Dipole Antennas: Theory, Numerical Analysis and Practice
Theory: current on a wire radiates
Overview
Antenna textbooks always start with the dipole. Is it actually used much in real products?
It's used widely in many guises. Two wires in line fed at the centre make a dipole; at half a wavelength the input resistance is about 73 Ω, easy to match to feed lines. The driven element of a TV Yagi, the insides of Wi-Fi router stick antennas and many printed wire antennas are dipoles or variants. And whatever antenna you analyse, checking that the solver reproduces dipole values validates the setup, so it matters as a reference too.
Radiation pattern
The far field for an assumed sinusoidal current along the wire. $\theta$ is the angle from the wire axis, $k = 2\pi/\lambda$ and $L$ the total length. Radiation is strongest broadside ($\theta$ = 90°) and zero along the axis — a doughnut pattern.
Radiation resistance and directivity
$P_{rad}$ is total radiated power, found by integrating the pattern over all directions. $R_{rad}$ is radiation resistance referred to the current maximum $I_0$, and $D$ the directivity — peak intensity over the all-direction average. When the feed isn't at a current maximum, input resistance differs from $R_{rad}$.
Hertz's apparatus was a dipole
In 1887–88 Hertz generated electromagnetic waves by sparking across a gap between two metal rods, and detected them by sparks jumping a tiny gap in a distant loop. The transmitting apparatus was precisely a dipole antenna. The experiment showed that Maxwell's predicted waves were real, and gave its name to the unit of frequency. Hertz reportedly said it was of no practical use — yet little more than a decade later Marconi achieved transatlantic radio.
Worked examples
Example 1: variation with length
Thin wire with sinusoidal current, obtained by numerically integrating the radiation pattern:
| Length | Input resistance | Directivity |
|---|---|---|
| 0.1 λ | 2.0 Ω | 1.50 (1.78 dBi) |
| 0.5 λ | 73.1 Ω | 1.64 (2.15 dBi) |
| 1.0 λ | Very high (zero current at feed) | 2.41 (3.82 dBi) |
| 1.25 λ | 213 Ω | 3.28 (5.16 dBi) |
| 1.5 λ | 105 Ω | 2.23 (3.48 dBi) |
Even a short dipole has nearly the same directivity.
Yes — 1.78 dBi at 0.1 wavelength, close to the half-wave 2.15 dBi; the pattern shape barely changes. What changes is input resistance, only about 2 Ω at 0.1 wavelength. With small radiation resistance, even small losses in matching networks or conductors cut efficiency sharply. Directivity peaks at 5.16 dBi around 1.25 wavelengths; longer, the beam splits and it falls. Longer isn't automatically higher gain.
Example 2: efficiency of a short dipole
At 10 MHz (30 m wavelength), a 3 m (0.1 λ) dipole of 1 mm-radius copper has 1.97 Ω radiation resistance against about 0.13 Ω conductor loss including skin effect, for about 94% efficiency. Not bad for the conductor alone, but the loading coil needed to cancel input reactance usually adds much more loss. Small-antenna design is a battle against these losses.
Example 3: real resonant length
| Frequency | Half wavelength | Typical resonant length (0.47–0.48 λ) |
|---|---|---|
| 433 MHz | 346 mm | 326–333 mm |
| 2.4 GHz | 62.5 mm | 59–60 mm |
Thicker wires have stronger end effects and shorter resonant lengths; printed antennas shrink further due to the substrate dielectric. Final length is set by computing input impedance with electromagnetic analysis and trimmed on prototypes.
How to approach the analysis
Methods and cautions
- Method of moments: efficient for wire antennas, solving for current on segmented wires; confirm the setup reproduces about 73 Ω for a half-wave dipole.
- Feed model: feed-gap size changes input impedance; keep it on the order of the wire diameter.
- Surroundings: enclosures, cables, the human body and ground change pattern and resonance greatly; analyse as installed.
- Baluns: dipoles are balanced; feeding directly with coax lets current flow on the outer conductor and distorts the pattern.
“The cable was an antenna too”
When measuring a dipole, input impedance and pattern change every time the cable moves — the classic symptom of current leaking onto the coax outer conductor, making the cable part of the antenna. A dipole is balanced, with symmetric currents, whereas coax is unbalanced with one conductor outside; connecting them directly upsets the balance. A balun or ferrite core near the feed point helps, and including the cable in the analysis reproduces the effect.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Cutting exactly half a wavelength | Resonance shifts low | Start at 0.47–0.48 λ and trim |
| Assuming longer means more gain | Beam splits | Check the pattern |
| Ignoring short-dipole losses | Efficiency overstated | Include matching losses |
| Direct coax feed | Cable radiates | Add a balun |
| Free-space analysis only | Changes once installed | Include enclosure and ground |
I'd like to try antenna calculations myself.
Try the antenna basics tool for basics, the radiation pattern simulator for radiation patterns and the antenna gain calculator for gain. Antennas developed from the dipole are covered in Yagi–Uda antennas, and related topics in impedance matching, radiation patterns and gain and directivity.
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