Dependent Sources in Circuit Analysis

Dependent Sources in Circuit Analysis

A dependent source, also called a controlled source, is an ideal source whose output is set not by a fixed number but by some other voltage or current elsewhere in the same circuit. Independent sources push a known quantity into the network; dependent sources push a quantity that is a function of a controlling variable. They are not laboratory curiosities. Every transistor, every operational amplifier, and every feedback loop is modeled with dependent sources, so understanding how they enter nodal and mesh analysis is essential before you can analyze any amplifier. This article defines the four types, explains the one extra rule they impose on your equation set, and works a full numerical nodal example with a voltage-controlled current source.

The Four Types

There are exactly four dependent sources, named by what they output and what controls them. The controlling variable is always a voltage $v_x$ across some branch or a current $i_x$ through some branch.

• VCVS — voltage-controlled voltage source. Its output voltage is a gain $\mu$ times a controlling voltage: $v = \mu v_x$. The gain is dimensionless (V/V).
• VCCS — voltage-controlled current source. Its output current is a transconductance $g$ times a controlling voltage: $i = g v_x$. The transconductance has units of siemens (A/V). This is the source inside every transistor small-signal model.
• CCVS — current-controlled voltage source. Its output voltage is a transresistance $r$ times a controlling current: $v = r i_x$. The transresistance has units of ohms (V/A).
• CCCS — current-controlled current source. Its output current is a gain $\beta$ times a controlling current: $i = \beta i_x$. The gain is dimensionless (A/A).

Why They Need a Controlling-Variable Equation

In a circuit with only independent sources, every source contributes a known constant to the right-hand side of your equations. A dependent source is different: its value is an unknown until you have solved for the controlling variable. That coupling is the whole subtlety. When you write a node equation that contains a dependent source, you must express the controlling variable in terms of the same node voltages you are already solving for, and substitute it back in. The controlled source therefore moves from the right-hand side (where independent sources live) to the left-hand side, becoming a term that depends on the unknowns.

For nodal analysis the recipe is: write KCL at each node treating the controlled quantity as a symbol, then write one auxiliary equation that defines the controlling variable from node voltages, and substitute. For mesh analysis the same idea applies with mesh currents and KVL. The number of independent equations does not grow if you substitute immediately; you simply have extra algebra to keep the unknown count equal to the number of node voltages.

How They Change Mesh Analysis

The same coupling appears in mesh analysis, just expressed through mesh currents instead of node voltages. When a controlled source sits on a branch shared by a loop, its value enters the KVL sum around that loop, and you must rewrite the controlling variable in terms of the mesh currents you are solving for. A current-controlled source is natural here because the controlling current is often simply a mesh current or a difference of two mesh currents, which you substitute directly. A voltage-controlled source requires one more step: you express the controlling voltage as a mesh current times the resistance of the branch it appears across, then substitute. Either way the rule is identical to the nodal case. Treat the controlled quantity as a symbol while writing the loop equations, add an auxiliary relation that ties the controlling variable to your chosen unknowns, and substitute so the final system has exactly as many equations as there are independent loops. Choosing nodal or mesh analysis for a circuit with dependent sources comes down to the same trade-off as for any circuit: prefer nodal when there are fewer nodes than loops, and prefer the method in which the controlling variable is most directly one of your unknowns, because that minimizes the substitution work.

Worked Example: Nodal Analysis with a VCCS

Consider a two-node circuit. An independent current source $I_s = 2\,\text{mA}$ drives node 1. A resistor $R_1 = 1\,\text{k}\Omega$ connects node 1 to ground. A resistor $R_2 = 2\,\text{k}\Omega$ connects node 1 to node 2. At node 2, a VCCS with transconductance $g_m = 2\,\text{mS}$ pushes current $g_m v_x$ from node 2 to ground, where the controlling voltage is the node-1 voltage, $v_x = V_1$. A load resistor $R_3 = 1\,\text{k}\Omega$ connects node 2 to ground. Ground is the reference, and the unknowns are $V_1$ and $V_2$.

Write KCL at node 1. The current into the node from the source equals the current leaving through $R_1$ and $R_2$:

Write KCL at node 2. Current arrives from node 1 through $R_2$, and leaves through $R_3$ and through the VCCS. Because the VCCS draws $g_m v_x = g_m V_1$ out of node 2:

Notice the controlling-variable substitution already happened: $v_x = V_1$, so the dependent term $g_m V_1$ sits on the left among the unknowns. Now substitute numbers, using conductances in millisiemens (1/kΩ = 1 mS) and the source in milliamps. Node 1 becomes:

That simplifies to:

Node 2, with $g_m = 2$ mS, becomes:

which collects to:

Substitute $V_2 = -V_1$ into the node-1 equation:

So $V_1 = 1\,\text{V}$ and $V_2 = -1\,\text{V}$. The negative node-2 voltage is the signature of the controlled source: the VCCS pulls current out of node 2 in proportion to $V_1$, dragging that node below ground even though no independent source touches it. Had you ignored the $g_m V_1$ term, you would have predicted a positive $V_2$ and been completely wrong about the sign. This sign inversion is exactly the small-signal behavior of an inverting amplifier.

Connection to Small-Signal Models

That last observation is the bridge to real devices. A transistor is, to a small signal, nothing but a transconductance source: the hybrid-pi model of a BJT places a VCCS of value $g_m v_{be}$ between collector and emitter, and the MOSFET model places a VCCS of value $g_m v_{gs}$ between drain and source. The node equations you just wrote are the same ones CircuitMath assembles when it analyzes an amplifier. If you can handle the VCCS by hand, you can read the machine's output and trust it. See the BJT small-signal model and the steps in nodal analysis step by step for the full procedure.

It is worth dwelling on why this matters pedagogically. Students often treat the small-signal transistor model as a separate, magical object with its own special rules, when in truth it is built entirely from the passive elements and dependent sources covered here. Once a transistor has been replaced by its resistance terms and its controlled current source, the resulting network is just a linear circuit, and every technique you already know applies unchanged: nodal analysis, superposition with the independent sources, Thevenin and Norton reductions, and so on. The dependent source is the one piece that links the input port to the output port, and it is what gives an amplifier its gain. If that source were absent, the input and output sides of the model would be electrically isolated and the device would do nothing. Seeing the controlled source as the heart of the model, rather than as an exotic add-on, is the conceptual step that turns transistor analysis from memorization into reasoning.

Common Mistakes

• Leaving the controlled term on the right. Treating $g_m V_1$ like an independent source and keeping it on the right-hand side gives a wrong, decoupled system. It depends on an unknown, so it belongs on the left.
• Forgetting the auxiliary equation. If the controlling variable is not itself a node voltage (for example a branch current), you must add one equation defining it, or you will have more unknowns than equations.
• Sign of the controlling variable. Reversing the assumed polarity of $v_x$ or the reference direction of $i_x$ flips the sign of the whole dependent term and inverts the answer.
• Unit confusion. Transconductance is siemens and transresistance is ohms; plugging a transconductance in where a gain belongs produces nonsense with the wrong dimensions.
• Killing a dependent source during superposition. Dependent sources stay active when you apply superposition; only independent sources are turned off.

Summary Table

Related Tutorials

• Nodal Analysis Step by Step
• BJT Small-Signal Model
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