UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

A comparison of a ferromagnetic core coil and an air core coil as the sensor head of the induction magnetometer 1975

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1975_A6_7 U33_3.pdf [ 3.17MB ]
UBC_1975_A6_7 U33_3.pdf
Metadata
JSON: 1.0085745.json
JSON-LD: 1.0085745+ld.json
RDF/XML (Pretty): 1.0085745.xml
RDF/JSON: 1.0085745+rdf.json
Turtle: 1.0085745+rdf-turtle.txt
N-Triples: 1.0085745+rdf-ntriples.txt
Citation
1.0085745.ris

Full Text

A COMPARISON OF A FESROMAGNETIC CORE COIL AND AN AIR CORE CCIL AS THE SEBSCB K E A E OI TEE INDUCTION MAGNETOMETER by Hajime Ueda B . S c , Tchoku U n i v e r s i t y , 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIRE PINTS FOR TEE EEGBEE OF MASTER OF SCIENCE i n the Department of Geophysics and Astronomy We accept " t h i s t h e s i s as c c n f c r n i n g tc the '» r e q u i r e d standard The U n i v e r s i t y Of B r i t i s h Cclumtia A p r i l , 1975 In present ing t h i s thes is in p a r t i a l fu l f i lment o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f r ee ly a v a i l a b l e for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or p u b l i c a t i o n of th is thes is for f i n a n c i a l gain sha l l not be allowed without my wri t ten permiss ion. Department of The Un ivers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 i i ABSTRACT An e x p e r i m e n t a l approach i s made to e l u c i d a t e the i n p u t - output c h a r a c t e r i s t i c s of a f e r r o m a g n e t i c core c o i l used as a sensor head of an i n d u c t i o n magnetometer. Analyses made here are mainly concerned with comparison of magnetograms o b t a i n e d by the permalloy core c o i l with r e s p e c t to the ene a c q u i r e d s i m u l t a n e o u s l y by a l a r g e open loop c o i l , or a i r core c o i l . R a t i o s of the c o r r e s p o n d i n g peak-to-peak v a l u e s are taken to examine d i s t o r t i o n s of the output amplitude. Blackman-Tukey*s method of power spectrum e s t i m a t i o n i s a p p l i e d t c a p o r t i o n of magnetograms, and s p e c t r a l peaks are compared. The r e s u l t s of these a n a l y s e s show t h a t no s i g n i f i c a n t d i s t o r t i o n s or harmonics are r e c o g n i z e d i n data o b t a i n e d by the permalloy core c o i l . T h i s suggests t h a t the response of the permalloy core c o i l s ensor i s l i n e a r p r o v i d i n g t h a t the a i r core c o i l sensor i s l i n e a r i n response. The demagnetization e f f e c t may be an e x p l a n a t i o n f o r the l i n e a r response of the permalloy core c o i l s e n s c r s . The demagnetization f a c t o r of the core m a t e r i a l depends on l y on the geometry of the core. The p e r m e a b i l i t y of the core m a t e r i a l i s suppressed i n i t s e f f e c t i f i t has so l a r g e a value t h a t i t s r e c i p r o c a l value i s much s m a l l e r than the demagnetization f a c t o r : under such c i r c u m s t a n c e s , the s e n s i t i v i t y of the c o i l does not depend on the p e r m e a b i l i t y of core but on the demagnetization f a c t o r . The demagnetization e f f e c t a l s o suggests t h a t . rhe l o n g e r the core, the h i g h e r the c o i l s e n s i t i v i t y w i l l be. i i i TABLE OF CONTENTS ABSTRACT , . . . i i LIST OF FIGURES V . LIST OF TABLES , v i ACKNOWLEDGEMENTS . v i i 1 . INTRODUCTION 1 2. CHARACTERISTICS OF A FERROMAGNETIC CORE COIL 3 2.1 Nonlinear Response Of Ferromagnetic Mater i a l . . . . . . . 3 2.2 Demagnetization Ef fect 4 2.3 Hysteres is Response . . . . . . . . . . . 9 3. EXPERIMENT 14 3.1 Instrumentation 14 3.2 Some Considerat ions On C o i l C h a r a c t e r i s t i c s 20 4. ANALYSIS 23 4.1 Least Sguares F i t 23 4.2 Amplitude Ratio 29 4.2.1 Relat ive S e n s i t i v i t y And Ratio Of Two Normal Variables . . . . 3 0 4.2.2 Noise Analys i s on P r o b a b i l i t y Graph 34 4 .2 .3 P r o b a b i l i t y Density Function Of A Ratio Of Two Normal Var iables . . . . . . . 3 7 4.3 E f f ec t ive Permeabil i ty Of Permalloy Core 40 4.4 Comparison Of Power Spectra 4 1 4.4.1 Confidence In terva l And Bandwidth Of The Tukey Window 42 4.4.2 Results Of Spectra l Analys i s 44 i v 5. SUHMiiRY AND CONCLUDING REMARKS 49 A P P E N D I C E S . 5 1 R E F E R E N C E S 59 V LIST OF FIGURES Figure Page 2.1 Apparent Permeability Of Rod With Respect To True Permeability And Ratio Of Length To Diameter 8 3.1 Schematic Diagram Of The Experimental Setup .......15 3.2 Frequency Response Of System A .17 3.3 Frequency Response Of System B .......18 3.4 Frequency Response Of System C ...............19 4.1 Magnetogram Of October 10, 1973 ......24 4.2 Least Squares F i t To System B Output 26 4.3 Least Squares F i t To System C Output 27 4.4 Ratio Of System B Output To System A Output .......32 4.5 Ratio Of System C Output To System A Output 33 4.6 Normal D i s t r i b u t i o n Of The Difference Between Signals By System A And B ..36 4.7 Normal D i s t r i b u t i o n Of The Difference Between Signals By System A And C .37 4.8 Maximum Likelihood Power Spectral Estimate Of System A Record Of October 10, 1973 ..45 4.9 Difference i n Power Spectra Of System B Record By Blackman-Tukey Method ..........46 4.10 Difference In Power Spectra Of System C Record By Blackman-Tukey Method .....47 vi LIST OF TABLES Table Page 2.1 Demagnetization Factors Of Bod And Ellipsoids Magnetized Parallel To Long Axis ....... 8 3.1 Coil Characteristics . . . . 1 6 4.1 Least Squares F i t Coefficients .......28 4.2 Comparison Of Power Spectra ....48 v i i ACKNOWLEDGEMENTS I express my appreciation and gratitude to my thesis advisor Dr. Tomiya Watanabe who directed and helped me to a great extent throughout this experiment and also encouraged me to analyze data and obtain the results which are presented in this thesis. I gratefully acknowledge not only encouragement but also the helpful discussions and suggestions of Dr. R. D. Russell. I appreciate also the generous help as well as the provision of the main equipment for the experiment, from Dr. B . Caner and Dr. L. Law, of the Victoria Geophysical Observatory. I thank Dr. T. Oguchi for the induction magnetometer system with permalloy core c o i l sensor. I also thank Mr. J. Walter for the provision of an experimental site in the University of British Columbia Research Forest. This research received financial support from the National Research Council of Canada under grant A-3564, and from the Defence Research Board under grant 9511-112. 1. INTRODUCTION There are many types of magnetometers used for observing the a c t i v i t i e s of the earth's magnetic f i e l d . They are the proton precession magnetometer, fluxgate magnetometer, induction magnetometer, etc. From the standpoint cf aeronomy, reguirements for these observation systems are such that they should cover the frequency band from UHz to 0.002Hz and should take the si g n a l l e v e l from the order of milligamma to gamma or ten gammmas. The induction magnetometer turns out tc be a type of instrument which s a t i s f i e s these two conditions. It consists of a c o i l with many windings as a sensor followed by a low noise a m p l i f i e r . Output signals are time derivatives of flu c t u a t i o n s of the magnetic f i e l d of the earth. An open loop c o i l , or a i r core c o i l , has a large diameter to gain as much e f f e c t i v e area as possible so that i t can save a great number of windings. On the other hand, some high magnetic permeability materials are used as a core of the sensor c o i l to reduce i t s si z e and weight. I t i s generally believed that magnetic devices using a ferromagnetic material for a sensor are not free from nonlinearity i n response due to hysteresis, eddy current l e s s , Barkhausen noise of the ferromagnetic material, etc. In sp i t e of these speculations, many magnetic stations are using a c o i l wound on a core which has extremely high magnetic permeability. The materials commonly used for these sensors are mumetal, permalloy, supermalloy, etc. As analysis of output wave forms, the i r spectra, amplitude attenuation as well as phase r e l a t i v e to other stations are a l l fundamental and important part of 2 research on geomagnetic micropulsations, i t cannot be allowed that the sensor head introduces nonlinearity at the beginning. It i s of i n t e r e s t to the author of t h i s thesis to investigate the d i s t o r t i o n s due to the nonlinearity of ferromagnetic core sensor through the analysis on magnetogram data. The approach made here i s to simultaneously operate magnetometers using d i f f e r e n t kinds of sensors for actual observation and to compare the output from the d i f f e r e n t systems. Ratios of the peak-to-peak values i n the output of the ferromagnetic core c o i l system r e l a t i v e to the corresponding peak-to-peak values by the a i r core c o i l system were analyzed. Also power spectra are compared between d i f f e r e n t systems. 3 2. CHAHACTERISTICS OF A FERROMAGNETIC CORE COIL There are many fac tors which should be considered i f the ferromagnetic core acts as a nonlinear response element i n a sensor conf igura t ion of a magnetometer. A l i s t given i n the sec t ion 2.1 shows the e f fec t s which are poss ib le sources of nonlinear response. On the other hand, one of the probable source of nonlinear response, demagnetization e f f ec t , may r e s u l t i n a main factor which dominates the c o i l response. In that case, the response becomes l i n e a r . I t requires a condi t ion which i s discussed i n the sect ion 2.2. Hysteres is response i s discussed in the sec t ion 2 .3 . I t i s shown that the response due to hys teres i s produces odd number harmonics. 2 . 1 . NONLINEAR RESPONSE OF FERROMAGNETIC MATERIAL Many poss ible problems with a ferromagnetic core c o i l have been speculated and discussed. Campbell (1967) l i s t s the f o l l o w i n g ; 1) E f f ec t ive permeabi l i ty var ie s depending upon the pos i t i on of c o i l windings because magnetic f lux leaks through the side wall of the core. 2) Magnetization caused by the primary magnetic f i e l d produces a secondary magnetic f i e l d which i s , i n most cases, opposite in d i r e c t i o n to the o r i g i n a l one. This i s re fer red to as 'demagnetization e f f e c t ' . 3) Some loss may be caused by eddy currents . 4) Barkhausen noise may be caused by microscopic magnetization of elements which form the core mater ia l . 5) Permeabi l i ty of a ferromagnetic mater ia l i s unstable against 4 thermal s t ress as wel l as mechanical shocks. 6) 'B-H • curve i s a hysteres i s response. 7) Some loss may be caused by completing the hys teres i s loop . 8) Operation i n a large bias such as the magnetic f i e l d of the earth produces second harmonics and also causes va r i a t i on of the permeabi l i ty . 9) Cross modulation may happen i n case of two or more s igna l s appl ied at the same time because of the nonlinear response of the ' B - H ' curve. The items 3, 4, 7, 8 and 9 are the ones which need d i scuss ion in terms of freguency. The problems 6, 7, 8 and 9 are the ones which o r i g i n a l l y stem from c h a r a c t e r i s t i c s of h y s t e r e s i s . Above a l l , the demagnetization e f fect plays an important ro le i n l a t e r d i scus s ion . 2.2. DEMAGNETIZATION EFFECT When a ferromagnetic body i s placed i n an ambient magnetic f i e l d , i t becomes magnetized. Further , t h i s magnetization produces secondary magnetic f i e l d whose d i r e c t i o n i s , i n the case of a s t r a ight rod , opposite to the magnetization so that i t tends to reduce the appl ied magnetic f i e l d . This secondary magnetic f i e l d i s c a l l e d 'demagnetization f i e l d ' . The s o l u t i o n of Lap lace ' s eguation describes the magnetization cf a body of homogeneous mater ia l placed i n a uniform and p a r a l l e l magnetic f i e l d , but i t i s d i f f i c u l t to ca l cu la te exact ly the strength of the magnetization except for a few simple cases. An e l l i p s o i d i s one such case. S trat ton (1941) obtained the magnetic p o t e n t i a l through Lap lace ' s equation. 5 The Laplace's equation becomes in e l l i p s o i d a l coordinates where where a, b and c are semiprincipal axes of ellipsoid along x, y and z coordinates respectively. If the primary f i e l d i s directed along the x-coordinate axis which i s parallel to the major axis, a, of the e l l i p s o i d , the potential of the applied magnetic f i e l d i s i C^-d^X^-^) ( (2.2.3) This i s also a solution of 2.2-1, and can be rewritten such as where 4=-Et {Cb-d^X^)} m3T (2.2-5) The induced potential i s expected to have the form such as <h = C>4-,&)J^<y)fi(ZJ (2.2-6) This i s substituted into 2.2-1, and Knowing (s|J)= \J^-f^(is a solution of 2.2-7, we get an independent solution. Then the potential of induced magnetization becomes 6 ^ ~ ^ c 2 j ? y ^ h ^ 2 _ 9 ) where C2 i s an undetermined constant . Outside the e l l i p s o i d , there fore , goes i n f i n i t e at 4=_c2, the po tent i a l within the body has the form, (2.2-11) The boundary condi t ions determine C2 and C3. These are where metr ica l c o e f f i c i e n t , JJ.Q i s the permeabi l i ty of ambient materia l and^C i s that of e l l i p s o i d a l body. The f i r s t of these leads to C^-hf-)^ (2.2-13) The second one gives -2 ' /AX D (2.2-14) Rith 2.2-3, the p o t e n t i a l at any i n t e r i o r point of the e l l i p s o i d i s = n * AJ—J*2— (2.2-15) The f i e l d i n t e n s i t y becomes ^ " ^ ^ ^ y ^ ^ (2.2-16) 7 Consequently, the f l u x d e n s i t y i n s i d e the e l l i p s o i d becomes / ^ ^ (2.2-18) from t h i s equation, i t i s c l e a r t h a t when i s very l a r g e , the e f f e c t i v e p e r m e a b i l i t y B/Ho reduces to a value c f 1/Nd. Thus the p e r m a b i l i t y i s determined by the g e o m e t r i c a l c o n s t a n t o n l y . T h e r e f o r e , i n t h i s case, o v e r a l l response of a f e r r o m a g n e t i c core c o i l i s f r e e from i t s n o n l i n e a r h y s t e r e s i s response. In the case of m a g n e t i z a t i o n of a s l i m e l l i p t i c c y l i n d e r along i t s l e n g t h , demagnetization f a c t o r Nd i s c a l c u l a t e d by Osborn (1945) . W-^{»^Afa-fifrT~)-l) (2.2-19, where m i s a r a t i o of i t s l e n g t h to the c r o s s - s e c t i o n a l diameter. I f m >>1, Nd= -j£r{JtL2»<-l } (2.2-20) Table 2.1 shows the demagnetization f a c t o r f o r a red, a p r o l a t e e l l i p s o i d and an o b l a t e e l l i p s o i d . Data f o r the rod given here i s o b t a i n e d through the experiment by Bozorth (1951). For a given core diameter, the longer the c o r e l e n g t h the higher i s the s e n s i t i v i t y of the sensor, because eguation 2.2-18 shows t h a t the r e c i p r o c a l value of m a g n e t i z a t i o n , or the e f f e c t i v e p e r m e a b i l i t y i n t h i s case, i s almost p r o p o r t i o n a l to the sguare of the l e n g t h . F i g . 2.1 f o r a rod shows c l e a r l y t h a t , f o r a longer core, higher p e r m e a b i l i t y i s r e g u i r e d to, 8 Dimensional Ratio Rod Prolate Oblate (length/diameter) Ellipsoid Ellipsoid 0 1.0 1.0 1.0 1 .27 .3333 .3333 . 2 .14 .1735 .2364 $ .040 .0558 .1248 10 .0172 .0203 .0696 20 .00617 .00675 .0369 50 .00129 .00144 • .01472 100 ' 200 .00036 .000430 .00776 .000090 .000125 .00390 500 .000014 .0000236 .001567 1000 .0000036 ' .0000066 .000784 2000 .0000009 .0000019 .000392 Table 2 .1 . Demagnetization fac tors of rod and e l l i p s o i d s magnetized p a r a l l e l to the long axis (Bozorth, .1951) . TRUE PERMEABILITY, fX F i g . 2 .1 . Chart for convert ing true to apparent permeabi l i ty of ferromagnetic c y l i n d e r s of given r a t i o of length to diameter (Bozorth, 1951). 9 secure the l i n e a r r e g i o n i n which the demagnetization e f f e c t dominates o v e r a l l response. C o n s i d e r i n g s u f f i c i e n t l y high p e r m e a b i l i t y values a t t a i n a b l e f o r some f e r r o m a g n e t i c m a t e r i a l s , there seems to be no d i f f i c u l t y - i n s u p p r e s s i n g n o n l i n e a r response i n h e r e n t i n the o r i g i n a l m a g n e t i c p e r m e a b i l i t y . 2.3. HYSTERESIS RESPONSE I f the 'B-H' r e l a t i o n dominates c o i l response r a t h e r than the l i n e a r response by v i r t u e of the demagnetization e f f e c t , some harmonics of the a l t e r n a t i n g primary f i e l d are expected to r e s u l t from the h y s t e r e s i s of the fe r r o m a g n e t i c m a t e r i a l . Peterson (1928) c a r r i e d out a mathematical a n a l y s i s of the h y s t e r e s i s response. I t i s assumed t h a t the a p p l i e d magnetic f i e l d i s pu r e l y s i n u s o i d a l and i t s i n t e n s i t y i s not s t r o n g enough to cause eddy c u r r e n t l o s s . When H i s the maximum amplitude, the magnetic f i e l d , h, becomes For one c y c l e of t h i s e x c i t a t i o n , each h y s t e r e s i s curve i s assumed t o be one complete l o o p , and the f l u x d e n s i t y E, i s a f u n c t i o n of the a p p l i e d f i e l d h, i t s maximum value H, and the s i g n of dh/dt. r> (2.3-1) 10 The s ign of dh/dt s p l i t s the hys teres i s loop in to two branches, top and bottom ones; B1 and B2, r e s p e c t i v e l y . Each branch i s expressed by the same form M=0 A=0 (2.3-2) where ffjMJtX,— (2.3-3, As B=0, at h=0 and H=0, Q.oo — 0 (2.3-4) The hys teres i s loop i s symmetric about the o r i g i n , and i t s each branch has the r e l a t i o n , 5iC^H)=~-fia.C-A/H> (2.3-5) Then, •4 (2. 3-6) (2.3-7) These branches meet at the loop t i p , P i C H / H ^ B i ^ / H ^ (2.3-8) Subs t i tu t ing 2.3-6 and -7 into -8, we obtain a r e l a t i o n up to t h i r d order of H. O.0lH + (^ + *c*)tf'+<A2\+*c*}tt*=O (2.3-9) Since t h i s r e l a t i o n holds for any value of H, d-tz = , <2a3=-#2f (2.3-10) 11 2.3-6 and -7 become (2.3-11) (2. 3-12) Let (2.3-13) Then, R> Gicasti&ti^-x+p cosofr+ txcas*zot + Tcog* cot (2.3-15) From these two eq u a t i o n s we see t h a t CX r e p r e s e n t s the remanence and Q an approximate p e r m e a b i l i t y . L e t t i n g A = - # , c—A, />—£ ,2.3-16, the e q u a t i o n s 2.3-14 and 2.3-15 can be modified f u r t h e r , 3 , < r r f ^ A / i , t f ) « A ^ £ ^ (2.3-17) (2.3-18) Both branches of h y s t e r e s i s curve are combined i n the form, Jk=\ (2.3-19) where 12 '70 Jo PL k « r - ^ / { B 6 A , W [ I+ H J * * 1 1} teS-hftdwt so t h a t ( 2 . 3 - 2 0 ) ( 2 . 3 - 2 1 ) ( 2 . 3 - 2 2 ) ( 2 . 3 - 2 3 ) The c o e f f i c i e n t s bc>r t>x, txt,' - , b-2*-' a l l become z e r o , because of the symmetry of the l o o p . The fundamental and the t h i r d harmonic components have c o e f f i c i e n t s , _ 7 ( 2 . 3 - 2 5 ) ( 2 . 3 - 2 6 ) ( 2 . 3 - 2 7 ) The v o l t a g e a c r o s s the output t e r m i n a l s i s 13 where N i s the number of turns of the winding, A i s the cross- sectional area, E"—/9"^V/4 (ivAi c&tvt-t-3toT^3 casStot • -iobi?tii0it—3a)tbB'Sir\Sa)t} ( 2 . 3 - 2 9 ) Therefore i t i s more l i k e l y that the third harmonic rather than the second harmonic signal w i l l be produced by the hysteresis response. 14 3. EXPERIMENT 3.1. INSTRUMENTATION Three i n d u c t i o n magnetometers have been operated at the U n i v e r s i t y of B r i t i s h Columbia Research F o r e s t , Maple Ridge, B r i t i s h Columbia s i n c e September 10, 1973. The setup of the experiment i s s c h e m a t i c a l l y shown on F i g . 3.1. System A c o n s i s t s of an a i r core c o i l as the sensor head and a semiconductor p a r a m e t r i c a m p l i f i e r which can take as high an output impedance as the 200 k-ohm of the sensor with a noise f i g u r e l e s s than 3db. System B uses a permalloy core c o i l sensor which i s connected to parametric a m p l i f i e r i d e n t i c a l t o t h a t of the system A . D i r e c t comparison i s p o s s i b l e between the outputs from the two systems A and B. System C has a permalloy core c o i l with FET chopper a m p l i f i e r . To o b t a i n n c i s e l e v e l s as low as 0.1 m i c r o - v o l t peak-to-peak, t h i s a m p l i f i e r employs i n t e n s i v e f i l t e r i n g throughout ( F i g . 3.4). Outputs of these systems are recorded on a 7 channel FM magnetic tape r e c o r d e r t o g e t h e r with the output of a f l u x g a t e magnetometer, c l o c k p u l s e s and a r e f e r e n c e FM c a r r i e r s i g n a l . S e n s i t i v i t y of t h i s r e c o r d e r i s a d j u s t e d to be 2 v o l t s per 40% freguency d e v i a t i o n of FM c a r r i e r s i g n a l . T h i s FM r e c o r d e r determines the dynamic range of systems and i t i s 40db i n rms base. C o i l c o n s t a n t s are g i v e n i n Table 3.1. The a i r core c o i l was c o n s t r u c t e d and c a l i b r a t e d by B. Caner (1970). Data of the f e r r o m a g n e t i c core c o i l are obtained by T. Watanabe from the freguency response of the c o i l . The s i g n a l from the o s c i l l a t o r i s i n t r o d u c e d to the c o i l through a r e s i s t o r which has a value r System A 0 . 1 2 8 m V / f f - H z System Li v 0 . 0 1 8 9 i u V / y - H z System C 45 db I : o . - . . l i conc luctor P ; . r a i . i o t r i c A m p l i f i e r 6'0 db • 0 . 0 1 3 9 m V / y - H z 7-channel FM. Tapo Smenri; l / l 6 l r > a Fig. 3 . 1 . Schematic diagram of experimental setup. A i r core c o i l Permalloy core c o i l Weight 175 kg 12 kg Dimension Diameter; 1.2? ni Core length; 1.0 m Cross-section; 2.0x2.0 cm Number of turns 16,000 turns 50,000 turns DG resi s t a n c e 5. ^5 k-ohm 158 ohm Resonance frequency 3*f Hz 400 Hz S e l f inductance 550 H 170 II Stray capacitance 1 6.7 nF 0.8 nF Damping r e s i s t o r 200 k-ohm 10 k-ohm Johnson noise (DC - kllz) 18.5 nV(RMS) 5.17 nV(Hl-iS) S e n s i t i v i t y 0.128 mV/ -Hz 0.0189 mV/ - H z Table 3.1. C o i l c h a r a c t e r i s t i c s 0.1 J UJ a g 0.0J J Si 0.001 J Amplitude Ugnnortsg Overall Response Coi l Response COIL INDUCTPNCC CAPACITANCE RESISTANCE DAMPING RESISTANCE O'.l S30 6.7 S.4S H . NANO-f K-OHM K-OHM "i i i iI 1 1.0 FREflUENCr(HZ) Fig. 3.2. Attenuation and phase response of system A. Coil response includes the input impedance of the preamplifier. H « . « —I O.I J a S o.oi J I 0.001 J Phase Overall Response Coll Response COIL INDUCTRNCE CAPACITANCE RESISTANCE DAMPING RESISTANCE 170 O.B CIS H. NANO-F K-OHM K-OHM 1.0 I HIM 10* Oil FREQUENCriHZ) Fig. 3.3. Attenuation and phase response of system B. Coil response includes the input impedance of the preamplifier. Amplitude Oil i'.o FREQUENCY(HZ) Fig. 3.4. Attenuation and phase response of system C. Coil response includes the input impedance of the preamplifier. H 20 much s m a l l e r than the r e s i s t a n c e of the c o i l windings. Then the c u r r e n t through t h i s r e s i s t o r i s measured as a f u n c t i o n of frequency. The frequency response of the c o i l i s s e n s i t i v e to the i n p u t impedance of the p r e a m p l i f i e r . The combined response of c o i l and a m p l i f i e r impedance i s shown t o g e t h e r with the o v e r a l l system response f o r each sensor c o i l s on F i g . 3.2, 3.3 and 3.4. 3.2 SOME CONSIDERATIONS ON COIL CHARACTERISTICS The sensor c o i l i s approximated to be an N t u r n winding whose r a d i u s i s c o n s t a n t . Corresponding to the u n i f o r m l y v a r y i n g magnetic f i e l d , the v o l t a g e induced i n the c e i l , Eo, i s where S i s the c r o s s - s e c t i o n a l area of the c o i l , JLL i s p e r m e a b i l i t y of the c o i l c e r e , and h i s the magnetic f i e l d component p e r p e n d i c u l a r to the c o i l c r o s s - s e c t i o n . U s u a l l y , p e r m e a b i l i t y i s measured as a normalized value t c the vacuum one, so t h a t S0=4.70 • /d^fee, A/sJL (3.2-2) A c c o r d i n g to data f o r the a i r core c o i l i n Table 3.1, N=16,000, S=1.28 mz and ^^=1. The magnetic f i e l d , h, i s measured c o n v e n t i o n a l l y i n milligammas which i s equal to 1/̂ x x 1 0 - 5 AT/m i n MKS u n i t s . The s e n s i t i v i t y of a i r core c o i l becomes Bo^A/ZZ^r/^r-ttfr (3.2-3) As the t o t a l g a i n of the a m p l i f i e r i s c a l i b r a t e d to 1.87x10 5, a magnetic f i e l d v a r i a t i o n of 1 milligamma peak g i v e s r i s e to a d e f l e c t i o n of 23.5 mV on the r e c o r d e r s . Campbell (1960) suggested a simple form of e q u i v a l e n t 21 c i r c u i t to approximate c h a r a c t e r i s t i c s of sensor c o i l . AW—nrsip*—i—o—,—> - r - C Where r i s the r e s i s t a n c e of windings, C i s the s t r a y c a p a c i t a n c e , L i s the s e l f - i n d u c t a n c e . S i s an o p t i o n a l r e s i s t o r which i s sh u n t i n g the output t e r m i n a l s of c o i l . T h i s r e s i s t o r damps the resonance peak of c o i l and, with minimal e f f o r t , p r o v i d e s maximal f l a t amplitude response. The t r a n s f e r f u n c t i o n of the above e q u i v a l e n t c i r c u i t i s (3.2-4) So the amplitude response i s M"H Htfo)h -£/isi<w <3- 2-5) where ^ NcV= r+fc u*/-Cg] + io\L -h r CRf)* ( 3 . 2 - 6 ) I t i s d e s i r a b l e t h a t t h e r e be no i r r e g u l a r peaks i n the amplitude response, f o r a l l tO . That i s , ^ { W/CzLC-r^A^, ( 3 . 2 - 8 ) where 2L > rC, which most sensors with many windings s a t i s f y . On the oth e r hand, f o r a c o i l wound with extremely t h i n , or h i g h l y r e s i s t i v e wire and with s m a l l inductance, the l o s s by the r e s i s t i v e component becomes so l a r g e t h a t no resonance peaks appear higher than the l e v e l a t DC. The maximum l i m i t value of R a l s o s a t i s f i e s the c o n d i t i o n t h a t the f i r s t and second 22 d e r i v a t i v e s of N(w) be p o s i t i v e , f o r a l l OU . The amplitude response damped by R, t h e r e f o r e , decreases m o n o t o n i c a l l y toward the h i g h e r freguency r e g i o n . As the damping r e s i s t o r shunts the output of the c o i l , i t produces a d d i t i o n a l thermal n o i s e . O L. where and ^ are rms value of thermal n o i s e of r e s i s t i v e components, r and R, r e s p e c t i v e l y . Using the c u r r e n t source r e p r e s e n t a t i o n , i t becomes L These two thermal n o i s e s are c o n s i d e r e d to be u n c c r r e l a t e d , and two c u r r e n t sources are r e p l a c e d by one source such t h a t •j f (3.2-9) S i m i l a r l y , a c u r r e n t source e q u i v a l e n t to the v o l t a g e induced by magnetic f i e l d v a r i a t i o n i s r - t - J i o L r+j&L ( 3 . 2 - 1 0 ) Thermal n o i s e , then, can be estimated as an e q u i v a l e n t magnetic v a r i a t i o n . r+JooL 23 4. ANALYSIS Outputs of the a m p l i f i e r s were recorded on a magnetic tape by an FM slow speed analog tape r e c o r d e r . The tape was then played back a t a speed 15 times f a s t e r than f o r the r e c o r d i n g . Then the data was d i g i t i z e d a t a r a t e of 29 p o i n t s per second a f t e r passing through a lowpass f i l t e r whose c u t o f f freguency i s 8 Hz and a t t e n u a t i o n s l o p e 24 db/oct. There was a p e r i o d of a c t i v e geomagnetic m i c r c p u l s a t i o n on October 10, 1973. The data of a 282 sec d u r a t i o n s t a r t i n g from 2 1 : 4 2 ' 0 6 " UT were analyzed mainly because the data i s f r e e of s p o r a d i c n o i s e i n h e r e n t to the parametric a m p l i f i e r s used i n t h i s experiment. As a p r e l i m i n a r y comparison of the three channels, l e a s t squares f i t s of these s i g n a l s were taken. Secondly, the amplitude r a t i o of the permalloy core c o i l s i g n a l to the a i r core c o i l s i g n a l was examined and f i n a l l y , power s p e c t r a of the t h r e e were compared. 4.1. LEAST SQUARES FIT A l e a s t squares f i t was t r i e d between the a i r core c o i l s i g n a l and each of the permalloy core c o i l s i g n a l s . Commencing on 21h 42m 06s UT, data of 2 min 21 sec d u r a t i o n was sampled from the r e c o r d of October 10, 1973. There i s a medium l e v e l of geomagnetic m i c r o p u l s a t i o n a c t i v i t y during t h i s p e r i o d and i t s major frequency components l i e i n the range 0 . 0 2 - 0 . 0 8 H Z . Let Yb and Yc be the output v o l t a g e of the permalloy core c o i l systems, B and C, r e s p e c t i v e l y . As mentioned a l r e a d y , the system B has a p a r a m e t r i c a m p l i f i e r immediately f o l l o w i n g the 23H 42H 055 UT OCT. 10/1973 FLUXGRTE MAGNETOMETER OflHHfl-HZ - o . o a J 0 o g_, PERMALLOY CORE COIL + PflRRMETRlCJWPLlFIER GRMMR-HZ -0.0SU Q Q a_, RIR CORE COIL • FRRflHETRIC RHPLIFJER 6HHMH-HZ -o .oaJ ~5a i i is H i !ul 5IB9 tas - bs4 S E C fa F i g . 4.1. Magnetograns of 21h 42m 06s UT on October 10. l<m 25 sensor c o i l and the system C has a FET chopper a m p l i f i e r . X i s used t o denote the output v o l t a g e of the a i r core c o i l system (system A). The f i t t n g curve i s g i v e n by fi-^O (4.1-1) From t h i s , the curve i s f i t t e d to Yb and Yc u s i n g the l e a s t squares f i t t i n g i . e . (4.1-2) i s minimized over the 141 sec d u r a t i o n of the r e c o r d . The g a i n d i f f e r e n c e between channels i s e s s e n t i a l l y k1. The higher order c o e f f i c i e n t s r e p r e s e n t the p o s s i b l e n o n l i n e a r e f f e c t s due to the f e r r o m a g n e t i c core as mentioned i n s e c t i o n 2.1 a l r e a d y . Although the amplitude response of the a m p l i f i e r s i n each systems i s taken to be f l a t enough over the frequency content of the s i g n a l , d i f f e r e n c e s i n phase are not n e g l i g i b l e . T h i s a f f e c t s the r e s u l t s of the l e a s t squares f i t method. The c h a r a c t e r i s t i c s of the c o i l response w i l l be p r o j e c t e d on the c o e f f i c i e n t s i f thes.e phase responses are compensated, i d e a l l y with f i l t e r s which have the complementary responses to the a m p l i f i e r s . Here i t i s done simply by r e a l i g n i n g the r e l a t i v e p o s i t i o n of the data of each channel and the s m a l l e s t sum of sguares i s o b t a i n e d . I f r e l a t i v e l a g i s ~C- sec between r e c o r d s , so t h a t t h i s o p e r a t i o n i s e q u i v a l e n t t o using a phase s h i f t e r which g i v e s a phase s h i f t p r o p o r t i o n a l to frequency. The best r e s u l t i s obtained when a l a g of -0.55 sec i s placed on the LERST SOURRE FIT Ei l U ~ S S i S 7L T i l 5Ti !l27 !l Fig. 4.2. Least squares f i t . Output of system A i s fitt e d to the one of system B. 0.081 LERST SOUARE FIT DIFFERENCE YC-VF= YD VOLTS -o.oaJ VOLTS -0.42J 2 Q ] _ OUTPUT OF AIR CORE CDIL= X VOLTS -L.B7. S TA ST ^2 11 71 H5 5i ~113 1 2 7 S E C \ Fig. 4.3. Least squares f i t . output of system A i s fi t t e d to the one of system C. • • System B System C n Kn Kn x 2 N Kn Kn x 2 N 1 0.14154 0 . 2 8 5 0 8 0.18365 0 . 3 6 7 3 0 2 (0.01532) (0 . 0 5 5 2 3 ) (-0.00733) (-0.02932) 3 (-0.00340) (-0.02720) 0..01111 0 . 0 3 8 3 8 k -O.OO56I -O.O8976 (0.00157) (0.02512) 5 (O.C0195) (0.. 06240) (-0 .00097) (-0.05104) * Table 4.1. Coefficients of the least sguares f i t . Air core c o i l signal i s f i t t e a to permalloy core c o i l signal. Bracketted values are not significant. 29 r e c o r d of system B with r e s p e c t t o t h a t of the system A. T h i s l a g i s e q u i v a l e n t to about 9.9 degrees l e a d f o r a s i g n a l of 0.05 Hz. In case of system C, i t was 3.3 sec l a g , which i s about 59.4 degree l a g f o r 0.05 Hz. The r e s u l t s are shown on F i g . 4.2 and F i g . 4.3. As the maximum amplitude of X i s 2 v o l t s , the terms - f c * x k c o n t r i b u t e the maximum amount of -^Cw2A. These maximum c o n t r i b u t i o n s are l i s t e d together with the c o e f f i c i e n t s -£CM- on Table 4.1. The l a r g e s t value appears f o r the f o u r t h order c o e f f i c i e n t i n the f i t t o the s i g n a l o f system B, and f o r the t h i r d order to the system C s i g n a l . The i n c o n s i s t e n c y of the magnitude of l e a s t squares f i t c o e f f i c i e n t s between the r e s u l t s by system B and the one by system C may be caused by the noise of the tape r e c o r d e r used i n t h i s experiment. The noise l e v e l of the FM tape r e c o r d e r i s 0.02 v o l t s rms, so that noise higher than 0.06 v o l t s o c c u r s with p r o b a b i l i t y l e s s than 0.12 %. But such noise i s s t i l l a s u b s t a n t i a l p a r t of many valu e s i n columns 3 and 5 of Table 4.1, so t h a t many of the c o e f f i c i e n t s are masked by tape n o i s e . 4.2. AMPLITUDE RATIO A s t r a i g h t f o r w a r d comparison of the s i g n a l amplitude i s made by t a k i n g the r a t i o of each permalloy core c o i l s i g n a l to the a i r core c o i l s i g n a l . I t i s a p p r o p r i a t e to use peak-to-peak r e a d i n g s . They are f r e e of dc o f f s e t and phase s h i f t . B e s i d e s , they are l e s s a f f e c t e d by noise because of r e l a t i v e l y higher S/N r a t i o s . Data commencing on 21h 42m 06s UT was taken from the d i s t u r b e d day r e c o r d , October 10, 1973. 30 To obtain peak-to-peak readings, maximum and minimum values of the s i g n a l waveform were c o l l e c t e d . Then, d i f ference between a succesive pa irs of values was obtained which gives a peak-to- peak reading. One maximum/minimum reading i s used for c a l c u l a t i n g two peak-to-peak values i n the s e r i e s . Conseguently, they are opposite i n p o l a r i t y . Readings of noise , mostly sporadic pulses caused by a m p l i f i e r s , are then delated. F i g . 4.4 and F i g . 4.5 show the r a t i o s of the readings by the permalloy core c o i l systems to the peak-to-peak reading of the a i r core c o i l system s i g n a l . Apparently the noise of every system af fects small peak-to-peak values . P lo t s f c r smaller values are more sca t tered . Studies of noise c h a r a c t e r i s t i c s as well as the p r o b a b i l i t y of the r a t i o of two normal var iables enable us to draw the d i s t r i b u t i o n range of r a t i o values with respect to the value af the output of system A. 4 . 2 . 1 . RELATIVE SENSITIVITY AND RATIO OF TWO NORMAL VARIAELES Let n1 be the noise i n a peak-to-peak reading on system A s i g n a l , Y1. Then, we have J/» = Kl (4.2.1-1) where x i s the s i g n a l , the true peak-to-peak reading. The corresponding peak-to-peak reading on system B channel , Y2 i s given by (4.2.1-2) 31 where S i s the gain of system B r e l a t i v e to system A. The r a t i o i s •X (4.2.1-3) I f x i s much larger than n1, so that (4. 2.1-4) X C R - S ^ « n-i- H.iS (4.2.1-5) I f noise of each channel i s assumed to follow a normal d i s t r i b u t i o n , the guantity x (R-S), follows approximately a normal d i s t r i b u t i o n , N (0 ,07* • ( C\ S) 2) ; the average value i s zero and the variance i s equal to 0~i 2+ ( rr̂  S) 2 where CJj" and r/^ are the standard deviations of n1 and n2, resp e c t i v e l y . Consequently, the mean value becomes <X(RS)> V>0 (4.2.1-6) or <XR> *.SX>S (4.2.1-7) That i s , <X> (4.2.1-8) Thus, an approximate value of the r e l a t i v e s e n s i t i v i t y can be obtained. The s e n s i t i v i t y of system B r e l a t i v e to the one of system A was found to be 0.1438. For system C, i t i s 0.1838. Moreover, knowledge of thi s s e n s i t i v i t y makes i t possible to check the assumption on the c h a r a c t e r i s t i c s of ncise, because the quantity, ^ - 5 ^ = ^ - 5 ^ 1 1 (4.2.1-9) must follow a normal d i s t r i b u t i o n , N (0 , <fe 2+ (<7T S) 2) , i f the F i g . 4 . 4 . R a t i o of system B output to system A output. R a t i o i s p l o t t e d v e r s u s system A output. The i n s i d e p a i r of d i s t r i b u t i o n l i m i t s correspond t o one standard d e v i a t i o n , 68.3%. The o u t s i d e ones are twice the s t a n d a r d d e v i a t i o n , 9 5 .5%, 2.00 _ 1.75 1.50 1.25 J a 1.00 J 0.75 J 0.50 STRN0RRO DEV1RTION OF NOISE SYSTEM Ri 0.27 MILtIGRMMR-HZ SYSTEM C; 3.81 HILLIGRMKR-HZ 25 PERK TO PERK REflDlNGCMILLIGRMHR-HZ) 130 145 160 ITS F i g . 4.5. Ratio of system C output to system A output. Ratio i s p lo t ted versus system A output. The ins ide pair of d i s t r i b u t i o n l i m i t s correspond to one standard' d e v i a t i o n , 68.3X. The outside ones are twice the standard d e v i a t i o n , 95.5?. VM 34 n o i s e on each channel f o l l o w s a normal d i s t r i b u t i o n . 4.2.2 NOISE ANALYSIS ON PROBABILITY GRAPH As d e f i n e d by the l e f t hand s i d e of the eguation 4.2.1-9, d i f f e r e n c e s between the c o r r e s p o n d i n g peak-to-peak r e a d i n g s on the two channels are taken and p l o t t e d on a p r o b a b i l i t y graph. The p r o b a b i l i t y , y, t h a t a d i f f e r e n c e be l e s s than x i s g i v e n as f o l l o w s . % )/27C V (4.2.2-1) P r o v i d i n g t h a t the d i f f e r e n c e s form a sample ensemble of normal d i s t r i b u t i o n with mean, m, and a standard d e v i a t i o n , CT ; t h a t i s , Ntm,^^). The p r o b a b i l i t y , y, i s r e w r i t t e n as f o l l o w s . where t^CZ-^)/<T (4.2.2-2) The range of sample val u e s x, i s d i v i d e d i n t o r e g u l a r i n t e r v a l s of c l a s s e s , and the cumulative p r o b a b i l i t y , y i , i s obtained on each c l a s s . For a l a r g e number of samples, the cumulative p r o b a b i l i t y should be c l o s e to t h a t of a normal d i s t r i b u t i o n . i h ' - T ^ r / ( , , 2 . 3 ) — As the p r o b a b i l i t i e s , y i ' s , are p l o t t e d along the a x i s of t where t i s r e l a t e d to x l i n e a r l y ; tc = CXt-r*L)/<T (4.2.2-4) The p l o t s should be a s t r a i g h t l i n e i f the sample ensemble f o l l o w s a normal d i s t r i b u t i o n . The reading at t=0 or y=0.5 35 g i v e s the mean g i v e s the mean v a l u e , m. The standard d e v i a t i o n i s o b t a i n e d by the f o l l o w i n g r e l a t i o n , t i - % J ^ C X . C - X j ) / q - (4.2.2-5) or <T= CZC -X-p/CVt—lj) (4.2.2-6) The s t a n d a r d d e v i a t i o n of noise f o r comparison between system A and system B i s found to be 13.59 milligamma-Hz from F i g . 4.6. In the case of comparison between system A and C, i t i s found to be 3.81 milligamma-Hz as shown i n F i g . 4.7. 4.2.3. PROBABILITY DENSITY FUNCTION OF A RATIO OF TWO NORMAL VARIABLES The p r o b a b i l i t y d e n s i t y f u n c t i o n f o r the r a t i o of two normal v a r i a b l e s was o b t a i n e d , f o r example, by Geary (1935). Let a and b be c o n s t a n t and a l s o y and x be random v a r i a b l e s which f o l l o w normal d i s t r i b u t i o n s with zero mean and s t a n d a r d d e v i a t i o n s p and (X , r e s p e c t i v e l y . The r a t i o v a lue becomes , X _ A + X (1.2.3-1) i f x and y are independent of each other, the j o i n t p r o b a b i l i t y , or t h e p r o b a b i l i t y of simultaneous occurrence of x and y, i s given by . ' (4.2.3-2) Changing the v a r i a b l e s (x,y) to (x,z) by 4.2.3-1, we get 21.BB 19.69 37.50 15.31 13.12 JO.93 8.74 • 6.55 4.36 2.17 £ -0.01 p -2.20 g-4.39 _j d -6.58 sr. -8.77 J -10.96 J -13.15 -15.34 -17.53 J -19.72 J -21.91 T= 001 0.01 O'.OS O'.l 0'.2 0.3 0.4 0.5 0.6 0'.7 O'.B CUMULATIVE PROBABILITY 0.95 0.99 0.999 Fig. 4.6. Normal distribution of the difference between signals of system A and B. Mean value i s found at t=0. Standard deviation appears as the slope of the line. Fig. 4.7. Normal distribution of the difference between signals of system A and C. Mean value i s found at t=0. Standard deviation appears as the slope of the line. 38 2.7CC*£ ^ r 0*"* (4.2.3-3) with y=(a+x)z-b and where |a+x| i s a p o s i t i v e value. The p r o b a b i l i t y f u n c t i o n P(z) becomes with y=(a+x)z-b. Let Q (z) and B(z) r e p r e s e n t the f i r s t and (4.2.3-4) second i n t e g r a l s on the r i g h t hand s i d e of t h i s e q u a t i o n . = + (4.2.3-5) The i n t e g r a t i o n i n Q(z) can be performed through a t r a n s f o r m of so t h a t \4d (4.2.3-7) Then P C\<ltht* ^7 V p W / (4.2.3-8) Q(z) becomes a normal d i s t r i b u t i o n f u n c t i o n by the tra n s f o r m of the v a r i a b l e V ^ T D ' i^i-Ct^ (4.2.3-9) P r a c t i c a l l y , Q (z) becomes a good approximation of the p r o b a b i l i t y d e n s i t y f u n c t i o n , P ( z ) . From 4.2.3-5, 39 -tf -V1 (4.2.3-10) The l e f t hand s i d e of the equation i s u n i t y . That i s the e r r o r , , by equating Q(z) to P(z) i s ,p0 (4. 2.3-1 1) s hd (4.2.3-12) In o t h e r words, £ , i s the p r o b a b i l i t y of f i n d i n g a d e v i a t i o n from the average i n a b s o l u t e value g r e a t e r than or egual to I t w i l l be noted t h a t ^-/z i s the p r o b a b i l i t y c f f i n d i n g a n e g a t i v e value of a + x. Since R (z) i s a p o s i t i v e f u n c t i o n of z, i t f o l l o w s t h a t z (4.2.3-13) f o r a l l values of z1 and z2. I f P (z) denotes the p r o b a b i l i t y of f i n d i n g a value of z between z1 and z2, i t f o l l o w s from the equation 4.2.3-10 t h a t (4.2.3-14) Even i f Ct/cZ , the c o e f f i c i e n t of v a r i a t i o n cf a+x, i s not g r e a t e r than 1/3, £ i s very minute and P(z1,z2) can be taken as e q u a l to Q(z1,z2). In the p a r t i c u l a r case of ^ # = 1 / 3 , £ = 0 . 0 0 2 7 . The e x p r e s s i o n 4.2.3-8 shows t h a t the v a r i a b l e , t , i s normally d i s t r i b u t e d with zero mean . and u n i t v a r i a n c e . T h e r e f o r e , | t |<1 g i v e s the range of 68.3% f o r the r a t i o value 40 d i s t r i b u t i o n . | t |<2 g i v e s the range of 95.5%. To draw these d i s t r i b u t i o n l i m i t s , the r e s u l t s of 4.2.1 are used t o g e t h e r with the s t a n d a r d d e v i a t i o n of noise of system A which i s es t i m a t e d from r e c o r d s of ge o m a g n e t i c a l l y g u i e t days. I t i s 0.27 milligamma-Hz. 4.3. EFFECTIVE PERMEABILITY OF PERMALLOY CORE In the s e c t i o n 4.2.1 r e l a t i v e s e n s i t i v i t i e s of the permalloy core c o i l systems with r e s p e c t tp the a i r core c o i l system were o b t a i n e d . With these v a l u e s , the e f f e c t i v e p e r m e a b i l i t y of each of the permalloy c o r e s i s c a l c u l a t e d as f o l l o w s . From s e c t i o n 3.2, output of the a i r core c o i l system w i l l be 0.0235 v o l t s per s i n u s o i d a l geomagnetic v a r i a t i o n s c f 1 milligamma-Hz. Then, output of a permalloy core c o i l system with r e l a t i v e s e n s i t i v i t y , S, i s 0.0235xS v o l t s . Knowing the t o t a l gain of the a m p l i f i e r of t h i s system, G, we get an estimate of the output v o l t a g e of the sensor c o i l as The r e l a t i o n 3.2-6 g i v e s the e f f e c t i v e p e r m e a b i l i t y such as The c o n s t a n t s g i v e n i n Tab l e 3.1 are used. The e f f e c t i v e p e r m e a b i l i t y of system B i s obtained as 152 while t h a t of system C as 146. Acc o r d i n g to F i g . 2.1, the apparent p e r m e a b i l i t y i s expected to be 800 f o r the core whose le n g t h to diameter r a t i o i s 50. The apparent p e r m e a b i l i t y of 150 can be ob t a i n e d when tr u e p e r m e a b i l i t y i s 150, but t h i s value i s e x t r a o r d i n a r i l y 41 s m a l l f o r a f e r r o m a g n e t i c m a t e r i a l such as permalloy. Besides, t h i s low apparent p e r m e a b i l i t y i s i n the n o n l i n e a r r e g i o n where sensor response depends more on the p e r m e a b i l i t y of the co r e . I t i s p o s s i b l e i n t h i s case t h a t the dynamic response f o r a b i a s e d f l u c t u a t i o n of the e a r t h ' s magnetic f i e l d i s s u b j e c t e d to much s m a l l e r p e r m e a b i l i t y , namely the i n c r e m e n t a l p e r m e a b i l i t y (Bozorth,1951). I t s response may appear l i n e a r w i t h i n a s m a l l amplitude of magnetic f i e l d f l u c t u a t i o n . 4.4 COMPARISON OF POWER SPECTRA The r e c o r d s by the t h r e e systems, 282 sec long d u r a t i o n s t a r t i n g at 21h 42m 06s UT on October 10, 1973, were taken and t h e i r power s p e c t r a were o b t a i n e d f o r another i n t e r c h a n n e l comparison. The c o n v e n t i o n a l Blackman-Tukey method as w e l l as the maximum l i k e l i h o o d method were employed f o r e s t i m a t i n g the power spectrum of each. The maximum l i k e l i h o o d power e s t i m a t o r i s taken f i r s t with an i n t e n t i o n of o b t a i n i n g a r e f e r e n c e f o r the Blackman-Tukey method because i t i s e x c e l l e n t f o r reproducing s p e c t r a l e s t i m a t o r s out of r e l a t i v e l y s m a l l number of data p o i n t s (Lacoss 1971). A d e r i v a t i o n of the maximum l i k e l i h o o d power estimate i s given i n the appendix. Various l e n g t h s of a u t o c o r r e l a t i o n were t r i e d with the Blackman-Tukey method u n t i l i t s s p e c t r a l e s timate becomes c l o s e s t to the one given by the maximum l i k e l i h o o d method. Then a co n f i d e n c e i n t e r v a l f o r the s p e c t r a l e s t i m a t o r as w e l l as the bandwidth was c a l c u l a t e d from the window shape and the l e n g t h of the a u t o c o r r e l a t i c n used. 42 4.4.1. CONFIDENCE INTERVAL AND BANDWIDTH OF THE TUKEY WINDOW Acc o r d i n g to the Blackman-Tukey method, power s p e c t r a are estimated by t a k i n g the F o u r i e r t r a n s f o r m o f the a u t o c o r r e l a t i o n . T h i s a u t o c o r r e l a t i o n i s c a l c u l a t e d d i r e c t l y from a sampled l e n g t h of time s e r i e s data and then i t i s t r u n c a t e d and/or weighted with a window f u n c t i o n . The Tukey window, or c o s i n e b e l l , used here f o r weighting the a u t o c o r r e l a t i o n i s given i n l a g window form as I O IUI > M (4.4. 1-1) where M i s the l e n g t h of the t r u n c a t e d a u t o c o r r e l a t i o n . S t h i s weighting i s e q u i v a l e n t to smoothing the raw s p e c t r a , which i s a d i r e c t F o u r i e r transform of data of time domain, through a bandpass f i l t e r . The bandwidth of t h i s s p e c t r a l window, b, i s d e f i n e d as / /, (4.4.1-2) = I w H v ) A » = ^ Q + ^ i c v f M E A T S ' 2-3) 'A That i s , the bandwidth i s r e l a t e d to K by £>~C^7£~/~/) ~' (4.4.1-4) The smoothed sample s p e c t r a C (f) estimated through the atove windowing i s (4.4.1-5) where R (u) i s the a u t o c o r r e l a t i o n and T i s the l e n g t h of the 43 time s e r i e s data. In the frequency domain, where , r WCft & (4.4.1-7) I f P (f) v a r i e s smoothly r e l a t i v e to the bandwidth, b, then sleep]- Kfif-wtpdz * Pep (4.4.1-8) Using P a r s e v a l ' s theorem, the v a r i a n c e of smoothed s p e c t r a l e s t i m a t o r may be d e s c r i b e d . ' ZiO (4.4.1-9) That i s , ^ 1 = / lV*CH)dU=l/fr (4.4.1-10) With t h i s mean and v a r i a n c e , C (f) can be approximated to random v a r i a b l e , CK ̂ Z , which f o l l o w s c h i - s q u a r e d d i s t r i b u t i o n with degrees of freedom, V . l > = -2 (B ICCpV fori CCp\ \ (4. 4.1-11) 4 E[cCpJ/\j _ (^.4.1-12) In o t h e r words, the random variable", \J i s d i s t r i b u t e d as a with degrees of freedom, V . Tc f i n d the 100(1-CV)% d i s t r i b u t i o n range of \J , we r e f e r to the t a b l e of c h i - s q u a r e d i s t r i b u t i o n which g i v e s — and ^\)(^") i n the r e l a t i o n . CX f^f) (4.4.1-13) _ CX where ft { ^ < ^ (J*-> ] = f " Then c o n f i d e n c e i n t e r v a l f o r P (f) becomes uu In l o g a r i t h m i c s c a l e , (a.a.1-15) T h e r e f o r e , from the value of l o g C ( f ) , l o g b̂ cfvO-̂r) i s the lower l i m i t and l o g V̂ v£"§f") i s t h e h i g h e r l i m i t of the i n t e r v a l i n which the t r u e value of the spectrum P ( f ) , c o u l d f a l l with p r o b a b i l i t y , 100 (l-cx/) These l i m i t s depend only on the c o n f i d e n c e and the degrees of freedom which are r e l a t e d to M by x ( a . a . 1 - 1 6 ) T h e r e f o r e M c o n t r o l s the bandwidth and c o n f i d e n c e l e v e l a t the same time. 4.4.2. RESULTS OF SPECTRAL ANALYSIS Tabl e a.2 show the numerical r e s u l t s of s p e c t r a l a n a l y s i s . Spectrum p a t t e r n s are g i v e n on F i g . a.8, F i g . a.9 and F i g . a. 10. These spectrum peaks occur i n a p o s i t i o n c o r r e s p o n d i n g w e l l to the ones o b t a i n e d by the maximum l i k e l i h o o d method. Three peaks at 0.110Hz, 0.138Hz and 0.166Hz seem to be n o i s e because of t h e i r low l e v e l , lower than -20db, with r e s p e c t t c the main peak. The d i f f e r e n c e i n power spectrum between the a i r core c o i l system and each of the permalloy core c o i l systems i s w e l l w i t h i n 96% c o n f i d e n c e i n t e r v a l s . Fig. 4.8. Maximum likelihood power estimate of system A record of October 10, 1973. (see Fig. 4.1.) o'.OO 0\02 OTM G\06 o!u8 O'.IO JA2 OYU GLIB CUB 5.20 FREQUENCY(HZ) F i g . 4.9. Dif ference in power spectra of system B record by Blackman-Tukey method. The hor i zonta l cross l i n e shows the bandwidth, and the v e r t i c a l ene i s the 96% confidence i n t e r v a l . A l so , the center pc int corresponds to zero di f ference of power spectra by the scale on the r i g h t hand s ide . Difference of pewer spectra i s taken a f te r each spectrum i s normalized to i t s peak. CLOD CL02 CUM o'.06 CUOB (TH) cTTlZ OLH O'.16 cTTlB 0.20 FREQUENCY IHZ) F i g . 4.10. Dif ference i n power spectra of system C record by Blackman-Tukey method. The h o r i z o n t a l cross l i n e shows the bandwidth, and the v e r t i c a l one i s the 96% confidence i n t e r v a l . Al so , the center point corresponds to zero di f ference of power spectra by the scale on the r i g h t hand s ide . Difference of power spectra i s taken a f ter each spectrum i s normalized to i t s peak. Peaks i n the power spectrum Difference from Permalloy of the a i r core c o i l s i g n a l core c o i l s i £nal Maximum l i k e l i h o o d Blackman--Tukey System B System C Hz db Hz db db db • 0.0106 -0.2053 0.0106 -0.^235 -0.9582 -0.2095 O.OJ518 0.0* 0.0518 0.0* 0.0* 0.0* 0.0566 -2.7992 C.0566 -2.856 -0.9299 0.1953 — 0.110 -22.9825 1.2^36 0.6705 - 0.138 -25.9197 2.286*f 0.571^ — 0.166 -27.3723 -O.O956 0.765^ 0.191 -28.697 0.191 -29.3180 0.0206 0.66^1 Table 4.2. Comparison of power spectra. Each spectra are normalized to i t s peak. That i s , peaks with are a r b i t r a r i l y set to O.Odb. 49 5. SUMMARY AND CONCLUDING REMARKS In the r e g i o n of high to mid geomagnetic l a t i t u d e , c o r r e l a t i o n s between geomagnetic m i c r o p u l s a t i o n data obtained a t v a r i o u s s t a t i o n s provide important i n f o r m a t i o n . For example, with the data along the geomagnetic meridian i t i s i n t e r e s t i n g to study the behavior of the plasma pause as w e l l as the e l e c t r o magnetic phenomena a s s o c i a t e d with t h i s boundary. The i n d u c t i o n magnetometer cove r s the e n t i r e frequency band o f the m i c r o p u l s a t i o n s i g n a l . For the f i e l d o b s e r v a t i o n , a f e r r o m a g n e t i c i r o n core c o i l i s a p r a c t i c a l c h o i c e of sensor because of i t s p h y s i c a l c o n s t a n t s . But, c o n t r a r y to t h e o r i e s about the l a r g e open loop c o i l , or a i r core c o i l , the r o l e of the f e r r o m a g n e t i c c o r e i s not f r e e from s p e c u l a t i o n s . Three geomagnetic m i c r o p u l s a t i o n d e t e c t i o n / r e c o r d i n g systems, one u s i n g an a i r core c o i l sensor and the other two permalloy core c o i l s e n s o r s , have been run s i m u l t a n e o u s l y s i n c e September 20, 1973. The a n a l y s i s c a r r i e d on the data of October 10, 1973 r e v e a l s t h a t the permalloy core c o i l sensors produce no s i g n i f i c a n t d i s t o r t i o n i n the output s i g n a l s compared to the a i r core c o i l system. The permalloy core c o i l sensors were designed to be p o r t a b l e whereas the a i r core c o i l was designed f o r use a t a permanent s t a t i o n . The r a t i o of two s i g n a l s , or more s p e c i f i c a l l y , a peak-to-peak r e a d i n g by each of the permalloy core c o i l systems d i v i d e d by the c o r r e s p o n d i n g value of the a i r core c o i l system, was found to f a l l i n the v i c i n i t y c f the value 1. The p l o t s are more s c a t t e r e d as the S/N becomes poorer, but i t i s s t i l l w i t h i n the p r e d i c t e d l i m i t s based on an e s t i m a t i o n of n o i s e . The power spectrum was estimated f o r each of the 50 t h r e e by Blackman-Tukey method as vwell as the maximum l i k e l i h o o d method. No s i g n i f i c a n t d i f f e r e n c e was r e c o g n i z e d among these s p e c t r a . The e x p l a n a t i o n f o r these r e s u l t s may be t h a t the n o n l i n e a r response i n h e r e n t t o any f e r r o m a g n e t i c m a t e r i a l i s suppressed by the demagnetization e f f e c t . The theory of the demagnetization e f f e c t a l s o suggests t h a t a f e r r o m a g n e t i c core c o i l i s more s e n s i t i v e i f i t i s l o n g e r . As the core becomes l o n g e r , p e r m e a b i l i t y of the core m a t e r i a l i s r e q u i r e d t c be higher to suppress the n o n l i n e a r response. Although, the emphasis i s p l a c e d on the demagnetization e f f e c t , other e f f e c t s of ferromagnetism should be taken i n t o account f o r the i n t e r p r e t a t i o n of the a c t u a l core response. The i n c r e m e n t a l p e r m e a b i l i t y , f o r example, seems to e x p l a i n c o n s i d e r a b l e r e d u c t i o n of the e f f e c t i v e p e r m e a b i l i t y of the core (see appendix 2 ) . A l s o , f u r t h e r a n a l y s e s should be done on the r e c o r d s of b e t t e r q u a l i t y , e s p e c i a l l y b e t t e r S/N. 51 APPENDICES 1. MAXIMUM LIKELIHOOD POWER ESTIMATOR The Blackman-Tukey method of power e s t i m a t i o n i s a procedure which i n f e r s from a f i n i t e l e n g t h of data a b e t t e r approximation of the a u t o c o r r e l a t i o n c a l c u l a t e d f o r an i n f i n i t e d ata. The weight f u n c t i o n , which i s a p p l i e d to the a u t o c o r r e l a t i o n , i s r a t h e r a r t i f i c i a l . The maximum l i k e l i h o o d i s based on a d i f f e r e n t concept. The power e s t i m a t i o n process i s designed to pass the s i g n a l through but to at t e n u a t e and minimize the r e s i d u a l power a t the output. Because t h i s m inimizing, or o p t i m i z a t i o n o f the pr o c e s s , i s done f o r i n d i v i d u a l power e s t i m a t o r , i t i s regarded as a data a d a p t i v e method. An e l e c t r i c a l analog of t h i s process i s a f i l t e r . Based on the above concept, t h i s f i l t e r i s designed as f o l l o w s . For an in p u t which c o n s i s t s o f s i g n a l and zero mean noise produced by a random pro c e s s . x^M c^ AM - f t * ,1-1, where k i s a time index and ̂  i s the sampling i n t e r v a l . Output of t h i s f i l t e r r e t a i n s the s i g n a l . With A - 1 , t h i s i s reduced to VU=\ (A- 3) In matrix n o t a t i o n , column v e c t o r s are d e f i n e d as f o l l o w s . CL= CJ&C/-Aa./ (A-4) 52 m - a t a . e ' * * , y * * ™ - ) ( s. 5 ) A - 3 becomes I = E* T ^ ( A - 6 ) The o t h e r c o n d i t i o n i s t h a t the v a r i a n c e of the noise i s to be minimized. The v a r i a n c e of output, y, i s <r*=<ty±-<^>f> ( A - 7 ) (A-9) > ( A - 1 0 ) ( A - 1 1 ) ( A - 1 2 ) ( A - 1 3 ) The c o r r e l a t i o n i s d e f i n e d as A - 1 0 becomes where R i s N by N c o r r e l a t i o n matrix. A - 1 3 i s to be minimized under the c o n s t r a i n t A - U . Taking Lagrange's m u l t i p l i e r method, we f i r s t r e w r i t e A - U . f-B \ -I --=o < * - i 6 ) Then, with =• CK^ -f JOijl y Sfl ( A - 1 7 ) 53 where & i s an undetermined m u l t i p l i e r . In matrix n o t a t i o n , = (A-19) The f i l t e r c o e f f i c i e n t v e c t o r becomes T h i s i s s u b s t i t u t e d i n t o a complementary equation to A-4, (A-21) which d e s c r i b e s the n e g a t i v e frequency r e g i o n . So t h a t , I = R~' E* '(»:»> E l i m i n a t i n g c9 with A-17, we get (A-23) The power e s t i m a t o r by the maximum l i k e l i h o o d method i s F i n a l l y , ^ (A-25) 2. INCREMENTAL PERMEABILITY The o r d i n a r y h y s t e r e s i s curve d e s c r i b e s the m o n o t o n i c a l l y i n c r e a s i n g or d e c r e a s i s n g m a g n e t i z a t i o n , or EC magn e t i z a t i o n between the s a t u r a t i o n l i m i t s . The h y s t e r e s i s response a l s o o ccurs when magnetic f i e l d c o n s i s t s of an a l t e r n a t i n g ' f i e l d and a r e l a t i v e l y l a r g e b i a s f i e l d . Bozorth (1951) reviewd t h i s phenomena. In the presense of a r e l a t i v e l y l a r g e b i a s f i e l d , the 54 a l t e r n a t i n g f i e l d produces magnetic induct ion which fe l lows minor hys te r s i s loops bounded to the major loop at ene cf i t s tips. In most cases the slope of each branch of these sublcops i s l e s s steep than the main loop. These are shown in F i g . A - 1 . •4 12 2 1 0 «0 « r ffi 6 0 ^ 2 (a) A H „v SLOPE, /IL-LB/c\H B ^ , ®% Is IRON u- 10 3XI6 io < 8 O Z ID t IO 3 X 6 *» 5 IU 10 10 5 « z O 3 (c) 4 - 7 9 MO PERMALLOY 'p.r*\Z,000 __| // s / x / • Is 6 0 IO3 X 16 <fl 10 3 < z ID -0.04 0 0.04 0-G3 0.12 0.18 0.20 H IN OERSTEDS (b) i 1 i • ^ S L O P E - A X 1 -Bn - B 0 " B 6 r p /l /l IRON (d) • ML Ba 1260 7000 T- 725 7600 *- 200 7850 SO 6 000 IRON 2 3 * s e H IN OE.RSTEDS F i g . A - 1 . Minor hys tere s i s loops taken under various condi t ions i n i r o n (a,b,d) and 4-79 Permalloy (c) . (from Eozorth 1951) B r i e f l y , the permeabi l i ty for the e x i t a t i o n of a v a r i a t i o n a l magnetic f i e l d becomes much l e s s than the one traced fcy CC magnetization. The incremental p s r m e a b i l i t y , jJJ , i s defined as the average of each branch. The r e v e r s i b l e permeabi l i ty , Jj,y , i s the asymptotic value when the amplitude cf the a l t e r n a t i n g 55 f i e l d , , i s increa sed , the incremental permeabi l i ty i s also increased i n a l i n e a r r e l a t i o n such as J^t~ J^f* til'  Hn̂ re F i g . A - 2 shows that 45 Permalloy has the i n i t i a l permeabi l i ty of 2.4x 10 3 , I t s nominal permeabi l i ty by CC magnetization i s about 5x 103 for a f i e l d of 0.5 oers ted . The incremental permeabi l i ty becomes 1x 10 3 at t h i s b ia s . In the case of pure i r o n , the d i f ference between the incremental permeabi l i ty and the one by DC magnetization appears even more pronounced (see F ig .A-1 (d)). BIASING FIELD STRENGTH, H b , IN OERSTEDS F i g . A - 2 . Incremental and r e v e r s i b l e permeab i l i t i e s of 45 Permalloy measured with various b ias ing f i e l d s and incremental f i e l d s t rength . (from Bozorth 1951) I t was suggested by GansH^IO) that the r e v e r s i b l e permeabi l i ty i s determined by the bias i n d u c t i o n , By • The 56 d e p a r t u r e from t h i s s i m p l i f i c a t i o n i s not l a r g e i n case of some m a t e r i a l s . Fig.A-3 shows the p l o t f o r 45 P e r m a l l o y . o -\ \ \ • *\ ) \\ \\ i 5 • X « 2 Ol A. A s 10 2 4 IS X BIASING INOUCTION, Bo, IN GAUSSES Fig.A-2. R e v e r s i b l e p e r m e a b i l i t y vs i n d u c t i o n f o r 45 P e r m a l l o y , (form Bozorth 1951) But, g e n a r a l l y , t h e r e are wide d i f f e r e n c e s i n the r e v e r s i b l e p e r m e a b i l i t y c u r v e s f o r d i f f e r e n t m a t e r i a l s . Four d i f f e r e n t c h a r a c t e r i s t i c s are shown i n Fig.A-4. The r e v e r s i b l e p e r m e a b i l i t y i s a measure of the f i r m n e s s with which the magnetic domains are h e l d i n p o s i t i o n by the b i a s i n g f i e l d . The magnetic domain i s d i s c u s s e d fcy models which depend on the c o m p o s i t i o n of the m a t e r i a l . These models a r e : i ) the magnetic domain has no p r i n c i p a l d i r e c t i o n c f easy m a g n e t i z a t i o n , i i ) the d i r e c t i o n of the domains are r e s t r i c t e d so t h a t they always l i e p a r a l l e l or a n t i p a r a l l e l to the f i e l d , 57 (a) MOLYBDENUM PERMALLOY -Ac , -f ' r \ a, H = 0 10 I21103 O IS 2 0 2 * X I 0 » (d) 65 PERMALLOY, HARD f ^ k H = 0 BIASING INDUCTION, Bfc,, IN CAUSSES 10 12«K>> Fig„A-4. R e v e r s i b l e p e r m e a b i l i t y as dependent on b i a s i n g i n d u c t i o n i n (a) Molybdenum Permalloy, (b) Vanadium Permendur, (c) Permalloy c o n t a i n i n g 74% n i c k e l and (d) hard Permalloy c o n t a i n i n g 65% n i c k e l . (from Eozorth 1951) Fig.A-5. Reduced r e v e r s i b l e p e r m e a b i l i t y vs reduced b i a s i n g i n d u c t i o n f o r s e v e r a l m a t e r i a l s , i n comparison with the t h e o r y : (a) Gans' r e l a t i o n , (b) i s o t r o p i c domains, (c) a n i s o t r o p i c domains (Drown). Broken curve shows the approximate change i n d u / d l l ( f o r weak f i e l d s ) as dependent on b i a s i n g i n d u c t i o n , observed f o r a v a r i e t y of m a t e r i a l s . (from Bozorth 1951) 58 i i i ) they l i e parallel or anti parallel to a single crystallographic direction in crystals that are oriented at random in the material. The third model i s designed for cobalt by Brown(1938). With these assumptions, the differences in the reversible permeability of different materials -are further coincided by plotting jULy/jJL0 against Bj>/ B5 , where y,0 i s the i n i t i a l permeability and B$\ i s the induction at saturation, in Fig.A-5. An interesting case of superposition of f i e l d occurs when a weak ^alternating f i e l d i s applied at right angles to a constant f i e l d . Fig.A-6 shows that the reversible permeability of Molybdenum Permalloy i s larger than the ordinary one except at the end points where these values must coincide. e j 6 ffi < •Sf * a UJ UJ _J a s 3 UJ > • UJ _ OC 2 t O O I 2 3 4 5 e 7 8 9 x PRINCIPAL COMPONENT CP INDUCTION, B b , IN G A U S S E S **-JL T O B b v i i T O e b \ \ \ \ \ \ Fig.A-6. E f f e c t of b i a s i n g i n d u c t i o n on r e v e r s i b l e p e r m e a b i l i t y when i t i s measured p a r a l l e l or p e r p e n d i c u l a r to the d i r e c t i o n of uieasurement. (from Eozorth 1951) 59 REFERENCES Bozorth, R. H. (1951) . £-112 ma^ ne t i sra, New York, Van Nostrand Co. , pp. 968. Brown, W. F. (1 938). Domain theory of f e r r o m a g n e t i c s under s t r e s s I I I . R e v e r s i b l e s u s c e p t i b i l i t y , P h y s i c a l Rayisw, 54, 279-299. Campbell, W. H. (1967). I n d u c t i o n loop antennas f o r geomagnetic f i e l d v a r i a t i o n measurement, ESSA T e c h n i c a l Report, ERL 123-ESL 6. Caner, B. ( 1970) . Design c o n s i d e r a t i o n s , i n d u c t i o n c o i l s f o r m i c r o p u l s a t i o n measurement at V i c t o r i a Magnetic Observatory, Unpublished r e p o r t , V i c t o r i a G e o p h y s i c a l Observatory, R. R. 7 V i c t o r i a , B. C. Gans, R. (1910) . Magnetic corresponding s t a t e s , F h v s k a l i s c h Ze i t s c h r i f t. 11 , 988-99 1. < Geary, R. C. (1930). The frequency d i s t r i b u t i o n of the q u o t i e n t of two normal v a r i a b l e s , J o u r n a l of Royal S t a t i s t i c s S o c i e t y , 93, 442-444. J e n k i n s , G. M. and D. G. Watts, (1 968). S p e c t r a l Ana l y s i s and I t s A ogl i c a t i o n t 5 an F r a n c i s c o , Holden Day Inc., pp. 525. K o l l a r , F. and R. D. R u s s e l l , (1966). Seismometer a n a l y s i s using an e l e c t r i c c u r r e n t a n a l o g . B u l l e t i n of the §i.i§!S2l23.i£S.£ S o c i e t y of America, 56, 1 193-1205. La c o s s , R. T. (197 1). Data a d a p t i v e s p e c t r a l a n a l y s i s methods, £§2Eiiy.§i£§» 3 6 , 66 1-675. Osborn, J . A. (1945). Demagnetizing f a c t o r s . of the g e n e r a l 60 e l l i p s o i d , P h y s i c a l Review, 67, 351-357. Peterson, E. (1928). Harmonic p r o d u c t i o n i n f e r r o m a g n e t i c m a t e r i a l s a t low frequency and low f l u x d e n s i t i e s , B e l l System T e c h n i c a l J o u r n a l , 7, 762-796. S t r a t t o n , J . A. (1941). I lectrcmacjne t i c The or j , New York and London, McGraw-Hill I n c . , pp.615. i

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
United States 10 0
Japan 2 0
France 2 0
China 1 10
City Views Downloads
Mountain View 4 0
Ashburn 4 0
Tokyo 2 0
Unknown 2 0
Matawan 1 0
Beijing 1 5
Boulder 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}

Share

Share to:

Comment

Related Items