Superposition Theorem: Principle and Worked Examples

The Idea Behind Superposition

The superposition theorem states that in any linear circuit with multiple independent sources, the response (any branch current or node voltage) is the sum of the responses caused by each independent source acting alone, with all other independent sources deactivated. It is a direct consequence of linearity: if a circuit's governing equations are linear, then the response to a sum of inputs equals the sum of the responses to each input. This lets you decompose a daunting multi-source problem into several single-source problems, each far easier to analyze, and then add the pieces back together.

The catch is the word linear. Superposition applies to circuits built from resistors, capacitors, inductors, and both independent and linear dependent sources. It does not apply to power, which depends on the square of voltage or current and is therefore nonlinear, nor to circuits containing nonlinear elements such as diodes operating across their knee. Within its domain, though, it is one of the most useful conceptual tools in linear circuit theory and pairs naturally with KVL and KCL for solving each single-source sub-circuit.

How to Kill a Source

Deactivating ("killing") a source means setting its value to zero, and the way you do that depends on the source type. Setting an ideal voltage source to zero volts makes it a wire (a short circuit), because a 0 V source enforces zero potential difference across itself. Setting an ideal current source to zero amps makes it an open circuit, because a 0 A source enforces zero current through itself regardless of the voltage. A clear table prevents the single most common superposition error.

That last row is critical. Dependent (controlled) sources are never killed. Their value tracks a controlling variable elsewhere in the circuit, so they remain active in every sub-circuit and you simply re-evaluate them for each case.

A Fully Worked Two-Source Example

Consider a node with voltage $V_x$ connected by $R_1 = 4\ \Omega$ to a 12 V source, by $R_2 = 6\ \Omega$ to ground, and fed by a 2 A current source injecting current into the node. We want $V_x$ using superposition.

Contribution from the 12 V source alone

Kill the 2 A current source by opening it. The circuit reduces to a simple voltage divider: 12 V across $R_1 = 4\ \Omega$ and $R_2 = 6\ \Omega$ in series, with $V_x$ taken across $R_2$:

Contribution from the 2 A source alone

Now kill the 12 V source by shorting it. With the voltage source shorted, $R_1$ and $R_2$ both connect from the node to ground, so they appear in parallel as seen by the current source:

The full 2 A flows into this parallel combination, producing:

Combine

By superposition the actual node voltage is the algebraic sum of the two contributions (both raise the node above ground, so both are positive):

As a check, solve the original circuit directly with a single node equation: $(V_x - 12)/4 + V_x/6 = 2$. Multiplying through by 12 gives $3(V_x - 12) + 2 V_x = 24$, i.e. $5 V_x - 36 = 24$, so $V_x = 60/5 = 12\ \text{V}$. The superposition result matches the direct solution exactly, confirming the method.

Why Power Does Not Superpose

It is tempting to compute the power from each source separately and add them, but this is wrong. Suppose a resistor carries $i'$ from one source and $i''$ from another. The total current is $i' + i''$, and the power is:

The cross term $2 i' i'' R$ is precisely what naive summation of individual powers omits. Always superpose the currents or voltages first, then compute power from the combined value.

Linearity, the Property That Makes It Work

Superposition is not an independent axiom; it is a theorem that follows from two properties of linear circuits. The first is homogeneity (scaling): if a source of value $x$ produces a response $y$, then a source of value $\alpha x$ produces $\alpha y$. The second is additivity: the response to $x_1 + x_2$ equals the response to $x_1$ plus the response to $x_2$. Together these define a linear system, and any circuit whose element relations are linear (resistors obeying $v = Ri$, capacitors and inductors with constant $C$ and $L$, and linear controlled sources) inherits both. The node and mesh equations of such a circuit form a linear system $A\mathbf{x} = \mathbf{b}$, and the solution depends linearly on the source vector $\mathbf{b}$. Splitting $\mathbf{b}$ into one nonzero entry at a time and summing the partial solutions is precisely what superposition does.

Seeing it this way clarifies the boundaries. A diode's exponential characteristic and a saturated transistor both break additivity, so superposition fails outright for them. But the moment you linearize such a device, replacing a transistor with its small-signal hybrid-pi model, for instance, you recover a linear circuit and superposition applies again to the small-signal quantities. This is exactly the trick that makes multi-source small-signal amplifier analysis tractable, and it is why the theorem remains central to analog design despite applying only to linear systems.

When to Reach for Superposition

Superposition shines when sources operate at different frequencies (a DC bias plus an AC signal, for instance), because each sub-circuit can be analyzed in its own domain and the results summed. It is also the conceptual backbone of phasor analysis and of small-signal amplifier models. For a single-frequency circuit with many sources, however, nodal or mesh analysis is usually faster because superposition multiplies the number of sub-circuits you must solve.

Superposition and Source Transformations

Once you are comfortable killing sources, a natural companion technique is the source transformation, which converts a voltage source in series with a resistance into an equivalent current source in parallel with the same resistance, and vice versa. The two are interchangeable from the terminals' point of view because both deliver the same $v$-$i$ relationship. In the worked example, the 12 V source behind $R_1 = 4\ \Omega$ could be redrawn as a 3 A current source ($12/4$) in parallel with 4 Ω, after which both sources are current sources feeding the same node and the problem becomes a single current-divider calculation. Source transformation does not replace superposition, but combining the two often shortens the arithmetic dramatically, and it reinforces the idea that a real source is characterized by an open-circuit voltage, a short-circuit current, and an internal resistance relating them. This same equivalence is the heart of Thevenin and Norton modeling, which you can think of as superposition's most practical offspring.

Common Mistakes

• Opening voltage sources or shorting current sources. It is exactly backwards. Voltage sources become shorts; current sources become opens.
• Deactivating dependent sources. Controlled sources stay active in every sub-circuit and must be re-evaluated each time.
• Adding individual powers. Power is nonlinear; sum the voltages or currents first, then square.
• Dropping signs when combining. Each contribution has a polarity relative to the chosen reference. Add them algebraically, not as magnitudes.
• Applying superposition to a nonlinear circuit. If a diode, saturating transistor, or other nonlinear element is present, the theorem simply does not hold.

Related Tutorials

• Nodal Analysis Step by Step — often the faster alternative for single-frequency circuits.
• Voltage Divider Equation — used to solve each single-source sub-circuit here.
• Kirchhoff's Current Law — the basis of the direct-solution check above.