In antenna arrays, what is the correct way to obtain the total far-field at a distant observation point from the contributions of all individual elements? (Assume the point is in the far field and mutual coupling is already accounted for in the element patterns/phases.)

Introduction / Context:
In antenna array theory, the radiation observed in the far field results from the superposition of the contributions of each element. Because fields are sinusoidal and have phase, the combination must preserve both amplitude and phase information, which is naturally handled by phasors.

Given Data / Assumptions:

• Observation point is in the far field (Fraunhofer region).
• Element excitations and relative positions are known.
• Mutual coupling effects, if any, are already reflected in the element patterns/phases.

Concept / Approach:

The electromagnetic field is a vector quantity with time-varying phase. Superposition therefore requires vector (phasor) addition, not scalar addition. The array factor arises from summing complex exponentials representing element phases and spacings.

Step-by-Step Solution:

Verification / Alternative check:

If you incorrectly add magnitudes first, you will lose interference effects (lobes/nulls). Correct phasor addition predicts constructive and destructive interference that match measured patterns.

Why Other Options Are Wrong:

• Algebraic sum of magnitudes: Ignores phase; cannot predict nulls/lobes.
• Arithmetic average of powers: Power is a squared quantity; averaging powers cannot reproduce interference.
• Sum of directivities: Directivity is a system property, not an element-level additive quantity.
• RMS sum of magnitudes: Still discards relative phase information.

Common Pitfalls:

Adding gains or powers instead of fields; forgetting that phase delays from spacing/feeding control the pattern.

Final Answer:

Phasor (vector) sum of the individual element fields