Superposition Theorem

This theorem is used to analyze linear circuits with multiple sources.

Statement

A linear circuit containing several sources obeys the principle of superposition. The current or voltage produced by the sources acts independently.

Rules for Source Elimination

• Replace all voltage sources with short circuits (wires)
• Replace all current sources with open circuits (breaks)

Example

$$I = I_1 + I_2$$

Calculation of $I_\alpha$

We replace the current source $I_2$ with an open circuit.

We analyze the circuit with the current source $I_1$ only.

$$I_{\alpha} = \frac{R_2}{R_1 + R_2} \cdot I_1$$

Calculation of $I_\beta$

We replace the current source $I_1$ with an open circuit.

We analyze the circuit with the current source $I_2$ only.

$$I_{\beta} = \frac{R_2}{R_1 + R_2} \cdot I_2$$

Conclusion for $I$

$$I = I_{\alpha} + I_{\beta}$$ $$I = \frac{R_2}{R_1 + R_2} \cdot I_1 + \frac{R_2}{R_1 + R_2} \cdot I_2$$ $$I = \frac{R_2}{R_1 + R_2} \cdot (I_1 + I_2)$$