Magnetic Loop Arithmetic

Magnetic (Small Transmitting) Loop Computation.

Some of the formulae behind magnetic loop design are included below. I am grateful to Reg G4FGQ for these. I have used Ln to refer to natural logs because I'm a Sinclair/BBC baby I suppose...

The inductance of a circular loop (L) in microHenrys may be found where:

P = The perimeter or circumference through the centre of the conductor (m)]
d=The diameter of the conductor (mm)

0.2*P*(Ln(4000*P/d)-2.451)

For a square loop the inductance is smaller for the same perimeter and the value 2.451 changes to 2.853 - the area within the loop being the main variable.

Above a few hundred kHz RF flows in the surface or skin of a conductor - the material for the loop can be hollow tube, or perhaps the outer braid of good quality co-axial feeder.

The reactance of the loop in Ohms where:

F=frequency in MHz
L=inductance in microHenrys

X = 2 * Pi * F * L

and the tuning capacitor setting in pico-farads is calculated:
=1000000/(2*pi*f*x)

The radiation resistance of the loop is

31171 ( PI to the fourth power x 320 ) multiplied by the Area squared (A*A), divided by the Wavelength to the fourth power (W**4)

Rr = PI **4 x 320 x A x A / W **4> </p>

<ul>

<li>Where A is the area enclosed by the loop in square metres</li>

<li>W is wavelength in metres</li>

</ul>

<p>For multiple turns N, multiply the above by the square of the

number of turns (N squared). </p>

<h4>The HF loss resistance of the loop in Ohms is</h4>

<p><img src=

P= Circumference of loop in metres
d= diameter of conductor in mm
F=Frequency in MHz

Radiating power efficiency of the loop is

Eff = Rrad / (Rrad + Rloss)

Rr=Radiation resistance
Rl=Loss resistance

In terms of decibels relative to a loss-less antenna,

Loss=4.343*Ln(1/Eff)

The above formulae apply only to small loops with perimeters not greater than 1/5 of a wavelength. They do not apply accurately to loops near to the ground, at a height less than about half a loop diameter, because heavy loop currents induce currents in the ground in a manner similar to electric induction-heating furnaces. This power loss is represented by an additional loss resistance inside the loop. Reg. G4FGQ

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