# Moving to a multi-turn loop

I have been hesitant to implement a multi-turn transmitting loop due to the reported measurements and experience of W8JI, who worked on the commercially-available MFJ loop antenna. His comments can be found here: https://forums.qrz.com/index.php?threads/magnetic-loops.241400/page-3#post-1961567

However, when evaluating such assertions as "multi-turn loops increase loss", it is always important to consider the assumptions underlying those assertions, and whether those assumptions hold for a specific case.

In my case, a multi-turn loop is beginning to look like the best option. My assumptions are as follows:

- The loop antenna should be 1 meter in diameter and should operate at 7 MHz.
- The loop antenna should be feasible for me to construct physically. For the loop conductor, thick copper tube (10 cm diameter) is not available for purchase locally. Even if it were available, I lack the equipment to solder or braze these tubes together.
- The loop antenna should be as lightweight as possible to enable easy setup and dismantling, and possible transport for field use. This means that the loop antenna should not be made of heavy, rigid materials. This also implies that the loop antenna may be subject to small shape deformations over time due to transport and repeated setup and dismantling.
- Transmitter power will be 5 watts.

Until now, the strategy for this project was to use a single-turn loop, and to construct a thick copper tube piecewise, by: (1)
changing the tube cross section from circular to square; (2)
conceptually separating the four sides of the square-cross-section tube with a small air gap,
yielding four flat copper sheets; (3) purchasing flat copper sheets and
gluing them onto the surface of a square-cross-section plastic former;
and (4) electrically joining several such tubes together to form a large loop (an octagon, or a square), by
soldering the flat copper sheets together.

However, as described in the last post (http://qrp-gaijin.blogspot.com/2022/02/a-1-meter-diameter-small-transmitting.html),
small deformations in the cross-sectional shape of a 10-cm-diameter
tube can cause unacceptably high losses in a single-turn, 1-meter-diameter loop at 7
MHz. Hand-assembly of such a copper tube will not yield a perfectly-shaped cross-section, and will contain small distortions, which may increase over time. Removing such distortions, and preventing distortions over time, would require more rigid and heavy materials, and more thorough and solid construction techniques, all of which go against the project assumptions for my case.

# Overcoming objections to a multi-turn loop

An alternative is to construct a multi-turn loop. W8JI's objections to a multi-turn loop are as follows (https://forums.qrz.com/index.php?threads/magnetic-loops.241400/page-3#post-1961567):

Three things kill the small loop's efficiency when it is multiple turns. The first is turn-to-turn capacitance. The turn-to-turn capacitance adds circulating currents across each turn of the loop, and this seriously de-Q's the system by increasing I^2 R losses.

The second problem was air. In order to not arc, the turns needed spaced. This means for a given volume of area more of it is useless air. The same amount of material in a thin single tube occupying the same area would have much more surface area and less resistance.

The final problem was current bunching or pushing, just like occurs in an inductor or even in a single conductor (skin effect). Current tends to push out from the most concentrated area of magnetic fields, and a larger amount of current flows in a smaller area of surface. There was a lot less of this effect in a round tube compared to a flat strip (like the AEA loop), and multiple turns made it act more like a flat strip.

Examining these objections in turn:

- Yes, coil self-capacitance will cause circulating currents "across" the turns instead of "along" the turns as desired. It is open for debate whether or not the concept of "turn-to-turn capacitance" is actually valid, because attempts to explain coil self-capacitance that are based on "turn-to-turn capacitance" generally fail (see e.g. https://www.g3ynh.info/zdocs/magnetics/appendix/self-res.html, which asserts that "
*The inter-turn capacitance approach is found to have no predictive power*").

If some form of inter-turn capacitance actually exists and can be explained by "capacitors" formed by opposing faces of adjacent turns, then this capacitance can likely be reduced by (1) increasing the turn spacing, hence increasing the distance between the "plates" of each "inter-turn capacitor", and hence reducing the "inter-turn capacitance"; and (2) using a thin strip conductor to minimize the conductor surface area between turns, so that any "capacitor plates" formed by adjacent turns have very small surface area (along the thin edge of the strip conductor).

Conclusion: This concern is valid, and can be mitigated. - Yes, it is true that turns must be spaced, introducing "useless air" into the volume of the loop. It is also true that "The same amount of material in a thin single tube occupying the same
area would have much more surface area and less resistance."

However, as described above, when the tube diameter approaches 10 cm, then avoiding deformations become critical to ensure low RF resistance along the tube. A perfect 10-cm-diameter single tube would be ideal, but is difficult to obtain and more difficult to construct.

While useless air does increase the total volume required for the loop antenna, it is not a critical requirement in my case. For a commercially-sold antenna like the MFJ loop, which must be shipped to customers via post, it is understandable that minimising volume was a concern.

Furthermore, the RF resistance -- including skin effect and proximity effect -- can be computed and analyzed by Finite-Element Methods, using the FEMM software. It is not necessary to rely on general statements about conductor geometry and resistance; it is instead possible, and much better, to compute and visualise the current distribution and the RF resistance, to evaluate a specific conductor configuration and also to distort the shape of that conductor configuration to test the effects of the distortion.

Conclusion: Useless space caused by multiple turns can be tolerated. The specific RF resistance can be computed in FEMM. - Yes, multiple turns will cause a skin-effect-like proximity effect; the current in the turns will not be exactly equal.

But we can easily compute the actual current distribution in either FEMM (which accounts only for magnetic effects, like proximity effects and skin effect) or in NEC-based Method-of-Moments simulators (which account for full electromagnetic field effects between wires, including both electric-field and magnetic-field effects, but which does not include proximity effects and skin effects on the surface of a single conductor).

Conclusion: The actual distribution of RF current can be computed, and its effect on efficiency evaluated, in FEMM or NEC.

# Simulation results of RF resistance of multiple turns in series

A future post may examine the simulations in detail, but here are some preliminary impressions:

- Analytically, a square-shaped loop antenna with 4 turns (in series) has a radiation resistance at 7 MHz of 148 milliohms. For an intuitive explanation of why series turns increase the radiation resistance, see https://owenduffy.net/blog/?p=16986 -- each turn carries the same current (since all are in series), and because the turns are physically arranged to drive this same current around the perimeter of the antenna, the result is increased current flowing around the loop antenna's perimeter, and hence increased radiation. Turns in parallel do not (contrary to some mistaken information online) cause a similar increase in radiation resistance due to the available current dividing simply itself among the turns, leading to no increase in current around the loop; again, see https://owenduffy.net/blog/?p=16986 for details.
- 4nec2 simulations of a 4-turn square loop with 1-meter sides yield a radiation resistance of 170 milliohms. This is in reasonable agreement with the analytical value. This was measured as the real part of feedpoint impedance at the highest-current point in the antenna, in free space, with wire loss set to zero, so that the only resistive component of the feedpoint impedance is the radiation resistance. The reactive part of the feedpoint impedance was zero, since the antenna was adjusted with a capacitance to be resonant at 7 MHz.

Due to the potential for calculation inaccuracies of the NEC-2 engine with small loops, the simulation was re-run with a double-precision port (by VK3DIP; see http://www.yagicad.com/Projects/mininec.htm) of the MININEC simulator, which uses a completely different implementation of the Method-of-Moments simulation and which therefore provides a valuable second opinion for sanity-checking the NEC-2 results. MININEC is also reported to be able to measure much smaller loops with high accuracy; the report at https://archive.org/details/DTIC_ADA181682, p. 51, Section 3.2 asserts that MININEC with single-precision floating point calculations can accurately model loops as small as 0.01 wavelengths in circumference -- which is much smaller than loop under investigation, which is about 0.10 wavelengths in circumference. The MININEC results, run with shorter segment lengths (46 segments per meter, or about 2.17 cm per segment), yielded a radiation resistance of 167.78 milliohms (measured at the feedpoint which was located at the wire segment with maximum current), which agrees well with the NEC-2 reported result of 170 milliiohms. The following is an excerpt of the MININEC output, run in free space with no resistive losses.******************** SOURCE DATA ********************

PULSE 359 VOLTAGE = (1,0J)

CURRENT = (5.960077 , -0.000552131 J)

IMPEDANCE = (0.1677831 , 1.554313E-05 J)

POWER = 2.980038 WATTS

---------- POWER BUDGET ---------

TOTAL POWER = 2.980038 WATTS

RADIATED POWER= 2.980038 WATTS

TOTAL LOSS= 0 WATTS

EFFICIENCY = 100.000 %

The following is another excerpt from the MININEC output, showing that the feedpoint location (pulse 359) is the segment of maximum-current, which is where the radiation resistance should be measured as the real part of the feedpoint impedance.WIRE NO. 9:

PULSE REAL IMAGINARY MAGNITUDE PHASE

NO. (AMPS) (AMPS) (AMPS) (DEGREES)

J 5.945504 -0.0007043754 5.945504 -0.006787942

345 5.94675 -0.0006978156 5.94675 -0.006723317

346 5.94794 -0.0006914593 5.94794 -0.006660743

347 5.949186 -0.0006847983 5.949186 -0.006595197

348 5.950457 -0.0006779634 5.950457 -0.006527976

349 5.951738 -0.0006709474 5.951738 -0.006459031

350 5.953006 -0.0006637438 5.953006 -0.006388322

351 5.954234 -0.000656331 5.954234 -0.006315673

352 5.955411 -0.0006486758 5.955411 -0.006240776

353 5.95651 -0.0006407233 5.95651 -0.006163129

354 5.957508 -0.0006323904 5.957508 -0.006081956

355 5.95838 -0.0006235364 5.95838 -0.005995925

356 5.959102 -0.0006139791 5.959102 -0.005903307

357 5.959646 -0.0006028565 5.959646 -0.005795837

358 5.959982 -0.0005922992 5.959982 -0.005694018

359 5.960077 -0.000552131 5.960077 -0.00530778

360 5.95989 -0.0005934233 5.95989 -0.005704913

361 5.959375 -0.0006051083 5.959375 -0.00581775

362 5.958478 -0.0006173661 5.958478 -0.005936494

363 5.957143 -0.0006280719 5.957143 -0.006040793

364 5.95532 -0.0006380953 5.95532 -0.006139076

365 5.953004 -0.0006476296 5.953004 -0.00623323

366 5.950294 -0.0006568312 5.950294 -0.006324671

367 5.947456 -0.0006658013 5.947456 -0.006414105

368 5.944841 -0.0006746035 5.944841 -0.006501761 - FEMM simulations of the RF resistance of 4 turns in series, with 1 meter diameter, yielded 130 milliohms for a perfectly-aligned series of turns, or 137 milliohms for a slightly-misaligned (by a few millimeters) series of turns.

The following image shows a conceptual diagram (not to scale) of the perfectly-aligned loop geometry. Four exactly identical loops are placed exactly above one another at exactly the same spacing, and the RF resistance is computed assuming the turns are in series.

The following image shows the computed RF resistance and the flux density for a perfectly-aligned, 4-turn loop (four turns of copper in series, each 0.1mm thick and 100mm wide, placed exactly vertically above one another, with 15mm spacing between turns).

The following image shows a conceptual diagram (not to scale) of the misaligned loop geometry. Four loops with slightly differing tilted cross-sectional profiles are approximately above one another at approximately the same spacing, and the RF resistance is computed assuming the turns are in series. This misalignment can capture some of the effects of distorted turns that are not exactly spaced and not exactly in line with one another. (Due to restrictions in the FEMM software, it is not possible to simulate distortions that alter the tilt within one turn, because FEMM only allows definition of a single cross-sectional profile for each turn, that is then uniformly rotated about the vertical axis.)

The following image shows the computed RF resistance and the flux density for misaligned conductors. The top and bottom ends of each conductor's cross-sectional profile have been slightly displaced in the x direction by a few millimeters, so that the turns are not exactly aligned with one another.

This simulation assumes a round loop shape -- a circle of 1 meter diameter, not a square with 1-meter sides -- and so a square loop shape will have slightly higher conductor resistance. This could be approximated by making the round loop diameter slightly larger in FEMM, so that the total conductor length of the 4 round loops corresponds to the total conductor length of 4 square loops. - From the above point 3, we see that small distortions in the geometry of the multi-turn loop do cause variations in RF resistance of several milliohms. However, unlike the single-turn loop, the multi-turn loop has higher radiation resistance. Therefore, the effect of several added milliohms of Rloss is insignificant. For example, using the figures above, the perfectly-aligned loop has an efficiency of Rrad/(Rrad+Rloss) = 148/(148+130) = ~53%, and the misaligned loop has an efficiency of 148/(148+137) = ~52%.

As stated above, the Rloss is actually higher due to the longer conductor path required for a square loop perimeter, but nevertheless it can be seen that the higher radiation resistance of the multi-turn loop makes it much more forgiving of milliohm-level increases in loss resistance caused by distortions or misalignments of the loop geometry.

The price we pay for this higher radiation resistance is the increased volume of the multi-turn antenna, needed due to the inter-turn spacing.

A rough sketch of how the multi-turn loop might be implemented looks as follows.

A tuning capacitor will be connected between the open ends of the loop. 4nec2 simulations of a similar loop indicate that, depending on the turn spacing and conductor diameter, the loop may become self-resonant below 7 MHz, which would make operation at or above 7 MHz impossible. 4nec2 simulations (using the NEC-2 engine) can only serve as a guideline here, since NEC-2 only models round conductors, but the actual antenna will use a flat strip conductor. The actual self-resonance of the structure as constructed with a flat strip will need to be measured in practice.

# Spiral winding: a possible (but lossy) alternative

An alternative to winding the turns in a solenoidal coil is to wind the turns into a flat spiral coil, with all the turns lying in a plane and with later turns being nested inside of earlier turns. This has the advantage of reducing the physical volume required for the antenna. The disadvantage is increased losses due to proximity effect and possibly due to inter-turn capacitance.

The current distribution and RF resistance of spiral-wound (concentrically-wound) turns can be simulated in FEMM by modeling each turn to lie inside of the previous turns. Conceptually, this would look like the following diagram, where 4 strip conductors are placed concentrically.

FEMM can then simulate the RF resistance, assuming the turns are connected in series. The result is an RF resistance of 250.57 milliohms -- almost twice as high as the 130-137 milliohms of the solenoidally-wound loop. Note that the simulation shown below uses perfectly-aligned conductors. No test was performed with misaligned conductors.

The reason for this high RF resistance is the proximity effect: with turns placed in proximity and carrying identical current flowing in the same direction, the current flows influence each other and cause redistribution of current, under-utilization of the conductor, and an increase in the RF resistance.

Furthermore, if we accept that the opposing faces of adjacent turns form "capacitors" that contribute to "inter-turn capacitance", then the spiral-wound configuration -- in stark contrast to the solenoidally-wound configuration -- has very large surface areas from adjacent turns that face one another, potentially increasing the "inter-turn capacitance" (if the concept itself is in fact valid), and hence potentially increasing unwanted current flow "across" turns (via capacitance, i.e. displacement current) instead of the desired current flow "along" the turns. Unfortunately, when FEMM simulates proximity effect and RF resistance, it cannot account for electric field effects like capacitance. Therefore, the effect of "inter-turn capacitance" cannot be seen in FEMM. It can, however, be seen in NEC-2-based simulations, but in NEC-2, proximity effects and skin effect are not modeled, and all wires are assumed to be round, not flat. Synthesizing results from both simulators will give the best insight -- but the preliminary result is clear: spiral winding is lossier than solenoidal winding.

# Self-resonance, capacitor current, and capacitor loss

An interesting aspect of a nearly-self-resonant antenna is that the required resonating capacitance is very small. This makes it feasible to construct a high-quality variable capacitor rather easily. Furthermore, the current flowing through the capacitor is quite low, making losses in the capacitor far less significant. Most of the current flows through the nearly-self-resonant structure, and little of the current flows through the capacitor. In the limiting case of self-resonance, no current flows through the capacitor, and all of the current flows through interior of the self-resonant structure. However, there is a disadvantage -- at or near self-resonance, we can expect that the entire structure will be very sensitive to detuning by its surroundings. Environmental changes such as people walking near the loop; the addition of moisture due to rain; or strong winds that change the loop's orientation relative to its surroundings or that cause fluttering of the individual turns or of the feedline may change the loop's self-capacitance and change the loop's tuning.

# Future work

Good agreement was established between the MININEC results and the NEC-2 results for a loop approximately 0.10-wavelengths in circumference. This is very encouraging, because the single-precision MININEC was validated to return valid results for much smaller loops down to 0.01 wavelengths; the double-precision port by VK3DIP can likely simulate even smaller loops. Future work can conduct further MININEC simulations of the radiation resistance of multi-turn loops with smaller loop diameters and with more turns.

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