MagLoop Efficiency Myths: Why the 40/80 m Shortcut Fails
MagLoop Efficiency Myths: Why the 40/80 m Shortcut Fails
A percentage of a resonant wire-loop length can give you a dimension. It cannot tell you how much power a compact transmitting loop radiates—or how hard its conductor, joints and tuning capacitor must work.
RF.Guru working definition: Common-mode current is the non-cancelling phasor-sum current in a specified set of conductors, evaluated at a defined cross-section and using a declared current-direction convention. In the intended differential transmission-line mode, the outgoing and return currents are equal and opposite, so their phasor sum is zero. When they do not cancel, the remaining current must close through another reference or return path—such as the outside of a coax shield, a mast, equipment chassis, station wiring, nearby structures, earth, the operator, or distributed coupling through the environment.
This broader working definition is especially useful in practical antenna systems. On transmit, non-cancelling current on the outside of the coax can make the feedline and connected structures part of the radiating antenna system unless that path is intentional, clearly defined and properly controlled—for example by providing the required return path and placing a suitable common-mode choke at the correct boundary.
A “magic magnetic loop formula” circulating online says to take 10% of the full-wave loop length on 40 m, do the same for 80 m, and average the two. The result is then presented as a maximum-efficiency magnetic loop. It sounds tidy. But the arithmetic has not included conductor loss, capacitor loss or even the loop's enclosed area.
That is the claim I am challenging—not the usefulness of a small transmitting loop. An average can choose a convenient length. It cannot optimise a quantity that never entered the calculation. The useful question is what that length does on each band, and what construction is needed to make the compromise worthwhile.
What the Percentage-and-Average Rule Actually Calculates
The familiar 1005/fMHz estimate gives a full-wave wire-loop length in feet. It is a starting-length heuristic for a resonant wire antenna, not the resonance equation of a small loop tuned with a capacitor.
For example, using 7.0 and 3.5 MHz, taking 10% gives about 14.36 ft and 28.71 ft. Their arithmetic mean is about 21.54 ft—effectively the 21.55 ft figure under discussion, allowing for the chosen frequencies and rounding. Nothing in that calculation specifies conductor diameter, joint resistance, capacitor ESR, shape, height or a target efficiency.
Do not confuse two different uses of “10%.” Keeping a loop's circumference small compared with a wavelength—often expressed by a rough one-tenth-wavelength guideline—helps justify an approximately uniform-current model. That is a model-validity guideline, not a maximum-efficiency design rule. Averaging two such dimensions supplies neither an optimum nor equal performance on both bands.
Safety boundary: a tuned transmitting loop can develop high circulating current and high RF voltage. Keep the loop and capacitor inaccessible while energised, prevent accidental transmission during adjustment, and use remotely operated tuning and suitable guards or interlocks where required. Component voltage, current, temperature and spacing must be qualified for the actual waveform and duty cycle.
A Tuned Small Loop Is Not a Shrunken Full-Wave Loop
The antenna at issue is a one-turn tuned small transmitting loop: a closed conductor resonated by a capacitor and coupled to a feed line. A resonant full-wave wire loop, halo, multiturn inductor, shielded receiving loop and tuned small transmitting loop do not share one size or current model.
Let as be the radius of the smallest sphere that encloses the antenna and k = 2π/λ. When kas is well below one, the electrically small, approximately uniform-current model is useful. As the structure grows electrically, current magnitude and phase vary around the loop; full-wave analysis replaces the simple area formula.
Also declare the power boundary. If the matching network is inside the antenna boundary, its loss belongs in antenna efficiency. Feed-line loss and mismatch outside that boundary should be reported separately. Moving the reference plane changes which losses the result includes.
Radiation Resistance and the Complete Loss Budget
For a one-turn electrically small loop with approximately uniform current in free space, the area approximation is:
Rrad ≈ 31,200(A/λ²)² Ω
ηrad = Rrad / (Rrad + Rloss)
A is enclosed area and every resistance must be referred to the same loop-current basis. For fixed geometry inside the small-loop approximation, Rrad scales approximately with the fourth power of frequency. That strong scaling explains why loss that appears modest in ohms can dominate on a lower band.
| Loss term | What controls it | Evidence needed |
|---|---|---|
| Conductor loss | Conductivity, circumference, tube or wire diameter, skin effect, proximity effect and current crowding. | Exact geometry, material data, RF model or measurement, and hot-state check. |
| Joint loss | Contact area, surface condition, pressure, solder or braze geometry, oxidation and temperature. | Construction record, milliohm-sensitive RF evidence or thermal localisation. |
| Capacitor loss | ESR, plate and contact current, tuning mechanism, dielectric loss and field concentration. | Manufacturer RF data or component measurement at the intended frequency, current and voltage. |
| Matching loss | Coupling-loop conductor, transformer or network loss, connectors and unintended common mode. | Loss referred to the declared antenna port and loop-current reference. |
| Ground and environmental loss | Height, orientation, soil, wet material, walls, metal, wiring and nearby people or equipment. | Installed full-wave model and controlled field, gain, current or loss measurements. |
Environmental coupling can also change radiation resistance, pattern and current distribution, so it is not always representable by one series resistor. The lumped loss model is a design tool; the installed antenna remains the final test object.
What Happens to the Quoted 21.55 ft Loop?
To examine that particular claim, assume the 21.55 ft is the conductor centre-line circumference of a circular one-turn loop. That is 6.568 m around, about 2.091 m in diameter, enclosing 3.433 m². The circle and the following loss value are explicit assumptions, not measurements of an identified antenna.
Applying the leading-order area formula at 3.5 and 7.0 MHz gives this comparison:
| Frequency | Circumference / wavelength | Approximate Rrad | Illustrative efficiency with Rloss = 0.100 Ω |
|---|---|---|---|
| 3.5 MHz, 80 m band | 0.0767 | 0.00683 Ω | 6.40% |
| 7.0 MHz, 40 m band | 0.1534 | 0.109 Ω | 52.2% |
These are first-order estimates, not predicted performance figures for a finished loop. The 7 MHz circumference exceeds the conservative one-tenth-wavelength guideline, so the actual current distribution and radiation resistance need a more complete model. Holding loss at 0.100 Ω on both bands deliberately isolates the frequency effect; real conductor, capacitor, joint and environmental loss change with frequency.
Even this favourable simplification exposes the problem with the shortcut: the same averaged piece of conductor does not produce the same radiation resistance on the two bands. In the uniform-current model, doubling frequency multiplies radiation resistance by sixteen. It does not multiply efficiency by sixteen, because loss remains in the denominator.
Nor is 0.100 Ω necessarily achievable in a particular build. On 80 m the illustrated radiation resistance is only a few milliohms. A small additional joint or capacitor loss is therefore a major part of the power budget, not an insignificant rounding error. The average has chosen a size; it has not solved the low-band loss problem.
A Separate Worked Example Shows the Electrical Stress
The circumference argument tells us why the band results differ. A smaller, explicitly defined circuit example shows the next issue the shortcut leaves out: circulating current and capacitor voltage.
Illustrative model, not a construction rating: use a circular one-turn loop with 1.000 m diameter, 25 mm outside-diameter round tubing and a 7.10 MHz sinusoidal operating point. The loss entries below are declared hypothetical inputs, not claims for a material, capacitor, product or installation.
Using the exact speed of light, λ = 42.2243 m. The loop centre-line radius is 0.500 m, its circumference is 3.1416 m, and its area is 0.78540 m². Circumference is 0.07440λ; including the tube radius gives an enclosing-sphere radius of about 0.5125 m and kas = 0.07626. The small-loop area formula gives:
Rrad = 31,200(0.78540 / 42.2243²)² = 0.006055 Ω
Now declare this complete series loss budget, with every term referred to loop current:
| Illustrative term | Equivalent series resistance |
|---|---|
| Conductor, including skin/proximity allowance | 0.045 Ω |
| Joints and current-carrying hardware | 0.010 Ω |
| Tuning capacitor ESR and contacts | 0.015 Ω |
| Matching structure referred to loop current | 0.010 Ω |
| Effective ground/environment loss for this model | 0.020 Ω |
| Total Rloss | 0.100 Ω |
The modelled radiation efficiency is therefore:
ηrad = 0.006055 / (0.006055 + 0.100) = 0.0571, or 5.71%.
If 50.0 W of sinusoidal power is accepted at the declared antenna boundary, the total series resistance is 0.106055 Ω and the resonant loop current is:
Iloop,rms = √(50.0 / 0.106055) = 21.7 A
Prad = I²Rrad = 2.85 W
Ploss = I²Rloss = 47.15 W
The power split closes to 50.0 W apart from rounding. It also shows why accepted power, radiated power and transmitter output must not be used as synonyms.
Tuning, Q, Bandwidth and Capacitor Voltage
For the worked geometry, the high-frequency thin-circular-conductor approximation
L ≈ μ0a[ln(8a/r) − 2]
with loop radius a = 0.500 m and conductor radius r = 0.0125 m gives L ≈ 2.368 µH. The actual gap, joints, capacitor plates and nearby material will change this value and should be measured on the finished assembly.
At 7.10 MHz, the corresponding first-order values are:
| Quantity | Calculated value | Boundary |
|---|---|---|
Inductive reactance XL = 2πfL
|
105.6 Ω | Thin circular-conductor approximation. |
Tuning capacitance C = 1/[(2πf)²L]
|
212 pF | Ideal series resonance before stray capacitance. |
Intrinsic resonator Q ≈ XL/Rtotal
|
996 | Includes radiation and declared series loss, before added coupling load. |
Intrinsic half-power bandwidth f/Q
|
7.13 kHz | Series-resonator current response, not matched-port SWR bandwidth. |
Capacitor voltage VC,rms ≈ IXC
|
2.29 kV RMS | Sinusoidal steady-state lumped model. |
| Capacitor peak voltage | 3.24 kV peak |
√2 times RMS for the declared sinusoid. |
Coupling adds loading, so loaded Q and observed SWR bandwidth can differ substantially from the intrinsic values. Stray capacitance, non-uniform current and distributed voltage also limit the lumped model. The capacitor must be selected and tested for RF peak voltage, RMS current, ESR, spacing, corona, tuning contacts, temperature, waveform and duty cycle—not merely nominal capacitance or a DC voltage label.
Loss Changes Efficiency and Stress Together
Keeping the same geometry, frequency and 50 W accepted power while changing only the total loss produces this sensitivity result:
| Total Rloss | Efficiency | Loop current | Capacitor peak voltage | Intrinsic Q |
|---|---|---|---|---|
| 0.030 Ω | 16.8% | 37.2 A RMS | 5.56 kV | 2,930 |
| 0.100 Ω | 5.71% | 21.7 A RMS | 3.24 kV | 996 |
| 0.300 Ω | 1.98% | 12.8 A RMS | 1.91 kV | 345 |
Lower loss improves efficiency but raises circulating current, capacitor voltage and Q at the same accepted power. Higher loss broadens the response and reduces circulating stress by converting more accepted power to heat. Bandwidth alone is therefore not an efficiency measurement.
Area, Circumference and Conductor Geometry Do Different Jobs
Area enters the ideal radiation-resistance approximation directly. Circumference sets conductor length and electrical size. Conductor diameter affects RF resistance, inductance, proximity effect, voltage concentration and mechanical construction. Shape changes area for a given perimeter and can concentrate current at bends or joints.
Increasing a loop's size usually raises radiation resistance while the small-loop assumptions remain valid, but it also changes conductor loss, inductance, capacitance, current distribution, pattern and environmental coupling. Once ka and circumference are no longer small, use a full-wave model with the real conductor and installation rather than extending the uniform-current formula.
That is why I would start an 80/40 m design at the lowest band that must genuinely work. Use the available area well, reduce resistive loss in the conductor and current-carrying joints, and choose a tuning capacitor that can carry the current and withstand the RF voltage. Then check that the same assembly remains tunable and behaves as intended on the upper band.
For a fixed geometry that stays in the small-loop regime, 40 m normally has a much more favourable radiation-resistance budget than 80 m. In the simple model, efficiency improves when frequency doubles provided total loss rises by less than the sixteenfold radiation-resistance increase. That is a useful design tendency—not a guarantee of stronger signals on every path or a claim that the loop's mode and pattern never change.
Efficiency, Gain and Match Are Separate Results
Radiation efficiency is the fraction of accepted antenna power that is radiated instead of dissipated. Directivity describes how that radiated power is distributed by angle. Gain combines directivity and radiation efficiency; realised gain additionally includes mismatch at the declared port. Feed-line loss belongs outside or inside the result according to the stated reference plane.
G(θ,φ) = ηradD(θ,φ)
Grealised(θ,φ) = (1 − |Γ|²)ηradD(θ,φ)
A low SWR proves neither high radiation efficiency nor useful gain in a required direction. Ground and nearby structures can reshape the pattern while also changing loss. A field-strength comparison at equal accepted antenna power addresses gain in the measured direction; a realised-gain comparison also includes the declared port mismatch. Keep those power references, geometry and uncertainty consistent—not just the SWR readings.
Make the Chosen Compromise Work
- Record the geometry. Document loop area and circumference, conductor material and cross-section, gap, joints, capacitor, matching structure, feed line, control wiring, height and surroundings.
- Fix the reference planes. Calibrate at the antenna port and separately characterise feed-line and matching loss. Refer every equivalent resistance to the same current.
- Measure the resonator at low level. Save complex impedance, resonant frequency, loaded response and the coupling state. Use a method that separates unloaded and externally loaded Q.
- Build the loss budget. Combine conductor modelling or measurement, joint evidence, capacitor ESR, matching loss and installed environmental effects. Include uncertainty and hot-state drift.
- Measure radiation performance. Use calibrated gain comparison, pattern integration or another recognised efficiency method. Control distance, alignment, polarisation, reflections and instrumentation.
- Check common mode. Measure feed-line and control-cable current at several positions; an unintended radiator changes both pattern and the apparent loss boundary.
- Increase power in controlled steps. Monitor current, tuning, capacitor and joint temperatures, match and signs of discharge using instruments suitable for the RF environment.
- Repeat in the installed environment. Compare controlled configurations after moving the loop, changing height or adding nearby objects. Report repeatability and uncertainty.
A useful compromise is still useful. A compact loop may be the antenna that fits the site and makes the required contacts. Judge that choice by its actual low-band efficiency, useful pattern, tuning range and safe operating limits. It does not need a fictional optimum to justify its place in the station.
The Average Has Not Done the Engineering
The shortcut promises to optimise 40 and 80 m by averaging two lengths. It cannot: radiation resistance depends strongly on area and frequency, while real loss depends on the construction and installation. Those dependencies do not disappear because the resulting circumference looks precise.
Use a length rule as a convenient starting dimension if it helps. But for the lowest band, put the effort where the mechanism says it belongs: available area, conductor and joint loss, capacitor quality, current and voltage limits. Then verify that the upper-band behaviour remains useful.
That is my objection to the “magic formula.” Physics, not averages, determines magnetic-loop efficiency. The loop can be a good engineering compromise; the claim that the arithmetic has made it optimal is the part that fails.
References and Further Reading
- NIST Technical Note 1506, Appendix C: Small Loop Antenna: the area-dependent small-loop radiation-resistance expression, equation C3.
- NBS Circular 544, formulas for capacitance and inductance: circular-conductor inductance methods and their geometry limits.
- Rosa and Cohen, On the Self-Inductance of Circles: the distinction between volume current and high-frequency surface current; equation 6 gives the thin-ring basis for the inductance approximation.
- IEEE Open Journal of Antennas and Propagation, circular-loop analysis: current distribution, input impedance and the transition beyond the simplest small-loop model.
- IEEE 145-2025: current antenna terminology and definitions.
- IEEE 149-2021: antenna pattern, gain and test-range measurement practice.
- ITU-T K.91 (2024): gain, efficiency, accepted power and feeding-loss relationships.
- ARRL QEX small-loop ground-effects model record: installed height, ground and capacitor-position effects on loop gain.
Mini-FAQ
- Does averaging 10% of the 40 m and 80 m full-wave loop lengths maximise efficiency? No. It chooses a circumference without accounting for area, conductor and capacitor losses, current distribution or installation. A small-circumference guideline is a model approximation, not an efficiency optimum.
- What determines small transmitting-loop efficiency? Radiation resistance divided by radiation resistance plus the complete loss resistance, with every term referred to the same loop-current and port boundary.
- Does a larger loop automatically have higher efficiency? Not automatically. More area usually raises radiation resistance while the small-loop model applies, but conductor loss, current distribution, pattern and environmental coupling also change.
- Can low SWR prove that a loop is efficient? No. Matching controls the impedance presented at the feed port; it does not separate radiated power from conductor, joint, capacitor, matching or environmental loss.
- Can bandwidth be converted directly into efficiency? Only with a valid resonator and coupling model plus known radiation resistance. Loaded Q, matching and loss all affect the observed bandwidth.
- Why can a small loop have high capacitor voltage? Resonance produces circulating current, and capacitor voltage is approximately current multiplied by capacitive reactance. The actual peak also depends on waveform and distributed effects.
- How should loop gain be verified? Use a calibrated antenna-measurement method with defined reference planes, accepted power, distance, orientation, polarisation, reflections, pattern directions and uncertainty.