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.
• 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:
[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.