Cambridge Underground 1990 pp 35-38,41


David Gibson M.A


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

nnumber of turns[]
Across sectional area of wire [m2]
rhomass density of wire [kgm-3]
sigmaelectrical conductivity of wire [ohm-1m-1]
rradius of loop[m]
Icurrent in wire[A]

from these we can derive:

llength of wire = n·2pir[m]
Rresistance of wire = l/sigmaA[ohm]
Mmass of wire = lArho[kg]
Ppower dissipation in wire = l2R [W]
Mmagnetic dipole moment = nI·pir2 [Am2]

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·pir2 (1)

I can be found from I2 = P/R = P·sigmaA/l = Psigma/l·M/lrho, thus I = 1/lsquare-root(MPsigma/rho), substituting this in the above, and writing I=n·2pir gives:

M = ½·square-root(MPsigma/rho)(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.

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

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

sigma rho square-root(sigma/rho)
106/ohmm103/kgm-3 m/square-rootohmkg
Tin (tetragonal)8.707.334.5

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

3.1 Application to time-varying dipole

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, delta, is the depth at which the surface current density has dropped to 1/e (37%) of its surface value note 2. The equation

delta = square-root(2/omegaµsigma)(3)

can be applied, but only where delta is a small fraction of the conductor depth. (µ is permeability, omega 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

Pr = Z0Mrms2k04 / 6pi(4)

where k0 is the wave number, 2pi/lambda, and Z0 is the characteristic impedance of free space, approx 120pi ohm. (This equation and other aspects of e-m theory will be derived in another paper note 3). A signal with an rms moment Mrms=70Am2 (ie. peak M=100Am2) will radiate 2µW mean power at 100kHz. For small loops and frequencies below 100kHz or so, the radiated power can be ignored.

3.2 Figure of merit for transmitter loop effective aperture

There is a need to be able to compare the performance of antenna systems. One method is to quote the magnetic moment, M = nIpir2 [Am2]. 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/square-rootP rather than M/P, and we can define a figure of merit, Phi, as

Phi =()
M = = [m2I/~] (5) p This has the dimensions of area//resistance and could be called the effective aperture of the antenna. This is similar to the concept of ~effective length' used to describe the noise performance of line antennas3, although I have chosen to make ~ relate to moment (ie current) rather than to power. ~ can also be written as = n I nr 2 /(12R) _ number ~ loop + ~ loop (6) - of turns area resistance So, provided that the loop resistance can be measured accurately, this provides an easy way to measure ~ avoiding such ambiguities as whether to measure peak or rms current. Note that the effective area of the loop does not say anything about the performance of the amplifier, and there is no point in having an optimum loop if it is poorly matched to an inefficient amplifier. 4. CURRENT DENSITY The current density in the wire, p;, ~Am~2], is simply I/A. Since 12/A2 = P/RA2 = P~o'/AI = P~~p/M we can write p; = /(po'P/M) (7) Normally current density would not be a problem, though for extended transmissions the temperature rise would need to ascertained by experiment. If current density is critical then it can be seen that it is better to increase the transmitter strength by increasing the mass rather than the power, the product /p~ is 720x103 [kg~m~2Q-~] for copper and 316x103 for aluminium. As a guide, copper hook-up wire is usually rated at around 1.5MA/m2 or 1.5A/mm2. A 1kg, lOW aerial has p; - 1.0 MAIm2 for aluminium and 2.3MA/m2 for copper Current density would be important if a superconducting aerial were to be used since a superconductor can only support a limited current density. In this case ~i would become one of the parameters by which the performance is measured (note that o'=~, and the above expression for M does not apply). The present generation of high temperature superconductors are not suitable for this purpose as the properties do not extend across crystal boundaries. 5. MATCHING THE LOOP TO AN AMPLIFIER We are not primarily concerned with the power dissipation in the loop, but the power drawn from the power supply. The quantity P in the above equations can represent this after adding a correcting factor that takes into account the matching and the type of amplifier. The basic relationship that m r/MP is unchanged. The expressions will be derived in a later paper note 3, and will show that a small degree of mismatching could be desirable. 6. RECEIVING AERIAL PERFORMANCE If ~he field strength at the receiver is H = /2H1coswt and the flux density is given by B = ~H then the signal voltage across a parallel-tuned resonant loop is dB U = -~n~A~Q = ()ji/2H.sin~t.nnr2Q dt ~ U. = ~~H.~nnr2Q (8) where Q is nominally equal to ~L/R = 1l~~CR. In practice the Q-factor may be much less than this. H is related to the transmitter magnetic moment by H. = MI'2nd3 where d is the distance between the two co-axial loops. This is only true for near-field measurements when 2nd (( ~ and d '< ~. In this case the attenuation due to the skin effect can probably be ignored. Following a similar analysis to that for the transmitting loop, we can write U. in terms of 'cost parameters' as U. - t~)~H.~~r/(Mo'/p)./R.Q = td~~/R~O~H. - G~OH. (9) where Gj is the 'receiving loop gain' ~ [V/Am~1]. ~ is the effective aperture of the loop, as derived earlier. The inclusion of ~ shows that the same design rules apply to the receiving loop as to the transmitting loop. The number of turns does not appear explicitly, but in the guise of resistance and 0 factor. More turns results in a higher value for R, and so a higher thermal noise, but this is counteracted by a larger induced signal. Increasing the turns (keeping the mass constant) increases L and R in proportion, so does not alter 0. The result is that the thermal noise performance is independent of number of turns. This is derived formally below. 7. NOISE SOURCES 7.1 Parameters Discussions of noise can be confusing, so it is as well to define a few terms to begin with. Thermal noise power is spread throughout the spectrum, so its 'density' can be measured in Watts/Hertz. This implies that noise voltage has the dimensions V//Hz. This is termed the noise voltage spectral density. It can be shown that a resistor generates a noise power of 4KT W/Hz. K is Boltzmann's constant (1.38xl0~23J/K), T is the temperature in Kelvin. (290K is 17~C - at this temperature 4KT is 1.6x10~2oJ ~W/Hz]). It is useful to represent amplifier noise in terms of the value of resistor needed to produce the same noise voltage density. This is the 'equivalent noise resistance' and is not to be confused with the ratio of noise BUGGER missed two pages out :-(

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