By Serge
Lacôte (France)
Introduction
As we know, the
electromagnetic field (EMF) associated with a plane
polarized wave can be modelled by two same frequency
plane fields (electric and magnetic) whose planes are
perpendicular. We have said « modelled by »
and not « constituted of », because if we
create and associate an electric field on the one hand
and a magnetic field of the same frequency on the other
hand, we will not obtain an EMF ! The electric field and
the magnetic one have no interaction, moreover, these
two fields decrease as the square of the distance
whereas a true EM field decreases as the distance. It is
the power
per surface unit of an EMF which decreases as the square
of the distance as it results from the formula of the
surface area of a sphere (if there is no loss and for an
isotropic transmission). However, among EMF
manifestations, it is possible to identify two fields
corresponding to those of the model (and pick them up).
To pick up the electric one we use an antenna and an
induction coil or a “loop” (sometimes improperly named
loopantenna) to pick up the magnetic one.
In the special case of
extremely low frequency (ELF) waves, i.e. for less than
40 Hz frequencies, the wave length is in the range of
tens or hundreds of thousands of kilometres (), and, ideally, the antenna length being
of the same order of magnitude, it's judicious to
attempt to use the earth as an antenna in order to pick
up the electric field by means of two electrodes plunged
into the soil. The exact nature of pickedup signals is
moot and it's not the place to raise concerns (on this
matter, read http://www.vlf.it/ed/earthprobes.html).
However,
it
must be observed that the method is used by professional
geophysicists (with non polarizable electrodes) to pick
up the E component, at least in the ELF range and
espescially in a lowpolluted environment (see for
example the Annales geophysicae publication called Schumann resonance frequency
variations observed in magnetotelluric data recorded
from Garhwal Himalayan region by R. Chand, M. Israil, and J.
Rai^{(2)}.
It's possible to find a description of such a system for
example at the URL http://www.elfradgroup.com/index.htm.
. It's also possible to use a Marconi antenna (much
shorter) as described at http://www.vlf.it/cumiana/livedata.html.
If we choose to pick up the
magnetic component, as we have said, there are two
possibilities for the sensor : the classic one, used by
universities, which consists of using a long coil wound
on a high permeability core such as mumetal or
permalloy. But these ferronickel alloys are expensive
and difficult to obtain. To make this classical version
in an economic way, see http://www.vlf.it/matteobruna/ULF_InductionCoil.htm
.
There remains the loop
solution. The general theory of loop antennas is given
at http://sidstation.loudet.org/antennatheoryen.xhtml
and we will refer to this extensive text, from which it
is necessary to extract the matter which interests us.
Given the wave lengths at stake and the extreme weakness of the magnetic field to pick up, in the range of
Spectrogram^{(4)} of an
hour's recording in a
forest in the North Limousin, France^{(5)}.
The 5 Schumann resonances are clearly visible^{(6)}.
But in real practice, we do
not know the nature of the numerous EM waves where we
pick up the magnetic component : some may be plane
polarized, other circular polarized, elliptic polarized,
or … not polarized. Even if we consider only plane
polarized waves, the polarization plane (which is the E
field plane) may be vertical, horizontal, or …
inclined.
Near the transmitter, the fields E and H are neither in phase, nor orthogonal, so that the H field direction does not allow us to know the wave propagation direction. More precisely, the space surrounding a transmitting antenna is divided into three zones :
There
exist
formulas which define approximatively these two zones as
spherical crowns centered at the middle of the antenna.
Particulary, the Fraunhofer zone is characterized by the
formula , where a is the antenna size
and R the distance between the current point and the
antenna centre. Unfortunately, it would be foolhardly to
apply this formula in the ELF case, because if we go
back to the proof^{(8)},
we can see that, as often in physics, various
approximations and hypothesis have been made to simplify
the calculations. In particular, it has been suposed
that and . It is sufficient to consider the value to see how wild our hope to use these
formulas were. Indeed, these formulas are, above all,
used in optics and in the RF domain.
However It would be useful to hold studies then the antenna size is small or very small with regard to the lengthwave, because all picked up ELF signals are not transmitted from the ionosphere or from the earth, but from various wires or humansized devices. Thanks to Renato without whom I would have forgotten to talk about that problem. In reality, this type of use holds little interest for us and we must abandon the hope to realize a transmitter finding system with one loop (or even two), principally because we do not know the pickedup wave nature and EFL sources are not ordinary transmitters with a classical antenna (the antenna may be the ionosphere). So, we must consider the magnetic field picked up in itself, and that is what we will study. If we want to rebuilt it, we must use three loops where the normals are respectively directed NS, EW and upward. If the loops and signal conditioners are identical, we will have the three components of the complex magnetic field picked up. Submitting these components separately to spectral analysis, it's possible to identify determined signals. From that, and knowing the three magnetic field spectral densities, it could be possible to haveapproximately as a vector for such a signal, and perhaps deduce some information about its source. For more information, cf. http://www.vlf.it/rdfatvlf/rdfatvlf.htm . I specify that I have not done it and for the moment I content myself, with only one loop where the normal is N/S directed (to pick up ) or E/W (to pick up ). The loop I tried to reconcile :
In fact, if all these
conditions were more or less reached, it certainly was
not at the first attempt ! My first loops, with wooden
frames, were neither sufficiently stable nor
sufficiently sensitive. I must thank Renato Romero,
whose famous page http://www.vlf.it/minimal/minimal.htm
constitutes an obliged passage for the beginner, for the
kindness, patience and competence with which he has
often lavished judicious councils, allowing me to
progressively improve my system. I need to say that if
several critical points are not correctly treated, the
desired results will not be achieved. Especially I had
underestimated the length of wire to wind and the
importance of the microphonic effect. The threshold of
sensitivity and the signal/noise ratio obtained by
(imperfect) calculation are perhaps more optimistic than
the real ones and it was necessary to lengthen the wound
wire by three times. It's not worth describing the first
imperfect versions. I will now describe the actual
efficient version.
Wire :
I have chosen 0.4 mm enamelled wire, which is relatively thick, to obtain a low resistance in order to minimize the thermal noise due to the loop and enhance the Q quality factor. This wire is sold in 400meterlong coils weighting 500 grams. Finding such coils at electronics suppliers' at a reasonable price is not always easy but possible^{(9)}. I've experimented that a length of at least 6000 m (i.e 7.5 kg) is necessary to obtain good results with a preamplifier mainly working in voltage^{(10)}. This represents 15 small coils of 400 m. It's easy to join these lengths end to end because the varnish is destroyed by the solder heat. However It is necessary to lightly sandpaper each extremity before twisting them together and wait a few seconds for the varnish to burn, which is visible by the emission of a pungent smoke ; a 50 W minimum soldering iron with a flat tip is necessary to bring and maintain a sufficient quantity of heat. To isolate the junction, I used a tiny piece of electrical adhesive tape but it's possible to use a special varnish. Concrete frame :
For evident reasons, any presence of iron or magnetic metal is prohibited, that's why, in particular, the concrete is not reinforced. We subsequently can see how to palliate this difficulty. This frame, which is a main part must be made with special care : the mould must at once be very precise, solid (to be able to resist any chocks and vibrations without deformation during the concrete filling) and designed with the problem of removal and shrinkage in mind. Indeed, during setting, the concrete retracts and its size decreases while the wood keeps its length (in the fibre direction) therefore the inner mould tends to prevent the retraction and may cause the concrete to break, so, remove the moulder with care a few days later (about a week) after the concrete filling without warping the frame. To overcome the absence of
steel reinforcements, I used :
It's necessary to (moderately)
use remoulding oil to prevent the concrete from sticking
to the wood. The concrete must be prepared with a
minimum of water (compatible with sufficient plasticity)
to minimize shrinkage and the mould repeatedly hit with
a hammer to remove air bubbles. Make sure that the
concrete penetrates everywhere. This operation needs to
be repeated several times at inbetween rest
periods.
After setting ( for a minimum
of three weeks, place the mould on a flat and fixed
support), glue the fibre glass textile after carefully
removing the remoulding oil rests and the thin layer of
the cement laite, which has no resistance. A resin
quantity of 800 g is a minimum requirement, although 1
kg is preferable. Proceed with placing the fibre glass
textile in strips side by side (as it's impossible to
bend it).
Spread the resin with a spatula or a small roller over the concrete, place the fibre glass textile strip, then press it with the spatula or the roller chasing out the air bubbles (important and not always easy; do it again several times, as the textile has a tendency to raise itself if air remains under it) and lay a new resin coat above. It would be preferable to use fibre glass textile specially adapted to concrete reinforcement, but confronted with the practical impossibility of finding any in small quantities, I used solid fibre glass wallpaper. I staggered this operation on several days. The weight of the frame without the coil is about 32 kg. The frame supports:
Also in fibrereinforced concrete, they are 14 cm thick and each of them weighs about 30 kg. There is no need to cover them with fibre glass textile. A layer of rubber is stuck to the bottom to receive the frame (which has also a rubber strip in the bottom). For recordings, they lie on the ground coated with a levelled sand bed. As the frame is a somewhat thick than the supports fences, I used wood wedges to block the frame. The loop coil itself When the epoxy resin is dry,
put the PVC trunking (30 x 20 mm) into its groove using
a polyurethane glue such as Sikaflex© and cover the
4 inner angles with a good electrical tape to avoid the
wound wires^{(11)}.
Make a wooden cross with a hole in the centre and fix it
to the concrete frame to wind the coil. The wire must be
stretched enough ; for this I used a kind of felt
presswire between the loop and the small cooper coils.
It's necessary to maintain the wires in the trunking
place to place with small electric adhesive strip
pieces. It's unprofitable to try to obtain joined turns
because it's almost impossible and it would increase the
distributed capacity. When the coil is achieved (about 8
h), the best would be to impregnate it with epoxy resin,
but I have not done so in order to recover the cooper in
case of necessity. I have used flat wood sticks hardly
jammed between the cooper and the trunking cover to
immobilize the turns.
I stuck thick rubber strips
onto the parts of the frame (outside the trunking) which
must be in contact with the concrete supports, in order
to limit the transmission of groundoriginated
vibrations. The best would be air cushions^{(12)} but I
did not succeed in finding any.
The preamplifier box (a PVC
waterproof derivation box) is glued to the exterior of
the frame with polyurethane glue.
The medium length of the side
coil is 0.96 m, its area A is 0.92 m² and its turn
number N is 1560.
The pure
resistance Inductance
and distributed capacity. N.B. 1°/ This method is approximative for two reasons : 1°/ The modelling of a real inductor by a pure inductor and a capacitor in parallel is questionable and sometimes gives aberrant results ; 2°/ the resonance point determination is lacking in precision. For these reasons, it's prudent to remain with only two significant digits. 2°/ It's interesting to compare this result with the one calculated with a formula. I have not found serious formulas giving the inductance of such a multilayer square coil (warning : there are fanciful formulas on the web !). Only to have a size order, I tried to apply the formula given by http://sidstation.loudet.org/antennatheoryen.xhtml : (e is the coil width – here 0.03 m , K the square side in meters and N the turns number) valid for a single layer coil^{(15)}. I obtained 7.3 H exactly as before, which shows that this value is probably near to truth. 3°/ Now, we can try to evaluate distributed capacity from the resonance frequency. This one, measured as before, is about^{(16)} 835 Hz. Hence, the distributed capacity is about. Impedance. If we model the real inductor by a capacitor C in parallel on a pure inductor L in series with a resistor R, the impedance Z of the real inductor is ^{(17)}; if C is negligible, we obtain . But, is it really pertinent to neglect C ? Here is Maple worksheet, which shows that the answer is yes in the frequency band [0, 40] if we content ourselves with some per cent precision. 
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> 
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(1) 
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(2) 
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(3) 
On the other hand, it's impossible to
neglect L in the loop impedance (except for very low
frequencies) :
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(1) 
> 
Afterwards, we will
adopt .
Cutoff
frequency and quality factor Q.
The loop modelled by an inductance L in series with a resistor R may be seen as a highpass filter whose cutoff frequency is . Supposing that the resonance frequency is 835 Hz, we deduce that the quality factor is . Loop
thermal noise.
The spectral density of noise voltage of a resistor R is given by the formula ^{(18)}, where R is the resistance, T the thermodynamic (absolute) resistor temperature () and the Boltzmann constant (). It's expressed in . If we apply this formula to our loop, we obtain, for and : . It's necessary to specify that voltages here considered are RMS voltages. Loop
properties as a voltage generator.
We must now make an incursion into the next section dedicated to the preamplifier. Considering the relatively low loop impedance, this one is inserted into the inverting input of an OPAMP inverting amplifier. We may schematize the circuit as below^{(19)} : We have voluntarily omitted the bias resistor R1 on the noninverting input which figures in the preamplifier schematic, because it has no effect on the present considerations. We know that the potential difference between the inputs 2 and 3 is nil. Hence the loop is shortcircuited and the current which goes through it is the quotient of the induced voltage by its impedance. Now, we will calculate the induced voltage, the loop current and its transfer function with regard to the magnetic field which goes through the loop.
We assume that : At every fixed time t, the
magnetic field is constant in all the
points of the loop section.
The magnetic field maintains a constant
direction when t varies.
If iis a unitary vector of arbitrarily fixed, is given versus time by the formula with and .
The
magnetic
field intensityis expressed in teslas (T)^{(20)}.
is a unitary vector
arbitrarily fixed, normal to the loop plane.
is the nonoriented
angle of and ; it's defined
by its cosine : (usual
euclidean scalar product).
A
is the loop area in m².
In these conditions, the
component of vector with respect to the unitary vectoris given by and the flux of the magnetic field through the loop N turns is . To determine the loop transfer
function, it's judicious to use complex amplitudes
because the classical Opamps DC formulas stay valid
in AC with complex formalism. Now, b is complex and
function of .
Here is the Maple worksheet
:

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> 
(1) 
> 
(2) 
> 
> 
(3) 
> 
(4) 
> 
(5) 
> 
(6) 
> 
(7) 
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>  
> 
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> 
Hence, the supraconductivity
interest. I recommend reading the interesting
considerations on that subject developed by Marco Bruno
at www.vlf.it/looptheo7/looptheo7.htm (although my attempts to
implement a very low impedance loop able to receive
SRs remained vain).
However, in spite of the handicap of the
loop response curve non horizontality, the practical
results obtained below 10 Hz remain good.

System aptitude to pull SRs out of noise
To know the SR levels, I
referred to the article : Schumann resonance
frequency variations observed in magnetotelluric data
recorded from Garhwal Himalayan region by R.
Chand, M. Israil, and J. Rai free (in Annales
geophysicae) available as a download on the WEB^{(21)}. The
level and the exact SR frequencies change according to
the day hour and the season, but we can see that the
value of the NS component magnetic field spectral
density peak of the first resonance, which is centred
about on 7.83 Hz (8 Hz to simplify), is about RMS andRMS for the fifth, which is centred on
about 26 Hz. I will evaluate the signal to Noise Ratio
(SNR) at the preamplifier input to see in what extent
the SR level exceeds the noise one.
The input noise (outside
exterior EM perturbations) has two sources : the loop
thermal noise and the preamplifier OPAMP noise. To
evaluate it I used the Analog Device study Operational
Amplifier Noise, available free as a download on
the WEB and the AD797 datasheet. The AD797 choice
pertinence appears on figures 4 and 13 considering the
input impedance and the frequency range. The datasheet
fig. 1 plots the AD797 input noise versus frequency. We
have a 1/f pink noise below 200 Hz. Thus that case
occurs for all SRs. Unfortunately, the plotting stops at
10 Hz but an approximate extrapolation shows that the
voltage noise spectral density at 8 Hz is about (without certitude). The data lacking is
the same regarding the current noise.
Evaluation
at 26 Hz.
Input voltage noise density
: (approximate
evaluation : we are in the pink noise OPAMP zone).
Voltage noise due to current
noise flowing in the loop. The AD797 datasheet gives a
2 pA current noise (measured at 1 kHz) and any other
precision. Then, the sought voltage noise density
is
Loop thermal noise density
: .
Total noise density : .
Signal RMS voltage density :
since the induced voltage and the magnetic field are
both sinusoidal, their RMS value is obtained by
application of the same coefficient to the peak value. The formula (3) of the
Maple worksheet above shows that the RMS induced voltage
is for . For f=26 we find : rms, which corresponds to a spectral
density of .
Signal to Noise Ratio (SNR) ,
which is here bandwidth independent : .
Evaluation
at
8 Hz. Conclusion.
The calculations confirm the experience : the SRs are
discernible with the system.
The preamplifier
The preamplifier is embedded
on the loop to avoid losses. The quality of this
preamplifier is a determining factor to “extract” the
signal from the noise. These noises are essentially the
resistor thermal noise (of the loop and of the
preamplifier), and the noise of the OPAMP used. I chose
an ultralow noise OPAMP (Analog Device AD797) and I
relatively limited the voltage gain of this stage to
about 207 (or 46 dB). As said and as it's visible on the
photo, the preamplifier box is glued onto the loop
frame side. The presence of any magnetic metal is
disallowed in the practical realization, in particular,
screws are in brass and I omitted the chock coils
traditionally inserted into the loop input circuit to
kill an eventual crossing modulation in order to avoid a
ferrite presence near the loop. Till now, their absence
has not been harmful in the places where I have
recorded, but it can be the case elsewhere. A 4 twisted
pairs FTP 5e Ethernet wire implements the junction with
a box containing the two 12 V batteries which powers the
system and the rest of electronical circuits. The shield
is not connected. I have tried a 2 m wire and a 30 m
one. The results have remained good with the 30 m wire,
but they have been a little lower than with the 2 m one.
Considering the weakness of
the picked signal and the modest gain of the stage, a 5V
supply voltage is more than sufficient^{(22)}, but
according to the AD797 datasheet, it's also the minimal
Vs voltage possible. With an 870 Ohms loop, a bias
resistor as shown in the AD797 datasheet fig. 40 is
necessary. Notice that the exact value of the bias
resistor R1 would be and not ...
Click here to download the full resolution scheme Notice that the feedback resistor R2 is decoupled by a 10 nF capacitor C2 to provide a first attenuation of “high” frequencies, since this couple acts as a first order lowpass filter with a cutoff frequency of . The 2350 uF liaison capacitor is implemented with two head to tail assembled 4700 uF electrolytic capacitors. These capacitors are useful because the R5 offset adjustment is temperature dependent, which may variate strongly since the preamplifier is located outside. The 100 Ohms impedance adaptation resistor R4 was determined with regard to the input impedance of the conditioner first stage after various experiments.
I have neglected the rolloff for frequencies approaching zero due to the 2350 uF liaison capacitor. This is the Maple worksheet: 
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(1) 
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(2) 
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(3) 
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N.B. This last formula will be used in part II to calculate the global transfer function of the receiver.
The system powering
All leds are lowconsumption ones (“2 mA”). Battery voltage monitoring is performed by the 10 bit A/D converter included in the recorder MCU. In fact, only one battery is monitored, the one which is the most solicited (the recorder uses only one battery). This system – nonindispensable  will be described in part 3. The global consumption measured is about 45/50 mA without the recorder and 80/85 when the recorder is active. Therefore, we have more than two days of autonomy with 7 Ah batteries.
End of part 1 NOTES: 2) Ann.
Geophys.,
27, 34973507, 2009. www.anngeophys.net/273497/2009/.
Without entering in controversy, it is easy to observe
that spectrograms obtained with electrodes are not
superimposable with those obtained with the induction
coil. Since there are only two components for the EMF
field, we are able to think that the pickedup signal
with electrodes is really the E component. Naturally, to
explain how the « earthantenna » works is
another problem, tackled by Renato in
http://www.vlf.it/ed/earthprobes.html.
4) The
FFT
tool used is Spectrum Lab V2.76 b13, nicely adapted by
Wolfgang Büscher to allow reading of my raw file
(cf. Part 3 for more details). The FFT resolution is
0.062 Hz (4096 points) and the effective sample rate is
254.5 sps (nominal : 256). Notice that such a quality
involves a weak level of ionospheric disturbances, in
particular absence of magnetic storm.
6) Several
horizontal
lines are due to car passing on a road a few hundred
meters away. We can state that the Schumann frequencies
are in accordance with the ones indicated in the classic
scientific literature. In particular, the first SR is
centred about on 8 Hz, and has NOT increased to 12.5 or
13 Hz as claimed by the New Age gnosis.
15) Considering its low value (see http://sidstation.loudet.org/antennatheoryen.xhtml)
and the approximations made besides, needless to
consider the wire inductance.
>
> (1) > > (2) > > (3) > (4) Which is equivalent to the above formula. 18) Deduced from the
JohnsonNyquist formula : in which is “the noise passband”,
V the “noise voltage on the frequency band ”. This
formula is very simple and classical but however
its term meaning must be well understood.
Under the effect
the thermal agitation of its electric charges, a
resistor in thermal equilibrium acts as a v(t)
voltage generator. For statistical reasons, the
mean value during a time of this
variable voltage v(t) is nil (the electrons don't
know privileged direction in their erratic
motions), but not the RMS voltage . This
voltage v(t) can be expanded into elementary
sinusoidal functions according to the Fourier
theory. It's generally admitted that all
frequencies are equally represented (white
noise). But, only the component frequencies
included in the frequency band which
interests us will be harmful to us.If we name V the
RMS value calculated on the frequency bandat the
resistor terminals, the JohnsonNyquist formula
occurs. It's the voltage which would be
indicated by a RMS voltmeter with an exact
– no more and no less  band pass connected to the R
terminals.
The term density is justified : for
all physical quantify (homogeneous), a density
is always the ratio between the measure of a
certain portion of this quantify and the measure
of another quantify on which it depends. For
example, the volumetric mass density of a solid
is the ratio of its mass
by its volume, or, its mass per volume unity and
is expressed in in the MKSA
system. In the electrical noise case – as
resistor thermal noise , which is bandwidth
dependent, the noise power spectral density is
the noise power per unit of bandwidth, i.e. per
Hz. It's homogeneous to a voltage square divided
by a frequency unit, and thus, expressed
in . If we take
its square root to give voltages instead of
powers, we obtain the voltage spectral density,
which is expressed in. Then, we
have .
