Losses in a helically-wound small transmitting loop antenna

In 2014, K4HKX published measurements showing that a helically-wound small transmitting loop (using 3-inch wide by 0.012-inch thick copper strap wound 40 times around an octagonal support frame) had poorer performance, as evidenced by lower signal strengths on the Reverse Beacon Network, than equivalently-sized loops made of 3-inch diameter copper pipe and 1-inch diameter copper pipe. Specifically, the helically-wound loop's signals were 2 dB below those of the 3-inch pipe loop, and 1 dB below those of the 1-inch pipe loop. Measurements were made on the 40 meter band (7.148-7.150 MHz). See https://www.qrz.com/db/k4hkx for details.

K4HKX's results established without doubt that in this context, a 3-inch strap, when helically wound, is worse than 3-inch diameter pipe or 1-inch diameter pipe. The purpose of this post is to further show, based on further analyses of K4HKX's data, that a helically-wound 3-inch strap is worse than a non-helically-wound 3-inch strap. All measurements below are assumed to have been performed at 7.148 MHz (the K4HKX page mentions both 7.15 MHz in the text and 7.148 MHz on the EZNEC graph).

1. K4HKX measured that an 18-foot perimeter octagon with 3-inch diameter pipe (his so-called "FatLoop") yields signal strength 6 dB (median value) below that of a dipole (at 70 feet). See https://www.qrz.com/db/k4hkx, "Section 7:  Magnetic Loop Comparisons vs My Reference 40M Dipole".
   
2. K4HKX simulated an 18-foot perimeter octagon with 3-inch diameter pipe, which yielded signal strength 6 dB below that of a dipole (at 70 feet) at 45 degrees above the horizon. See the EZNEC graph at https://www.qrz.com/db/k4hkx, "Section 6:  Magloop Version VI:  40 Meter FatLoop".
   
   I could reproduce these simulation results in 4nec2. For the dipole, I modeled an inverted-V using 0.3-mm diameter copper wire, with the antenna apex at 21.336 meters (70 feet), and with arms sloping down at 23 degrees, simulated above real ground with ground conditions specified as "Rocky, steep hills". The resulting far-field graph for the dipole matches very closely the K4HKX simulation, with a maximum gain of about 7.11 dBi (compared with K4HKX's simulated maximum gain of 7.08 dBi), and minimum gain about 7.2 dB below the maximum gain (about the same as K4HKX's simulated minimum gain). The gain of the dipole at 45 degrees above the horizon is -0.77 dB. We will use this value for comparison later.
   
   
   
   
   
   
   For the loop antenna, I used the same ground conditions, used a 3-inch diameter copper conductor, formed an octagon with eight 27-inch segments, and added 15 milliohms of loss for the tuning capacitor, located at the bottom of the loop. The bottom of the loop was 8 feet above the ground, corresponding to K4HKX's report of his original loop height of 8 feet. Again, the resulting graph closely matches K4HKX's graph. Compare the below graph, which overlays my simulated dipole and fat loop, with K4HKX's EZNEC graph linked above (direct link: https://cdn-bio.qrz.com/x/k4hkx/20140926_FatLoop_vs_Dipole.jpg). My simulated loop gains at 15, 30, 45, and 60 degrees above the horizon are 7.11 dB, 7.81 dB, 6.19 dB, and 3.28 dB below the dipole's gain, compared to K4HKX's simulated results of 7 dB, 8 dB, 6 dB, and 2.5 dB below the dipole's gain. So my simulation setup is reasonably close to K4HKX's simulation setup. At 45 degrees, the loop gain of -6.96 dB is 6.19 dB below the dipole gain at 45 degrees of -0.77 dB; the 6.19 dB poorer performance of the loop matches K4HKX's calculated and simulated FatLoop performance of 6 dB below that of the dipole.
   
   
   
   
   
   
   
3. Next, let us consider helical winding. K4HKX measured that an 18-foot perimeter octagon with 3-inch wide x 0.012-inch thick strap wound helically with 40 turns yields signal strength 8 dB (median value) below that of a dipole (at 70 feet height). See https://www.qrz.com/db/k4hkx, "Section 1:  Helically Wound Magloop", which includes a photograph of the helical loop.
4. I modeled an equivalently-performing non-helical loop (which I call the "equivalent-helical" loop below) in 4nec2 as follows. An 18-foot perimeter octagon with 0.43-inch diameter pipe (routed non-helically in the shortest direct path around the octagon's perimeter) yields signal strength 8.23 dB below that of a dipole (at 70 feet height) at 45 degrees above the horizon. This simulated performance is essentially identical to the K4HKX-measured performance of the helically-wound octagon (8 dB below the dipole, and 2 dB below the FatLoop). Therefore, we can say that the helically-wound octagon constructed with 3-inch wide strap (0.012 inch thickness) has the same performance as an identically-sized and non-helically-wound octagon constructed with 0.43-inch diameter pipe.
   
   The simulation results for the equivalent-helical octagonal loop (constructed with a non-helically-wound 0.43-inch diameter pipe) are shown below. The gain at 45 degrees above the horizon of the equivalent-helical loop, shown in blue, is -9 dB.
   
   
   
   
   
   Recall that the dipole had gain of -0.77 dB at 45 degrees above the horizon.
   
   
   
   
   Therefore, the equivalent-helical loop has gain at 45 degrees that is 8.23 dB below that of the dipole at 70 feet, which corresponds with K4HKX's measured result that the helically-wound loop had signal strength 8 dB below that of the dipole at 70 feet.
   
   Also, the equivalent-helical loop has gain at 45 degrees (with a gain value of -9 dB relative to the dipole) that is 2.04 dB below that of the 3-inch pipe FatLoop (which has a gain value of -6.96 dB relative to the dipole, as shown above). Again, this corresponds to the actual K4HKX measurement that the helically-wound loop had signal strength 2 dB below that of the 3-inch pipe FatLoop.
   
5. Having now established that the loop constructed with helically-wound 3-inch strap (0.012 inch thickness) has equivalent performance to a non-helically-wound loop constructed with 0.43-inch diameter pipe, next we can extrapolate a similar result for a non-helically-wound strap. Using W9CF's formulas for the RF resistance of rectangular cross-section conductors, we can calculate that a 0.43-inch diameter pipe with round cross-section has equivalent RF resistance to a 1.44-inch wide strap having rectangular cross section and 0.012-inch thickness. If we denote the strap's width as w=1.44 inches and its thickness as t=0.012 inches, then the equivalent diameter is found by the following W9CF formula, which yields a 0.43-inch diameter for a round-cross-section pipe with equivalent RF resistance.
   
   
   
   
   
6. From #5, we can say that an 18-foot perimeter octagon with 0.43-inch diameter pipe has equivalent RF resistance to an 18-foot perimeter octagon with 1.44-inch wide strap (0.012-inch thickness) routed non-helically in the shortest direct path around the octagon's perimeter.
7. From #6 we can say that an 18-foot perimeter octagon with 1.44-inch wide strap (0.012-inch thickness), routed non-helically in the shortest direct path around the octagon's perimeter, will achieve the same signal strength as the same-sized loop constructed with 0.43-inch diameter pipe, which from #4 we computed to be 8.23 dB below that of a dipole (at 70 feet height) at 45 degrees above the horizon.
8. From #7 and #3, we can say that the K4HKX 18-foot perimeter octagonal loop with 3-inch strap wound in a long helical path has the same performance as an identically-sized loop constructed with 1.44-inch wide strap that is routed in a shorter and non-helical path straight around the loop's perimeter. This is the first important result.
   
9. Next, consider K4HKX's helically-wound loop using 3-inch wide (0.012 inch thick) strap, and let us see what happens if we undo the long and helical winding of the 3-inch strap, and instead route the same 3-inch strap in a shorter, straight, and direct path around the octagon's perimeter. Again using W9CF's formulas for the RF resistance of rectangular cross-section conductors, we can compute that a 3-inch wide strap (with 0.012-inch thickness) has equivalent RF resistance to a 0.809-inch diameter pipe.
10. Adjusting the conductor diameter in the simulation from #4, we find that an 18-foot perimeter octagon with 0.809-inch diameter pipe (routed non-helically in a straight path around the octagon's perimeter) yields signal strength 7.24 dB below a dipole (at 70 feet) at 45 degrees above the horizon.
    
    The image below shows the gain of the octagonal loop constructed with 0.809-inch diameter pipe, equivalent to the non-helically-wound 3-inch strap, in blue. The gain at 45 degrees above the horizon is -8.01 dB.
    
    
    
    
    
    Again, recall that the dipole had gain of -0.77 dB at 45 degrees above the horizon.
    
    
    
    
    Therefore, the non-helically-wound strap loop has gain at 45 degrees that is 7.24 dB below that of the dipole at 70 feet. The loop constructed with a non-helically wound strap has higher gain (having simulated gain 7.24 dB below that of the dipole) than the loop constructed with a helically wound strap (having measured gain 8 dB below that of the dipole from K4HKX's report). This is the second important result.
    

To summarize the results:

• From #3 and #8 we see that helically winding the 3-inch strap, in a long and helical path around the support tubes of the loop perimeter, has reduced its performance to be on par with that of a narrower 1.44-inch strap routed non-helically in the shortest direct path around the octagon's perimeter.
• From #3 and #10 we see that removing the helical winding and routing the same 3-inch strap in a shorter and non-helical direct path around the loop perimeter will increase signal strength by about 1 dB.

In conclusion, there is no performance benefit to be gained by helically winding the conductor in a small transmitting loop antenna; on the contrary, this technique incurs additional resistive loss. This has been stated over and over again since the idea was first introduced 2011, but unfortunately many persons still mistakenly believe that there is some performance benefit to be gained by helically winding the conductor of a small transmitting loop. There is no such performance benefit. 

For the best performance, it is better to run the loop conductor in the shortest direct and non-helical path around the loop's perimeter, minimizing the conductor length while maximizing the loop's enclosed area. 

As with all engineering, the antenna designer must decide whether or not the performance loss caused by helical winding is outweighed by other factors such as ease of construction or mechanical sturdiness.