Antenna Arrays and Modern Topics
A single dipole spreads its energy across a wide, fixed pattern — fine for coverage, useless for reaching far or for aiming. The remedy is not a bigger antenna but many small ones, fed in concert. By controlling the amplitude and, above all, the phase of each element, an array can sharpen its beam, point it anywhere, and re-aim in microseconds with nothing moving. That principle — the array factor and pattern multiplication — powers radar, 5G beamforming, and satellite links. This closing chapter develops it, then surveys the modern antennas built on every idea in this book, from the printed patch in your phone to the metasurfaces now reshaping how we steer waves.
- Why an array of small elements outperforms one large antenna for shaping and steering.
- The two-element array factor and the phase term \(\psi=\beta d\cos\theta+\alpha\).
- The principle of pattern multiplication.
- The N-element uniform array factor and its main beam.
- Broadside, end-fire, and electronically steered (phased) arrays.
- Array directivity, and a survey of modern antennas.
Why Combine Elements?
The half-wave dipole of Chapter 29 has a directivity of only 1.64 and a pattern fixed by its geometry — you cannot make it narrower or aim it without physically rebuilding or rotating it. An array sidesteps both limits. Place several identical elements near one another and feed each with its own amplitude and phase; in some directions their fields add, in others they cancel. The result is a pattern far sharper than any single element, and — crucially — one whose direction can be changed purely by adjusting the feed phases, with no moving parts.
The mathematics is gratifyingly simple, because the elements are identical. Their combined field separates into the field of one element times a factor that depends only on the array's geometry and excitation — the array factor.
The Two-Element Array
Start with two identical elements separated by a distance \(d\), driven with equal current magnitudes but a progressive phase difference \(\alpha\). In a far-field direction making angle \(\theta\) with the array axis, the wave from the second element travels an extra distance \(d\cos\theta\), adding a path phase \(\beta d\cos\theta\) on top of the feed phase \(\alpha\). The two fields add as phasors, giving the array factor:
The magnitude \(|\text{AF}|=2|\cos(\psi/2)|\) is maximum (equal to 2) when \(\psi=0\) — the elements add fully in phase — and zero when \(\psi=\pm\pi\). By choosing \(d\) and \(\alpha\), the designer decides which direction \(\theta\) makes \(\psi=0\), and therefore where the beam points.
Pattern Multiplication
The single most useful theorem in array theory is that the total radiation pattern of an array of identical elements is simply the product of the element's own pattern and the array factor:
This separates the problem cleanly. The element pattern (a broad \(\sin\theta\) for dipoles) provides the gentle backdrop; the array factor, sharp and adjustable, carves the actual beam out of it. Design the array factor for the beam you want, then multiply by whatever element you happen to use. Everything that follows is really just the study of the array factor.
The N-Element Linear Array
Extend to \(N\) identical elements, equally spaced \(d\) apart, with uniform amplitude and a constant phase step \(\alpha\) between neighbours. Each contributes a phasor \(e^{jn\psi}\), and the sum is a geometric series with a famous closed form. Normalized to its peak, the array factor is:
The main beam sits where \(\psi=0\), and it is flanked by smaller side lobes. As \(N\) grows the main beam narrows and the directivity rises — more elements, a tighter beam. The price is more side lobes and more feed complexity, but the trade is overwhelmingly worth it: this single formula underlies essentially every high-performance antenna system in use.
Broadside, End-Fire, Steering
The beam direction is set by where \(\psi=0\), i.e. by \(\beta d\cos\theta+\alpha=0\). Three choices of the phase step \(\alpha\) give the three classic behaviours:
| Array type | Phase step \(\alpha\) | Beam direction |
|---|---|---|
| Broadside | \(\alpha=0\) | \(\theta=90^\circ\) — perpendicular to the axis |
| End-fire | \(\alpha=-\beta d\) | \(\theta=0^\circ\) — along the axis |
| Steered (phased) | \(\alpha=-\beta d\cos\theta_0\) | \(\theta_0\) — any chosen angle |
The steered case is the phased array, and it is transformative. Solving \(\beta d\cos\theta_0+\alpha=0\) gives the beam angle \(\cos\theta_0=-\alpha/\beta d\): change \(\alpha\) electronically and the beam swings to a new \(\theta_0\) in microseconds, scanning the sky with no mechanical motion. One caution — if the spacing is too large (\(d\ge\lambda\) for broadside), a second full-strength beam, a grating lobe, appears in an unwanted direction, so practical arrays keep \(d\le\lambda/2\) when scanning widely.
Array Directivity
Stacking elements concentrates power, so directivity climbs roughly in proportion to the array's electrical length. For a long uniform broadside array of \(N\) elements spaced \(d\), the directivity and the beamwidth follow simple rules of thumb:
Doubling the number of half-wave-spaced elements roughly doubles the directivity (adds 3 dB) and halves the beamwidth. This is why a radar face carries thousands of elements and a satellite dish — effectively a continuous aperture — reaches tens of thousands. The aperture relation \(A_e=(\lambda^2/4\pi)G\) from Chapter 29 still holds: a bigger effective area, whether one dish or many elements, means a higher gain.
Modern Antennas
Every modern antenna is an application of the fundamentals in this book. A few define the present landscape:
- Microstrip (patch) antennas — a metal patch printed on a grounded dielectric, resonant at roughly half a guide-wavelength. Flat, cheap, and easily arrayed, they fill phones, GPS receivers, and satellite terminals.
- Phased arrays & beamforming — the steered array of this chapter, scaled to thousands of elements. Massive-MIMO base stations and modern radar form and aim many beams at once, entirely in software.
- Reflector (dish) antennas — a feed at the focus of a parabola, turning a small source into a huge effective aperture for deep-space links and radio astronomy.
- Horns — flared waveguide openings, used as standards and as reflector feeds for their clean, predictable patterns.
- Metasurfaces & reconfigurable intelligent surfaces — engineered surfaces that reshape or redirect incident waves, an active research frontier for controlling propagation itself.
All of them rest on the same chain you have now followed end to end: Maxwell's equations, the wave they predict, its behaviour when guided or reflected, and its release into space as radiation.
Worked Examples
Problem. Two isotropic elements are spaced \(d=\lambda/2\) with \(\alpha=0\). Find the array-factor maximum and null directions.
Solution. \(\psi=\pi\cos\theta\); \(|\text{AF}|=2|\cos(\tfrac{\pi}{2}\cos\theta)|\). Max at \(\psi=0\), null at \(\psi=\pm\pi\):
Problem. Two elements are spaced \(d=\lambda/4\). What phase step \(\alpha\) makes the array end-fire (beam at \(\theta=0\))?
Solution. End-fire needs \(\psi=0\) at \(\theta=0\): \(\alpha=-\beta d\):
Problem. Estimate the directivity (in dBi) of a 10-element broadside array with \(d=\lambda/2\).
Solution. Use \(D\approx 2N(d/\lambda)\):
Problem. A phased array has \(d=\lambda/2\). What phase step \(\alpha\) steers the main beam to \(\theta_0=60^\circ\)?
Solution. Use \(\alpha=-\beta d\cos\theta_0\):
Problem. For the two-element broadside array of Example 1, find the normalized array factor at \(\theta=60^\circ\).
Solution. \(\psi=\pi\cos60^\circ=\pi/2\); normalized \(|\text{AF}|/2=|\cos(\psi/2)|\):
Problem. For a broadside array, what element spacing keeps the only maximum at broadside (no grating lobes)?
Solution. A second maximum appears when \(\psi=\pm2\pi\) is reachable; with \(\alpha=0\) this needs \(\beta d\ge2\pi\), i.e. \(d\ge\lambda\):
Chapter Summary
Many elements shape and steer a beam a single antenna cannot.
Two-element: \(|\text{AF}|=2|\cos(\psi/2)|\), \(\psi=\beta d\cos\theta+\alpha\).
Total pattern = element pattern × array factor.
\(|\text{AF}|=\left|\dfrac{\sin(N\psi/2)}{N\sin(\psi/2)}\right|\); beam at \(\psi=0\).
Broadside \(\alpha=0\); end-fire \(\alpha=-\beta d\); steer with \(\cos\theta_0=-\alpha/\beta d\).
\(D\approx 2Nd/\lambda\); patches, phased arrays, reflectors, metasurfaces.
Problems
For each item, write \(\psi=\beta d\cos\theta+\alpha\) and find where \(\psi=0\) (the beam) and \(\psi=\pm\pi\) (the nulls). Keep spacings in wavelengths. Difficulty rises down the list.
- Two isotropic elements, \(d=\lambda/2\), \(\alpha=180^\circ\). Find the beam direction.
- For \(d=\lambda/4\), \(\alpha=0\), find the array-factor maximum and null directions.
- What phase step gives an end-fire beam for \(d=\lambda/2\)?
- State the pattern-multiplication theorem and why it makes array design modular.
- A uniform array has \(N=8\), \(d=\lambda/2\). Estimate its broadside directivity in dBi.
- How many half-wave-spaced elements give a directivity of about 16 dBi?
- A \(d=\lambda/2\) phased array steers to \(\theta_0=45^\circ\). Find \(\alpha\).
- For the array in problem 7, find the normalized two-element AF at \(\theta=90^\circ\).
- Why do grating lobes appear when \(d\ge\lambda\), and how does small spacing prevent them?
- A patch antenna on \(\varepsilon_r=2.2\) substrate resonates at \(2.4\,\text{GHz}\). Estimate its length (\(L\approx\lambda_g/2\), \(\lambda_g\approx\lambda_0/\sqrt{\varepsilon_r}\)).
- Explain how a phased array re-aims its beam without moving, and one advantage over a mechanically steered dish.
- Trace one example signal path through this course: from a source current, name the stages by which energy reaches a distant receiver.