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The calibration of a portable induction magnetometer system Zambresky, Liana 1977

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THE CALIBRATION OF A PORTABLE INDUCTION MAGNETOMETER SYSTEM by Liana Zambresky B.Sc, University of Redlands, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Geophysics and Astronomy We accept this thesis as conforming to the required standard The University of British Columbia June 1977 © Liana Fran ces> Zamb resky, 1977 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ABSTRACT An investigation made concerning the characteristics of a sensor coil for an induction magnetometer shows that it is feasible to make the first stage 60 Hertz rejection filter of the Butter-worth type. This is an improvement in design over the Twinr-T filter which is sometimes used as a first stage filter as the number of electrical components is reduced and there is no pos sibility of ringing between the coil inductor and the filter capacitors. Two methods of relative calibration for the induction magne tometer system give reliable response curves. One method uses a Wheatstone bridge. The sensor is one arm of the bridge and it is shown that the effect of the signal generator is the same as if the coil was excited by a natural event. The second method involves exciting the sensor by a field created by a small secon dary coil. The agreement between the two methods is good. An experimental approach to the absolute calibration is suc cessfully carried out by comparing the output from the uncalibra ted system to an air core system which has been previously cali brated. A theoretical approach is used to give a good indication of the sensitivity of the sensor coil. The sensitivity is depen dent primarily upon the turn number and the length of the coil. iii TABLE OF CONTENTS ABSTRACT Ii LIST OF FIGURES v LIST OF TABLES viiACKNOWLEDGEMENTS ix CHAPTER I GENERAL INTRODUCTION 1 CHAPTER II-1 THE 60 HERTZ REJECTION PROBLEM 6 II-l Introduction 6 II-2 Theory of Twin-T filter 7 II-2.1 Computer Results 15 II-3 Theory of Butterworth filter 17 II- 3.1 Laboratory Results 22 CHAPTER III THEORY OF THE BRIDGE METHOD 25 III- l Introduction 2III-2 An Intuitive Approach Using the Norton Equivalent 26 III-2.1 Theory at Higher Frequencies 29 III-3 Theory of the Bridge Method 33 III-3.1 Case of Observation 37 III-3.2 Case of CalibrationIII-3.3 Computer and Laboratory Results 40 III-4 The Case of Observation with and Without the Bridge 46 iv. CHAPTER IV THEORY OF THE SECONDARY COIL METHOD 52 IV-1 Introduction 5IV-2 Theory of Operation 53 IV- 3 Theory of Calibration 56 CHAPTER V THE ABSOLUTE CALIBRATION 66 V- l A Laboratory Approach 6V-2 A Theoretical Approach to the Absolute Sensitivity 74 CHAPTER VI SUMMARY AND CONCLUDING REMARKS 81 APPENDIX 1 Methods of Determining the Inductance and Capacitance of a Coil with Finite Resistance 83 APPENDIX 2 The Transfer Function of the Amplifier System 88" APPENDIX 3 The Transfer Function for the Third Order Low Pass Filter 91 APPENMxRIiESEFCSS COT^eMbstolute Calibration According to -- 'J the Bridge Method 93 LIST OF REFERENCES CONSULTED 95. V List of Figures Chapter I Fig. 1-1 Equivalent circuit of sensor coil 2 Chapter II Fig. 2.2-1 The symmetric Twin-T filter 7 Fig. 2.2-2 Block diagram of sensor coil/Twin-T network 12 Fig. 2.2.1-1 Computer simulation of the frequency response of the coil/Twin-T sensing system 16 Fig. 2.3-1 An analog Butterworth filter 17 Fig. 2.3-2 The transfer function for the Butter-worth filter 20 Fig. 2.3.1-1 Laboratory set-up for the Butterworth filter 2 Fig. 2.3.1-2 Butterworth filter characteristics of the sensor coil system 24 Chapter III Fig. 3.2-1 The Wheats tone bridge 26 Fig. 3.2-2 Norton equivalent of the sensor coil 28 Fig. 3.2.1-1 Wheatstone bridge at higher frequencies 29 Fig. 3.2.1-2 Norton equivalent of the Wheatstone bridge 30 Fig. 3.2.1-3 Circuit used to derive a new Norton equivalentFig. 3.3-1 The bridge circuit with parameters as they are defined for circuit analysis 34 VI Fig. 3.3.3-1 Computer simulation to determine the effect of the conditions R^» and R^>> R2 on the response of the magne tometer system when using the bridge method 41 Fig. 3.3.3-2 The laboratory bridge circuit 43 Fig. 3.3.3-3 The normalized amplitude response of the magnetometer system according to the bridge method 44 Fig. 3.3.3-4 The phase response of the magnetometer system according to the bridge method 45 Fig. 3.4-1 Diagram of sensor coil and load im pedance 47 Fig. 3.4-^2 The essential elements of the bridge circuit 50 Fig. 3.4-3 The effect of the resistance R^ on the frequency response as obtained from the bridge method 51 Chapter-jLV Fig. 4.2-1 The secondary coil method at the time of operation 53 Fig. 4.3-1 The secondary coil method at the time of calibration 56 Fig. 4.3-2 The secondary coil method in>the lab oratory 60 Fig. 4.3-3 The normalized amplitude response of the magnetometer system according to the secondary coil method 62 Fig. 4.3-4 The phase response of the magnetometer system according to the secondary coil method 63 Fig. 4.3-5 A comparison of the frequency response curves obtained in the laboratory from the secondary coil method and the bridge method 64 vii Fig. 4.3-6 Chapter V Fig. 5.1-1 Fig. 5.1-2 Fig. 5.1-3 Fig. 5.1-4 Fig. 5.2-1 Appendix 1 Fig. A.1-1 Fig. A.1-2 Fig. A.1-3 Fig. A.1-4 Appendix 2 Fig. A.2-1 Fig. A.2-2 Appendix 3 Fig. A.3-1 A comparison of the phase response curves obtained in the laboratory from the secondary coil method and the bridge method 65 Circuit diagram for the air core system 69 Computer simulated frequency response of the air core coil system 71 Method to determine the inner radius of a large coil 72 The micropulsation event used for the absolute calibration 73 The dependence of the geometric per meability M\ upon the ratio of the lengths of the semiprinciple axes of a prolate spheroid, a/b 80 Equivalent circuit of a sensor coil 83 First method of determining L 84 Second method of determining L 85 The anti-resonance point 86 Mu-metal core amplifier system 89 Transfer function for filters of the Mu-metal core amplifier system 90 The third order low pass filter 91 List of Tables Chapter I Table 1-1 Coil specifications ix ACKNOWLEDGEMENTS I would like to express my sincere gratitude and appreciation to Dr. Tomiya Watanabe for his continuous help and guidance as my research advisor throughout the work presented in this thesis. I gratefully acknowledge Dr. R.D. Russell for many helpful discussions and ideas. I would like to express many thanks to Dr. K. Hayashi and Dr. T. Oguti for their patient guidance in the electronics laboratory. I would like to acknowledge the Victoria Geophysical Observa tory, Dept. of Energy, Mines and Resources for equipment which we borrowed. For the operation at Churchill, Manitoba, I would like to thank Mr. C.R. Barrett, superintendent and his stafft.at the Churchill Re search Range of NRC. This research received financial support from the following grants : NRC A-3564 NRC E-2923 DRB 9511-112 UBC Arctic and Alpine Research Committee 65-0444 NRC D-6409 UBC Summer Research Scholarship 1 CH&PTEB I GENERAL INTRODUCTION Recently, a portable induction magnetometer system was designed by the aeronomy group at the University of British Columbia., It is the long term intent of this group to establish a number of observing stations across northern Canada for micropulsation research., It was felt that a system could be built at a cost significantly less than commercial magnetometers. The reguirements of this instrument are such that it should cover the freguency band from 0.002 Hz to 4 Hz and take the signal level from the order of milligamma to ten gammas. The sensor was constructed with a Mu-metal core in order to reduce the physical size from that of an air core sensor. Mu-metal is a high permeability alloy of nickel and iron with a small amount of chromium and molybdenum. Ueda (1975) demonstrated that the Mu-metal core does not cause significant distortions or harmonics to the signal contrary to what some earlier investigators had thought. The equivalent circuit which is used for the sensor coil throughout this research is shown in Fig- 1-1. The capacitance effect of the windings, noted as early as 1910 from general antenna use, Campbell (1969), gives rise to the capacitor which is parallel to the resistance and inductance of the coil., The voltage generator, V, arises from the current which is induced in the coil by the changing magnetic flux. 2 It is difficult to measure the coil inductance and capacitance using a bridge which is designed for measuring pure inductance and capacitance., For this reason^ somewhat laborious methods must be put into practice. A compilation of such methods used for this research is given in Appendix A-1., These methods lend a great amount of credibility to the equivalent circuit concept because the coil was found to behave exactly as the circuit theory predicts in each method. R V L . pAAA/W ©—WW—| • lie Fig. 1-1 Eguivalent Circuit of Sensor Coil Further support for the eguivalent circuit occurs from observations that the output of the sensor coil with a parallel resistor and capacitor of appropriate values has characteristics of a Butterworth filter. The theory of the Butterworth filter is developed in detail according to circuit theory. Laboratory results concur with theory and give more support to Fig. 1-1.. A useful improvement in the design of a 60 Hz rejection filter results from this analysis which is discussed in detail in Chapter II. The primary intent of this thesis is to investigate the practicality of two methods of relative calibration for 3 finding the freguency response of the induction magnetometer. One of these methods is called the bridge method in which one arm of a Hheatstone bridge circuit is the sensor coil of the magnetometer. It will be shown that emf generated in the sensor coil by an external time-varying magnetic field can be simulated by driving the bridge with a signal generator.. The other method, called the secondary coil method, is one in which a small secondary coil is wound on the core coaxially with the sensor coil and creates magnetic flux to be detected by the sensor coil. The secondary coil method has the disadvantage that an extra cable is needed between the secondary coil and the amplifier system. In the field, the coils and the amplifier electronics are separated by approximately 100 yards in order to prevent spurious noise generated by the electronics from being mixed with the natural signal. Thus, it would be better to have one cable instead of two, not only from the standpoint of cost and convenience, but also because the possibility of undesirable cable effects would be reduced, i.e. "cross-^talk**. This problem makes the bridge method more desirable as a calibration procedure, An absolute calibration must be carried out in order to determine the level of the relative freguency response curve. This is done by comparing the output of the Mu—metal core system to another system which is already calibrated. This second system is an air core magnetometer. Sensitivity of an air core coil can be calculated exactly knowing the 4 geometry of the coil and its turn number. The two magnetometers are set up in the field, approximately 100 yards apart, and record micropulsations. The assumption is made that such a global event will not change over this distance. Some micropulsation events are recorded as sinusoidal signals and it is this type of event which is used for the comparison. Readings can be made over a number of cycles so that the error of this measurement is small. Finally, a theoretical approach is taken towards the sensitivity of a coil in order to elucidate which parameters are significant. The parameters of interest are the coil dimensions, the gauge of the wire out of which the windings are made, and the permeability of the core., Such an approach would be helpful when designing new sensors. The specifications of all of the coils used for this research are given in Table 1-1. These specifications will be referenced throughout this thesis. Turn Number R (fi) L (H) C Inner Diam. (cm) Outer Diam. (cm) Length (cm) Air Core Coil 5000 5130 120 .50 yf 149.02 151.40 4.76 Mu-metal Core Coil (1976) 50,000 1831 930 .188 yf 3.17 7.30 r45\72 Secondary Coil 20 .011 3.17 Calibration Coil 1000 11 17.14 241.3 Mu-metal Core Coil (19 75) 50,000 2230 1050 .048 nf 4.00 9.5 35.56 Table 1-1 Coil Specifications 6 CHAPTER II THE 60-HERTZ REJECTION PROBLEM II-J Introduction The rejection of 60 Hz noise is a problem which reguires careful consideration for a magnetometer that is being designed to detect micropulsations. At low noise sites which are carefully chosen in the field, the noise is considerably reduced from what it would be in a city. It is necessary to reduce the 60 Hz noise by at least a factor of 100 before allowing the signal to be amplified. Otherwise the amplifier may be overloaded or the micropulsation information lost due to the extremely low signal to noise ratio. One method which has been used for some previous magnetometers is to place a Twin-T filter, a notch filter with center freguency at 60 Hz, directly between the sensor coil and the amplifier. It will be shown in the discussion which follows that the freguency response of the signal output from the Twin-T filter is dependent upon the input impedance of the amplifier. This is not a desirable effect. There is a different approach to the first stage of the 60 Hz rejection problem that is better for two reasons. One reason is that the frequency response in the micropulsation range is ensured to be flat. The other is that only one resistor and capacitor are needed instead of six carefully matched components for a Twin-T filter. This approach 7 requires making the sensor coil and the parallel RC combination into an analog Butterworth filter. I1-2 Theory of Twin-T Filter The filter of interest is the symmetric Twin-T. A circuit diagram is shown in Pig. 2.2-1. Ll R R 2 R -A/WW 2C < X v. Fig. 2.2-1 The symmetric Twin-T filter To aide making a circuit analysis, the circuit is generalized to that of: 8 This is a parallel connection of two networks and may be depicted schematically as: Each network may be described by a matrix equation in current, admittance and voltage. To describe the network as a whole, consider the figure: Then: i * i> r v - y = v~ This leads to the desired result: This last eguation is important because it means that the admittance matrix of a number of networks connected in 9 parallel is the sum of the admittance matrix of each individual network. In the case of the parallel-T network, the entire network can be broken down into two simpler networks of the type: The mesh eguations for this network are: V, ZbI, + (2b+2c)rz The matrix representation which follows is: It is of interest to have the matrix eguations in the form I=¥V. The I matrix, or the admittance matrix, is the inverse of the Z. matrix. Thus, I=YV implies: 10 I III v, JZ| is the determinant of the %_ matrix. It is necessary to determine the Y matrix for each of the parallel-T sub-networks. The first network to consider is: R 2jwC The resulting Y matrix is: i w 2j ui C The second network to consider is: JwC _JL_ jwC i * R 2 11 This has the Y matrix: C 2. \utC - JL z R z 2. The Y matrix for the entire parallel-T network as depicted in Fig. 2.2-1 is then Y=Y'+Y' This matrix is: Y Y Y Y, 2.2-1 where: V - 1 » (ju>CR )* 2R0 + The ultimate aim of this discussion is to find the freguency response of the Twin-T network when it has a signal from the sensor as input and when it is terminated by the finite impedance of the amplifier system. Schematically, this is depicted by Fig 2.2-2. The matrix equation describing this system is: V 12 / A' A, 8t + 8,0, C, B, + D, Dj "1 Al Bl Cl Dl 2 A2 B2 C2 D2 3 4 1 V2 y Sensor Coil Twin-T Network R. Fig. 2.2-2 Prom the matrix equation, it follows that V, =A' ?3 +B' IJ and since 1% =?3 /R.t , the resulting transfer function is: V, R; A' R; + B' The matrices depicted above are not in the form of the Y matrix. In the terminology of electrical engineering these have the form of the F matrix. The reguired transformation is: F --/ V *1X m Y», i * i where: Yn Ytl ~ Xi,. YZ, 13 To compute the F matrix for the sensor coil, consider the following circuit diagram: From the RLC loop, an equation can be derived for I, From the potential drop across R and L: I, = V - V R + juiL This leads to the expression for V, : V,-- (l-"*LC +iu,CR)vt+ (R +ju,L)l. The sensor coil matrix is then: l-wHC + iwCR R*jwL V l w C Sow all of the guantities to determine the complete transfer function of the system in Fig. 2.2-2 are known.. The final result is: 14 V. A' R- + B ' 2.2-2: where: A - A,Al+B,Cl Bv' A, Bx + B, A,-- 1- u,lLC + 310CR 8, - R+ 0. 1L Yx \ Y,x Y,t - Y„ Y.» Y» Y, »' 1 Ml are defined by 2.2-1 15 11-2.1 Computer Besnlts A computer program was written to compute the transfer function of eguation 2.2-2-. The results of this computing may be seen in Pig. 2,2.1-1. The striking feature to notice is how the shape of the transfer function changes as a function of the terminating impedance of the Twin-T network. This terminating impedance is the input impedance to the amplifier system. If the amplifier is an integrated circuit, then its input impedance could easily be of the order of 10Mix and a severe ringing effect will occur near 5 Hz. This ringing is the result of coupling between the coil inductor and the Twin-T capacitors. If the amplifier input impedance can be reduced, then the ringing effect Dill lessen. The curve for an input impedance of 7.5 Ksi. shows no ringing at all and has a smooth drop off,. However, as described earlier, such a response curve can be obtained by exploiting a Butterworth filter. It would reguire only one resistor and capacitor in parallel to the coil.. The,Twin-T would reguire six carefully matched resistors and capacitors, and in the case of balanced input, twelve elements would have to be matched instead of six. In practice, this may be a troublesome requirement.. It should also be noticed that as the terminating impedance is reduced, the sensitivity at the lower frequencies is reduced. This is not a desirable effect. 16 FREQUENCY (HER^Z) Fig. 2.2.1-1 Computer simulation of the frequency response of the coil/Twin-T sensing system 17 II~3 Theory, of the Butterworth Filter The theory will now be developed for an analog Butterworth filter. The circuit to be analyzed is shown in Fig. 2.3-1. A resistor and capacitor are placed parallel to the output of the sensor coil. L -rSinnnnr-v. $ l vwvww R :R. v Fig.2.3-1 An Analog Butterworth Filter To determine V0 , first find the equivalent impedance of the three parallel quantities C, C, and R, . 1 + iu»U + O R, The equivalent circuit becomes: L © -MMAAr R V can now be found from the voltaqe divider method. R •* jw L + H 18 After substituting for Z, a considerable amount of algebra leads to the result: 2.3-1 ~1 This equation for V„ can be put into a non-dimensional form. The following substitutions will be made: 1. Let oc represent the ratio of d.c. resistance of the sensor coil to the terminating resistance. R 2.3-2 2. Let f? represent the inverse Q factor of the S-L-C series circuit. Q. 4 2.3-3 3. Let fr represent the resonance freguency of the B-L-C series circuit. 2-tr ] L (c *0 Now equation 2.3-1 can be re-expressed as: 2.3-4 v. V; e 2.3-5 19 where -Q~ -- tan j In order for the circuit 2.3-1 to respond as a Butterworth filter, the term {f/fr)2 must be made to approach zero. A Butterworth filter of any order is present when the transfer function contains the following expression; A plot of the transfer function for the Butterworth filter of orders n=1,2,3 can be seen in Fig 2.3-2- It is evident that in the case of equation 2.3-5, this is a Butterworth filter of order 2 since it is of the form: i 11 2.3-6 if the second term in the denominator can be made negligable. It can be seen from the circuit diagram 2.3-1 that it would be desirable to have S, large compared to R as this would increase the sensitivity. Thus it can be expected that *<1.. , In the case that oc»0, the second term vanishes for: 0 - ^ In the case that 0<«<1, the second term vanishes for either of the following two values of p : Fig. 2.3-2 The transfer function for the Butterworth filter 21 If 0««1 ., the binomial series expansion may be used to show that the above two values are approximately equal to: P • «/nr In the case that rt>1, there is no p which can make the second term vanish. This is another reason why * is desired to be small. In practice, it is best to determine P first and then make * equal to that value which will nullify the second term. As the sensor will be used to detect micropulsations, it is desirable that it have a flat freguency response over the range from . -002 Hz - 4 Hz . Also, since there is a considerable amount of man made noise at 60 Hz, it is necessary to optimize the gualities of the Butterworth filter in order to reduce the amount of noise picked up by the sensor. The amplitude of the 60 Hz noise can be dropped by a factor of 100 if the cut-off freguency is made egual to 6 Hz. Thus, C, can be determined from eguation 2.3-4 provided the coil parameters R, L and C are known. This leads to: c = J_ - r U> L (2irfvr ^ 2.3-7 P can now be found from eguation 2.3-3. The value of o( which will nullify the second term in 2.3-5 is: If P>JT , no «C can make the second term vanish. 0 for this type of sensor is typically of the order of 10-4-10-2 , Oeda 22 and watanabe (1975) , so there is not any threat that this condition might occur., Thus, B can be found from the eguation: R = R 2.3-8 II-3.1 Laboratory Jesuits The circuit of Fig.,2.3.1-1 was set up in the laboratory to determine whether the theory of the Butterworth filter is correct for the coils. -© wvvvv -VVVvVVV1-v out I 1 ma rms Fig. 2.3.1-1 Laboratory Set-up for the Butterworth Filter A sinusoidal magnetic field was created inside a large calibration coil,, The intensity of the field at the center of the coil is |^71: Gamma from the rms current of 1 ma that flows through its windings. The sensor coil was placed inside the large coil.. The time-varying magnetic field induced a 23 sinusoidal emf in the sensor coil and the output from the coil was measured across a parallel HC load. From eguation 2.3-7, the corner frequency of 6 Hz and the coil specifications of the flu-metal core coil (1975), {see Table 1-1), the value of the parallel capacitor was determined to be .67 yuf. Then, from equation 2.3-3, 0 =.0565. This leads to the value B, =28KA as determined from equation 2.3-8. The results of the laboratory experiment can be seen in Fiq. 2.3.1-2. The frequency response is flat at the lower frequencies and begins to drop off at 4 Hz. The amplitude has dropped by nearly a factor of 100 at 60 Hz. It should be noted that vou+ in Fig 2.3.1-1 has been divided by freguency because the emf induced in a coil by a changing magnetic flux is proportional to freguency and the amplitude of the magnetic field. 24 Fig. 2.3.1-2 Butterworth filter characteristics of the sensor coil system 25 CHAPTER III THEORY OF THE BRIDGE METHOD III-1 Introduction The original idea to calibrate the induction magnetometer by using a Hheatstone bridge was conceived by Dr. R.D. Russell at the University of British Columbia. This idea stemmed from a previously successful undertaking to calibrate an electromechanical seismometer using a Maxwell bridge, Kollar and Russell (1966). The first part of this chapter will be concerned with intuitively analyzing the Hheatstone bridge according to Norton's theorem, otherwise known as the current source model.. The remainder of the chapter will present a detailed analysis according to Kirchoff*s laws. The bridge system will be analyzed for three configurations. The first will be when the magnetometer is making observations and the bridge is a part of the electronics., It is hoped that the bridge can always remain a part of the magnetometer electronics so that it will not need to be wired into the system every time a calibration is made. The second configuration will be the proposed calibration procedure in which the bridge is driven by an electrical oscillator. The third will be to compare the case of observation when the bridge is not present to when it is. This last step is required in order to determine how the presence of the bridge could distort data. 26 III-2 An Intuitive Approach Dsinq the Norton Equivalent It is the intent of the developement which follows to show that a rate of change of magnetic flux, <ij , can be simulated in the sensor coil by driving a Hheatstone bridge with an electrical oscillator. The sensor coil will be one arm of the bridge and the equivalent circuit which will be used for it has already been given in Fig.,, 1-1./ The Hheatstone bridge is shown in Fig. 3.2-1. Fig.,3.2-1 The Hheatstone bridge As the sensor is to be used for micropulsation research, frequencies of interest will be between .002 - ,4 Hz, so that the capacitive reactance of Fig. 1-1 is negligable. Also, it is assumed that the amplitude of the signal from the electrical oscillator will be much larger than any signal which could be induced in the coil by a natural magnetic 27 event. Therefore E»V. Two further assumptions which are made concerning the magnitudes of the bridge components are: 3.2-1 Also, the bridge is balanced at d.c.. This balancing condition is expressed as: RR3- RZR^ 3.2-2 In order to simplify the analysis, the Norton or current source equivalent is to be found between the points a and b of Fig. 3.2-1. This is done by removing L and short circuiting the two points a and b. The current which flows between these points would be: 3.2-3 Next, consider the impedance when looking in from the two terminals a and b. The impedance would be B*RL*Rt if Hlt»RL Then, the equivalent circuit which follows is: and R3 »R2 R -M/VVW 28 For a coil of N turns immersed in an average magnetic flux per one turn of the coil, <t> , the total flux through the coil is N<|>. Thus, the emf induced in the coil toy the changing flux is: E = "-3t(N<J)) 3.2-4 But, it is also true that: E = - (Li) ^ui; 3.2-5 Applyiag Norton's theorem, the eguivalent source current is : 3.2-6 Using the result of 3.2-6, the Norton eguivalent of the sensor becomes: R —'WWW— L Fig. 3.2-2 Norton eguivalent of the sensor coil If the currents through the inductor in the cases of 3.2-3 and 3.2-6 are egual, then an important result exists between the amplitude of the driving voltage and the flux through the coil: A.. ~ I 3.2-7 29 III-2.1 Theory at Higher Frequencies If the Hheatstone bridge is going to be used at higher frequencies, then the capacitance of the sensor must be taken into account. To compensate for this, an inductor L3 is added to the fij arm of the bridqe. The bridge circuit which follows is shown in Fig. , 3. 2. 1-r 1. , Fig..3.2.1-1 Hheatstone bridge at higher freguencies The Norton equivalent circuit which results from using the condition 3.2-1 is shown in the following figure. 30 Fig. 3.2.1-2 Norton eguivalent of the Wheatstone bridge In order to arrive at a new Norton eguivalent, short the two points a and b, , Then calculate the current which flows through S. Currents are defined according to Fig. , 3. 2. 1-3. E ± E R R, R3+303L3 Fig. 3.2.1-3 Circuit used to derive a new Norton equivalent Let Z, be the parallel composite impedance of R and C. R 3.2-8 applying Kirchoff's law to the R-ZL-RZ loop, the eguation which results is: 3.2-9 31 Using the condition for the d.c. balanced bridge; R R3 = Rz R^ and another condition on L, that: CR - 3.2-10 eguation 3.2-9 simplifies to: R. 3.2-11 Then, the current it flowing through R is given by: Therefore, the eguivalent circuit becomes: R4 (1+JOJCR) R R„ If i, and i of equation 3.2-6 are equal, then the result is; Ru. Ci+^cR) L 3.2-12 The importance of the results 3.2-12 and 3.2-7 are that they predict that a rate of change of magnetic flux, <l) , can be simulated in the sensor coil by driving a Hheatstone bridge 32 with an electrical oscillator. In practice, it is much simpler to simulate a magnetic flux in this manner than it is to immerse the coil in a uniform calibrating field. A freguent method which is used to create an artificial field is to put the sensor coil inside a larger calibration coil. Besides being physically cumbersome so that an investigator would not want to bring a calibration coil along to the field, it costs nearly as much to build as the sensor coil. By using the Hheatstone bridge method, the calibration can be carried out at any time and with very little expense. At this point, the problem of the absolute calibration is not worked out too well. The total flux N<|> is related to the external field B by the relation; According to Dr. B.D. Bussell at DBC, the calibration of sensors is reduced to finding expressions for 4> in terms of H, and for L in terms of the coil geometry. The problem of absolute calibration will be discussed in detail in Chapter V. s 33 III-3 Theory of the Bridge Method A detailed circuit diagram for the bridge is shown in Fig. 3.3-1. In practice Ze ,Z2 and Z^ are pure resistors. It was originally hoped that Z3 would consist of the SLC configuration sketched below. AVVWV-Theoretical Z, In this way, both an a.c. and a d.c. balance could be achieved using the bridge. Theoretically and experimentally, this configuration would work except for the fact that a large inductor, of the order of 10 Henry, could not be found which had negligable resistance. Even a resistance of 50XL was large enough to render the a.c. balancing condition useless. A sketch of the laboratory Z3 is shown below. R -MAAAAr -AA/VVW—nfwrr-Experimental Z. Hhat follows is a circuit analysis of Fig. 3.3-1 which will show that this type of bridge arrangement will truly reflect the freguency response of the induction magnetometer. Let fi, ,L, and C, be the d.c. resistance, self-inductance and capacity of the sensor. For convenience, introduce the 35 quantities: X, = R,+ ju,L, 3.3-1 X.+ Y, 3.3-3 E0 will be the emf caused by the signal generator. E, will be the emf generated by the time variations of the magnetic field of the earth. A system of equations can now be defined by applying Kirchoff's law to current loops. The X,~ZZ-ZS loop yields: E, = X^, - 2,1, • 2,1, The Y,-Zz-Zj. loop yields: The Z0 -Zz -Z3 loop yields: eo= z.Cl-U-Virl,)- 22i2 The Zy-Z^-Zj loop yields: O - 2,1,-2,^1-1,) -23(1,-13) These eguations can be arranged, regarding I,#Ia,I3 and I as the independent variables. The corresponding matrix equation is: 36 X, Z, o -U„*z^z,} o -z. 2, (2, + Let 3.3-4 D be the determinant which is defined by the coefficients. It can be evaluated as follows: where + Z^(Z2Z3 + Z3Z,-757J 3.3-5 The current I3 is the key to the freguency response because it flows through the load impedance Zs. If this method is going to properly determine the freguency response of the sensor, then it must be shown that calibration, =Const * <J3 > observation ' The observation condition implies that I3 is the result of currents induced in the sensor by a fluctuating magnetic field and Eo=0. The calibration condition means that I3 is caused by the emf of the signal generator and E0»E,. Therefore, the system of simultaneous linear algebraic eguations must be solved with respect to I3 for the two cases. 37 III-3.1 Case of Observation For this case, E,#0 and Eo=0. The solution of tae matrix equation 3.3-4 qives: D The natural magnetic field fluctuations may be defined in the following manner, where B is the amplitude of the field changes and S is the absolute sensitivity of the sensor coil: E x = 1 ]u> S B e>U> 3.3.1-2 III-3.2 Case of Calibration -For this case, E0»E, . The solution of 3.3-4 gives for E4 =0 and E„ #0: The numerator of this last equation can be reduced to a simplified form by making some substitutions and approximations. First, substitute for Z, according to eguation 3.3-3., Then: 38 (XT + Y,)( Z.Z.-Z^ZJ* X.Y.Z,- (X,*Y,) 22Z, Hext, assume that the branches ZT ,Z3 and 2^. are all pure resistors. That is Zl=R2 , Z3=a3 aBd Z^=RV. Then: (y(*Y,Hz,Z,-21Zj* X,Y,R3- U.+ YJR^ How substitute for X, and I, according to eguations 3.3-1 and 3.3-3. If the bridge is balanced for the d.c. calibration signal, then the condition which is met is: This leads to the result: (y(fY,Kzlz3-2l7^- (~- 3-L,R,R3 as L,ai103Hr C,«i10-7F and B,«* 103XL , the condition is well satisfied that: — >> R,1 3.3.2-3 The eguation reduces to: (xt+Y(Kz,VZJ,) - T RB " >-L'R.R3 39 For micropulsation research, the frequency range of interest is .002 Hz_i< f < 4 Hz-, The quotient 1/CR*10*. Then, a final assumption can be made that: I CJ << 3.3.2-4 It should be noted that this last condition will break down at higher frequencies., Whereas it is well satisfied at the lower frequencies, below 1 Hz, at a frequency of 10 Hz, the condition is really not too well met. Substituting C, =.2 /xF and R =2Ka: 3 4-0 Thus, the final reduction of the numerator of equation 3.3.2-1, keeping in mind that f < 5 Hz, is: (X,+ Y,K Z.l^H, ±- R3 3.3.2-5 Now the ratio (I3)COli /{I )tt\a% can be calculated using 3.3.1-1, 3.3.2-1 and 3.3.2-5: Osing 3.3.1-2, the final result, keeping in mind conditions 3.3.2-3 and 3.3.2-4, is: -- il.R, io_ 3 3 2_6 {U)0^ Z0(R3+R^^+ RUf(R1+R3) SB Thus, the important conclusion is that for the d.c. balanced no bridge: (i3)c*l " 3.3.2-7 The ratio /(X,')0v>s is independent of freguency. Therefore, the freguency response of (I3 )co.i faithfully reflects that of the induction magnetometer. III-3.3 Computer and Laboratory Results The success of the bridge method will depend upon how well two conditions are satisfied. These conditions are: B^»R, 3.3.3-1 R3»RZ If these conditions are poorly met, then the output from the bridge may differ considerably from the case when these conditions are met,. a computer program was written to precisely determine how the response curve for the sensor coil would be affected by the ineguality.. The eguation programmed was V=(I3)ob<. %s, where (I3)0v,t is defined by 3-3-1-1- These results may be seen in Fig 3.3.3-1., The program was written with R3=R,. The curve which results from Rl = .5R, and R^ =2R , differs substantially from the curve when Hj_=.02R, and R^SOR^ . There is not much difference between the curve for R1=.05R, and RIV=20R, and the curve when ai=.02H, and R^=50R-, . The conclusion is that the condition Rlfr»Rl implies Rlt>20R, and R3»R1 implies Ra<. 05R3 . 41 ,2 .3 .4 £ ,8 10 2 3 4 6 8 • 10 Frequency (Hertz) KEY: V .5^ V2Ri V .lR^ R =10R. 4 1 V .051^ R=20R1 4 1 V .02R! R =5QR., 4 1 Fig. 3.3.3-1 Computer simulation to determine the effect of the conditions R^» R^ and R^» R2 on the response of the magnetometer system when us ing the bridge method 42 A laboratory experiment was performed to determine the freguency response of the magnetometer system using the bridge method.. The circuit diagram is shown in Fig. 3.3.3-2. The bridge resistors were determined according to the conditions 3.3.2- 2 and 3.3.3-1 . To satisfy the condition 3.3.2-2, R3 was made approximately egual to R( , R^-R,/20 and the bridge was adjusted to zero d.c. output by the variable resistor R^. Also, it must be noted that the first stage of the amplifier system is a chopper amplifier (see Appendix 2) which requires a balanced input signal. This is the reason >. for using the inverting amplifier as part of the input signal electronics to the bridge., The results of the laboratory test can be seen in Fig. 3.3.3- 3., For convenience, the data has been normalized to the value at 2 Hz. The dots represent data points obtained in the laboratory. The smooth curve is the result of a computer analysis. The eguation which was programmed is: V (X^)eks* 2f *• Ttju/) 3.3.3-2 T(jw) is the transfer function for the amplifier electronics and is derived in Appendix 2. As can be seen, the agreement between theory and the laboratory is excellent. The phase is shifted by 90° in the low freguency range. This is because the eiaf is induced in the coil by Faraday*s law of induction which states that the emf around a stationary loop is proportional to the rate of change of flux through the loop. ^ .1 .2 .3 .5 .7 1.0 2.0 3.0 5.0 Frequency (Hertz) Fig. 3.3.3-3 The normalized amplitude response of the magnetometer system according to the bridge method 45 180 135 90 45 0 -45 -90 1 1 X T T i i I i III 1 1' II ill 1 III l ML 1 1 i 1 :i ill I 1 III j 1 j U. i I T ll II 1 it 1' !il j 1(1 Ii! 1 1 -Ll -1 1 11 1 II H II i !|l 1 11 1 i lu. 1 T 1 T tt 1 1 i ll \) 1 ' ! 1 i j 1 1 U 1 i II I ii 1 - I 1 ! 1 n i I '! !' i FI | l I t 1 ll 1 III 1 1 1 • Laboratory results 1 _ i I H nt iiii i f\ 1 1 Computer results l i i| 1 HI i i 1 1 1 I 1 I1 1 II TTiy rt i ' ' ! 1 1 /i I i 1 1 I 1 1 mi i 1 /'• •J illl T i il III /' i 1 ill1 mil i n in ' i ! 1 111 I11! ! '1 III 1 1 1 •1 JLiiL i 1 il 1 1 I' i "1 1nr f 1 |l t- - -i- 4- i i i • 1 lihi i i| l|i i T j 1 Illl i 11 --4J444-1 jm i I i ' 1 l 11 tt Tit f\ 1 i i 1 i i ll II ' ' ' O i i I i r j J l| nil 1 ! ' 1 1 I l 11 ill! ii i! III 1 1 i ! 1 i UJ a) nl 11 Ji 11 Illl I HI 1 1 ! 1 111 11"1 I i JlJiiil -I ill 1 i 1 11 U T i I ; nil i i|i I i 1 i 00 i i +trrr i 11 Till 11 I CD T3 v y _ IT 1 I 1 | 1 111 I 1 1 i 11 1 1 i i i i i i i I' i / I l JJ 11 i 1 1 i Ii fA \ 1 I T Ijl l al 1 I 1 1 i I i 1 I t TH lili CO I j "i I i 1 i i I ill I 1 I i 11 i i 11 i ct) I 1 i 1 1 ! II i A 1 1 I I I Ti'ltt ni IIII 1 fu 1 ; II 1 I i 1 II i • i 1 i i i 111 i 1 iih ] ll j i 1 ll i i »' 1 I 1 i in 1 1 i i i I i 1 j 1 i \ Pl 1 1 i 11 1 i 1 1 1 1 1 I 1 i 1 1 I 1 i in 1 11 : | 1 i 1 1 i / I i i i l j 11 1 11 1 i I 1 1 Lj_ H 1 I 1 ""TI I 1 i 1 i IT ij 1 1 \V 1 1 i 1 1 I YJ 1 i ii i 1 i */ '-,\ III 1 1 i n i j 1 III ' I 1 i II i 11 1 y 1 I 1 II 1 i II 1 i 1 i II I 11 II i ""tT" llj II I 1 ll i n i II • inn 1 1 II TT i H i Ji III | ..Iii i 1 '1 n i 1 1 1 ! 1 1 1 l.Ji_L I 1 11 1 .11. 1 ill i iii M i IMII TTllH I 1 ill i I III TT I i HI I Frequency (Hertz) I i i > r MI 1 -JJ.-L t I . iT 1 i M l i i I 1 ; j j 1 TTTiiir j i ; i • ! i 1 i 1 I'll i 1 ! i 1 1 1 II ill' 1 ii11 1 ! 1 1 i i T •; 1 1 | i I 1 .1 .2 .3 /'-P^5 .7 1.0 2.0 3.0 5.0 Fig. 3.3.3-4 The phase response of the magnetometer system according to the bridge method , , 46 For a field which is normal to the loop: B-n dA -- BA Let B=B0e Then : f » - ju>BA Thus, it is expected that the emf will lag by 90° at the lower freguencies. it the higher freguencies, the phase will be affected by the low pass filters of the amplifier electronics, of which there are seven orders. It is no longer such a simple problem to predict what the phase response will be, as in the low freguency case. Ill—4 The Case of Observation With and Without the Bridge The voltage output as measured across Z5 in the case of observation with the bridge can be found using eg. 3.3.3-It is desired that the result 3.4-1 concur with the case of observation without the bridge so that it will not be necessary to have the bridge as a permanent part of the electronics. If the sensor coil is to be operating in the field without the bridge , ., then a diagram representing the "2, It is given by Vobt = (I3)ol)S*Z5. Vob< = 3.4-1 0 47 input voltage to the amplifier system is shown in Fig. 3.4-1. REAL ^ ^ E (£) I REAL Fig. 3.4-1 Diagram of sensor coil and load impedance X, and Y, are given by eguations 3.3-1 and 3.3-2 respectively. The composite impedance of I, and Zs is given by: Z -- Y.Zr Then the current I is: I E/(X,*Z) The potential drop across Zy is: V 2 r-RCAL *,+ z The ratio of the output voltage to the input voltage is: V REAL By making the substitution for Z, this last result becomes: J_ 2,Zy E X,' (Z.+Z*) 3.4-2 48 If the bridge method is going to be used for calibration purposes, then it is necessary to show that Vobs » V^EAL . The eguation for Vots is much more complicated than the one for VREAL , but by carefully considering orders of magnitude of the guantities which are involved in each term, it can be shown that V0fe« is essentially identical to VJ£flL .As a starting point to evaluate these orders of magnitude, let Z0=10fl., Z3=R, Z1=R/20 and Zlf=20R, where R is the resistance of the sensor coil and taken to be approximately 2000^. Also let Z5=10000^t. By evaluating all of the terms of D according to eguation 3.3-5, and retaining only the largest terms, it can be shown that: D ~ 2^2,+ 2,2^2, •Z,Z,ZS 3.4-3 The error of this approximation is of the order of 6%. By evaluating the numerator of 3.4-1 with the same substitutions as were made for D, it is apparent that: Z0(Z3+2j + 2^(2^2,) » ZjZ^. 3.4-4 This approximation is of the order of 5%., 8ith all of these reductions of terms and noting that for pure resistances Z3=R3 and ZH.=R,V, 3.4-1 becomes: V. obs . Y, R,Zg  3.4-5 49 Upon rearranging terms: VL « Z? . r 3.4-6 1 + ~ ( ) If can be made large enough, then the radical in the denominator containing R^ becomes small and reduces to: V ~ J_ Z'Z* E 3.4-7-oks ~ X," ( Zl+Zf) * This is identical to 3.4-2- It must be noted that R^ of 3-4-6 cannot be increased to a very high value without reconsidering the approximations of 3.4-3 and 3.4-4., If R^ is to be made larger than 20 times the sensor coil resistance, then it is imperative that Zz and ZQ be small, certainly less than 100XL and preferably of the order of 10XL . Otherwise, the approximations leading to 3.4-7 will no longer be valid and the voltage output as measured across Zs in the case of observation with and without the bridge will no longer be comparable. The physical significance of 3.4-6 is interesting in itself. What it means is that the bridge circuit of Fig. 3.3-1 with Eo=0 effectively reduces to the following: 50 E Pig. 3.4-2 The essential elements of the bridge circuit The effect of the term containing fi^ has been observed from computer simulations. The results of the computer analysis can be seen in Fig. 3.4-3. Eguation 3.3.3-1 was programmed with and without the 8^ term as given by 3.4-6. The same values of bridge components were used as shown in Fig. 3.3-3-2. It can be seen that the effect of Z^. is to cause a small separation between the two curves with the maximum separation occurring at 2 Hz. By comparing Fig. 3.4-1 and 3.4-2, the final conclusion reached is that VoVj5s VRHL only if the effect of can be reduced by making it a high value. This infers that the comparison of observations between two systems, one with the bridge as a permanent part of the electronics and another without the bridge, may have a slight discrepancy which can be attributed primarily to acting as a parallel load across the output of the sensor system. Frequency (Hertz) Fig. 3.4-3 The effect of the resistance R, on the frequency response as obtained from the bridge method 52 CHAPTER IV THEORY QF THE SEC-QNDA3Y- COIL fiETflOD-IY--1 Introduction When using this method to determine the freguency response of the magnetometer system, a time varying magnetic field is created by a secondary coil. The secondary coil is aligned coaxially with the sensor coil and at the center of the coils is a Mu-metal core.. The field is detected by the sensor coil.. It is the purpose of the derivations which follow to show that the freguency response can be accurately determined using this approach.. As in the case of the bridge method, the input voltage to the amplifier system will be derived when the system is being calibrated and also when it is in its normal observational mode.. Finally, a comparison will be made between the freguency response which is obtained from the secondary coil method to the one obtained from the bridge method. 53 IV-2 Theory of Operation Secondary Sensor coil coil Fig.,4.2-1 The secondary coil method at the time of operation The circuit which is representative of the magnetometer system when it is in the observational mode is depicted in Fig. 4.2-1. , The parameters C2 , 82 and L2 are the secondary coil constants., I is defined as: Y -- jwM 4-2-1 where M is the mutual inductance between the two coils. H0 is the load on the secondary coil that is adjusted to remain constant when the signal generator is added for calibration purposes (see Figs., 4.2-1 and 4.3-1).. The sensor coil constants are R, , L, and C,.. Because of the geometry of the two coils and the fact that they have a Mu-metal core in common, there is a possibility that the response of the sensor coil in the observational mode could be distorted by mutual inductance between itself and the secondary coil. It is the purpose of this section to explore this problem in detail., Let Z, = R, + jio L, 4>2_2 54 J 1 R, 4.2-4 4.2-5 Then, the equivalent circuit at the time of operation for observation becomes:. ( E -0" Applyinq Kirch offs theorem, the two circuit equations are: ( z. + z,)!, - E:-YI2 The solution for I, is: I.- ( zl + z5)(z^zj- Y* 4.2-6 The input voltage to the amplifier is Vg =i( z3. In order to make I, independent of any effects from the secondary coil, the following condition must be satisfied: I (*.•*,)( Z,.Z.)|» lYl1 4.2-7 An upper limit to the magnitude of I can be found by putting a bound on H., If the self inductance of the two coils are known and it is assumed that all the magnetic lines of force set up by the first coil cut all the turns of the second coil, then 55 the mutual inductance M is given by: M = \l,Lj_ 4.2-8 It is not certain what percentage of magnetic lines set up by one coil will cut the turns of the other coil, so it is safer to make this last eguation into an ineguality. M ^ JUT 4'2"9 Then^ it is of interest to show that: >> 1 For the coils used, L,.-1000 H and La=.01 H. , Also, Cg =4.5Mf and 89 =7.5 Ka., The coil constants are given in Table 1-1. A reasonable worst case result, which would be at the highest possible freguency that one might expect to observe micropulsations, would be at 10 Bz. Substituting these numbers into the last ineguality: (Z,^3XV2j 720,000 u1-L, L-j. loo Thus, the condition 4.2-7 is easily satisfied., The final result for the input voltage to the amplifier system is given by Vow =Z3I,: vok< -- E. 4.2-10 obs Z + Z, The conclusion is that the induction magnetometer at the time of operation for observation is unaffected by the presence of the secondary coil. , 56 IV-3 Theory of Calibration In this section the calibration procedure will be discussed., A time varying magnetic field will be induced into the sensor coil by a signal from the secondary coil. It sill be assumed that this signal is much larger in magnitude than any natural magnetic fluctuations so that £ as defined in Fig. 4.2-1 can be considered negligable. A circuit diagram for the calibration is shown in Fig. ,4.3-1., Secondary Sensor coil coil Fig.,4.3-1 The secondary coil method at the time of calibration In order to simplify the picture for algebraic calculations, the following substitutions will be made: Z, = R, +• ju»L;j; ^1 K9 57 The equivalent circuit diagram at the time of calibration becomes: J0 © L L2 -4-There are three loop equations vhich result. (Z.'Z.U.- -YI2 Hc(l.-I,)-I5R0 * - E. Solving this system of eguations for I,: I. - ~ Y where RoZc *.*zc Zc/(Zt •"B0)=1/(1*jwClR0) is approximately egual to 1 if | wC,.R.| C< 1* The laboratory value for R0 is 6 Kru Cx includes the capacitance of the secondary coil and that of the cable connecting the secondary coil to the signal generator. The cable is normally long, frequently 100 meters or more.. The capacitance of the cable is of the order of 10 nf., & typical cable capacitance is 50 pf/ft. Then, the total capacitance for 100m is approximately 15 nf. The capacitance of the secondary * This is not a necessary restriction, but a convenience. 58 coil should be much smaller than this., Therefore, at 10 Hz, wCzR0« (20-IT) (6 K) {1.5x10-8)=5.65x10-3«1. It has already been shown that: Kz.+z.Kz^+zjl » Y2 Therefore, the simplified result for the current I, is: i, - Y The resulting voltage drop across the input to the amplifier system when the magnetometer is being calibrated is Vcal =1 z3 : V Y Zs r 4.3-1 The ratio Vobs /VC0l| can now be found using eguations 4.2-10 and 4.3-1. V*L , - _Ii*l±._L 4.3-2 Y E, The emf induced in the sensor coil by natural magnetic field fluctuations can be described by: S D E = i- wBe 4.3-3 2.TT where S is the sensitivity coefficient and B is the amplitude of the oscillating magnetic field. Substituting 4.3-3 and 4.2-1 into 4.3-2, the result is: 59 The final observation to make is that the a.c. signal from the signal generator is given by: Ee - Es e >"* Then : - — — (R^ju-L^R.) 4.3-6 As shown previously, wClBt>«1 and this implies that Z^=B0. , In order to make 4.3-6 independent of freguency, the following condition must be met: (Rx+ Ra) >> uuLz 4.3-7 Again, the worst case would be at the high freguency end, at 10 Hz. , Substitute the values Bt= llxj. , B, =6 U and L =.01 H. Then : Rx+ R„ & * lo' The condition 4.3-7 is easily satisfied,. The final result is: 60 - _L.il Ul*R.) 4.3-8 21T MES Thus, it is seen that the voltage output at the time of calibration is directly proportional to the output at the time of observation. It is expected that this method of calibration will correctly produce the freguency response of the magnetometer system., The laboratory circuit diagram for the secondary coil method is shown in Pig. ,4.3-2. 6K -AWWV 1 Mu-metal core M-+.3 K 10K 0 W.4- pi Magnetometer amplifiers and filters Fig. 4.3-2 The secondary coil method in the laboratory The results from the laboratory analysis can be seen in Fig. 4.3-3 which shows the normalized freguency response and in Fig.,4.3-4 which shows the phase response.. In both cases, the laboratory data is plotted as sguares and the smooth curve represents the results of a computer analysis., The eguation programmed is : V - T(iu,) 4.3-9 where X, and Z, are defined by 3.3-1 and 3.3-3 and : 61 £ - R +. ' - ('OKX^^ I This follows from eg., 3.4-7 and from the derivation of the amplifier system transfer function given in the appendix. The agreement is r .'good between the data obtained experimentally and the results predicted from the computer analysis., a plot with laboratory data can now be made which is analogous to the computer plot of Fig.,3.4-3. This can be done simply by comparing the laboratory results obtained from the bridge method and the secondary coil method. On the same graph appears the laboratory data of Fig., 3.3.3-3 and Fig. 4.3-3. This is shown in Fig.,4.3-5. The agreement between the computer analysis of Fig. 3.4-3 and the laboratory analysis of Fig. „JI.3-5 is excellent. a comparison of the results of the phase analysis from the bridge method and the secondary coil method is shown in Fig., 4.3-6. The agreement between the two methods is quite good. .1 .2 .3 .5 .7 1.0 2.0 3.0 5.0 Frequency (Hertz) Fig. 4.3-3 The normalized amplitude response of the magnetometer system according to the secondary coil method ON ho 63 Fig. 4.3-4 The phase response of the magnetometer system according to the secondary coil method Frequency (Hertz) Fig. 4.3-5 A comparison of the frequency response curves obtained in the laboratory from the secondary coil method and the bridge method (see Fig. 3.4-3 and discussion on page 50 about the separation between the, curves) . 65 ui in hi! 180 iii II! Ill i 135-in TTT ill Wi + Secondary coil method • Bridge method _!___ •II; TTiTt i I hi MI TT i ! 4444 UU rmiT M iu m I I ! Ill III TT in Mi. iltr ! i TTT iii 90 Iii i ill Mil i i 111 nu ! I to cu cu u 60 01 CU CD n) X III! ill iii mi 11 iii Mil III! 45 i 11 Mi Mi! iii i ! ii I I I ! i I Ui iii i-U-L Tl~[ U i I ! I III hil III! Illl m i i i. i illl I 1 MM illl ilii TTT MIL -45 i i 111 111 I ! I rn" in Ui! II! III!! Ml ill. m UM TTTfi mua. iiii II! iiii _LL i ! I mm Ttffl Tl TTtti TTT ill! Ill UU iii ! I I I I -90 .1 ilii III 445 Ul I ! I III! Illl llllll .3 .5 1.0 2.0 3.0 Frequency (Hertz) Fig. 4.3-6 A comparison of the phase response curves obtained in the laboratory from the secondary coil method and the bridge method 66 CHAP-TE1 V-1SI ABSOLUTE-CAL-IBRATIOH--V-1 A Laboratory Approach Now that the relative sensitivity of the Mu-metal core system is well known, it is necessary to determine the absolute sensitivity., Conceptually, the simplest way to do this would be to put the sensor coil in a known, uniform, sinusoidally varying magnetic field and record the response of the system., In practice, it is not easy to create an artificial field which would he uniform over a volume large enough to accomodate a three foot long sensor coil., A logical solution is to use the earth*s natural magnetic field when a sinusoidal micropulsation event is occuring. An air core coil magnetometer system is used to precisely determine the absolute amplitude of the micropulsation event., The air core coil and the Mu-metal core coil are located about 100 yards apart and it is assumed that the natural event is uniform over this distance., The response of the Mu-metal core system can then be compared to that of the air core system, thus determining the absolute sensitivity of the Mu-metal core magnetometer., It is necessary to do this at only one frequency as the relative freguency response is already known and linearity is assumed., The flux, <$ ,through :a circular coil is ; AB 5.1-1 where A=irR2*N is the average cross-sectional area times the 67 number of turns and B is the magnetic field strength... The electromotive force induced by the changing flux is: VtMf i£ . A i§. 5.1-2 dt dt The output voltage from the amplifier system, with a d.c. gain G and an input impedance B- , which would be observed is given by 5.1-3. .. H{f) is the transfer function normalized to the d.c. value and R is the d.c. resistance of the coil., Vottt - Vew, * G - HlfW 5.1-3 It is convenient for micropulsation research to express electromotive force in micro-volts and magnetic field strength in milli-gammas. / This leads to the following two eguations where Ve„,f is in volts, Vgmf is in micro-volts, B is in tesla and B' is in milli-gammas., V^f - I0b V.mf 5.1-4 8' - to"" B 5.1-5 Substituting these eguations into 5. 1-2 leads to the result: 'vLf - (^IO") ii' 5>1_6 If B' is a sinusoidal field, then B'=B0sin2irft. Substituting this into 5.1-6 yields : VeWf = ( 21r A * \0'b ) • f • B„ cos 2lr-f+ or vLf = S^'f- 8e o>s lirft 5.1-7 Sa=2ira*10-6 is called the absolute sensitivity when Vemf is 68 measured in micro-volts, B0 in milli-gammas and freguency in Hertz., it a frequency of 1 Hz, a field of 1 milli-gamma C10—a Gauss) will induce a potential of 1 micro-volt and the sensitivy will be 1 Caner., 1 Caner = 1,uV/(my*Hz) How two eguations can be written, one for the air core coil system and the other for the Mu-metal core coil system by substituting 5.1-7 into 5.1-3. The superscript "1" is for the air core and n2n is for the Mu-metal core., V0(°- • Hl'V) 5.1-8 i V.w>* Si"- f • B. « *** • 6". -ggf^ • H~(«) An important result occurs when the ratio of these eguations is taken.. This is given by : as the agreement between computer analysis and laboratory analysis has been very good for the work concerning the bridge method and the secondary coil method, the freguency response of the air core coil system is determined by programming the circuit parameters on the computer. The circuit diagram for this system is shown in Fig. , 5. 1-r 1..,, H<*>{f) can be expressed as the product of two transfer functions. , The first one is Fig. 5.1-1 Circuit diagram for the air core system 70 H,<lJ(f)» approximately of a Butterworth type. The second is the transfer function of the third order low pass filter, H^Uf). In order to determine H,<*>(f),- consider the following figure : R From this diagram R* where: Z = I + ju> Cx Rx Normalizing 7X /E to one at d.c., H,*1*^) becomes Ha<*5(f) is derived in Appendix 3. Therefore, the transfer function for the air core coil system, normalized to one at d. c. is : HH(f) = H?»m. Href) This result was programmed on the computer. The freguency response curve which was determined is shown in Fig.,5.1-2.. As the freguency response curves for both the air core and the Mu-metal core systems are flat in the low freguency range, the terms H<J*(f) and Ht2,{f) in 5.1-10 are both egual to one. 71 Fig. 5.1-2 Computer simulated frequency response of the air core coil system 72 The absolute sensitivity of the air coil, S<4->^ is easily determined from its geometry and turn number. In order to find the inner radius of the coil, the following method was used. A Fig. 5.1-3 Method to determine the inner radius of a large coil Three measurements were taken along the inner circumference of the coil as shown in Pig. ,5. 1-3. From the law of cosines, the angle C is given by : C = co.-'/'-'+b'-c1) ^ 2cb 1 Then, an eguation was used for a circumscribed triangle in order to find the radius. This eguation is : o- b c 2-mA " 2-<nB 2 «.»vC After finding the inner radius, the outer radius was determined by adding the thickness of the coil to it. From this approach, fi. =74.511 cm and R_„+„_ =75.702 cm. Now the *••* * inner ouTer absolute sensitivity can be found.. K 73 where r. and ro are the inner and outer radii, respectively. . Substituting in the numbers leads to S^<1>=.0557 Caner ± .2%. The ratio V0<2>/V0<»> can be determined by measuring the amplitude of the signal from the air core and the Hu-metal core systems when a sinusoidal micropulsation event occurs., The output signals from the event used is shown in Fig. 5.1-4. Feb. 17, 19 77' 21h 27m Air core Mu-metal core Z comp. Fig. 5.1-4 The micropulsation event used for the absolute calibration Peak-to-peak measurements were made from microfilm by using a travelling microscope. .. The ratios were taken with the result of : —°— = 1.4-1 + .747. .24-5- H2 t l._7„ Finally, the remaining parameters of 5.1-10 have the following values : 74 G <1 > = 5:09 7'* 10* tU R.C O = i|.0 K„ t. 135 RO. = 5130 si t. 1$ G< z> = 4.01*10* ±.% R; «2> = 7.21 Kxi ±4% R<2) = 1831 -v •. 1% Note that Rtt2> is different from R, =7.5 K__ (p. 55) because another amplifier unit was used for the field observation. The value of S,.<z> according to 5.1-10 is .0548 Caner ±6.4%. V-2 A Theoretical Approach to the Absolute Sensitivity-The absolute sensitivity can be dealt with from a theoretical point of view. Consider a prolate spheroid with semiprinciple axes a,b immersed in a magnetic field of strength H0. The Mu-metal core of the sensor is in reality a cylinder, but approximating the cylinder as a prolate spheroid is reasonably accurate and saves a great amount of mathematical difficulty. The results of an analysis shown in the book by Stratton (1940) indicates that the magnetic field strength anywhere inside the cavity is given by : 75 where: A - ~~T~7 ( ~ 2 6 H n — ) cv i-e ' The magnetic field follows from B=/u/u„H where /* is the relative permeability and the permeability of free space. In - HKS units, /*„=4,rr*10-7 Henry/meter, /u- is non-dimensional and is the permeability of the metal from which the core is made. By letting Ba = /A,H0., it follows that : B - rr — Bc 5.2-2. 2 ' The total flux $ through the coil follows from 5.1-1. $ TT • B • N Or, substituting for B according to 5.2-2 : 5.2-3 |+ 2^ Cyu-l) A Let ,i , S ^ M-^b-N 5.2-4 7. Then $=SB„. The electromotive force generated by the changing flux would be : 76 Comparing this to 5.1-7, it is seen that S represents a theoretical sensitivity. , Expressing S in units of Caner, the sensitivity becomes : For a Mu-metal core, /x is of the order of 10s. If a restriction is put on the ratio of the lengths of the semiprinciple axes: a 200 then the denominator of the expression 5.2-4 for S can be reduced to a simpler form,, | + f___L ( u. ,) A « o-b V A 5.2-6 Z ' 2 The error of this approximation is less than 8%. The expression for sensitivity reduces to : S - mi 5.2-7 OLA The expression 5,2-1 for A can also be simplified. If ? ~ , then e-1 and A becomes ; A - ^ [ " 2* In 2 - In 5.2-8 By the binomial expansion : 0 ' a- Z aa Then 5.2-8 reduces to : A -- In 5.2-9 a' e. ba where ea =2.71828. The final simplified expression for 77 sensitivity follows from 5.2-9, 5.2-5 and 5.2-6. , SaPP-.„ = * 10 Caner 5.2-10 The importance of this expression is that the sensitivity of the sensor coil is primarily dependent upon the number of windings, ti, and the length of the coil, a. The overall weight of the sensor could be reduced by using lighter gauge wire and a thinner core. The length and turn number could be adjusted to obtain the desired level of sensitivity. In spite of many approximations, the agreement amongst the theoretical and experimental results of 5.2-10, 5.2-5 and S^2* of 5.1-10 is guite good,. The values are shown below : S0,<2> = .0548 Caner + 6.4% S+K»,r =-0583 Caner Sapp-** = -0587 Caner These results imply that the approximate theoretical expression for the absolute sensitivity, eg. 5.2-10, can be reliably used to aide in the design of future sensor coils. Another interesting observation which has been made as a result of this analysis is that the ratio a/b cannot be increased without bound unless an undesirable effect begins to take place.. Hhen ^>100, S+Ka(,r and Safpro» start to diverge by about 2% and the agreement becomes worse as a/b becomes larger., There is an important reason behind this which can be * For absolute calibration according to the bridge method, see Appendix IV. 78 unfolded by onee again taking a look at 5-2-2. B can be re-expressed in terms of a demagnetization coefficient Na. ,. B -- ^ B. 5.2-11 l+Nd (/*-•) Nd - Of A It can be seen that the inverse of the demagnetization coefficient is the permeability which leads to the sensitivity given by eq- ,5-.2-10.'„- Writing yU, = — r • — 5.2-12 Na ab^A it is seen with the help of 5.2-9 : yU, - ! 5.2-13 e„b Note that this is dependent upon the geometry of the core but not the properties of the metal., yu, is dependent upon the ratio a/b*, By making the substitution 5-2-12 into 5.2-13, it is seen that : B * M&f$ 80 5.2-14 where: M ~r r -For: the most recent Mu-metal core coil used for this research, a=:>45-.:r7-2cm and b<?i,/Q~ cm {see Table 1-1). This 79 leads to a value for /u., of 595'-Since yu&IO5, the ineguality fx »/JLX is satisfied. , Then by 5.2-14, yu-eff a yu., . , This is a desirable result because the effective permeability of the core is dependent upon its geometry. When a/b*100, the ineguality yu»yu, begins to weaken., When a/b=1000, //, = 1.5*10s and /~€ff is no longer dependent principally upon yu, . The permeability of the metal, /x, becomes egually as important., This is not a desirable result as the permeability of a metal is not always constant., It may change significantly according to environmental conditions such, as temperature and stress upon the metal. & plot of yu, as a function of a/b can be seen in Fig. 5.2-1. / The conclusion of this analysis is that eg. „ 5.2-10 will give a good indication of the sensitivity of a cylindrically shaped sensor coil provided s;> I iL ^ ISO . b Note: The value of b at the bottom of p. 78 is not simply the inner diam eter of the coil windings. It actually denotes an effective radius result ing from the cross-sectional area of the Mu-metal core itself. The core consists of approximately 48 rectangular strips. The cross-sectional area of one strip is (3/4")(.014")=(1.905 cm)(.03556 cm)=.06774 cm2. If there 2 are 48 strips, then the total cross-sectional area is 3.252 cm . The ef fective, radius fective radius for this area is ^3.252/lT = 1.02 cm. 80 1,000,000 500,000 300,000 4-1 •rl rH •rl •8 p. o •rl S-l 4J OJ e o 0J 60 100,000 50,000 30,000 '10,000 5000 3000 1000' 500 300 100 10 50 100 500 1000 a. b. Fig. 5.2-1 The dependence of the geometric permea bility /x, upon the ratio of the lengths of the semiprinciple axes of a prolate spheroid,a/b 81 CHAPTER-VI-SUMMARY AND CONCLODINS HIARK-S-Many useful results have been obtained from the investigations carried out in this thesis.v These results fall into two categories. One category is concerned with improvements in the design of the magnetometer system. As described in Chapter II, use can be made of a Butterworth filter rather than a notch filter to reduce 60 Hz noise.. This guarantees a flat response in the low freguency range and reguires only two matched electrical components rather than six carefully matched components. In Chapter V, which was concerned with the absolute calibration, it was shown that the sensitivity of the sensor coil is primarily dependent upon the number of windings and the length of the coil. , The overall weight of the sensor could be reduced by using lighter gauge wire and a thinner core. However, for optimum performance, the ratio of the length to the diameter of the core must be within a specific range.. One of the reasons for not being able to increase sensitivity ad infinitum is that thermal agitations at the atomic level induce small currents in the sensor and other electrical components.. This is known as Johnson noise. The second category concerns the calibration procedure. Two methods were investigated in great detail, the bridge method and the secondary coil method. It was determined that both methods can produce reliable relative freguency response 82 curves. An advantage of the bridge method over the secondary coil method is that only one cable is needed between the sensor coil and the remaining electronics, rather than two. For this reason, the bridge method will probably be used for future calibrations. The absolute calibration was successfully performed by comparing the Mu-metal core system to a previously calibrated air core system., A theoretical approach to the absolute calibration was discussed which agreed well, considering all of the approximations made, with the laboratory results. This theoretical approach may be used to obtain a good indication of the sensitivity of a cylindrically shaped coil before a laboratory analysis is performed. 83 APPENDIX 1 METHODS OF DETERMINING THE INDOCTANCE AND CAPACITANCE OF A COIL WITH FINITE RESISTANCE It is not a trivial problem to determine the inductance and capacitance of a coil which has a finite resistance. The sensor coils used with the magnetometer system all have the eguivalent circuit of Fig. A.1-1. R V L —nM/wl—©—Tomr—i— lie Fig. A.1-1 Eguivalent circuit of a sensor coil A meter which is normally used to measure an inductance or a capacitance operates on the principle that R=0. For the sensor coils, R is of the order of a few thousand ohms. Because of this, other less straight forward techniques must be employed to determine L and C. This appendix is a compilation of those technigues which were used in the laboratory. It is safe to measure coil resistance by using an ohm meters Methods of Determining L The most reliable method of determing L follows from wiring a signal generator, the coil and a load resistor in 84 series. r~ C J Fig* a.1-2 First method of determining L The signal generator must produce a sinusoidal signal of a freguency which is low enough such that : >> |R+i<-L| A. 1-1 Then the presence of the capacitor can be ignored in the circuit analysis. „. For the R-L -R loop, the following eguation is true: Thus, a plot of (V/Vjj)2 versus w2 is a straight line with slope (L/R,)2. Another circuit which can be used for determining L is shown in Fig. A.1-3. This is identical to the previous circuit except that the signal generator produces a sguare Rearranging this into a different form : A. 1-2 85 wave instead of a sinusoid., 1 c i i i I ! Fig.,A.1-3 Second method of determining L In order for this method to produce the desired results, there are two conditions which must be met. These are : A. 1-3 -!=- >> CRq Then the scope picture across RA will show an increasing and a decreasing exponential. By measuring the time which it takes for VJJ to obtain one half of the final value, the inductance can be found from : A. 1-4 This method of determining L is reliable, but in practice it is difficult to obtain better than one significant figure of accuracy because the time interval of r,,t is not easy to measure using available scopes or chart recorders. 86 Methods of Determining C-The most reliable method to determine C follows by again using the circuit of Fig. a.1-2, but this time at high freguencies. The condition which needs to be met is ; Also, another necessary condition is : a. 1-5 a. 1-6 Bhen the freguency becomes high enough, IjwLJ becomes so large that nearly all the current passes through the capacitor., If V4 is the voltage drop across Rt , the capacitance C follows from A. 1-'7. , V R a. 1-7 On the plot of log w versus log lt , this eguation is valid on the 45° sloping line to the right of the resonance point. o loq U Fig. A.1-i+ The anti-resonance point wr is called the anti-resonance freguency.„ The coil 87 inductance and capacitance are related to wr according to A. 1-8. This expression actually defines the resonance freguency, but the resonance point and the anti-resonance point are in this case essentially identical. ur -- I / JIT A. 1-8 Thus, if wr and L are known, then C can be determined using A. 1-$. , If a capacitor is wired parallel to the coil in the circuit of Fig., A.1-2, then the resonance point will be shifted to a lower freguency according to A. 1-IT" where Ca is the additional capacitor. Both L and C can be determined by shifting the resonance point as A.1-8 and A.1-90 are two eguations with two unknowns. A practical difficulty with this approach is that as the resonance point is shifted to the lower freguencies, the trough of Fig. A. 1-4; becomes shallower and wr more poorly defined. A more complete review of these techniques for determining L and C may be found in Oeda and Hatanabe (1975). 88 APPENDIX 2 THE TRANSFER FUNCTION OF THE AMPLIFIER SYSTEM-Both methods of calibration have in common the amplifier system which filters and amplifies the signal from the sensor coil. In order to make computer simulations of freguency response curves, the transfer function for the amplifier electronics must be determined. A schematic for the amplifier is shown in Fig. A. 2-1- The transfer function for each stage is given in Fig. A.2-2. The transfer function for all of the electronics of the amplifier system is then : A. 2-1 where : wt - lirr I—WWV—I I.8J KA = input impiJUmc* i HH-.3 K r output in>p«donce< \0-fT--AAAAAA*—i 5 K AAAAAr 2.5SK o—o—wwv (00 si 1 Out»&t •\5V 200 ja so XL Fig. A.2-1 Mu-metal core amplifier system Voltage Divider VS - IS Fig. A.2-2 Transfer functions for filters of the Mu-metal core amplifier sys tem An com amplifier ° i 1 ° (jw) + wN * RC u»ft« 12TT • st * Butterworth filter of 1— degree. Corner frequency = 6 Hz. 91 APPENDIX 3 THE TRANSFER FONCTION FOR THE THIRD ORDER LOW PASS FILTER The circuit diagram for the third order low pass filter which is used as a part of the air core magnetometer system electronics is shown in Fig. A.3-1. R _L_C. Fig. A.3-1 The third order low pass filter Four fundamental eguations which can be written are given ~. by eguations A.3-1 through A.3-4. V.-- Kc", Z, ^ K-2 A. 3-1 A. 3-2 A. 3-3 A. 3-4 For these eguations, Z,, Zt and Z3 are the complex impedances of C,, C. and C,. Using the top three eguations, the matrix eguation that results is : 92 2. -2. -2, o \ I A. 3-5 The solution for I, is : I.- ZRV.Z,- VeRl + Vs?.2_ A. 3-6 Baking the substitution for Vo from A.3-* into A.3-6 and then solving for the ratio V„ /Vs leads to : Vs " (R*Z,K 2*^ + 2,2,+ Rl + 2,R)- RZ.KUZ. + R^+R^Z^R+Z,) A. 3-7 The final substitutions to make are for Z(, Zx and Z^ into A.3-7 where : After rearrangement of terms, this leads to the result of A.3-8 where s=jw. K \J% ' JVCACJR** sX{2R*CFCCt4. R^Cl-^C.C^ 2R*C,CS} + S£3RC,* 2R(l-K)C2 + RC, ] + I*] A. 3-8 Let H2 be the transfer function of the third order low pass filter normalized to one at d.c. Then : u _ JL Vo A. 3-9 93 APPENDIX 4-121 ABSOLUTE CALIBRATION ACCORDING TO- THE- BRIDGE ~METHOD Using the developement of the bridge method as discussed in III-2, a relationship can be derived between the time varying external magnetic field, B , and the emf generated in the sensor coil,, This is a calculation of the absolute sensitivity as discussed in Chapter V. Eg. 3.2-7 is a relationship between the driving voltage of the bridge and the flux through the coil. This result is: It is necessary to express <fr in terms of H, and L in terms of coil geometry. , For a solenoid of length 'J. and cross-section area A such that end effects are negligable, and closely wound with N turns of thin wire so that the winding resembles a current sheet, the two expressions for 4 and L are : E A.4-1 L A. 4-3 A. 4-2 where ju.0 is the permeability of free space and k is the relative permeability. Substituting these two eguations into A.4-1, the result is : APPENDIX 4 THE ABSOLUTE CALIBRATION ACCORDING TO THE BRIDGE METHOD Using the developement of the bridge method as discussed in III-2, a relationship can be derived between the time varying external magnetic field, B , and the emf generated in the sensor coil. This is a calculation of the absolute sensitivity as discussed in Chapter V. Eg. 3.2-7 is a relationship between the driving voltage of the bridge and the flux through the coil. This result is: — - A.4-1 % L It is necessary to express <fr in terms of H, and L in terms of coil geometry. For a solenoid of length JL and cross-section area A such that end effects are negligable, and closely wound with N turns of thin wire so that the winding resembles a current sheet, the two expressions for 4 and L are : 4> ~- ^iAH A-4~2 L ~- M. & M'A/i A.4-3 where ju0 is the permeability of free space and k is the relative permeability. Substituting these two eguations into A.4-1, the result is : 94 The sensor used for the field observation (see Fig- 5.1-4) was not available to test in the laboratory so that a direct comparison cannot be made between the sensitivity as obtained from A.4-4 and the results on page 77. The different sensors are nearly identical, so the best that can be done at this point is to make the sensitivity calculation using another sensor. At a freguency of 1 Hertz, an input signal to the bridge of .75 mV p-p caused an output from the amplifier of 13 V p-p. For N=50,000, l=.9144m and R =38787a, A.4-4 indicates that B=/*0H, where /*„ =4irx10*, is 13.3x10* m!f. , The output from the bridge or the input to the amplifier is 13.0V/2x10s=65/«.V. , To find the emf in the sensor coil, use can be made of eg.,3.4-6. The following values are used: R, = 1.83 Ko. L,=930 H C, =4.4juf Zf = 44.3 -Ka II 10 Kfl.= 8.158 Kn. R% = 38.8 K*a This indicates that the ratio of the output from the bridge to the emf in the sensor coil is .771. Therefore, the emf in the sensor coil is 65/uV/. 771=84.3/*.V. The sensitivity is then : B IS.BMcfm* This is a reasonable result when it is compared to the results on page 77. LIST OF REFERENCES CONSULTED Campbell, W.H. (1967) Induction Loop Antennas for Geomagnetic Field Variation Measurement, ESSA Technical Report, ERL 123-ESL 6. Kanasewich, E.R. (1973) Time Sequence Analysis in Geophysics, Edmonton, The University of Alberta Press, pp 170-186. Kollar, F. and Russell, R.D. (1966) Seismometer Analysis Using an Electric Current Analog, BSSA, 56, 1193-1205. Lewis, W.E. and Pryce, D.G. (1965) The Application of Matrix Theory to Electrical Engin eering, London, E. & F.N. Spon Ltd, pp 141-171. Schwartz, M. (19 72) Principles of Electrodynamics, San Francisco, McGraw-Hill Inc. Slurzberz, M. and Osterheld, W. (1944) Electrical Essentials of Radio, McGraw-Hill Inc., pp. 269-303. Stratton, J.A. (1941) Electromagnetic Theory, New York and London, McGraw-Hill Inc. Ueda, H. (1975) University of British Columbia, Geophysics, M.Sc. Thesis. Ueda, H. and Watanabe, T. (1975) Comments on the Anti-Resonance Method to Measure the Circuit Constants of a Coil Used as a Sensor of an Induc tion Magnetometer, The Science Reports of the Tohoku University, Series 5, Geophysics, Vol. 22, No. 3-4, pp 129-135. Ueda, H. and Watanabe, T. (1975) Several Problems about Sensitivity and Frequency Response of an Induction Magnetometer, The Science Reports of the Tohoku University, Series 5, Geophysics, Vol. 22, No. 3-4, pp. 107-127. 

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