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Experimental investigations of a recent fluxgate theory Carter, Matthew 1988

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EXPERIMENTAL INVESTIGATIONS OF A RECENT FLUX6ATE THEORY by MATTHEW CARTER B.Sc, Western Washington University, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Geophysics and Astronomy We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1988 © Matthew Carter, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) ii ABSTRACT A recent theory describes the fluxgate magnetometer as a modulated inductor. In that theory, hysteresis and demagnetization are implicitly incorporated in the sense-coil inductance, an easily measured quantity. In this thesis, the principle equation of that theory is experimentally tested. Expressions relating the open-circuit and short-circuit output from a fluxgate magnetometer to the magnetic field are derived from the principle equation. In order to test the proposed relations, the writer assembled a ring-core fluxgate. a computer-controlled current source to drive the fluxgate, and circuits required to monitor the open-circuit and short-circuit output signals, initial tests showed that the integrated open-circuit output voltage from the fluxgate is proportional to the magnetic field. The constant of proportionality is simply the product of the length-to-tums ratio of the sense-coil and the maximum change in the sense-coil inductance caused the drive current. This result was correctly predicted by the aforementioned fluxgate equation. Test results from the short-circuit experiment were also correctly predicted by the fluxgate equation. Therefore, experimental data is provided that supports the validity of the fluxgate equation. The same fluxgate theory is used to predict specific values of drive current parameters that maximize the fluxgate output signal. The computer-controlled current source was used to generate a bipolar square-pulse waveform with an adjustable amplitude, frequency, and duty cycle. A sinusoidal drive waveform was also used. Experimental data confirm the validity of all the predicted relations, and thus, provide substantial support for theoretical work that has been recently published. As a final application, the fluxgate theory was used to quantify the behavior of a ring-core fluxgate immersed in a magnetic fluid. A fluxgate was put in magnetic fluid in an attempt to discover if the fluxgate responds primarily to the ambient flux density, and consequently, to determine whether the output signal could be enhanced by simply placing the sensor in a container fi l led with magnetic fluid, The experiment was terminated when inductance measurements taken on the immersed sensor showed that stray flux from the toroidal drive-coil significantly altered the permeability of the magnetic fluid, and thus altered the calculated values of flux density in the magnetic fluid. i i i TABLE OF CONTENTS Abstract ii Table of Contents i i i List of Tables v List of Figures vi Acknowledgement v i i i 1 Introductory Material 1 1.1 Introduction to the Thesis 1 1.2 Equation of State for the Fluxgate Magnetometer 3 2 Relations Predicted by the Equation of State 9 2.1 Integrated Open-Circuit Voltage 10 2.2 Maximizing the Integrated Open-Circuit Voltage 12 2.2.1 Using a Regularly Spaced, Alternating Square-Pulse Drive Waveform 13 2.2.2 Using a Sinusoidal Drive Waveform 16 2.3 Short-Circuit Current 18 2.4 Summary of Relations Presented in Chapter 2 20 3 Equipment Used in the Experimental Work 22 3.1 The Drive-Current Circuitry 22 3.1.1 The Computer-Controlled Current Source 23 3.1.2 The LCR Drive-Current Circuit 28 3.2 The Fluxgate Sensor 30 3.2.1 Sensor Materials 30 3.2.2 Sensor Construction 31 3.2.3 Sensor Calibration 32 3.3 The Output Circuitry 38 3.3.1 Voltage Follower and Integrator 38 3.3.2 Calibration of the Follower and Integrator 41 3.3.3 Short-Circuit Apparatus 42 3.3.4 Calibration of the Short-circuit Apparatus 44 iv 4 Experiment 1: Tests of the Relations Predicted by the Equation of State 4 9 41 Tests Involving the Integrated Open-Circuit Voltage 50 42 Tests Involving the Short-Circuit Current 57 43 Summary of the Experimental Results 63 5 Motivations for Immersing a Fluxgate Sensor in a Magnetic Fluid 66 5.1 B and H in a Body of Magnetizable Material 68 5.2 Magnetic Fluid 70 6 A Fluxgate Sensor Immersed in a Magnetic Fluid 74 6.1 The IGCMagfluid 74 6.2 Inductance of a Fluxgate Immersed in Magf luid 79 6.3 Summary and Conclusions for the Magf luid Experiment 84 7 Summary and Conclusions 86 BIBLIOGRAPHY 89 V LIST OF TABLES Table 3.1 Percentage of second harmonic in the SDK-85 signal 27 Table 3.2 Calibration data for the short-circuit apparatus , 48 Table 4.1 Integrated open-circuit output voltage 51 Table 4.2 First harmonic of the integrated open-circuit voltage 54 Table 6.1 Susceptibility of the IGC Magfluid 77 _____ " ' v i . -, .. ' V . LIST OF FIGURES Chapter 1 FI6. 1.1 Equivalent circuit for the fluxgate magnetometer 4 Chapter 2 Fig. 2.1 Relation between the drive signal, sense-coil inductance, and the output signal 14 Fig. 2.2 Square pulse drive signal and the inductance waveform 15 Fig. 2.3 Sinusoidal drive signal and the inductance waveform 17 Chapter 3 Fig. 3.1 Schematic diagram of the SDK-85 signal generator 24 Fig. 3.2 Circuit used to synthesize the SDK-85 signal into a drive current waveform 26 Fig. 3.3 The LCR drive circuit 29 Fig. 3.4 Fluxgate ring-core and the sense-coil former 31 Fig. 3.5 Output current as a function of load resistance 33 Fig. 3.6 Output current for a smaller range of load resistance 34 Fig. 3.7 Circuit used to calibrate sensor MC-1 35 Fig. 3.8 Response of sensor MC-1 to magnetic fields 36 Fig. 3.9 Sense-coll inductance as a function of drive amplitude 37 Fig. 3.10 Circuit used to obtain the open-circuit output voltage 39 Fig. 3.11 Short-circuit apparatus and calibration circuit 43 vii Chapter 4 Fig. 4.1 Integrated open-circuit voltage and its relation to the SDK-85 and LCR drive signals 53 Fig. 4.2 Parabolic curves fitted to the data in Table 4.2 55 Fig. 4.3 Integrated open-circuit output as a function sinusoidal drive current amplitude 56 Fig. 4.4 Short-circuit current and its relation to the SDK-85 signal.... 59 Fig. 4.5 Semi-log plots of the short-circuit data 61 Fig. 4.6 Short-circuit current as a function of drive frequency 63 Chapter 6 Fig. 6.1 Sense-coil inductance of MC-1 in air and in Magfluid 81 viii ACKNOWLEDGEMENT I would like to express sincere thanks to Professor R. D. Russell for the support and encouragement that he provided during the course of my endeavors at the University of British Columbia. A large portion of the work presented herein drew inspiration, both directly and indirectly, from Russell and his colleagues. Theoretical relations derived in Sections 1.2 and 2.1 were first published in 1983 by R. D. Russell, B. Barry Narod. and Frank Kollar. The analysis presented in Section 2.2.1 was suggested to me by Professor Russell, as were the relations presented in Section 2.3. A substantial portion of the thesis is devoted to describing the construction and calibration of instrumentation that was used by the writer to investigate the fluxgate theory of Russell, Narod, and Kollar. Although the instrumentation work was done entirely by the writer, initial suggestions as to the type of instrumentation to construct came primarily from Russell. Barry Narod also made his expertise on fluxgate magnetometry avilable to me, and I am very grateful for his suggestions. Walter Urbanski of the intermagnetics General Corporation is thanked for donating the magnetic fluid that was used in my research. I would also like to thank Bi l l Siep, Dieter Schreiber, Harry Verwoerd, and Peter Michalow for their assistance and friendship during the past two years. This work was financed by the NSERC A-0720 grant held by R. D. Russell. 1 CHAPTER 1: introductory Material SECTION 1.1: Introduction to the Thesis In 1983, Russell, Narod, and Kollar presented a theoretical study o f the transient behavior of a capacitively loaded fluxgate magnetometer. In that publication, the authors introduced a new and simpler equation o f state for the fluxgate. The equation of state is expressed in terms o f easily measured sensor parameters, and suggests that the fluxgate can be described as a modulated inductor. A synopsis of the theoretical foundations upon which Russell et al. based that equation is presented in Section 1.2. Narod and Russell (1984) used that equation as the starting point in a theoretical study of the steady-state behavior of a capacitively loaded fluxgate. Again, starting with the same equation of state, Gao and Russell (1987a,b) extended the theory to Include the fluxgate gradiometer. These four publications form a sound theoretical basis for the quantitative study of fluxgate magnetometers. However, they provide no direct experimental evidence to support the equation of state on which they are based. Therefore, as a thesis project, the writer conducted a suite o f tests designed to examine the validity of the aforementioned equation of state. As a consequence of the results obtained from theoretical and experimental endeavors, the writer also examined the possibility of using a ferromagnetic fluid to concentrate magnetic flux in the immediate vicinity of the fluxgate sensor, thereby effectively increasing the sensitivity of the magnetometer. The experimental work was completed in two phases. The f irst phase was carried out during the Summer and Fall of 1987. This phase, 2 henceforth referred to as Experiment I, tested the validity of the fluxgate equation of Russell. Narod, and Kollar by examining relations (between sensor parameters, a magnetic field, and fluxgate input/output signals) that can be predicted from that equation. The predicted relations are derived and presented in Chapter 2. In order to examine these relations, the writer assembled the necessary instrumentation which included the following: the drive circuitry required to operate the fluxgate, the fluxgate sensor, and the circuitry used to obtain the desired forms of fluxgate output. Details on the materials used, methods of construction, and instrument calibrations are presented Chapter 3. Chapter 4 contains an outline of the experimental procedure followed in Experiment 1, as well as a presentation and discussion of the results. The second phase of the experimental work was originally conceived by the writer as a means of determining whether the aforementioned fluxgate equation should be cast in terms of the intensity of the magnetic field (H), as proposed by Russell et al. (1983), or, in terms of the magnetic induction (B). In other words, an attempt was made to answer the question, "Does a fluxgate magnetometer respond primarily to B, or to H?". Theoretical considerations presented in Chapter 5 show that the magnitude of an ambient B-field increases, and the H-field decreases, as one moves from the outside to the inside of a body made of magnetizable material. Thus, it was expected that the question posed above could be answered conclusively by observing the change in fluxgate output created by placing the fluxgate sensor in a container filled with a magnetic fluid. Consequently, it occurred to the writer that a simple and 3 Inexpensive flux concentrator could be designed from a container of magnetic fluid, if the sensor responds primarily to B (and if the sensor is able to function properly while it is immersed in such a fluid). During the Summer of 1988, the writer tested these hypotheses. Chapter 6 contains an outline of the procedure followed, as well as a presentation and dicussion of the results. There are three principle topics presented in this thesis. The first is a synopsis of the theoretical foundations upon which R. D. Russell, B. B. Narod, and F. Kollar based a quantitative description of the fluxgate mechanism. The second ia a presentation of the results of experimental tests designed to determine the validity of that description. The third is documentation of the behavior of a ring-core fluxgate that was immersed in a highly magnetic medium, thus answering some basic questions regarding the feasibility of constructing a flux concentrator from a container filled with magnetic fluid. SECTION 1.2: Equation of State for the Fluxgate Magnetometer Russell, Narod, and Kollar (1983) proposed a linear differential equation of state for the fluxgate magnetometer. The physical basis for that equation is the essential mechanism of the fluxgate — the modulation of magnetic flux, and the resulting changes in sense-coil inductance. Gao (1985, Ch. 1) presented a case for the historical significance of this theory. Introducing their theory, Russell et al. (1983) write: "We have chosen to base our theory on a variational principle, expressed in the form of Lagrange's equations.". The Lagrangian equations are used to describe the dynamical properties of a system. They are a result of the 'principle 4 of least action': the unifying principle behind the equations of motion for any dynamical system (Lanczos,1962; Feynman et al., 1966). As explained Immediately below. Russell et al. use Lagrange's equations to derive an appropriate equation for the fluxgate system, expressed In terms of the motion of electrical charge (current) in the sense-coil of a ring-core fluxgate magnetometer. AAA Figure 1.1: Equivalent circuit for a capacitively loaded fluxgate sensor that is immersed in a magnetic field H. L(t) is the sense-coil inductance and 1/N is its effective length-to-turns ratio. R is the combined resistance of the sense-coil and the load. C is the capacitive load. Mutual inductance between the drive and sense coils has been assumed negligible in this diagram. After Narod and Russell (1984). Narod and Russell (1984) illustrate the application of Lagrange's equations to the fluxgate magnetometer by presenting an equivalent circuit for a fluxgate sensor that is loaded with a resistor and capacitor in series (Figure 1.1), and then applying Lagrange's equations to that system. The first step in applying Lagrange's equations to a physical system is to choose so-called 'generalized coordinates' — a generalization 5 of the original coordinate concept of Descartes. As is often done in texts dealing with the application of Lagrange's equations to electric circuits (Karman and Blot, 1940; Page and Adams, 1940; Margenau and Murphy, 1943; Jeans, 1948), Russell et al. (1983) chose q, the charge on the capacitor, to be the generalized coordinate. The circuit shown in Figure 1.1 represent the dynamical fluxgate system with one degree of freedom, and thus, Lagrange's equations are reduced to a single equation in a single variable. For the circuits shown in Figure 1.1, the appropriate equation is [1.1] -d-dt d [T - V] aq _ a [ T - v ] . _iM + e dq dq The first time derivative of the generalized coordinate (q) is equal to the current flowing through the sense-coil of the fluxgate (1). The symbols T, V, and D represent the kinetic, potential, and dissipatlve energies in the system, respectively, and e is an optional back EMF from the load. The next step is to determine expressions for T, V, and D, and substitute them into [1.1]. The dissipative term (D) comes from the resistance in the circuit and is given by [1.2] D = J-Ri 2 2 This term represents one-half the energy that is transformed into heat per unit time (Karman and Blot, 1940). The potential energy of a system (V) is always a function of the generalized coordinate, (Margenau and Murphy, 1943; Jeans, 1948). For the circuit shown in Figure 1.1, 6 This term represents the electrostatic energy of the system (Karman and Blot, 1940). The kinetic energy of a system (T) is a function of the first derivative of the generalized coordinate (Margenau and Murphy, 1943; Jeans, 1948). Russell et al. (1983) present three terms for the kinetic energy of the fluxgate system, so that [1.4] T = l L 1 2 + M11d + l H L i 2 N The first of these terms represents the energy stored in the magnetic field generated by current (i) flowing in the fluxgate's sense-coil. The second term in [1.4] represents the energy associated with the interaction between leakage flux from the drive current (id) and the field generated by the sense-coil. Thus, M is the mutual Inductance between the drive-coil and the sense-coil. The third term in [1.4] represents the energy associated with the interaction of an ambient magnetic field (H) and the field generated by 1. For this term, Russell et al. (1983) made use of the fact that the quantity 1/N is precisely equal to that current, which when flowing through an ideal solenoid of infinite length, will produce unit magnetic field in it. Hence, multiplying this quantity by 1 gives an expression that represents the 7 magnetic field generated by such a coil. The sense-coil Inductance (L) in the third term implicitly incorporates shape' factors which must be included to account for the fact that the sense-coil is not an ideal solenoid of infinite length. This attribute of the inductance alleviates the need for detailed descriptions of demagnetization factors and hysteresis associated with the fluxgate. In fact, there are a total of six terms in the kinetic energy budget of the fluxgate system, three of which are shown in 11.41. There are three terms associated exclusively with the sources of magnetic field in the system: the ambient field, the field generated by the sense-coil current, and the leakage flux from the toroidal drive-coil; and there are three terms that arise from the interaction of those fields. However, of these six terms, only the three included in [1.4] contain the generalized coordinate (q) or its first derivative (i), and thus are the only ones that will contribute when substituted Into equation [1.1], Substituting [ 1.2], [ 1.3], and [ 1.4] into [1.1] yields [1.5] J L [ L i M * i R = e - l f [HL]-£[Mi d] dt C N dt dt This is the equation of state for the fluxgate magnetometer as introduced by Russell et al. (1983), and it is the starting point for theoretical dicussions subsequently published by Narod and Russell (1984), and Gao and Russell (1987a, b). The mutual inductance between the drive-coil and sense-coil (M in [1.5]) is usually taken to be zero, not as a theoretical necessity, but as a physical characteristic of well designed sensors. It 8 can be seen that equation [1.5] is expressed in terms of easily measured sensor parameters, and thus represents a powerful tool for quantitative studies of the fluxgate magnetometer. Also, it should be noted that even though the analysis presented above used a ring-core fluxgate as a model, the results are expected to be valid for any fluxgate in which the inductance is well-defined and periodic. The validity of equation [1.5] has been thoroughly tested in the laboratory by the present writer, and the results of those tests are presented later in this thesis. 9 CHAPTER 2: Relations Predicted by the Equation of State The purpose of this chapter is to present theoretical relations between an ambient magnetic field, physical parameters of a fluxgate magnetometer, and electrical Input/output signals associated with the sensor. In particular, expressions relating the "open-circuit" and "short-circuit" fluxgate output to the magnitude of the ambient magnetic field and parameters of the drive current will be derived. The open-circuit and short-circuit expressions will be derived from the fluxgate equation presented in Section 1.2 (equation [1.5]) that is based on an application of Lagrange's equations. The relations derived and described below form the theoretical structure upon which Experiment 1 is founded. In the context of the experiments presented In this thesis, "fluxgate output" simply refers to the electrical signal (voltage and/or current) available at the terminals of the sense-coil when the magnetometer is operating in the presence of a magnetic field. The nature of that signal is a function of the load circuitry to which the terminals are connected. The simplest form of load is one containing no reactive components. For the case of a purely resistive load, the electrical signal measured at any point in the load circuit is proportional to the signal appearing at the terminals of the sense-coil. Hence, the physics of the system is simplified, as are the equations that describe that physics. It is for this reason that theoretical work presented below deals with a resistlvely loaded fluxgate magnetometer. 10 SECTION 2.1: Integrated Open-Circuit Output Voltage In Section 1.2, Russell, Narod, and Kollar's equation of state for the fluxgate magnetometer (equation [1.5]) was derived by applying Lagrange's equation to the energy budget of the fluxgate system. That equation is linear in i(t), the current flowing through the sense-coll at time t. Therefore, the fluxgate can be described as a linear system. The fluxgate output is completely specified by a voltage/current pair (output = {v,i}). Since the fluxgate is a linear system, its behavior for any arbitrary load is calculable, if the output is completely specified for two different loading conditions. The two loading conditions that are expected to lead to the conceptually simplest models (and those involving the simplest mathematics), are the cases for which i a 0 (output = {v,0}), and the one for which v a o (output = (0,0). The first case is considered below by placing the sense-coil in an "open-circuit" configuration. For the case of a purely resistive load and no mutual Inductance, [1.5] reduces to The open-circuit output voltage Is equal to that voltage which, when applied to the terminals of the sense-coil, results in a flow of zero current from the terminals. Thus, with the sense-coil in an open-circuit configuration, i is equal to zero, and [2.1] becomes: [2.1] e = Li + dt L [2.2] dt L N J 11 The left-hand-side of [2.2] symbolizes the open-circuit voltage available at the terminals of the sense-coil. If the sensor is aligned with Its axis parallel to the ambient magnetic field (1 II H), and if the field is constant over the period of time in which it is measured, then [2.2] can be rewritten as: [2.3] e 0 C = lH<*L N dt Therefore, the open-circuit voltage is proportional to the magnitude of the ambient magnetic field (H), and to the time rate of change in the sense-coil inductance (dL/dt). The self-inductance of the sense-coil (L) is a function of the coil's geometry and proportional to the effective permeability of the material enclosed by the coil. The permeability is periodically affected by alternating current flowing through the drive coil (the drive current), thus creating a non-trivial dL/dt. Therefore, the open-circuit voltage is affected by the drive current waveform. Since L is proportional to the effective permeability of the material enclosed by the sense-coil, extremes in L occur when the drive current amplitude is zero, and when it is above the core saturation level (see Figure 2.1a and b, page 13). Let tj and t 2 represent instants in time just before and just after a transition between the two states of inductance. Integrating [2.3] from tj to t 2 yields: 12 [2.4] .N dtJ N [2.5] e o c dt = J-H AL N Equation [2.5] relates a particular form of fluxgate output (the integrated open-circuit voltage) to the magnitude of the ambient magnetic field, and to parameters of the fluxgate sensor that can be easily measured in the laboratory. Specifically, the integral of the open-circuit output voltage is predicted to be equal to the product of the magnetic field strength, the length-to-turns ratio of the sense-coil, and the sense-coil inductance. This equation was first presented by Russell et al. (1983) as their equation (6). SECTION 2.2: Maximizing the Integrated Open-Circuit Voltage 6ao and Russell (1987b) define the sensitivity of a fluxgate sensor as the ratio of the amplitude of the fundamental harmonic of the sensor's output signal to the magnitude of the ambient magnetic field. Accepting such a definition and using the integrated open-circuit voltage ((e o cdt) as sensor output, it becomes important to optimize the fundamental harmonic of Jencdt. Since Je^dt is proportional to the sense-coil Inductance, the challenge reduces to considering means by which the fundamental harmonic of the time-dependent Inductance can be maximized. The inductance waveform is determined by the current waveform used to drive 13 the fluxgate. It follows that the fundamental harmonic of the inductance, and thus the sensor's output, can be optimized by adjusting parameters of the drive current. Two distinct types of drive current waveforms are considered below. The first type is that used for the majority of the experimental work presented in this thesis. It consists of a series of regularly spaced pulses of alternating sign. The second type considered is a sinusoid, one commonly used to in the fluxgate literature. SECTION 2.2.1: Using a Regularly Spaced. Alternating Square Pulse Drive  Waveform Figure 2.1a displays one period of a practical drive current waveform. It consists of bipolar square pulses, each of which has a finite rise and fall time, as well as an amplitude greater than that required to saturate the core of high permeable material inside the sense-coil. This current drives the core into a state of magnetic saturation during each rise time, and brings it out of saturation during each fall time. Figure 2.1b represents the time-dependent permeability of the material enclosed by the sense-coil. Since the inductance of the sense-coil is proportional to the permeability of the material it encloses, Figure 2.1b also represents the waveform followed by the sense-coil inductance. The amount of magnetic flux linking the sense-coil is altered whenever the permeability of the material enclosed by the coil is in a state of transition. Thus, as described by Faraday's Law, an electromotive force (equal to the open-circuit voltage: e^) is induced in the sense-coil 14 creating a spiky, bipolar waveform with a frequency equal to twice that of the drive current. Figure 2.1 c represents an idealized open-circuit voltage waveform for the fluxgate magnetometer. a JSaturation Time^ Drive Current Permeability and Inductance i Open-Circuit Voltage J Integrated Open Circuit Voltage 1 \ [Saturation, \ L e v e ' * ] ! Time,. I Time Timet Figure 2.1: a) Drive current waveform with peak amplitude larger than that required to saturate the ring-core, b) Variations In the ring-core permeability, and thus the sense-coil inductance, that are caused by the drive current, c) Idealized open-circuit voltage waveform induced in the sense-coil, d) Waveform that results from the integration of b, To integrate the open-circuit voltage, the appropriate circuitry is attached to the terminals of the sense-coll. An Ideal Integrator of e o c produces output equal to the area under the pulses of e^. Hence, the Integral of e^ climbs as area under a positive pulse increases, stays constant while the area under e^ is constant, and then drops as "negative 15 area" under a negative pulse increases. Therefore, if the areas enclosed by the positive and negative pulses are equal, the Integral of e^ is expected to be a series of unipolar square pulses with frequency twice that of the drive current (Figure 2.Id). Figure 2.Id also demonstrates the relation between the sense-coll Inductance and the Integrated open-circuit voltage that is expected to exist. Drive Current t Time T / 2 Time T / 2 Figure 2 . 2 : Idealized waveforms of a square pulse drive current and the variable sense-coil inductance L( t) that it creates. The waveform followed by the integrated open-circuit voltage is identical to that of L ( t ) . To simplify matters, assume that the transitions in L are rapid enough to consider the unipolar pulses truly square. Such a waveform is 16 displayed in Figure 2.2. The period is denoted by T, the pulse duration by T, and the duty cycle by T / T . The arbritrary coordinate axes are chosen such that L(t) is an even, periodic function and (excluding a D.C. component) can be represented as: [2.6] L[t]= I ancos2DffiL n= 1 T The a n in [2.6] are amplitude coefficients of the harmonics of L(t) defined by the relation p T / 2 r / « n/2 L(t) cos (2Dffit)j dt = cos [2.7] a n = - ^ -T/2 f Jo 2nrtt T dt [2.8] an = 2^Ls in n ti n * i T Therefore, the fundamental harmonic of L (n=l), and hence of jeo cdt, is expected to be a maximum for a duty cycle (T/T) near 50%. This is consistent with the theoretical estimates presented in Figure 7 of Gao and Russell (1987b). SECTION 2.2.2: Using a Sinusoidal Drive Waveform Figure 2.3a represents a sinusoidal drive current with maximum amplitude (A) greater than the minimum current (S) required to saturate the fluxgate core. During any half-period of the drive current, A is greater than S for a time denoted by L. (time saturated), and less than S for a time denoted by t u s (time unsaturated). It is easily seen that t s and t u s are determined by the magnitude of A. 17 Figure 2.3: a) Sinusoidal drive current with peak amplitude A. The amount of current required to saturate the ring-core is denoted by S. b) The expected waveform of the sense-coil inductance (and the integrated open-circuit voltage, also). During t s, the sense-coil inductance (L) is held at a constant, minimum value. However, L is variable and always greater than its minimum during t u s . An idealized waveform for L(t) (and hence fe^dO that would result from a sinusoidal drive current is shown in Figure 2.3b. A straight-line approximation to the sine wave has been assumed for values of t near a zero crossing. Reasonable deviations from this assumed behavior will not greatly affect the outcome of the analysis. The width of the triangular base (equal to t u s ) is denoted by T, and the period of the waveform (equal to t u s + t s) by T. The fundamental harmonic component of the triangle pulse 18 waveform shown in Figure 2.3b is altered by changing the ratio T / T (the duty cycle). By Fourier analysis it can be shown that the amplitude of the fundamental is maximized for a duty cycle near .46. The final step in this analysis is to establish a relation between the drive current amplitude (A) and the "optimum" duty cycle of the sense-coil Inductance (as it has been defined above). By Inspection of Figure 2.3a, It Is seen that [2.9] S = As1n[(o^-Note that a = 2n/T, and that T B t u s . Thus, [2.9] can be rewritten as [2.10] S = A sin 15. i ] . TJ Therefore, substituting the "optimum" value of .46 for the quantity T / T in [2.10], we arrive at the final conclusion that the fundamental harmonic of the sense-coil inductance (and thus of le^dt) is optimized when the drive current amplitude (A) is approximately 1.01 times greater than the minimum current required to saturate the fluxgate core. This demonstrates a fundamental difficulty associated with using a sinusoidal waveform to drive low noise sensors, which are generally considered to require being driven far Into saturation (Scouten, 1970; Primdah1,1979; Narod et al., 1985). SECTION 2.3: Short-Circuit Current So far, one distinct output voltage/current pair has been considered, namely where the output • [eoc,0). To examine a second case, 19 consider the situation in which i is defined to be the current that flows through the sense-coil when the voltage across the coil's terminals is maintained at zero. This occurs when the sense-coil terminals are shorted together (the "short-circuit" condition). For the short-circuit condition, the voltage across the sense-coil terminals is zero, and the only resistance in the load circuit Is from the the sense-coil itself (Rg) . Thus, assuming the sensor is aligned parallel to the ambient magnetic field (1 II H), equation [1.5] can be rewritten as: Since i and R s represent physical quantities, they must be single-valued (at a point in time) and finite. Thus, the left-hand-side of [2.11 ] must also be f inlte and single-valued. This implies that must be smooth and continuous. Therefore, at a transition in L, it is predicted that: [2.11] JL u +1HL = - iRs dt L N J [2.12] Li + -LHL N [2.13] The (-) subscript implies limiting values approaching the transition from the left (on the time axis), and the (+) subscript implies quantities approaching the transition from the right. Thus, rearranging [2.13]: 20 [2.14] U U - U - = - l H [ U - L - ] N [2.15] 1] = - J- H AL N It should be emphasized that equations [2.13] to [2.15] assume that the transition in L is rapid enough to be considered a step function. This can be well approximated in practice by using a drive current waveform that has rise and fall times that are much shorter than the circuit time constant (L_/R or L+/R, whichever is smaller), as well as an amplitude much greater than that required to saturate the fluxgate core. The system of equations that lead to [2.15] were suggested to me by R. D. Russell. SECTION 2.4: Summary of Relations Presented in Chapter 2 In conclusion, four theoretical relations have been proposed in this chapter. The first two are specific relations between the magnitude of a magnetic field and electrical output from a fluxgate magnetometer. The latter two are relations between the electrical input and output signals associated with the fluxgate. Specifically, the following relations are predicted to exist: [a] [b] A[Li] = - J- H AL N [c] If the fluxgate is driven by a current waveform as shown in Figure 2.1a, the amplitude of the fundamental harmonic of the integrated 21 open-circuit voltage (a^ Is optimized when the duty cycle of the drive current is near 50%; [d] If the fluxgate is driven by a sinusoidal current, is optimized when the amplitude (0-P) of the drive current is set to be approximately equal to the minimum current required to saturate the fluxgate core. The validity of these proposed relations was examined In detail during the course of Experiment 1. Equipment that was designed for that experiment is presented in the next chapter. The experimental procedure and the results of that experiment are presented and discussed in Chapter 4. 22 Chapter 3: Equipment Used in the Experimental Work Theoretical relations between the electrical input/output signals associated with a fluxgate magnetometer and an ambient magnetic field were established in Chapter 2. In order to examine the validity of the proposed relations, a fluxgate system was constructed and specific tests were conducted. The three basic components of the fluxgate system used were the drive circuitry, the fluxgate sensor, and the output circuitry. The writer assembled the equipment used for each of these components. Information on the materials used, methods of construction, and instrument calibration procedures are presented in this chapter. SECTION 3.1: The Drive-Current Circuitry The fluxgate magnetometer requires an input (drive) current to operate. A drive-current source with adjustable frequency, duty cycle, and amplitude was needed to test the theoretical relations proposed in Chapter 2. Acting on a suggestion from R. D. Russell, I constructed a computer controlled current source that used an Intel SDK-85 microprocessor development kit to drive a programmable power supply in a transconductance mode. This turned out to be an excellent current source that provided a waveform that was well-suited to drive the fluxgate and whose parameters were easily adjusted. However, it was desired to extend the scope of the experimental data by testing the fluxgate system with a drive-current frequency that was beyond the capabilities of the computer-controlled current source. To accomplish this, a second drive circuit was assembled. It was an oscillating LCR circuit that operated at a frequency ten times greater than the upper frequency limit of the computer-controlled current source. Details regarding these two drive 23 circuits are presented below. SECTION 3.1.1: The Computer-Controlled Current Source The microcomputer used as a signal generator was an expanded version of the Intel SDK-85 System Design Kit. The SDK-85 is a microcomputer with input/output, programable memory and timer, and CPU interrupt capabilities. The SDK-85 centers around the Intel 8085 microprocessor which serves as the Central Processing Unit (CPU), Clock Generator, and System Controller. The kit was originally assembled by Harry Verwoerd of the U.B.C. Dept. of Geophysics and Astronomy. For details on the System Design Kit and the 8085 microprocessor, see Intel Corporation (1978 and 1979, respectively). The SDK-85 was expanded to facilitate its use as a signal generator in June, 1987. An additional RAM/IO/Timer chip (Intel 8155) with a battery supported memory was connected to the system by Verwoerd so that a program could be deposited in RAM and remain there when the SDK-85 was turned off. The writer made a wiring change on the SDK-85 so that it could be used as a signal generator. The output pin from the timer onboard the 8155 was connected to the "RST 7.5" interupt input pin on the 8085. The RST 7.5 is the second highest priority vector interupt on the 8085 (Intel Corp., 1979). The highest priority interupt was already connected to the RAM/IO/Timer chip originally on the SDK-85, and it was decided to leave that connection intact for possible future use. The 8085 notes the occurrence of the RST 7.5 interupt the instant a rising edge appears on the RST 7.5 input line, and subsequently vectors program execution to a fixed memory location. 24 a) b) Microprocessor (8085) Clock IN CPU Clock OUT RST 7.5 t RAM, I/O, and Timer (8155) Timer IN Timer OUT Addresses/Data Output Port O O Bit 1 Bit 2 t , t 2 t, t 2 t , t2 T i m e ^ T i m e 1 ^ Figure 3.1: a) Block diagram showing components of the SDK-85 used to generate the drive current signal, b) Voltage signals created at Bits 1 and 2 of the SDK-85 output port. 25 The method by which the SDK-85 was made to generate a signal at one of its output ports can be seen in the,block diagram of Figure 3.1a. A machine language program was deposited into RAM via a keypad on the SDK-85. The program instructs the CPU to place a particular value (equal to t, in Figure 3.1b) into the timer, and to switch Bit 1 on. The timer counts down from that value to zero at a rate regulated by the clock on the 8085. The program in RAM instructs the CPU to wait for further instructions while the timer is counting. The timer sends a signal to the RST 7.5 interupt pin when the value in the timer reaches zero. When the CPU receives the Interupt signal, program execution branches to a location in RAM that instructs the CPU to load a new value into the timer (t 2 in Figure 3.1b), and to switch Bit 1 off. Repeating the sequence, the timer counts to zero, the CPU is Interupted, the timer Is set to t| again, but this time Bit 2 is switched on. During the last section of program execution, Bit 2 is switched off and no signal appears at the output port. This describes one complete cycle of program execution and results in the signals (shown in Figure 3.1b) appearing in Bits 1 and 2 of an output port on the SDK-85. The voltage signal at the output port was synthesized into a drive-current waveform by passing it through a series of amplifiers (Figure 3.2a). Bit 1 was connected to an inverting operational amplifier (LF353). The signals from Bit 1 and Bit 2 were combined in another LF353 amplifier configured to add two signals. This synthesized voltage waveform was then passed to a KEPCO power supply (model BOP 72-1.5) set In a transconductance mode. The KEPCO power supply transforms the voltage signal from the SDK-85 into a fluxgate drive current. Figure 3.2: a) Series of amplifiers used to synthesize the voltage waveforms at Bits 1 and 2 into a current waveform that was used to drive the fluxgate. b) An idealized representation of the resultant drive current waveform. 27 Through the circuitry represented In Figures 3.1a and 3.2a, a current waveform (shown In Figure 3.2b) with adjustable frequency (f), duty cycle (T), and amplitude (A) was created. The frequency and duty cycle can be adjusted by programming appropriate values for t| and t 2 (Figure 3.1b) Into the SDK-85. The duty cycle Is the ratio between the pulse duration (tj) and the half period (t| + t 2). The amplitude can be adjusted by altering the value of the feedback resistor on the KEPCO (Figure 3.2a). By using the SDK-85 system, the drive-current waveform can be maintained as a series of bipolar, square pulses with zero D.C offset for the following range of parameter values: 0 s f s 1680 Hz; 0 s T s 1.00; 0< A s 500 mA (0-P). The drive-current was Input to a spectrum analyser (Spectral Dynamics, Model SD345 Spectrascope III). Table 3.1 shows the amplitude ratio of the first to second harmonic components of the drive-current for a variety of frequencies and duty cycles. It Is Important that the second harmonic component of the drive current be small because It will contaminate the sensor's output if there is any coupling between the drive-coil and sense-coil. Drive Frequency Duty Cycle 252 502 75X 500 0.0124 0.0125 0.0082 1000 0.0158 0.0148 0.0101 1500 0.0195 0.0168 0.0107 Table 3.1: Amplitude (P-P) ratio of the second harmonic to the first for the current waveform generated by the SDK-85 drive circuitry. The drive current amplitude was set at 100 mA(O-P). 28 The upper limit of the current amplitude is imposed by the KEPCO power supply. The amplitude required to operate the fluxgate used in Experiment 1 is well within the range provided. The upper limit of the frequency is set by the finite amount of time required by the SDK-85 to execute program instructions. The machine language computer program represents the most efficient means of accomplishing the desired task that the writer was able to devise. However, it should be emphasized that a significant reduction in the number of instruction cycles required by the SDK-85 program would not extend the frequency range to that used in low-noise fluxgates (up to 20 kHz: Narod, 1988). The frequency range of the SDK-85 is sufficient to document frequency-dependent behavior in the fluxgate magnetometer. SECTION 3.1.2: The LCR Drive-Current Circuit The SDK-85 provided the means by which a fluxgate magnetometer could be driven at a limited range of frequencies (0-1680 Hz). This frequency range was adequate for the writer's experiments, but it was desired to have at least one set of measurements at a frequency well above the SDK-85 limit in order to extend the scope of the experimental data. Therefore, a different drive circuit was used to produce a current waveform with a higher frequency. The circuit used (Figure 3.3a) generates a large current pulse for a short period of time, producing the waveform displayed in Figure 3.3b. The impedence provided by the inductance of the drive-coil (L) is large when the permeability of the ring-core inside the coil is large, and this allows the capacitor (C) to accumulate charge from the input generator. The 29 impedence of the drive-coil drops drastically as the permeability of the ring-core is reduced by current flowing in the coll, and the capacitor is allowed to discharge through the drive-coil. Thus, the circuit in Figure 3.3a is a non-linear oscillating LCR circuit with its natural frequency determined by the core saturation flux capacity, number of turns in the drive-coil, and the initial voltage across the capacitor (Acuna, 1974). The LCR drive circuit was used to produce a current with a frequency that could be set between 10K Hz and 20K Hz. The circuit shown in Figure 3.3 was presented by Acuna (1974). Drive Current Waveform Figure 3.3: a) The LCR drive current circuit, b) The current waveform generated by the LCR drive circuit. 30 SECTION 3.2: The Fluxgate Sensor The writer constructed three ring-core fluxgate sensors in October 1986 in order to supplement the number of sensors available to students taking the U.B.C. Geophysical Instrumentation course (Geop. 422). One of these sensors (I.D. *MC-1) was used for the experimental work presented in this thesis. The sensor consists of four basic components: a toroidal drive-coil former, a sense-coil former, and the associated coils. SECTION 3.2.1: Sensor Materials The drive-coil former (Figure 3.4) is a steel toroidal shell containing seven layers of .004" thick Supermalloy tape wrapped azlmuthally around Its Interior. Supermalloy is a trade name for a material composed of 80% Ni, 15% Fe, and 5% Mo (Arnold Engineering Co., 1978). The purpose of the Supermalloy tape is to provide a medium with high magnetic permeability that can be periodically driven into a state of uniform magnetic saturation by current flowing through the drive-coll. Henceforth, the toroidal shell and the Supermalloy tape are simply referred to as the "ring-core". The sense-coil former (Figure 3.4) Is made of Plexiglas and was machined by Dieter Schreiber of the U.B.C. Geophysics Department. The sense-coil former is a hollow body of rectangular cross section with two end pieces that buttress the sense-coil. The hollow space In the sense-coil former Is used to house the ring-core and drive-coll. The drive-coil and sense-coil are made from standard magnet wire, 28 Gauge and 30 Gauge, respectively. These particular sizes of wire were chosen to maximize the similarity to the wire used In existing Geop. 422 31 sensors, and on the basis of their availability in the department supplies. Figure 3.4: Coil formers used in the ring-core fluxgate magnetometer. The toroidal drive-coil former contains Supermalloy tape in its interior. The section of the sense-coil former that holds the coil determines the effective length of the sense-coil. SECTION 3.2.2: Sensor Construction The ring-core was wrapped with 28 Gauge magnet wire, thus forming a toroidal drive-coil. A Plexiglas bobbin was used to wrap the drive-coil. The wire was first wound onto the bobbin, and then onto the ring-core. The bobbin made it easy to thread the wire through the center of the toroidal ring-core. The ring-core was wound with approximately 150 turns of wire. The sense-coil was wound upon its former with the aid of a lathe. A wedge was fashioned to fit tightly inside the sense-coil former, and this wedge was clamped to the lathe spindle. The lathe spindle was manually Effective Length of Sense-Coll Drive-Coil Former Sense-Coil Former 32 rotated while the sense-coil was fed onto the coil former. Also, a rotation counter was coupled to the system so that the number of turns on the sense-coil could be accurately recorded. The sense-coil was wound with 310 turns, using four layers of wire along the effective length of the coil former (2.49 cm). After the drive-coil and sense-coil had been formed, the ring-core was placed inside the hollow region of the sense-coil former. The sense-coil former was then attached to the inside of a Plexiglas display case machined by Bill Siep of the U.B.C. Geophysics Department. The coil former was attached to the display case by applying Ethylene Dichloride to the common surface with a syringe and needle. The chemical causes the molecules of the two plastic surfaces to join through covalent bonding in a polymerization reaction (Breck et al., 1981). Next, leads from the drive-coil and sense-coil were soldered to appropriately labelled banana plug receptacles on the display case. Finally, the sensor was protected with a clear acrylic paint spray (Krylon, No. 1301). SECTION 3.2.3: Sensor Calibration A series of measurements were made in order to document characteristics of the fluxgate sensor MC-1. The sensor was calibrated with the following objectives in mind: 1) Document the relation between the amount of resistance attached to the terminals of the sense-coil ("load resistance": RL) and the amount of current flowing in the sense-coll (i); 2) Investigate the linearity of the sensor's response to magnetic fields; 3) Determine the relation between sensor output (i) and strength of the 33 ambient magnetic field, thereby establishing a calibration constant (K) for the sensor; 4) Document the relation between the amplitude of the drive current and the magnitude of the sense-coil inductance (L). The purpose behind each of these objectives, the means by which they were achieved, and the calibration results are presented in the following paragraghs. The sensor MC-1 was calibrated in December 1986 and again in June 1987. For the first calibration test, the terminals of the sense-coil were loaded with an adjustable resistance (RL), and the current flowing through this resistance (i) was plotted as a function of RL. The goal here is to find a region on the graph where i is independent of RL, implying the sensor is behaving as a field-dependent current source that is independent of the load. This situation is indicated by regions on the graph where the plot of i vs. RL is a horizontal line. Figure 3.5: Output current from a resistively loaded fluxgate as a function of load resistance. Data was generated by the circuit shown in Figure 3.7. OH • 1 1 . 1 0 1000 2000 3000 Load Resistance (Ohms) 34 A plot of i vs. RL for a large range of RL is displayed in Figure 3.5. Figure 3.6 shows a plot of the same data, but for a smaller range of RL. The largest output current existing in a region that was reasonably load-independent occurred for values of RL between 250 Q and 350 Q. The next calibration test required sensor MC-1 to be a load-independent current source, so a value for RL was chosen from this range (300 Q). Figure 3.6: Subset of the data presented in Figure 3.5. A smaller range of load resistance is displayed so that a 'load independent' region can be seen. Load Resistance (Ohms) Now that a value of RL had been chosen that allowed the sensor to behave as a load-independent current source, a test was conducted in order to investigate the linearity of the sensor's response to magnetic fields. The sensor was positioned with its axis perpendicular to the laboratory's average magnetic field, thereby allowing the sensor to produce minimum output. An adjustable magnetic field was created parallel to the sensor's axis by sending a direct current through the sense-coil via a calibration circuit (Figure 3.7). The magnitude and polarity of the magnetic field was altered by varying the calibration current (ic). Figure 3.7: Circuit used to calibrate the fluxgate sensor. The D.C. power source was used to send direct current through the sense-coil, thus creating a magnetic field that was used to calibrate the sensor. The drive and sense coils are separated in the diagram for clarity. After Figure 2, on page 46 of the U.B.C. Geop. 422 Lab Manual 36 The intensity of the magnetic field generated by i c is directly proportional to the magnitude of i c . Thus, if the sensor responds in a linear fashion to the intensity of the magnetic field, a plot of i vs. i c is expected to result in a straight line, the slope of which indicates the sensitivity of the sensor as a field-dependent current source. The response of the sensor is very linear (Figure 3.8). A slight nonlinearlty in the sensor is noticeable in Figure 3.8. Figure 3.8: Response of sensor MC-1 to a magnetic field. -6 H — ' — I ' — I — 1 1 — ' — I — ' — I •—1 -6 -4 -2 0 2 4 6 Calibration Current (mA) As a third calibration test, the relation between the amplitude of the drive current and the magnitude of the sense-coil inductance was Investigated due to the significance of this property In the theoretical aspects of the fluxgate magnetometer. Briefly, the performance of the fluxgate as a magnetic field sensor depends on the ability of the drive current to periodically saturate the ring-core, thus modulating the magnetic flux passing through the sense-coil. The modulation of the flux is associated with the modulation of the sense-coil inductance. 37 Figure 3.9: Sense-coil inductance as a function of drive amplitude. 2.0-1 1.8-0.8 T 1 1 1 1 1 1 1 1 r -0 50 100 150 200 250 300 350 400 450 Drive Current (mA) The terminals of the sense-coll were connected to a digital LCR meter (Hewlett-Packard, model HP 4262A). The drive circuitry required to operate the sensor was removed and a direct current source (Universal Power Source, model 6050A) was attached to the terminals of the drive-coil. The sense-coll Inductance was monitored as the magnitude of the direct current was varied from 0 to 475 mA. The sense-coil inductance is proportional to the effective permeability of the material it encloses. Thus, from Figure 3.9, it appears that this permeability is essentially at a minimum for a drive current amplitude greater than about 50 mA, and that the maximum change in sense-coil inductance is approximately 0.95 mH. To conclude the sensor calibration procedure, a "calibration constant" (K) was calculated. The function of this constant is to define the amount of sensor output (i) expected for a given magnitude of ambient 38 magnetic field. The units of K are "current/field intensity", which in the MKS system reduces to "meters/turn". As explained in Section 1.2, the calibration constant (apart from possible geometric factors) is simply the length of the sense-coil (1 = 2.49 cm) divided by the number of turns of wire it contains (N = 310 turns). Therefore, [3.1] K= 1/N = 8.03 X 10" 5 meters/turn. This suggests that approximately 3.5 mA is expected to flow in the sense-coil (in the absence of feedback) when the fluxgate is operated in a 55,000 nanoTesla field (a typical value for the magnetic field at the Earth's surface). SECTION 3.3: The Output Circuitry The final component of the fluxgate system used in Experiment 1 was the output circuitry. The experiment required two forms of sensor output: the integrated open-circuit voltage and the short-circuit current. The circuits used to synthesize these types of output are described in the following sections. Also presented are circuit calibration procedures and the results of those calibrations. Section 3.3.1: Voltage Follower and Integrator The circuit used to obtain the integrated open-circuit voltage is shown in Figure 3.10. The terminals of the sense-coll were connected to an operational amplifier (LF353) configured as a voltage follower. Thus, the signal coming from the sense-coil encountered the (ideally) infinite resistance of the "op amp" input terminals, and a replica of the 39 open-circuit voltage appeared at the output terminals of the voltage follower. By using the voltage follower, the open-circuit signal could be manipulated while the sense-coil continued to experience infinite resistance at its terminals. Figure 3.10: Circuit used to obtain the integrated open-circuit output voltage for the fluxgate. The open-circuit signal appearing at the output terminals of the voltage follower (e^) was passed to a simple RC low-pass filter. The low-pass filter in Figure 3.10 can function as an integrator under certain conditions. The process by which the filter can perform integration on an electrical signal is described with the aid of Figure 3.10. At an instant in time, [3.2] V o u t = q/C, where q is the charge on the capacitor (C). If no current flows between the output terminals of the filter, then the current (1) flowing through 40 resistor R is equal to the current flowing through the capacitor C. Thus, we can write [3.3] " - ' d q q . j j d t . j . d t [3.4] V o u t = J-1 1 dt If V o u t « e^, then i = e^/R, and we arrive at the desired relation [3.5] Vout = -M e o c dt RC I Therefore, the output amplitude of the RC filter is equal to the integral of the open-circuit voltage only when V 0 u t « e^ ,. This requirement can be quantified by the filter's frequency response. Let the frequency response of the RC filter be denoted by [3.6] A B VfiaL e0c where the " A " symbol implies quantities in the frequency domain. Also, [3.7] Vout - - L J(DC e0c = + R 13.8] A = J 1 + jtoRC [3.9] |A|-Vl + [a>RC]; 41 When the quantity wRC is equal to one, the frequency response is at the half-power point. Noting that w a 2nf, and defining the quantity f j to be the frequency at which the response of the f i lter Is at the half-power point, [3.9] can be rewritten as The quantity f is the frequency of the signal entering the filter, and f j is equal to 1/2tiRC. An ideal integrator has a frequency response that is inversely proportional to the frequency of the signal (Jones, 1985). Therefore, by inspection of [3.10], the extent to which the quantity departs from unity is the error In the integrator output amplitude. With these considerations in mind, the resistance of the f i lter was set at 10 KQ, and the capacitance at 149 nF. Thus, the half-power frequency was 107 Hz, and the RC f i l ter was expected to integrate signals of 500 Hz, or higher, with an amplitude error of 2%, or less. SECTION 3.3.2: Calibration of the Follower and Integrator There were two objectives in calibrating the follower/integrator circuit shown in Figure 3.10. First, a simple test was done to ensure that the signal appearing at the output terminals of the follower was identical to the input signal. Second, output from the RC f i l ter was checked to verify that the input signal had been integrated, and that the amplitude response of the f i lter was as expected (equation [3.10]). [3.10] lAl - '-j—^ 42 The performance of the follower was examined by applying a square wave (from a Wavetek function generator: model 182) to the input terminals. A square wave was used because the higher harmonics of the wave provided a good test of the follower's frequency response. The follower was able to pass a I Volt (P-P), 100 kHz square wave without any noticeable change to the waveform. Therefore, it was concluded that the voltage follower would perform adequately in Experiment 1. The RC integrator was also tested with a square wave. It is generally known that passing a square wave to an integrating circuit w i l l produce a triangle wave. The test signal input to the RC low-pass f i lter was a 1000 Hz, 1 Volt (P-P) square wave. As mentioned above, the values of R and C were set at 10 KQ and 149 nF, respectively. Thus, the predicted output (from equation [3.10]) was a 1000 Hz, 106 mV (P-P) triangle wave. The actual output from the RC f i l ter was a 1000 Hz, 111 mV (P-P) triangle wave. Therefore, the RC low-pass f i l ter performed admirably as an integrator. SECTION 3.3.3: Short-Circuit Apparatus The circuit used to obtain the short-circuit current is shown in Figure 3.11a. The virtual ground existing at the input terminal of the operational amplifier allowed a current, equal to the short-circuit current, to flow in the sense-coil. An operational amplifier (LF353) was used so that an oscilloscope could monitor the short-circuit signal without altering it. By inspection of Figure 3.1 la, It is seen that [3.11] - i = Vout* R, where i represents the short-circuit current. 43 998 £2 a) Sense Coll out Generator Output Resistance Pulse Generator b) OUULr Coil Inductance Shunt Resistance Rc AAA 998 Q AAA Coll Resistance 0 0 Figure 3.11: a) C i rcu i t used to obtain the shor t -c i rcu i t current. b) C i rcu i t used to cal ibrate the short -c i rcu i t apparatus shown in a. 44 SECTION 3.3.4: Calibration of the Short-Circuit Apparatus Theoretical behavior of the short-circuit current was discussed in Section 2.3. In that discussion, it was assumed that the EMF induced in the sense-coil of an operating fluxgate would be distributed across the coil's resistance and inductance. Thus, it was predicted that the fluxgate behaves as an RL circuit when the sense-coil terminals are shorted together. The apparatus used to monitor the short-circuit current was designed so that the sensor would behave as if the sense-coil terminals were shorted. Therefore, a calibration test was conducted on that circuit with the objective of determining whether (or not) the shorted fluxgate can be described in terms of an RL circuit. The means by which this objective was accomplished was to apply a voltage to the sense-coil and monitor the resulting short-circuit current. The time-dependent behavior of current flowing in an RL circuit can be predicted if the following are known: the voltage input to the circuit, and the circuit's resistance and inductance. Predicted values for the short-circuit current were obtained from RL circuit theory. The predicted and observed values of current were compared to determine if the shorted fluxgate behaves as an RL circuit. The calibration circuit is shown in Figure 3.1 lb. A pulse generator (Data Pulse, model 101) was used to send a series of square voltage pulses (V1n) into the sense-coil, approximating the EMF Induced in an operating fluxgate. The shunt resistance (Rs) was put In the circuit to reduce the effective output impedence of the pulse generator, thus reducing its impact on the system. The symbols R,. and L represent the resistance and 45 Inductance of the sense-coll, respectively. By inspection of Figure 3.1 lb, it is seen that [3.12] V i n = R p R s +RC Rp + Rs i • L dt The sense-coil inductance was expected to be constant during this calibration test, therefore, a complete description of the current predicted to flow in the calibration circuit is given by the following two equations: [3.13] i=^in-[l -e-b'^} R 1 [3.14] i = W e-fR'Jt In [3.13], t is defined to be zero when the input pulse is switched on. In [3.14], t is defined to be zero when the input pulse is switched off. Also, the symbol R in [3.13] and [3.14] represents a resistance such that: R * A f ! _ + R c Rp + Rs The predicted peak value of short-circuit current ( i m a x in equation [3.14]) was calculated by using a value for t in equaion [3.13] that was equal to the duration of the input voltage pulse. Using equation [3.14], it was predicted that i would decay from its peak value at a rate determined by the resistance and inductance of the circuit. Hence, a semi-log plot of i versus time was expected to result in a straight line with slope equal to -R/L. The parameters R and L were measured before the calibration test was conducted, allowing this aspect of the circuit's behavior to be scrutinized. 46 Several circuit parameters were adjusted in the calibration test to roughly approximate conditions encountered by an operating fluxgate. The first being the inductance of the sense-coil. From calibration tests on the fluxgate sensor MC-1 (see Section 3.2.3) it was expected that, when driven by the SDK-85, the sense-coil inductance would be about 1.0 mH during a drive current pulse, and 1.9 mH between pulses. Hence, the short-circuit calibration was done with the sense-coil inductance set at these two values. The sense-coil inductance was brought down to 1.0 mH by using a Universal Power Supply (model 6050A) to send 120 mA (D.C.) through the drive-coil. Another parameter considered was the duration of the input voltage pulse. In Section 2.2, it was predicted that the duration of a voltage pulse induced in the sense-coil of an operating fluxgate is equal to the time required by the drive current to cause a maximum change in the coil's inductance. By Inspection of the rise and fall times of the current waveform generated by the SDK-85, it was known that an upper limit on this time would be 10 to 30 microseconds. Thus, the pulse generator was set to produce voltage pulses of 10, 20, and 30 microseconds, The final input parameter taken into consideration was the amplitude of the input voltage pulse. The voltage distributed across the sense-coil of a shorted fluxgate is equal to the voltage appearing at the terminals of the coil when it is in an open-circuit configuration (e ,^). It was proposed in Section 2.2 that 47 j e o c d t = l H A L This is the area under a voltage pulse induced in the sense-coil of an operating fluxgate. This condition was roughly approximated in the calibration test by noting that if V|n and t p represent the height and duration of the input voltage pulses, respectively, then it was desired to have [3.15] VIN t p = l H A L and thus, [3.16] V i n = -LH^ N t p The maximum change in sense-coil inductance (AL) and the length-to-turns ratio of the sense-coil (1/N) were measured and found to be 0.9 mH and .08 turns/mm, respectively. A reasonable value for the ambient magnetic field (H) of 45 amp-turns/meter was arbitrarily chosen. Therefore, using values of t p between 10 and 30 usee, it was decided that Vj n should be between 105 and 315 mV. A value of 360 mV was used because it was the smallest value that could be accurately generated by the pulse generator. Table 3.2 summarizes the input parameters and the results of the calibration test. The data was collected over two days. Data using the 1.9 mH inductance was collected on the first day, data using the 1.0 mH inductance on the second. The values of circuit resistance were obtained by in situ measurements just prior to the data collection. 48 Pulse Time (microsecs.) Input (volts) R (measured] (Ohms) L (measured) (mHenries) R/L (observed) (seconds ) i (predicted) (mAmps) i (observed) (mAmps) 10 0.36 18 1.90 -9586 1.81 1.90 20 0.36 18 1.90 -9305 3.45 3.62 30 0.36 18 1.90 -9083 4.95 5.06 10 0.36 17 1.00 -16835 3.31 3.43 20 0.36 17 1.00 -17040 6.10 6.20 30 0.36 17 1.00 -17314 8.46 8.54 Table 3.2: Calibration data from the output circuit shown in Figure 3.1 lb. The measured values of sense-coil inductance (L) were obtained from an LCR meter. The observed values of R/L and peak short circuit current (i) were obtained from semi-log plots of current versus time. The predicted values of 1 were calculated from equation [ 3.15] The predicted and observed values of the peak short-circuit current (1) match extremely well, as do the values of R/L. The slight drift seen in the measured values of R/L is known to be time-dependent. The pulse generator was set to produce pulses of 30, 20, and 10 useconds (in that order) on the first day. Starting where I left off, the order was reversed on the second day. Hence, the observed drift is believed to be caused by a slight drift in the circuit resistance (R), and not by changes in inductance (L) caused by sense-coil current. This demonstrates that the shorted fluxgate can be described as an RL circuit. In addition, it is apparent that the inductance is not significantly altered by the levels of sense-coil current that are created under these conditions. This conclusion supports an explicit assumption in the fluxgate theory of Russell et al. (1983). 49 CHAPTER 4 : Experiment 1: Tests of the Relations Predicted by Experiment 1 was a suite of tests designed to examine the validity of four theoretical relations presented in Chapter 2. Recalling the summary of Chapter 2, the following relations were predicted to exist: [c] When the fluxgate is driven by a current (i d) with the type of waveform created by the SDK-85, the first harmonic ( a ^ of |e o c dt is optimized when the duty cycle of i d is near 50%; [d] When the fluxgate is driven by a sinusoidal drive current, aj is optimized when the amplitude (O-P) of the drive current is about equal to the minimum current (S) required to saturate the fluxgate core; Referring to parameters listed above, the quantities 1, N, AL, and S were recorded when the fluxgate sensor MC-1 was built and calibrated. The parameters H, A(Li), J e ^ d t , and a, were recorded during the course of Experiment 1. References are made to relations [a], [b], [c], and [d] in the remainder of this chapter. Al l tests were conducted in Room 160 of the U.B.C. Geophysics Building. the Equation of State [a] A[L1] = - J -HAL N In order to establish a value for H in relations [a] and [b], the 50 magnetic field in the vicinity of sensor MC-1 was determined. The first measurements were made in June of 1987. At that time the most convenient means of measuring magnetic fields was with a McPhar proton precession magnetometer (model GP-81). However, the magnetic gradient in Room 160 prevented the operation of that instrument in the immediate vicinity of sensor MC-1. Therefore, in June of 1987 the McPhar was used to measure the field just outside Room 160, and the results of those measurements (along with known values of 1, N, and AL) were used in relations [a] and [b] to compute predicted values of |eocdt and A(L1). In June of 1988, the tests of Experiment 1 were repeated. The data collected in 1988 confirmed the results obtained in 1987. However, in 1988 an improved method of determining H was used. A three-component ring-core fluxgate magnetometer built by Barry Narod of the U.B.C. Dept. of Geophysics and Astronomy was placed directly beside sensor MC-1. Output from Narod's instrument was monitored while tests were conducted on sensor MC-1. The output from Narod's fluxgate was calibrated with the McPhar proton precession magnetometer prior to using it in Experiment 1. SECTION 4.1: Tests Involving the Integrated Open-Circuit Voltage Having a value of H at hand, relation [a] was examined by connecting the input terminals of the fluxgate sensor MC-1 to the computer-controlled current source and monitoring the signal appearing at the output terminals of the integrating circuit. Since relation [a] is independent of drive current parameters such as frequency and duty cycle, several values of drive current frequency and duty cycle were used in order to test its validity. Frequencies between 500 Hz and 1500 Hz were 51 used, as well as duty cycles of 25%, 50%, and 75% (at each of those frequencies). A single test of relation [a] was conducted at 16,700 Hz using the LCR drive circuit. The results obtained when using the LCR drive were identical to those obtained when using the SDK-85 drive. Table 41 displays observed values of (e^dt for various drive frequencies and duty cycles using the SDK-85 as a drive source. The values presented in Table 4.1 were obtained by multiplying the voltage measured at the output terminals of the integrating circuit ( V o u t ) by the product of the circuit resistance and capacitance (RC = 1.48 X 10" 3 seconds). This was done because Je^dt = RC V o u l (see equation [3.9]). Drive Current Freauency (Hz) Duty Cycle 15% 5 0 * 75* 250 3.1 3.4 3.3 500 3.1 3.1 3 750 2.8 3.3 3.4 1000 3.3 3.3 3.1 1250 3.1 3.1 3 1500 3 3 2.8 Table 4.1: Integrated open-circuit output voltage presented in units of jiV-s. The fluxgate was driven by the SDK-85 drive circuitry. The predicted value (see relation [a], p. 49) was 3.15 u.V-s. The observed data have a mean value of 3.12 uV-s, and a standard deviation of 0.18. The predicted value of je^dt was calculated from the following parameter values: 1 = 0.025 meters, N = 310 turns, H = 43.4 Amp-turns/meter, and AL = 0.9 mH. Thus, the predicted value of fe^dt was 3.15 uV-s. The observed data has a mean value of 3.12 uV-s, and a standard deviation of 0.18. Therefore, It appears that the validity of equation [2.5] (relation [a]) has been experimentally confirmed. This 52 provides substantial support to the theoretical work first presented by Russell et al. (1983) in which the authors proposed a functional description of the fluxgate mechanism that makes use of sensor parameters that are easily measured. Specifically, the results presented in Table 41 demonstrate that the magnetic field sensed by the fluxgate Is directly proportional to the Integrated open-circuit output voltage. The proportionality factor is equal to the product of the length-to-turns ratio of the sense-coil (1/N) and the maximum change in sense-coil inductance caused by the drive current (AL). In Section 2.2, It was predicted that the sense-coil inductance waveform, and thus the Integrated open-circuit voltage, would be a series of unipolar square pulses (if the SDK-85 or LCR circuitry was used to drive the fluxgate). In order to obtain a visual record of the integrated open-circuit signal and its relation to the SDK-85 and LCR drive signals, a Tektronics camera (model C-30) was attached to a Jlwatsu oscilloscope (model SS-5212), and photographs were taken. Examples of these photographs are displayed on the following page in Figure 4.1. These photographs confirm the expected nature of the integrated open-circuit signal. The ringiness seen in the output signal of Figure 4.1b is caused by the harder drive signal and higher frequency associated with the LCR drive current waveform. Relation [c] predicted that, when the fluxgate was driven by the SDK-85, the first harmonic of |eocdt would be maximized for a drive current duty cycle near 50%. In order to test relation [c], the sensor was connected to the SDK-85, and the (e^dt signal was passed to a Spectral 53 Figure 4.1: a) Drive current generated by the SDK-85 (lower trace) and the Integrated open-circuit output voltage signal (upper trace). Some 60 Hz noise Is visable In the output signal, b) Drive current generated by the LCR circuit (upper trace) and the integrated open-circuit signal (lower trace). The fluxgate sensor has been rotated 180*, and thus the output signal is inverted with respect to the output signal shown in a. 54 Dynamics spectrum analyzer (model Spectrascope III). Table 4.2 shows the amplitude coefficients of the first harmonic of |eocdt as a function of SDK-85 drive current frequency and duty cycle. Drive Current Frequency (Hz) Duty Cycle Optimum Duty Cycle 25* 50* 75* 250 2.8 3.8 2.5 48.4* 500 2.8 4 2.5 48.6* 750 3 3.8 2.4 46.6* 1000 3 3.8 2.4 46.6* 1250 3.1 3.8 2.2 45.1* 1500 3 3.7 2.1 45.1* Table 4.2: Amplitude coefficients for the first harmonic of the integrated open-circuit output voltage obtained when the sensor was driven by the SDK-85 circuitry. Data are presented in units of jiV-S. The data clearly show larger values occurring for duty cycles near 50*. as predicted. The data presented in Table 4.2 show that the first harmonic of {eocdt is substantially larger for a drive current duty cycle of 50% than for 25% or 75%. Thus relation [c] was verified. A parabola was fit to the three data points associated with each drive current frequency in order to determine the duty cycle that is expected to produce the peak output. Those "optimum duty cycles" are also presented In Table 4.2. The optimum duty cycles presented in Table 4.2 are not precisely equal to 50% and they show a slight frequency dependence because the (e^dt signal consists of pulses that are not precisely square. Three of the parabolic curves used to obtain the optimum duty cycles are presented in Figure 4.2. The results shown in Figure 4.2 are conslstant with theoretical estimates presented in Figure 7 of 6ao and Russell (1987b). 55 0.2 0.3 0.4 0.5 0.6 0.7 0 .8 Duty Cycle Figure 4.2: Parabolic curves that were f i t to the 5 0 0 Hz, 1000 Hz, and 1500 Hz data presented in Table 4.2. These curves were used to determine the values of drive current duty cycle that are expected to result in peak sensor output. The behavior of the first harmonic of J e ^ t was also examined when the fluxgate was driven with a sinusoidal current (i.e. relation [d]). It was predicted (Section 2.2) that when the fluxgate is driven by a sinusoidal current, the amplitude coefficient of the first harmonic of the je^dt signal (ap Is a function of the drive current amplitude (A). Specifically it was shown that a, is expected to experience a maximum value when A is about equal to the minimum current required to saturate the fluxgate core. Relation [d] was tested by using a Wavetek function generator (model 182) to feed a sinusoidal signal Into a transconductance amplifier. The output signal from the transconductance amplifier was used to drive the fluxgate. The amplitude of the sine-wave drive was varied from 25 mA to 250 mA (0-P), and the spectrum of the signal appearing at the terminals of the Integrator was monitored. 56 2.0 0.8 H—• i • i • i • i • i • i • i • i • 25 50 75 100 125 150 175 200 225 250 Current Amplitude (mA) Figure 4.3: Plot showing the integrated open-circuit output voltage (first harmonic) as a function of (peak) sinusoidal drive current amplitude. Figure 4.3 demonstrates the severe impact that the amplitude of a sinusoidal current has on the integrated open-circuit output signal. The output is obviously maximixed for a very specific value of drive current. Since It was predicted that the optimum sinusoidal drive current, amplitude is about equal to the current required to saturate the ring-core, it appears that the ring-core of MC-1 is effectively saturated when the drive current is greater than about 42 mA (O-P). This conclusion is supported by the data presented in Figure 3.9 (page 37). Data presented in Figure 4.3 strongly suggest that if one is using a demodulator that is sensitive to the amplitude spectrum of the integrated open-circuit signal, then a sinusoidal drive current should be avoided. The sinusoidal drive current amplitude that produces an optimal output signal is only slightly greater than the (minimum) saturation current. However, 57 in order to reduce Barkhousen noise in the sensor, a drive current must create a state of deep and uniform magnetic saturation in the fluxgate core. To accomplish this, the amplitude of the drive current must be much larger than the minimum saturation current. SECTION 4.2: Tests Involving the Short-Circuit Current The final test in Experiment 1 was to examine relation (bj: A[U] = - J -HAL N The objective was to record values of L and i occurring just before and just after a transition in L, and thus obtain the quantity A(Li). The sense-coil inductance (L) is proportional to the integrated open-circuit voltage. Results presented in the previous section (Figure 4.1) show that the SDK-85 drive current caused the open-circuit output signal to be a series of unipolar square pulses. Therefore, L(t) is also a series of unipolar square pulses. This implies that, as a first order approximation, L experiences only two values: L ) O W and LNJGN. From calibrations done on senor MC-1 (Section 3.2), it was known that LH L G H is about 1.9 mH and L,ow is about 1.0 mH. Since values of L occurring on either side of a transition were known, the only parameter that needed to be recorded was the short-circuit current (i). The short-circuit tests were first conducted In August of 1987. At that time the short-circuit apparatus (Figure 3.11, p. 43) was connected to the output terminals of the fluxgate, and 1 was observed. The sensor was driven by the SDK-85 at frequencies ranging 58 from 50 Hz to 250 Hz, at a duty cycle of 50%. The reason that such low frequencies were used was that by doing so, the sense-coil current had time to decay to zero before each transition in L. Therefore, as can be seen by inspection of equation [2.14] (p. 20), the quantity A(L1) reduces to L+i+ and relation [b] can be rewritten as [41] i + = - ± H 4 L N U The quantity L + represents the value of sense-coil inductance occurring just after a transition, and i+ represents the peak value of sense-coil current. , The photographs presented on the following page (Figure 44) show the drive waveform generated by the SDK-85 (lower traces) and the short-circuit current that results (upper traces). The output signal associated with the leading edge of a drive current pulse (Figure 4.4b) is larger than that associated with the lagging edge (Figure 4.4c). This phenomenon is caused by the difference in transition times between the leading and lagging edges. Figure 44 clearly shows that the short-circuit current achieves peak values immediately after a transition (i +), and then decays exponentially. Results of the short-circuit tests that were obtained in 1987 were very discouraging. The right-hand-side of [41] was used to calculate the predicted values of peak short-circuit current. Since the open-circuit tests (a high impedence equivalent to the short-circuit tests) had worked out precisely as predicted, it was expected that the short-circuit tests would also. However, recorded values of i+ were consistantiy 25% to 35% 59 b) c) Figure 4.4: a) Drive current generated by the SDK-85 (lower trace) and the short-circuit output current (upper trace). The output pulse occurs at a transition in the drive current, and then decays at a rate determined by the circuit time constant, b) Short-circuit current (upper trace) and the leading edge of a negative drive current pulse (lower trace), c) Short-circuit current (upper trace) and the lagging edge of a negative drive current pulse (lower trace). 6 0 smaller than the values calculated from [41]. Thus, a process of trouble shooting began. The circuit used to monitor the short-circuit current (Figure 3.1 la) was calibrated. The calibration procedures and the results of the calibration tests were presented in Section 3.3. Those tests showed that the shorted fluxgate could be described as an RL circuit. The peak values of short-circuit current recorded in the calibration tests matched the predicted values, so there was an obvious discrepancy between the calibration data and the short-circuit data collected in Experiment 1. R. D. Russell suggested that I examine the short-circuit current waveform by making semi-log plots of i vs. time. These plots allowed the sense-coil inductance to be examined by Inspection of the circuit time constants. Also, the transitions in L(t) from L l o w to L n j g h were not instantaneous (an implicit assumption in [4.1]), but occurred over finite time intervals. Therefore, a critical function of the semi-log plots was to provide a means of extrapolating values of 1 back to the start of the inductance transition (t = 0). A typical example of such a plot is displayed in Figure 4.5. The values of 1(0) obtained by the extrapolation of data on the semi-log plots brought the experimental results closer to the values predicted by equation [4.1]. However, the experimental data continued to be 20% to 25% below the predicted values. In June of 1988, the SDK-85 drive circuitry was reassmbled and the short-circuit tests were repeated at drive frequencies around 200 Hz. The procedure outlined above was used as a starting point to record the 61 0 50 100 150 200 t i m e (microseconds) Figure 4.5: Semi-log plots of the short-circuit current (using a 200 Hz drive signal) after a transition in sense-coil Inductance, Two data sets are plotted. The first, "current (lead)", is the output associated with the leading edge of a drive current pulse. The second data set is from the output associated with the lagging edge of a drive current pulse. The (least squares) best fit line for each data set was extrapolated back to t = 0. The slope of the lines are determined by the circuit time constant (R/L). The predicted values of i(0) for the first and second data sets (using equation [4.1 ]), were 3.15 mA and 1.66 mA. Therefore, the experimental values were low by about 21*. short-circuit current. The poor results obtained during the previous summer were essentially reconfirmed. Slight Improvements in the results were obtained by careful alignment of the onset of a drive current pulse (t = 0) and the onset of the short-circuit signal. However, the experimental values (extrapolated back to t = 0) continued to be at least 20% below the predicted values. As was pointed out earlier, the equation that was used to predict the peak value of short-circuit current assumes a step transition in the sense-coil inductance. The transition actually takes a finite length of time, and in that time, as the short-circuit current (i) rises toward its peak value, the EMF that is ultimately responsible for the existance of i 62 decays at a rate determined by the circuit time constant. It was also assumed that the square-pulsed drive current waveform of the SDK-85 caused transitions in L that were rapid enough to be considered (effectively) instantaneous. As it turns out, it was the duration of the transition time that caused the experimentally determined values of i to differ from the predicted values. The SDK-85 drive circuitry was removed from the fluxgate and the LCR circuitry (Figure 3.3, p. 29), with its high frequency capabilities, was used to drive the sensor. The pulses in the higher frequency signal cause a more rapid transition in L. The drive frequency was set at 16,000 Hz. Initial tests showed that the circuit originally devised to monitor the short-circuit current (Figure 3.11a) was not able to deal with the high frequency signal that was produced. On a suggestion from Barry Narod, a Tektronix current probe (model P6016) was used to monitor the short-circuit current. The current probe worked well. The observed values of A(L1) obtained when using the LCR drive (at frequencies greater than about 14,000 Hz) matched the predicted values. In order to confirm that it was the drive frequency that made the difference and not the output circuitry, the current probe was used to monitor the short-circuit current generated when the SDK-85 was used to drive the sensor. Again, the observed values of A(L1) were well below the predicted values, but there was an improvement when the drive signal was increased from 200 Hz to 1000 Hz. Figure 4.6 shows the observed values of MLi) as a function of drive current frequency as well as the predicted value of MU). 63 3.0 2.8 e 2 6 w 2.4 4 2.2-2.0 J-I I I I 20K predicted 200 IK 10K 12K 14K 16K 18K D r i v e Frequency (Hz) Figure 4.6: Observed values of A(L1) as a function of drive current frequency. The data at 200 Hz and 1000 Hz were obtained by using the SDK-85 drive. All other data were obtained by driving the sensor with the LCR circuit. The predicted value of A(Li) is presented in the right-most column. Values of A( Li) are in units of microHenry-Amps. The results presented in Figure 4.6 show that equation [2.15] (relation [b]) is valid as long as the transitions in L are extremely short in comparison with the time constant of the output circuit. For the shorted fluxgate, the time constant is determined by the resistance and inductance of the sense-coil. SECTION 4.3; Summary of the Experimental Results Experiment 1 was a series of tests designed to examine the validity of four theoretical relations proposed in Chapter 2. Results obtained during the course of Experiment 1 confirmed all four relations. Specifically, experimental results presented in Chapter 4 support the following conclusions: 1) The integrated open-circuit output voltage of a ring-core fluxgate 64 sensor is proportional to the magnetic field in which the sensor is immersed. The constant of proportionality is the product of two easily measured sense-coil parameters: the length-to-turns ratio and the maximum change in inductance caused by the drive current. Specifically, 2) When the drive current is such that it creates a sense-coil inductance waveform that (effectivley) assumes only two values (L j o w and L h j g n), the following relation holds: Thus, the short-circuit current (i) is also related to the magnetic field through the length-to-turns ratio of the sense-coil (1/N) and its inductance (L). 3) Knowledge of the sense-coil inductance waveform created by the drive current can be used to optimize the fluxgate output signal. The validity and usefulness of this concept was demonstrated by two examples, one of which is presented in *4, below. 4) The spectrum of the integrated open-circuit output is optimized in a ring-core fluxgate that is driven by a sinusoidal current when the peak drive amplitude (O-P) is about equal to the minimum current required to saturate the ring-core. This represents a fundamental problem for 4 U ] = - - LH A L N 65 using a sinusoidal current to drive low-noise sensors because they require a drive with amplitude many times larger than the saturation current. 66 Chapter 5: Motivations for Immersing a Fluxgate Sensor in Magnetic Fluid Experimental data presented in the previous chapter confirmed the validity and usefulness of the fluxgate equation originally presented by Russell et al. (1983). In that equation, Russell, Narod, and Kollar chose to express the magnetic field, as did Serson and Hannaford (1956), in terms of the field intensity H rather than the flux density B. Proclaiming the validity of this choice, Narod and Russell (1984) state: "a flux gate element senses the ambient magnetic field H and not the flux density B'. However, Lowes (1974), in an article entitled "Do Magnetometers Measure B or H?", proposed that the fluxgate mechanism responds primarily to B. Unfortunately, none of these authors presented experimental data to support their claims, hence Lowes' question remains unanswered. In this chapter, the writer presents a simple argument that suggests a means by which one could possibly determine whether a fluxgate responds primarily to B, or to H. An SI system of magnetic units is used In the following discussion. The notation that I have adopted is based on the Sommerfeld system as presented by Crangle (1975). In this system, B In a magnetizable material Is given by the relation [5.1] B a u0[H + M] The symbol M represents the magnetization per unit volume. The quantity JX0 is the permeability of free space and is equal to 4 n X 10~7 H/m. Equation [5.1 ] can be written such that 67 [5.2] B = Ho H 1 + Mo J t [MoHJ The quantity M/ j i 0 H is the volume susceptibility and will, henceforth, be denoted by the symbol K . Thus, [5.3] B = j ioH [ l + Mo*] In this form, the volume susectibility is exactly 107 times larger than the volume susceptibility in cgs units. The term 1 + J I 0 K is called the relative permeability and is denoted by the symbol J L Therefore, [5-4] B = HJJI0H The two materials in which geophysical measurements are most likely to be made are air and water. Lowes (1974) pointed out that in air and water, the quantity JJL0K (approximately equal to 5 XI0"6 and -1 X 10~4, respectively) can generally be ignored. Hence, output from the magnetometer can easily be calibrated In units of either B or H by the relation B = j i 0 H. Thus, Lowes concluded that from the instrumentation aspect, the distinction between B and j* 0H does not matter. Contrary to this conclusion, theoretical considerations presented below demonstrate that determining whether a fluxgate responds to B or H can have practical consequences for instrumentation. Specifically, it is shown that the output signal from the fluxgate may be enhanced by 68 immersing the sensor in a container of magnetizable material, j l the sensor responds primarily to B. SECTION 5.1: B and H In a Body of Magnetizable Material Consider a magnetically permeable body that has been placed in a uniform magnetic field. This field causes an alignment of the magnetic dipoles in the permeable material, thus magnetizing it. The magnetization of the body corresponds to a splitting of the internal magnetic field into two components: one resulting from the ambient field and the other caused by the field of the internal magnetic dipoles (Oatly, 1976). Thus, the magnetization creates a difference between the ambient field and the field existing Inside the permeable material. As a starting point for quantitative considerations, a description of the magnetic field in and around a magnetically permeable body is needed. Due to symmetry, the sphere presents the simplest case for which an analytic solution can be obtained. The solution for the field associated with a permeable sphere placed In a uniform magnetic field can be found in several texts (Plonsey and Collin, 1961: Whitmer, 1962; Jefimenko, 1966; Jackson, 1975). It can be shown that for a sphere with relative permeability us, placed in a medium with relative permeability JA that supports a uniform field H (and B), the following relations exist: [5.5] B is. = 3 Ms B 2ji + jis [5.6] Hs. = _ l l H H 2^j» s 69 The symbols H. and Bs represent the field Intensity and flux density Inside the sphere. Therefore, It can be seen that if jis > ji, then H s < H and B s > B. Qualitatively, the flux density and field strength have this behavior about any magnetized body (Whitmer, 1962). Equations [5.5] and [5.6] suggest a (conceptually) simple means of determining whether a fluxgate magnetometer responds primarily to B or H. Suppose that a fluxgate was used to measure the ambient field. Next, place the fluxgate inside of a spherical container located in the ambient field. Let the permeability of the material medium in the container be greater than the permeability of the ambient medium. If the output signal from the sensor goes down, one would conclude that the sensor responds primarily to H; If the output signal goes up, then the sensor responds to B. Therefore, if the fluxgate responds primarily to B, it may be feasible to enhance the output signal by simply immersing the sensor in a container filled with a highly permeable material. The difference between the ambient field and the field existing inside a material body depends on the geometry of that body and the permeability contrast at the boundary between the body and the ambient medium. As can be seen by [5.5], the largest ratio for the B-fields that can be obtained by using a spherical body occurs when jis » JA, thus producing a situation where Bs/B = 3. If the fluxgate does respond primarily to the B-field, then by considering container geometries that maximize the internal flux density, one could produce a well designed flux concentrator that would effectively Increase the sensitivity of the fluxgate magnetometer. 70 One could also set up an experiment in which the magnetic field intensity (H) was concentrated in a well-defined region. This would be advantageous if the fluxgate responds primarily to H. A container filled with a medium in which the permeability was less than the permeability of the ambient medium would concentrate the H-field inside the container. Assuming the ambient medium Is air (relative permeability slightly greater than 1), one would want to have a purely diamagnetic (relative permeability less than 1) material in the container. However, diamagnetism is a very weak magnetic property and it is not likely that a large enough permeability constrast could be achieved so that an effective concentrator for H could be built by simply immersing the sensor in a diamagnetic material. This is not to say that an effective H-field concentrator couldn't be devised. One example is an air gap transformer. Both B and H are heavily concentrated in the air gap. However, I have chosen to conduct an experiment in which a fluxgate is immersed in a highly permeable medium on account of the nature of the magnetic fields in such a medium (B increases and H decreases), and because It provides a simple means of significantly concentrating B. Section 5.2: Magnetic Fluid All materials have magnetic properties to some degree. However, to test the hypothetical situation described above, one needs a material that has several special properties. Two very desirable characteristics are high magnetic permeability and low viscosity (preferably existing near standard temperature and pressure). High permeability is needed in order 71 to create a significant difference between the ambient field and the field inside the material, thus allowing the difference to be measured. The need for low viscosity enters the picture when one attempts to immerse the sensor in the permeable material. These two characteristics (low viscosity and high magnetic permeability) can be obtained in a fluid mixture made from a stable suspension of very small ferromagnetic particles, i.e. a 'ferrofluid'. Describing modern ferrofluids, Rosensweig (1975) writes: "Typically, ferrofluid comprises a colloidal dispersion of finely divided magnetic particles of subdomain size whose liquid condition is remarkably unaffected by the presence of an applied magnetic field, and which particles resist settling under the influence of gravitational, centrifugal, magnetic or other force fields." A full description of the physical properties of ferrof luids is beyond the scope of this thesis, though several properties are discussed in the next chapter. Readers interested in the general properties of ferrofluids are directed to reviews by Kaiser and Mlskolczy (1970), Rosensweig (1979), Charles and Popplewell (1980), and Chlkazumi et al. (1987). Three exhaustive bibliographies of ferrofluid literature are available: pre-1980 material was compiled by Zahn and Shenton (1980), 1980-1983 material by Charles and Rosensweig (1983), and 1983-1986 material by Kamiyama and Rosensweig (1987). Titles provided in these bibliographies were compiled from the World Patent Index, Claims/U.S. Patents, INSPEC data base, NTIS file, Magazine Index, Chemical Abstracts, Engineering Index, and the Science Citation Index. 72 I attempted to make a ferrofluld by (loosely) following one of the earliest methods of ferrofluld manufacturing (Bond!, 1956). Powdered magnetite was obtained from Bernard Kline of the Coal Processing facility at the University of British Columbia. I used equipment at Coal Processing to seive the powder, thus eliminating particles larger than 38 microns. Next, the sieved powder was dispersed in water and allowed to settle. This procedure served to differentiate the magnetite particles by size, and permitted a large portion of the dirt and clay to be removed. In order to reduce the average particle size, the powder was then subjected to a grinder located in the U.B.C. Geological Sciences building. On advice from Kline, the magnetite powder was ground in the presence of small quantities of Calgonite dish detergent. The detergent was used in an attempt to coat the individual magnetite particles, thus preventing floatation, After several hours of grinding, the powder was dispersed in various liquids in the hope of obtaining a reasonably stable magnetic liquid. Numerous fluids were tested as carriers for the magnetite including various vegetable and mineral oils, corn syrup, and glycerol. None worked adequately; the magnetite particles failed to stay in suspension for a reasonable period of time. Eventually, I came into contact with the Intermagnetics General Corporation (P.O Box 566, Guilderland, New York 12084; (518) 456-5456). Among other things, IGC markets instruments that use ferrofluld ("Magfluid") for the magneto-gravitational separation of material, Walter Urbanski of IGC kindly donated 450 ml of the Magf luid to me so that I could 73 carry out experimental tests on the hypotheses presented in the previous section. 74 CHAPTER 6 : A Fluxgate Sensor Immersed in Magnetic Fluid In the previous chapter, It was proposed that an experiment be conducted in which magnetic measurements are made inside a container filled with a magnetic fluid. It was pointed out that by doing so, one could possibly determine whether a fluxgate responds primarily to B, or to H. Also, It was noted that the output signal from a fluxgate might be enhanced by Immersing the sensor in a container filled with magnetic fluid. In June of 1988, the writer obtained a commercial ferrofluid ("Magfluid" from Intermagnetics General Corporation), and thus, it became possible to test these hypotheses. Physical properties of the Magfluid that may affect the behavior of a fluxgate are discussed in Section 6.1. Preceding chapters presented the fluxgate theory of Russell, Narod, and Kollar, and demonstrated how (through their fluxgate equation) a knowledge of the sense-coil inductance can be used to understand, describe, and predict the sensor's output signal. In Section 6.2, the behavior of a ring-core fluxgate immersed in Magfluid is examined, and the results are presented in terms of the sense-coil inductance. Section 6.1: The IGC Magnetic Liquid: "Magfluid" In June of 1988, I received a sample of "Magfluid" (and specifications on that fluid) from Walter Urbanskl of the Intermagnetics General Corporation (IGC). The fluid contains magnetite grains that are suspended In a solution of water and Llgnosulphonate. The Lignosulphanate is a surfactant and dispersing agent (Eckroth, 1985), thus it helps to prevent Individual particles of magnetite from agglomerating. According 75 to IGC data, the Magf luid has an absolute viscosity of approximately 14 centipoise at 25° C (similar to olive oil at about 60° C), and a saturation magnetization of approximately 100 Gauss. These specifications pertain to the Magfluld In its most concentrated form, as it is dilution stable in water. Unfortunately, data regarding susceptibility was not sent with the Magf luid and Urbanski was not able to provide me with this information. I needed to know the susceptibility so that the ratios Bs/B and Hs/H (see equations [5.5] and [5.6]) could be calculated for a spherical container filled with Magfluld. Thus, I began to examine the Magfluid. With assistance from John Knight of the Geological Sciences Department at U.B.C, I was able to examine a sample of the Magfluid with a scanning electron microscope. Distinct images of the magnetite particles were not obtained due to a lack of instrument resolution, but the results clearly showed that the particles are well dispersed and significantly less than 100 nanometers (100 X 10"9 meters) In diameter. A phone conversation with Urbanski revealed that the average diameter of the magnetite particles is about 50 Angstroms (50 X 10~10 meters). Therefore, the magnetite particles in Magfluid are sub-domain size and are expected to exibit "superparamagnetic" behavior in that they have no remnance or coercivity (Kaiser and Miskolczy, 1970). Superparamagnetic grains in a ferrofluid have thermal vibrations (at room temperature) that have energies of the same order of magnitude as their magnetic energies (Charles and Popplewell, 1980). Hence their 76 magnetization is continually undergoing thermal reorientation. However, in the presence of an applied field the grains are caused to have an overall magnetic alignment, and thus the fluid becomes magnetized. Susceptibility defines the ability of a material to be magnetized. The susceptibility of superparamagnetic grains is much larger than that of an equivalent amount of single-domain or multidomain particles (Thompson andOldfield, 1986). Tests on the susceptibility of a ferrofluid containing superparamagnetic particles were reported by Soffge and Schmidbauer (1981). They used an a.c susceptibility bridge to show that the susceptibility of a ferrofluid is dependent on the strength of the ambient field (H), the frequency of the bridge test signal, and on the temperature of the fluid. It was found that, at a constant temperature and test signal frequency, the susceptibility was very nearly proportional to log(H~2). The susceptibility decreased by about 50% as the temperature of the fluid was Increased from 150 °K to 375 °K (at a test signal of 1100 Hz and zero ambient field). Soffge and Schmidbauer also showed that the susceptibility obtains a peak value at a particular temperature, and the peak shifts toward higher temperatures as the test signal frequency increases. Menear et al. (1983), Chantrell and Wohlfarth (1983), and Menear et al. (1984) showed that this phenomenon can be attributed to field-induced particle agglomeration. These findings suggest that there may be fundamental problems associated with operating a magnetometer in a ferrofluid if the instrument produces a significant magnetic field, or if it generates a substantial amount of heat. 77 I used four SOILTEST a.c. susceptibility bridges (model MS-3) to evaluate the susceptibilty of the IGC Magfluid concentrate. Manufacturer's specifications provided with the susceptibility bridges suggest that the instruments are capable of measuring absolute values of volumetric susceptibility to an accuracy of 5-10% (SOILTEST, 1966). Output from the susceptibility bridges Is calibrated In cgs units of emu/cm3. The results of the susceptibility measurements are presented in Table 6.1. Instrument Id. Number *1067 *2896 *2803 *2897 Average Value Susceptibility 0.031 0.025 0.026 0.025 0.027 Table 6.1: Susceptibility of the IGC Magfluid as measured by a.c. susceptibility bridges. The values of susceptibility are presented, as they were recorded, in units of emu/cc. The data show no obvious dependence on test signal frequency since two bridge units (*1067 and *2896) produce a 1000 Hz test signal, and the other two (*2803 and *2897) produce a 2000 Hz test signal. The writer also used these instruments to obtain susceptibility measurements on numerous rock samples. For all rock samples, as well as the Magfluid, a larger value of susceptibility was obtained with instrument * 1067. This suggests a difference between the calibration of *1067 and the other three bridge units. The susceptibility was used to calculate a value of relative permeability. I converted the susceptibility data to an SI system of magnetic units based on Crangle (1975). This sytem was described in some detail in Chapter 5. In order to convert the susceptibility data obtained from the MS-3 bridges to SI units, the values were multiplied by 107. Next, a value of relative permeability (ji) was calculated from the relation LA = 1 + J A 0 K , where j i 0 is equal to 4 n X 10~7, and K IS the SI susceptibility. Data presented in Table 6.1 imply that the relative permeability of the Magfluid (at 'room temperature' and in a field of about 45 A/m) is approximately 1.34. A knowledge of the relative permeability, along with equations [5.5] and [5.6], provide a means of calculating the ratios rt/H and Bs/B In a spherical container filled with Magfluld. The terms Hs and Bs are the field strength and flux density in the Magfluld, and H and B represent the ambient field strength and flux density. If we let the relative permeability of the Magfluld be denoted by J A s and use a value of 1 for the relative permeability of air (JA), then [5.5] and [5.6] yield tk 2 go is. a i 20 H B These ratios indicate that the field strength in the sphere is expected to be about 10% less than in the ambient field, and the flux density about 20% greater. Hence, if the fluxgate responds to the field strength, the output should decrease by about 10% when the sensor is placed in a spherical container of Magfluid. However, the output should increase by about 20% if the sensor responds to the flux density. Therefore, whether the sensor reponds to B or H, it appears that the difference between the ambient field and the B and H fields inside a spherical container of Magfluid should be large enough to be detected by a fluxgate magnetometer. 79 SECTION 6.2: Inductance of a Fluxgate Immersed In a Magnetic Fluid The next step in the experiment was to examine the sense-coil inductance of a fluxgate sensor Immersed in Magfluid. It was demonstrated in Section 41 that the field measured by a ring-core fluxgate is equal to The quantity N/l Is the turns-to-length ratio of the sense-coll, AL is the maximum change In sense-coll Inductance caused by the drive current, and e o c Is the open-circuit output voltage from the sensor. When the fluxgate sensor MC-1 was operated in air, AL (created by a change in the ring-core permeability) was measured to be about 0.9 mH. Ideally, flux generated by current in the toroidal drive-coll surrounding the ring-core is contained by that coil. Therefore, though values of L should increase due to the permeability of Magfluid, AL should continue to be created by a change In the ring-core permeability and not be affected by the presence of Magfluid. Hence, the quantities N, 1, and AL would be known and an increase or decrease in the sensor output could be attributed to a proportional change in the measured field. Obviously, the number of turns on the sense-coil (N) and the effective length of that coil (1) are Independent of the medium in which the sensor Is operated. However, the quantity AL is independent of the medium only if the permeability of the medium is not altered by the fluxgate. To determine whether AL was independent of the medium, the inductance had to be monitored when the sensor was immersed in Magfluid. 80 In Figure 3.9 (page 37), data collected in June of 1987 showed the sense-coil inductance of sensor MC-1 as a function of the drive current. That data was used to determine the quantity AL when the sensor was operated in air. The same type of test was repeated on sensor MC-1 in July, 1988. However, this time the sensor was immersed in the IGC Magfluid. The sensor was placed in a Plexiglas display case when it was constructed, and it was this display case that was used as a container for the Magfluid. The display case was used as a container because it provided a sealed environment that would hold the Magfluid, and all the necessary electrical connections between the sensor (fixed to the inside of the case) and the drive and output circuitry were already established. One alteration had to be made to the system in preparation for the Magfluid test. The wire from which the drive and sense coils are made is insulated (standard magnet wire). However, the insulation at the ends of the colls had been removed during sensor construction so that the coils could be soldered to the binding posts of banana plug receptacles imbedded in the display case. For the Magfluid test, the binding posts were sprayed with several layers of an insulating material (START: "PC-101 Protective Coating"), and then covered with a thick layer of candle wax. The need for taking this action was revealed by preliminary tests on the fluxgate. At first, the sense-coil inductance decreased when the sensor was moved from an air medium to a Magfluid medium. Also, when the sensor was immersed in Magfluid (and driven by the SDK-85 current source) a substantial amount of drive signal was able to get from the drive-coil terminals of the sensor to the sense-coil terminals through the 81 Magfluld. Both situations were corrected by Insulating the terminals of the sense and drive coils from the conductive Magfluid. Data showing the sense-coil Inductance (L) as a function of drive current, collected when the sensor was Immersed in air and when in Magfluid, are presented in Figure 6.1. The data show that when the sensor was in air, L varied from a maximum of 1.9 mH to a minimum of 0.98 mH. However, when immersed in Magfluld, the L varied from 2.13 mH to 1.03 mH. Therefore, AL increased by about 20% when the sensor was immersed in Magfluid, i.e. from 0.9 mH in air to 1.1 mH in Magfluid. It is clear from Figure 6.1 that values of L obtained in the two media began to converge as the drive current was increased. Therefore, it can be concluded that the permeability of the Magfluid was altered by stray flux generated by current in the toroidal drive-coll. z e 2.15 2.05 1.95 1.85 1.75 1.65 1.55 1.45 1.35 1.25 1.15 1.05 0.95 |_J in Air • in Magfluid Hi 1 ' P ' -ilflfr 1 J r l rl rl r l r l J J J 10 20 30 40 50 60 70 80 90 100 125 150 175 200 Drive Current (mA) Figure 6.1: Inductance of fluxgate sensor MC-1 obtained from a Hewlett-Packard digital L C R meter (model 4262A). Measurements were taken with the sensor in air and then repeated with the sensor immersed in the IGC Magfluid. The "drive current" was from a Universal d.c. power supply (model 6050A). 82 The fact that stray flux from the drive current Is able to affect the permeability of Magfluid demonstrates a fundamental problem for using this material. Analytic solutions for the ratios of the ambient field to the Internal field of a permeable sphere filled with Magfluid have been presented. However, those solutions were based on a known, fixed value of Magfluid permeability. Unfortunately, the permeability of the Magfluid enclosed by the fluxgate's sense-coil is not single-valued. Therefore, the empirical results predicted by equations [5.5] and [5.6] cannot be observed by immersing a ring-core fluxgate in Magfluid (until a means of substantially reducing stray flux from the drive-coil is provided). I would also like to point out a discrepancy between the susceptibllty data in Table 6.1 and the inductance data presented in Figure 6.1. The susceptibility data Implied that the relative permeability of the Magfluid is approximately 1.34 Since the relative permeability of air is essentially equal to 1, the relative permeability of Magfluid is about 34% greater than that of air. The inductance of a coil Is proportional to the permeability of the material enclosed by that coil. Therefore, the inductance of fluxgate sensor MC-1 should have increased by about 34% when it was immersed In Magfluid. As can be seen in Figure 6.1, the inductance increased by only 12.1% (when no current was sent through the drive-coll). Therefore, there appears to be a discrepancy of about 22% regarding the permeability of the Magfluid. As mentioned earlier, It was noticed that the measured values of sense-coil inductance actually decreased until the sense-coil terminals were insulated. The Inductance reading produced by the LCR meter is a 83 function of the impedence encountered by the test signal that it generates. The values obtained by the LCR meter decreased because the test signal was partially dissipated into the Magfluid, decreasing the apparent impedence of the sense-coil. This phenomenon was verified by removing the ends of the sense-coil from their binding posts on the display case, immersing the binding posts into the Magfluid, and using a Simpson multimeter (model C464) to measure the resistance between the binding posts. It was found that the resistance between the binding posts, separated by 3/4" of Magfluid, was less than 5 Q. The resistance of the sense-coll is 8.2 a. To ensure that efforts to insulate the sense-coil terminals from the Magfluid had been successful, the process was repeated after the binding posts had been treated. Resistance measurements indicated that the insulated terminals (immersed in Magfluid) were seperated by at least 20 M Q. The data presented in Figure 6.1 were obtained after the sense-coil terminals were insulated. The test signal produced by the LCR meter can affect Inductance measurements in another way. The LCR meter uses a "constant" current as a test signal. The magnitude of the current is automatically set by the LCR meter. Current sent through the sense-coil by the LCR meter creates a magnetic field, and this field, if strong enough to alter the permeability of the Magfluld, could alter the measured Inductance. The field (H) can be approximated by the relation H = N1/1, where N/1 is the turns-to-length ratio of the sense-coil and i is the current. The LCR meter produced a 1 mA current. This implies that the field it generated was about 12.4 A/m, 84 a quantity equal to about 1/3 of the ambient field in which the inductance readings were taken. It is extremely unlikely that such a field would significantly alter the permeability of the Magfluid. I do not have an explanation for the discrepancy between the value of Magfluid permeability suggested by the susceptibility data and the value suggested by the inductance data. If this project is continued at some later date, this issue will have to be addressed. SECTION 6.3: Summary and Conclusions for the Magfluid Experiment Measurements that were obtained with an a.c. susceptibility bridge indicated that the relative permeability of the IGC Magfluid Is approximately 1.34. By substituting that value of permeability into equations [5.5] and [5.6], it was predicted that in a spherical container filled with Magfluid, the field strength would be about 10% less than in the ambient field, and the flux density about 20% greater. It was demonstrated in Section 4.1 that the Integrated open-circuit output voltage from a ring-core fluxgate is proportional to the magnetic field in the Immediate vicinity of the sensor. Therefore, by Immersing the fluxgate in a spherical container filled with Magfluid, the Integrated open-circuit output from the sensor would (Ideally) increase by about 20% If the sensor responded to B, or decrease by 10% If the sensor responded to H. Published material discussed in Section 6.1 suggested that the permeability of the Magfluld might be altered by the magnetic field 85 generated by the fluxgate drive current, as well as any changes In fluid temperature. A test on the sense-coil inductance of a ring-core fluxgate Immersed in Magfluid showed that stray flux from current in the toroidal drive-coil reduced the permeability of the Magfluid. Thus, it was concluded that the relations proposed above would not be verified by constructing a spherical container, filling it with Magfluid, and placing a fluxgate in the sphere. Therefore the experiment was terminated. The results of the Magfluid experiment were negative in the sense that the objective (determining whether a fluxgate responds primarily to B or H) was not realized. However, tests that were conducted demonstrated the applicability of the fluxgate theory of Russell, Narod, and Kollar to quantitative studies of the fluxgate mechanism, it was shown that a knowledge of the sense-coil inductance could be used to evaluate the behavior of a fluxgate immersed in a highly magnetic medium. 86 CHAPTER 7: SUMMARY AND CONCLUSIONS In 1983 Russell, Narod, and Kollar presented an equation of state for the fluxgate magnetometer that was based on the notion that the fluxgate can be described as a modulated inductor. The terms appearing In that fluxgate equation are well-defined, easily measured sensor parameters such as the length-to-turns ratio of the sense-coll, Its inductance, and the load circuit parameters. Thus, the fluxgate theory of Russell et al. circumvents the complications traditionally introduced by discussions of hysteresis and demagnetization of the fluxgate core, and therefore, represents a powerful tool for quantitative studies of the fluxgate mechanism. As pointed out by Russell et al. (1983) and Narod and Russell (1984), the Integrated open-circuit output voltage is expected to be directly proportional to the ambient magnetic field. The constant of proportionality Is simply the product of the length-to-turns ratio of the sense-coil (1/N) and the maximum change in sense-coil Inductance (AL) caused by the drive current. R. D. Russell suggested to me that, when the Inductance L(t) is made to be a series of rectangular pulses, a simple expression relating the short-circuit output to the ambient magnetic field can also be presented in terms of the inductance and the quantity 1/N. Therefore, the fluxgate theory of Russell et al. has been used to produce simple expressions (presented in terms of easily measured sensor parameters) that predict the open-circuit and short-circuit behavior of the fluxgate magnetometer. A derivation of these relations was presented In this thesis. 87 The writer constructed a ring-core fluxgate sensor, as well as the required input and output circuitry in order to Investigate the open-circuit and closed-circuit relations predicted by Russell and his colleagues. A computer-controlled current source was constructed that produced a current waveform with adjustable amplitude, frequency, and duty cycle. High and low impedence output circuits were fashioned from operational amplifiers, though short-circuit tests performed at high frequencies required the use of a current probe. Experimental results that were obtained verified the validity of the predicted open-circuit and short-circuit relations. Specifically, experimental data presented in this thesis demonstrated that The symbol e o c represents the open-circuit output voltage and H is the ambient magnetic field. The quantity A(Li) is the change in the product of the sense-coil inductance and the short-circuit current across a transition in L. This result effectively confirms the validity of the fluxgate equation of Russell, Narod, and Kollar. A knowledge of the sense-coil inductance (L) and Its relation to drive current parameters can easily be used to optimize the fluxgate output. Theoretical aspects of this phenomenon have been extensively documented by 6ao (1985) and Gao and Russell (1987a, 1987b). Experimental data presented in this thesis add substantial support to the 88 recently published theoretical work. It was demonstrated that a bipolar square-pulse drive current caused L(t) to be a series of unipolar square pulses. Since the integrated open-circuit output voltage is proportional to L(t), a Fourier anlysis Indicated that the integrated open-circuit signal would be near a maximum when the drive current was set for a duty cycle of 50%. Experimental data confirmed this. The same method was used to demonstrate that, when using a sinusoidal drive current, the peak output occurs when the drive amplitude (O-P) is approximately equal to the minimum current required to saturate the ring-core. This represents a fundamental problem with using sinusoidal currents to drive low-noise sensors. The behavior of a ring-core fluxgate Immersed in a magnetic fluid was investigated by applying the fluxgate theory of Russell, Narod, and Kollar. This was done in an attempt to discern whether a fluxgate responds primarily to the intensity of the magnetic field (H), or the flux density (B). Theoretical considerations suggested that a simple form of flux concentrator might be fashioned by Immersing a fluxgate in a container filled with magnetic fluid, H the sensor responded to the flux density. The experiment was discontinued when It was discovered that stray flux from the toroidal drive-coll was significantly altering the permeability of the magnetic fluid, thus altering predicted relations between B and H. 89 BIBLIOGRAPHY Acuna, M. H., (1974): "Fluxgate Magnetometers for Outer Planets Exploration", IEEE Transactions on Magnetics, Vol. MA6-10, No. 3, p. 519-523. Arnold Engineering Company, (1978): Pamphlet No. TC-101C, "Tape Wound Cores", p. 21 -22. Bondi, A. A., (1956): "Magnetic Fluids", United States Patent 2-751-352, June 19, 6 pages. Breck, W. G., Brown, R. J. C, and McCowen, J. D., (1981): Chemistry for  Science and Engineering. McGraw-Hill Ryerson Ltd., Toronto, 721 pages. Chantrell, R. W. and Wohlfarth, E.P., (1983): "Dynamic and Static Properties of Interacting Fine Ferromagnetic Particles", Journal of Magnetism and Magnetic Materials, Vol. 40, p. 1-11. Charles, S. W. and Popplewell, J., (1980): "Ferromagnetic Liquids", In Ferromagnetic Materials. Vol. 2, Chapter 8, North-Holland Publishing Company, Amsterdam, p. 509-559. Charles, S. W. and Rosensweig, R. E., (1983): "Magnetic Fluids Bibliography", Journal of Magnetism and Magnetic Materials, Vol. 39, p. 190-220. Chikazumi, S., Taketomi, S., Ukita, M., Mizukami, M., Miyajlma, K, Setogawa, M., Kurihara, Y., (1987): "Physics of Magnetic Fluids", Journal of Magnetism and Magnetic Materials, Vol. 65, p. 245-251. Crangle, J., (1975): "SI Units in Magnetism", Physics Bulletin, Vol. 26, p. 539-540. Eckroth, D.: Editor, (1985): Kirk-Othmer Concise Encyclopedia of Chemical  Technology. John Wiley and Sons, New York, page 699. 90 Feynman, R. P., Leighton, R. B., Sands, M., (1966): The Feynman Lectures on  Physics. Volume 2, Addison-Wesley Publishing Company, Reading, Massachusetts. Gao, I., (1985): "The Theory of Fluxgate Sensor Systems", Ph.D Thesis, University of British Columbia, Vancouver, B.C., Canada. Gao, Z. and Russell, R. D., (1987a): "Fluxgate Sensor Theory: Stability Study", IEEE Transactions on Geoscience and Remote Sensing, Vol. GE-25, No. 2, p. 124- 129. Gao, Z. and Russell, R. D., (1987b): "Fluxgate Sensor Theory: Sensitivity and Phase Plane Analysis", IEEE Transactions on Geoscience and Remote Sensing, Vol. GE-25, No. 6, p. 862 - 870. Intel Corporation, (1978): SDK-85 System Design Kit User's Manual. Manual Order Number 9800451B. Intel Corporation, (1979): MCS-80/85 User's Manual. Jackson, J. D., (1975): Classical Electrodynamics. Second Edition, John Wiley and Sons, New York, 848 pages. Jeans, J. H., (1948): The Mathematical Theory of Electricity and Magnetism. Cambridge University Press, Cambridge, 652 pages. Jefimenko, 0. D., (1966): Electricity and Magnetism: An Introduction to  Electric and Magnetic Fields. Appleton-Century-Crofts, New York, 591 pages. Jones, M. H, (1985): A Practical Introduction to Electronic Circuits, second edition, Cambridge University Press, Cambridge, 278 pages. Kamiyaka, S. and Rosensweig, R. E., (1987): "Magnetic Fluids Bibliography", Journal of Magnetism and Magnetic Materials, Vol. 65, p. 401-439. Karman, T. V. and Biot, M., (1940): Mathematical Methods in Engineering. McGraw-Hill Book Company, Inc., New York, 505 pages. Lanczos, C, (1962): The Variational Principles of Mechanics, second edition, University of Toronto Press, Toronto, 367 pages. 91 Lowes, F. J., (1974): "Do Magnetometers Measure B or H?", Geophysical Journal of the Royal Astronomical Society, Vol. 37, p. 151-155. Margenau, H. and Murphy, G. M., (1943): The Mathematics of Phvsics and  Chemistry. D. Van Nostrand Company, Inc., New York, 581 pages. Menear, S., Bradbury, A., and Chantrell, R. W., (1983): "Ordering Temperatures In Ferrofluids", Journal of Magnetism and Magnetic Material, Vol. 39, p. 17-20. Menear, S., Bradbury, A., and Chantrell, R. W., (1984): "A Model of the Properties of Colloidal Dispersions of Weakly Interacting Fine Ferromagnetic Particles", Journal of Magnetism and Magnetic Material, Vol. 43, p. 166-176. Narod, B. B. and Russell, R. D., (1984): "Steady-State Characteristics of the Capacitively Loaded Fluxgate", IEEE Transactions on Magnetics, Vol. MAG-20, No. 4, p. 592 - 597. Narod, B. B., Bennest, J. R., Strom-Olsen, J. 0., Nezel, F., and Dunlap, R. A., (1985): "An evaluation of the noise performance of Fe, Co, Si, and B amorphous alloys in ring-core fluxgate magnetometers", Canadian Journal of Physics, Vol. 63, No. 11, p. 1468-1472. Narod, B. B., (1988): Personal communication, University of British Columbia, Vancouver, B.C., Canada. Oatly, C, (1976): Electric and Magnetic Fields. Cambridge University Press, Cambridge, 262 pages. Page, L. and Adams, N. I., (1940): Electrodynamics. D. Van Nostrand Company, Inc., New York, 506 pages. Plonsey, R. and Collin, R. E., (1961): Principles and Applications of  Electromagnetic Fields. McGraw-Hill Book Company, New York, 554 pages. Primdahl, F., (1979): "The fluxgate magnetometer", Journal of Physics E: Scientific Instruments, Vol. 12, p. 241-253. 92 Rosensweig, R. E., (1975): "Ferrofluid Compositions and Process of Making Same", United States Patent 3-917-538, Nov. 4, 12 pages. Rosensweig, R. E., (1979): "Fluid Dynamics and Science of Magnetic Liquids", in Advances in Electronics and Electron Physics, Vol. 48, p. 103-199. Russell, R. D., Narod, B. B., and Kollar, F., (1983): "Characteristics of the Capacitively Loaded Fluxgate", IEEE Transactions on Magnetics, Vol. MAG-19, No. 2, p. 126-130. Scouten, D. C, (1970): "Barkhousen Discontinuities in the Saturation Region", IEEE Transactions on Magnetics, Vol. MAG-6, p. 383-385. Serson, P. H., and Hannaford, W. L. W., (1956): "A Portable Electrical Magnetometer", Canadian Journal of Technology, Vol. 34, p. 232-243. Soffge, F. and Schmidbauer, E., (1981): "A.C. Susceptibility and Static Properties of an Fe 30 4 Ferrofluid", Journal of Magnetism and Magnetic Materials, Vol.24, p. 54-66. SOILTEST, (1966): "MS-3 Magnetic Susceptibility Bridge Operation and Application Manual", SOILTEST, Inc., 2205 Lee Street, Evanston Illinois 60202. Whitmer, R. M., (1962): Electromagnetics, second edition, Prentice-Hall, Inc., Englewood Cliffs, N.J., 357 pages. Zahn, M. and Shenton, K. E., (1980): "Magnetic Fluids Bibliography", IEEE Transactions on Magnetics, Vol. MAG-16, p. 387-415. 

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