---
title: "Superposition theorem in antennas"
subject: "Propagation and Antenna"
chapter: 1
key: "semester_6/propagation_and_antenna:superposition-theorem"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/superposition-theorem
site: BCT Engineering
---

# Superposition theorem in antennas

Propagation and Antenna · Chapter 1: Radiation and Antenna Fundamentals

- Asked in: 2080 Asoj, 3 marks; 2076 Bhadra, 3 marks; 2072 Magh, 2 marks (asked 3 times)
- Video: [1.6 Antenna Theorems | IOE | Antenna and propagation](https://www.youtube.com/watch?v=Ns8n2fwZaq8) — Yugal Nyoupane
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/superposition-theorem
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-1

## Question

State and explain the superposition theorem used in antenna, with necessary diagrams.

## Short answer

In a linear system, the total response is the phasor sum of responses due to each source acting alone. For an array, E_total=E1+E2+...+En, including path-difference phase. This yields array pattern = element pattern times array factor, and holds only for linear media.

## Answer

**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

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

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

```text
         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:

```text
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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does superposition say about the total field from several sources in a linear system?
   A. It is the largest single contribution
   B. It is the sum of magnitudes only
   C. It is the product of the individual fields
   D. It is the vector (phasor) sum of the responses of each source acting alone
   Answer: D. The total response is the phasor sum, each with its own amplitude and phase. (Key point: Statement)

2. For two equal in-phase isotropic sources with ψ = k d cosθ, what is the total field?
   A. 2E0 cos(ψ/2)
   B. 2E0 sin(ψ/2)
   C. E0 cos ψ
   D. E0 + ψ
   Answer: A. E0 e^(jψ/2) + E0 e^(-jψ/2) = 2E0 cos(ψ/2). (Key point: Two-source result)

3. Which array result follows from superposition?
   A. Gain equals directivity
   B. Array pattern = element pattern × array factor
   C. Array pattern = element pattern + array factor
   D. Array pattern is always omnidirectional
   Answer: B. Pattern multiplication follows from adding the element fields with their phases. (Key point: Array application)

4. When is superposition valid?
   A. For any nonlinear medium
   B. Only in vacuum with no sources
   C. Only for linear media and linear circuit elements
   D. Only for DC
   Answer: C. It is valid only for linear media and linear elements. (Key point: Validity)

## Related in Propagation and Antenna

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## Related in other subjects

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