"Science, Faculty of"@en . "Earth, Ocean and Atmospheric Sciences, Department of"@en . "DSpace"@en . "UBCV"@en . "Cover, Keith Sean"@en . "2010-06-20T00:35:18Z"@en . "1986"@en . "Master of Science - MSc"@en . "University of British Columbia"@en . "The fluxgate electric field meter is a new solid-state instrument capable of measuring slowly varying electric fields equally well in an insulating or conducting medium with high resolution. The field meter determines the strength of an electric field by measuring the potential difference between two insulated electrodes immersed in the electric field. The electrode insulation is important because it eliminates the contact potential noise between the conductive material of the electrode and a conductive medium. The capacitance of the insulated electrodes is of the order of 10pF; therefore measurement of the potential difference between the electrodes requires an electrometer with extremely low bias current and high input resistance. A novel electrometer was concieved and designed to accomplish this task. The new electrometer is a solid-state analogue of the vibrating-capacitor electrometer and is claimed to have zero bias current. MOS capacitors are substituted for the vibrating capacitor. The noise level for the crude, proof-of-concept electrometer was 220\u00CE\u00BCVHz[sup -1/2] at 1Hz. This noise level is only 20 dB worse than the best commerical device I could locate. A crude, proof-of-concept electric field meter constructed with the electrometer had a high frequency cutoff of 150Hz and low frequency cutoff of 100\u00CE\u00BCHz. Possible applications of the fluxgate electric field meter include measurement of atmospheric and interplanetary electric fields, measurement of static charge buildup, and geophysical surveys."@en . "https://circle.library.ubc.ca/rest/handle/2429/25865?expand=metadata"@en . "The F luxga te Electr ic F i e l d M e t e r A Feasibil i ty S tudy by K e i t h Sean Cover B.Sc, The University of Waterloo, 1983 A Thesis S u b m i t t e d in P a r t i a l Fulf i lment of The Requirements for the Degree of M a s t e r of Science in The Facul ty of Gradua te Studies Depar tment of Geophysics and A s t r o n o m y We accept this thesis as conforming to the required standard The Unive r s i ty of B r i t i s h C o l u m b i a October, 1986 @ Keith Sean Cover, 1986 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of (J eofiAyS/C5 j- $s fre/lp/H y The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date )E-6 (3/81) Abstract The fluxgate electric field meter is a new solid-state instrument capable of measur-ing slowly varying electric fields equally well in an insulating or conducting medium with high resolution. The field meter determines the strength of an electric field by measuring the potential difference between two insulated electrodes immersed in the electric field. The electrode insulation is important because it eliminates the contact potential noise between the conductive material of the electrode and a con-ductive medium. The capacitance of the insulated electrodes is of the order of 10pF; therefore measurement of the potential difference between the electrodes requires an electrometer with extremely low bias current and high input resistance. A novel elec-trometer was concieved and designed to accomplish this task. The new electrometer is a solid-state analogue of the vibrating-capacitor electrometer and is claimed to have zero bias current. MOS capacitors are substituted for the vibrating capacitor. The noise level for the crude, proof-of-concept electrometer was 220/tiVHz _ 1 / 2 at 1Hz. This noise level is only 20 dB worse than the best commerical device I could locate. A crude, proof-of-concept electric field meter constructed with the electrometer had a high frequency cutoff of 150Hz and low frequency cutoff of 100/iHz. Possible appli-cations of the fluxgate electric field meter include measurement of atmospheric and interplanetary electric fields, measurement of static charge buildup, and geophysical surveys. ii Table of Contents Abs t r ac t 1 1 Table of Contents \u00C2\u00BB\u00C2\u00BB Lis t of Figures v Acknowledgements V 1 1. In t roduc t ion 1 2. Theory \u00E2\u0080\u00A2 3 2.1 History 4 2.2 Nonideal Amplifiers 5 3. The Var iab le -Capac i to r M o d u l a t o r 6 3.1. Equation of the Modulation Circuit 9 3.2. The Fluxgate Mechanism 10 3.3. Electrical Properties of the Sensor 13 4. Rea l i za t ion \u00E2\u0080\u00A2 ^ 4-1. The Variable Capacitor 17 4.2. A Nonlinear Capacitor ; 20 4.3. Feedback Voltage 22 5. E x p e r i m e n t a l Results 23 5.1. Feedback Linearity 23 5.2. Sensitivity 24 5.3. Directionality of Instrument 25 5.4. Drift Rate 2 5 5.5. Potential of the Instrument 26 5.6. Instrument Noise 27 6. The Var iab le -Capac i to r Elec t rometer 27 i i i 7. Conclus ions 28 B i b l i o g r a p h y 30 A p p e n d i x A . Elec t ronic C i r c u i t r y of the Instrument 65 A p p e n d i x B Deta i l s of the Var iab le Capaci tor 74 iv Lis t of Figures 1 Collection capacitor in ambient electric field 32 2 Equivalent circuit of the electric field meter 34 3 Variable-capacitor modulation circuit 36 4 Fluxgate magnetometer operation 38 5 Variable capacitor configuration 41 6 Voltage-versus-capacitance curves for M O S O M capacitors 43 7 Variation of the time varying capacitor's capacitance 45 8 M O S O M nonlinear capacitor design . \u00E2\u0080\u0094 47 9 M O S O M nonlinear capacitor operation 49 10 Variable-capacitor modulator with feedback 51 11 Feedback Linearity 53 12 Sensitivity-versus-load resistance curve 55 13 Directionality of instrument 57 14 Drift rate 59 15 Potential of the instrument 61 16 A fluxgate electrometer ..63 A l Clock and drive circuit 68 A2 Variable-capacitor modulation circuit 70 A3 Synchronous detector 72 v Acknowlegdements I would like to thank John Bennest for his instruction and help with the elec-tronics for the fluxgate electric field meter and for his explanation of the fluxgate magnetometer. To Dr. B. Barry Narod I owe thanks for numerous discussions on the principles and theory of the fluxgate magnetometer and for help with the theory of the fluxgate electric field meter. I would like to thank my supervisor, Dr. R. Don Russell, for pointing out the similiarities between the vibrating-capacitor electrome-ter and the fluxgate electric field meter. I am especially thankful to Dr. Russell for allowing me the freedom and providing the support for me to follow my own ideas. I am also greatful to Dr. L. Young, Dr. P. Janega and Dr. D. L. Pulfrey, all of the Department of Electrical Engineering at the University of British Columbia, for their assistance in the fabrication of the semiconducting wafers used for constructing the fluxgate electric field meter. To D. Schreiber I owe thanks for his help in construction of equipment to test the proof-of-concept electric field meter. The Natural Sciences and Engineering Research Council of Canada was the pri-mary source of funds for the research on the fluxgate electric field meter under the Operating Grant A0720 and the Strategic Grant G1050. Energy, Mines fc Resources Canada, Earth Sciences also assisted under Research Agreement No. 166. vi 1. In t roduct ion Of what use is an electric field meter which can measure electric fields equally well in an insulator and a conductor? As Alfven (1981) pointed out, it is natural to present the results of space exploration with pictures of the magnetic field configuration because magnetic fields are far easier to measure than electric fields. The ease of measurement of magnetic fields in space is primarily due to the ruggedness, small size, low power consumption and high resolution of modern fluxgate magnetometers (Acuna 1974). To be useful for measurements of electric fields in space an electric field meter must have noise level of less than 1 0 / x V m _ 1 H z _ 1 / ' 2 over the frequency bandwidth from 1Hz up to 2MHz (Shawhan et al 1981). (Throughout this thesis I will be specifying noise by amplitude spectral density which is covered in many publications including Motchenbacher and Fitchen (1973).) It also must function at frequencies lower that 1Hz but these specifications are not well determined. Another example of a potentially useful application of an electric field meter is measurent of electric fields in the ocean. The majority of present methods require electric contact with sea water (Filloux 1974). The electrochemical reactions between the sea water and the contacting electrode generate noise which significantly contam-inates the signal. Since measurement of electric fields in sea water by an electric field meter would not require electric contact with the sea water, the noise due to contact potential should be eliminated. According to Filloux, the power spectrum, P{f), of the average natural electric activity at the sea floor in the 300nHz to 3mHz band roughly follows the trend P = k/f , where f is the frequency in Hertz, and k is 1 0 _ 2 / i V 2 m - 2 . Present methods using electric contact with sea water require 1 a separation between electrodes on the order of 100m or more to overcome contact noise and to achieve the desired 1% resolution. Conventional uses of electric field meters include measuring electric field in clouds (Rust 1978) and measuring electrostatic buildup in locations, such as oil tanker holds and on the surfaces of plastics, to aid in the prevention of explosions and fires (Haenen 1977). Other possible uses of a high resolution wide bandwidth electric field meter include electric field measurement in geophysical surveys such as resistivity, induced polarization and magnetotellurics. For my M . Sc. thesis in geophysical instrumentation, I conceived, designed, constructed and tested a solid-state fluxgate electric field meter. The instrument consisted of a novel electrometer measuring the potential difference between a pair of insulated electrodes. The prototype was constructed primarily to prove that the concept worked and little effort was made to obtain good specifications in noise or other parameters. None the less, measurements of the noise and other parameters were made to determine a worst case performance of the instrument. The prototype constructed was sensitive to frequencies from 150Hz down to 100/xHz . The dominant noise source of the electric field meter was the noise of the electrometer. At 1Hz the electrometer had a noise of 220/xVHz - 1 / 2 . For the electric field meter prototype the separation between the electrodes was 6.6mm. The resulting noise level of the electric field meter at 1Hz is 3 3 m V H z _ 1 / ' 2 m _ 1 . The noise of the electric field meter can be reduced by increasing the separation between the electrodes or reducing the noise of the electrometer. The instrument is called an \"electric field meter\" as it can measure slowly varying electric fields equally well in an insulating or conducting medium. The field meter is 2 referred to as \"solid-state\" because it has no moving parts. If the fluxgate electric field meter is analysed in a certain way its mechanism for measuring the vector component of an electric field is similiar to that used by the fluxgate magnetometer (Primdahl 1979); thus the modifier \"fluxgate.\" 2. Theory Electric field meters, such as the fluxgate, which are able to measure electric fields with no current flowing along the field lines cannot extract energy from the field to generate a signal and must extract the required energy from another source. Consider two electrodes in an ambient electric field as shown in Figure 1. The two electrodes together form a capacitor which will be called the collection capacitor and has capacitance Cc. The electric field in which the collection capacitor is immersed induces a potential difference, V e , across the electrodes. Ignoring no edge effects, the value of Vc will be equal to X \u00E2\u0080\u00A2 E ; where X and E are vector quantities. The vector X has an amplitude that is the distance between the electrodes and a direction that is normal to the surface of the electrodes. The vector E describes the strength and direction of the ambient electric field. If an electrometer with zero input bias current and infinite input resistance is connected to the collection capacitor electrodes Ve can be measured accurately. Assuming X is known, the vector component of E parallel to X is calculable. But no one has invented an electrometer with both these ideal characteristics. Finite input resistance and a nonzero input bias current will cause the value of Ve to drift, interfering with the measurement of the ambient electric field. The finite input resistance will dissipate the energy stored in the capacitor until there is zero potential 3 drop across the capacitor. The energy dissipated by the finite input resistance must somehow be replaced. 2.1. History Once the collection capacitor has been discharged it must somehow be recycled before it is again possible to measure the electric field. Flipping the capacitor upside down relative to the electric field induces a new potential difference across the elec-trodes of \u00E2\u0080\u00942E \u00E2\u0080\u00A2 X and accomplishes the recycling. The effect is the same as if the direction of the electric field had been reversed. The energy the collection capacitor supplies as a charge flow comes from the work done in flipping the electrodes over in the electric field. Russelvedt in 1925 invented a instrument using this principle and called! it an arigmeter (Chalmer 1967). An arigmeter has the major disadvantage of requiring a mechanical instrument to flip the electrodes. Another mechanical method for measuring an electric field without loading the field was described by de Saussure in 1779 (Chalmer 1967). If one of the electrodes of the collection capacitor is moved away from the other along the field lines, a new charge difference is induced between the electrodes which can again be measured by an electrometer. Moving the electrodes back and forth generates a signal proportional to the ambient electric field. The energy to generate the signal comes from the energy supplied to move the electrodes. Imianitov (1949) suggested a design for a solid-state electric field meter using a ferroelectric material. The suggested field meter, as described by Chalmer (1967), in-volved using a strong electric field to saturate the dielectric constant of a ferrolelectric material. The changing dielectric constant of ferroelectric was intented to modulate the ambient electric field the ferroelectric was immersed in. The modulated ambient 4 electric field could then be detected with capacitive electrodes. In the early stages of my research into the fluxgate electric field meter, before I had found the suggestion by Imiantov, I conducted experiments with this type of field meter and I obtained no useful results. I can only surmise Imiantov arrived at the same conclusion as I could find no other references to this suggestion. 2.2. Nonideal Amplifiers Since I was interested in designing a high resolution, rugged, low power electric field meter, I chose to use a solid-state electrometer for the electrometer in Figure 1. Figure 2 shows a more realistic schematic diagram of the electric field meter with additional components that correspond to the nonideal characteristics of the elec-trometer. The electrometer circuit I selected for the fluxgate electric field meter involves a modification of the circuit in Figure 2; but its nonideal operating charac-teristics can be handled mathematically in a similar manner. The aim of this section is to determine what specifications of nonideal electrometers a solid-state electric field meter requires for satisfactory performance. Simple circuit analysis will give the effects of input offset voltage, Vos , input bias current, , and finite input impedance of the electrometer on the performance of the electric field meter; therefore only the results will be given. By measuring the performance of the electric field meter it is possible to determine the electrometer's input characteristics. An initial potential difference induced across the collection capacitor will decay exponentially. The time constant of the decay will be RinCc. The output voltage of the electrometer, Vout , will approach the value hRin + Vos at large time. In an ideal world hRin + o^s would remain constant. Unfortunately, in real electrometers it varies with time by an amount that is a significant part of 5 its value. This variation is a source of noise. In calculating the relationship between the ambient electric field, E , and Vout an ideal electrometer is normally assumed; therefore the ambient electric field is assumed to be related to the output voltage by - E - X = V o u t . As an example, consider a collection capacitor with an separation, X , of 8mm and a capacitance, Cc , of 15pF. These are the parameters of the collection capacitor used by the prototype. For resolution of the signal in the ocean bottom electric field problem mentioned in the introduction, a low frequncy cut off of 300nHz is necessary (Filloux 1974). Treating the circuit formed by the collection capacitor and the input resistance of the electrometer as a simple resistor-capacitor circuit, to achieve the necessary cutoff frequency an input resistance of greater than 4 X 10 1 6O is needed. The highest input resistance commerieal solid-state electrometer that I could find is the Keithly 642. The guarded M O S F E T input stage of the 642 has an input resistance of 10 1 6fi and a noise of 20/ iVHz~ 1 / 2 at 1Hz. It also has a input bias current of 5x 10~ 1 7 A within the temperature range of 22C to 24C. An amplifier with these specifications would result in an electric field meter, using the above collection capacitor, with a 3 decibel low frequency cutoff of IfiHz and an IbRin + Vos \u00C2\u00B0f 500mV. These specifications, while they may be useful for some electric field meter applications, are not good enough for many electric field meter applications. In particular, the large voltage drop across the input resistance due to the bias current will cause serious drift. Also, the limited temperature range of the instrument would cause problems. An electrometer with better specifications is desirable. 6 3. The Var i ab l e -Capac i to r M o d u l a t o r Figure 3 show a simplified schematic of the modulation part of the electrometer used to measure the voltage across the collection capacitor. The modulation part of the electrometer consists of a time varying capacitor and a load resistor placed in series with the collection capacitor. Strictly speaking, the collection capacitor is not part of the electrometer but is a voltage source. But the electrometer as shown will only function with capacitive voltage sources. Section 6 of this paper will cover the electrometer configuration which will handle noncapacitive sources. An A C amplifier across the load resistor and a synchonous detector fed by the A C amplifier are the two parts of the electrometer not shown in Figure 3. The modulation circuit, including the collection capacitor, will be called the variable-capacitor modulator. The electrometer will be called the fluxgate electrometer or simply the electrometer. If the capacitance of the time varying capacitor is initially very much smaller than collection capacitor's all the charge will remain on the collection capacitor. But when the time varying capacitor's capacitance approachs the capacitance of the collection capacitor, charge will flow from the collection capacitor through the load resistor and on to the time varying capacitor. The flow of charge through the load resistor will be proportional to the initial voltage cross the collection capacitor. Therefore, by Ohm's Law the voltage drop across the load resistor will be proportional to the initial voltage across the collection capacitor. Varying the capacitance of the time varying capacitor periodically with time will generate a periodic signal across the load resistor. Hence, the amplitude of the periodic signal across the load resistor will be proportional to the initial voltage drop across the collection capacitor. 7 For the fluxgate electric field meter the time varying capacitor consists of two nonlinear capacitors. At this point it is useful to outline the definitions I will use to describe linear and nonlinear capacitances. The definitions I have chosen are those used by Locherer and Brandt (1982). A capacitor is linear if it obeys the relationship VC = q ; where V is the voltage across the capacitor, q is the charge stored in the capacitor and C is the capacitance of the capacitor and is independent of the value of V . A nonlinear capacitor is a capacitor which does not have a linear relationship between charge and voltage. The definition of the capacitance of a nonlinear capacitor is ambiguous. In this thesis I will use the differential definition of capacitance. By this definition the capacitance is defined as C{V) \u00E2\u0080\u0094 [dq / dv)v=v . The node between the collection and time varying capacitor is only capacitivley coupled to the rest of the circuit; therefore conservation of charge dictates that the charge on the collection capacitor when the capacitance of the time varying capacitor is very small must be equal to the sum, at any time, of the charge on the collection and variable capacitors. For convience, this sum of charges will be called the induced charged. A leakage path for electrons from the node between the collection and variable capacitors will allow the induced charge to leak away and cause a signal drift. The resistance the collection and variable capacitors must be very large to keep the leakage of the induced charge very small. The variable-capacitor modulator is almost identical to that of the vibrating-capacitor electrometer (Palevsky et al 1947); but, there are two important differences. One difference is that the variable-capacitor electrometer uses a solid-state modulator while the vibrating-capacitor electrometer uses a mechanical one. Solid-state modu-lation has the advantages, among others, of needing less power, of being more rugged and of supporting modulation to a higher frequency. A solid-state modulator should 8 be capable of modulation at the megahertz range. The prototype electric field meter constructed uses a carrier of 32 KHz . High modulation frequency is valuable because the power generated by the variable-capacitor modulation circuit increases with in-creased carrier frequency. The extra power will reduced the effect of the thermal noise of the load resistor as well as most other noise sources in the circuit. The second important difference between the modulator used in the vibrating-capacitor electrometer and the variable-capacitor modulator is the way in which the charge is delivered to the collection capacitor. In the vibrating-capacitor electrometer the charge is delivered via a high resistance resistor to the node between the vari-able and collection capacitors (Palevsky et al 1947). The end of the high resistance resistor not attached to the node between the variable and collection capacitors is attached to the voltage source of which we are interested in measuring the poten-tial. This configuration will be covered in more detail in Section 6 on the fluxgate electrometer. In contrast, the fluxgate electric field meter has its charged induced on the collection capacitor by the potential difference induced between the collection capacitor electrodes by the ambient electric field. S.l. Equation of the Modulation Circuit The differential equation describing the variable capacitor modulator follows eas-ily by application of Thevenin's theorem and Kirchhoff's Law (Millman and Halkias 1967). Besides the collection capacitor, C c , the variable capacitor, Cv(t), the load resistance, R, and the potential drop across the unloaded collection capacitor due to the ambient electric field, Ve, the other required parameters are the charges on the collection and variable capacitors, qc and qv respectively. Thevenin's theorem allows Ve to be placed in series with the collection capacitor to represent the effect 9 of the ambient electric field. The induced charge is accounted for by the Vc term; therefore, since charge must be conserved, the charge on the collection and variable capacitors must be equal. In other words qv \u00E2\u0080\u0094 qc . Applying Kirchhoff's law to Figure 3, taking into account Thevenin's theorem, gives the equation ( i ) Equation (l) can be simplified slightly by the change of variables yielding The mathematical difficulty in solving equation (3) depends on the choice of function o,{t). Up to this point it as be assumed that the only source of charge has been the induced charge caused by the ambient electric field. If there is a charge on the collection or variable capacitor when the ambient electric field is zero it will show up in the electric field measurement as an apparent electric field offset. The additional charge could be due to static charge buildup or some other cause. With time, the additional charge will drain off through the leakage resistance and contribute to the drift of the electric field meter. 10 3.2. The Fluxgate Mechanism You may be wondering why I chose to call this instrument a \"fluxgate\" electric field meter. The reason is that the electric field meter's method of modulating the ambient electric field developed from the fluxgate magnetometer. To understand this analogy it is useful to understand how the fluxgate magnetometer senses magnetic The object of magnetic fluxgating is to make a slowly varying magnetic field mea-sureable without mechanical motion of a sense coil. Primdahl (1979) gives a brief description of the method. It was explained to me in more detail by John Bennest, a technician in my lab. By Faraday's law, inducing an electromotive force in a conduc-tor requires relative motion between the conductor and the ambient magnetic field. Since the sense coil must remain stationary, the ambient magnetic field must change to induce a signal in the sense coil. The change in the magnetic field is accomplished by modulating the permeabililty of a ferromagnetic core. The modulated perme-abililty in turn distorts the magnetic field lines near the core. Figures 4(a) and 4(b) are diagrams of the field line distortions. The time varying distortion of the field lines induces a signal in the sense coil. The induced signal will show up as a voltage drop across the load resistor which in turn can be measured. The equation describing the resistively loaded fluxgate magnetometer is an ana-logue of the equation for the fluxgate electrometer given in equation (3). Russell et al (1983) gave the equation describing the fluxgate magnetometer as fields. 0 + R = ~AHL(t))i n at (4) Lit) l l where is the magnetic flux in the sense coil, l/n is a geometeric constant of the sense coil, R is the load resistor across the sense coil and L(t) is the time varying inductance of the sense coil. The inductance of the sense coil is time varying because of the time varying permeability of the ferromagnetic core. It is useful to remember that magnetic flux, , is equal to L(t)i; where t is the current through the sense coil. When the inductance of the sense coil changes the flux is conserved. In the fluxgate electric field meter when the capacitance of the variable capacitor changes conservation of charge applies. Therefore, the flux of the fluxgate magnetometer is analogous to the charge of fluxgate electric field meter. The method normally used in the fluxgate magnetometer for modulating the fer-romagnetic core is straight forward. A second coil is placed around the ferromagnetic material of the core. This second coil is referred to as the drive coil. The drive coil is used to generate a periodic very intense magnetic field which saturates the ferro-magnetic core thus modulating the permeability. The drive and sense coil are usually positioned relative to each other so their mutual inductance is small. The small mu-tual inductance prevents the very intense field from the drive coil from dominating the signal in the sense coil caused by the ambient magnetic field. The fluxgate electric field meter measures a slowly varying ambient electric field in the same way as the fluxgate magnetometer measures a slowly varying ambient magnetic field\u00E2\u0080\u0094 by modulating it. The modulation is accomplished by the time varying capacitor. The time varying capacitor forces the induced charge to move back and forth between the variable and collection capacitors. The motion of the induced charge generates a time varying electric field around the collection capacitor electrodes. Instead of measuring the modulated electric field with a second set of electrodes, I have chosen to measure the rate of charge flow from the variable to 12 collection capacitors which is also proportional to the ambient electric field. Measur-ing the rate of charge flow between the variable and collection capacitors makes the fluxgate electric field meter easier to construct. S.S. Electrical Properties of the Sensor The important properties of any electrical sensor are sensitivity, impedance, up-per and lower limits on bandwidth, and noise level. The lower limit on bandwidth is determined by drift and noise. Sensor drift was covered during the treatment of nonideal electrometers earlier in this thesis. The prediction of the noise level is diffi-cult at best and will not be a concern of this thesis. But the sensitivity, impedance and upper limit of the bandwidth follow from the solution of equation (3). Choosing C it) _ J \u00C2\u00B0i f o r ntp allows for a closed-form solution of equation (3); where tv is the period of one com-plete cycle and t\ is the time after the starting transition of a cycle at which the second transition occurs. Equation (3), using the time varying capacitance given in equation (5), is also easy to implement with electrical circuitry to a good approxi-mation. An upper limit on the bandwidth follows from the homogeneous solution to equa-tion (l), denoted q\u00E2\u0080\u009E[t) \u00E2\u0080\u00A2 A method of finding the homogeneous response of linear differential equations with periodic coefficients was first published by Floquet (1883); a modern reference is Nayfeh and Mook (1979). Russell et al (1983) applied the tech-nique to the fluxgate magnetometer. Floquet showed the homogeneous solution has the form of a periodic function times an exponential decay. The time constant of the 13 exponential decay determines the transient response time and thus the upper limit on bandwidth of the variable-capacitor modulation circuit. Using the time varying capacitance described in equation (5), the form of the homogeneous solution depends upon the fact that if q\u00E2\u0080\u009E(t) is a solution to equation (3) then so must qv{t + tp) due to periodicity and linearity; therefore, the ratio q\u00E2\u0080\u009E{t) j q^(t + tp) is a constant and hence, the transient response time, 7 , is 7 = ~ ~ t p (6) The upper limit on the bandwidth is then 1/(2^7) . Calculating qv{tp)/q\u00E2\u0080\u009E{0) is straightforward for the Cs(t) function given in equa-tion (5). The discontinuities in Ca(t) break equation (3) into 2 separate equations which apply during alternate time periods. The conservation of charge is the conti-nuity condition. The two parts of equation (3) yield the homogenous solutions q${t) = A n e ~ t n R C l ) for ntp < t < ntp + i i ; n = 0 , 1 , 2 . . . (7) q%(t) = B n e ~ t n R \u00C2\u00B0 2 ) for ntp + ti < t < {n + l)tp. The continuity conditions at the transition points are given by conservation of charge, thereby giving the value of the coefficients, A n and Bn . The only exception is An which is determined by the initial condition qv(0) . Solving equation (7) for 14 (\u00C2\u00B0) produces + tp \u00E2\u0080\u0094 t\ RC2 (8) \u00E2\u0080\u0094 exp \u00E2\u0080\u0094 Substituting into equation (6) yields 7 = (9) the transient response time, which leads directly to the upper limit on the bandwidth. Before it is possible to calculate the sensitivity of the fluxgate electric field meter, it is necessary to define the sensitivity. In this thesis I will define the sensitivity, S , of the fluxgate electric field meter to be the root-mean-square average of the modulation frequency voltage across the load resistor at divided by the strength of the electric field normal to the collection capacitor electrodes. The root-mean-square average of the modulation frequency voltage across the load resistor will be called the \"second harmonic voltage.\" The second harmonic voltage is noted by the symbol V2nd . The reason for the modifier \"second harmonic\" will become clear later in this thesis. This definition of sensitivity ties together well with the definition of \"conversion efficiency\" defined by Palevsky et al (1947) for the vibrating-capacitor electrometer. They defined conversion efficiency, a , as V2nd divided by the input voltage. The corresponding definition for the fluxgate electric field meter would be a = V2nd/Ve; where Ve is the potential drop across the unload collection capacitor due to the ambient electric field. Thus, the sensitivity and the conversion efficiency are related by the equation S = aX ; where X is the amplitude of the vector X . Because X 15 is a measure of the separation between the collection capacitor electrodes and can easily be measured only a method to calculate the conversion efficiency is needed. Calculation of the conversion efficiency of the field meter involves calculating the steady-state solution of equation (3). A method of finding a steady-state solution, for this type of problem, was presented by Narod and Russell (1984). To obtain the steady-state solution, one needs the particular solution to equation (3) (Boyce and DiPrima 1977). Equation (3), using the function for Cv(t) defined in equation (5), yields the two-part particular solution q\{t) = VtOx for ntp +12VI 15p IK I M H i 15p >10Mi O / J I S 1K1 (LL-Black) 1> Godj (LL-Blut) \u00E2\u0080\u00A2 100K 1M5I 6SK (AA) -t> +12V 68 K; IWi (LL-Purplt) -wv* 1> v -1% \u00E2\u0080\u009E (LL-Grey)! \u00E2\u0080\u009E (BB) 4> God O -12V| 2F; Clock and Drive Circuit Figure A l : Figure A 2 Variable-capacitor modulation circuit. It includes the collection ca-pacitor, C c , and the nonlinear M O S O M capacitors, C + and C_ . The complete circuit shown was immersed in the ambient electric field in the test chamber. A l l other circuitry was located outside the test chamber and connected to modulation circuit via a 8m long wires. A l l resistances are in ohms and all capacitances are in farads. National Semiconductor LF353 operational amplifier was used exclusively. 70 (LL-Grey) A A Y- ^/ (LL-Purpie) - (IlJWhite) 100 (LL- Red) -v W \u00E2\u0080\u0094 \u00E2\u0080\u00A2 U3+ 1 2 V .240 (LL-Orange) - V W ^ 1> Sig (LL-Black) - "Thesis/Dissertation"@en . "10.14288/1.0052979"@en . "eng"@en . "Geophysics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "The fluxgate electric field meter : a feasibility study"@en . "Text"@en . "http://hdl.handle.net/2429/25865"@en .