The performance of a small, very low frequency, loop antenna is derived in terms of the transmitted field strength obtained in return for cost factors such as mass and power. It is shown that the number of turns is not a contributory factor. It is also shown that aluminium performs better than copper in this respect. For a receiving antenna the number of turns does not affect the signal to thermal noise ratio. The paper is part of a longer and more rigorous analysis of VLF techniques for underground radio which will be published in due course.

The moment of a static magnetic dipole is given by number of turns × current × loop area. Doubling the number of turns doubles the magnetic moment, but it also doubles the mass. Given a certain mass of wire the designer must choose between many turns of thin wire or a few turns of thick wire, (Bedford, 1986). 'Number of turns' is thus a difficult parameter to quantify - it is better to relate the field strength to factors such as aerial mass and power dissipation, to both of which we can assign practical limits. The magnetic moment of a static dipole is analysed and the results extended to include the time varying case, and amplifier parameters. A similar analysis shows that the signal to thermal ('Johnson') noise of a receiving loop can also be written in terms of cost factors.

We define some basic parameters: ^{note 1}

n | number of turns | [] |

A | cross sectional area of wire | [m^{2}] |

mass density of wire | [kgm^{-3}] | |

electrical conductivity of wire | [^{-1}m^{-1}] | |

r | radius of loop | [m] |

I | current in wire | [A] |

from these we can derive:

l | length of wire = n·2r | [m] |

R | resistance of wire = l/A | [] |

M | mass of wire = lA | [kg] |

P | power dissipation in wire = l^{2}R |
[W] |

M | magnetic dipole moment = nI·r^{2} |
[Am^{2}] |

A circumflex (^) above a symbol will denote that it represents an rms value. Further parameters are listed in §7.1.

The performance of the aerial will be defined as the magnetic moment
achieved for given values of mass and power, eliminating as many other
factors as possible. Proceeding either by a dimensional analysis, or a
haphazard mess of algebra we begin with an expression for *M*, thus:

M = nI·r^{2} |
(1) |

I can be found from I^{2} = P/R = P·A/l = P/l·M/l, thus I = 1/l(MP/), substituting
this in the above, and writing I=n·2r gives:

M =
½·(MP/) | (2) |

Thus the transmitter 'strength' is proportional to the aerial radius, square root of mass and square root of power dissipation. There are a number of points to observe.

- The number of turns and the wire diameter do not appear in the expression, so they are not of primary concern.
*M*is proportional to radius and not area of the loop.- Given constraints on mass and power aluminium performs almost 50% better than copper.

This last point can be seen by considering the factor (/). This is tabulated below for selected metals. The values are at 20°C and taken from Hansson, 1972.

metal | conductivity | density | merit |
---|---|---|---|

(/) | |||

10^{6}/m | 10^{3}/kgm^{-3}
| m/kg | |

Sodium | 21.3 | 0.97 | 148 |

Potassium | 14.9 | 0.86 | 132 |

Beryllium | 30.3 | 1.84 | 128 |

Calcium | 23.3 | 1.55 | 123 |

Aluminium | 37.0 | 2.7 | 117 |

Magnesium | 21.7 | 1.74 | 112 |

Copper | 58.1 | 8.93 | 80.7 |

Silver | 62.5 | 10.5 | 77.2 |

Zinc | 16.1 | 7.1 | 47.6 |

Gold | 43.5 | 19.3 | 47.5 |

Nickel | 12.8 | 8.9 | 37.9 |

Iron | 9.52 | 7.86 | 34.8 |

Tin (tetragonal) | 8.70 | 7.3 | 34.5 |

Obviously most of these are unsuitable because of reactivity, toxicity or cost.

The above expression relates to a static dipole. For a time-varying dipole
the skin and proximity effects need to be taken into account, and there is
little point is using wire of a diameter much larger than the skin depth.
'Skin depth' refers to the fact that electromagnetic fields do not travel
within conductors, but 'seep' in from the surface in an exponential fashion,
(the diffusion equation rather than the wave equation applies). The skin
depth, , is the depth at which the surface current
density has dropped to 1/e (37%) of its surface value ^{note 2}. The equation

= (2/µ) | (3) |

can be applied, but only where is a small fraction of the conductor depth. (µ is permeability, is angular frequency).

The skin depth at 100kHz is around 0.2mm.

A small, slowly varying dipole does not radiate any significant power, The radiated power is given by

P_{r} =
Z_{0}M_{rms}^{2}k_{0}^{4}
/ 6 | (4) |

where k_{0} is the wave number, 2/, and Z_{0} is the
characteristic impedance of free space, 120 . (This equation and other aspects
of e-m theory will be derived in another paper ^{note 3}). A signal with an rms moment
*M*_{rms}=70Am^{2} (ie. peak *M*=100Am^{2})
will radiate 2µW mean power at 100kHz. For small loops and frequencies
below 100kHz or so, the radiated power can be ignored.

There is a need to be able to compare the performance of antenna systems.
One method is to quote the magnetic moment, *M* = nIr^{2}
[Am^{2}]. This is obviously better than just quoting physical size
and number of turns but it does not take into account the mass of the aerial,
nor give an indication of the electrical power required. A more suitable
'figure of merit' might be the magnetic moment per unit of power dissipation
in the aerial. The dimensions of the units require that we actually use
M/P rather than M/P, and we can define a figure of
merit, , as

= | () |

Index to Cambridge Underground

Table of Contents for Cambridge Underground 1990

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