5.3 Superposition theorem

Linear Circuits and Sources

Linearity and Sources

Linearity means a circuit's output is directly proportional to its input. If you double the input voltage, the output voltage doubles too. This proportionality is what makes superposition work: because each source contributes independently and proportionally, you can analyze them one at a time and add the results.

There are two categories of sources you need to know:

• Independent sources provide a fixed voltage or current no matter what the rest of the circuit is doing.
  • A voltage source (like a battery) maintains a constant voltage across its terminals.
  • A current source maintains a constant current through its terminals.
• Dependent sources have their output controlled by some other voltage or current in the circuit. The four types are:
  • Voltage-controlled voltage source (VCVS): output voltage depends on a voltage elsewhere
  • Current-controlled current source (CCCS): output current depends on a current elsewhere
  • Voltage-controlled current source (VCCS) and current-controlled voltage source (CCVS)

When applying superposition, you only turn off independent sources. Dependent sources stay active in every sub-circuit because their value depends on circuit conditions, not on a fixed setting.

Source Transformation

Source transformation lets you convert between equivalent voltage and current source configurations, which can simplify a circuit before you apply superposition.

• A voltage source $V$ in series with a resistor $R$ can become an equivalent current source in parallel with that same resistor. The equivalent current is:

$I = \frac{V}{R}$

• A current source $I$ in parallel with a resistor $R$ can become an equivalent voltage source in series with that resistor. The equivalent voltage is:

$V = IR$

The key point: the circuit behaves identically at its external terminals after the transformation. Only the internal arrangement changes, which can make analysis easier.

Applying Superposition

Circuit Decomposition

To use superposition, you break a multi-source circuit into sub-circuits, each with only one independent source active. Here's how you "turn off" the other sources:

• Voltage sources are replaced by a short circuit (a wire connecting their two terminals), because an ideal voltage source with $V = 0$ is just a wire.
• Current sources are replaced by an open circuit (a break in the wire), because an ideal current source with $I = 0$ passes no current, as if the branch doesn't exist.

Each sub-circuit is now simple enough to solve with Ohm's law, voltage/current dividers, or Kirchhoff's laws.

Partial Solutions and the Superposition Principle

Each sub-circuit gives you a partial solution: the voltage across or current through a particular element due to that one source alone. The superposition principle says the actual voltage or current in the original circuit equals the algebraic sum of all the partial solutions.

"Algebraic sum" matters here. Some contributions may be positive and others negative depending on polarity and current direction, so pay attention to signs.

Steps to apply superposition:

1. Identify all independent sources in the circuit.
2. Pick one source to keep active. Replace every other independent voltage source with a short circuit and every other independent current source with an open circuit. Leave all dependent sources in place.
3. Solve the resulting sub-circuit for the voltage or current you need. Record this partial solution with its sign.
4. Repeat steps 2–3 for each remaining independent source, one at a time.
5. Add all partial solutions together (respecting signs) to get the total voltage or current in the original circuit.

For example, in a circuit with two voltage sources, you'd solve two sub-circuits. If source A contributes $3\,\text{mA}$ through a resistor and source B contributes $-1\,\text{mA}$ (opposing direction), the total current is $3 + (-1) = 2\,\text{mA}$.

Superposition is especially handy when you want to see how much each source individually contributes to a particular voltage or current. It won't save you time on every problem, but for circuits with several independent sources, it turns one hard problem into several straightforward ones.