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Mathematical Analysis of Super Low Frequency Ground Loop Receiving Antennas

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Mathematical Analysis of Super Low Frequency Ground Loop Receiving Antennas Appl. Math. Inf. Sci. 7, No. 3, 1087-1092 (2013) 1087 Applied Mathematics & Information Sciences An International Journal ⃝c 2013 NSP Natural Sciences Publishing Cor. Mathematical Analysis of Super Low Frequency Ground Loop Receiving Antennas Hui Xie1, Ming Yang2 and Long Ji3 1Electronics Engineering College, Navy University of engineering, Wuhan 430033, China 2Unit 91404 of PLA, Qinhuangdao 066000, China 3702 repair factory of Navy, Shanghai 200434, China Received: 20 Sep. 2012, Revised: 30 Dec. 2012, Accepted: 14 Jan. 2013 Published online: 1 May 2013 Abstract: A compact super low frequency receiving antenna in the form of a loop can be built for which the induced atmospheric noise voltage dominates both the antennas thermal noise voltage and preamplifier noise. Air-core and ferromagnetic-core ground loop receiving antennas were introduced, and mathematical calculation methods to analyse the ENF spectrum of the atmospheric noise and thermal noise were derived. The research identified the roles played by the various contributing factors such as antenna dimensions, wire conductivity, ferromagnetic-core permeability, etc. One air-core and three ferromagnetic-core antennas were designed, and a compromise core design was recommend, which takes advantage of the improvement that ferromagnetic cores can provide while largely avoiding the disadvantages. Keywords: super low frequency, loop receiving antenna, air-core loop, ferromagnetic core loop 1. Introduction factor of self-resonance may increase the output signal voltage, this is more than offset by a concomitant increase The two most conventional super low frequency (SLF) in the resistance of the antenna [4]. Usually, therefore, at receiving antennas are the whip and the loop. The SLF, it is better to arrange for the self-resonant frequency extremely high input impedance of the whip, however, of the loop to be large compared with the operating causes the problems of sensitivity variation and noise due frequency. Then the simple flux linkage relationship is to static electricity [1,2]. These, coupled with the fact that accurate. This may mean that the output signal voltage is it is suitable only for above-surface reception make it not high enough compared with the noise voltage of the unattractive compared with the loop. preamplifier. In that case, a signal transformer can be used The loop receiving antenna is the receiving to raise the voltage level. If the transformer were perfect, counterpart of the transmitting loop antenna. The received it would not change the figure of merit of the antenna. signal voltage across the terminals of the loop is given Since it is not perfect, some increase in thermal ENF simply by the time rate of change of the signal flux occurs which must be balanced off against the relative density linking the turns of the loop [3]. If the stray decrease in amplifier ENF. For geophysical applications, capacitance of the loop is large enough, however, or if the the frequency of interest is usually so low that the self-inductance of the loop is large enough, the dominant noise is the amplifier noise. Then, increase the self-resonant frequency of the loop may be low enough to number of turns to bring the antenna to self-resonance at be commensurate with the operating frequency. Then the the frequency of interest results in a net decrease in the simple flux-linkage formula is inaccurate. But since total ENF, even though the thermal ENF is increased [5]. appreciable stray capacitance lead to a non-uniform This effect is less likely in SLF. It also has less appeal current distribution along the wire of the loop winding, because a large front-end bandwidth may be required to the signal induced in one turn is not perfectly in phase take full advantage of the impulsive nature of atmospheric with that induced in another. The result is a reduced figure noise. of merit. Thus even though the voltage amplification ∗ Corresponding author e-mail: [email protected] ⃝c 2013 NSP Natural Sciences Publishing Cor. 1088 H. Xie, M. Yang, L. Ji: Mathematical Analysis of Ground... sc 2. Air-core loop antenna To obtain this expression, the ratio s has been d 2 identified with the ratio d 2 by multiplying both numerator The simplest loop antenna is the air-core loop. The c wm d resistance Ra of the antenna is given by and denominator by 2 0 and denoting by c the skin depth of the conductor used to wind the loop antenna. 2pan2 Substituting into formula (5), the dimensions R = (1) 2 a Ac = 0:01m , a = 0:5m, the maximum possible filling scCwAc factor of unity and the frequency 100Hz, one finds where a is the average turn radius, n is the total number of Re = 0:038 if the winding is with copper wire Ra turns, sc is the conductivity of the wire conductor, Ac is 8 (sc = 0:580 × 10 S/m ) and the antenna is immersed in the cross-sectional area available for the winding and Cw ocean water (s = 4 S/m ).Since ocean water is a typical is the filling factor. (Cw is the factor, less than unity, by worst case and since, also, the antenna example chosen is, which the actual total conductor cross section is smaller at a weight of 280 kg, atypically large for this type of Ac than the winding area. Thus Cw = , where Aw is the nAw receiving antenna, the external resistance would usually cross-sectional area of a single wire.) If the winding area be even less than the already small 3.8% of the winding Ac is square of side length s, the inductance La of the loop resistance. is given by [6] The absolute magnitude V of the voltage induced in m an2 p L = 0 (2) the loop by an external uniform magnetic field strength H a 2 is, by Faradays law, where p depends on the former factor m f of the loop, s V = Awm0Hn (6) defined as m f = 2a , for m f < 0:2, the formula for p is Assuming the loop axis is parallel to the magnetic field m2 f 8 2 − lines and where A is the average turns area. Thus, since the p = (1 + )ln( 2 ) + 0:41m f 1:70 (3) 6 m f open-circuit thermal noise voltage spectral density Sv(w) is This square cross-section coil is compact, and the Sv(w) = 4kbTkRa (7) computation of its inductance is simple. However, it has × −23 the disadvantage that its self-inductance and stray Where kb is Boltzmanns constant (1:38 10 J/K ) capacitance are both larger than they need be, making the and Tk is the absolute temperature in Kelvins, the spectral w self-resonant frequency lower than it would be were the density Sh( ) of the magnetic equivalent noise field is 4kbTkRa winding spread out further. The spreading could be done given by 2 . By virtue of formula (1), this can be (Awm0n) axially, to obtain a solenoid winding, or radially, in which rewritten as case a pancake type of coil would result. For these, and 4pk T d 2a S (w) = b k c other, more general shapes, the inductance may be h 2 (8) wm0CwAcA calculated from Grovers [7] formulas and Tables. 2 2 The radiation resistance of the loop in air is so small in A =m =Hz, provided Re is negligible compared with −2 2 compared with the resistance of the winding that it is Ra. For the dimensions (Ac = 10 m ;a = 0:5m ), wire 8 completely negligible. On the other hand, since the conductivity (sc = 0:58 × 10 S/m ), filling factor (1) and wavelength is so long, the loop antenna is unlikely ever to frequency (100 Hz) used in the earlier example, and with be so far from the earth or other conducting media for the Tk = 300K, this works out to be w × 19 2 2 radiation resistance to be larger than the mutual resistance Sh( ) = 2:34 10 A =m =Hz . This is several orders of between the loop and the closest conducting medium. magnitude smaller than atmospheric noise and justifies However, even when it is completely buried in natural the statement made earlier that this loop antenna is Earth media, for which the skin depth is large compared atypically large. with the loop radius, the total external resistance of the If the loop is being used to receive the magnetic field air-core loop would still usually be small compared with component of a plane wave in a conducting medium, the its winding resistance. electric ENF component of a plane wave in a conducting w For example, if the external resistance Re of the loop is medium, the electric ENF spectrum Se( ) corresponding given by [8] to formula (8) is given by multiplying formula (8) by the wm0 8n2 a4 squared magnitude s of the wave impedance in the Re = (4) medium. Thus 3as d 4 Where s and d are the conductivity and skin depth of the 2pwm k T d 2d 2a S (w) = 0 b k c (9) surrounding medium. Thus the ratio Re can be written, e 2 Ra CwAcA from (1) and (4), as in V2=m2=Hz. On the other hand, if it is being used at, or R 4A a2C above, the surface for the reception of long-range SLF e = c w (5) pd 2d 2 transmissions, the electric ENF spectrum can be Ra 3 c ⃝c 2013 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 7, No. 3, 1087-1092 (2013) / www.naturalspublishing.com/Journals.asp 1089 expressed either in the manner of formula (9) (if the The SLF loop antenna is versatile, in that it operates horizontal electric field is taken as the reference) or by equally well above or below the surface.
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