A dependant voltage source (standardized as dependent in US engineering texts) is an active circuit element whose output voltage is mathematically controlled by another voltage or current elsewhere in the same network, rather than maintaining a fixed output. In practical installations, this concept changes how we model gain, feedback, and sensor interfaces, shifting our analysis from simple Kirchhoff voltage loops to systems of coupled equations. Beginners commonly confuse it with an independent source (like a 9V battery or a bench power supply) or mistakenly assume it can supply infinite power, ignoring real-world saturation and rail limits.
The Core Mechanics: VCVS vs. CCVS
In circuit theory, dependent sources are the mathematical bridge between abstract equations and physical semiconductor behavior. While there are four types of dependent sources (including current outputs), we focus strictly on the two voltage-output variants here:
- Voltage-Controlled Voltage Source (VCVS): The output voltage is a multiple of an input voltage. The multiplier is a dimensionless gain factor ($\mu$ or $A_v$). The classic physical embodiment is an operational amplifier in a closed-loop configuration.
- Current-Controlled Voltage Source (CCVS): The output voltage is proportional to a controlling current. The multiplier is a transimpedance value measured in ohms ($\Omega$). This is the foundational model for current shunt monitors and Hall-effect sensor interfaces.
Transimpedance, in this context, is the ratio of the output voltage to the input current ($Z_m = V_{out} / I_{in}$). It defines how effectively your circuit translates electron flow into a readable voltage signal.
Worked Numeric Example: Building a CCVS for Motor Sensing
Let’s move off the whiteboard and onto the bench. Suppose you need to monitor the current draw of a 12V DC motor that peaks at 2.5A, and you want to feed a scaled 0–3.3V signal into an ESP32 ADC pin. We will build a physical CCVS using a shunt resistor and an instrumentation amplifier.
Step 1: The Shunt Resistor (Sensing the Current)
We place a $0.01\Omega$ (10 m$\Omega$) shunt resistor in series with the motor's ground path. At peak current (2.5A), the voltage drop across the shunt is:
$V_{shunt} = I \times R = 2.5A \times 0.01\Omega = 0.025V$ (25 mV).
Bench check: Power dissipated in the shunt is $P = I^2R = (2.5)^2 \times 0.01 = 0.0625W$. A standard 0805 SMD resistor (rated for 125mW) will survive, but a 1206 package (250mW) is preferred to minimize thermal drift and keep the resistance stable.
Step 2: The Dependent Source (Amplification)
We need to amplify that 25 mV signal to roughly 2.5V. This requires a gain of 100. We select the AD8421 instrumentation amplifier. Its gain is set by a single resistor ($R_G$) using the formula $G = 1 + (10,000 / R_G)$.
To get $G = 101$, we install a standard $100\Omega$ precision resistor across the $R_G$ pins.
Step 3: The Output Calculation
Our dependant voltage output is now:
$V_{out} = V_{shunt} \times G = 25mV \times 101 = 2.525V$.
This 2.525V is strictly dependent on the motor's current. If the motor stalls and draws 5A, $V_{shunt}$ becomes 50mV, and $V_{out}$ attempts to reach 5.05V. However, if your op-amp is powered by a 3.3V single supply, the output will hard-saturate at the positive rail (~3.1V), clipping your data. This brings us to real-world limitations.
Where You Meet This in Practice
You rarely see a diamond-shaped dependent source symbol outside of a university textbook, but you use physical implementations of them in almost every analog design:
- Op-Amp Buffers and Amplifiers: Every time you wire an LM358 or OPA333 as a non-inverting amplifier, you are building a VCVS. The output voltage depends entirely on the voltage at the non-inverting input pin.
- Transistor Small-Signal Models: When analyzing the AC behavior of a BJT or MOSFET, engineers use the hybrid-pi model. This model replaces the physical transistor with a VCVS or VCCS to calculate voltage gain and output impedance without dealing with non-linear semiconductor physics.
- Current Shunt Monitors: Dedicated ICs like the INA226 or AD8218 are packaged CCVS devices. They take a differential voltage across a shunt (representing current) and output a single-ended voltage referenced to ground.
Decision Tree: Selecting the Right Dependent Source IC
Choosing the right physical IC to act as your dependent source depends on your common-mode voltage, required precision, and budget. Use this decision matrix to terminate your part selection.
| If your design requires... | Then you need this model... | Concrete IC Pick |
|---|---|---|
| High-side current sensing with high common-mode voltage (e.g., 48V battery pack) | CCVS (High Common-Mode) | AD8218 (Handles up to 80V common-mode, fixed 20V/V gain) |
| Low-side precision shunt amplification for microcontroller ADCs | CCVS (Precision Instrumentation) | AD8421 (Low noise, high bandwidth, gain set by single resistor) |
| High-gain, single-ended voltage buffering for high-impedance sensors (e.g., piezo) | VCVS (Zero-Drift) | OPA333 (Micro-power, auto-zeroing, minimal offset voltage) |
| General-purpose, low-cost voltage scaling where millivolt precision isn't critical | VCVS (General Purpose) | LM358 (Cheap, ubiquitous, but suffers from crossover distortion and rail limits) |
Real-World Non-Ideal Behaviors and Saturation
In circuit theory, a dependant voltage source can output 1,000V if the controlling variable and gain dictate it. On your workbench, physics intervenes. When designing with physical dependent sources, you must account for three non-ideal realities:
- Rail Saturation: An op-amp cannot output a voltage higher than its positive supply rail or lower than its negative rail. Furthermore, older parts like the LM358 cannot swing all the way to the positive rail; they typically lose 1.5V of headroom. If powered by 5V, your maximum VCVS output is only ~3.5V.
- Bandwidth and Slew Rate: The gain-bandwidth product (GBWP) limits your source. If your AD8421 is set to a gain of 100, its bandwidth drops significantly compared to a gain of 10. If the controlling current changes faster than the op-amp's slew rate (measured in V/$\mu$s), your dependant voltage will distort into a triangle wave.
- Output Impedance: An ideal dependent voltage source has zero output impedance. A real op-amp has an open-loop output impedance of a few dozen ohms. While negative feedback drives this down to milliohms at DC, at high frequencies, the output impedance rises, making your voltage susceptible to loading effects if you connect a low-impedance load directly to the output.
Frequently Asked Questions
Can a dependant voltage source supply power to a circuit?
Yes, in theoretical models, dependent sources can deliver power. In physical implementations (like an op-amp), the power actually comes from the DC supply rails connected to the IC. The dependent source merely acts as a valve, modulating the rail power based on the input signal.
Why do textbooks use a diamond shape for dependent sources?
The diamond shape is purely a drafting convention established to visually separate them from independent sources (which use circles). It reminds the analyst during Kirchhoff's Voltage Law (KVL) walks that the value inside the diamond is a variable, not a constant, and requires a second equation to resolve.
Is a transformer a dependent voltage source?
When analyzing or building circuits, always treat the dependant voltage source as a mathematical relationship first, and a physical component second. Define your controlling variable, calculate your required gain, and then select an IC from the decision tree above that respects your supply rails and bandwidth limits. For 90% of maker-level current sensing and voltage buffering tasks, the AD8421 (for CCVS) or the OPA333 (for VCVS) will provide the precision and headroom needed to bridge theory and reality without signal clipping.
