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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 i n the Department of Geophysics and Astronomy We accept this thesis as conforming to the required standard The University of B r i t i s h Columbia June 1977 © Liana Fran ces> Zamb resky, 1977 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ABSTRACT An i n v e s t i g a t i o n made concerning the c h a r a c t e r i s t i c s of a sensor c o i l f o r an induction magnetometer shows that i t i s f e a s i b l e to make the f i r s t stage 60 Hertz r e j e c t i o n f i l t e r of the Butter-worth type. This i s an improvement i n design over the Twinr-T f i l t e r which i s sometimes used as a f i r s t stage f i l t e r as the number of e l e c t r i c a l components i s reduced and there i s no pos-s i b i l i t y of ringing between the c o i l inductor and the f i l t e r capacitors. Two methods of r e l a t i v e c a l i b r a t i o n f o r the induction magne-tometer system give r e l i a b l e response curves. One method uses a Wheatstone bridge. The sensor i s one arm of the bridge and i t i s shown that the e f f e c t of the s i g n a l generator i s the same as i f the c o i l was excited by a n a t u r a l event. The second method involves e x c i t i n g the sensor by a f i e l d created by a small secon-dary c o i l . The agreement between the two methods i s good. An experimental approach to the absolute c a l i b r a t i o n i s suc-c e s s f u l l y c a r r i e d out by comparing the output from the uncalibra-ted system to an a i r core system which has been previously c a l i -brated. A t h e o r e t i c a l approach i s used to give a good i n d i c a t i o n of the s e n s i t i v i t y of the sensor c o i l . The s e n s i t i v i t y i s depen-dent p r i m a r i l y upon the turn number and the length of the c o i l . i i i TABLE OF CONTENTS ABSTRACT I i LIST OF FIGURES v LIST OF TABLES v i i i ACKNOWLEDGEMENTS i x CHAPTER I GENERAL INTRODUCTION 1 CHAPTER II-1 THE 60 HERTZ REJECTION PROBLEM 6 I I - l Introduction 6 II-2 Theory of Twin-T f i l t e r 7 II-2.1 Computer Results 15 II-3 Theory of Butterworth f i l t e r 17 I I - 3.1 Laboratory Results 22 CHAPTER I I I THEORY OF THE BRIDGE METHOD 25 I I I - l Introduction 25 III-2 An I n t u i t i v e 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 C a l i b r a t i o n 37 III-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 52 IV-2 Theory of Operation 53 IV- 3 Theory of C a l i b r a t i o n 56 CHAPTER V THE ABSOLUTE CALIBRATION 66 V- l A Laboratory Approach 66 V-2 A Theoretical Approach to the Absolute S e n s i t i v i t y 74 CHAPTER VI SUMMARY AND CONCLUDING REMARKS 81 APPENDIX 1 Methods of Determining the Inductance and Capacitance of a C o i l with F i n i t e Resistance 83 APPENDIX 2 The Transfer Function of the Amplifier System 88" APPENDIX 3 The Transfer Function f o r the Th i r d Order Low Pass F i l t e r 91 APPENMxRIiESEFCSS COT^eMbstolute C a l i b r a t i o n According to -- 'J the Bridge Method 93 LIST OF REFERENCES CONSULTED 95. V L i s t of Figures Chapter I Fi g . 1-1 Equivalent c i r c u i t of sensor c o i l 2 Chapter II F i g . 2.2-1 The symmetric Twin-T f i l t e r 7 F i g . 2.2-2 Block diagram of sensor coil/Twin-T network 12 F i g . 2.2.1-1 Computer simulation of the frequency response of the coil/Twin-T sensing system 16 Fi g . 2.3-1 An analog Butterworth f i l t e r 17 F i g . 2.3-2 The transfer function f o r the Butter-worth f i l t e r 20 Fi g . 2.3.1-1 Laboratory set-up f o r the Butterworth f i l t e r 22 F i g . 2.3.1-2 Butterworth f i l t e r c h a r a c t e r i s t i c s of the sensor c o i l system 24 Chapter I II F i g . 3.2-1 The Wheats tone bridge 26 F i g . 3.2-2 Norton equivalent of the sensor c o i l 28 F i g . 3.2.1-1 Wheatstone bridge at higher frequencies 29 F i g . 3.2.1-2 Norton equivalent of the Wheatstone bridge 30 F i g . 3.2.1-3 C i r c u i t used to derive a new Norton equivalent 30 Fi g . 3.3-1 The bridge c i r c u i t with parameters as they are defined f o r c i r c u i t analysis 34 V I F i g . 3.3.3-1 Computer simulation to determine the e f f e c t of the conditions R^» and R^ >> R2 on the response of the magne-tometer system when using the bridge method 41 Fi g . 3.3.3-2 The laboratory bridge c i r c u i t 43 Fi g . 3.3.3-3 The normalized amplitude response of the magnetometer system according to the bridge method 44 Fi g . 3.3.3-4 The phase response of the magnetometer system according to the bridge method 45 Fi g . 3.4-1 Diagram of sensor c o i l and load im-pedance 47 Fi g . 3.4-^ 2 The e s s e n t i a l elements of the bridge c i r c u i t 50 Fi g . 3.4-3 The e f f e c t of the resistance R^ on the frequency response as obtained from the bridge method 51 Chapter-jLV F i g . 4.2-1 The secondary c o i l method at the time of operation 53 Fi g . 4.3-1 The secondary c o i l method at the time of c a l i b r a t i o n 56 F i g . 4.3-2 The secondary c o i l method in>the lab-oratory 60 Fi g . 4.3-3 The normalized amplitude response of the magnetometer system according to the secondary c o i l method 62 Fi g . 4.3-4 The phase response of the magnetometer system according to the secondary c o i l method 63 F i g . 4.3-5 A comparison of the frequency response curves obtained i n the laboratory from the secondary c o i l method and the bridge method 64 v i i F i g . 4.3-6 Chapter V F i g . 5.1-1 F i g . 5.1-2 F i g . 5.1-3 F i g . 5.1-4 F i g . 5.2-1 Appendix 1 F i g . A.1-1 F i g . A.1-2 F i g . A.1-3 F i g . A.1-4 Appendix 2 Fi g . A.2-1 F i g . A.2-2 Appendix 3 F i g . A.3-1 A comparison of the phase response curves obtained i n the laboratory from the secondary c o i l method and the bridge method 65 C i r c u i t diagram f o r the a i r core system 69 Computer simulated frequency response of the a i r core c o i l system 71 Method to determine the inner radius of a large c o i l 72 The micropulsation event used f o r the absolute c a l i b r a t i o n 73 The dependence of the geometric per-meability M\ upon the r a t i o of the lengths of the semiprinciple axes of a prolate spheroid, a/b 80 Equivalent c i r c u i t of a sensor c o i l 83 F i r s t method of determining L 84 Second method of determining L 85 The anti-resonance point 86 Mu-metal core amplifier system 89 Transfer function f o r f i l t e r s of the Mu-metal core amp l i f i e r system 90 The t h i r d order low pass f i l t e r 91 L i s t of Tables Chapter I Table 1-1 C o i l s p e c i f i c a t i o n s i x ACKNOWLEDGEMENTS I would l i k e to express my sincere gratitude and appreciation to Dr. Tomiya Watanabe for h i s continuous help and guidance as my research advisor throughout the work presented i n this t h e s i s . I g r a t e f u l l y acknowledge Dr. R.D. R u s s e l l for many h e l p f u l discussions and ideas. I would l i k e to express many thanks to Dr. K. Hayashi and Dr. T. Oguti f o r t h e i r patient guidance i n the e l e c t r o n i c s laboratory. I would l i k e to acknowledge the V i c t o r i a Geophysical Observa-tory, Dept. of Energy, Mines and Resources for equipment which we borrowed. For the operation at C h u r c h i l l , Manitoba, I would l i k e to thank Mr. C.R. Barrett, superintendent and h i s stafft.at the C h u r c h i l l Re-search Range of NRC. This research received f i n a n c i a l support from the following grants : NRC A-3564 NRC E-2923 DRB 9511-112 UBC A r c t i c 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 B r i t i s h Columbia., I t i s the long term intent of t h i s group to es t a b l i s h a number of observing stations across northern Canada fo r micropulsation research., It was f e l t that a system could be b u i l t at a cost s i g n i f i c a n t l y l e s s than commercial magnetometers. The reguirements of t h i s instrument are such that i t should cover the freguency band from 0 . 0 0 2 Hz to 4 Hz and take the s i g n a l l e v e l from the order of milligamma to ten gammas. The sensor was constructed with a Mu-metal core i n order to reduce the physical s i z e from that of an a i r core sensor. Mu-metal i s a high permeability a l l o y of nickel and i r o n with a small amount of chromium and molybdenum. Ueda (1975) demonstrated that the Mu-metal core does not cause s i g n i f i c a n t d i s t o r t i o n s or harmonics to the s i g n a l contrary to what some e a r l i e r investigators had thought. The equivalent c i r c u i t which i s used for the sensor c o i l throughout t h i s research i s shown i n Fig- 1-1. The capacitance e f f e c t of the windings, noted as early as 1910 from general antenna use, Campbell (1969), gives r i s e to the capacitor which i s p a r a l l e l to the resistance and inductance of the c o i l . , The voltage generator, V, arises from the current which i s induced in the c o i l by the changing magnetic f l u x . 2 I t i s d i f f i c u l t to measure the c o i l inductance and capacitance using a bridge which i s designed for measuring pure inductance and capacitance., For t h i s reason^ somewhat laborious methods must be put into practice. A compilation of such methods used for t h i s research i s given in Appendix A - 1 . , These methods lend a great amount of c r e d i b i l i t y to the equivalent c i r c u i t concept because the c o i l was found to behave exactly as the c i r c u i t theory predicts i n each method. R V L . pAAA/W © — W W — | • lie Fig. 1-1 Eguivalent C i r c u i t of Sensor C o i l Further support f o r the eguivalent c i r c u i t occurs from observations that the output of the sensor c o i l with a p a r a l l e l r e s i s t o r and capacitor of appropriate values has c h a r a c t e r i s t i c s of a Butterworth f i l t e r . The theory of the Butterworth f i l t e r i s developed in d e t a i l according to c i r c u i t theory. Laboratory r e s u l t s concur with theory and give more support to F i g . 1-1.. A useful improvement i n the design of a 60 Hz rejec t i o n f i l t e r r e s u l t s from t h i s analysis which i s discussed i n d e t a i l i n Chapter I I . The primary intent of t h i s thesis i s to investigate the p r a c t i c a l i t y of two methods of r e l a t i v e c a l i b r a t i o n for 3 finding the freguency response of the induction magnetometer. One of these methods i s c a l l e d the bridge method i n which one arm of a Hheatstone bridge c i r c u i t i s the sensor c o i l of the magnetometer. I t w i l l be shown that emf generated i n the sensor c o i l by an external time-varying magnetic f i e l d can be simulated by driving the bridge with a sig n a l generator.. The other method, c a l l e d the secondary c o i l method, i s one in which a small secondary c o i l i s wound on the core c o a x i a l l y with the sensor c o i l and creates magnetic flux to be detected by the sensor c o i l . The secondary c o i l method has the disadvantage that an extra cable i s needed between the secondary c o i l and the amplifier system. In the f i e l d , the c o i l s and the amplifier e l e c t r o n i c s are separated by approximately 100 yards i n order to prevent spurious noise generated by the electronics from being mixed with the natural s i g n a l . Thus, i t would be better to have one cable instead of two, not only from the standpoint of cost and convenience, but also because the p o s s i b i l i t y of undesirable cable e f f e c t s would be reduced, i . e . "cross-^talk**. This problem makes the bridge method more desirable as a c a l i b r a t i o n procedure, An absolute c a l i b r a t i o n must be carried out i n order to determine the l e v e l of the r e l a t i v e freguency response curve. This i s done by comparing the output of the Mu—metal core system to another system which i s already calibrated. This second system i s an a i r core magnetometer. S e n s i t i v i t y of an a i r core c o i l can be calculated exactly knowing the 4 geometry of the c o i l and i t s turn number. The two magnetometers are set up i n the f i e l d , approximately 100 yards apart, and record micropulsations. The assumption i s made that such a global event w i l l not change over t h i s distance. Some micropulsation events are recorded as sinusoidal signals and i t i s t h i s type of event which i s used for the comparison. Readings can be made over a number of cycles so that the error of t h i s measurement i s small. F i n a l l y , a t h e o r e t i c a l approach i s taken towards the s e n s i t i v i t y of a c o i l i n order to elucidate which parameters are s i g n i f i c a n t . The parameters of interest are the c o i l dimensions, the gauge of the wire out of which the windings are made, and the permeability of the core., Such an approach would be h e l p f u l when designing new sensors. The s p e c i f i c a t i o n s of a l l of the c o i l s used f o r th i s research are given i n Table 1-1. These s p e c i f i c a t i o n s w i l l be referenced throughout t h i s thesis. Turn Number R (fi) L (H) C Inner Diam. (cm) Outer Diam. (cm) Length (cm) A i r Core C o i l 5000 5130 120 .50 yf 149.02 151.40 4.76 Mu-metal Core C o i l (1976) 50,000 1831 930 .188 yf 3.17 7.30 r45\72 Secondary C o i l 20 .011 3.17 Cal i b r a t i o n C o i l 1000 11 17.14 241.3 Mu-metal Core C o i l (19 75) 50,000 2230 1050 .048 nf 4.00 9.5 35.56 Table 1-1 C o i l S p e c i f i c a t i o n s 6 CHAPTER II THE 60-HERTZ REJECTION PROBLEM I I - J Introduction The r e j e c t i o n of 60 Hz noise i s a problem which reguires c a r e f u l consideration for a magnetometer that i s being designed to detect micropulsations. At low noise s i t e s which are c a r e f u l l y chosen in the f i e l d , the noise i s considerably reduced from what i t would be in a c i t y . I t i s necessary to reduce the 60 Hz noise by at least a factor of 100 before allowing the s i g n a l to be amplified. Otherwise the amplifier may be overloaded or the micropulsation information l o s t due to the extremely low s i g n a l to noise r a t i o . One method which has been used for some previous magnetometers i s to place a Twin-T f i l t e r , a notch f i l t e r with center freguency at 60 Hz, d i r e c t l y between the sensor c o i l and the amplifier. It w i l l be shown i n the discussion which follows that the freguency response of the signal output from the Twin-T f i l t e r i s dependent upon the input impedance of the amplifier. This i s not a desirable e f f e c t . There i s a d i f f e r e n t approach to the f i r s t stage of the 60 Hz rejec t i o n problem that i s better for two reasons. One reason i s that the frequency response i n the micropulsation range i s ensured to be f l a t . The other i s that only one r e s i s t o r and capacitor are needed instead of six c a r e f u l l y matched components for a Twin-T f i l t e r . This approach 7 requires making the sensor c o i l and the p a r a l l e l RC combination into an analog Butterworth f i l t e r . I1-2 Theory of Twin-T F i l t e r The f i l t e r of interest i s the symmetric Twin-T. A c i r c u i t diagram i s shown i n Pig. 2.2-1. L l R R 2 R -A/WW 2C < X v. Fig. 2.2-1 The symmetric Twin-T f i l t e r To aide making a c i r c u i t analysis, the c i r c u i t i s generalized to that of: 8 This i s a p a r a l l e l connection of two networks and may be depicted schematically as: Each network may be described by a matrix equation i n 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 r e s u l t : This l a s t eguation i s important because i t means that the admittance matrix of a number of networks connected i n 9 p a r a l l e l i s the sum of the admittance matrix of each i n d i v i d u a l network. In the case of the p a r a l l e l - T network, the entire network can be broken down into two simpler networks of the type: The mesh eguations for t h i s network are: V, Z b I , + (2 b +2c)r z The matrix representation which follows i s : It i s of int e r e s t to have the matrix eguations i n the form I=¥V. The I matrix, or the admittance matrix, i s the inverse of the Z. matrix. Thus, I=YV implies: 10 I III v, JZ| i s the determinant of the %_ matrix. I t i s necessary to determine the Y matrix for each of the p a r a l l e l - T sub-networks. The f i r s t network to consider i s : R 2jwC The res u l t i n g Y matrix i s : i w 2j ui C The second network to consider i s : 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 p a r a l l e l - T network as depicted in Fig. 2.2-1 i s then Y = Y ' + Y ' This matrix i s : Y Y Y Y, 2.2-1 where: V - 1 » (ju>CR )* 2 R 0 + The ultimate aim of t h i s discussion i s to f i n d the freguency response of the Twin-T network when i t has a signal from the sensor as input and when i t i s terminated by the f i n i t e impedance of the amplifier system. Schematically, t h i s i s depicted by Fig 2.2-2. The matrix equation describing t h i s system i s : V 12 / A ' A, 8 t + 8,0, C, B , + D, Dj "1 A l B l C l D l 2 A 2 B 2 C 2 D 2 3 4 1 V 2 y Sensor C o i l Twin-T Network R. F i g . 2.2-2 Prom the matrix equation, i t follows that V, =A' ?3 +B' I J and since 1% =?3 /R.t , the r e s u l t i n g transfer function i s : V, R; A' R; + B' The matrices depicted above are not i n the form of the Y matrix. In the terminology of e l e c t r i c a l engineering these have the form of the F matrix. The reguired transformation i s : F --/ V *1X m Y», i * i where: Y n Y t l ~ X i , . Y Z, 13 To compute the F matrix for the sensor c o i l , consider the following c i r c u i t 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)v t+ (R +ju,L) l . The sensor c o i l matrix i s then: l-wHC + iwCR R*jwL V l w C Sow a l l of the guantities to determine the complete transfer function of the system i n Fig. 2.2-2 are known.. The f i n a l result i s : 14 V. A' R- + B ' 2.2-2: where: A - A , A l + B , C l Bv' A, B x + B, A,-- 1- u, lLC + 3 1 0 C R 8 , - R + 0. 1L Yx \ Y,x Y , t - Y „ Y . » Y » Y, »' 1 M l 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 re s u l t s of t h i s computing may be seen i n Pig. 2,2.1-1. The s t r i k i n g feature to notice i s how the shape of the transfer function changes as a function of the terminating impedance of the Twin-T network. This terminating impedance i s the input impedance to the amplifier system. I f the amplifier i s an integrated c i r c u i t , then i t s input impedance could e a s i l y be of the order of 10Mix and a severe ringing e f f e c t w i l l occur near 5 Hz. This ringing i s the re s u l t of coupling between the c o i l inductor and the Twin-T capacitors. If the amplifier input impedance can be reduced, then the ringing e f f e c t D i l l lessen. The curve for an input impedance of 7.5 Ksi. shows no ringing at a l l and has a smooth drop o f f , . However, as described e a r l i e r , such a response curve can be obtained by exploiting a Butterworth f i l t e r . I t would reguire only one r e s i s t o r and capacitor i n p a r a l l e l to the c o i l . . The,Twin-T would reguire s i x c a r e f u l l y matched r e s i s t o r s and capacitors, and i n the case of balanced input, twelve elements would have to be matched instead of s i x . In practice, t h i s may be a troublesome requirement.. I t should also be noticed that as the terminating impedance i s reduced, the s e n s i t i v i t y at the lower frequencies i s reduced. This i s not a desirable e f f e c t . 16 FREQUENCY (HER^Z) F i g . 2.2.1-1 Computer simulation of the frequency response of the coil/Twin-T sensing system 17 II~3 Theory, of the Butterworth F i l t e r The theory w i l l now be developed for an analog Butterworth f i l t e r . The c i r c u i t to be analyzed i s shown i n Fig. 2.3 -1. A r e s i s t o r and capacitor are placed p a r a l l e l to the output of the sensor c o i l . L -rSinnnnr-v. $ l vwvww R :R. v Fig.2.3 -1 An Analog Butterworth F i l t e r To determine V0 , f i r s t f i n d the equivalent impedance of the three p a r a l l e l quantities C, C, and R, . 1 + iu»U + O R, The equivalent c i r c u i t becomes: L © - M M A A r R V can now be found from the voltaqe divider method. R •* j w L + H 18 After substituting for Z, a considerable amount of algebra leads to the r e s u l t : 2.3-1 ~1 This equation for V„ can be put into a non-dimensional form. The following substitutions w i l l be made: 1. Let oc represent the r a t i o of d.c. resistance of the sensor c o i l to the terminating resistance. R 2.3-2 2. Let f? represent the inverse Q factor of the S-L-C series c i r c u i t . Q. 4 2.3-3 3. Let f r represent the resonance freguency of the B-L-C series c i r c u i t . 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~ -- t a n j In order for the c i r c u i t 2.3-1 to respond as a Butterworth f i l t e r , the term { f / f r ) 2 must be made to approach zero. A Butterworth f i l t e r of any order i s present when the transfer function contains the following expression; A plot of the transfer function for the Butterworth f i l t e r of orders n=1,2,3 can be seen i n Fig 2.3-2- I t i s evident that in the case of equation 2.3-5, t h i s i s a Butterworth f i l t e r of order 2 since i t i s of the form: i 1 1 2.3-6 i f the second term i n the denominator can be made negligable. I t can be seen from the c i r c u i t diagram 2.3-1 that i t would be desirable to have S, large compared to R as t h i s would increase the s e n s i t i v i t y . Thus i t can be expected that *<1.. , In the case that oc»0, the second term vanishes f o r : 0 - ^ In the case that 0<«<1, the second term vanishes for either of the following two values of p : F i g . 2.3-2 The trans f e r function f o r the Butterworth f i l t e r 21 I f 0 « « 1 ., the binomial series expansion may be used to show that the above two values are approximately equal to: P • « / n r In the case that rt>1, there i s no p which can make the second term vanish. This i s another reason why * i s desired to be small. In practice, i t i s best to determine P f i r s t and then make * equal to that value which w i l l n u l l i f y the second term. As the sensor w i l l be used to detect micropulsations, i t i s desirable that i t have a f l a t freguency response over the range from . -002 Hz - 4 Hz . Also, since there i s a considerable amount of man made noise at 60 Hz, i t i s necessary to optimize the g u a l i t i e s of the Butterworth f i l t e r i n 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 i f the cut-off freguency i s made egual to 6 Hz. Thus, C, can be determined from eguation 2.3-4 provided the c o i l parameters R, L and C are known. This leads to: c = J_ - r U> L ( 2 i r f v r ^ 2.3-7 P can now be found from eguation 2.3-3. The value of o( which w i l l n u l l i f y the second term i n 2.3-5 i s : If P>JT , no «C can make the second term vanish. 0 f o r th i s type of sensor i s t y p i c a l l y of the order of 10 - 4-10 - 2 , Oeda 22 and watanabe ( 1 9 7 5 ) , so there i s not any threat that t h i s condition might occur., Thus, B can be found from the eguation: R = R 2.3-8 II-3.1 Laboratory J e s u i t s The c i r c u i t of Fig.,2.3.1-1 was set up i n the laboratory to determine whether the theory of the Butterworth f i l t e r i s correct for the c o i l s . -© wvvvv -VVVvVVV1-v out I 1 ma rms Fig. 2.3.1-1 Laboratory Set-up for the Butterworth F i l t e r A sinusoidal magnetic f i e l d was created inside a large c a l i b r a t i o n c o i l , , The intensity of the f i e l d at the center of the c o i l i s |^71: Gamma from the rms current of 1 ma that flows through i t s windings. The sensor c o i l was placed inside the large c o i l . . The time-varying magnetic f i e l d induced a 23 sinusoidal emf i n the sensor c o i l and the output from the c o i l was measured across a p a r a l l e l HC load. From eguation 2.3-7, the corner frequency of 6 Hz and the c o i l s p e c i f i c a t i o n s of the flu-metal core c o i l (1975), {see Table 1 - 1 ) , the value of the p a r a l l e l capacitor was determined to be .67 y u f . Then, from equation 2 . 3 - 3 , 0 =.0565. This leads to the value B, = 2 8 K A as determined from equation 2 . 3 - 8 . The r e s u l t s of the laboratory experiment can be seen i n Fi q . 2.3.1-2. The frequency response i s f l a t 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. I t should be noted that v o u + i n Fig 2.3.1-1 has been divided by freguency because the emf induced i n a c o i l by a changing magnetic f l u x i s proportional to freguency and the amplitude of the magnetic f i e l d . 24 F i g . 2.3.1-2 Butterworth f i l t e r c h a r a c t e r i s t i c s of the sensor c o i l system 25 CHAPTER III THEORY OF THE BRIDGE METHOD III-1 Introduction The o r i g i n a l idea to c a l i b r a t e the induction magnetometer by using a Hheatstone bridge was conceived by Dr. R.D. Russell at the University of B r i t i s h Columbia. This idea stemmed from a previously successful undertaking to c a l i b r a t e an electromechanical seismometer using a Maxwell bridge, Kollar and Russell (1966). The f i r s t part of t h i s chapter w i l l be concerned with i n t u i t i v e l y analyzing the Hheatstone bridge according to Norton's theorem, otherwise known as the current source model.. The remainder of the chapter w i l l present a detailed analysis according to Kirchoff*s laws. The bridge system w i l l be analyzed for three configurations. The f i r s t w i l l be when the magnetometer i s making observations and the bridge i s a part of the electr o n i c s . , I t i s hoped that the bridge can always remain a part of the magnetometer el e c t r o n i c s so that i t w i l l not need to be wired i n t o the system every time a c a l i b r a t i o n i s made. The second configuration w i l l be the proposed c a l i b r a t i o n procedure i n which the bridge i s driven by an e l e c t r i c a l o s c i l l a t o r . The t h i r d w i l l be to compare the case of observation when the bridge i s not present to when i t i s . This l a s t step i s required i n order to determine how the presence of the bridge could d i s t o r t data. 26 III-2 An I n t u i t i v e Approach Dsinq the Norton Equivalent I t i s the intent of the developement which follows to show that a rate of change of magnetic f l u x , <ij , can be simulated i n the sensor c o i l by driving a Hheatstone bridge with an e l e c t r i c a l o s c i l l a t o r . The sensor c o i l w i l l be one arm of the bridge and the equivalent c i r c u i t which w i l l be used for i t has already been given i n Fig.,, 1 - 1 . / The Hheatstone bridge i s shown in F i g . 3.2 - 1 . Fig.,3.2 - 1 The Hheatstone bridge As the sensor i s to be used for micropulsation research, frequencies of i n t e r e s t w i l l be between .002 - ,4 Hz, so that the capacitive reactance of Fig. 1 - 1 i s negligable. Also, i t i s assumed that the amplitude of the s i g n a l from the e l e c t r i c a l o s c i l l a t o r w i l l be much larger than any s i g n a l which could be induced in the c o i l 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 i s balanced at d.c.. This balancing condition i s expressed as: R R 3 - RZR^ 3.2-2 In order to si m p l i f y the analysis, the Norton or current source equivalent i s to be found between the points a and b of F i g . 3.2 - 1 . This i s done by removing L and short c i r c u i t i n g the two points a and b. The current which flows between these points would be: 3.2-3 Next, consider the impedance when looking i n from the two terminals a and b. The impedance would be B*RL*Rt i f H l t » R L Then, the equivalent c i r c u i t which follows i s : and R3 » R 2 R -M/VVW 28 For a c o i l of N turns immersed in an average magnetic flux per one turn of the c o i l , <t> , the t o t a l flux through the c o i l i s N<|>. Thus, the emf induced i n the c o i l toy the changing flux i s : E = "-3t(N<J)) 3.2-4 But, i t i s also true that: E = - ( L i ) ^ u i ; 3.2-5 Applyiag Norton's theorem, the eguivalent source current i s : 3.2-6 Using the r e s u l t of 3.2-6, the Norton eguivalent of the sensor becomes: R —'WWW— L F i g . 3.2-2 Norton eguivalent of the sensor c o i l I f the currents through the inductor i n the cases of 3.2-3 and 3.2-6 are egual, then an important r e s u l t exists between the amplitude of the driving voltage and the flux through the c o i l : A.. ~ I 3.2-7 29 III-2.1 Theory at Higher Frequencies If the Hheatstone bridge i s going to be used at higher frequencies, then the capacitance of the sensor must be taken i n t o account. To compensate for t h i s , an inductor L 3 i s added to the fij arm of the bridqe. The bridge c i r c u i t which follows i s shown i n F i g . , 3. 2. 1-r 1. , Fig..3.2.1-1 Hheatstone bridge at higher freguencies The Norton equivalent c i r c u i t which r e s u l t s from using the condition 3.2-1 i s shown i n the following f i g u r e . 30 Fig. 3.2.1-2 Norton eguivalent of the Wheatstone bridge In order to a r r i v e at a new Norton eguivalent, short the two points a and b, , Then calc u l a t e the current which flows through S. Currents are defined according to Fig . , 3. 2. 1-3. E ± E R R, R 3+303L 3 F i g . 3.2.1-3 C i r c u i t used to derive a new Norton equivalent Let Z, be the p a r a l l e l composite impedance of R and C. R 3.2-8 applying Kirchoff's law to the R-ZL-RZ loop, the eguation which r e s u l t s i s : 3.2-9 31 Using the condition for the d.c. balanced bridge; R R 3 = R z R^ and another condition on L, that: CR - 3.2-10 eguation 3.2-9 s i m p l i f i e s to: R. 3 . 2 - 1 1 Then, the current i t flowing through R i s given by: Therefore, the eguivalent c i r c u i t becomes: R 4 (1+JOJCR) R R„ I f i , and i of equation 3.2-6 are equal, then the r e s u l t i s ; Ru. Ci+^cR) L 3.2-12 The importance of the re s u l t s 3.2-12 and 3.2-7 are that they predict that a rate of change of magnetic f l u x , <l) , can be simulated i n the sensor c o i l by driving a Hheatstone bridge 32 with an e l e c t r i c a l o s c i l l a t o r . In practice, i t i s much simpler to simulate a magnetic flux in t h i s manner than i t i s to immerse the c o i l i n a uniform c a l i b r a t i n g f i e l d . A freguent method which i s used to create an a r t i f i c i a l f i e l d i s to put the sensor c o i l inside a larger c a l i b r a t i o n c o i l . Besides being physically cumbersome so that an investigator would not want to bring a c a l i b r a t i o n c o i l along to the f i e l d , i t costs nearly as much to b u i l d as the sensor c o i l . By using the Hheatstone bridge method, the c a l i b r a t i o n can be carr i e d out at any time and with very l i t t l e expense. At t h i s point, the problem of the absolute c a l i b r a t i o n i s not worked out too well. The t o t a l flux N<|> i s related to the external f i e l d B by the r e l a t i o n ; According to Dr. B.D. Bussell at DBC, the c a l i b r a t i o n of sensors i s reduced to finding expressions for 4> i n terms of H, and for L i n terms of the c o i l geometry. The problem of absolute c a l i b r a t i o n w i l l be discussed i n d e t a i l i n Chapter V. s 33 III-3 Theory of the Bridge Method A detailed c i r c u i t diagram f o r the bridge i s shown i n F i g . 3.3-1. In practice Z e ,Z 2 and Z^ are pure r e s i s t o r s . I t was o r i g i n a l l y hoped that Z 3 would consist of the SLC configuration sketched below. AVVWV-T h e o r e t i c a l Z, In t h i s way, both an a.c. and a d.c. balance could be achieved using the bridge. Theoretically and experimentally, t h i s 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 Z 3 i s shown below. R -MAAAAr -AA/VVW—nfwrr-Experimental Z. Hhat follows i s a c i r c u i t analysis of F i g . 3.3-1 which w i l l show that t h i s type of bridge arrangement w i l l t r u l y r e f l e c t 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 E 0 w i l l be the emf caused by the signal generator. E, w i l l be the emf generated by the time variations of the magnetic f i e l d of the earth. A system of equations can now be defined by applying Kirchoff's law to current loops. The X,~ZZ-ZS loop y i e l d s : E, = X ^ , - 2 ,1 , • 2 , 1 , The Y,-Z z-Zj. loop y i e l d s : The Z0 -Zz -Z 3 loop y i e l d s : e o = z . C l - U - V i r l , ) - 2 2 i 2 The Zy-Z^-Zj loop y i e l d s : O - 2 , 1 , - 2 , ^ 1 - 1 , ) - 2 3 ( 1 , - 1 3 ) These eguations can be arranged, regarding I , # I a , I 3 and I as the independent variables. The corresponding matrix equation i s : 3 6 X, Z, o -U„*z^z,} o - z . 2, ( 2 , + Let 3.3-4 D be the determinant which i s defined by the c o e f f i c i e n t s . I t can be evaluated as follows: where + Z ^ ( Z 2 Z 3 + Z 3 Z , - 7 5 7 J 3.3-5 The current I 3 i s the key to the freguency response because i t flows through the load impedance Zs. I f t h i s method i s going to properly determine the freguency response of the sensor, then i t must be shown that calibration, = C o n s t * <J3 > observation ' T h e observation condition implies that I 3 i s the res u l t of currents induced i n the sensor by a f l u c t u a t i n g magnetic f i e l d and Eo=0. The c a l i b r a t i o n condition means that I 3 i s caused by the emf of the s i g n a l generator and E 0 » E , . Therefore, the system of simultaneous l i n e a r algebraic eguations must be solved with respect to I 3 f o r the two cases. 37 III-3.1 Case of Observation For t h i s case, E,#0 and Eo=0. The solution of tae matrix equation 3.3-4 qives: D The natural magnetic f i e l d fluctuations may be defined i n the following manner, where B i s the amplitude of the f i e l d changes and S i s the absolute s e n s i t i v i t y of the sensor c o i l : E x = 1 ]u> S B e>U> 3.3.1-2 III-3.2 Case of Calibration -For t h i s case, E 0 » E , . The solution of 3.3-4 gives f o r E4 =0 and E„ #0: The numerator of t h i s l a s t equation can be reduced to a s i m p l i f i e d form by making some substitutions and approximations. F i r s t , substitute f o r Z, according to eguation 3.3-3., Then: 38 ( X T + Y , ) ( Z . Z . - Z ^ Z J * X . Y . Z , - ( X , * Y , ) 2 2 Z , Hext, assume that the branches ZT ,Z 3 and 2^ . are a l l pure r e s i s t o r s . That i s Z l=R 2 , Z 3 = a 3 a B d Z^=RV. Then: (y (*Y,Hz ,Z,-2 1Zj* X , Y , R 3 - U . + Y J R ^ How s u b s t i t u t e f o r X, and I, ac c o r d i n g t o egu a t i o n s 3.3-1 and 3.3-3. I f t he b r i d g e i s balanced f o r the d.c. c a l i b r a t i o n s i g n a l , then the c o n d i t i o n which i s met i s : T h i s l e a d s t o the r e s u l t : (y(fY,Kzlz3-2l7^- ( ~ - 3-L,R,R 3 as L,ai10 3H r C,«i10-7F and B,«* 10 3 XL , the c o n d i t i o n i s w e l l s a t i s f i e d t h a t : — >> R, 1 3.3.2-3 The eguation reduces t o : (xt+Y(Kz,VZJ,) - T R B " > - L ' R . R 3 39 For micropulsation research, the frequency range of i n t e r e s t i s .002 Hz_i< f < 4 Hz-, The quotient 1/CR*10*. Then, a f i n a l assumption can be made that: I C J < < 3.3.2-4 I t should be noted that t h i s l a s t condition w i l l break down at higher frequencies., Whereas i t i s well s a t i s f i e d at the lower frequencies, below 1 Hz, at a frequency of 10 Hz, the condition i s r e a l l y not too well met. Substituting C, =.2 /xF and R =2Ka: 3 4 - 0 Thus, the f i n a l reduction of the numerator of equation 3.3.2-1, keeping i n mind that f < 5 Hz, i s : (X,+ Y ,K Z . l ^ H , ± - R 3 3.3.2-5 Now the r a t i o ( I 3 ) C O l i /{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 f i n a l r e s u l t , keeping i n mind conditions 3.3.2-3 and 3.3.2-4, i s : -- i l . R , i o _ 3 3 2 _ 6 {U)0^ Z 0 ( R 3 + R ^ ^ + R U f ( R 1 + R 3 ) S B Thus, the important conclusion i s that for the d.c. balanced no bridge: ( i 3 ) c * l " 3.3.2-7 The r a t i o /(X,')0v>s i s independent of freguency. Therefore, the freguency response of (I 3 )co.i f a i t h f u l l y r e f l e c t s that of the induction magnetometer. III-3.3 Computer and Laboratory Results The success of the bridge method w i l l depend upon how well two conditions are s a t i s f i e d . These conditions are: B^»R, 3.3.3-1 R 3 » R Z I f these conditions are poorly met, then the output from the bridge may d i f f e r considerably from the case when these conditions are met,. a computer program was written to precisely determine how the response curve for the sensor c o i l would be affected by the ineguality.. The eguation programmed was V=(I 3) o b <. % s , where (I 3) 0v, t i s defined by 3-3-1-1- These r e s u l t s may be seen i n Fig 3.3.3 - 1 . , The program was written with R3=R,. The curve which r e s u l t s from Rl = .5R, and R^ =2R , d i f f e r s s u b s t a n t i a l l y from the curve when Hj_=.02R, and R^SOR^ . There i s not much difference between the curve f o r R1=.05R, and RIV=20R, and the curve when ai=.02H, and R^ =50R-, . The conclusion i s that the condition R l f r»R l implies Rlt>20R, and R 3 » R 1 implies Ra<. 05R3 . 41 ,2 .3 .4 £ ,8 10 2 3 4 6 8 • 10 Frequency (Hertz) KEY: V . 5 ^ V 2 R i V .lR^ R =10R. 4 1 V . 0 5 1 ^ R = 2 0 R 1 4 1 V . 0 2 R ! R =5QR., 4 1 F i g . 3.3.3-1 Computer simulation to determine the e f f e c t of the conditions R ^ » R^ and R ^ » R 2 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 c i r c u i t diagram i s shown in F i g . 3.3.3-2. The bridge r e s i s t o r s were determined according to the conditions 3.3.2- 2 and 3.3.3-1 . To s a t i s f y 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 r e s i s t o r R^. Also, i t must be noted that the f i r s t stage of the amplifier system i s a chopper amplifier (see Appendix 2) which requires a balanced input s i g n a l . This i s the reason >. for using the inve r t i n g amplifier as part of the input s i g n a l e l e c t r o n i c s to the bridge., The r e s u l t s of the laboratory test can be seen i n F i g . 3.3.3- 3., For convenience, the data has been normalized to the value at 2 Hz. The dots represent data points obtained i n the laboratory. The smooth curve i s the r e s u l t of a computer analysis. The eguation which was programmed i s : V ( X ^ ) e k s * 2f *• Ttju/) 3.3.3-2 T(jw) i s the transfer function for the amplifier e l e c t r o n i c s and i s derived in Appendix 2. As can be seen, the agreement between theory and the laboratory i s excellent. The phase i s s h i f t e d by 90° i n the low freguency range. This i s because the eiaf i s induced i n the c o i l by Faraday*s law of induction which states that the emf around a stationary loop i s 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) F i g . 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 l l II 1 it 1' !il j 1(1 Ii! 1 1 - L l -1 1 1 1 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 i i 1 - I 1 ! 1 n i I ' ! ! ' i FI | l I t 1 ll 1 III 1 1 1 • Laboratory r e s u l t s 1 _ i I H nt iiii i f\ 1 1 Computer r e s u l t s 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 i l l l T i il III / ' i 1 i l l 1 mil i n in ' i ! 1 111 I11! ! 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II i A 1 1 I I I T i ' l t t ni IIII 1 fu 1 ; II 1 I i 1 II i • i 1 i i i 111 i 1 i i h ] ll j i 1 ll i i »' 1 I 1 i in 1 1 i i i I i 1 j 1 i \ P l 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" l l j II I 1 ll i n i II • inn 1 1 II TT i H i J i III | ..Iii i 1 '1 n i 1 1 1 ! 1 1 1 l . J i _ L I 1 11 1 .11. 1 ill i iii M i I M I I 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 T T T i i i r 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 F i g . 3.3.3-4 The phase response of the magnetometer system according to the bridge method , , 46 For a f i e l d which i s normal to the loop: B-n dA - BA Let B=B0e Then : f » - ju>BA Thus, i t i s expected that the emf w i l l lag by 90° at the lower freguencies. i t the higher freguencies, the phase w i l l be affected by the low pass f i l t e r s of the amplifier e l e c t r o n i c s , of which there are seven orders. I t i s no longer such a simple problem to predict what the phase response w i l l be, as i n the low freguency case. I l l — 4 The Case of Observation With and Without the Bridge The voltage output as measured across Z 5 i n the case of observation with the bridge can be found using eg. 3.3.3-It i s desired that the r e s u l t 3.4-1 concur with the case of observation without the bridge so that i t w i l l not be necessary to have the bridge as a permanent part of the e l e c t r o n i c s . I f the sensor c o i l i s to be operating i n the f i e l d without the bridge , ., then a diagram representing the "2, I t i s given by V o b t = ( I 3 ) o l ) S * Z 5 . Vob< = 3.4-1 0 47 input voltage to the amplifier system i s shown i n F i g . 3.4-1. REAL ^ ^ E (£) I REAL F i g . 3.4-1 Diagram of sensor c o i l 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 i s given by: Z -- Y.Zr Then the current I i s : I E/(X,*Z) The potential drop across Z y i s : V 2 r-RCAL *,+ z The r a t i o of the output voltage to the input voltage i s : V REAL By making the substitution for Z, t h i s l a s t r e s u l t becomes: J_ 2 ,Z y E X,' (Z.+Z*) 3.4-2 48 If the bridge method i s going to be used for c a l i b r a t i o n purposes, then i t i s necessary to show that V o b s » V^ E A L . The eguation for V o t s i s much more complicated than the one f o r V R E A L , but by c a r e f u l l y considering orders of magnitude of the guantities which are involved i n each term, i t can be shown that V0fe« i s e s s e n t i a l l y i d e n t i c a l to V J £ f l L . A s a st a r t i n g point to evaluate these orders of magnitude, l e t Z0=10fl., Z3=R, Z1=R/20 and Zlf=20R, where R i s the resistance of the sensor c o i l and taken to be approximately 2000^. Also l e t Z5=10000^t. By evaluating a l l of the terms of D according to eguation 3.3-5, and retaining only the largest terms, i t can be shown that: D ~ 2 ^ 2 , + 2,2^2, • Z , Z , Z S 3.4-3 The error of t h i s approximation i s of the order of 6%. By evaluating the numerator of 3.4-1 with the same substitutions as were made f o r D, i t i s apparent that: Z 0 ( Z 3 + 2 j + 2 ^ ( 2 ^ 2 , ) » ZjZ^. 3.4-4 This approximation i s of the order of 5%., 8ith a l l 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,Z g  3.4-5 49 Upon rearranging terms: V L « Z ? . r 3.4-6 1 + ~ ( ) If can be made large enough, then the r a d i c a l i n the denominator containing R^  becomes small and reduces to: V ~ J_ Z ' Z * E 3 . 4 - 7 -o k s ~ X," ( Z l + Z f ) * This i s i d e n t i c a l 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 . , I f R^ i s to be made larger than 2 0 times the sensor c o i l resistance, then i t i s imperative that Zz and Z Q be small, c e r t a i n l y l e s s than 1 0 0 X L and preferably of the order of 1 0 X L . Otherwise, the approximations leading to 3 . 4 - 7 w i l l no longer be v a l i d and the voltage output as measured across Zs in the case of observation with and without the bridge w i l l no longer be comparable. The physical s i g n i f i c a n c e of 3 . 4 - 6 i s int e r e s t i n g i n i t s e l f . What i t means i s that the bridge c i r c u i t of Fig. 3.3-1 with E o = 0 e f f e c t i v e l y reduces to the following: 50 E Pig. 3.4-2 The e s s e n t i a l elements of the bridge c i r c u i t The e f f e c t of the term containing fi^ has been observed from computer simulations. The r e s u l t s of the computer analysis can be seen i n F i g . 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. I t can be seen that the effect of Z^ . i s 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 f i n a l conclusion reached i s that V o V j5s V R H L only i f the e f f e c t of can be reduced by making i t a high value. This i n f e r s that the comparison of observations between two systems, one with the bridge as a permanent part of the e l e c t r o n i c s and another without the bridge, may have a s l i g h t discrepancy which can be attributed primarily to acting as a p a r a l l e l load across the output of the sensor system. Frequency (Hertz) F i g . 3.4-3 The e f f e c t 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 t h i s method to determine the freguency response of the magnetometer system, a time varying magnetic f i e l d i s created by a secondary c o i l . The secondary c o i l i s aligned c o a x i a l l y with the sensor c o i l and at the center of the c o i l s i s a Mu-metal core.. The f i e l d i s detected by the sensor c o i l . . I t i s the purpose of the derivations which follow to show that the freguency response can be accurately determined using t h i s approach.. As i n the case of the bridge method, the input voltage to the amplifier system w i l l be derived when the system i s being c a l i b r a t e d and also when i t i s in i t s normal observational mode.. F i n a l l y , a comparison w i l l be made between the freguency response which i s obtained from the secondary c o i l method to the one obtained from the bridge method. 53 IV-2 Theory of Operation Secondary Sensor c o i l c o i l F i g . , 4 . 2 - 1 The secondary c o i l method at the time of operation The c i r c u i t which i s representative of the magnetometer system when i t i s i n the observational mode i s depicted in Fig. 4 . 2 - 1 . , The parameters C 2 , 8 2 and L 2 are the secondary c o i l constants., I i s defined as: Y -- jwM 4- 2- 1 where M i s the mutual inductance between the two c o i l s . H0 i s the load on the secondary c o i l that i s adjusted to remain constant when the signa l generator i s added for c a l i b r a t i o n purposes (see Figs., 4.2 - 1 and 4 . 3 - 1 ) . . The sensor c o i l constants are R, , L, and C,.. Because of the geometry of the two c o i l s and the fact that they have a Mu-metal core i n common, there i s a p o s s i b i l i t y that the response of the sensor c o i l i n the observational mode could be distorted by mutual inductance between i t s e l f and the secondary c o i l . I t i s the purpose of t h i s section to explore t h i s problem in d e t a i l . , Let Z, = R, + jio L, 4 > 2 _ 2 54 J 1 R, 4.2-4 4.2-5 Then, the e q u i v a l e n t c i r c u i t at the time of o p e r a t i o n f o r o b s e r v a t i o n becomes:. ( E - 0 " A p p l y i n q K i r c h o f f s theorem, the two c i r c u i t e q u ations are: ( z. + z,)!, - E : - Y I 2 The s o l u t i o n f o r I, i s : I.- ( z l + z 5 ) ( z ^ z j - Y * 4.2-6 The i n p u t v o l t a g e t o the a m p l i f i e r i s Vg =i ( z 3 . In order to make I, independent of any e f f e c t s from the secondary c o i l , the f o l l o w i n g c o n d i t i o n must be s a t i s f i e d : I (*.•*,)( Z , . Z . ) | » l Y l 1 4.2-7 An upper l i m i t t o the magnitude of I can be found by p u t t i n g a bound on H., I f the s e l f i n d u c t ance of the two c o i l s are known and i t i s assumed t h a t a l l the magnetic l i n e s of f o r c e s e t up by the f i r s t c o i l c u t a l l the t u r n s of the second c o i l , then 55 the mutual inductance M i s given by: M = \l,Lj_ 4.2-8 I t i s not c e r t a i n what percentage of magnetic l i n e s s e t up by one c o i l w i l l cut the t u r n s of the other c o i l , so i t i s s a f e r t o make t h i s l a s t eguation i n t o an i n e g u a l i t y . M ^ J U T 4 ' 2 " 9 Then^ i t i s of i n t e r e s t to show t h a t : >> 1 For the c o i l s used, L,.-1000 H and L a=.01 H. , A l s o , C g =4.5 Mf and 8 9 =7.5 Ka., The c o i l c o n s t a n t s are given i n Table 1-1. A reasonable worst case r e s u l t , which would be a t the h i g h e s t p o s s i b l e freguency t h a t one might expect t o observe m i c r o p u l s a t i o n s , would be at 10 Bz. S u b s t i t u t i n g these numbers i n t o the l a s t i n e g u a l i t y : ( Z , ^ 3 X V 2 j 7 2 0 , 0 0 0 u 1 - L , L-j. l o o Thus, the c o n d i t i o n 4.2-7 i s e a s i l y s a t i s f i e d . , The f i n a l r e s u l t f o r the i n p u t v o l t a g e t o the a m p l i f i e r system i s given by V o w = Z 3 I , : v o k < -- E . 4.2-10 o b s Z + Z, The c o n c l u s i o n i s t h a t the i n d u c t i o n magnetometer a t the time of o p e r a t i o n f o r o b s e r v a t i o n i s u n a f f e c t e d by the presence of the secondary c o i l . , 56 IV-3 Theory o f C a l i b r a t i o n In t h i s s e c t i o n the c a l i b r a t i o n procedure w i l l be di s c u s s e d . , A time v a r y i n g magnetic f i e l d w i l l be induced i n t o the sensor c o i l by a s i g n a l from the secondary c o i l . I t s i l l be assumed t h a t t h i s s i g n a l i s much l a r g e r i n magnitude than any n a t u r a l magnetic f l u c t u a t i o n s so t h a t £ as d e f i n e d i n F i g . 4 . 2 - 1 can be c o n s i d e r e d n e g l i g a b l e . A c i r c u i t diagram f o r the c a l i b r a t i o n i s shown i n F i g . , 4 . 3 - 1 . , Secondary Sensor c o i l c o i l F i g . , 4 . 3 - 1 The secondary c o i l method at the time of c a l i b r a t i o n In order t o s i m p l i f y the p i c t u r e f o r a l g e b r a i c c a l c u l a t i o n s , the f o l l o w i n g s u b s t i t u t i o n s w i l l be made: Z, = R , +• j u » L ; j ; ^ 1 K 9 57 The equivalent c i r c u i t diagram at the time of c a l i b r a t i o n becomes: J0 © L L2 -4-There are three loop equations vhich r e s u l t . (Z . ' Z . U . - - Y I 2 H c ( l . - I , ) - I 5 R 0 * - E . Solving t h i s system of eguations for I , : I. - ~ Y where RoZc *.*zc Z c/(Z t •"B0)=1/(1*jwClR0) i s approximately egual to 1 i f | w C , . R . | C< 1* The laboratory value f o r R0 i s 6 Kru Cx includes the capacitance of the secondary c o i l and that of the cable connecting the secondary c o i l to the s i g n a l generator. The cable i s normally long, frequently 100 meters or more.. The capacitance of the cable i s of the order of 10 nf., & t y p i c a l cable capacitance i s 50 p f / f t . Then, the t o t a l capacitance for 100m i s approximately 15 nf. The capacitance of the secondary * This i s not a necessary r e s t r i c t i o n , but a convenience. 58 c o i l should be much smaller than t h i s . , Therefore, at 10 Hz, wC zR 0« (20-IT) (6 K) { 1 . 5 x 1 0 - 8 ) = 5 . 6 5 x 1 0 - 3 « 1 . I t has already been shown that: K z . + z . K z ^ + z j l » Y 2 Therefore, the s i m p l i f i e d result for the current I, i s : i , - Y The resu l t i n g voltage drop across the input to the amplifier system when the magnetometer i s being calibrated i s Vca l =1 z3 : V Y Zs r 4.3-1 The r a t i o V o b s /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 i n the sensor c o i l by natural magnetic f i e l d fluctuations can be described by: S D E = i - wBe 4.3-3 2.TT where S i s the s e n s i t i v i t y c o e f f i c i e n t and B i s the amplitude of the o s c i l l a t i n g magnetic f i e l d . Substituting 4.3-3 and 4.2-1 into 4.3-2, the result i s : 59 The f i n a l observation to make i s that the a.c. s i g n a l from the sig n a l generator i s given by: E e - E s e >"* Then : - — — (R^ju-L^R.) 4 . 3 - 6 As shown previously, w C l B t > « 1 and t h i s implies that Z^=B0. , In order to make 4.3-6 independent of freguency, the following condition must be met: ( R x + R a) >> u u L z 4 . 3 - 7 Again, the worst case would be at the high freguency end, at 10 Hz. , Substitute the values Bt= l l x j . , B, =6 U and L =.01 H. Then : Rx+ R„ & * lo' The condition 4.3-7 i s e a s i l y s a t i s f i e d , . The f i n a l r e s u l t i s : 60 - _ L . i l U l * R . ) 4.3-8 21T MES Thus, i t i s seen that the voltage output at the time of ca l i b r a t i o n i s d i r e c t l y proportional to the output at the time of observation. I t i s expected that this method of ca l i b r a t i o n w i l l c orrectly produce the freguency response of the magnetometer system., The laboratory c i r c u i t diagram f o r the secondary c o i l method i s shown i n Pig. ,4.3-2. 6K - A W W V 1 Mu-metal core M-+.3 K 10K 0 W.4- pi Magnetometer amplifiers and f i l t e r s F i g . 4.3-2 The secondary c o i l method in the laboratory The re s u l t s from the laboratory analysis can be seen i n 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 i s plotted as sguares and the smooth curve represents the r e s u l t s of a computer analysis., The eguation programmed i s : V - T ( i u , ) 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 i n the appendix. The agreement i s r .'good between the data obtained experimentally and the resu l t s predicted from the computer analysis., a plot with laboratory data can now be made which i s analogous to the computer plot of Fig. , 3 . 4 - 3 . This can be done simply by comparing the laboratory r e s u l t s obtained from the bridge method and the secondary c o i l method. On the same graph appears the laboratory data of Fig. , 3.3.3-3 and F i g . 4.3-3. This i s shown i n Fig. ,4.3 -5 . The agreement between the computer analysis of F i g . 3.4-3 and the laboratory analysis of Fig. „JI.3-5 i s excellent. a comparison of the results of the phase analysis from the bridge method and the secondary c o i l method i s shown i n Fig., 4.3-6. The agreement between the two methods i s 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 F i g . 4.3-4 The phase response of the magnetometer system according to the secondary c o i l method Frequency (Hertz) F i g . 4.3-5 A comparison of the frequency response curves obtained i n the laboratory from the secondary c o i l method and the bridge method (see F i g . 3.4-3 and discussion on page 50 about the separation between the, curves) . 65 u i in h i ! 180 iii II! Ill i 135-i n TTT i l l Wi + Secondary c o i l method • Bridge method _!___ •II; TTiTt i I hi MI-TT i ! 4444 UU rmiT M iu m I I ! I l l III T T in M i . iltr ! i TTT i i i 90 Iii i i l l M i l i i 111 nu ! I to cu cu u 60 01 CU CD n) X I I I ! i l l i i i mi 11 i i i M i l III! 45 i 11 M i Mi! iii i ! i i I I I ! i I Ui i i i i - U - L T l ~ [ U i I ! I III hil I I I ! I l l l m i i i . i i l l l I 1 M M i l l l i l i i T T T MIL -45 i i 111 111 I ! I rn" i n Ui! II! I I I ! ! M l i l l . m UM TTTfi mua. iiii II! iiii _LL i ! I mm Ttffl Tl TTtti TTT i l l ! Ill UU iii ! I I I I -90 .1 ilii III 445 Ul I ! I III! Illl l l l l l l .3 .5 1.0 2.0 3.0 Frequency (Hertz) F i g . 4.3-6 A comparison of the phase response curves obtained i n the laboratory from the secondary c o i l method and the bridge method 66 CHAP-TE1 V-1SI ABSOLUTE-CAL-IBRATIOH--V - 1 A Laboratory Approach Now that the r e l a t i v e s e n s i t i v i t y of the Mu-metal core system i s well known, i t i s necessary to determine the absolute s e n s i t i v i t y . , Conceptually, the simplest way to do t h i s would be to put the sensor c o i l in a known, uniform, sinusoi d a l l y varying magnetic f i e l d and record the response of the system., In practice, i t i s not easy to create an a r t i f i c i a l f i e l d which would he uniform over a volume large enough to accomodate a three foot long sensor c o i l . , A l o g i c a l solution i s to use the earth*s natural magnetic f i e l d when a sinusoidal micropulsation event i s occuring. An a i r core c o i l magnetometer system i s used to precisely determine the absolute amplitude of the micropulsation event., The a i r core c o i l and the Mu-metal core c o i l are located about 1 0 0 yards apart and i t i s assumed that the natural event i s uniform over t h i s distance., The response of the Mu-metal core system can then be compared to that of the a i r core system, thus determining the absolute s e n s i t i v i t y of the Mu-metal core magnetometer., It i s necessary to do t h i s at only one frequency as the r e l a t i v e freguency response i s already known and l i n e a r i t y i s assumed., The f l u x , <$ ,through :a c i r c u l a r c o i l i s ; AB 5.1-1 where A=irR2*N i s the average cross-sectional area times the 67 number of turns and B i s the magnetic f i e l d strength... The e l e c t r o m o t i v e f o r c e induced by the changing f l u x i s : V t M f i£ . A i§. 5.1-2 dt dt The output v o l t a g e from the a m p l i f i e r system, with a d.c. gain G and an i n p u t impedance B- , which would be observed i s gi v e n by 5.1-3. . H{f) i s the t r a n s f e r f u n c t i o n normalized t o the d.c. value and R i s the d.c. r e s i s t a n c e of the c o i l . , Vottt - V e w, * G - H l f W 5.1-3 I t i s convenient f o r m i c r o p u l s a t i o n r e s e a r c h t o express e l e c t r o m o t i v e f o r c e i n m i c r o - v o l t s and magnetic f i e l d s t r e n g t h i n milli-gammas. / T h i s l e a d s t o the f o l l o w i n g two eguations where V e „ , f i s i n v o l t s , V g m f i s i n m i c r o - v o l t s , B i s i n t e s l a and B' i s i n milli-gammas., V ^ f - I 0 b V.mf 5.1-4 8' - to"" B 5.1-5 S u b s t i t u t i n g these eguations i n t o 5. 1-2 l e a d s to the r e s u l t : 'vLf - ( ^ I O " ) i i ' 5 > 1 _ 6 I f B' i s a s i n u s o i d a l f i e l d , then B'=B 0sin2irft. S u b s t i t u t i n g t h i s i n t o 5.1-6 y i e l d s : V e W f = ( 21r A * \0'b ) • f • B„ cos 2lr-f+ or v L f = S^'f- 8e o>s lirft 5.1-7 S a=2ira*10 - 6 i s c a l l e d the a b s o l u t e s e n s i t i v i t y when V e m f i s 68 measured i n m i c r o - v o l t s , B 0 i n milli-gammas and freguency i n Hertz., i t a frequency o f 1 Hz, a f i e l d of 1 milli-gamma C10— a Gauss) w i l l induce a p o t e n t i a l of 1 m i c r o - v o l t and the s e n s i t i v y w i l l be 1 Caner., 1 Caner = 1,uV/(my*Hz) How two eguations can be w r i t t e n , one f o r the a i r core c o i l system and the other f o r the Mu-metal core c o i l system by s u b s t i t u t i n g 5.1-7 i n t o 5.1-3. The s u p e r s c r i p t " 1 " i s f o r the a i r core and n 2 n i s f o r the Mu-metal c o r e . , V 0 ( ° - • H l'V) 5 . 1 - 8 i V.w>* S i " - f • B. « *** • 6". -ggf^ • H ~ ( « ) An important r e s u l t occurs when the r a t i o of these eguations i s taken.. T h i s i s g i v e n by : as the agreement between computer a n a l y s i s and l a b o r a t o r y a n a l y s i s has been very good f o r the work concerning the bridge method and the secondary c o i l method, the freguency response of the a i r core c o i l system i s determined by programming the c i r c u i t parameters on the computer. The c i r c u i t diagram f o r t h i s system i s shown i n F i g . , 5. 1-r 1..,, H<*>{f) can be expressed as the product of two t r a n s f e r f u n c t i o n s . , The f i r s t one i s F i g . 5.1-1 C i r c u i t diagram for the a i r core system 70 H, < l J(f)» approximately of a Butterworth type. The second i s the transfer function of the t h i r d order low pass f i l t e r , H ^ U f ) . In order to determine H,<*>(f),- consider the following figure : R From t h i s diagram R* where: Z = I + ju> Cx R x Normalizing 7 X /E to one at d.c., H,*1*^) becomes H a <*5(f) i s derived i n Appendix 3. Therefore, the transfer function for the a i r core c o i l system, normalized to one at d. c. i s : H H ( f ) = H?»m. H r e f ) This re s u l t was programmed on the computer. The freguency response curve which was determined i s shown i n Fig.,5.1-2.. As the freguency response curves f o r both the a i r core and the Mu-metal core systems are f l a t in the low freguency range, the terms H< J*(f) and H t 2 , { f ) i n 5 .1 -10 are both egual to one. 71 F i g . 5.1-2 Computer simulated frequency response of the a i r core c o i l system 72 The absolute s e n s i t i v i t y of the a i r c o i l , S<4->^  i s e a s i l y determined from i t s geometry and turn number. In order to fi n d the inner radius of the c o i l , the following method was used. A Fig. 5 . 1 - 3 Method to determine the inner radius of a large c o i l Three measurements were taken along the inner circumference of the c o i l as shown i n Pig. ,5. 1-3. From the law of cosines, the angle C i s given by : C = co.-'/'-'+b'-c 1) ^ 2cb 1 Then, an eguation was used f o r a circumscribed tr i a n g l e in order to f i n d the radius. This eguation i s : o- b c 2-mA " 2-<nB 2 « . » v C After finding the inner radius, the outer radius was determined by adding the thickness of the c o i l to i t . From t h i s approach, fi. = 7 4 . 5 1 1 cm and R_„+„_ = 7 5 . 7 0 2 cm. Now the * • • * * i n n e r o u T e r absolute s e n s i t i v i t y can be found.. K 73 where r. and r o are the inner and outer r a d i i , respectively. . Substituting i n the numbers leads to S^<1>=.0557 Caner ± .2%. The r a t i o V 0 < 2 > / V 0 < » > can be determined by measuring the amplitude of the s i g n a l from the a i r core and the Hu-metal core systems when a sinusoidal micropulsation event occurs., The output signals from the event used i s shown i n Fig. 5.1-4. Feb. 17, 19 77' 21 h 27 m A i r core Mu-metal core Z comp. F i g . 5.1-4 The micropulsation event used for the absolute c a l i b r a t i o n Peak-to-peak measurements were made from microfilm by using a t r a v e l l i n g microscope. .. The r a t i o s were taken with the result of : — ° — = 1.4-1 + .747. .24-5- H 2 t l._7„ F i n a l l y , the remaining parameters of 5.1-10 have the following values : 74 G < 1 > = 5:09 7'* 10* t U 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 R tt 2> i s d i f f e r e n t from R, =7.5 K__ (p. 55) because another amplifier unit was used for the f i e l d observation. The value of S,.<z> according to 5.1-10 i s .0548 Caner ±6.4%. V-2 A Theoretical Approach to the Absolute S e n s i t i v i t y -The absolute s e n s i t i v i t y can be dealt with from a t h e o r e t i c a l point of view. Consider a prolate spheroid with semiprinciple axes a,b immersed i n a magnetic f i e l d of strength H 0. The Mu-metal core of the sensor i s i n r e a l i t y a cylinder, but approximating the cylinder as a prolate spheroid i s reasonably accurate and saves a great amount of mathematical d i f f i c u l t y . The r e s u l t s of an analysis shown i n the book by Stratton (1940) indicates that the magnetic f i e l d strength anywhere inside the cavity i s given by : 75 where: A - ~~T~7 ( ~ 2 6 H n — ) cv i -e ' The magnetic f i e l d follows from B= / u / u„H where /* i s the r e l a t i v e permeability and the permeability of free space. In - HKS units, /*„=4,rr*10-7 Henry/meter, /u- i s non-dimensional and i s the permeability of the metal from which the core i s made. By l e t t i n g Ba = /A,H0., i t follows that : B - r r — Bc 5.2-2. 2 ' The t o t a l flux $ through the c o i l follows from 5 . 1 - 1 . $ TT • B • N Or, substituting f o r 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 t h i s to 5.1-7, i t i s seen that S represents a th e o r e t i c a l s e n s i t i v i t y . , Expressing S i n units of Caner, the s e n s i t i v i t y becomes : For a Mu-metal core, /x i s of the order of 10 s. I f a r e s t r i c t i o n i s put on the r a t i o of the lengths of the semiprinciple axes: a 200 then the denominator of the expression 5.2-4 f o r S can be reduced to a simpler form,, | + f___L ( u. ,) A « o-b V A 5.2-6 Z ' 2 The error of t h i s approximation i s l e s s than 8%. The expression for s e n s i t i v i t y reduces to : S - mi 5.2-7 OLA The expression 5,2-1 for A can also be s i m p l i f i e d . I f ? ~ , then e-1 and A becomes ; A - ^ [ " 2* In 2 - In 5.2-8 By the binomial expansion : 0 ' a- Z a a Then 5.2-8 reduces to : A -- In 5.2-9 a ' e. ba where e a =2.71828. The f i n a l s i m p l i f i e d expression for 77 s e n s i t i v i t y follows from 5 . 2 - 9 , 5 . 2 - 5 and 5 . 2 - 6 . , S a P P - . „ = * 10 C a n e r 5.2-10 The importance of t h i s expression i s that the s e n s i t i v i t y of the sensor c o i l i s primarily dependent upon the number of windings, ti, and the length of the c o i l , a. The o v e r a l l weight of the sensor could be reduced by using l i g h t e r gauge wire and a thinner core. The length and turn number could be adjusted to obtain the desired l e v e l of s e n s i t i v i t y . In s p i t e of many approximations, the agreement amongst the t h e o r e t i c a l and experimental r e s u l t s of 5 . 2 - 1 0 , 5 . 2 - 5 and S^2* of 5 . 1 - 1 0 i s guite good,. The values are shown below : S0,<2> = . 0 5 4 8 Caner + 6.4% S + K » , r = - 0 5 8 3 Caner Sapp-** = - 0 5 8 7 Caner These r e s u l t s imply that the approximate t h e o r e t i c a l expression for the absolute s e n s i t i v i t y , eg. 5 . 2 - 1 0 , can be r e l i a b l y used to aide in the design of future sensor c o i l s . Another i n t e r e s t i n g observation which has been made as a r e s u l t of t h i s analysis i s that the r a t i o a/b cannot be increased without bound unless an undesirable e f f e c t begins to take place.. Hhen ^ > 1 0 0 , S + K a (, r and S a f p r o » s t a r t to diverge by about 2% and the agreement becomes worse as a/b becomes large r . , There i s an important reason behind t h i s which can be * For absolute c a l i b r a t i o n according to the bridge method, see Appendix IV. 78 unfolded by onee again t a k i n g a look at 5-2-2. B can be r e -expressed i n terms of a demagnetization c o e f f i c i e n t N a. ,. B -- ^ B. 5.2-11 l + N d (/*-•) Nd - Of A I t can be seen t h a t the i n v e r s e o f the demagnetization c o e f f i c i e n t i s the p e r m e a b i l i t y which l e a d s t o the s e n s i t i v i t y given by eq- ,5-.2-10.'„- W r i t i n g yU, = — r • — 5.2-12 N a a b ^ A i t i s seen with the h e l p of 5.2-9 : y U , - ! 5.2-13 e„b Note t h a t t h i s i s dependent upon the geometry of the c o r e but not the p r o p e r t i e s of the metal., yu, i s dependent upon the r a t i o a/b*, By making the s u b s t i t u t i o n 5-2-12 i n t o 5.2-13, i t i s seen t h a t : B * M&f$ 8 0 5.2-14 where: M ~r r -F o r : the most r e c e n t Mu-metal core c o i l used f o r t h i s r e s e a r c h , a=:>45-.:r7-2cm and b<?i,/Q~ cm {see Table 1 - 1 ) . T h i s 79 l e a d s t o a value f o r /u., of 5 9 5 ' - S i n c e yu&IO5, the i n e g u a l i t y fx »/JLX i s s a t i s f i e d . , Then by 5.2-14, yu-eff a yu., . , T h i s i s a d e s i r a b l e r e s u l t because t h e e f f e c t i v e p e r m e a b i l i t y of the core i s dependent upon i t s geometry. When a/b*100, the i n e g u a l i t y yu»yu, begins t o weaken., When a/b=1000, //, = 1.5*10 s and / ~ € f f i s no longer dependent p r i n c i p a l l y upon yu, . The p e r m e a b i l i t y of the metal, /x, becomes e g u a l l y as important., T h i s i s not a d e s i r a b l e r e s u l t as the p e r m e a b i l i t y of a metal i s not always constant., I t may change s i g n i f i c a n t l y a c c o r d i n g to environmental c o n d i t i o n s such, as temperature and s t r e s s upon the metal. & p l o t of yu, as a f u n c t i o n of a/b can be seen i n F i g . 5.2-1. / The c o n c l u s i o n o f t h i s a n a l y s i s i s that eg. „ 5.2-10 w i l l g i v e a good i n d i c a t i o n of the s e n s i t i v i t y of a c y l i n d r i c a l l y shaped sensor c o i l provided s;> I i L ^ ISO . b Note: The value of b at the bottom of p. 78 i s not simply the inner diam-eter of the c o i l windings. I t act u a l l y denotes an e f f e c t i v e radius r e s u l t -ing from the cro s s - s e c t i o n a l area of the Mu-metal core i t s e l f . The core consists of approximately 48 rectangular s t r i p s . The cross-sectional area of one s t r i p i s (3/4")(.014")=(1.905 cm)(.03556 cm)=.06774 cm2. If there 2 are 48 s t r i p s , then the t o t a l c r o s s - s e c t i o n a l area i s 3.252 cm . The ef-fective, radius f e c t i v e radius for this area i s ^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. F i g . 5.2-1 The dependence of the geometric permea-b i l i t y /x, upon the r a t i o of the lengths of the semiprinciple axes of a pro l a t e spheroid,a/b 81 CHAPTER-VI-SUMMARY AND CONCLODINS HIARK-S-Many useful r e s u l t s have been obtained from the investigations c a r r i e d out i n t h i s t h e s i s . v These r e s u l t s f a l l into two categories. One category i s concerned with improvements i n the design of the magnetometer system. As described i n Chapter I I , use can be made of a Butterworth f i l t e r rather than a notch f i l t e r to reduce 60 Hz noise.. This guarantees a f l a t response in the low freguency range and reguires only two matched e l e c t r i c a l components rather than six c a r e f u l l y matched components. In Chapter V, which was concerned with the absolute c a l i b r a t i o n , i t was shown that the s e n s i t i v i t y of the sensor c o i l i s primarily dependent upon the number of windings and the length of the c o i l . , The o v e r a l l weight of the sensor could be reduced by using l i g h t e r gauge wire and a thinner core. However, for optimum performance, the r a t i o of the length to the diameter of the core must be within a s p e c i f i c range.. One of the reasons f o r not being able to increase s e n s i t i v i t y ad infinitum i s that thermal agitations at the atomic l e v e l induce small currents in the sensor and other e l e c t r i c a l components.. This i s known as Johnson noise. The second category concerns the c a l i b r a t i o n procedure. Two methods were investigated i n great d e t a i l , the bridge method and the secondary c o i l method. I t was determined that both methods can produce r e l i a b l e r e l a t i v e freguency response 82 curves. An advantage of the bridge method over the secondary c o i l method i s that only one cable i s needed between the sensor c o i l and the remaining electronics, rather than two. For t h i s reason, the bridge method w i l l probably be used f o r future c a l i b r a t i o n s . The absolute c a l i b r a t i o n was successfully performed by comparing the Mu-metal core system to a previously calibrated a i r core system., A t h e o r e t i c a l approach to the absolute c a l i b r a t i o n was discussed which agreed well, considering a l l of the approximations made, with the laboratory r e s u l t s . This t h e o r e t i c a l approach may be used to obtain a good in d i c a t i o n of the s e n s i t i v i t y of a c y l i n d r i c a l l y shaped c o i l before a laboratory analysis i s performed. 83 APPENDIX 1 METHODS OF DETERMINING THE INDOCTANCE AND CAPACITANCE OF A COIL WITH FINITE RESISTANCE It i s not a t r i v i a l problem to determine the inductance and capacitance of a c o i l which has a f i n i t e resistance. The sensor c o i l s used with the magnetometer system a l l have the eguivalent c i r c u i t of Fig. A. 1 - 1 . R V L —nM/wl—©—Tomr—i— lie F i g . A.1-1 Eguivalent c i r c u i t of a sensor c o i l A meter which i s normally used to measure an inductance or a capacitance operates on the p r i n c i p l e that R= 0 . For the sensor c o i l s , R i s of the order of a few thousand ohms. Because of t h i s , other le s s straight forward techniques must be employed to determine L and C. This appendix i s a compilation of those technigues which were used i n the laboratory. I t i s safe to measure c o i l resistance by using an ohm meters Methods of Determining L The most r e l i a b l e method of determing L follows from wiring a sig n a l generator, the c o i l and a load r e s i s t o r i n 84 s e r i e s . r~ C J Fig* a.1-2 F i r s t method of determining L The s i g n a l generator must produce a sinusoidal signal of a freguency which i s low enough such that : >> |R+i<-L| A. 1-1 Then the presence of the capacitor can be ignored in the c i r c u i t analysis. „. For the R-L -R loop, the following eguation i s true: Thus, a plot of (V/Vjj)2 versus w2 i s a straight l i n e with slope (L/R,) 2. Another c i r c u i t which can be used for determining L i s shown i n F i g . A.1-3. This i s i d e n t i c a l to the previous c i r c u i t except that the si g n a l generator produces a sguare Rearranging t h i s i n t o a d i f f e r e n t 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 t h i s method to produce the desired r e s u l t s , there are two conditions which must be met. These are : A. 1-3 -!=- >> CRq Then the scope picture across RA w i l l show an increasing and a decreasing exponential. By measuring the time which i t takes for VJJ to obtain one half of the f i n a l value, the inductance can be found from : A. 1-4 This method of determining L i s r e l i a b l e , but i n practice i t i s d i f f i c u l t to obtain better than one s i g n i f i c a n t figure of accuracy because the time i n t e r v a l of r,,t i s not easy to measure using available scopes or chart recorders. 86 Methods of Determining C-The most r e l i a b l e method to determine C follows by again using the c i r c u i t of F i g . a . 1 - 2 , but t h i s time at high freguencies. The condition which needs to be met i s ; Also, another necessary condition i s : a. 1-5 a. 1-6 Bhen the freguency becomes high enough, IjwLJ becomes so large that nearly a l l the current passes through the capacitor., I f V4 i s 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 , t h i s eguation i s v a l i d on the 45° sloping l i n e to the r i g h t of the resonance point. o l o q U F i g . A.1-i+ The anti-resonance point wr i s c a l l e d the anti-resonance freguency.„ The c o i l 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 t h i s case e s s e n t i a l l y i d e n t i c a l . u r -- I / J I T A . 1-8 Thus, i f wr and L are known, then C can be determined using A. 1-$. , If a capacitor i s wired p a r a l l e l to the c o i l i n the c i r c u i t of Fig., A.1-2, then the resonance point w i l l be shif t e d to a lower freguency according to A. 1-IT" where C a i s the additional capacitor. Both L and C can be determined by s h i f t i n g the resonance point as A.1-8 and A.1-90 are two eguations with two unknowns. A p r a c t i c a l d i f f i c u l t y with t h i s approach i s that as the resonance point i s shifted to the lower freguencies, the trough of Fi g . 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 i n Oeda and Hatanabe (1975). 88 APPENDIX 2 THE TRANSFER FUNCTION OF THE AMPLIFIER SYSTEM-Both methods of c a l i b r a t i o n have i n common the amplifier system which f i l t e r s and amplifies the si g n a l from the sensor c o i l . In order to make computer simulations of freguency response curves, the transfer function for the amplifier e l e c t r o n i c s must be determined. A schematic f o r the amplifier i s shown in F i g . A. 2-1- The transfer function for each stage i s given i n F i g . A.2-2. The transfer function f o r a l l of the el e c t r o n i c s of the amplifier system i s then : A. 2-1 where : w t - l i r r I—WWV—I I.8J K A = i n p u t i m p i J U m c * i HH-.3 K r o u t p u t i n > p « d o n c e < \0-fT--AAAAAA*—i 5 K AAAAAr 2 . 5 S K o—o—wwv (00 si 1 O u t » & t •\5V 200 ja so XL F i g . A.2-1 Mu-metal c o r e a m p l i f i e r s y s t e m Voltage Divider VS - IS F i g . A.2-2 T r a n s f e r f u n c t i o n s f o r f i l t e r s o f t h e Mu-metal c o r e a m p l i f i e r s y s tem An com a m p l i f i e r ° i 1 ° (jw) + wN * RC u»ft« 1 2 T T • s t * Butterworth f i l t e r of 1 — degree. Corner frequency = 6 Hz. 91 APPENDIX 3 THE TRANSFER FONCTION FOR THE THIRD ORDER LOW PASS FILTER The c i r c u i t diagram f o r the t h i r d order low pass f i l t e r which i s used as a part of the a i r core magnetometer system elec t r o n i c s i s shown i n F i g . A . 3 - 1 . R _L_ C . F i g . A.3-1 The t h i r d order low pass f i l t e r Four fundamental eguations which can be written are given ~. by eguations A.3-1 through A.3-4. V.-- K c " , Z , ^ K - 2 A. 3-1 A. 3-2 A. 3-3 A. 3-4 For these eguations, Z,, Z t and Z 3 are the complex impedances of C,, C. and C,. Using the top three eguations, the matrix eguation that r e s u l t s i s : 92 2. -2. -2, o \ I A. 3-5 The solution for I, i s : I . - Z R V . Z , - V e R l + V s ? . 2 _ A. 3-6 Baking the substitution for Vo from A.3-* int o A.3-6 and then solving for the r a t i o V„ /Vs leads to : V s " (R*Z,K 2 * ^ + 2 , 2 , + R l + 2 , R ) - R Z . K U Z . + R^+R^Z^R+Z,) A. 3-7 The f i n a l substitutions to make are for Z (, Zx and Z^ into A.3 -7 where : After rearrangement of terms, this leads to the resu l t of A . 3 -8 where s=jw. K \J% ' JVCACJR** sX{2R*CFC t4. R^Cl-^C.C^ 2R*C,CS} + S£3RC,* 2 R ( l - K ) C 2 + R C , ] + I*] A. 3-8 Let H 2 be the transfer function of the t h i r d order low pass f i l t e r 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 i n III-2, a r e l a t i o n s h i p can be derived between the time varying external magnetic f i e l d , B , and the emf generated i n the sensor c o i l , , This i s a c a l c u l a t i o n of the absolute s e n s i t i v i t y as discussed i n Chapter V. Eg. 3.2-7 i s a relationship between the d r i v i n g voltage of the bridge and the flux through the c o i l . This r e s u l t i s : I t i s necessary to express <fr i n terms of H, and L i n terms of c o i l geometry. , For a solenoid of length 'J. and cross-section area A such that end effects are negligable, and c l o s e l y 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 i s the permeability of free space and k i s the r e l a t i v e permeability. Substituting these two eguations into A.4-1, the re s u l t i s : APPENDIX 4 THE ABSOLUTE CALIBRATION ACCORDING TO THE BRIDGE METHOD Using the developement of the b r i d g e method as d i s c u s s e d i n I I I - 2 , a r e l a t i o n s h i p can be d e r i v e d between the time v a r y i n g e x t e r n a l magnetic f i e l d , B , and the emf generated i n the sensor c o i l . T h i s i s a c a l c u l a t i o n of the a b s o l u t e s e n s i t i v i t y as d i s c u s s e d i n Chapter V. Eg. 3.2-7 i s a r e l a t i o n s h i p between the d r i v i n g v o l t a g e of the b r i d g e and the f l u x through the c o i l . T h i s r e s u l t i s : — - A.4-1 % L I t i s necessary to express <fr i n terms of H, and L i n terms of c o i l geometry. For a s o l e n o i d o f l e n g t h JL and c r o s s - s e c t i o n area A such t h a t end e f f e c t s are n e g l i g a b l e , and c l o s e l y wound with N t u r n s of t h i n wire so that the winding resembles a c u r r e n t sheet, the two e x p r e s s i o n s f o r 4 and L are : 4> ~- ^ i A H A- 4~ 2 L ~- M. & M ' A / i A.4-3 where ju0 i s the p e r m e a b i l i t y of f r e e space and k i s the r e l a t i v e p e r m e a b i l i t y . S u b s t i t u t i n g t h e s e two e g u a t i o n s i n t o A.4 - 1 , the r e s u l t i s : 94 The sensor used for the f i e l d observation (see Fig- 5.1-4) was not available to test i n the laboratory so that a di r e c t comparison cannot be made between the s e n s i t i v i t y as obtained from A.4-4 and the r e s u l t s on page 77. The dif f e r e n t sensors are nearly i d e n t i c a l , so the best that can be done at t h i s point i s to make the s e n s i t i v i t y c a l c u l a t i o n using another sensor. At a freguency of 1 Hertz, an input signal to the bridge of . 7 5 mV p-p caused an output from the amplifier of 1 3 V p-p. For N=50,000, l=.9144m and R =38787a, A.4-4 indicates that B=/*0H, where /*„ = 4 i r x 1 0 * , i s 13.3x10* m!f. , The output from the bridge or the input to the amplifier i s 13.0V/2x10s=65/«.V. , To find the emf i n the sensor c o i l , use can be made of eg.,3.4-6. The following values are used: R, = 1.83 Ko. L , = 9 3 0 H C, =4.4juf Z f = 44.3 -Ka II 10 Kfl.= 8.158 Kn. R% = 38.8 K*a This indicates that the r a t i o of the output from the bridge to the emf i n the sensor c o i l i s . 7 7 1 . Therefore, the emf in the sensor c o i l i s 65/uV/. 771=84.3/*.V. The s e n s i t i v i t y i s then : B I S . B M c f m * This i s a reasonable r e s u l t when i t i s compared to the results on page 77. LIST OF REFERENCES CONSULTED Campbell, W.H. (1967) Induction Loop Antennas for Geomagnetic F i e l d V a r i a t i o n Measurement, ESSA Technical Report, ERL 123-ESL 6. Kanasewich, E.R. (1973) Time Sequence Analysis i n Geophysics, Edmonton, The University of A l b e r t a Press, pp 170-186. K o l l a r , F. and R u s s e l l , R.D. (1966) Seismometer Analysis Using an E l e c t r i c Current Analog, BSSA, 56, 1193-1205. Lewis, W.E. and Pryce, D.G. (1965) The Application of Matrix Theory to E l e c t r i c a l Engin-eering, London, E. & F.N. Spon Ltd, pp 141-171. Schwartz, M. (19 72) P r i n c i p l e s of Electrodynamics, San Francisco, McGraw-H i l l Inc. Slurzberz, M. and Osterheld, W. (1944) E l e c t r i c a l E s s e n t i a l s 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 B r i t i s h Columbia, Geophysics, M.Sc. Thesis. Ueda, H. and Watanabe, T. (1975) Comments on the Anti-Resonance Method to Measure the C i r c u i t Constants of a C o i l 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 S e n s i t i v i t y and Frequency Response of an Induction Magnetometer, The Science Reports of the Tohoku Uni v e r s i t y , Series 5, Geophysics, Vol. 22, No. 3-4, pp. 107-127. 

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