Designing Small Loop Antennas: Radiation Resistance, Gain, and Bandwidth Demystified

Why Small Loop Antennas Deserve a Closer Look

Small loop antennas — sometimes called magnetic loops — occupy an interesting niche in HF antenna design. They're compact, can be mounted indoors or in restricted spaces, and offer surprisingly good performance when designed carefully. The catch? Their radiation resistance is extremely low, which means conductor losses and tuning bandwidth become critical design parameters.

Unlike a dipole or a quarter-wave vertical, where the antenna dimensions are a significant fraction of a wavelength, a small loop has a circumference well under $\lambda / 10$. This makes the analysis tractable with closed-form equations, but it also means every milliohm of loss resistance matters. That's exactly why having a reliable calculator at hand is so valuable — open the Loop Antenna Calculator to follow along.

The Key Equations Behind the Calculator

For a circular loop of circumference $C$ operating at a frequency where $C \ll \lambda$, the radiation resistance is given by:

where $A = \pi (D/2)^2$ is the area of the loop with diameter $D$, and $\lambda$ is the free-space wavelength. This is often written equivalently as:

Notice the fourth-power dependence on the electrical size $C/\lambda$. Double the loop diameter at a fixed frequency and your radiation resistance goes up by a factor of 16. This is the fundamental reason small loops are inefficient — $R_r$ drops precipitously as the loop shrinks relative to wavelength.

The loss resistance $R_L$ comes primarily from the conductor's ohmic resistance, which depends on the skin depth $\delta$, the conductor circumference, and the conductor diameter $d$:

where $\sigma$ is the conductivity of the conductor material (for copper, $\sigma \approx 5.8 \times 10^7$ S/m). Larger conductor diameters reduce $R_L$ because current flows on a larger skin-depth "strip" around the tube's circumference.

The antenna gain relative to an isotropic radiator (for a small loop with a figure-eight pattern) is:

or in dBi:

The factor of 1.5 (1.76 dBi) is the directivity of a small loop — identical to a short dipole. Efficiency $\eta = R_r / (R_r + R_L)$ is what makes or breaks the design.

Worked Example: 1-Meter Loop on 14 MHz

Let's design a copper loop antenna for the 20-meter band (14 MHz) with a loop diameter of 1 meter and a conductor diameter of 22 mm (common copper tubing).

Step 1 — Wavelength and circumference:

The electrical size is $C/\lambda = 3.14 / 21.43 = 0.147$, which is just under the $0.1\lambda$ small-loop threshold but still in the regime where these approximations are reasonable.

Step 2 — Radiation resistance:

That's 92 milliohms — tiny, but not hopeless.

Step 3 — Loss resistance:

Skin depth of copper at 14 MHz: $\delta = \sqrt{1/(\pi f \mu_0 \sigma)} \approx 17.6\ \mu\text{m}$.

Step 4 — Efficiency and gain:

That's actually quite respectable for a compact antenna. The 22 mm copper tubing keeps the loss resistance well below $R_r$.

Step 5 — Bandwidth:

The $-3$ dB bandwidth of a tuned small loop is governed by the loaded Q. With a high-Q vacuum or air-spaced capacitor, the bandwidth is approximately:

For our loop, $L \approx \mu_0 (D/2)[\ln(8D/d) - 2] \approx 1.87\ \mu\text{H}$, giving:

This narrow bandwidth is characteristic of magnetic loops — you'll need to retune when moving more than about 10 kHz across the band. It's a trade-off for the compact size.

Design Trade-offs to Keep in Mind

Loop diameter vs. frequency: Moving down to 3.5 MHz (80 m) with the same 1-meter loop drops $R_r$ by a factor of roughly $(0.147/0.037)^4 \approx 256$. Efficiency collapses unless you scale the loop up significantly — typically 2–3 meters in diameter for 80 m operation. Conductor diameter matters a lot. Switching from 22 mm tubing to 2 mm wire roughly doubles $R_L$, cutting efficiency on lower bands where $R_r$ is already marginal. Always use the fattest conductor you can afford. Tuning capacitor losses are not captured in the basic model but can dominate in practice. A capacitor with an equivalent series resistance (ESR) of even 20 milliohms adds meaningfully to $R_L$. High-voltage vacuum variable capacitors are preferred for transmitting loops for exactly this reason. At higher frequencies (28 MHz and above), the same 1-meter loop becomes electrically larger ($C/\lambda \approx 0.29$), and radiation resistance climbs rapidly. Efficiency approaches 90%+ with good conductors, making small loops very practical on 10 meters.

Comparing Across the HF Bands

*Values for 1 m diameter loop, 22 mm copper conductor.*

The table tells the story clearly: a 1-meter loop is excellent on 10 m, good on 20 m, marginal on 40 m, and essentially unusable on 80 m without scaling up.

Try It

Plug in your own loop dimensions and target frequency to see exactly where you land on the efficiency curve. Open the Loop Antenna Calculator and experiment with different conductor sizes and loop diameters — it's the fastest way to find the sweet spot between size constraints and performance for your next magnetic loop build.