Chapter 3. The Twisted Copper Ring: Flux Cancellation and SWR

Copper rings one cubit in circumference are often made from two strands of wire twisted together. This chapter sets out the author’s standard construction, calculates the cut length that makes the finished ring exactly one cubit round, explains when such a ring cancels its own magnetic field and when it does not, and turns the measurement plan of the sample edition into version 1.0.

The standard construction

Evidence

The rings made in the workshop of sacredcubit.com and tensorcoil.com follow one specification.

Wire: annealed, 99.9% pure electrolytic copper (alloy C11000), in two common diameters: 1.6 mm (close to 14 AWG) and 2.0 mm (close to 12 AWG).
Forming: a single length of wire is folded in two so that the strands lie side by side, and the pair is twisted at a fixed pitch, that is, a fixed number of twists per cubit. Rings are made with both directions of twist (clockwise and anticlockwise).
Length: the cut length is chosen so that, after twisting, the mean circumference of the finished ring is 523.5 mm. Twisting shortens the pair, so the cut length is corrected by a twist contraction factor.
Closing: the two ends of the twisted pair are joined with lead-free silver-bearing solder to close the ring.

The twist contraction factor

When two touching wires of diameter $d$ are twisted together, the centre of each wire follows a helix of radius $d/2$ around the axis of the pair. Over one pitch $p$ (the length of one full twist along the axis), each wire travels a distance $\ell = \sqrt{p^2 + (\pi d)^2},$ which is longer than $p$ . The twist contraction factor is the ratio $k = \frac{p}{\sqrt{p^2 + (\pi d)^2}},$ and a strand that must cover an axial length $C$ has to be cut to $C/k$ .

Calculation

For the two standard wire sizes and three pitches (one twist per palm, two per palm, one per finger):

import math

C = 523.5                         # target ring circumference along the pair's axis, mm
for d in (1.6, 2.0):              # wire diameter, mm (14 AWG, 12 AWG)
    for n in (7, 14, 28):         # twists per cubit (one per palm, two per palm, one per finger)
        p = C / n                 # pitch, mm
        k = p / math.hypot(p, math.pi * d)   # twist contraction factor
        print(f"d = {d} mm, {n:2d} twists: pitch {p:5.1f} mm, k = {k:.4f}, "
              f"strand cut length {C / k:6.1f} mm")
# d = 1.6 mm,  7 twists: pitch  74.8 mm, k = 0.9977, strand cut length  524.7 mm
# d = 1.6 mm, 14 twists: pitch  37.4 mm, k = 0.9911, strand cut length  528.2 mm
# d = 1.6 mm, 28 twists: pitch  18.7 mm, k = 0.9657, strand cut length  542.1 mm
# d = 2.0 mm,  7 twists: pitch  74.8 mm, k = 0.9965, strand cut length  525.3 mm
# d = 2.0 mm, 14 twists: pitch  37.4 mm, k = 0.9862, strand cut length  530.8 mm
# d = 2.0 mm, 28 twists: pitch  18.7 mm, k = 0.9479, strand cut length  552.3 mm

Each strand of the folded wire is cut to the length shown, plus the allowance needed for the fold and the solder joint; the total length of wire is twice that. At one twist per palm the correction is about a millimetre; at one twist per finger with 2 mm wire it is nearly 29 mm, and a ring cut without it would come out more than 5% short.

Note

The direction of the twist (clockwise or anticlockwise, an S or a Z twist) does not change the contraction factor: the helix is the same length either way. Nor does it change the ring’s behaviour at the frequencies discussed in this book. A helix responds differently to the two senses of circular polarisation only when its dimensions are comparable to the wavelength; here the helix radius is under a millimetre and the wavelengths are tens of centimetres or more.

Two strands, opposite currents

Evidence

When two conductors lie close together and carry equal currents in opposite directions, the magnetic field of each is largely cancelled by the other outside the pair. Engineers use this in two familiar ways. A bifilar (non-inductive) winding doubles the wire back on itself so that a coil has almost no net magnetic field, which is how precision wire-wound resistors are made. A twisted pair swaps the positions of the two conductors at every half twist, so that interference picked up in one half twist is cancelled in the next; Alexander Graham Bell patented the idea for telephone lines in 1881 (Bell 1881). Both rest on the same physics: the far magnetic field of a current loop is set by its magnetic dipole moment, the current times the area it encloses (Grover 1946).

Calculation

For a ring one cubit in circumference made of 1.6 mm wire, compare a single loop with two strands 2 mm apart (centre to centre) carrying opposite currents:

import math

mu0 = 4e-7 * math.pi
ell = 0.5235                 # wire length around the ring = one cubit, m
R = ell / (2 * math.pi)      # ring radius, m
a = 0.8e-3                   # wire radius, m (1.6 mm copper wire)
s = 2.0e-3                   # centre-to-centre spacing of the two strands, m

# single loop: inductance of a circular ring of round wire
L_loop = mu0 * R * (math.log(8 * R / a) - 2)
# two strands carrying opposite currents: a two-wire line of length ell
L_pair = mu0 / math.pi * math.acosh(s / (2 * a)) * ell
# far-field strength follows the magnetic dipole moment m = I x area
ratio = (ell * s) / (math.pi * R ** 2)
print(f"ring radius {R*1000:.1f} mm")
print(f"single loop:  L = {L_loop*1e6:.3f} uH, moment per amp = {math.pi*R**2*1e4:.1f} cm^2")
print(f"opposed pair: L = {L_pair*1e6:.3f} uH, moment per amp = {ell*s*1e4:.1f} cm^2")
print(f"far field of the pair relative to the loop: {ratio:.3f} ({20*math.log10(ratio):.1f} dB)")
# ring radius 83.3 mm
# single loop:  L = 0.495 uH, moment per amp = 218.1 cm^2
# opposed pair: L = 0.145 uH, moment per amp = 10.5 cm^2
# far field of the pair relative to the loop: 0.048 (-26.4 dB)

The opposed pair’s magnetic moment is only the narrow strip between the strands, about 5% of the loop’s, so its far field is about 26 dB weaker. Twisting cancels most of that remainder too, because each half twist reverses the sign of the strip.

When does a closed ring actually cancel?

This is the most important point in the chapter, and it depends not on the length of the ring but on how its ends are connected. A twisted ring can carry current in two ways.

Common mode: both strands carry current in the same direction around the ring. The pair then acts like a single thick wire, and the ring behaves like an ordinary loop with the full magnetic moment.
Differential mode: the strands carry current in opposite directions. This is the mode in which the fields cancel.

Which modes a ring can support is decided by three facts about its construction.

Hypothesis · not established

1. Are the strands insulated from each other? Bare copper strands that touch are in electrical contact at many points along the twist. They then behave as one stranded conductor: there is no differential mode to speak of, only common mode. Opposed currents need enamelled (magnet) wire or another insulating layer between the strands. A multimeter set to resistance, applied between the two free ends before the ring is closed, shows at once whether the strands are touching electrically.

2. Where does the solder joint connect? If the ring is closed by soldering the fold end and the two free ends together into one joint, every strand begins and ends at that joint. The two strands are then two loops in parallel around the same area; a magnetic field passing through the ring induces current in both in the same direction, and the ring responds like a single loop.

3. Is there an insulated gap? If the two free ends are soldered to each other but kept insulated from the fold (for example by a sleeve where the ends meet), the only closed path runs around the ring on one strand and back on the other. Current induced by a field through the ring must flow in opposite directions on the two strands, and the induced voltages cancel: this ring is non-inductive to an external field.

So a ring of bare C11000 wire closed with one solder joint behaves, toward an external magnetic field, like an ordinary single-turn loop of thick wire. The opposed-current cancellation calculated above describes the differential mode, which such a ring cannot carry. A ring of insulated wire with an insulated gap is a genuine non-inductive (bifilar) ring. The measurement plan below tests both.

Measuring with standing-wave ratio

The cleanest way to see what a cubit ring or rod does electrically is to measure how it reflects or couples radio-frequency energy. This is the everyday work of a radio amateur tuning an aerial, and the instrument, a vector network analyser (VNA), is now inexpensive (Chapter 5).

Evidence

When a signal travels down a cable of characteristic impedance $Z_0$ (usually 50 Ω) and meets a load of impedance $Z_L$ , part of it is reflected. The reflection coefficient is $\Gamma = (Z_L – Z_0)/(Z_L + Z_0)$ . Reflected and forward waves add along the cable into a standing wave whose maximum-to-minimum ratio is the standing-wave ratio, $\text{SWR} = (1 + |\Gamma|)/(1 – |\Gamma|)$ ; the return loss is $-20\log_{10}|\Gamma|$ dB (Balanis 2016). A load that matches the cable reflects nothing and has SWR 1. An ideal quarter-wave monopole has a resistance of about 36.5 Ω at resonance.

Calculation

import math

def swr(z_load, z0=50.0):
    gamma = abs((z_load - z0) / (z_load + z0))
    return (1 + gamma) / (1 - gamma), -20 * math.log10(gamma) if gamma else float("inf")

for z in (36.5, 50, 75, 25 + 30j):
    s, rl = swr(z)
    print(f"Z = {z!s:>9} ohm: SWR {s:4.2f}, return loss {rl:5.1f} dB")
# Z =      36.5 ohm: SWR 1.37, return loss  16.1 dB
# Z =        50 ohm: SWR 1.00, return loss   inf dB
# Z =        75 ohm: SWR 1.50, return loss  14.0 dB
# Z =  (25+30j) ohm: SWR 2.87, return loss   6.3 dB

A one-cubit monopole should therefore show an SWR minimum of about 1.4 near 136 MHz when fed directly from a 50 Ω analyser over a good ground plane.

Measurement plan 1.0

Hypothesis · not established

What will be measured. Five objects, each made with one royal cubit (523.5 mm) as its working length:

a straight 1.6 mm wire mounted vertically on a 50 × 50 cm aluminium ground plane (the reference quarter-wave monopole);
a single closed ring of 1.6 mm bare wire, 523.5 mm in circumference;
a standard twisted ring of bare 1.6 mm C11000 wire, closed with one solder joint, at seven twists per cubit;
the same as 3 but with fourteen twists per cubit and the opposite direction of twist;
a twisted ring of 1.6 mm enamelled copper wire with an insulated gap between the fold and the joined free ends (a true bifilar ring).

How. A calibrated NanoVNA sweeps 1–900 MHz (Chapter 5). For object 1 the analyser feeds the wire against the ground plane and records S11. For objects 2 to 5, two small single-turn probe loops, 20 mm in diameter, are placed 20 mm from the ring on opposite sides, and the transmission between them (S21) is recorded with and without the ring. Each measurement is repeated three times, with the object removed and replaced, and the room temperature is recorded.

Predictions, written down before measuring.

Object 1: SWR minimum between 130 and 143 MHz, with minimum SWR below 2.
Objects 2, 3 and 4: at low frequencies, nearly the same change in probe coupling as each other; a first loop resonance where the circumference is close to one wavelength, below the ideal 572.7 MHz because of the wire’s thickness, and lower for 3 and 4 than for 2 because the twisted strands are longer than the ring. No difference between 3 and 4 beyond what the different strand lengths explain; in particular no effect of the direction of twist.
Object 5: at low frequencies, a change in probe coupling at least 20 dB smaller than objects 2 to 4, because the induced currents cancel.

Publication. The raw sweep files (Touchstone .s1p/.s2p), calibration files, photographs and analysis code will be published as a new version of this book, with its own DOI, whether or not the predictions hold.

Note

Health claims. Copper rings of cubit length are sometimes sold with claims that they harmonise energy, purify water, relieve pain or improve sleep. There is no reliable evidence for any of these. The measurements in this chapter can show only what a ring does to radio-frequency signals in a test set-up; they cannot show, and this book does not claim, any effect on the body.

Summary

The standard ring uses annealed 99.9% C11000 copper, 1.6 or 2.0 mm, folded, twisted at a fixed pitch and closed with lead-free silver-bearing solder.
Twisting shortens the pair by the factor $k = p/\sqrt{p^2 + (\pi d)^2}$ ; strands are cut to $523.5/k$ mm plus allowances, from about 525 mm at one twist per palm to over 550 mm at one twist per finger in 2 mm wire.
Opposed currents cancel the far field by more than 20 dB, but a closed ring supports opposed currents only if its strands are insulated and its joint leaves an insulated gap; a bare ring closed at one joint behaves like a single loop.
Measurement plan 1.0 compares five objects and states its predictions in advance; the results will be published whatever they are.

References

Balanis, Constantine A. 2016. Antenna Theory: Analysis and Design. 4th ed. Wiley.

Bell, Alexander Graham. 1881. Telephone-Circuit. US Patent 244,426.

Grover, Frederick W. 1946. Inductance Calculations: Working Formulas and Tables. Van Nostrand.

Fan, Y. (2026). The Sacred Cubit and Resonant Geometry: Measuring the Egyptian Royal Cubit, the Quarter Wave and the Bifilar Ring [神圣肘尺与谐振几何] (Version 1.0.0). 5D Library. https://doi.org/10.5281/zenodo.23666560 · Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0).