Measuring the Efficiency of a
QRP Small Loop Antenna
The efficiency of an antenna is an indicator of how well it radiates. It's the amount of radiated power divided by the power flowing into the antenna. Efficiency is lowered by resistive losses in the antenna, ground, and any conductors close enough to couple to the near fields.
This article describes two ways for amateurs to measure the efficiency of a QRP small transmitting loop.
The Resistance Method
In Reference 1, the authors use this definition of efficiency, based on the equivalent circuit for a loop:
Efficiency = RRAD / RTOT Eq. 1
RTOT is the feedpoint resistance measured at the loop, and RRAD is the radiation resistance. The authors explain how to use a small toroidal ferrite-cored transformer to measure RTOT, and they calculate RRAD using a formula for an electrically small loop:
RRAD = 31171 * (loop area)2 / (wavelength)4 Eq. 2
Length is measured in meters.
By the ARRL Antenna Book definition, an electrically small loop has a circumference less than a tenth of a wavelength. For a 1 meter diameter loop this corresponds to frequencies below 9.6 MHz. When a loop is electrically small, the amplitude of the current is nearly constant around the loop, so RRAD is much easier to calculate.
There is a formula for loop efficiency in the ARRL Antenna Book and online calculators that differs from Eq. 1:
[Design] Efficiency = RRAD / RCOND Eq. 3
RCOND is the resistive loss in the loop conductor. This definition of efficiency is always higher than the actual efficiency (Eq. 1), because Eq. 3 does not include resistive losses in the capacitor, ground, and nearby conductors, which are difficult to estimate accurately. 
The Bandwidth Method
In Reference 2, K9CHP and KE4PT explain how to use the quality factor (Q) of a loop to calculate the efficiency:
Efficiency = Q / QRAD Eq. 4
As shown below, you can calculate Q using the half-power VSWR bandwidth at resonance. QRAD is the Q of a theoretical loop in free space with no resistive loss in the loop structure or the capacitor. There is a formula for QRAD in Ref. 2, as well as a description of how to compute it using an NEC model.
The bandwidth method is not limited to electrically small loops.
Two QRP Loops
Loop 1: The dashed line in Figure 1 shows data from Ref. 1 for the efficiency of a 1 m (3.3 ft) diameter loop of 22 mm (0.87 inch) copper tubing. The authors use the resistance method to study loss mechanisms of small loops. They minimized resistive losses in the ground and capacitor by suspending the loop 6 m (19.7 ft) high and by tuning it with low loss porcelain fixed capacitors (resistance 2 - 20 milliohms).
Figure 1. Dashed Line: RRAD / RTOT for an electrically small copper loop (Ref. 1). Solid Line: Q / QRAD for an Alex Loop (Ref. 2)
Loop 2: The solid line in Figure 1 (above) shows the efficiency of an Alex Loop—a commercially produced 0.9 m (3 ft) diameter loop of RGC-213 coax, tuned with a small split-stator capacitor.  The measurements are from Ref. 2, where they are used to demonstrate the bandwidth method. For example, on 20 meters the measured half-power frequencies from the VSWR curve for the loaded loop are F1 = 14.188 MHz and F2 =14.146 Mhz. These give the loaded
Q = [SQRT( F1 * F2)] / ( F1 – F2) = 337, and QRAD = 1703, so the efficiency from Eq. 4 is Q / QRAD = 0.198 = -7.03 dB.
The loaded bandwidth (F1–F2 = 42 kHz) is twice the unloaded bandwidth measured with an antenna analyzer, because the total resistance of the loop plus an impedance-matched load is twice the resistance of an unloaded loop.
Figure 1 shows that the efficiency of small loops increases rapidly with frequency.
Two Higher Power Loops
Tests of the two loops in this section show that the bandwidth method does not apply to them at high frequencies.
Loop 3: Figure 2 shows Q / QRAD for a 3 ft. diameter loop of RG-8 coax, tuned with a butterfly capacitor. This antenna was built by Dr. Carol Milazzo, KP4MD. Construction details and test results are on her web site.  To calculate Q / QRAD, I divided KP4MD's measurements of unloaded Q by 2 to convert them to loaded Q, and calculated QRAD using the formula in Ref. 2.
Figure 2. KP4MD Loop Q/ QRAD and NEC efficiency limit.
The dashed line in Figure 2 shows efficiency computed using an EZNEC model of KP4MD's loop in free space.  (NEC Efficiency = RRAD / RTOT) Resistive losses in the ground and capacitor are not included in the model, so this limit is higher than the actual efficiency. The measurements shown in Figure 2 extend above the NEC limit, indicating that Q / QRAD is not a good measure of efficiency for this loop at high frequencies.
Loop 4: Figure 3 shows Q / QRAD for my antenna—a 3.3 ft. diameter loop of 5/8 inch diameter copper tubing, tuned with an MFJ-19 butterfly capacitor. It works on the 15, 17, and 20 meter bands. The figure also includes a measurement at 24.8 MHz, which is just below the 12 meter band. At high frequency the curve extends above the NEC limit to an impossible efficiency greater than 100%. A table of the measurements and calculations is on a separate page, as well as photos and details about the loop, and how I measured bandwidths.
Figure 3. AE7PD Loop Q / QRAD and NEC efficiency limit.
Figure 4 (below) shows that the Q / QRAD curves for both loops diverge from the efficiency curve of the Alex Loop as the frequency increases.
Figure 4. Q / QRAD and the Alex Loop efficiency curve (dashed line).
The next section explains why the Q/QRAD measurements for Loops 3 and 4 are too high at high frequencies.
Radiation by the Capacitor
Eq. 4 does not apply to a loop if the capacitor radiates a significant amount of power. Kai Siwiak, KE4PT, co-author of Ref. 2, explained in an email that the bandwidth method is based on a model with an infinitesimally small capacitor.
In a real loop the capacitor can be a radiating element—an electric dipole. In a typical QRP loop the capacitor is physically small, so the power it radiates is insignificant. KE4PT points out that the electric dipole moment of a physically big capacitor can cause the antenna to radiate better. In that case Eq. 4 does not apply.
Based on web site photos, the Alex Loop capacitor is smaller than the butterfly capacitors in Loops 3 and 4. The plates in KP4MD's capacitor are 1.5 inch in diameter. The voltage rating is 2 - 5 kV, and arcing occurs at 50 watts peak power, while the Alex loop is limited to 20 peak watts in SSB mode. The stack of plates in the MFJ-19 capacitor on my loop is bigger: 3.8 inches long and 2.5 inches in diameter, with a 5.9 kV peak voltage rating.
Also, the electric dipole moment of a capacitor depends on the distribution of charges on the plates, which is different in the “2-gang” capacitor of the Alex Loop.
Even a loop antenna with an infinitesimal capacitor has a small electric dipole moment.[6,7] Whether the capacitor is big or small, the electric dipole moment (and the amount of power radiated) is greater at higher frequencies, where the RF current is less uniform.
The slopes of the KP4MD and Alex Loop curves in Figure 4 are similar below 25 MHz, suggesting that radiation by the capacitor becomes significant above that frequency. The AE7PD loop curve diverges from the Alex Loop above 15 MHz, suggesting that the radiation from the biggest capacitor in these comparisons is significant above that frequency, so the bandwidth method does not apply to this loop.
The resistance method is limited to QRP loops for the same reason. The calculations of RRAD (Eq. 2) are for a loop with an infinitesimal capacitor.
QRP Loop Efficiency Calculator
The formula for QRAD is lengthy, so I used a spreadsheet to calculate efficiency. The inputs are the loop dimensions, resonant frequency, and unloaded bandwidth. Details are on a separate page. Two versions are available to download. One requires the LibreOffice Calc spreadsheet program (Windows, Mac OS X, and Linux), and the other runs in Excel.
Amateurs can measure the efficiency of a QRP loop using the resistance method or the bandwidth method. The resistance method works for electrically small loops (frequencies below 9.6 MHz for a 1 meter loop). The bandwidth method is not limited to electrically small loops.
Both methods are for loops with a physically small capacitor that does not radiate a significant amount of power.
1. B.A. Austin, et al, “Loss Mechanisms in the Electrically Small Loop Antenna” IEEE Antennas and Propagation Magazine. Vol. 56, No. 4. August 2014, 142-147.
2. A. Findling and K. Siwiak, “How Efficient is Your QRP Small Loop Antenna?” The QRP Quarterly. Summer 2012, 22-23.
4. www.qsl.net/kp4md/magloophf.htm and
6. A. Boswell, “Loop Antennas in the 3-30 MHz Band” Eighth International Conference on HF Radio Systems and Techniques, 2000. IEE Conf. Publication No. 474, 33-36.
7. K. Siwiak, "Ionospherica," The QRP Quarterly, Fall 2014, 24-24.
Thanks to Kai Siwiak, KE4PT, for his articles on small loops and for helpful answers to my questions. Thanks also to Dr. Carol Milazzo, KP4MD, for publishing her research.
Peter DeNeef, AE7PD, is an Extra Class amateur radio operator in the U.S. This Web site has no ads or conflicts of interest.
Email: HamRadioAndVision "at" gmail "dot' com.