According to the superposition theorem for linear circuits in phasor (AC) analysis, how should multiple independent sources be handled when determining currents or voltages at a given element?

Introduction / Context:
The superposition theorem enables stepwise analysis of linear circuits containing multiple independent sources. It is especially useful in AC phasor analysis where each source may have distinct magnitude and phase. By evaluating one source at a time while properly deactivating the others, we can sum the resulting phasor contributions to find the total response.

Given Data / Assumptions:

• The circuit is linear and time-invariant.
• We are performing AC phasor analysis (complex impedances).
• Multiple independent sources (voltage and/or current) exist.

Concept / Approach:
Superposition states that the total response equals the sum of individual responses due to each independent source acting alone. When considering one source, all other independent voltage sources are replaced by their internal impedance (ideal voltage source → short circuit; finite source resistance/impedance remains). Independent current sources are opened (ideal current source → open circuit; any internal impedance remains). Dependent sources, if present, are not deactivated because they depend on circuit variables.

Step-by-Step Solution:

Verification / Alternative check:
As a check, you can analyze the full circuit with all sources active at once (e.g., by nodal analysis). The resulting voltages and currents will match the phasor sum obtained via superposition if the circuit is linear.

Why Other Options Are Wrong:

• 'all sources are considered independently': Too vague; the method requires specific deactivation rules using internal impedance.
• 'all sources are considered simultaneously': That is a valid direct solve, but not the definition of superposition.
• 'internal resistance': In AC phasor analysis, elements contribute complex impedance; limiting to resistance omits reactance and is incorrect.

Common Pitfalls:

• Deactivating dependent sources (should remain active).
• Forgetting that ideal voltage source → short, ideal current source → open.

Final Answer:
the sources are considered one at a time with all others replaced by their internal impedance