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A Viscosity and Density Sensor Based on Diamagnetically Stabilized Levitation Stefan Clara, Hannes Antlinger, Wolfgang Hilber, and Bernhard Jakoby, Senior Member, IEEE Abstract— We investigate the feasibility of viscosity and density measurements using diamagnetically stabilized levitation of a floater magnet on pyrolytic graphite. This principle avoids any clamping structures in the measurement chamber and is, therefore, not suffering under unknown mounting conditions and is furthermore easy to integrate into microfluidic systems. The only part that has to be in contact with the liquid is the floater magnet. Immersing it in a liquid, buoyancy forces will come into play. Keeping the levitation height of the floater magnet constant in different liquid surroundings by accordingly adjusting the lifter magnet, the buoyancy force and, therefore, the density of the fluid can be determined from these adjustments. For more accurate results, a magnetic field modeling was used to determine the levitation height of the floater magnet out of the superposed magnetic fields of both magnets. For viscosity measurements, we add an additional ac-driven coil to the setup, which yields a superposed alternating force on the floater magnet causing periodic vibrations of the floater magnet. The vibrations are damped according to the viscosity of the surrounding fluid. By performing a frequency sweep, the frequency response of the damped spring mass resonator can be obtained where the resonance frequency for our setup is around 6 Hz. Furthermore, the influence of the levitation height on the resonance characteristics was examined by studying the resonance frequency and quality factor for different lifter magnet positions. Index Terms— Diamagnetically stabilized levitation, pyrolytic graphite, resonant fluid property sensor, viscosity, density
I. I NTRODUCTION
R
RECENTLY presented resonant viscosity sensors for inline process monitoring operate at relatively high frequencies [1] (from 1 kHz up to several MHz). All this principles suffer to a certain extent from the influence of the mounting structures. Lately we introduced a concept for an alternative viscosity measurement principle based on a diamagnetically levitated permanent magnet using pyrolytic graphite [1], [2]. The general concept of diamagnetic levitation is discussed, see in [3] and [4]. As the floater magnet is freely levitated no springs or other fixings influence the measurement. There are also no mechanical or electrical connections
Manuscript received May 6, 2014; revised October 2, 2014; accepted November 1, 2014. Date of publication November 10, 2014; date of current version January 21, 2015. This work was supported in part by the Research Network entitled Industrial Methods for Process Analytical Chemistry under Grant 843546 and in part by the Linz Center of Mechatronics within the framework of the COMET-K2 Program. This is an expanded paper from the IEEE SENSORS 2013 Conference. The associate editor coordinating the review of this paper and approving it for publication was Prof. Massood Z. Atashbar. (Corresponding author: Stefan Clara.) The authors are with the Institute for Microelectronics and Microsensors, Johannes Kepler University Linz, Linz 4040, Austria (e-mail:
[email protected];
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSEN.2014.2368983
into the measurement chamber necessary, only the floater magnet gets in contact with the liquid and its position can be sensed from outside the device. The system can be treated as a spring mass resonator, where the damping of the system is dependent on the viscosity of the surrounding fluid. The resonance frequency of our system is in the order of 6 Hz. Due to its very low resonance frequency, the sensing principle is distinguished from most other resonant principles. The gained viscosity values are more comparable to the viscosity measured by standard laboratory equipment. But the viscosity is not the only parameter which can be obtained. Immersing the floater magnet in a liquid, buoyancy forces will act on the floater magnet. Keeping the levitation height of the floater magnet constant in different liquid surroundings by accordingly adjusting the lifter magnet, the buoyancy force and therefore the density of the fluid can be determined from these adjustments. In contrast to other principles independent sensing of both, viscosity and density is possible. The system is applicable for virtually all types of liquids except magnetic and highly conductive fluids. The resonance characteristics are influenced by the levitation height of the system which has to be taken into account for the viscosity measurements. The range for the viscosity measurements using the described resonant principle goes up to 20 mPa s. In this contribution we briefly review the setup introduced previously [2] and present a model for the magnetic fields allowing to determine the levitation height leading to an analytical model for the sensor characteristics. Furthermore, we introduce preliminary results illustrating the feasibility of viscosity and density measurements and provide an accuracy analysis of the correlation between density and modeled B-field for densities from 1 kg/m3 to 1300 kg/m3 . II. S ETUP The levitation setup is shown in Fig. 1: a small floater magnet is floating above a piece of pyrolytic graphite, where the levitation force is partly provided by a lifter magnet above the setup. The other part of the levitation force is caused by the diamagnetic characteristics of the pyrolytic graphite, which moreover maintains a stable levitation point of the floater magnet. For the viscosity measurement also an AC-driven coil is necessary (see Fig. 1) [5]. The position of the lifter magnet influences directly the force acting on the floater magnet and therefore controls the levitation height of the floater magnet. To adjust the levitation height, the position of the lifter magnet has to be variable. The levitation height can be measured over the vertical magnetic flux density by a Hall sensor placed underneath the pyrolytic graphite. This sensor measures mainly the magnetic field of the floater magnet such that
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height constant by rearranging the position of the lifter magnet or by superposing the magnetic field by an additional constant field of the coil, allows calculating the buoyancy force out of the displacement of the lifter magnet or of the coil current. In this measurement procedure no changing (AC) magnetic fields are necessary to achieve steady conditions. B. Viscosity Measurement The viscosity measurement is based on the principle of determining the quality factor of a viscously damped oscillator. For this the position of the floater magnet is sinusoidally modulated with small amplitudes (to guarantee that the nonlinearity of the spring constant can be neglected, see (8)). This is achieved by means of the AC-driven coil which superposes the constant magnetic field of the lifter magnet with a very small alternating field. This leads to an alternating force acting on the floater magnet in vertical direction and therefore to a periodic change of the position of the floater magnet. Note that due to the small changes in the B-field due to the AC-driven coil and due to the low periodic displacement of the floater magnet at low frequency cause no thermal problems due to eddy currents. For the measurement procedure, the resonance characteristic is measured by performing a frequency sweep. The resonance frequency and the quality factor are obtained by a fit algorithm. The viscosity is related to the quality factor. IV. T HEORETICAL BACKGROUND Fig. 1.
Sensor setup based on diamagnetically stabilized levitation.
the distance to the latter can be determined (where the nonuniformity of the magnetic flux density is taken into account). The influences of the lifter magnet and the coil are small due to the large distance to the Hall sensor. The output signal is composed of two parts, a DC-component, which is related to the distance to the floater magnet, and an AC-component which is related to its oscillation. Due to the small vibration amplitudes, the AC-part of the Hall sensor’s output signal has to be high-pass filtered and amplified. Both magnets are made of NdFeB (Neotexx), where the lifter magnet is a ring magnet with an inner diameter of di L = 15 mm, an outer diameter of doL = 45 mm and a height of h L = 4 mm. The floater magnet has a cylindrical shape with a height of h F = 1.5 mm and a diameter of d F = 3 mm. The distance between the lifter and the floater magnet is in the order of 165 mm. The pyrolytic graphite is 18 × 18 × 2 mm3 and is placed directly above the Hall sensor (A1324LUA-T from Allegro MicroSystems). The dimensions of the whole sensor setup are relatively large, but the sensitive area is concentrated on the surrounding of the floater magnet. Therefore the necessary sample volume can be kept low (below 100 μL). III. R EADOUT AND M EASUREMENT P RINCIPLE A. Density Measurement If the lifter magnet is immersed into a liquid, a buoyancy force is acting on it. This additional force clearly influences the levitation height of the floater magnet. Keeping the levitation
A. Diamagnetic Levitation Diamagnetism is a property of all materials and will always make a weak contribution to the response of the material to an external magnetic field. For materials that show some other form of magnetism (such as ferromagnetism or paramagnetism), the diamagnetic contribution often becomes negligible. Substances that dominantly display diamagnetic behavior are termed diamagnetic materials or diamagnets and include water, most organic compounds and many metals including copper, gold and bismuth [6]. Diamagnetic materials have a relative magnetic permeability μr that is slightly less than one, and therefore a magnetic susceptibility χ which is less than 0 since the susceptibility is given by χ = μr − 1. This means that diamagnetic materials are repelled by magnetic fields. However, since diamagnetism is such a weak property, its effects are most often not observable in everyday life. For example, the magnetic susceptibility of diamagnets such as water is χ = −9.05E−6. The strongest isotropic diamagnetic material is bismuth featuring χ = −1.66E−4. Pyrolytic carbon features a strong anisotropic diamagnetism with an even higher susceptibility of χ = −4.00E−4 in one plane. Diamagnetic levitation originally denotes the effect of free levitation of a diamagnetic material in a non-uniform magnetic field. The feasibility of diamagnetic levitation was first predicted by William Thomson in 1847 [7] and later verified experimentally by Werner Braunbek in 1939 with the levitation of small pieces of bismuth (Bi) and graphite (C) in a non-uniform magnetic field realized with a strong electromagnet [8]. Nowadays with strong superconducting magnets different diamagnetic materials and objects can be easily
CLARA et al.: VISCOSITY AND DENSITY SENSOR BASED ON DIAMAGNETICALLY STABILIZED LEVITATION
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levitated such as water droplets and even living frogs and mice [9]. A variant of diamagnetic levitation is the diamagnetically stabilized levitation, where diamagnetic materials are used to stabilize the free levitation of a permanent magnet. In this configuration a permanent magnet is attracted by a lifter magnet and floats freely above or between stabilizing diamagnetic materials. The integration of a diamagnetic material into this configuration provides a contribution to the total potential energy function which facilitates the stable levitation of the floater magnet at certain positions [10]. Diamagnetically stabilized levitation was first demonstrated by Boerdijk [4]. Today levitation gaps of up to 3 mm can be realized in such a setup with NdFeB magnets and pyrolytic graphite. The potential energy of a floating magnet with magnetic moment M (magnitude M) in the magnetic field (flux density) B (magnitude B) of a lifter magnet is
U = −M · B + mgz = −M B + mgz,
(1)
where mgz is the gravitational energy with the mass m, the gravitational acceleration g, and the distance of the magnet orthogonal to earth surface z. Because of magnetic torques, the magnet will align with the local field direction and hence the energy is only dependent on the magnitude of the magnetic field, and not on the individual field components. Expanding the field magnitude of the lifter magnet around the levitation point in polar coordinates and adding two new terms C z z 2 and Cr r 2 which represent the influence of the diamagnet, the potential energy of the floating magnet can be written as [3] 1 mg z + B z2 U = −M B0 + B − M 2 2 1 B + − B r 2 + · · · + C z z 2 + Cr r 2 , (2) 4 2B0 where ∂ Bz z and B = ∂z ∂z 2 At the levitation point, the expression in the first curly brackets, representing the z-component of the total force at the levitation point (z = 0), must be zero, i.e. the forces due to the non-uniform magnetic field balance the force of gravity mg . (3) B = M Furthermore, (2) can be used to derive the conditions for vertical stability B =
Cz −
∂2 B
1 M B > 0, 2
and for horizontal stability 2 1 B 1 m 2 g2 = Cr + M B − Cr + M B − >0 4 2B0 4 2M 2 B0
(4)
(5)
These conditions are necessary to ensure a local minimum of U at the point of equilibrium and thus enable stability.
Fig. 2. Duffing effect of the setup at larger actuation amplitudes in air. The ratio between the amplified output voltage of the hall sensor and the constant actuation voltage amplitude is ploted over the frequency.
The evaluation of the C z and Cr diamagnetic terms will not be reproduced here, instead the interested reader is referred to [10]. If both equations (4) and (5) are fulfilled, stable levitation is possible if MB’=mg, see (3). Hence, it is possible to adjust the field gradient or the weight of the floating magnet to match this condition. B. Damped Spring Mass Resonator The equation of motion of a damped spring-mass resonator can be written as m
∂z ∂2z + kz = Fω cos (ωt), + γ ∂t 2 ∂t
(6)
where z is the position of the mass m, γ is the damping coefficient, k is the spring constant, and Fω is the magnitude of the forcing term at frequency √ ω. With the definition of the resonant frequency ω0 = k/m and the quality factor Q = ω0 m/γ the solution of (6) can be written as
Fω m |z (ω)| = . (7)
2 ω2 − ω02 + (ωω0 /Q)2 Due to the nonlinear character of the spring constant k at the levitation point, see (2) and (3), the resonator features an amplitude dependent resonant frequency and exhibits the so-called Duffing effect [11] (see Fig. 2). This means that, for large vibration amplitudes, the resonator shows a hysteresis behavior when performing a frequency sweep. In general, the nonlinear spring force can be written as F = −kz − k1 z 2 − k2 z 3 + O(z 4 ),
(8)
where k is the linear spring constant, k1 and k2 are the first and second order corrections, respectively. Inserting (8) in (6) and performing a perturbation analysis around the
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linear oscillation frequency ω0 yields an amplitude dependent resonance frequency ω0 ω0 = ω0 + κZ02 ,
(9)
5k 2
2 1 where κ = 3k 8k ω0 − 12k 2 ω0 and Z 0 is the solution (oscillation amplitude) of the harmonic resonator according to
z0 = Z0 cos (ω0 t).
(10)
V. M AGNETIC F IELD M ODELING For the modeling of the magnetic flux density of the floater and the lifter magnet, both of these permanent magnets can be represented by an approximate model in terms of equivalent cylindrical surface currents (which are related to the remanent flux density of the magnets), see also [3], was used. Specifically, for the floater magnet the flux density on the z-axis can be written as ⎡⎛ ⎞⎤ dF + hF BrF ⎣⎝ dF ⎠⎦ (11) − BF = 2 2 2 2 2 d +R (d + h ) + R F
F
F
F
Fig. 3. Modeled and measured flux density of the cylindrical floater magnet measured along the z-axis. The first measurement point was excludet from the fit, because the highly nonuniform flux density nearby the magnet and the finite size of the Hall-Sensor were not taken into account for this model.
F
where Br F, d F, h F, R F, is the remanent flux density of the floater magnet, distance to the bottom of the magnet, height of the magnet, radius of the magnet, respectively. As the lifter magnet is ring shaped the flux density along the z-axis can be written as the difference field of two cylindrical magnets, i.e. ⎞ ⎡⎛ BrL ⎣⎝ dL dL + hL ⎠ − BL = 2 2 2 2 2 d L + RLo (dL + hL ) + RLo ⎛ ⎞⎤ + h d d L L L ⎠⎦. (12) − −⎝ 2 2 2 2 (d L + h L ) + RLi dL + RLi Here Br L , d L , h L , R Lo , R Li , is the remanent flux density of the lifter magnet, distance to the bottom of the magnet, height of the magnet, outer radius and inner radius of the ring shaped magnet, respectively. Both models were fitted with respect to the actually measured flux density (GM05 Gaussmeter from Hirst Magnetic Instruments) adding an offset for the distance (to compensate for the not exactly known position of the hall sensor) and an offset for the measured flux density (to compensate for the offset of the instrument’s Hall-sensor). The geometrical parameters were set to the values given above, and the remanent flux density of both magnets was set to 1.33 T according to the data sheets. Figs. 3 and 4 show the measured values and the magnetic flux density obtained from the model for both magnets. Both measurements fit very well to the theoretical model except for the first measurement point of the floater magnet which was taken close to the magnet. Here the finite active area of the Hall-sensor influences the measurement result very close to the magnet surface. The measured value therefore lies visibly below the theoretical value and was excluded for the offset fit.
Fig. 4. Modeled and measured flux density of the ring-shaped lifter magnet along the z-axis.
VI. P OSITION M EASUREMENTS For the viscosity and density measurements described below a standard Hall-Sensor IC (A1324LUA-T) from Allegro MicroSystems was implemented in the setup (see also Fig. 1) to determine the position of the floater magnet. The output from this IC was adjusted to match the measurements of the GM05 Gauss meter (see Fig. 5). The levitation height is important for the viscosity and density measurements, and will be estimated from the measured magnetic flux density of the Hall-Sensor placed underneath the pyrolytic graphite. Therefore, a calibration measurement was performed in air using a camera (VKT Photron Fastcam SA4) to determine the levitation height of the floater magnet for different lifter magnet positions. In order to determine the levitation height, the calculated contribution of the lifter magnet (see Fig. 4) was subtracted
CLARA et al.: VISCOSITY AND DENSITY SENSOR BASED ON DIAMAGNETICALLY STABILIZED LEVITATION
Fig. 5. B-Field measured by the Hall-Sensor IC (A1324LUA-T) matched to the measurements of the GM05 Gaussmeter (on which the models for the magnetic fields are based).
Fig. 6. Calibration measurement for the levitation height (in air) (levitation height h L = d F − d H P , where dHP is the distance between the Hall sensor and pyrolytic graphite sheet) calculated from the Hall-Sensor and compared to measurements performed with a camera.
from the measured flux density. The obtained value should thus correspond to the field of the floater magnet. Using this value and (11), the distance d F from the Hall-sensor (placed underneath the pyrolytic graphite) to the bottom of the floater magnet can be obtained, see Fig. 6. However, as we are interested in the distance to the pyrolytic graphite (which is closer to the magnet) we have to subtract the distance between the Hall sensor and pyrolytic graphite sheet (d H P ), i.e. we have: levitation height h L = d F − d H P . As d H P is not exactly known (the sensor chip is embedded in a housing) we determine it by fitting the data from Fig. 6 minus the value d H P (fit parameter) to the distance floater magnet – pyrolitic graphite obtained from measurements using the camera. Due to the limited resolution of the camera, the obtained data is noisy but matches the trend obtained from the measurement data.
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Fig. 7. Measurement for the levitation height (in 2-Propanol) calculated from the Hall-Sensor and compared to camera measurements.
Fig. 8. Measurement of the frequency response of the floater magnet immersed in isopropanol, for different lifter magnet positions (levitation heights).
To verify the accuracy of the approach, the same measurement was repeated for the floater magnet surrounded by 2-Propanol (see Fig. 7). Now, due to the buoyancy forces the levitation height is larger than for the previous air measurements (for the same position of the lifter magnet). The model together with the fit parameters determined before was now used to estimate the levitation heights. The latter were also determined using the camera and both characteristics are well matched confirming the validity of the approach and the model. Note that this also justifies neglecting the influence of the diamagnetic material on the field in the model. For the stabilization of the equilibrium position of the floater magnet, its small influence is essential, though.
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Fig. 9. Nyquist plot for different levitation height of the floater magnet immersed in 2-propanol.
VII. F EASIBILITY FOR V ISCOSITY S ENSING Due to the nonlinear characteristics of the magnetic and diamagnetic forces, the resonance frequency of the setup depends on the levitation height of the floater magnet. Figs. 8 and 9 show the resonance curves of a lifter magnet immersed into 2-propanol for different positions of the lifter magnet and therefore also for different levitation height of the floater magnet. The frequency responses represent the AC-component of the hall-sensor’s signal when superposing the lifter magnets field by an AC field generated by an actuation coil (see Fig. 1). The resonance frequency is decreasing with increasing levitation height of the floater magnet. This effect comes from the strong dependence on the restoring force induced by the diamagnetic material on the levitation height. In particular, this force decreases with an increasing levitation height, see also [3]. It is also clearly visible that the damping factor increases with decreasing levitation height which is mostly caused by the damping associated with the squeezing of the fluid between the floater magnet and the diamagnetic material. Also the Nyquist plot clearly shows this effect, the radius of the resonance curve is decreasing with decreasing levitation height. To obtain the resonance frequency and the quality factor Q for each resonance characteristic, the model of a second order response function was fitted to the data points using the algorithm described in [12]. Fig. 10 shows the obtained resonance frequency which is increasing with the decreasing levitation height of the floater magnet, due to the aforementioned increasing restoring force of the diamagnetic material. Fig. 11 shows the quality factor obtained out of the resonance curves. As observed above, the quality factor is strongly decreasing with decreasing levitation height due squeeze film damping. Interestingly enough the quality factor is relatively constant above a certain levitation height. Therefore it is important to keep the levitation height above this level to guarantee robust viscosity measurements. For accurate viscosity measurements all these effects have to be taken into account. In particular, a well-defined levitation height should be used to
Fig. 10. Resonance frequency versus lifter magnet position for 2-propanol as surrounding fluid.
Fig. 11. Quality factor versus lifter magnet position (measured in 2-propanol).
yield the desired clear dependence of resonance frequency and quality factor on the viscosity. Spurious changes in levitation height would also affect these measurement parameters. VIII. D ENSITY AND V ISCOSITY M EASUREMENTS A. Density For the density measurement we utilize the effect that upon immersion of the floater magnet in a fluid, besides diamagnetic repulsion, magnetic attraction of the lifter magnet, and gravity, also buoyancy forces come into play and enter the equilibrium condition of balanced forces. If the levitation height is set to a defined position by accordingly shifting the lifter magnet, the buoyancy force (and thus the density of the fluid) can be calculated from the position of the lifter magnet based on the model for the magnetic field developed before.
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Fig. 12. Density measurement, density related to the nominal density for different levitation height measured with the Hall sensor (influence of the lifter magnet considered using the abowe described modelling). TABLE I R ELATIVE E RRORS
To determine the magnetic force on the floater magnet due to the lifter magnet, FLz , the gradient of the B-field of the lifter magnet in z-direction has to be considered as this force is given by [13] → − → − → − (13) FL = ( M · ∇) · B is the magnetic moment of the floater magnet where M and ∇ is the del operator. This force is balanced by the gravitational force, the forces due to the diamagnetic sheet, and the buoyancy forces (which we are interested in). Neglecting the small contribution of the diamagnetic material and considering that the gravitational force is constant, we expect FLz to linearly change with the buoyancy force and thus also the density of the liquid. The gradient ∂ B/∂z (which is proportional to FLz ) can be calculated from (12) for a known levitation height. For several liquids, we adjusted the lifter magnet position such that the levitation height took a prescribed value. For these values we calculated ∂ B/∂z. Fig. 12 shows the according relation of ∂ B/∂z to the fluid density. As expected, the relation is almost linear. Fig. 12 also shows that the relation is more linear for higher levitation positions. To give an indication of the accuracy of the linear relation (which could be used as a simple calibration curve) we show the relative deviation of actual densities from this linear fit for different levitation heights in Table I. This method is applicable for a large range of densities from below 1 kg/m3 to 1300 kg/m3 and beyond if the setup is adapted accordingly.
Fig. 13. Resonance curves of different liquids, the frequency shift depends on the distance to the floater magnet and also on the viscosity and density of the surrounding fluid. Both characteristics have been measured in an up- and a down-sweep yielding slightly different characteristics for the H2 O case due to non-linear effects. (Measured with the setup described in [2]).
Note that the deviation increases with decreasing levitation height. For larger levitation heights, the levitation point is more sensitive to the position of the lifter magnet, and therefore to the gradient of the magnetic flux density in z-direction such that other, spurious effects are less pronounced. The deviation increases for smaller levitation heights as the levitation setup is not exactly symmetric and the surface of the pyrolytic graphite not precisely horizontally aligned. Due to this the floater magnet tends to be spuriously diverted in radial direction. Therefore, the field estimation is no longer correct and leads to deviations in the density calculation. B. Viscosity For determining the viscosity, oscillations around the equilibrium positions are required. In order to achieve sufficiently strong sensor signals, the oscillation amplitude should be preferably as large as possible. Then, however, the Duffing effect (associated with nonlinear restoring forces around the equilibrium point) may become significant as it is shown in Fig. 2. Hence, to avoid the Duffing effect, sufficiently small oscillation amplitudes are necessary. As also discussed above, the resonance frequency depends on (i) the position of lifter magnet, (ii) the levitation height of the floater magnet, and (iii) on the viscosity of the surrounding fluid. The influence of different viscosities on the resonance curve is shown in Fig. 13, where the measurements have been made for the same levitation height. The quality factor Q decreases with an increasing dynamic viscosity of the surrounding fluid. Due to the varying liquid density, a constant levitation height (chosen for each measurement series) of the floater magnet can only be achieved by adjusting the position of the lifter magnet. Note, however, that there are a number of spurious effects influencing damping and resonance frequency in a different
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manner, even though the position of the floater magnet does not change. As an example we name the change in restoring forces due to the varying distance to the lifter magnet. The resonant viscosity measurement principle leads to best results for low viscosities in the range below 20 mPa s. Hence, for an accurate evaluation of the viscosity from these measurement data, a refined model is required, which, however, is beyond the scope of the current work. IX. C ONCLUSION We present two new measurement principles for fluid parameters (density and viscosity) based on diamagnetically stabilized levitation of a floater magnet on pyrolytic graphite. We review the theoretical background of the principle and demonstrate the feasibility of the approach with first experimental results. For the density measurements, the relation of the position of the lifter magnet and the buoyancy force is used. To increase the accuracy of the measurements we used a magnetic field modeling, and calculated the levitation height of the floater magnet from the measured flux density obtained from a hall sensor placed below the pyrolytic graphite. Using this principle, a very linear relation between the density and the magnetic flux density for constant levitation height was achieved. For the viscosity measurement an additional magnetic AC-field is created causing oscillations of the floater magnet. These oscillations feature a distinct resonant behavior. A surrounding viscous liquid predominantly leads to an increased damping of the resonance curve indicating the feasibility of the setup for viscosity sensing. We identified some of the issues which have to be taken into account to implement accurate viscosity measurements and discussed the spurious influences of the levitation height on the measurement principle. In particular, these effects would have to be considered in a refined model. R EFERENCES [1] B. Jakoby et al., “Miniaturized sensors for the viscosity and density of liquids-performance and issues,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 57, no. 1, pp. 111–120, Jan. 2010. [2] S. Clara, H. Antlinger, W. Hilber, and B. Jakoby, “Viscosity and density sensor principle based on diamagnetic levitation using pyrolytic graphite,” in Proc. IEEE SENSORS, Nov. 2013, pp. 1–4. [3] M. D. Simon, L. O. Heflinger, and A. K. Geim, “Diamagnetically stabilized magnet levitation,” Amer. J. Phys., vol. 69, no. 6, pp. 702–713, 2001. [4] A. H. Boerdijk, “Levitation by static magnetic fields,” Philips Tech. Rev., vol. 18, pp. 125–127, 1956. [5] W. Hilber and B. Jakoby, “A magnetic membrane actuator utilizing diamagnetic levitation,” in Proc. IEEE Sensors, Taipei, Taiwan, Oct. 2012, pp. 1–4. [6] B. D. Cullity and C. D. Graham, Diamagnetism and Paramagnetism, in Introduction to Magnetic Materials, 2nd ed. Hoboken, NJ, USA: Wiley, 2008. [7] W. Thomson, “On the forces experienced by small spheres under magnetic influence; and on some of the phenomena presented by diamagnetic substances,” Cambridge Dublin Math. J., May 1847. [8] W. Braunbek, “Freies schweben diamagnetischer Körper im magnetfeld,” Zeitschrift Phys., vol. 112, nos. 11–12, pp. 764–769, 1939.
[9] Y. Liu, D.-M. Zhu, D. M. Strayer, and U. E. Israelsson, “Magnetic levitation of large water droplets and mice,” Adv. Space Res., vol. 45, no. 1, pp. 208–213, Jan. 2010. [10] M. D. Simon, L. O. Heflinger, and A. K. Geim, “Diamagnetically stabilized magnet levitation,” Amer. J. Phys., vol. 69, no. 6, pp. 702–713, 2001. [11] I. Kovacic and M. J. Brennan, The Duffing Equation: Nonlinear Oscillators and Their Behaviour. New York, NY, USA: Wiley, 2011. [12] A. O. Niedermayer, T. Voglhuber-Brunnmaier, J. Sell, and B. Jakoby, “Methods for the robust measurement of the resonant frequency and quality factor of significantly damped resonating devices,” Meas. Sci. Technol., vol. 23, no. 8, p. 085107, 2012. [13] T. H. Boyer, “The force on a magnetic dipole,” Amer. J. Phys., vol. 56, no. 8, pp. 688–692, 1988.
Stefan Clara was born in Bruneck, Italy, in 1985. He received the Dipl.-Ing. (M.Sc.) degree in mechatronics from Johannes Kepler University Linz, Linz, Austria, in 2010, where he has been a Research Assistant with the Institute for Microelectronics and Microsensors since 2011. His focus is on fluid properties sensors, in particular, high viscosities.
Hannes Antlinger received the Dipl.-Ing. (M.Sc.) degree in mechatronics from Johannes Kepler University Linz, Linz, Austria, in 2001. He was a Hardware Design Engineer of Embedded Systems with Keba AG, Linz, for several years, before he joined the Institute for Microelectronics and Microsensors, Johannes Kepler University Linz in 2010, where he is currently a Researcher in Viscosity Sensors.
Wolfgang Hilber, photograph and biography not available at the time of publication.
Bernhard Jakoby (SM’98) received the Dipl.-Ing. (M.Sc.) degree in communication engineering, the Ph.D. degree in electrical engineering, and the Venia Legendi degree in theoretical electrical engineering from the Vienna University of Technology (VUT), Vienna, Austria, in 1991, 1994, and 2001, respectively. He was a Research Assistant with the Institute of General Electrical Engineering and Electronics, VUT, from 1991 to 1994. Subsequently, he stayed as an Erwin Schrödinger Fellow with the University of Ghent, Ghent, Belgium, performing research on the electrodynamics of complex media. From 1996 to 1999, he was a Research Associate and later an Assistant Professor with the Delft University of Technology, Delft, The Netherlands, where he was involved in microacoustic sensors. From 1999 to 2001, he was with the Automotive Electronics Division, Robert Bosch GmbH, Reutlingen, Germany, where he conducted the development projects in automotive liquid sensors. In 2001, he joined the Industrial Sensor Systems Group at VUT as an Associate Professor. In 2005, he was appointed as a Full Professor of Microelectronics with Johannes Kepler University Linz, Linz, Austria. He is currently working in the field of liquid sensors and monitoring systems. Prof. Jakoby served as the Technical Co-Chair, Co-Chair, and General Chair for various conferences (including the IEEE Sensors Conference and I2MTC). He is currently an Editorial Board Member of the IEEE S ENSORS J OURNAL and Measurement Science and Technology. He was elected as a Eurosensors Fellow in 2009 and received the Outstanding Paper Award for a paper in the IEEE T RANSACTIONS ON U LTRASONICS , F ERROELECTRICS AND F REQUENCY C ONTROL in 2010.
