Superposition theorem in antennas

Superposition theorem: In a linear system with several sources, the total response (current, voltage or field) at any point is the vector (phasor) sum of the responses due to each source acting alone, with the other sources replaced by their internal impedances.

In antennas

Because free space is linear, the total field at a far point P from several radiating elements is

E_total = E1 + E2 + ... + En      (phasor sum)

each term having its own amplitude and phase (due to path difference and current phase).

         P (far point)
        /|\
       / | \
   E1 /  |E2\ E3
     /   |   \
   [1]  [2]  [3]   elements of an array
    |<-d->|<-d->|

Example – two isotropic sources spaced d with equal currents in phase:

E = E0 e^(jψ/2) + E0 e^(−jψ/2) = 2E0 cos(ψ/2)
ψ = k d cosθ

Uses:

• Array theory: array pattern = element pattern × array factor (pattern multiplication follows from superposition).
• Finding the field of a long wire or dipole by adding the fields of many infinitesimal current elements (integration).
• Finding the effect of ground using an image antenna.
• Circuit analysis of antenna feed networks with several sources.

It is valid only for linear media and linear circuit elements.