EQUATIONS USED

Notes:
The Magloop Antenna Calculator was developed to predict the characteristics of a small-transmitting-loop (aka "STL", "magnetic loop" or "magloop") antenna, given physical dimensions entered via slider widgets.
It supports:

• circular, octagonal, hexagonal and square-shaped loops
• main loops made from either hollow round anodised-copper or aluminium conductors
• metric and imperial units
• magloops with 1-to-8 turns

I developed this multi-turn capable magloop calculator to take advantage of the touch-screens and high-speed of modern mobile phones, to allow users to get realtime feedback of the predicted behaviour of a magloop antenna.
-- 73 de VK3CPU

Inputs via the slider and radio widgets:

• ⌀a : Conductor diameter in millimeters (mm) or inches ("). (Measured between opposing conductor outer surfaces.)
• ⌀b : Loop diameter in meters (m) or feet ('). (Measured between the conductor centers.)
• N : Number of turns or loops. Sweeping left-to-right for parallel multiloop, then single loop, then series multiloop configurations. A "(P)" indicates a multiloop antenna configured with parallel main loops.
• c/a : is the spacing ratio; based on 'c' being the inter-winding spacing for multi-turn loops measured between conductor centers, and 'a' is the conductor diameter. (Must be >= 1.1) A low-value will increase the resistance due to the proximity effect. (Ignore for single-turn loops.)
• Tx : The transmit power in Watts. This affects the predicted voltage across the capacitor (Vcap), and the RMS loop current (Ia).
• Re : Additional resistance due to external losses, due mainly from capacitor contact resistance and proximity-to-ground effects. Use Re=0.0 to assume the loop is in free-space with no capacitor losses (i.e. ideal conditions, with loop-related losses only). Adding Re will reduce antenna efficiency, Q, Vcap and Ia, while increasing antenna BW. According to [1] and [2], a 1 m diameter loop of 22 mm copper tubing at a height of 1.5 m above the ground operating at 7 MHz had a calculated capacitor contact resistance of ~190 mΩ and an additional ground proximity loss resistance of ~30 mΩ. Note that true ground losses are dependent on both frequency and height-above-ground.
• Metric or Imperial : selects the measuring system.
• Cu or Al : selects the type of metal conductor (annealed copper or aluminum).
• Circ, Oct, Hex or Sqr : selects the shape of the magloop.

Calculated parameters:

• L : Inductance in microhenries. Equations from [3].
• A : Loop area in square meters or square feet.
• C : Effective capacitance of the loop in picofarads.
• peri : Perimeter of the main loop in meters or feet.
• c : Distance between windings, measured from the conductor centers in mm or inches.
• cond : Total required conductor length in meters or feet.
• Tuning Cap (pF): The capacitance required to bring the loop into resonance at the given frequency. Value in picofarads.
• Vcap (kV): The predicted voltage across the capacitance given the desired transmit power.
• BW (kHz): The predicted 3dB bandwidth of the magloop antenna.
• Efficiency (%): The percentage of input energy that is actually radiated and not lost as heat. This efficiency figure is based on conductor losses of the main loop only. Missing, are losses due to the capacitor, feedline and the near-field environment. These losses can be added via the Re slider.
• R-radiation (Ω): The calculated radiation resistance of the loop in ohms.
• R-loop (Ω): The calculated resistance of the loop in ohms, due to the combination of material conductance, conductor length, skin-effect and proximity effects.
• Reactance (jΩ): The inductive reactance of the loop in ohms.
• Q : The antenna Q (quality) factor.
• Ia (A): The RMS loop current in amps.
• Perimeter (λ): Antenna perimeter size relative to the wavelength.
• Skin depth (μm): Conductor skin depth in micrometers.

Usage hints:

Tap on legend items to disable or enable an output parameter. This can be used to declutter the chart.

Tap on a chart 'dot' to display a tooltip containing calculated output parameters for that frequency or band.

Other VK3CPU calculators:

• RF Toroid Calculator
• RF Inductor Calculator
• Coil-loaded Dipole Antenna Calculator

References:
[1]: B. Austin, A. Boswell and M. Perks, "Loss Mechanisms in the Electrically Small Loop Antenna" , IEEE Antennas and Propagation Magazine, 56, 4, August 2014, pp. 143.
[2]: A. Boswell, A. J. Tyler and A. White, "Performance of a Small Loop Antenna in the 3 - 10 MHz Band" , IEEE Antennas and Propagation Magazine, 47, 2, April 2005, pp. 5 1 -56.
[3]: F W Grover, "Formulas and Tables for the Calculation of the Inductance of Coils of Polygonal Form" Scientific Papers of the Bureau of Standards, Vol 18, p753

Change history:
[21-Dec-23] - V10.4
* Added CB frequencies (11m, 26.965 - 27.855 MHz).
[26-Aug-23] - V10.3
* Added support for URL parameters. Some code-base clean-up.
[13-May-23] - V10.2
* Changed radiation and loss resistance to logarithmic. Capacitance limited to 5-5000pF.
[8-May-23] - V10.1
* Stopped autoscaling on all axes. Changed Vcap, Icap to logarithmic. This makes it easier when comparing different configurations.
[8-May-23] - V10
* Added conductor skin depth as a new calculated parameter.
[23-Apr-23] - V9
* Changed the frequency list to include VHF (2m) and UHF (70cm) bands.
* Reduced minimum loop diameter to 4 cm.
[12-Feb-22] - V8
* Changed the frequency list to include top and bottom frequencies of each band. This is to highlight the range of capacitance required between the top and bottom of each band.
* Increased max c/a ratio from 10 to 30.
[13-Jan-22] - V8
* Added support for parallel conductor magloop antennas.
[21-Nov-21] - V7
* Upgrade Chart.js to the latest version, v3.5.1.
* Tooltips are now justified, (using monospace fonts) and support changing metric prefix.
[27-Sep-21]
* Added band wavelength to tooltip display. Changed 60 m band from 5.0 to 5.3 MHz..
[26-Sep-21]
* Added metal type to schematic display.
* Reduced display precision for Cap and Q in tooltip.
* Changed Tuning Cap scale max from 2000 to 1000 pF.
[23-Sep-21]
* Changed Q equation back to the original Xl/Rtot. Changed max Q to 4000.
* Introduced a new slider "Re" to inject external losses to account for the combined losses due to capacitor contact resistance and ground losses.
* Renamed R-loss to R-loop to avoid confusion, as loop resistance is no longer the only resistance that contributes to losses. The other being Re.
* Changed to V6 to capture the significant changes.
[22-Sep-21]
* Added antenna perimeter size in wavelength to the chart display as a new item.
* Changed maximum spacing ratio c/a from 4.0 to 10.0. Values higher than 4 have no further effect on proximity resistance, but does reduce coil inductance which drives up the SRF.
[21-Sep-21]
* Added distributed capacitance calculation and display for the single turn loop.
[19-Sep-21]
* Increased supported conductor diameter to 80 mm. (3.15 inches)
[18-Sep-21]
* Updated to V5; Added support for octagon, hexagon and square shaped loops. Moved and hyperlinked equations-used to a separate page for clarity.
[16-Sep-21]
* Updated to V4; Updated equation used for Q to match the one use in the ARRL Antenna Book. This will affect predictions for V_cap, I_loop and BW. (Based on Q equation D.1 used in "Impedance, Bandwidth, and Q of Antennas" by A D Yaghjian, IEEE Transactions on Antennas and Propagation, April 2005.)
* Added equation graphics for V_cap, I_loop and BW formulas.
* Flipped the main-loop graphic to have the capacitor above the coupling loop.
[12-Sep-21]
* Set maximum values to Q, Vcap and I axes to stop autoscaling. Max Q set to 2000, Vcap to 20 kV and I to 100 A.
* Added formula/equation graphics in Notes section. A few more complex ones, such as effective capacitance and SRF, are still needed.
* Fixed minor error in calculation of resistive loss due to proximity effect.
[11-Sep-21]
* Added visual cues for all slider-controlled parameters to highlight which parameter is being modified in the graphic representation.
[10-Sep-21]
* Added c/a display to graphic representation. Moved N from center to left.
[30-Aug-21]
* Added SRF calculation and display for multi-loop antennas.
[28-Aug-21]
* Added support for imperial units and for aluminum metal.
[27-Jul-21]
* Added total conductor length display.
[24-Jul-21]
* Added loop circumference display.