Measurement of magnetic fields using the voltage generated by a vibrating wire
MMeasurement of magnetic fields using the voltage generatedby a vibrating wire
Jason Gilbert a and Cameron Baribeau Canadian Light Source, University of Saskatchewan, Saskatoon, Canada
August 25, 2020
Abstract
A vibrating wire may be used as an instrument with a variety of applications, one ofwhich is the measurement of magnetic fields. Often, the magnetic fields are determined bymeasuring the amplitude of the wire vibration under the action of a Lorentz force. Thoughgenerally adequate, this approach may be inconvenient in certain circumstances. One ofthese occurs when it is necessary to measure the amplitude of high-frequency vibration, asthe amplitude is expected to decrease linearly with frequency, and thus becomes harder tomeasure. Another example may be found in situations where the sensor must operate over awide range of vibration frequencies. In this case the sensor will be unresponsive to specificfrequencies of wire vibration, which are determined by the placement of the sensor. Thismeans that for the instrument to be robust, the sensor must be precisely mobile, or multiplesensors must be used.Here a technique which may be used to supplement the displacement sensor is described.This technique makes use of the voltage generated by the motion of the wire in the mag-netic field under measurement. It is predicted that the technique may be more suitable formeasurements requiring high frequency vibration, and is sensitive to all frequencies of vibra-tion. Measurements of a magnetic field obtained using this technique are compared to thosefound using only a displacement sensor, and the benefits and drawbacks of the technique arediscussed. a Corresponding author. Electronic mail: [email protected] a r X i v : . [ phy s i c s . acc - ph ] A ug Introduction
In the field of accelerator physics the vibrating wire technique [1] is utilized to measure the magneticfield of a variety of devices. These include quadrupoles [2, 3, 4, 5], undulators [6], and wigglers[7, 8, 9, 10, 11].This technique has recently been used at the Canadian Light Source [11] in an attempt to mea-sure the magnetic field along the centerline of an insertion device, a hybrid in-vacuum wiggler witha period of 80 mm. While taking these measurements, it was noted that the electrical impedanceof the wire appeared to be changing. It was proposed that this change may be explained by anelectromotive force (emf) generated by the motion of the wire. To test the validity of this expla-nation, the hypothetical emf generated by the wire was calculated, and in doing so, it was foundthat it may be possible to obtain useful information about the magnetic field from measurementsof the emf. For an example of this principle applied to the measurement of fluid viscosity, see Ref.[12].This contribution is organized as follows. In Section 2 the theory of the vibrating wire tech-nique is briefly explained; after which, a description of the emf generated by the vibration of awire, in terms of this theory, is developed. In Section 3 the application of this emf for magneticmeasurement is discussed. In Section 4 methods used to test the practicality of this approach,and the results of these tests, are discussed. Finally, the conclusions drawn from this work aresummarized in Section 5.
To provide context for the measurement method being proposed, the conventional method mustfirst be described. This description will be minimal, and those who are interested in more detailmay refer to Ref. [1].The principle of the vibrating wire technique is that by measuring the displacement of a wiredue to a Lorentz force acting along its length, the magnetic field which causes the Lorentz force maybe inferred. It is well known that the transverse displacement of a taut wire under the influenceof some driving force is described by
T u zz − γu t + µu tt = F . (2.1)Here u = u ( t, z ) is the transverse displacement of the wire, and F = F ( t, z ) is the force per unitlength acting on the wire; t is the time at which the wire is observed, and z is the direction whichthe wire extends. Subscripts are used to denote differentiation with respect to the subscriptedsymbol. The coefficients of the differential terms are T the tension in the wire, µ the mass perunit length of the wire, and γ the damping coefficient.In the context of magnetic field measurement, this differential equation relates a Lorentz force F ( t, z ) acting on the wire to the transverse displacement u ( t, z ) of the wire; if the current inthe wire is known, the magnetic field acting on the wire may be determined by measuring thedisplacement of the wire. A diagram which defines the coordinate system, depicting an insertiondevice, may be found in Figure 2.1. 2 𝒙 𝒚 Girders
Wire
Figure 2.1: A cross section of the measurement apparatus. The wire extends in the z direction,perpendicular to the page. The magnetic field produced by the girders is in the y − direction. Themotion of the wire is ideally confined to the plane y = 0.It has been shown [1] that if the wire is held fixed at points which are outside the influence ofthe magnetic field, a solution to the above differential equation may be obtained in the form of asine-series. This means that the magnetic field may be expressed as B ( z ) = ∞ (cid:88) n =0 B n sin (cid:16) nπL z (cid:17) , (2.2)where the series coefficients B n are related to the displacement of the wire, and L is the distancebetween the points where the wire is held fixed. If the wire is underdamped, and the current inthe wire is of the form I ( t ) = I e iωt , (2.3)which is meant to represent a sinusoidal signal with amplitude I and angular frequency ω , writtenin the form of a complex exponential for convenient mathematical manipulation, then the seriescoefficients are related to the vibration of the wire by u n ( t, z ) = B n sin (cid:16) nπL z (cid:17) I cos ( ωt − φ ) µ [( γ (cid:48) ω ) − ( ω − ω n ) ] / . (2.4)Here u n is the displacement of the wire when driven at a frequency near the n th harmonic of thefundamental frequency of the wire; ω n is the frequency of the n th harmonic, which corresponds tothe n th series coefficient, and is given by ω n = nπL (cid:115) Tµ ; (2.5)3 (cid:48) = γµ − , and φ is the phase shift between the driving current and the wire vibration, defined astan φ = γ (cid:48) ωω n − ω . (2.6)Examining equation (2.4), it can be seen that the measured amplitude of the wire vibration isgiven by the time-independent component of the equation, A n = B n sin (cid:16) nπL z s (cid:17) I µ (cid:2) ( γ (cid:48) ω ) − ( ω − ω n ) (cid:3) − / , (2.7)where z s is the position of the displacement sensor. From this the series coefficients may beobtained by fitting the following equation to measurements of vibration amplitude, A n = a n (cid:2) c n ω − ( ω − b n ) (cid:3) − / , (2.8)assuming that T , µ , and γ are effectively constant for the curve. From which the series coefficientsmay be found to be | B n | = a n µI sin (cid:16) nπL z s (cid:17) − . (2.9)One should note that equation (2.7), from which the series coefficients are calculated, is strictlyof one sign, as determined by the sensor position z s . This means that it can not by itself beused to calculate the corresponding magnetic field via equation (2.2), as it must be known whichof the coefficients are positive and which are negative. This information may be obtained frommeasurements of the phase shift between the wire vibration and the driving current, as was donein [1].To briefly elaborate on the need for assuming the wire is underdamped in the derivation ofequation (2.4), it is required so that the curves of equation (2.4) are sufficiently localized, so thateach u n is near zero in the vicinity of u n ± . If this is not the case, equations (2.7)-(2.9) will bebetter represented as a sum over n , and a more complicated approach to fitting may be required. In the previous section it was explained how an electrical signal may be used to induce motion ina wire by means of a Lorentz force, and that this motion acts as a signal which carries informationabout the strength of the magnetic field along the length of the wire. Here an alternative approachwill be explored. In this case, the motion of the wire will be considered to be a way of generatingan electrical signal, and this electrical signal will be shown to carry information about the magneticfield.The fact that an electrical signal is generated by the vibration of the wire may be explainedby Faraday’s Law. The circuit containing the wire is a loop of conducting material, meaningthe vibration of the wire causes the area of, and therefore the magnetic flux through, the circuitto vary over time, giving rise to a voltage via ε = − dΦ / d t . This means that a voltage whichopposes the driving current will be generated across the wire, so that the wire appears to resistthe driving current. In terms of an electric motor, the signal we mean to discuss may be referredto as ”counter-electromotive force” or ”back emf”. Here it will simply be referred to as the emf.At this point a mathematical description of ε , the emf generated by a vibrating wire, will bederived for a wire located at the center of an insertion device. In the case of a wire positioned4t x = y = 0, and a magnetic field directed along the vertical direction y , such as is depicted inFigure 2.1, the magnetic field acting on the wire may be written B y ( x, z ) = B ( x, , z ), where thesubscript indicates the component of the magnetic field. If the motion of the wire is confined tothe plane y = 0, then the emf can be expressed in terms of the flux enclosed by the circuit as ε = − dΦd t = − dd t (cid:90) L (cid:90) u −∞ B y ( x, z ) d x d z , (2.10)where u is the displacement of the wire. Since u = u ( t ), by the fundamental theorem of calculus ε = − (cid:90) L ∂u∂t B y ( u, z ) d z . (2.11)If the displacement of the wire takes the form of a standing wave this becomes ε = − ∂ Λ ∂t (cid:90) L sin (cid:16) nπL z (cid:17) B y ( u, z ) d z , (2.12)where Λ = Λ( t ) is the amplitude of the wire displacement at time t . Noting that u = u ( z ), meaning B y ( u, z ) = B y ( z ), and recalling the definition of the sine-series, B ( z ) = ∞ (cid:88) n =0 B n sin (cid:16) nπL z (cid:17) and B n = 2 L (cid:90) L B ( z ) sin (cid:16) nπL z (cid:17) d z , (2.13)then it becomes clear that ε = − B n L ∂ Λ ∂t . (2.14)At this point, one should recall that these results are obtained for a driving force in the form of F = F ( z ), meaning that it is assumed that the magnetic field does not vary appreciably in eitherof the transverse directions within the span of the wire vibration.If the motion of the wire is described by equation (2.4), then u n = Λ sin (cid:0) nπL z (cid:1) , and ε ( t, ω ) = B n I L µ sin ( ωt − φ ) ω (cid:104) ( γ (cid:48) ω ) − (cid:0) ω − ω n (cid:1) (cid:105) − / ; (2.15)comparing the time-dependent factors of equations (2.15) and (2.4), it can be seen that thephase difference between the emf and the driving current will be φ ε = φ + π/
2, so thattan φ ε = ω n − ω γ (cid:48) ω . (2.16)The series coefficient from equation (2.2) may then be obtained using a curve fitting approachsimilar to that described in the previous section, by measuring the amplitude of the emf as opposedto the amplitude of the wire vibration. Applied to equation (2.15), this gives ε n ( ω ) = a n ω (cid:2) c n ω − ( ω − b n ) (cid:3) − / , (2.17)and | B n | = (cid:114) a n µI L . (2.18)5hese results suggest that measurements of the voltage across the wire may be used to obtaininformation about the magnetic field acting on the wire, in a manner similar that described in theprevious section.There is, however, one key difference between the two which one must take note. This isthe fact that measurement of the voltage can only be used to determine the magnitude of theseries coefficients. In mathematical terms, this is evidenced by the relationship ε n ∝ B n , which isindependent of the sign of B n . In physical terms, one may consider the fact the voltage generatedby the wire depends only on the speed of the wire, meaning it is independent of the direction ofthe wire moves with respect to the driving current. In this section the application of the theory described in Section 2.2 will be discussed.To begin, the shortcomings of using emf as a signal will be considered. The first is that it may beunwieldy as a measurand for non-planar magnetic fields. Though the derivation of equation (2.18)explicitly treats only the case of planar motion in a field of one component, equation (2.11) shouldbe valid for any wire displacement in any magnetic field if B y ( x, z ) is replaced with B ( x, y, z ).It must be remarked that in the case that there is a significant field in both of the directionstransverse to the motion of the wire, each component of the field will contribute to the voltagewhich is produced. This would mean that the emf would carry information regarding both fieldcomponents, and special care would need to be taken to interpret the meaning of the signal. Ifoptical displacement sensors are used, the contribution of each field component may be easier todetermine, as the motion in each direction may be measured independently.Next, it should be reiterated that measurement of the amplitude of the emf is not by itselfsufficient to calculate the magnetic field distribution. Unlike those results, in this case the signalbeing measured is proportional to the square of the series coefficient. This means that the sign ofthe coefficient may not be determined from measurement of the emf alone, making it unsuitable fordetermining the field distribution over space via equation (2.2). Despite this, applications which donot require the sign of the coefficients still exits. One such is the determination of the magnitudeof coefficients related to specific frequencies of vibration, which may be used to characterize errorfields such as in Ref. [10].Turning now to the advantages of measuring the emf, the first is that it may be used tocomplement some of the shortcomings of a displacement sensor. For example, the sensor maybe positioned near the node of one of the harmonics, meaning that it would be unable to detectthe vibration of the wire at that frequency. To elaborate on the problematic nature of the sensorresponse, consider equation (2.9) in the form B n = a n µI sin (cid:16) nπL z s (cid:17) − ∝ sin ( x ) − , (3.1)and apply simple calculus-based error analysis to obtain ∂B n ∂x ∝ B n cot ( x ) . (3.2)This shows that for ( x mod π ) (cid:28)
1, or nz s /L near an integer, the uncertainty in the series coeffi-cient value becomes very large. This uncertainty becomes significant when many series coefficientsare needed, and the distribution of these coefficients is difficult to predict, such as in the detailed6easurement of insertion device fields [11]. To compensate for this, the sensor would need to bemobile so that it could be moved away from the nodes at these problematic frequencies, see Ref.[13] for example, or multiple sensors would need to be used to reduce the number of cases wherethere is no detectable motion. In these cases, the voltage produced by the motion of the wire maystill be detectable, despite the insensitivity of the displacement sensor.Another benefit to measuring the emf is that it may be better suited to determine the magnitudeof the series coefficients corresponding to higher harmonics of vibration, and therefore to higherdriving frequency. Examining the expression for wire displacement, equation (2.4), it can be seenthat u n ∝ ω − when ω ≈ ω n , meaning that the amplitude of the wire vibration will becomesmaller as the driving frequency increases and, consequently, harder to measure. In comparison,the expression for emf, equation (2.15), predicts the amplitude of the emf to be independent ofthe driving frequency. This implies that the emf may be better suited for measurements at highdriving frequency.Finally, it may be possible to use this technique for quadrupole fiducialization. By moving thecurrent-carrying wire relative to the quadrupole, the magnetic center may be found by measuringthe position where no emf is produced by the wire. A similar approach is described in in Ref.[14], in which the motion of the wire is mechanically driven. Another attempt at using a vibratingwire for this purpose may be found in Ref. [2], in which the displacement of the wire is measuredinstead of the voltage.In the interest of predicting the sensitivity of the emf to magnetic field strength, one mayconsider that the derivative of the signal d ε/ d B n ∝ B n , meaning that the sensitivity should varylinearly with the magnitude of the coefficient being measured. A more meaningful approach maybe to compare the emf signal to that of the vibration amplitude, the sensitivity of which hasalready been compared to other magnetic measurement techniques (Ref. [15]). To predict therelative sensitivity of each technique to the strength of the magnetic field, one may consider theratio of one measurand to the other. Comparing equations (2.7) and (2.15), it can be found thatmax | ε n | = B n ωL (cid:16) nπL z s (cid:17) − A n , (3.3)which, for driving frequencies near a resonant frequency, can be written asmax | ε n | = (cid:32) nπ (cid:115) Tµ (cid:33) sin (cid:16) nπL z s (cid:17) − ωω n B n A n ≈ (cid:32) nπ (cid:115) Tµ (cid:33) sin (cid:16) nπL z s (cid:17) − B n A n (3.4)upon substitution of equation (2.5). From this it can be seen that when the same magnet ismeasured using both methods, and the same measurement setup, the emf will be scaled withrespect to the vibration amplitude by two factors. The first is a factor proportional to the drivingfrequency, as has already been noted, and the second is the magnitude of the coefficient beingmeasured, which is the projection of the field onto that particular mode of vibration. This impliesthat measurement of the emf is less effective for small-magnitude coefficients, unless the frequencyof vibration is sufficiently high. In general, equation (3.4) may be used as a crude way of estimatingthe response of one approach compared to the other.7 Measurement and Analysis
This section contains a description of the approach used to measure the emf generated by the wire,as well as a discussion of the measurements taken. Due to a shortage of time with the equipment, itwas only possible to measure 9 harmonics, and only one measurement per harmonic was possible.In order to test the hypothesis that the emf generated by the vibrating wire may be used todetermine the magnetic field acting on the wire, the voltage drop across the wire was measuredas a function of driving frequency. This voltage is related to the emf generated by the wire byKirchoff’s Law, V = ε + IR , (4.1)where V is the measured voltage across the wire, I is the drive current, and R is the resistance ofthe wire. A plot of the predicted form of the measurable quantities, voltage and phase (equations(4.1) and (2.16), respectively), can be found in Figure 4.1.The amplitude of the wire voltage, as well as the phase offset with respect to the driving voltage,was measured using a Signal Recovery model 7265 lock-in amplifier which took the voltage acrossthe wire as a direct input. For these measurements, the magnetic field of an in-vaccum wigglerwith a period of 80 mm and a length of about 1 .
41 m was used. For the measurement a beryllium-copper wire with a diameter of 0 . .
516 m, was used, and tension was appliedusing a hanging mass. Parameters of the setup can be found in Table 1. The data collected maybe found in Figure 8.2.It should be noted that the measurements presented were taken using two different currentamplitudes to drive the wire. One set of data, for modes 61 through 70, were collected usinga 20 mA drive current, while a 1 mA current was used for modes 85 through 90. The choice ofcurrent amplitude was made based on the observed amplitude of the vibration near the resonantfrequency being considered. The driving force had to be sufficient to produce a clear signal butat the same time not be so large as to excite non-linear phenomena, meaning that frequencies atwhich the wire was particulary sensitive to driving had to be measured using smaller currents.Those interested in details concerning non-linear phenomena in the motion of vibrating wires mayrefer to [16, 17], and especially [18].Parameter Symbol ValueDensity µ .
74 mg · m − Length L .
516 mTension T .
022 NFundamental Frequency f . .9 0.95 1 1.05 1.1 Driving Frequency / n N o r m a li z ed W i r e V o l t age V / m a x | V | -100-50050100 P ha s e [ deg ] Figure 4.1: Voltage drop across the wire, predicted using equation (4.1) with IR = 1 . The measurements of the wire voltage were then compared to the theory outlined in Section 2.2by fitting the following altered version of equation (2.17). In order to account for variations in thedriving voltage of the wire, such as those caused by a change in ambient temperature, a DC offsetparameter was introduced to the curve-fitting equation so that ε n ( ω ) = a n ω (cid:2) c n ω − ( ω − b n ) (cid:3) − / + d n . (4.2)The result of this curve fitting process can be found in Figure 8.3. A measure of the quality of thefit for each data set can be found in Table 2. The quantity R found there is the sum of squaredifferences between the data set and the fit, normalized by the sum of squared measurements. Thisnormalization is applied to ensure the measure is independent of the scale of the data. Expressedmathematically, R = (cid:32)(cid:88) k y k (cid:33) − (cid:88) k [ y k − f ( ω k )] , (4.3)where y k is the k th element of the data set and f ( ω k ) is the fit-function evaluated at the k th frequency measured.Examining the data illustrated in Figure 8.2, it can be seen that, in general, the measuredcurves are of the same shape as the predicted curves found in Figure 4.1. However, there aresome features which appear unique to the driving current used. In particular, the measurementstaken using a drive current of 20 mA appear to have an irregular phase shift, in that the phase isbound between ± ◦ while the measurements taken using a drive current of 1 mA have a phasebound between ± ◦ . Referring to Figure 4.1, it appears the latter case agrees more closely with9heory. This implies either an unaccounted for dependence on the drive current, or an experimentalerror. It seems likely that this difference is due to measurement error related to the difference insignal strength. Typically, the curves measured at a smaller drive current have heights between 0.5and 1 volts, while the other curves are an order of magnitude smaller, and the latter are notablynarrower than the former. If measurement error is responsible for this discrepancy, it could possiblybe rectified by sampling the curve at a higher frequency resolution.To check the accuracy of the series coefficients calculated using measurements of the emf B nε ,the values of which may be found in Table 2, comparison was made to those calculated frommeasurements of wire displacement B nA . In theory, the two approaches to measurement shouldgive the same series coefficients. It should be noted that this comparison makes use of data setscollected several weeks apart. In this time the apparatus was used intermittently, and the samewire remained strung. Based on observations made throughout the use of the vibrating wire, itdoes not seem likely that the time interval between measurements will affect the values of theseries coefficients.A comparison of the two sets of coefficients may be found in Table 2, and is visualized inFigure 8.1, which shows the relative difference between the two sets of coefficients, taking B nA as the reference. These figures show close agreement of coefficients 87, 89, and 91, as well as theapparent relationship between driving current and measurement discrepancy. To elaborate on thelatter claim, comparison of the two sets of coefficients showed that B nε measured at 20 mA differedfrom B nA . It is noteworthy that these B nε correspond to data sets with irregular phase shifts,seen in Figure 8.2. In addition, the B nε for n = 85 is exceptional in that it was measured usinga different drive current than the B nA it is being compared to. If there is an unaccounted fordependence on current, this comparison may be invalid.Examining Figure 8.2, it is interesting to note that in the cases of n = 89 and n = 91 somesigns of non-linear oscillation, resulting from excessive vibration amplitude, may be seen. The mainfeature which implies non-linear oscillation is the asymmetry of the amplitude curves, namely thatthere appears to be a sharp drop on the high-frequency side of the curve associated with the collapseof transverse whirling motion [17]. Even though the motion of the wire appears to be outside theregime described by equation (2.1), the series coefficients calculated using either method are inclose agreement. This would indicate that non-linear wire motion may have either a similar, orminor, impact on the values of the series coefficients. In the least, this would mean that themeasurement of emf is no more sensitive to this problematic oscillation than the measurement ofdisplacement. 10 I ε (mA) I A (mA) B nε (G) B nA (G) δB n (%) R (10 − )61 20 20 175.1 ± ± ±
21 -429.0 27 2867 20 20 62.7 ± ± ±
170 -42850 8 1187 1 1 73810 ±
260 71640 3 5.589 1 1 73310 ±
290 -73280 0 7.391 1 1 37550 ±
180 38590 3 13Table 2: Table of drive currents used for each measurement set, coefficients calculated for eachset, and corresponding percent difference between B nε and B nA . B nA are reported to arbitraryprecision. R is a measure of the quality of the curve fit, normalized to be independent of the scaleof the data. In Section 2.2 a theory of an alternative way of measuring magnetic fields using the vibrating wiretechnique was developed. This alternative makes use of the electromotive force generated by themotion of the wire in the field. The advantages, disadvantages, and potential applications of thistheory were discussed in Section 3. In Section 4 experimental tests of the theory are reported, theresults of which were found to agree with the conventional method for 3 of 9 test cases. The caseswhich did not agree used a different drive current than the others, and produced considerablysmaller signals with narrower resonance curves. Based on this it is thought that the discrepancybetween the remaining 6 cases may be due to a much smaller measured signal, the amplitude ofthe driving current used, or to the frequency resolution at which the curves were measured.For comparison, the advantages and disadvantages of the technique with respect to the conven-tional method will be summarized. Among the advantages is the lack of need for a displacementsensor, which means simpler experimental design and setup. Another is, theoretically, a strongersignal at high frequencies of vibration. The disadvantages include a lack of information on thesign of the series coefficient, making the method unsuitable for determining field distributions, aswell as the fact that the orientation of the field under measurement is ambiguous. In addition, thesensitivity of the technique is predicted to be more complicated than the conventional technique,in that, comparatively, the sensitivity of the emf is worse for small-magnitude coefficients, butbecomes better as the magnitude increases.To conclude, based on the results presented here, it would appear that the theory developed inSection 2.2 is valid. This means that, in principle, it may be possible to determine magnetic fieldthrough measurements of the voltage across the vibrating wire instead of the amplitude of the wirevibration. A more thorough analysis will be needed to assess the practicality of this technique, aswell as identify unaccounted for factors, such as non-linear current dependence.
The authors are grateful to the following people. Jon Stampe for valuable discussion and aid inexperimental setup. Tor Pederson, Grant Henneberg, Garth Steel, Bruce Wu and Carl Jansen for11id in developing the measurement system.Research at the Canadian Light Source was funded by the Canada Foundation for Innovation,the Natural Sciences and Engineering Research Council of Canada, the National Research CouncilCanada, the Canadian Institutes of Health Research, the Government of Saskatchewan, WesternEconomic Diversification Canada, and the University of Saskatchewan.
JG planned and carried out measurements, developed the model, analysed the data, and wrotethe manuscript. CB contributed to the analysis of the data used as reference, and to the editingof the final manuscript. 12
Figures
61 63 65 67 70 85 87 89 91
Harmonic Number - | B n / B n A | I = 20 mAI = 1 mA I = 2 mAI = 1 mA Figure 8.1: Plot of relative error in emf coefficients. Bars coloured to indicate data set. I (cid:15) indicatesdriving current used for voltage measurements, while I A indicates that used for displacementmeasurements. For those labelled by I , I (cid:15) = I A . Black bars represent the uncertainty in thevalue. (If viewed in grayscale, Table 2 on p.11 may be used as reference).13
130 3135 3140 3145 3150 3155 3160 3165
Frequency [Hz] W i r e V o l t age [ V ] -0.500.511.5 P ha s e [ deg ] n = 61 I = 20 mA VoltagePhase
Frequency [Hz] W i r e V o l t age [ V ] P ha s e [ deg ] n = 63 I = 20 mA VoltagePhase
Frequency [Hz] W i r e V o l t age [ V ] -2-101234 P ha s e [ deg ] n = 65 I = 20 mA VoltagePhase
Frequency [Hz] W i r e V o l t age [ V ] P ha s e [ deg ] n = 67 I = 20 mA VoltagePhase
Frequency [Hz] W i r e V o l t age [ V ] -60-40-200204060 P ha s e [ deg ] n = 85 I = 1 mA VoltagePhase
Frequency [Hz] W i r e V o l t age [ V ] -80-60-40-20020406080 P ha s e [ deg ] n = 87 I = 1 mA VoltagePhase
Frequency [Hz] W i r e V o l t age [ V ] -80-60-40-20020406080 P ha s e [ deg ] n = 89 I = 1 mA VoltagePhase
Frequency [Hz] W i r e V o l t age [ V ] -60-40-200204060 P ha s e [ deg ] n = 91 I = 1 mA VoltagePhase
Figure 8.2: Measurements of wire voltage for several resonant frequencies. n is the harmonicnumber, and I is the amplitude of the driving current.14
130 3135 3140 3145 3150 3155 3160 3165
Frequency [Hz] C oun t e r- E M F [ m V ] n = 61 I = 20 mA Frequency [Hz] C oun t e r- E M F [ m V ] n = 63 I = 20 mA Frequency [Hz] C oun t e r- E M F [ m V ] n = 65 I = 20 mA Frequency [Hz] C oun t e r- E M F [ m V ] n = 67 I = 20 mA Frequency [Hz] -0.100.10.20.30.40.5 C oun t e r- E M F [ V ] n = 85 I = 1 mA Frequency [Hz] C oun t e r- E M F [ V ] n = 87 I = 1 mA Frequency [Hz] C oun t e r- E M F [ V ] n = 89 I = 1 mA Frequency [Hz] -0.100.10.20.30.40.5 C oun t e r- E M F [ V ] n = 91 I = 1 mA Figure 8.3: Measurements of emf with theoretical curve, equation (4.2), fit to data. n is theharmonic number, and I is the amplitude of the driving current.15 eferences [1] Alexander Temnykh. Vibrating wire field-measuring technique. Nuclear Instruments andMethods in Physics Research Section A: Accelerators, Spectrometers, Detectors and AssociatedEquipment , 399(2-3):185–194, 1997.[2] Alexander Temnykh. The magnetic center finding using vibrating wire technique.
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