Sensitivity of Multi Turn Receiving Loops
ABSTRACT:
Multi turn wire loops are often used as low frequency receiving antennas. Applications such as geophysical research, oil exploration and survivable communications require maximum sensitivity of receiving loop antennas. The loop sensitivity decreases as frequency decreases, becoming a formidable problem below 1 Hz. Basic electromagnetic theory is developed here as it relates to electrically small multi-turn loops at low frequencies. Simple algebraic expressions are produced describing the sensitivity of loops in simple geometries. The concept of antenna factor (effective aperture) is introduced, which allows comparison of different loops, and conversion of observations to common magnetic units of measure. It is hoped this work will be a useful reference to geophysical researchers, and to anyone designing loops for low frequencies.
INTRODUCTION :
Magnetism is manifested as a ‘field of vectors’, that is, any point in the magnetic field has not only a magnitude, but a direction in space. The four Maxwell equations describe how electric and magnetic vector fields behave and interact. These "fields" are actually primordial root forces and motions of our spacetime continuum. It is well said that all the laws of physics can be derived from the Maxwell equations, given here in integral form:
(a) ò c B · n da = qm = 0 No magnetic monopoles
(b) ò c D · n da = qe Sum of flux is electric charge.
(c) ò H ´ n da = ¶ /¶ t V Curl of magnetic field due to electric flux change
(d) ò E ´ n da = ¶ /¶ t I Curl of electric field due to magnetic flux change
According to Maxwell, an electric field cannot change without creating a magnetic field, and a magnetic field cannot change without creating an electric field. Any change in one force field creates a vortex or wake in spacetime appearing as the other aspect of the force. Electromagnetic waves have both electric (E) and magnetic (H) components, and propagate as ripples in the fabric of our continuum. The E and H aspects appear 90 degrees apart in space dimensions and in phase in the time dimension. Loops of wire are often used as antennas to interact with and detect the magnetic aspect of the electromagnetic force.
Suppose we have a varying magnetic field 'out there' which we want to detect and measure. This field may originate naturally or artificially. To make the analysis more tractable, the loop is assumed to be electrically small, the dimensions being much smaller than a wavelength of the frequencies of interest. We also take the distance to the source as being much larger than the loop dimensions. These conditions are usually well satisfied in geophysics. We will use vector calculus to derive from first principles the response of such a loop. Those unfamiliar with this branch of mathematics may skip down to equation (10), where the going gets easier.
THEORY OF MAGNETIC LOOPS :
Magnetic field intensity, H, expressed in units of amperes per meter, produces a magnetic flux density , B, expressed in volt seconds per square meter. Flux is proportional to applied field.
(1) B = m H
m , expressed in henrys per meter, is the magnetic permeability of the medium, the analog of electric permitivity. We will let m equal mo, the permeability of a vacuum (spacetime itself). This assumption is well justified for air core loops surrounded by non magnetic media, including air, water, dirt, vegetation, etc.
The total magnetic flux, F, in volt-seconds, threading an area is the flux density integrated over the area. The vector n denotes a unit vector normal to da, the element of the surface being integrated over.
(2) F = ò B · n da
Voltage around a loop is proportional to the rate of change of the amount of flux threading the loop area. When multiple turns are in series, the total voltage is the sum of the individual turns.
(3) V = N dF /dt
Notice from (3) that a motionless loop in a constant dc field produces no voltage. Combining these three equations gives an expression for the terminal voltage of a multiturn wire loop. The vector normal component of the H field is integrated over the loop area, and differentiated by time.
(4) V = m0 N d/dt ò H · n da
When the H field is uniform over a planar loop, we can take H out of the integral and express its vector normal component as the magnitude times the cosine of the angle between the H vector and the loop axis.
(5) V = m0 N cosq d/dt |H| ò da
and the integral becomes simply the loop area.
(6) V = m0 N A cosq d/dt |H|
Most of the calculus is solved, but the time derivative of H remains. We can reduce it to simple algebra by examining a discrete frequency (wt) component of H, with peak amplitude H0.
(7) H = H0 sin(wt)
Which transforms equation (6) into:
(8) V = m0 N A cosq d/dt (H0 sin(wt))
So we now rid ourselves entirely of calculus:
(9) V = m0 N A cosq H0 w cos(wt)
Taking the magnitude of the signal, we get loop terminal voltage as a straight algebraic product of six terms;
(10) V = 2pm0 N A H0 f cosq
Where:
2pm0 is a constant.
N is the number of turns
A is the loop area, in square meters.
H0 is the applied magnetic field, in amperes per meter.
f is the frequency, in Hertz.
cosq is the cosine of the angle between the loop axis and the field.
The persistent product of N and A are the only remaining terms which describe characteristics of the loop itself. This product suggests a figure of merit for loop antennas, the "effective aperture", Ae , which is the physical area times the number of turns.
We can now express the on-axis sensitivity of a loop, which is the terminal voltage divided by the applied magnetic field, as the product of only three terms:
(11) V/H0 = 2pm0 f Ae
Where:
V/H0 is the output voltage per unit magnetic field strength applied
2pm0 is a constant = 7.89 x 10-6.
Ae is the loop effective aperture, in square meters.
f is the frequency, in Hertz.
Equation (11) clearly shows the problem of loops at low frequencies: as f approaches zero, so does the loop voltage! Although we can’t do much to change 2p, we might try increasing m above m0 by using a ferrite loop core, but this becomes impractical with large areas. Our only recourse is to increase the effective aperture.
EFFECTIVE APERTURE :
We now have a rigorously derived expression for loop antenna sensitivity, reduced to the simple product of three terms, a constant, the frequency, and the effective aperture, which is the antenna factor. By knowing the effective aperture, we can relate the loop output back to the magnetic field strength. We can also compare the sensitivities of different loops, making possible the correlation of data from researchers using different loops. This effective aperture is simply the loop area times the number of turns, expressed in square meters.
The areas of some common loop geometries are:
MAXIMIZING Ae :
For maximum sensitivity, we want maximum effective aperture. Practical limitations dictate the effective aperture we can achieve. For example, we might be limited to 5lb loop mass of copper wire, and can handle wire as small as #30 AWG. What is the maximum effective aperture we can achieve?
From the NIST copper wire tables we get: wire diameter = 0.010 in, length = 16435 ft, and resistance = 526 ohms. For a circular loop, the wire length and loop area are:
(12) N = l / (p d)
(13) A = p/4 d2
So the antenna sensitivity for a fixed wire length is :
(14) Ae = N A = l d
4
The d term appearing in the numerator tells us to deploy a fixed length of wire as a single turn for maximum sensitivity. The diameter will then be :
(15) d = p/l = 5230 ft
With an antenna factor of :
(16) Ae = N A = A = 21,500,000 ft2 = 2,000,000 m2
A large effective aperture but our trepidation in handling a one mile diameter loop of #30 AWG wire leads us to now limit the loop diameter to ten feet. Equations (12) through (14) give us:
the turns N = 523
the area A = 78.5 ft2
the aperture Ae = 41,100 ft2, or about 3800 m2.
This is a manageable structure, but sensitivity has been reduced 523 times. Some rules of thumb for loop sensitivity are:
For a fixed number of turns:
Sensitivity goes up as loop diameter squared, and up as wire length squared.
For a fixed wire length:
Sensitivity goes up as the loop diameter, and down inversely as the number of turns.
For a fixed loop diameter:
Sensitivity goes up as number of turns, and up as wire length.
Which shows that "Turns are good, but size is better!" and "Use as much wire as you can!"
PRACTICAL EXAMPLES AND CONSIDERATIONS :
The ‘octoloop’ is an easily built, well shielded, VLF loop, small enough to gimbal, which was my primary design goal. The design files for the ‘octoloop’ are on the LWCA BBS. The octoloop characteristics are:
A = 3.42 m2
N = 50 turns
Ae = 171 m2
I also built a fixed loop of six pair telephone wire 160 feet in diameter in the backyard:
A = 1865 m2
N = 12 turns
Ae = 22,381 m2
Obviously, the fixed loop is more sensitive, by a factor of about 130. In antenna terms, this is a gain increase of about 42 db, a substantial improvement! However, if by gimballing the octoloop, I can get a 50 db deep null in interference, and stay above my receiver noise floor, the octoloop still has an 8 db advantage. On the other hand, with the fixed loop, if I filter out the power grid interference, I can go 130 times lower in frequency before falling below the thermal noise floor.
The octoloop is more useful for sferics and OMEGA reception, but the fixed loop is capable of deep infrasonic frequencies and geomagnetic work. Below some point in the spectrum, one must forego gimballing and portability to gain very large antenna effective apertures. Larry Grant’s "Life at 1200 Turns" loop probably has an aperture near Ae = 2000 m2, approaching the practical limit for portable loops. In oil exploration, loops of several hundred feet of multiconductor cable are transported by rolling them up on spools.
OTHER FACTORS:
If an electric current flows in the loop, the terminal voltage and the sensitivity will be modified from that derived above. Current may be drawn by resistively loading the loop output, which will decrease the available voltage. Parasitic capacitance, as well as external capacitance will also cause a current flow, but one which is leading in phase. Capacitance neutralizes the lagging phase of the loop inductance and causes a frequency resonance, increasing the aperture while reducing the bandwidth. The magnitude of these tuning effects are maximum when loop resistance is minimum. Capacitively tuned loops are useful for their sensitivity to a single discrete frequency.
Mechanical motion of a conductor in a steady DC field induces a voltage, leading to microphonic effects. Microphonics may be reduced by structural stiffening and damping, to reduce vibrational resonances and shift them out of the frequency bands of interest. Many loop structures will have an axis of minimum vibrational response, which may be aligned with the local field to further reduce microphonics.
Temperature effects in dissimilar metallic junctions cause Seebeck voltages to be produced, which generally have time constants as long or longer than the thermal cycle. Temperature also causes voltage drift in high gain DC coupled amplifiers. Thermal effects may be controlled by isolating amplifiers and metallic junctions from temperature changes, by DC blocking, or by chopper stabilizing DC amplifiers.
DIRECTION OF FUTURE WORK:
The fixed loop was originally intended for OMEGA (the 10 to 14 kHz squeal) reception, but Larry Grant and Bob Confrey have me interested in geomagnetics. Presently, I am working out an improved preamplifier design for geomagnetic frequencies. I am convinced the best way to go is a fixed moderate gain first stage at the loop, using a biomedical instrumentation amplifier such as Analog Devices AD620, and then put more gain with adjustments, filtering, etc in a separate indoor unit. For large aperture loops, the preamp must tolerate very high 60 Hz hum levels without desensing or intermodulation.
Also I am looking for yet more antenna aperture. Just today, I screeched my truck to a halt and leaped into a muddy excavation wearing my good pants because I believed I saw an abandoned length of 600 pair telephone trunk cable. The area of my backyard is about 6300 m2, which enclosed with 1200 turns (600 pair) would give Ae = 7,500,000 m2. This would be the most sensitive loop I know how to make here, having three times the aperture of the hypothetical one mile turn of wire discussed above, and should be useful to below 0.01 Hz.
A DC block below 0.001 Hz or so will be required to remove the Seebeck potentials from 1200 spliced joints, and the antenna may be buried to reduce microphonics. I am at a loss for a feasible method of removing seismic microphonics, which I believe will appear as the next envelope boundary, although seismic microphonics may in themselves be a worthwhile study.
CONCLUSION:
The sensitivity of loop antennas at low frequencies has been mathematically derived, and expressed in practical terms. The concept of effective aperture, and how to maximize it has been presented. It is my heartfelt recommendation that researchers calculate and report the effective apertures of the loops they use, and refer their measurements to loop terminal voltage. In this way, all geomagnetic observations can be converted to a common unit of measure.
OTHER INTERESTING EFFECTS: INSECT WING SOUNDS
A very interesting electrical effect easily observed with the WR-3/3E are insect wing sounds caused when insects such as bees, flies, and mosquitoes fly within a couple of feet of the WR-3/3E whip antenna. The resulting sound is a buzzing sound very similar to what can be heard by ear, however, this effect is caused by electrostatic discharges each time the insect's wings flap. It is thought that electrostatic charges (static electricity) are collected on the insect's wings and then and dumped during each wing beat, creating a "modulated" electrical field around the insect at the same frequency as the wings beat.
Large insects, such as wasps, Yellowjackets, Bumblebees and honeybees, make particularly strong buzzing sounds in the WR-3/3E headphones-easily heard when those insect fly within 3-4 feet (1 meter) of the WR-3/3E antenna. High-pitched Mosquito wing beat sounds can be heard the small insects fly within a few inches of the WR-3/3E Receiver's whip antenna. Certain kinds of flies and other insects have much more electrostatic "buzz" from them than other kinds - Bees and Horse Flies, in our observations, have the loudest "buzz" in the headphones! This may also have something to do with the composition of the insect's wing, with certain type of insect wings more prone to static electricity accumulation and subsequent discharge. There may also be insect bodily electrical discharges generated within the insect's wing muscles that contribute to this effect, and a some believe that the insect's carapace ( outer body shell) has a piezo-electric effect similar to quartz crystals, but not much is understood about this phenomenon yet and further studies are encouraged
ABOUT HEADPHONES:
The WR-3/3E's output jack requires a pair of STEREOheadphones (for dual-ear monaural sound). The most convenient and portable kind arethe lightweight "personal-stereo" type of stereo headphones that have a 3-conductor 1/8 inch (3.5 mm) plug. These kinds of headphones have an impedance of 8 or 16 ohms, are widely available, lightweight, match very well with the WR-3/3E and are highly recommended. While the output of the receiver is not true "stereo" but "dual-ear monaural" sound, we designed the WR-3/3E to match with these very commonly available headphones. Not all stereo headphones work the same with the WR-3/3E, and ones with 32 ohms impedance or those that are rather "cheap" may have too low of a volume to be very satisfactory. It is highly recommended that you try out several pairs of good quality for greatest volume level output and sound quality.
Beware of the kind of headphones which are small, "plug"-like types designed for bass-boost listening -- these can emit excessive/harmful audio output from sferics and other sounds and they may also cause deafening audio feedback. Some of these kind, however, work fantastically with the WR-3/3E when used properly. But...these are the words of one regular WR-3E listener of natural VLF radio and is very good advice: Do not use "earplug" phones with soft, flexible seals that fit tightly within the ear to improve "bass" audio effects, which might inflict serious hearing damage on very powerful, sharp "cracks" due to strong, local discharges. The cheap kind of phones with the openly vented foam pads are acoustically perfect and pretty safe. Other kinds of headphones such as full-size 8-16 Ohm stereo headphones designed for use with stereo component systems will work very well, though you may need an adapter to convert their 1/4 inch plug (7 mm) to the 1/8 inch (3.5 mm) plug.
The audio volume from the WR-3/3E may be considerably louder with full-size headphones compared to some mini-stereo style headphones - plenty for most listeners-especially if their impedance is 8 or 16 ohms. Full-size headphones also give you the advantage of enclosing your ears better than mini-headphones, thus shutting out exterior noises and enabling you to hear natural VLF phenomena better. High-impedance (600 to 1000 ohm) audiophile headphones will not work with theWR-3/3E because the volume is too low and inadequate due to mismatch with theWR-3/3E's headphone amplifier. However, you could employ an audio transformer to convert the 8-ohm output of the WR-3/3E to 1000 ohms for far better results with high-impedance headphones if you're willing. Whatever headphones you choose to use, they MUST have a 3-conductor (stereo) plug.
Two-conductor (monaural) plugs will not work conveniently without an adapter to convert the 2 conductor (mono) plug to the 3-conductor (stereo) plug, although in a pinch you can use headphones with a mono plug (or patch-cord) by plugging them in about half way into the WR-3/3E headphone jack, perhaps with the assistance of a non-metallic washer and tape to hold the mono plug out at the proper distance for it to make good contact with the WR-3/3E's headphone jack. If in doubt about the type of plug on the headphones, it should have three metal bands separated by two thin black plastic insulators. If it only has two metal bands separated by one plastic insulator (mono plug), it won't work without an adapter unless you plug it in only half way-enough to make contact.
BATTERY POWER:
A 9-volt (rectangular snap-type) PP9 ALKALINE battery is best recommended for use with the WR-3/3E and will last from 30 to 40 hours of use depending on the audio gain level. Other types of 9 volt batteries such as "General Purpose" types will not have as long a life as Alkaline types but will last quite a long time anyway (about 10-15 hours of listening).
Rechargeable Ni-Cad 9-volt batteries can be used, but are not recommended because of their limited discharge life-which is shorter than an alkaline battery, and also due to their lower voltage (8.4 volts). Battery replacement is achieved by removing the four screws that secure the receiver's top cover and snapping on a new battery to the battery clip and carefully re-fitting the cover. Be sure the WR-3/3E is switched off to avoid circuit damage if you accidentally connect the battery backwards, ESPECIALLY is you are using a Ni-Cad or other newer rechargeable types which have high current.
TAPE RECORDING:
WR-3: A tape recorder with an aux./line-level input can be used to record from the WR-3 headphone jack, but for MICROPHONE inputs, an attenuating patch cord will be required-as the audio level present in the headphone jack is not matched for mic. inputs on most tape recorders. Use an attenuating patch cord and adapters which will convert a stereo (3-conductor) 1/8 inch jack to the input jack style your tape recorder requires. Failure to use an attenuating patch-cord will result in recorder input overload and terrible audio quality.
WR-3E: The WR-3E has a microphone-level RCA-type output jack. This output has been designed to work well with 600-1K ohm MICROPHONE inputs (anything from small micro-cassette and standard analogue cassette recorders to DAT recorders). BOTH units: The output level of the WR-3 and WR-3E from its headphone jack is compatible with LINE-LEVEL inputs however, and can be patched directly into the "auxiliary" or line-level input of recorders having this input option, using appropriate adapters. Use SHIELDED coaxial-style audio patch cords only, whether standard or attenuating type for the finest results. Additional receiver grounding may be needed if a tape recorder is used to prevent audio feedback from the tape recorder into the WR-3/3E whip antenna. Better results are obtained if the WR-3 or WR-3E audio gain is kept at lower levels and not at full gain.
ANTENNA
Normally, the telescoping whip antenna should be used fully extended for maximum receiver sensitivity and held vertically at arm's-length away from you, with the receiver at chest or shoulder level. In very quiet locations far removed from powerline hum, up to (but NO more than) 6 ft/2 M of additional antenna wire may be clipped onto the whip antenna screw post and suspended vertically (via wooden pole or on the outside of a white PVC pipe), if the wire is away from obstacles or foliage. Alternatively, a longer whip antenna can be substituted in place of the 33 inch antenna if it has the same type of threaded base. The WR-3/3E will not match well with longer wires or with ones laid out on the ground, on bushes or in trees, as its antenna input circuit has been optimized for a 3-10 foot (1-3 meter) vertical "whip" antennae. The telescoping whip antenna needs only to be screwed on LIGHTLY to its screw-post mount. Over tightening the antenna may cause the antenna base and or/ screw-post mount to become loose.
SAVE YOUR EARS
Please avoid turning on the WR-3/3E VLF Receiver indoors with the headphones on your ears because this will result in a loud "hum" or "buzz" being heard in the headphones due to close proximity to 50/60 Hz AC lines, and this loud volume can be harmful to hearing. (Again, it's best to operate the WR-3/3E at least 1/4 mile/500 meters from AC power poles to avoid excessive hum pickup). The difference in sound level of the loudest lightning sferics to the most subtle Natural Radio sounds can be substantial. Resist the temptation to "crank up" the volume level. Excessive headphone volume, whether from the WR-3/3E VLF Receiver or other audio devices, can result in eventual hearing loss. Enjoy your WR-3/3E and the sounds of VLF "Natural Radio." Listening to the sounds of whistlers, tweeks, chorus, and other Natural Radio sounds under a star filled sky or while watching aurora or sunsets and sunrises increase one's wonder and appreciation of the natural beauty of Earth -- the WR-3/3E is simply a tool to enhance sensory awareness of Earth's natural beauty further and into another "realm," whether for research purposes or aesthetic enjoyment. Awareness of WHY these VLF radio sounds happen and their origin, much of it gained through scientific study and learning, helps to satisfy our curiosity about them. The necessity of taking the WR-3/3E out of the electrically "polluted" urbanized areas and into more open areas further into nature adds to the enjoyment and appreciation of the natural environment. Happy Listening!
WR-3 & WR-3E VLF RECEIVER SPECIFICATIONS:
Receive frequency range: 0.1-13 kHz (100 - 13,000 Hz)
WR-3: peak frequency approx. 1.0 kHz with roll-off below 400 Hz and above 2 kHz. RFI protected to reduce LF-VHF broadcast and utility station overload and IMD. Audio Output: Maximum 100 mW into 16-Ohm stereo headphones WR-3E: Broad-filter mode: peak freq. approx. 1.3 kHz. Switchable to approx. 3.5 kHz peak in high-pass mode. Additional filter for further peaking 1.5 to 2 kHz range while reducing below-180 Hz audio. Headphone jack: 1/8 inch (3.5 mm) stereo (3-conductor) audio jack Size: 4.5 x 2.5 x 1.2 in. (11.5 x 6.5 x 3.7 cm) not including antenna length Weight: Approx. 8 oz. (230 g) with battery Antenna: 33-inch (84 cm) telescoping whip (detachable) Antenna input impedance: Approx. 20 Megohms @ 1 kHz Power: Use a 9 volt battery for 15-40 hours of listening time depending on type (@7-20 mA current consumption) Acceptable Headphone Impedance: 8-32 Ohms (16 Ohm mini-stereo type recommended) Replacement telescoping whip antennas may be purchased for $8.00 each (includes S&H) Additional References and recommended reading for those interested in additional information about natural VLF phenomena:
The Lowdown, published by the Longwave Club of America (LWCA), is a monthly publication for people interested in this part of the radio spectrum. In addition to articles on Natural Radio sounds including writings by Mike Mideke, the publication covers lowpower Low and Medium Frequency experimental transmitting of voice and data, receiving techniques, radio wave propagation, and articles about controversial topics such as military radio transmissions, etc. Membership is $18.00 per year ($26.00 U.S. Overseas) from: LWCA, 45 Wildflower Road, Levittown, PA 19057. Ionospheric Radio Propagation, by Kenneth Davies, National Bureau of Standards, Monograph 80 (1965/1990). Excellent text on radio propagation including verylowfrequencies. The old 1965 edition is nearly impossible to find but a new 1990 edition (priced at about $65.00) is available. Write to: Space Environment Services Center, NOAA R/E/SE2, 325 Broadway, Boulder, CO 80303-3328. Phone (303) 4975127. Robert A. Helliwell ("Father" of VLF research), Whistlers and Related Ionospheric Phenomena, Stanford University Press, 1965.
A very comprehensive introduction to whistler research before space flight plus the beginnings of "spaceage" research documenting early satellite data. It is accurate and concise with numerous diagrams and illustrations. The book has more information than a beginner would likely immediately use or pursue, but is fascinating nonetheless. ***RARE BOOK*** It is difficult to find copies of this book. Check large libraries. They may be able to obtain the book using the interlibrary loan system. Syun-Ichi-Akasofu, The Dynamic Aurora, Scientific American, May 1989, pp. 90-94. Describes the functioning and structure of the magnetosphere and causative factors in the generation of Aurora. Examines the magnetosphere as an electrical "generator," field-aligned currents, electrojets and substorms, why Aurora are "curtain" shaped. While not directly examining VLF natural emissions, the article helps in understanding the magnetosphere and lends insight into its role in naturally occurring VLF emissions. Jeremy Bloxham and David Gubbins,
The Evolution of the Earth's Magnetic Field, Scientific American, December 1989, pp. 68-75. Examines geologic processes which generate Earth's magnetic field, describes the shape and functioning of the geo-magnetic field, magnetic field drift, magnetic lines-of-force, magnetic field polarity shifts, and so on. George John Drobnock, Radio Waves from a Meteor?, Sky & Telescope, March 1992, pp. 329-330. Reports on the author's experiences with a homemade H-field "loop" VLF receiver and a noted "hiccup in the background noise" as well as a "swoosh" sound correlating with the sighting of a meteor on two separate occasions. Examines the possibility that large meteors may disrupt the magnetic field because of their ionized "wakes" created by swift passages into the upper atmosphere, causing VLF radio emissions.
An under-researched and still-controversial premise demanding further attention. Russ Sampson, Fire in the Sky, Astromony, March 1992, pp. 38-43. Beautifully illustrated article about the phenomenon of Aurora and the geo-physical processes responsible. Written more for the layman than the Syun-Ichi-Akasofu article above, but essentially covers the same topics including the influence of solar activity on the magnetosphere. The article is replete with excellent amateur photographs of various colors of aurora. Since Natural Radio emissions arise from the same processes, this is also recommended reading to gain understanding about the magnetosphere.