictable, solvable linear equations for small-signal AC analysis.

Beginners frequently confuse voltage dependent sources with physical voltage regulators or variable power supplies. A voltage regulator (like an LM7805) is a physical, standalone power delivery component designed to maintain a steady output. A voltage dependent source (like a VCVS or VCCS) is primarily a mathematical model used in circuit theory and SPICE simulation to represent the internal gain mechanisms of active semiconductor devices.

The Core Concept: VCVS vs. VCCS

When we map physical silicon to schematic theory, we split voltage dependent sources into two distinct categories based on what they output:

• VCVS (Voltage-Controlled Voltage Source): The output is a voltage that is a multiple of a controlling voltage elsewhere. The proportionality constant is a dimensionless voltage gain ($A_v$ or $\mu$). This is the foundational model for an ideal operational amplifier.
• VCCS (Voltage-Controlled Current Source): The output is a current that is a multiple of a controlling voltage. The proportionality constant is a transconductance ($g_m$), measured in Siemens (S) or Amps per Volt (A/V). This is the foundational model for field-effect transistors (MOSFETs, JFETs) and Operational Transconductance Amplifiers (OTAs).

Worked Numeric Example: Modeling an OTA

Let us look at a real-world VCCS: the Texas Instruments LM13700 Operational Transconductance Amplifier (OTA). Unlike a standard op-amp that outputs a voltage, the LM13700 outputs a current based on the differential voltage at its inputs, and its gain is controlled by an external bias current ($I_{ABC}$).

The Scenario: You are designing a voltage-controlled filter (VCF) for an analog synthesizer. You set the Amplifier Bias Current ($I_{ABC}$) to $500 \mu A$ (0.5 mA) using a current mirror circuit. You then apply a tiny AC audio signal of $10 mV$ peak-to-peak across the differential inputs.

1. Calculate $g_m$: $g_m = 19.2 \times 0.5 \text{ mA} = 9.6 \text{ mA/V}$ (or $9.6 \text{ mS}$).
2. Calculate Output Current ($I_{out}$): $I_{out} = g_m \times V_{in}$.
3. Result: $I_{out} = 9.6 \text{ mA/V} \times 10 \text{ mV} = 96 \mu A$ peak-to-peak.

If you place a $10 k\Omega$ load resistor at the output, Ohm's law converts this dependent current back into a voltage: $V_{load} = 96 \mu A \times 10 k\Omega = 0.96 V$ peak-to-peak. By changing the DC bias current ($I_{ABC}$), you change the $g_m$, effectively creating a voltage-controlled amplifier or filter.

Where You Meet This in Practice

You will rarely see a diamond-shaped dependent source symbol on a physical wiring diagram, but you will encounter the concept constantly in these three areas:

1. SPICE Circuit Simulation

In SPICE engines like LTspice, dependent sources are first-class citizens. If you want to simulate a VCVS, you use the E prefix in your netlist (e.g., E1 out 0 in+ in- 100 creates a voltage source with a gain of 100). For a VCCS, you use the G prefix. These behavioral sources are vital for creating macromodels of complex ICs without simulating every single internal transistor.

2. Small-Signal Transistor Models

When analyzing a standard 2N3904 BJT in a common-emitter amplifier, we use the hybrid-pi model. The core of this model is a VCCS where the output collector current ($i_c$) is dependent on the base-emitter voltage ($v_{be}$). At room temperature, if the DC collector bias is $1 mA$, the transconductance $g_m$ is roughly $38.5 mA/V$. This single VCCS parameter dictates the maximum theoretical voltage gain of your amplifier stage.

3. Analog Audio and Synthesizers

In physical hardware, OTAs (VCCS) and VCAs (Voltage Controlled Amplifiers, which act as specialized VCVS circuits) are the engines behind guitar pedal compressors, analog synth oscillators, and studio dynamics processors. They allow a slow-moving DC control voltage (like an envelope generator) to dictate the amplitude of a fast-moving AC audio signal.

Decision Tree: Picking the Right Physical Component

When moving from theory to the workbench, you need to select a physical IC that embodies the dependent source behavior you need. Use this decision matrix to select your part.

Common Pitfalls and Troubleshooting

Working with physical components that act as dependent sources introduces specific failure modes that pure theory ignores.

• The OTA Input Clipping Trap: In theory, a VCCS is perfectly linear. In practice, the input differential pair of an LM13700 OTA will severely distort if the input voltage ($V_{in}$) exceeds about $20 mV$. Fix: You must use a voltage divider at the input, or rely on the chip's internal linearizing diodes (pins 1 and 16) to inject a small bias current that cancels out the transistor's natural exponential curve.
• SPICE Floating Node Errors: When using E or G behavioral sources in LTspice, the controlling nodes must have a DC path to ground. If you try to measure a voltage across a floating capacitor to control a dependent source, the simulation will fail or yield gibberish. Fix: Always place a high-value resistor (e.g., $1 G\Omega$) from the controlling nodes to ground to provide a DC operating point.
• Ignoring Output Compliance: A theoretical VCCS will push its designated current through an infinite resistance. A physical OTA has a compliance voltage limit. If your load resistor is too large, the output voltage will hit the supply rail and the current will flatline. Fix: Calculate $V_{load} = I_{out} \times R_{load}$ and ensure it stays at least 2V away from your positive and negative supply rails.