3. Thin Linear Wire Antennas // Antenna and Wave Propagation // bhaswanth

The simplest antenna is a wire. The wire is fed at its center (or sometimes its end) by an alternating current that pushes electrons back and forth. Different wire lengths give different patterns.

3.1 The infinitesimal dipole (Hertzian dipole)

Take a tiny wire, much shorter than $\lambda$ , carrying a uniform current $I_0$ . This is the Hertzian dipole or infinitesimal dipole. It is mathematically the simplest possible radiator.

The far-field expressions, derivable from Maxwell's equations:

$E_\theta = j \frac{\eta_0 k I_0 L}{4\pi r} \sin\theta \, e^{-jkr}$

$H_\phi = j \frac{k I_0 L}{4\pi r} \sin\theta \, e^{-jkr}$

The pattern is $\sin\theta$ , peaking broadside to the wire (90° to the axis) and going to zero along the axis (the wire ends). The 3D pattern looks like a doughnut wrapped around the wire, with the wire passing through the hole.

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       Pattern of an infinitesimal dipole
             (slice through the axis)
 
                _______
              /         \
             /           \
            |  doughnut   |
   wire ----|  (E radiates|---- wire
            |  outward)   |
             \           /
              \_________/

Quantities:

Radiation resistance: $R_r = 80\pi^2 (L/\lambda)^2$ Ω.
Directivity: 1.5 (1.76 dBi).
Beamwidth: about 90°.

For $L = 0.01\lambda$ , $R_r \approx 0.08$ Ω. Hopelessly inefficient, because conductor losses easily exceed 0.08 Ω. The infinitesimal dipole is a theoretical building block, not a real antenna you build.

3.2 The short dipole

A short dipole has length $L$ between $\lambda/50$ and $\lambda/10$ , with a triangular current distribution (peak at center, zero at the ends). Compared to the Hertzian:

$R_r = 20\pi^2 (L/\lambda)^2$ (a factor of 4 less than the uniform-current case).
Same directivity: 1.5.
Same pattern: $\sin\theta$ .

Used in: low-frequency mobile antennas (where $\lambda$ is too large for a true half-wave), some HF whips, AM-receiver loops (though those are loop antennas, not dipoles).

3.3 The half-wave dipole: the standard reference

A half-wave dipole is a center-fed wire of total length $L = \lambda/2$ . The current distribution is approximately a half-cosine peaking at the center and going to zero at the ends, naturally setting up a standing wave that radiates efficiently.

$E_\theta(r, \theta) = \frac{j 60 I_0}{r} \cdot \frac{\cos\left(\frac{\pi}{2}\cos\theta\right)}{\sin\theta} \, e^{-jkr}$

The factor $\cos((\pi/2)\cos\theta)/\sin\theta$ is the half-wave-dipole pattern factor. It is almost but not exactly $\sin\theta$ . Plot it on a polar diagram and you see a slightly squished doughnut, slightly more directional than the infinitesimal dipole.

Quantities:

Length: $\lambda/2$ exactly (in practice 0.47 to 0.49 of $\lambda$ to account for end-effect).
Radiation resistance at center: 73 Ω.
Reactance at exact resonance: ~0 (purely real).
Directivity: 1.64 (2.15 dBi).
HPBW: 78°.
Bandwidth: a few percent (it is a high-Q resonant structure).
Pattern: doughnut, peak broadside, nulls at the ends.

The half-wave dipole is the universal reference antenna. Gain is sometimes quoted in dBd (dBi minus 2.15) for this reason: "5 dBd" means "5 dB more than a half-wave dipole, which is 7.15 dBi." Many measurement standards use a half-wave dipole as the gold standard, partly because its gain is calculable from first principles and partly because it is robust and reproducible.

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   Half-wave dipole pattern (E-plane)
 
              θ = 90°
                ▲
              ╱   ╲
            ╱       ╲
          ╱           ╲
       ─ ─ ─ ─ ─ ─ ─ ─ ─ ─  axis (ends)
       ─ feed─point ─ ─ ─
          ╲           ╱
            ╲       ╱
              ╲   ╱
                ▼
              θ = 270°

Used in: TV "rabbit ears," every Yagi has one as the driven element, dipole arrays for FM/AM broadcast, reference antennas in EMC chambers.

3.4 The quarter-wave monopole and image theory

Now we cut a half-wave dipole in half and stand the bottom half on a perfectly conducting ground plane. The result is a vertical wire of length $\lambda/4$ above ground. By image theory, the ground plane acts as a mirror: from the antenna's perspective, there is an "image" wire below the ground that exactly mimics what the lower half of the dipole would have done. The radiation above the ground is identical to a half-wave dipole's upper half.

Image theory in a sentence: a vertical current above a perfect electric conductor is equivalent to that current plus an identical vertical image current below the conductor.

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   Real:                  Equivalent:
   
   ┃ λ/4 wire             ┃ λ/4 wire
   ━━━━━━━━━ (ground)     ┃ image (λ/4)
                          (no ground)

Properties of the quarter-wave monopole over an infinite ground plane:

Radiation resistance: ~36.5 Ω (half of the dipole's 73 Ω, because only half the radiation pattern exists).
Directivity: 5.15 dBi (3 dB more than the dipole, because all the energy goes into the upper hemisphere).
Pattern: omnidirectional in azimuth, peak at the horizon, null straight up.

In practice, real ground planes are finite. A car body, a copper-clad PCB, even a network of radial wires (a "ground plane" of thin radials at the base of a vertical antenna) all approximate the ideal infinite ground. Imperfect grounds cause the pattern to lift off the horizon (this is called "high-angle radiation") and reduce gain.

Used in: AM/FM whips on cars, mobile phone antennas (often folded down for compactness, the PIFA), VHF/UHF radios, base stations, the "rubber duck" on every walkie-talkie.

3.5 The folded dipole

A folded dipole is two parallel half-wave dipoles connected at the ends, with one driven at the center. Mechanically simple, electrically clever:

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   ┌─────────────────────┐
   │                     │
   ├──────●●──── feed ───┤
   │                     │
   └─────────────────────┘

The current divides equally between the two parallel wires, but the voltage at the feed is the same as a single dipole would see. This means the impedance is multiplied by 4: from 73 Ω (regular half-wave dipole) to about 292 Ω (folded). Useful when feeding from old TV-style 300 Ω twin-lead, or to broaden the bandwidth.

The pattern is identical to a single dipole. The wider bandwidth (because it is effectively two parallel resonators) and the impedance step-up are the reasons it is used.

Used in: FM and TV antennas, folded-dipole feed elements in many Yagis.

3.6 The sleeve dipole

A sleeve dipole is a vertical antenna with a coaxial sleeve over the lower half. The sleeve becomes an electrical "image" of the upper element, removing the need for a ground plane. Often used for portable VHF/UHF antennas where there is no convenient ground plane (think of a BNC-mounted antenna sticking out of a handheld radio).

3.7 Radiation pattern of a general center-fed dipole

For a dipole of arbitrary total length $L$ with sinusoidal current distribution, the far-field pattern is

$F(\theta) = \frac{\cos\left(\frac{kL}{2}\cos\theta\right) - \cos\left(\frac{kL}{2}\right)}{\sin\theta}$

For $L = \lambda/2$ this gives the half-wave pattern. For $L = \lambda$ it gives a sharper pattern with side lobes. For $L = 1.25\lambda$ ("extended double Zepp") even more directivity, but with side lobes that interfere with neighboring channels. As $L$ grows beyond a few wavelengths, the pattern fragments into a forest of lobes, none of which is much bigger than a doughnut.

This is why the half-wave is the workhorse. It is the longest single dipole that radiates almost purely in one main lobe.