---
title: "Need for arrays and two-element patterns"
subject: "Propagation and Antenna"
chapter: 2
key: "semester_6/propagation_and_antenna:array-need-two-isotropic-radiators"
source: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/array-need-two-isotropic-radiators
site: BCT Engineering
---

# Need for arrays and two-element patterns

Propagation and Antenna · Chapter 2: Antenna Parameters and Arrays

- Asked in: 2072 Magh, 2+6 marks
- Video: [Two Elements Antenna Array (Phase Difference, Electric Field & Array Factor) Explained](https://www.youtube.com/watch?v=7eOQ2foOxxc) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/propagation_and_antenna/array-need-two-isotropic-radiators
- Whole chapter: https://www.bctengineer.com/md/semester_6/propagation_and_antenna/chapter-2

## Question

Why do we need antenna array rather than changing the length of single antenna? Derive the expression and draw the pattern for an array of two isotropic radiators with: i) Equal amplitude and phase ii) Equal amplitude and opposite phase

## Short answer

Single antenna longer than ~1.25λ causes beam splitting, cannot be steered. Arrays provide high directivity, narrow beam, low sidelobes, electronic steering. Two isotropic sources equal amplitude & phase: E = 2E0 cos[(βd/2)cosφ] (broadside). Opposite phase: E = 2E0 sin[(βd/2)cosφ] (end‑fire).

## Answer

#### Why an array instead of a longer single antenna
- Making a single wire longer than about 1.25λ does not keep increasing gain: **current phase reversals** appear and the beam splits into many lobes (e.g. 3λ/2 dipole has lower directivity than a λ dipole).
- One element gives a fixed, broad pattern; its beam cannot be **steered** without moving it.
- An array of elements fed with controlled amplitude and phase gives **high directivity, narrow beam, low side lobes, nulls towards interference and electronic beam steering**, while each element stays a convenient size (e.g. λ/2).
- Power is shared among elements, and failure of one element only slightly degrades performance.

#### Two isotropic radiators
Consider two isotropic point sources 1 and 2 spaced a distance d apart on the x-axis, with the reference (origin) at the midpoint. A far point P is in direction φ measured from the array axis. Each source alone gives field E0 at P.

```text
                    to far point P
                   /
                  / φ
   (1)-----------O-----------(2)
    |<--- d/2 --->|<--- d/2 --->|
   path from (2) shorter by (d/2)cosφ
   path from (1) longer  by (d/2)cosφ
```

Path difference between the two waves = d cosφ, so the phase difference due to path is

```text
ψ = β d cosφ,      β = 2π/λ
```

With respect to the midpoint, wave 2 leads by ψ/2 and wave 1 lags by ψ/2.

#### i) Equal amplitude and phase
Currents equal in magnitude and **in phase**. By superposition:

```text
E = E0 e^(−jψ/2) + E0 e^(+jψ/2)
  = 2E0 cos(ψ/2)
E = 2E0 cos[(βd/2) cosφ]
```

For **d = λ/2** (βd/2 = π/2), normalized: E_n = cos[(π/2) cosφ].
- **Maxima:** (π/2)cosφ = 0 → cosφ = 0 → φ = 90°, 270° (broadside).
- **Minima:** (π/2)cosφ = ±π/2 → cosφ = ±1 → φ = 0°, 180°.
- **Half power:** cos[(π/2)cosφ] = 1/√2 → cosφ = ±1/2 → φ = 60°, 120° → HPBW = 60°.

```text
        φ=90°
        .-.
       (   )
  (1)---O---(2)   φ=0°  (null on axis)
       (   )
        '-'
       φ=270°
  Broadside figure-of-eight
```

#### ii) Equal amplitude and opposite phase
Currents equal in magnitude but **opposite in phase** (180°). By superposition:

```text
E = −E0 e^(−jψ/2) + E0 e^(+jψ/2)
  = 2jE0 sin(ψ/2)
|E| = 2E0 sin[(βd/2) cosφ]
```

For **d = λ/2** (βd/2 = π/2), normalized: E_n = sin[(π/2) cosφ].
- **Maxima:** (π/2)cosφ = ±π/2 → cosφ = ±1 → φ = 0°, 180° (along the axis, end-fire).
- **Minima:** (π/2)cosφ = 0 → cosφ = 0 → φ = 90°, 270°.
- **Half power:** sin[(π/2)cosφ] = ±1/√2 → cosφ = ±1/2 → φ = 60°, 120°, 240°, 300° → HPBW = 120° per lobe.

```text
            φ=90° (null)
                |
     .--.       |       .--.
 180°(    )---(1)O(2)---(    ) 0°
     '--'       |       '--'
                |
  End-fire figure-of-eight
```

## Check yourself

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

1. Why does lengthening a single wire not keep increasing directivity?
   A. Because longer wires radiate less power
   B. Beyond about 1.25λ, current phase reversals split the beam into many lobes
   C. Because efficiency becomes exactly zero
   D. Because the wire becomes isotropic
   Answer: B. A 3λ/2 dipole has lower directivity than a λ dipole. (Key point: Why not a longer antenna)

2. Which is an advantage of arrays over one element?
   A. Beam steering, narrow beam, nulls and graceful degradation if one element fails
   B. A fixed broad pattern
   C. Lower directivity
   D. No need for phase control
   Answer: A. Each element stays a convenient size, like λ/2. (Key point: Array advantages)

3. For d = λ/2, equal amplitude and phase, where are the maxima?
   A. At 0° and 180°
   B. At 45° and 135°
   C. At φ = 90° and 270° (broadside)
   D. At every angle
   Answer: C. E_n = cos[(π/2)cosφ] is largest when cosφ = 0. (Key point: In-phase pair)

4. For d = λ/2 and 180° out of phase, the pattern is
   A. Broadside with maximum at 90°
   B. Omnidirectional
   C. Unidirectional cardioid
   D. End-fire with maxima at 0° and 180° and a null at 90°
   Answer: D. E_n = sin[(π/2)cosφ]. (Key point: Opposite-phase pair)

## Related in Propagation and Antenna

- [Two isotropic sources, equal amplitude, opposite phase](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Atwo-sources-opposite-phase) — Propagation and Antenna, chapter 2, asked 3×
- [Antenna arrays and two-element expression](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Adefine-array-two-isotropic) — Propagation and Antenna, chapter 2
- [Opposite-phase two-source array maxima and minima](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Atwo-sources-opposite-maxima-minima) — Propagation and Antenna, chapter 2, asked 2×
- [Two isotropic sources, same and opposite phase](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Atwo-sources-same-and-opposite-phase) — Propagation and Antenna, chapter 2, asked 3×
- [Two-element array and end fire pattern](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Atwo-isotropic-endfire-pattern) — Propagation and Antenna, chapter 2
- [Pattern lobes and 180° out-of-phase array](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Alobes-importance-opposite-phase-array) — Propagation and Antenna, chapter 2

## Related in other subjects

- [Array overflow/underflow; add two arrays](https://www.bctengineer.com/question.md?key=semester_1%2Fcomputer_programming%3Aarray-overflow-underflow-add) — Computer Programming, chapter 7