Small Transmitting Loops: Efficiency, Current and Capacitor Voltage
Small Transmitting Loops: Efficiency, Current and Capacitor Voltage
A compact tuned loop is a high-current resonator. Its performance follows from radiation resistance, every loss at a common reference, the matching boundary, the installation and the resulting current, voltage, bandwidth and radiation pattern.
There is no universal efficient circumference. Area and wavelength set the ideal small-loop radiation resistance; conductor geometry, joints, capacitor, matching network and surroundings set the competing loss. A credible result names all of them and states where accepted power is measured.
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.
Define the Antenna and Its Reference Plane
This guide concerns 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.
Worked Example: A 1.0 m Loop at 7.10 MHz
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.
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. Compare loops by calibrated realised gain or field strength in defined directions, with equal accepted power, stable geometry and measurement uncertainty—not by match alone.
How to Qualify a Built Loop
- 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.
Decision rule: accept a loop design only when its measured efficiency or realised gain meets the mission, its bandwidth and tuning behaviour are usable, and every current, voltage, temperature, mechanical and installation limit remains satisfied under the worst credible operating state.
Primary Sources Checked
- NIST electrically small antenna treatment: small-loop area, radiation resistance and model assumptions.
- NBS Circular 544, formulas for capacitance and inductance: circular-conductor inductance methods and their geometry limits.
- 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
- 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.