KIMURA, Koichi

KIYONO, Takeshi

KOBAYASHI, Keiichi

English

Kyoto University Research Information Repository

RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Study on the Induction Method ofElectrical ProspectingKIYONO, Takeshi; KIMURA, Koichi; KOBAYASHI,KeiichiKIYONO, Takeshi ...[et al]. Study on the Induction Method of Electrical Prospecting.Memoirs of the Faculty of Engineering, Kyoto University 1954, 16(3): 147-1561954-09-25http://hdl.handle.net/2433/280304147 Study on the Induction Method of Electrical Prospecting By Takeshi KIYONO, Koichi KIMURA and Keiichi KOBAYASHI Department of Mining Engineering (Received April, 1954) 1. Introduction There are many kinds of electromagnetic prospecting methods, but the funda-mental quantities to be measured in these methods are the amplitude, phase angle and direction of the magnetic field produced by an.orebody in the alternating field of an exciting loop. In practice, the resultant vector composed of the primary field of loop and the secondary field due to orebody is observed. However, to iµterprete these results, it is necessary to know the electromagnetic behaviours of orebodies in the primary field. In this paper, some of the experimental and theoretical results concerning the electromagnetic induction method are reported. The effects of the electromagnetic properties of spherical orebody on the secondary magnetic field are discussed. The distribution of the field produced by a disc is also clarified theoretically ar-d compared with the experiment. 2. Theoretical Considerations (1) Secondary field due to a spherical body. If we assume that (1) the primary field surrounding the sphere is uniform and vertical, (2) the electromagnetic property of the medium surrounding the sphere is similar to that of the air, and (3) the displacement current in the medium and sphere can be ignored for low frequencies, then the three components Hr, H8 and H4, of the secondary field at an arbitrary point P(r, fl, rp) outside the sphere beome as follows (see Fig. 1) : a3 ' H,. = -QHo 3 cos fl cos (wt+iJ), l H9 = - ~ Q;0 :: sin O cos (wt+iJ), J H4,=0, (1) X Fig. 1. 148 Takeshi KIYONO, Koichi KIMURA and Keiichi KOBAYASHI where w denotes the angular frequency of the field, a, a and µ are radius, conductiv-ity and permeability of the sphere respectively. And the primary field is expressed by H0 cos wt. The factor Q and phase angle a in Eq. (1) are given by and where and X= - ![Cl-µ) (1-2µ)g (cosh 2p--cos2p)-p(sinh2p+sin2P) + p 2 (cosh 2p+cos 2p) }-p3 (2+ µ)(sinh 2p-sin 2P) +2p4 (cosh 2p-cos 2P)], Y = 31 2 { (cosh 2p-cos 2p)-p(sinh 2p+sin 2P)}, (2) (3) (4) (5) j = (l-µ) 2g (cosh 2p-cos 2p)-p(sinh 2P+sin 2p) +P 2 (cosh 2p+cos 2p)} -2(1-µ)p 3 (sinh 2p-sin 2p) +2P4 (cosh 2p-cos 2P), ( 6) p = (a/c)-,/2rcwµa • (7) From Eq. (1), we obtain the following expressions for the horizontal and vertical components of the secondary field : where and Hp = - "; 3pd, r (8) (9) (10) (11) Fig. 2 shows the distribution of the secondary field calculated from Eqs. (8) and (9). From the above expressions, it can be seen that the secondary field by the sphere is equal to that of the magnetic dipole which is placed at the center of the sphere and the moment of which is m in Eq. (11). Substituting Eqs. (2) and (3) into (11), we get Study· on the Induction Method of Electrical Prospecting Fig. 2. Secondary field due to a spherical consuctor. 0 2 .5 .o :,.. "' 1 0 -ns _, .0 I µ=10 -....[', '\ X ---y P=U01987a.-,u,'/ p ,1 ~._~ µ=1' ~~ I tk -µ=1001'\ I'\ ~ Ir\ .:'\ ;-,:::,., mt r-µ=1 Fig.13. 149 il µ=100 '\ r\ ~ \ 10 r\ I ,lJ "\ --µ=Ir/Jr- ::,,./ r,oo·" I (12) As the frequency, conductivity and permeability are included in X and Y, we can assume their effects on the secondary field as the effects on the function X and Y. Fig. 3 shows the real part X and imaginary part Y of Qet8 as functions of p and for various values of permeability µ. (2) Secondary field due to an oblate spheroid. The assumptions made in the treatment of the field of a sphere hold also for a spheroid. In addition, we assume, for the sake of simplicity, that the conductivity of the spheroid is infinite. In such a case the expressions for the components of secondary field at the point (A, µ, <jJ) become as follows '· where A, µ, <jJ are the oblate . spheroidal coordinates as shown in Fig. 4, and ).0 is the parameter for the surface of spheroidal conductor. The relation between A, µ, <p (13) and the rectangular coordinates x, y, z is given by A=const X = c✓Iz+I ✓l-µ2 cos <p, y=c✓A2 +1 ✓l-µ2 sinrp, (14) z=dµ, 2a where c is the distance between the center and a focus. Fig. 4. 150 Takeshi KIYONO, Koichi KIMURA and Keiichi KOBAYASHI Let 2a and 2b be major and minor axes of the spheroid, then c = ✓a2 -b2 ,} Ao= b/c. (15) If we let Ao be zero, the spheroid degenerates to a disc of radius a. For this limiting case, Eq. (13) becomes as follows : 'H - H ~ µ - Q1(. ') t ,\ - - 0 ✓ l2 2 1 Ill COS (J) , 71: /l + µ H,,. = -H0 ~ /~ 2- µ: Q1 (iA) cos wt. -n: 1/ /l +µ l J The Legendre functions of the second kind included in these equations, are Q1(iA) = A cot- 1A-1, Ql(iA) = ✓A2 +1 cot- 1J- ✓)+l' } (16) (17) The horizontal and vertical components of the field produced by a disc can be deduced from Eq. (16) : (18) Figs. 5 and 6 show the radial distribution of the horizontal and vertical com-poent of the secondary field produced by a horizontal disc of perfect conductor at various depths. r'\. f .• ~ \ 4 i \ 2 ~ A -~ i Ii 1:---2 -1.5 - -0.5 0 v5 .' / >---2 \-1z1H. ' \, I I -4 If--\ -6 ~ l.---' JhJM;,;;;;;;;;;;;;;;;; ) ;;;;;;;;;; 2~·~] Fig. 5. Secondary field due to a disc of perfect conductor (1). -2 "'""""",, '" ',,,, "'t'' ,,,,,, d:a.,,z 2a Fig. 6. Secondary fiield due to a disc of perfect conductor (2). 3. Experimental Research In this experiment, the relative amplitude and phase angle of the vertical com-ponent of resultant field are measured. These quantities are obtained on the basis Study on the Induction Method of Electrical Prospecting 151 of the primary field at the center of a square loop which produces the exciting field. The frequencies used in this experiment are 250, 500, 1,000, 2,000, 5,000 and 10,000 c/s, and the models for orebodies are as follows: Material Shape Radius in mm Thickness in mm Copper Disc 56 0.5, 1.0, 2.5, 4.0 Aluminium Disc 56 1.0 Iron Disc 56 1.0 Lead Disc 56 1.0, 4.0 Brass Sphere 56 Cast Iron Sphere 56 Fig. 7 shows the experimental setup. The exciting current flowing in two loops· is about 50 mA, and it produces the primary field in the exciting loop, and the standard field at the center of the reference loop. Measurements are made by the use of an AC potentiometer shown in Fig. 8. The amplitudes of two electromotive forces induced in two equal search coils are compared by means of two potentiometers, and the phase difference between these two electromotive forces is measured by two phase shifting networks using four triodes of the same type. Exciting Loop AC Source Fig. 8. In Fig. 8, let µ and r,, be the amplification constant and internal resistance of these triodes, and let r and L be the resistance and self-inductance of search coils respectively, then the voltage Va' appearing at the grid of triode Ta', can be expressed as follows: (19) where (20) 152 Takeshi KIYONO, Koichi KIMURA and Keiichi KOBAYASHI 8 _ t -1 wL a--an --r+R' Oa' = 2tan-• __ l_ +tan- 1 µ+ 2+a . wCaRa wCa[(µ+2+a)Ra+2rv] (21) (22) (23) In practice, we can make Ra)> rv, and 1/wCa )> rv, so Eqs. (19) and (23) may be written as follows: and On the other hand, the input voltage Vb' of triode T/ is given by where and fl t -1 wL fl b=-an -R= a, r+ If we make Rva =Rea= Rvb = Reb, then a= {j. -For the balance condition, Va'= Vb', then we obtain the relation: (24) (25) (26) (27) (28) (29) (30) So that, the amplitude ratio of the field Ha to the reference field Hb, and the phase difference between them are expressed by following equations : (31) 2 (t -1 1 . 1 ) <p = an wCbRb -tan-• wCaRa · (32) Eq. (32) is obtained assuming that rv is very small compared with Ra, Rb, 1/wCa and 1/wCb, but in practice, the error owing to this approximation is less than 0°45' within the frequency range of our experiment. The values of resistances and capacitances used in the potentiometer of Fig. 8 are: Study on the Induction Method of Electrical Prospecting Rea = R:va = Reb = R:vb = 3 ko, Ra= 30-50 ko, Rb= 41.8ko, Ca = Cb = 3.8/ f µF , where f is the frequency in c/s. 153 The range of amplitude measurement is 0.1 to 1.9, and that of the phase measurement is -20° to +20°. 4. Results of Measurements The amplitude and phase of the secondary field can be calculated from the resul-tant field by subtracting the primary field vectorially. The secondary field means the anomaly in magnetic field caused by an orebody, and it is affected by the depth and property of the orebody and the frequency of the field. So, we shall classify the results of observations in three groups for the sake of convenience. (1) The amplitude and phase difference as functions of distance from the center of the exciting loop when the depths of the orebody are varied at a constant frequency (f = 1,000 c/s). Figs. 9 and 10 show the fields for copper and lead discs, and Figs. 11 and 12 are for brass and iron spheres. In these figures s means the distance from the center of the loop to the top of the model orebody. The difference between Figs.· 9 and 10 shows the effects of conductivity of orebodies, and the difference between Figs. 11 and 12 shows the influence of per-meability on the secondary magnetic field. The magnetic property of material of a thin disc which lies perpendicular to the uniform primary field does not have much effect upon the secondary magnetic field, because the part of magnetic path occupied by the disc is very short. (2) The same as (1), but for various frequencies and a constant depth (s/a=l/4). These results are shown in Fig~. 13, 14, 15 and 16. It is found that the iron sphere behaves as a magnetic body at lower frequencies, but at higher frequencies it approaches a non-magnetic conductor. (3) The same as (1), but for various materials and a constant depth (s/a=l/4). Figs. 17 and 18 show the effects of properties of materials on the secondary field. As the frequency becomes higher, the differences between curves for different materials become smaller. Some of the results of these experiments can be compared with the theoretical curves. In Figs. 19 and 20, the theoretical distributions of vertical anomalies are compared with the experimental data. The primary field produced by a square loop is not perfectly uniform at about the sphere. So that we must take the value at 154 Takeshi KIYONO, Koichi KIMURA and Keiichi KOBAYASHI l2 I.I --"' Primary Field SIQ.::::: i/,4 S/a::: 1/2 ....... s1a= 1 ,o 12 14 -12 -16 (a) Relative amplitude (b) Phase angle Fig. 9. Resultant field due to a copper disc of thickness 1 mm (/ = 1,000c /s). L2 I.I OQi 12 14 c---- Primary field -+-Sia= f/2 -- Sia= I :~4 6 • . · 10· 12 c4 ... -8 (a) Relative amplitude (b) Phase angle Fig. 11. Resultant field due to a brass sphere (/=l,OO0c/s). (a) Relative amplitude (b) Phase :mgle Fig. 13. Resultant field due to a copper disc of thickness 1 mm (s/a = 1/4). I.I Q9 QB ___ : __ Primary fielJ -o-- S 1 Q;:: V4 _.,_ s,a;.VZ -- Sia= I ·JO 12 I (a) Relative amplitude (b) Phase angle L4 Fig. 10. Resultant field due to a lead disc of thickness 1 mm (/ = 1,000 c/s). 14 119 08 -o- S/Q = 14 --- S/a~/i2 _._..S/Q=_l -16 (a) Relative amplitude (b) Phase angle Fig. 12. Resultant field due to an iron sphere (f = 1,000 c/s). l2 I.I _,.P ----- Primary F<C!d _,._ 2SO CIS ~ 500 Cl,! ~ r.ooo us ~ 7.000 CI.S -<>-\QOOOCI., 4 8 10 12 I Q4 (a) Relative amplitude (b) Phase angle Fig. 14. Resultant field due to a lead disc of thickness 1 mm (s/a = 1/4). Study on the Induction Method of Electrical Prospecting 155 L2 . ~;,'; LI ,,. 1,0 8 ·10 12 14 14 Q9 ------ ftimary fi~ld _._ ·,.ooo c;s -12 --- 7,000 CIS -o- 10.000 C/J (a) Relative amplitude (b) Phase angle Fig. 15. Resultant field due to a brass sphere (s/a = 1/4). 24 (a) Relative amplitude (b) Phase angle Fig. 17. Resultant field due to discs of various materials ( / = 1,000 c/ s ). 0.4 ~-~' ~?~~ , 8 10 I 14 16 Theoretic~! Curve o Experlmenta.l Value Fig. 19. Vertical component of the secondary field due to a brass sphere. 14 ----- l)imory field - ]SOCIS ·s ro , 14 -I -16 (a) Relative amplitude (b) Phase angle Fig. 16. Resultant field due to an iron sphere (s/a = 1/4). (a) Relative amplitude (b) Phase angle Fig. 18. Resulant field due to discs of various materials ( f = 10,000 c/ s ). o Experimental Value Fig. 20. Vertical component of the secondary field due to a disc. 156 Takeshi KIYOJ'.O, Koichi KIMURA and Keiichi KOBAYASHI the center of the sphere, not at the center of the loop, as the average primary field in order to obtain a good agreement between the measured and calculated secondary fields. In Fig. 20, the experimental values ars taken from the measure-ment at 10,000 c/s. Some of the discrepancies between these points and the theoretical curve may be due partly to the fact that the measurements are conducted by using a copper disc of 1 mm thickness, whereas the theoretical curve is for a very thin disc of perfect conductor, but it is quite conceivable that they are mainly due to the experimental errors. O 100 Experimental ~ue. -1111--+µ;tn 1--+-+-t-l+H-tt--+-l-+H-H'tl o Real Component -U2 • JJMBir.;;y Conim,nt ·C4 .. LJ~R~;.i~""'~• '-~,~~~ Fig. 21. Real and imaginary parts of the func-tion Qe1B for a brass sphere. 100 Q6 Q5 04 (l3 Q2 (lj 0 -al ~a2 -Q3 ~ '~ -000. 10.000 I 00,000 C.(j 1111 • Exi,e rimenrdJ Value al Component L,,,.gina,y~r oRe • Iii/ 1111 \. "1::1 I ,1 Theoretl"'' Curve ~R, ~~:;.,...,) I)( V yr "" .-· --1.,.iro,y~. T , .... lr---.... 1·1 Fig. 22. Real and imaginary parts of the func-tion QetB for an iron sphere. Figs. 21 and 22 show the experimental values and theoretical curves of the func-tion X and Y in Eqs. (4) and (5) for brass and iron sphere. It is assumed that the resistivities of brass and cast iron are 20 x 10- 5 and 10 x 10- 5 o cm respectively, and the permeability of cast iron is about 450. By these assumptions, the most of measured points lie on the theoretical curves. ·0.1 -(12 -Q3 -Q4 -QS -Q,6 -QJ ---f:l ~ \ ' ~ I 000 ~" \ \ ~ .._ ! 10,000 Cl< ' "' \ )~ ' ' I Ii I I --0-----"' Copper Disc Aluminium Disc Lead Disc Iron Disr Fig. 23. Real part of the secondary field due to various discs. 0 -0.1 100 !. -Q2 -Q.3 --r-111 1000 /". vi---~D l/ Vj I'-. "'1 ",__ -Q.4 , __ L_LLL ~J ) I 000 c,r /,,.,. V _._ Copper Disc ~Aluminillm Disc --Lead Disc --Iron Disc Fig. 24. Imaginary part of the secondary field due to various discs. Figs. 23 and 24 show the function X and Y for various discs as functions of frequencies. No theoretical expression of the field due to an oblate spheroid of finite con-ductivity and permeability has been obtained, but the curves shown in these figures are similar to the theoretical curves for µ = 1 in Fig. 3 obtained for spherical conductor. 

Study on the Induction Method of Electrical Prospecting

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Study on the Induction Method of Electrical Prospecting

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