Small Loop Antennas: Resistance, Gain & Bandwidth

Why Small Loop Antennas Are Worth Your Time

Small loop antennas — you'll also hear them called magnetic loops — sit in this really interesting spot in HF antenna design. They're compact enough to mount indoors or squeeze into a balcony, and when you get the design right, they actually perform surprisingly well. But here's the thing: their radiation resistance is absurdly low. We're talking milliohms. That means every bit of conductor loss and the tuning bandwidth become absolutely critical to whether your antenna works or just heats up your shack.

Think about a dipole or quarter-wave vertical for a second. Those antennas have dimensions that are a decent chunk of a wavelength. A small loop? Its circumference sits well under $\lambda / 10$. The upside is that the math becomes manageable — we can use closed-form equations instead of reaching for numerical solvers. The downside is that every single milliohm of resistance in your conductor matters. A lot.

This is exactly why I built a calculator for this stuff — trying to optimize these things by hand gets tedious fast. Open the Loop Antenna Calculator if you want to follow along with actual numbers.

The Math That Makes It Work

For a circular loop with circumference $C$, operating at a frequency where the loop is electrically small ($C \ll \lambda$), the radiation resistance shakes out to:

Here $A = \pi (D/2)^2$ is just the loop area for a diameter $D$, and $\lambda$ is the free-space wavelength. You'll sometimes see this written another way:

Notice that fourth power on the electrical size ratio $C/\lambda$. This is brutal. If you double your loop diameter while staying at the same frequency, your radiation resistance jumps by a factor of 16. This is fundamentally why small loops fight an uphill battle — as the loop shrinks relative to wavelength, $R_r$ absolutely plummets.

Now for the loss resistance $R_L$. This comes mostly from the ohmic resistance of your conductor, which depends on skin depth $\delta$, how long your conductor is, and the conductor diameter $d$:

where $\sigma$ is the conductivity of whatever metal you're using. For copper, that's around $5.8 \times 10^7$ S/m. Bigger conductor diameter helps you here because the RF current spreads out over a larger circumference at the skin depth, reducing resistance.

The antenna gain relative to an isotropic radiator works out to:

or if you want it in dBi:

That 1.5 factor (1.76 dBi) is the directivity of a small loop — it's actually identical to a short dipole's pattern. The efficiency $\eta = R_r / (R_r + R_L)$ is what really determines whether your design is going to work or just warm up the copper.

Let's Build a Real One: 1-Meter Loop on 20 Meters

I'll walk through designing a copper loop for the 20-meter band (14 MHz). Let's say we're using a 1-meter diameter loop and 22 mm copper tubing — that's pretty standard stuff you can get at a hardware store.

Step 1 — Figure out the wavelength and circumference:

So our electrical size is $C/\lambda = 3.14 / 21.43 = 0.147$. That's just under the usual $0.1\lambda$ threshold people quote for "small" loops, but we're still in the ballpark where these approximations hold up pretty well.

Step 2 — Calculate radiation resistance:

So we're looking at 92 milliohms. That's tiny, but it's not game over — we can work with this.

Step 3 — Now for the loss resistance:

The skin depth of copper at 14 MHz works out to about $\delta \approx 17.6\ \mu\text{m}$.

That's 36 milliohms of loss. Not great, but manageable.

Step 4 — What's our efficiency and gain?

Honestly? That's pretty respectable for a compact antenna that fits in a square meter. The 22 mm tubing is doing its job — it keeps the loss resistance well below the radiation resistance, which is exactly what you want.

Step 5 — What about bandwidth?

Here's where small loops get annoying. The $-3$ dB bandwidth of a tuned small loop depends on the loaded Q. If you're using a high-Q vacuum or air-spaced variable capacitor (and you should be), the bandwidth approximates to:

For this loop, the inductance is roughly $L \approx \mu_0 (D/2)[\ln(8D/d) - 2] \approx 1.87\ \mu\text{H}$, which gives us:

Yeah, about 11 kHz of usable bandwidth. That's the classic magnetic loop characteristic — narrow as hell. Move more than 10 kHz or so across the band and you'll need to retune. It's the price you pay for cramming an HF antenna into a meter-wide circle.

The Trade-offs You Need to Think About

Loop diameter versus frequency is everything. Take that same 1-meter loop down to 3.5 MHz (80 meters). The radiation resistance drops by roughly $(0.147/0.037)^4 \approx 256$ times. Your efficiency just fell off a cliff. On 80 meters, you'd typically need a loop diameter of 2–3 meters minimum to get anything approaching reasonable performance. Most hams skip 80m loops for exactly this reason. Conductor diameter is not optional. If you swap out that 22 mm tubing for 2 mm wire — maybe because it's cheaper or easier to work with — you roughly double your loss resistance. On the lower bands where $R_r$ is already marginal, that kills you. Always use the fattest conductor you can reasonably afford and mount. I've seen people try to cheap out here and regret it later. Your tuning capacitor can ruin everything. The basic equations I showed you don't account for capacitor losses, but in the real world, they can dominate. Even a seemingly small equivalent series resistance (ESR) of 20 milliohms adds meaningfully to $R_L$ when your total resistance budget is measured in milliohms. This is why serious transmitting loops use high-voltage vacuum variable capacitors — the ESR is negligible compared to cheaper alternatives. Higher frequencies change the game completely. Move that same 1-meter loop up to 28 MHz (10 meters). Now your electrical size is $C/\lambda \approx 0.29$, and the radiation resistance climbs rapidly. With decent conductors, you can hit 90%+ efficiency. Small loops are actually really practical on 10 meters — they work great.

How This Plays Out Across HF

Here's what happens to that 1-meter loop with 22 mm copper conductor as you move across the HF bands:

The story here is crystal clear: this loop is excellent on 10 meters, quite good on 20 meters, barely functional on 40 meters, and effectively useless on 80 meters unless you scale everything up significantly. The physics just doesn't give you much choice.

Go Experiment With Your Own Design

The best way to figure out what works for your situation is to plug in your actual constraints — how big can you make it, what conductor can you get, what frequency do you care about most. Open the Loop Antenna Calculator and start playing with different combinations. Try varying the conductor diameter and watch how dramatically it affects efficiency on the lower bands. It's the fastest way to find that sweet spot between "fits in my space" and "actually radiates RF instead of just heating up." Most people are surprised by how much the conductor diameter matters once they see the numbers.