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Measure of the capacitance coefficients of square electrodes using capacitance-to-digital converters
Journal of Electrostatics 87 (2017) 26e31
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Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
Measure of the capacitance coefficients of square electrodes using capacitance-to-digital converters Enrico Cataldo a, Alberto Di Lieto b, Francesco Maccarrone a, *, Giampiero Paffuti b a b
University of Pisa, Physics Department, Largo Pontecorvo 7, Pisa, Italy University of Pisa, Physics Department and INFN Pisa, Largo Pontecorvo 7, Pisa, Italy
a r t i c l e i n f o
a b s t r a c t
Article history: Received 31 January 2017 Accepted 7 March 2017
We measure the coefficients of self capacitance C11 and of interelectrode capacitance C12 of a pair of square parallel plates, by using commercial models of capacitance to digital converters. The measured coefficient depends on which charge is sampled, that of the excited electrode or that of the other one. The analysis of the data at large distances is in agreement with the asymptotic formulas for the capacitance coefficients. The different leading behaviour as a function of the distance gives coherent independent estimations of the intrinsic capacitance of a single plate, in good agreement with the calculated one. © 2017 Elsevier B.V. All rights reserved.
Keywords: Coefficients of capacitance Asymptotic formulas for capacitance Capacitance-to-Digital-converters Capacitive sensors PACS: 41.20.Cv 84.32 Tt
1. Introduction Since several years, the readout of a capacitive sensor (CS) has been highly simplified through the application of Capacitance-toDigital Converter (CDC). They directly convert an unknown capacitance to a digital word. Nowadays, monolithic CDCs offer direct interface of analog sensors to digital systems. Currently, they are the first choice for experimenters working with instrumentation based on CSs, both from the point of view of the cost and for the limited needs of signal conditioning. Over the past few years, a large number of papers have dealt with apparata in which the CDC readout of a CS is used in applications considered typical such as with capacitive accelerometers or dynamometers [1,2], liquid level sensors [3] or proximity detectors and also in more fundamental studies ranging from Earth's gravimetry [4] to the detection of tiny traces of chemical substances adsorbed to one of the electrodes forming the CS [5]. In most of the experimental practice, an accurate accordance between the ideal model and the real device is not mandatory, since much of the interest resides into the relative changes of the capacitance and an appropriate calibration of the sensor is
* Corresponding author. E-mail address: [email protected] (F. Maccarrone). http://dx.doi.org/10.1016/j.elstat.2017.03.002 0304-3886/© 2017 Elsevier B.V. All rights reserved.
commonly feasible. On the other hand, the design of a real sensor often requires a complete knowledge of the absolute capacitance of a set of conductors having a given geometry. Over the past 30 years, a new generation of CSs and actuators has grown in the framework of microelectromechanical systems (MEMS) technology. A large market of MEMS devices has been developed offering many instruments able to sense and regulate very small displacements and forces through the measure and the control of a microscopic capacitor. In that kind of devices the dimensions of the microelectrodes are generally comparable to their separation and the field distribution around the electrodes must be considered in order to foresee the transfer function of the sensors. Although the analytical calculation of the capacitance of a system of conductors has been a challenging task since long time [6,7] and exact solutions are available only for a small variety of geometries [8,9], accurate numerical techniques are largely applied for many typical arrangements of the electrodes [10e13]. From the point of view of measurement technology, the changes in the capacitance value are detected by connecting the capacitive sensor to an electric circuit and by applying an excitation signal, e.g a static or variable voltage, between the electrodes of the capacitor and by registering the change of the amplitude, phase or frequency of some electrical quantity such as current, voltage, charge at appropriate test points of the circuit [14].
E. Cataldo et al. / Journal of Electrostatics 87 (2017) 26e31
In this paper we present the results of a comparison of numerical modelisation of a capacitor with parallel plane geometry with the capacitance measurements of a parallel pair of metal plates, the basic geometry adopted in the description of a CS. We used two models of the most popular CDC device available in the market, manufactured by Analog Device [15]. The aim of the study is to clarify the principle of operation of these instruments from the point of view of the effective physical quantities detected. Particularly, we show that it is possible to measure separately the capacitance coefficients Cij of a couple of conductors. We checked these properties measuring the self capacitance C11 and the interelectrode capacitance C12 of two equal conducting plates as a function of the distance. We compared the experimental data with the numerical results and with the asymptotic behaviour of the two quantities at large distances [16].
2. Materials and methods 2.1. Switched capacitor CDC Many modern CDC are based on the technology of the S D ADC. A particular implementation of the balancing process typical of the S D converters, named switched capacitor modulation, is the basis for using that technology for the capacitance measurement [17]. A reference voltage Vref is applied periodically to one plate of an unknown capacitor Cx and the charge induced on the conductor itself or on the other plate of the capacitor is summed, using an op-amp integrator, with the charge present on a second known capacitor excited with the opposite voltage Vref . The feedback mechanism tends to balance the charge on the two capacitors, increasing or decreasing the charge on the sampled plate by reversing the sign of Vref . The ratio of charging to the total cycles is used, in accordance of a built-in calibration procedure, to translate the ADC scheme into that of a CDC. Among these instruments based on the principle of S D conversion technique, the series of low-cost monolithic devices launched in 2005 by Analog Device became very popular in the last few years. In that family the AD7475 model is designed to be used with a floating capacitor while AD7747 modelmeasures a capacitor with a grounded leg. We show that the different internal circuitry of the two devices makes it possible to measure separately the capacitance coefficients of a capacitor made by two conductors of any shape.
2.2. Capacitance coefficients It is well known that in a two-conductors system, the charge Q1 on conductor 1 depends linearly on the potential V1 and V2 applied, respectively, to the conductor 1 itself and to the conductor 2 through:
C¼
2 C11 C22 C12 C11 þ C22 þ 2C12
27
(1)
having taken into account that C12 ¼ C21 . A system made of two plane conductors of the same shape and area A placed parallel to each other and separated by a distance d, is the pedagogical prototype of a two conductors capacitor and also the simplest model of the basic capacitive sensor. The parallel plate capacitor is very interesting from a practical point of view as similar configurations are quite common in sensor devices in a large variety of applications. For example, small displacement are detected through the variation of the capacitance of the two plates induced by the relative movement of one plate with respect to the other fixed to a testing body, e.g. in the direction orthogonal to the planes of the electrodes. In other sensors the change in capacitance is driven by the variation of the pressure acting on one deformable plate so that a capacitance measurements gives information on the force/pressure acting onto a body. So, the principle of operation of those devices is very simple and if one restricts the examination to the ideal case in which the electric field is confined in the volume between the two plates, it is also expressed in an elementary form. In fact, in these conditions, the value of the capacitance is approximated by the ratio of the area to the plate separation, according to the well-known formula:
C0 ¼ ε0
A d
(2)
In a real capacitor the “fringe effect” occurs due to the extension of the field lines outside of the volume between the plates. Its magnitude depends on the aspect ratio k which, for parallel square plates pffiffiffigeometry, is just the ratio of the plate separation to the side L ¼ A of the square:
k¼
d L
Increasing the aspect ratio generally results in a value of the capacitance larger than the ideal case and, for the parallel square plate geometry, the correction is lower than a few per cent when k is smaller than 0.01. Let us consider the quantity effectively measured with the Analog Device CDCs when the two electrodes of a capacitor are connected to the input channels of each CDC and a square wave excitation signal Vref is applied to one of either conductors. In the AD7745 model the charge induced on the other armature of the capacitor is compared with that of an internal reference capacitor and the converted value is returned. In this condition, the measured charge is Q2 and the CDC reads the capacitance coefficient C12 . Differently, using the AD7747, the excitation signal is applied to the same electrode which is connected to the comparator, while the other plate is grounded, and the measure of Q1 is used to derive the value of the capacitance coefficient C11 .
Q1 ¼ C11 V1 þ C12 V2 3. Measurements where the coefficient Cij are called capacitance coefficients. A similar relation holds for the charge Q2 on conductor 2:
Q2 ¼ C21 V1 þ C22 V2 The quantity to which we refer commonly as the capacitance C of a two conductors system applies when the charge of the two conductors is opposite: Q1 ¼ Q and Q2 ¼ Q . Then, the relative capacitance C is the ratio between the charge Q and the difference of potentials V1 V2 . From the two last equations, it is easy to calculate the relation of C with the capacitance coefficients:
In this section we present four experiments, which clearly highlight the operating mode of the two CDCs. The comparison between measured and expected results confirms the correctness and the accuracy of the principle of measurement described in the previous section. 3.1. Coefficients of capacitance of parallel square plates The parallel-plates capacitor is composed by a couple of square
28
E. Cataldo et al. / Journal of Electrostatics 87 (2017) 26e31
plates, realized in copper with a side L ¼ 25 mm and a thickness t ¼ 0.3 mm. Electrical connections are made with coaxial cables a few tens of centimetres long with the central conductor plunged into a small connector soldered on a corner of each plate. One of the two plates is fixed vertically while the other, parallel to the first one, is mounted on a translational stage having both fine and coarse micrometer movement, spanning many centimetres with tenths of millimetres accuracy. In the first experiment we fix the distance so that the aspect ratio is k ¼ 0.08, and the fringe effects are not negligible. When we use the AD7745, one plate is connected to the input CINðþÞ linked to the internal comparator and the other to the pin EXCA carrying the reference waveform, the measure returned by the device was Cm ¼ 2925 fF. If the measure is interpreted as that of the relative capacitance given by eq. (2), neglecting the fringe effect, the expected value should be C0 ¼ 2763fF. At first sight, the measured value is in partial agreement with the expected one, but one must take in account at least the fringe effect. The numerical treatment of the case of two bidimensional square plates is wellknown in literature. For example, in Ref. [10] the numerical results are presented as a multiplying factor g greater than one which depends on the aspect ratio. For the considered value of the parameters, g 1:25 and the corrected value of the capacitance is C ¼ gC0 ¼ 3453 fF, in severe disagreement with the measure. A similar discrepancy is observed also using AD7747. We put one of the plate at ground and the other in the CINðþÞ. With these electrical connections the measured value is Cm ¼ 3904 fF a value not compatible again with the expected one. In order to solve the problem let us refer to the discussion at the end of section 2. In the first experiment, i.e. using the AD7745 and with the plates floating with respect to ground, we are measuring the absolute value of the interelectrode capacitance coefficient C12 , which value is actually negative. On the other hand, with AD7747, the measured coefficient is C11 . The coefficients of mutual induction are equal by definition C12 ¼ C21 and the symmetry of the plates gives C22 ¼ C11 . With these conditions the relative capacitance C of eq. (1) simplifies to:
C¼
C11 C12 2
Substituting the measured values of the two capacitance coefficients and taking care to change the sign of the value of C12 one finds C ¼ 3414 fF in good agreement with the expected value. This result, easily reproducible, gives the first evidence of the principle of operation alleged in the previous section. 3.2. Capacitance of a monolithic capacitor Both models of CDC can be used to measure a commercial capacitor such as a multilayer ceramic chip (MLCC) or a monolithic disk ceramic capacitor of few pF of capacitance. The two devices return the same value within the declared accuracy, in close agreement with the nominal capacitance. In fact, those capacitors are realized in a quite small packages where the electrodes are very close together. Let us consider, for the sake of simplicity, two equal conductors having therefore C11 ¼ C22 and by definition C21 ¼ C12 placed at a tiny distance d. In order to ascertain the behaviour of the system when d/0, let us assume that the two plates are placed at the same potential V. As the conductors are identical one finds that each plate brings the same charge;
Q ¼ C11 V þ C12 V Let us now detach the plates from the potential sources so that the charges can not change and reduce the distance toward the
limit d/0. By symmetry, the two bodies stay equipotential. A finite electric field in the near limit would involve a discontinuity in the charge surface densities, but the electric field between the two very close surfaces at equal V is infinitesimal, so that both the charges and the potential do not diverge at the moment of the contact (d ¼ 0) at which the capacitors becomes a single equipotential conductor. Then we have that the limit
lim C11 þ C12 ¼ a
d/0
is finite and the divergence of the two coefficients of capacitance are equal and opposite. Denoting with S the divergent part of C12 we have:
C12 ¼ S þ O ðdÞ
C11 ¼ S þ a þ O ðdÞ ¼ C12 þ a þ O ðdÞ
and the relative capacitance C of the two conductors is for this special case of identical bodies:
C¼
C11 C12 ¼ C11 þ a ¼ C12 þ a 2
For very close plates [9], the diverging term overcomes a, which can be neglected in the near limit where
C ¼ C11 ¼ C12 This explains why the readout of the CDC for a monolithic capacitor having a gap very narrow coincides with the nominal value of the relative capacitance in both mode of operation. 3.3. Capacitance coefficients as a function of distance The experiments reported in the two previous subsections strengthen the statement that AD7745 measures the absolute value of the interelectrode capacitance C12 and AD7747 the selfcapacitance C11 . This is not a limitation when one of the two instruments is applied to a sensor where the two electrodes are very close. In that case the two coefficients tend to the same diverging value, independently from the shapes of the electrodes. On the other hand, the different value returned by the CDC when the interelectrode distance grows must be taken into account either if an accurate evaluation of the capacitance is needed or the sensitivity of the sensor must be maximized. In this subsection we describe a more systematic test reporting the measurements of the capacitance coefficients of the parallel plates as a function of their distance. Incidentally, it is worth noticing that we are not aware of similar experimental data in literature. By using the same plates described above, we collect data in the interval of interelectrode distance between 1 mm and many cm. The minimum distance is limited by the full scale value of both the converters. In the first few tens of millimetres the read value was recorded at intervals of one tenths of mm and the steps grow up to one mm in the far region. Particular care is taken in removing any external conductor from the measured system, consisting in the CDC, the two cables and the plates. The entire system is held up by plastic holders, maximizing the distance between the apparatus and all the other bodies, comprising those of the experimenters. But, even if the two plates are very far from any other major source of electric influence, there is an inevitable contribution from connection wires and the chip mount itself. Particularly, as the measurement of C12 consists in sampling the stationary charge on the plate connected to the input channel of the CDC caused by the charging cycles on the other plate, the influence of the cables can be reduced by maintaining the excitation wire as
E. Cataldo et al. / Journal of Electrostatics 87 (2017) 26e31
29
far as possible from the sampled side. Nevertheless, the two cables end to the very close chip pins, so a discrepancy between the measured coefficient and that of the pair of isolated conductors is expected. Induction of the exciting wire on the sampled part of the circuit is reduced by using coaxial cables with grounded shields. With that care the length of the wires has a minor influence on the readout. In the measure of C11 the excited plate is also the sampled one while the other side of the system is at ground potential. Then, a particular attention should be paid to the excitation cable: it changes the sensed quantity, being a part of the conductor 1. The self-capacitance of an unshielded wire grows up with the length so that it is possible to use only very short unipolar cables. If the plate 1 is placed distant from the CDC a coaxial cable must be used mandatorily together with a compensating mechanism, included in the design of the AD7747 CDC. Nevertheless, we observe that if the coaxial cable is too long a systematic negative offset is introduced. The magnitude of the shift must be taken into account if an accurate measure is needed. In our experiment we estimated that the discrepancy due to this effect is less than 50 fF. It is well known that the two coefficients C11 and C12 , which tend to equalize in the limit d/0 (see the last subsection), have a different asymptotic behaviour when the distance grows [18]. In particular, C12 tends to vanish and C11 goes to a constant value. Physically, these limits are expected. In fact, C12 measures the electrostatic induction of the conductor 1 on the conductor 2 and that mutual influence must vanish at large distances. On the other hand, the coefficient C11 is the parameter describing the selfcapacitance, i.e. the charge induced on the plate 1 when it is brought at a potential V1 , possibly in the presence of the second conductor. When the latter is very far C11 tends to the intrinsic capacitance C1 of the plate 1 which is simply the ratio of the charge Q1 to the potential V1 of the body very far from any other conductor. The two behaviours are shown in Fig. 1. In the absence of an analytic expressions valid in the entire range of investigated distances, it is only possible a comparison between computed and experimental data. Actually, several suitable numerical algorithms apply to the calculation of the coefficients of capacitance, particularly for 2D plates. We have implemented a numerical procedure taking into account also the
thickness (0.3 mm) of our plates, which is important for having accurate numerical results at very small distances. Even if a high accuracy is time-consuming, it has been possible to make a precise quantitative comparison between the measured quantities and those calculated with boundary element methods [19]. Details on our numerical algorithm will be discussed elsewhere. Fig. 1 shows a macroscopic discrepancy between measured and calculated values: raw data tend at large distances to a value r different from the calculated one. Roughly speaking, a typical decrease of few hundreds of fF is observed with respect to the numerical calculation when the cable bringing the excitation signal is few tens of cm long and the value varies with the position of the connections. Quantitatively, the value of C11 should converge to the intrinsic capacitance C1 , which for the plates considered is 1044 fF. On the contrary, the asymptotic value extrapolated from the data is only 720 fF. A similar offset is observed also measuring C12 although of minor magnitude and with different sign. Those systematic behaviours are attributable to the fact that the two conductors are composed by the plates and the connections, mainly the wires and their attachments. In fact the magnitude of the discrepancy depends mainly the length of the cables, their shielding and the relative position of the attachments. As the scope of our actual experiments is to clarify the primary physical quantities measured by the CDCs, we leave the problem of the absolute calibration of the capacimeter to a more refined version of the apparatus. It is worth to note that the principal effects of these multiple causes of deviation for each coefficient can be taken into account correcting the data with a uniform offset. In Fig. 2 the same data reported in the Fig. 1 are shifted uniformly in the vertical direction, minimizing the square residuals between the corrected and the calculated data. For a number of different geometries of the wires and of the connections the offset is of few hundreds of fF for the C11 and some tens of fF for C12. The right panels of the figure, showing the residual of the fitting procedure show a very good agreement, as in the entire range examined the discrepancies do not exceed two per cent. These results show that the CDCs are suitable to be used to measure with good accuracy the coefficients of capacitance of a pair of plates. Of course, the same experimental procedure applies also to more complex conductor bodies. In the hypothesis that the offset
Fig. 1. Coefficients of capacitance C11 (o) and C12 (þ) of two square copper plates with side L ¼ 25 mm, as a function of distance. The raw data show clearly that the two coefficients converge to the same value at very small gaps and that the asymptotic trends at large distances are compatible with a constant value for C11 and to zero for C12 . The continuous lines are the numerical calculation of the two coefficients, using a method of moments and taking into account the thickness of the plates.
Fig. 2. Left panel: same graphs as in figure, applying to the raw data uniform corrections consisting in a shift of the measured values of the capacitances. Here, DC11 ¼ 370fF DC12 ¼ 137fF. Right upper panel: Residual for C11 calculated as the difference between the unbiased and the calculated data. Right lower panel: Residual for the data of C12 .
30
E. Cataldo et al. / Journal of Electrostatics 87 (2017) 26e31
4. Discussion
Fig. 3. The data at distances d > 2L are fitted with the two models C11 ¼ a þ b=d2 and C12 ¼ h þ k=d of eq. (3). In the figure the curve fitted and the data, showing a very good agreement. As b≡C13 and k ¼ C12 we evaluate C1 in the two fitting procedures finding, in the fit of C12 , C1 ¼ 1117fF and in that of C11 C1 ¼ 1124fF. It is noticeable that, in the same interval of distances, the two coefficients undergo a much different change in accordance with the different power-dependence on 1=d in eq. (3).
in the converted datum is fixed by the geometry of the connections it seems possible to use the well-known and reproducible parallel plate geometry to calibrate the instruments and, after that connect it to a generic pair of conductor, obtaining with a little effort an experimental accurate knowledge of the complete electrostatic properties of the system. 3.4. Modelling the far limit When the conductors are very near or very far it is possible to write analytical models reproducing the asymptotic trends. At the far limit the coefficients of capacitance of two generic conductor can be expressed by the first terms of a power expansion in 1=d involving as parameters the intrinsic capacitances of the two conductors and their polarizabilities and quadrupole moments [16]. In the simplified geometry of two equal parallel plates C11 ¼ C22 ¼ C1 and taking the leading term in 1=d one finds:
C11 ¼ C1 1 þ
C2 C12 ¼ 1 d
C12 d2
!
(3)
As the two asymptotic trends are expressed, for identical sides of the capacitor, in term of the same parameter C1 we fit, with the two simple functions of eq. (3), the raw data for distances larger than the side of the plates. The graphical result of the fit is reported in Fig. 3, where a very good agreement is shown. It is worth to note that only the constant term in the model function is influenced by the offset. The other two parameters represent C13 and C12 for identical plates and wires. The value of C1 determined in this way are consistent with each other and with the, offset-dependent, horizontal asymptote of C11 . This check confirms the procedure of data analysis and also is a clear test of the formulas in the far limit for the coefficient of capacitances which at the leading level are valid for generic bodies. Moreover the evaluation of C1 is also independent from the absolute estimate of the offset.
Usually, the technical presentations of the operation of commercial CDCs and their specification documents emphasize the precision of the data conversion, the accuracy of the measurement and the sampling frequency. Indeed, as these devices are largely applied for the readout of capacitive sensors the primary goal is to obtain a high sensitivity to the variations of the charges on the capacitor plates resulting from a change of the relative position, the approach of a third body, the deposition of impurities which vary the dielectric constant of the space occupied by the electric field around the capacitor and so on. Little attention and little information is given on the physical quantities characterizing the system and on their influence on the returned data. A common architecture of the CDC produced by the primary manufacturers is based on the measure of the charge accumulated on one of the sensor plates during repeated cycles of charge and discharge. The measure consists in the comparison of the charge to be sampled on the external sensor with that filed on an embedded calibrated capacitor. A sigma-delta ADC converts the measures in digital words. The capacitive sensor is generally realized with two conductors and the measure can consist in exciting and sampling the charge on the same conductor, the other being connected, typically, to ground, as in the CDC AD7747, manufactured by Analog Device. A second scheme consists in the excitation of one of the two plates and in the sampling of the other (e.g. AD7745). The two techniques read a different capacitance coefficient. Moreover, our physical description of the principle of operation shed light on the effects of the connections between the main conductor bodies and the IC input ports and of the bodies surrounding the apparatus. With respect to the first issue, it is clear that the cable and the connection to the excited plate contributes in both mode of operation at the same level with the plate. But the magnitude of the effect is different for the measures of C11 and of C12 . In the first case, the wire is excited with the same waveform of the plate and it is charged at the same time. As the intrinsic capacity of the wire contributes with hundreds of fF already for length of few centimetres it is impractical to use long unipolar cables to connect the CDC measuring C11 . As a matter of fact, the AD7747 CDC is equipped with a second exciting signal which can be applied to the shield of a coaxial cable which internal conductor is connected to the conductor 1 when the measure is made. An explanation of the role of this connection can be given on a physical basis. Let us now consider that only the shielded cable is connected and the core and the shield are brought to the same potential so that the electric field between the wire and the shield is zero. This condition is realized when the inner wire has no charge. This condition is perturbed when the plate is connected and the electrical induction between the coaxial cable and the plate is responsible for the negative offset observed. A major difficulty to keep this effect under control is that one expects that the magnitude of the phenomenon could depend, even for the same cable, on the shape of the body connected to it. The other cable is not directly excited by the signal generated by the CDC circuit and it is maintained at constant potential in both modes of operation. Moreover it is further away from the excited one and from the first body so that the induction effects are weaker, mainly when the two principal bodies under test are far from each other and the two cable are placed maximizing their distance for the major part of their path. Nevertheless, an influence of the second cable on the readout of C11 and of C12 is unavoidable and could be reduced using also in the side 2 a coaxial cable with the shield grounded. We observe that when measuring C12 the inner conductor of the coaxial cable is not at the same potential of the grounded shield so that it is charged with plate 2 by induction of the side 1 and an increase of the sampled charge is expected, as the observed positive offset signals.
E. Cataldo et al. / Journal of Electrostatics 87 (2017) 26e31
5. Conclusions The experiments reported in the paper demonstrate the physical mechanism underlying the capacitance-to-digital converters operating through the sampling of the charges accumulated on one of a pair of conductor. We have shown that the returned data when two parallel plate are connected to the CDC does not coincide with what we intend usually as capacitance C. Really, the value returned is the coefficient of self-capacitance C11 or the induction capacitance C12 . We have explained also that, due to the convergence of C11 and C12 to the same value when the distance between the plates tends to vanish both the mode of operation of the CDCs return, in this condition, the value of relative capacitance C and this is the result when a small monolithic capacitor is tested. The results show that the instruments, basically applied for the ability to sense a change in sensor capacitance, are able to perform with a little effort accurate measure of the basic properties of an electrostatic system. We checked with a very simple apparatus the dependence of the coefficients of capacitance of two square plates up to a value of the distance several times the side of the plates and we made a strict comparison with the calculated numerical results finding a very good agreement both on the global range of the recorded data, when the effect of the connections, consisting primarily in an overall offset of the measured data, are taken into account properly. This quantitative experimental determination of the fringe effects of a parallel plate capacitor up to a very high value of the aspect ratio d=L is not common in the technical literature. On the other hand, it should be easy to investigate with the same apparatus other systems such as a couple of circular disks or two rectangular plates in the strip configuration. Moreover a very good agreement is found with the asymptotic behaviour of the coefficient of capacitance. The fitting of the well known minimal model for the expected trend give accurate value of the intrinsic capacitance of the plates used, and as the model resulting from eq. (3) is general the procedure should apply with minor modifications to conductors of generic shapes and relative disposition. The fact that sub-leading terms in the higher power of 1=d involve, as demonstrated in Ref. [16], other interesting parameters such as the polarizability of the conducting bodies makes interesting the possibility to refine
31
the apparatus and the data analysis in order to get deeper information on less symmetric systems of two conductors.
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Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
Measure of the capacitance coefficients of square electrodes using capacitance-to-digital converters Enrico Cataldo a, Alberto Di Lieto b, Francesco Maccarrone a, *, Giampiero Paffuti b a b
University of Pisa, Physics Department, Largo Pontecorvo 7, Pisa, Italy University of Pisa, Physics Department and INFN Pisa, Largo Pontecorvo 7, Pisa, Italy
a r t i c l e i n f o
a b s t r a c t
Article history: Received 31 January 2017 Accepted 7 March 2017
We measure the coefficients of self capacitance C11 and of interelectrode capacitance C12 of a pair of square parallel plates, by using commercial models of capacitance to digital converters. The measured coefficient depends on which charge is sampled, that of the excited electrode or that of the other one. The analysis of the data at large distances is in agreement with the asymptotic formulas for the capacitance coefficients. The different leading behaviour as a function of the distance gives coherent independent estimations of the intrinsic capacitance of a single plate, in good agreement with the calculated one. © 2017 Elsevier B.V. All rights reserved.
Keywords: Coefficients of capacitance Asymptotic formulas for capacitance Capacitance-to-Digital-converters Capacitive sensors PACS: 41.20.Cv 84.32 Tt
1. Introduction Since several years, the readout of a capacitive sensor (CS) has been highly simplified through the application of Capacitance-toDigital Converter (CDC). They directly convert an unknown capacitance to a digital word. Nowadays, monolithic CDCs offer direct interface of analog sensors to digital systems. Currently, they are the first choice for experimenters working with instrumentation based on CSs, both from the point of view of the cost and for the limited needs of signal conditioning. Over the past few years, a large number of papers have dealt with apparata in which the CDC readout of a CS is used in applications considered typical such as with capacitive accelerometers or dynamometers [1,2], liquid level sensors [3] or proximity detectors and also in more fundamental studies ranging from Earth's gravimetry [4] to the detection of tiny traces of chemical substances adsorbed to one of the electrodes forming the CS [5]. In most of the experimental practice, an accurate accordance between the ideal model and the real device is not mandatory, since much of the interest resides into the relative changes of the capacitance and an appropriate calibration of the sensor is
* Corresponding author. E-mail address: [email protected] (F. Maccarrone). http://dx.doi.org/10.1016/j.elstat.2017.03.002 0304-3886/© 2017 Elsevier B.V. All rights reserved.
commonly feasible. On the other hand, the design of a real sensor often requires a complete knowledge of the absolute capacitance of a set of conductors having a given geometry. Over the past 30 years, a new generation of CSs and actuators has grown in the framework of microelectromechanical systems (MEMS) technology. A large market of MEMS devices has been developed offering many instruments able to sense and regulate very small displacements and forces through the measure and the control of a microscopic capacitor. In that kind of devices the dimensions of the microelectrodes are generally comparable to their separation and the field distribution around the electrodes must be considered in order to foresee the transfer function of the sensors. Although the analytical calculation of the capacitance of a system of conductors has been a challenging task since long time [6,7] and exact solutions are available only for a small variety of geometries [8,9], accurate numerical techniques are largely applied for many typical arrangements of the electrodes [10e13]. From the point of view of measurement technology, the changes in the capacitance value are detected by connecting the capacitive sensor to an electric circuit and by applying an excitation signal, e.g a static or variable voltage, between the electrodes of the capacitor and by registering the change of the amplitude, phase or frequency of some electrical quantity such as current, voltage, charge at appropriate test points of the circuit [14].
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In this paper we present the results of a comparison of numerical modelisation of a capacitor with parallel plane geometry with the capacitance measurements of a parallel pair of metal plates, the basic geometry adopted in the description of a CS. We used two models of the most popular CDC device available in the market, manufactured by Analog Device [15]. The aim of the study is to clarify the principle of operation of these instruments from the point of view of the effective physical quantities detected. Particularly, we show that it is possible to measure separately the capacitance coefficients Cij of a couple of conductors. We checked these properties measuring the self capacitance C11 and the interelectrode capacitance C12 of two equal conducting plates as a function of the distance. We compared the experimental data with the numerical results and with the asymptotic behaviour of the two quantities at large distances [16].
2. Materials and methods 2.1. Switched capacitor CDC Many modern CDC are based on the technology of the S D ADC. A particular implementation of the balancing process typical of the S D converters, named switched capacitor modulation, is the basis for using that technology for the capacitance measurement [17]. A reference voltage Vref is applied periodically to one plate of an unknown capacitor Cx and the charge induced on the conductor itself or on the other plate of the capacitor is summed, using an op-amp integrator, with the charge present on a second known capacitor excited with the opposite voltage Vref . The feedback mechanism tends to balance the charge on the two capacitors, increasing or decreasing the charge on the sampled plate by reversing the sign of Vref . The ratio of charging to the total cycles is used, in accordance of a built-in calibration procedure, to translate the ADC scheme into that of a CDC. Among these instruments based on the principle of S D conversion technique, the series of low-cost monolithic devices launched in 2005 by Analog Device became very popular in the last few years. In that family the AD7475 model is designed to be used with a floating capacitor while AD7747 modelmeasures a capacitor with a grounded leg. We show that the different internal circuitry of the two devices makes it possible to measure separately the capacitance coefficients of a capacitor made by two conductors of any shape.
2.2. Capacitance coefficients It is well known that in a two-conductors system, the charge Q1 on conductor 1 depends linearly on the potential V1 and V2 applied, respectively, to the conductor 1 itself and to the conductor 2 through:
C¼
2 C11 C22 C12 C11 þ C22 þ 2C12
27
(1)
having taken into account that C12 ¼ C21 . A system made of two plane conductors of the same shape and area A placed parallel to each other and separated by a distance d, is the pedagogical prototype of a two conductors capacitor and also the simplest model of the basic capacitive sensor. The parallel plate capacitor is very interesting from a practical point of view as similar configurations are quite common in sensor devices in a large variety of applications. For example, small displacement are detected through the variation of the capacitance of the two plates induced by the relative movement of one plate with respect to the other fixed to a testing body, e.g. in the direction orthogonal to the planes of the electrodes. In other sensors the change in capacitance is driven by the variation of the pressure acting on one deformable plate so that a capacitance measurements gives information on the force/pressure acting onto a body. So, the principle of operation of those devices is very simple and if one restricts the examination to the ideal case in which the electric field is confined in the volume between the two plates, it is also expressed in an elementary form. In fact, in these conditions, the value of the capacitance is approximated by the ratio of the area to the plate separation, according to the well-known formula:
C0 ¼ ε0
A d
(2)
In a real capacitor the “fringe effect” occurs due to the extension of the field lines outside of the volume between the plates. Its magnitude depends on the aspect ratio k which, for parallel square plates pffiffiffigeometry, is just the ratio of the plate separation to the side L ¼ A of the square:
k¼
d L
Increasing the aspect ratio generally results in a value of the capacitance larger than the ideal case and, for the parallel square plate geometry, the correction is lower than a few per cent when k is smaller than 0.01. Let us consider the quantity effectively measured with the Analog Device CDCs when the two electrodes of a capacitor are connected to the input channels of each CDC and a square wave excitation signal Vref is applied to one of either conductors. In the AD7745 model the charge induced on the other armature of the capacitor is compared with that of an internal reference capacitor and the converted value is returned. In this condition, the measured charge is Q2 and the CDC reads the capacitance coefficient C12 . Differently, using the AD7747, the excitation signal is applied to the same electrode which is connected to the comparator, while the other plate is grounded, and the measure of Q1 is used to derive the value of the capacitance coefficient C11 .
Q1 ¼ C11 V1 þ C12 V2 3. Measurements where the coefficient Cij are called capacitance coefficients. A similar relation holds for the charge Q2 on conductor 2:
Q2 ¼ C21 V1 þ C22 V2 The quantity to which we refer commonly as the capacitance C of a two conductors system applies when the charge of the two conductors is opposite: Q1 ¼ Q and Q2 ¼ Q . Then, the relative capacitance C is the ratio between the charge Q and the difference of potentials V1 V2 . From the two last equations, it is easy to calculate the relation of C with the capacitance coefficients:
In this section we present four experiments, which clearly highlight the operating mode of the two CDCs. The comparison between measured and expected results confirms the correctness and the accuracy of the principle of measurement described in the previous section. 3.1. Coefficients of capacitance of parallel square plates The parallel-plates capacitor is composed by a couple of square
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plates, realized in copper with a side L ¼ 25 mm and a thickness t ¼ 0.3 mm. Electrical connections are made with coaxial cables a few tens of centimetres long with the central conductor plunged into a small connector soldered on a corner of each plate. One of the two plates is fixed vertically while the other, parallel to the first one, is mounted on a translational stage having both fine and coarse micrometer movement, spanning many centimetres with tenths of millimetres accuracy. In the first experiment we fix the distance so that the aspect ratio is k ¼ 0.08, and the fringe effects are not negligible. When we use the AD7745, one plate is connected to the input CINðþÞ linked to the internal comparator and the other to the pin EXCA carrying the reference waveform, the measure returned by the device was Cm ¼ 2925 fF. If the measure is interpreted as that of the relative capacitance given by eq. (2), neglecting the fringe effect, the expected value should be C0 ¼ 2763fF. At first sight, the measured value is in partial agreement with the expected one, but one must take in account at least the fringe effect. The numerical treatment of the case of two bidimensional square plates is wellknown in literature. For example, in Ref. [10] the numerical results are presented as a multiplying factor g greater than one which depends on the aspect ratio. For the considered value of the parameters, g 1:25 and the corrected value of the capacitance is C ¼ gC0 ¼ 3453 fF, in severe disagreement with the measure. A similar discrepancy is observed also using AD7747. We put one of the plate at ground and the other in the CINðþÞ. With these electrical connections the measured value is Cm ¼ 3904 fF a value not compatible again with the expected one. In order to solve the problem let us refer to the discussion at the end of section 2. In the first experiment, i.e. using the AD7745 and with the plates floating with respect to ground, we are measuring the absolute value of the interelectrode capacitance coefficient C12 , which value is actually negative. On the other hand, with AD7747, the measured coefficient is C11 . The coefficients of mutual induction are equal by definition C12 ¼ C21 and the symmetry of the plates gives C22 ¼ C11 . With these conditions the relative capacitance C of eq. (1) simplifies to:
C¼
C11 C12 2
Substituting the measured values of the two capacitance coefficients and taking care to change the sign of the value of C12 one finds C ¼ 3414 fF in good agreement with the expected value. This result, easily reproducible, gives the first evidence of the principle of operation alleged in the previous section. 3.2. Capacitance of a monolithic capacitor Both models of CDC can be used to measure a commercial capacitor such as a multilayer ceramic chip (MLCC) or a monolithic disk ceramic capacitor of few pF of capacitance. The two devices return the same value within the declared accuracy, in close agreement with the nominal capacitance. In fact, those capacitors are realized in a quite small packages where the electrodes are very close together. Let us consider, for the sake of simplicity, two equal conductors having therefore C11 ¼ C22 and by definition C21 ¼ C12 placed at a tiny distance d. In order to ascertain the behaviour of the system when d/0, let us assume that the two plates are placed at the same potential V. As the conductors are identical one finds that each plate brings the same charge;
Q ¼ C11 V þ C12 V Let us now detach the plates from the potential sources so that the charges can not change and reduce the distance toward the
limit d/0. By symmetry, the two bodies stay equipotential. A finite electric field in the near limit would involve a discontinuity in the charge surface densities, but the electric field between the two very close surfaces at equal V is infinitesimal, so that both the charges and the potential do not diverge at the moment of the contact (d ¼ 0) at which the capacitors becomes a single equipotential conductor. Then we have that the limit
lim C11 þ C12 ¼ a
d/0
is finite and the divergence of the two coefficients of capacitance are equal and opposite. Denoting with S the divergent part of C12 we have:
C12 ¼ S þ O ðdÞ
C11 ¼ S þ a þ O ðdÞ ¼ C12 þ a þ O ðdÞ
and the relative capacitance C of the two conductors is for this special case of identical bodies:
C¼
C11 C12 ¼ C11 þ a ¼ C12 þ a 2
For very close plates [9], the diverging term overcomes a, which can be neglected in the near limit where
C ¼ C11 ¼ C12 This explains why the readout of the CDC for a monolithic capacitor having a gap very narrow coincides with the nominal value of the relative capacitance in both mode of operation. 3.3. Capacitance coefficients as a function of distance The experiments reported in the two previous subsections strengthen the statement that AD7745 measures the absolute value of the interelectrode capacitance C12 and AD7747 the selfcapacitance C11 . This is not a limitation when one of the two instruments is applied to a sensor where the two electrodes are very close. In that case the two coefficients tend to the same diverging value, independently from the shapes of the electrodes. On the other hand, the different value returned by the CDC when the interelectrode distance grows must be taken into account either if an accurate evaluation of the capacitance is needed or the sensitivity of the sensor must be maximized. In this subsection we describe a more systematic test reporting the measurements of the capacitance coefficients of the parallel plates as a function of their distance. Incidentally, it is worth noticing that we are not aware of similar experimental data in literature. By using the same plates described above, we collect data in the interval of interelectrode distance between 1 mm and many cm. The minimum distance is limited by the full scale value of both the converters. In the first few tens of millimetres the read value was recorded at intervals of one tenths of mm and the steps grow up to one mm in the far region. Particular care is taken in removing any external conductor from the measured system, consisting in the CDC, the two cables and the plates. The entire system is held up by plastic holders, maximizing the distance between the apparatus and all the other bodies, comprising those of the experimenters. But, even if the two plates are very far from any other major source of electric influence, there is an inevitable contribution from connection wires and the chip mount itself. Particularly, as the measurement of C12 consists in sampling the stationary charge on the plate connected to the input channel of the CDC caused by the charging cycles on the other plate, the influence of the cables can be reduced by maintaining the excitation wire as
E. Cataldo et al. / Journal of Electrostatics 87 (2017) 26e31
29
far as possible from the sampled side. Nevertheless, the two cables end to the very close chip pins, so a discrepancy between the measured coefficient and that of the pair of isolated conductors is expected. Induction of the exciting wire on the sampled part of the circuit is reduced by using coaxial cables with grounded shields. With that care the length of the wires has a minor influence on the readout. In the measure of C11 the excited plate is also the sampled one while the other side of the system is at ground potential. Then, a particular attention should be paid to the excitation cable: it changes the sensed quantity, being a part of the conductor 1. The self-capacitance of an unshielded wire grows up with the length so that it is possible to use only very short unipolar cables. If the plate 1 is placed distant from the CDC a coaxial cable must be used mandatorily together with a compensating mechanism, included in the design of the AD7747 CDC. Nevertheless, we observe that if the coaxial cable is too long a systematic negative offset is introduced. The magnitude of the shift must be taken into account if an accurate measure is needed. In our experiment we estimated that the discrepancy due to this effect is less than 50 fF. It is well known that the two coefficients C11 and C12 , which tend to equalize in the limit d/0 (see the last subsection), have a different asymptotic behaviour when the distance grows [18]. In particular, C12 tends to vanish and C11 goes to a constant value. Physically, these limits are expected. In fact, C12 measures the electrostatic induction of the conductor 1 on the conductor 2 and that mutual influence must vanish at large distances. On the other hand, the coefficient C11 is the parameter describing the selfcapacitance, i.e. the charge induced on the plate 1 when it is brought at a potential V1 , possibly in the presence of the second conductor. When the latter is very far C11 tends to the intrinsic capacitance C1 of the plate 1 which is simply the ratio of the charge Q1 to the potential V1 of the body very far from any other conductor. The two behaviours are shown in Fig. 1. In the absence of an analytic expressions valid in the entire range of investigated distances, it is only possible a comparison between computed and experimental data. Actually, several suitable numerical algorithms apply to the calculation of the coefficients of capacitance, particularly for 2D plates. We have implemented a numerical procedure taking into account also the
thickness (0.3 mm) of our plates, which is important for having accurate numerical results at very small distances. Even if a high accuracy is time-consuming, it has been possible to make a precise quantitative comparison between the measured quantities and those calculated with boundary element methods [19]. Details on our numerical algorithm will be discussed elsewhere. Fig. 1 shows a macroscopic discrepancy between measured and calculated values: raw data tend at large distances to a value r different from the calculated one. Roughly speaking, a typical decrease of few hundreds of fF is observed with respect to the numerical calculation when the cable bringing the excitation signal is few tens of cm long and the value varies with the position of the connections. Quantitatively, the value of C11 should converge to the intrinsic capacitance C1 , which for the plates considered is 1044 fF. On the contrary, the asymptotic value extrapolated from the data is only 720 fF. A similar offset is observed also measuring C12 although of minor magnitude and with different sign. Those systematic behaviours are attributable to the fact that the two conductors are composed by the plates and the connections, mainly the wires and their attachments. In fact the magnitude of the discrepancy depends mainly the length of the cables, their shielding and the relative position of the attachments. As the scope of our actual experiments is to clarify the primary physical quantities measured by the CDCs, we leave the problem of the absolute calibration of the capacimeter to a more refined version of the apparatus. It is worth to note that the principal effects of these multiple causes of deviation for each coefficient can be taken into account correcting the data with a uniform offset. In Fig. 2 the same data reported in the Fig. 1 are shifted uniformly in the vertical direction, minimizing the square residuals between the corrected and the calculated data. For a number of different geometries of the wires and of the connections the offset is of few hundreds of fF for the C11 and some tens of fF for C12. The right panels of the figure, showing the residual of the fitting procedure show a very good agreement, as in the entire range examined the discrepancies do not exceed two per cent. These results show that the CDCs are suitable to be used to measure with good accuracy the coefficients of capacitance of a pair of plates. Of course, the same experimental procedure applies also to more complex conductor bodies. In the hypothesis that the offset
Fig. 1. Coefficients of capacitance C11 (o) and C12 (þ) of two square copper plates with side L ¼ 25 mm, as a function of distance. The raw data show clearly that the two coefficients converge to the same value at very small gaps and that the asymptotic trends at large distances are compatible with a constant value for C11 and to zero for C12 . The continuous lines are the numerical calculation of the two coefficients, using a method of moments and taking into account the thickness of the plates.
Fig. 2. Left panel: same graphs as in figure, applying to the raw data uniform corrections consisting in a shift of the measured values of the capacitances. Here, DC11 ¼ 370fF DC12 ¼ 137fF. Right upper panel: Residual for C11 calculated as the difference between the unbiased and the calculated data. Right lower panel: Residual for the data of C12 .
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E. Cataldo et al. / Journal of Electrostatics 87 (2017) 26e31
4. Discussion
Fig. 3. The data at distances d > 2L are fitted with the two models C11 ¼ a þ b=d2 and C12 ¼ h þ k=d of eq. (3). In the figure the curve fitted and the data, showing a very good agreement. As b≡C13 and k ¼ C12 we evaluate C1 in the two fitting procedures finding, in the fit of C12 , C1 ¼ 1117fF and in that of C11 C1 ¼ 1124fF. It is noticeable that, in the same interval of distances, the two coefficients undergo a much different change in accordance with the different power-dependence on 1=d in eq. (3).
in the converted datum is fixed by the geometry of the connections it seems possible to use the well-known and reproducible parallel plate geometry to calibrate the instruments and, after that connect it to a generic pair of conductor, obtaining with a little effort an experimental accurate knowledge of the complete electrostatic properties of the system. 3.4. Modelling the far limit When the conductors are very near or very far it is possible to write analytical models reproducing the asymptotic trends. At the far limit the coefficients of capacitance of two generic conductor can be expressed by the first terms of a power expansion in 1=d involving as parameters the intrinsic capacitances of the two conductors and their polarizabilities and quadrupole moments [16]. In the simplified geometry of two equal parallel plates C11 ¼ C22 ¼ C1 and taking the leading term in 1=d one finds:
C11 ¼ C1 1 þ
C2 C12 ¼ 1 d
C12 d2
!
(3)
As the two asymptotic trends are expressed, for identical sides of the capacitor, in term of the same parameter C1 we fit, with the two simple functions of eq. (3), the raw data for distances larger than the side of the plates. The graphical result of the fit is reported in Fig. 3, where a very good agreement is shown. It is worth to note that only the constant term in the model function is influenced by the offset. The other two parameters represent C13 and C12 for identical plates and wires. The value of C1 determined in this way are consistent with each other and with the, offset-dependent, horizontal asymptote of C11 . This check confirms the procedure of data analysis and also is a clear test of the formulas in the far limit for the coefficient of capacitances which at the leading level are valid for generic bodies. Moreover the evaluation of C1 is also independent from the absolute estimate of the offset.
Usually, the technical presentations of the operation of commercial CDCs and their specification documents emphasize the precision of the data conversion, the accuracy of the measurement and the sampling frequency. Indeed, as these devices are largely applied for the readout of capacitive sensors the primary goal is to obtain a high sensitivity to the variations of the charges on the capacitor plates resulting from a change of the relative position, the approach of a third body, the deposition of impurities which vary the dielectric constant of the space occupied by the electric field around the capacitor and so on. Little attention and little information is given on the physical quantities characterizing the system and on their influence on the returned data. A common architecture of the CDC produced by the primary manufacturers is based on the measure of the charge accumulated on one of the sensor plates during repeated cycles of charge and discharge. The measure consists in the comparison of the charge to be sampled on the external sensor with that filed on an embedded calibrated capacitor. A sigma-delta ADC converts the measures in digital words. The capacitive sensor is generally realized with two conductors and the measure can consist in exciting and sampling the charge on the same conductor, the other being connected, typically, to ground, as in the CDC AD7747, manufactured by Analog Device. A second scheme consists in the excitation of one of the two plates and in the sampling of the other (e.g. AD7745). The two techniques read a different capacitance coefficient. Moreover, our physical description of the principle of operation shed light on the effects of the connections between the main conductor bodies and the IC input ports and of the bodies surrounding the apparatus. With respect to the first issue, it is clear that the cable and the connection to the excited plate contributes in both mode of operation at the same level with the plate. But the magnitude of the effect is different for the measures of C11 and of C12 . In the first case, the wire is excited with the same waveform of the plate and it is charged at the same time. As the intrinsic capacity of the wire contributes with hundreds of fF already for length of few centimetres it is impractical to use long unipolar cables to connect the CDC measuring C11 . As a matter of fact, the AD7747 CDC is equipped with a second exciting signal which can be applied to the shield of a coaxial cable which internal conductor is connected to the conductor 1 when the measure is made. An explanation of the role of this connection can be given on a physical basis. Let us now consider that only the shielded cable is connected and the core and the shield are brought to the same potential so that the electric field between the wire and the shield is zero. This condition is realized when the inner wire has no charge. This condition is perturbed when the plate is connected and the electrical induction between the coaxial cable and the plate is responsible for the negative offset observed. A major difficulty to keep this effect under control is that one expects that the magnitude of the phenomenon could depend, even for the same cable, on the shape of the body connected to it. The other cable is not directly excited by the signal generated by the CDC circuit and it is maintained at constant potential in both modes of operation. Moreover it is further away from the excited one and from the first body so that the induction effects are weaker, mainly when the two principal bodies under test are far from each other and the two cable are placed maximizing their distance for the major part of their path. Nevertheless, an influence of the second cable on the readout of C11 and of C12 is unavoidable and could be reduced using also in the side 2 a coaxial cable with the shield grounded. We observe that when measuring C12 the inner conductor of the coaxial cable is not at the same potential of the grounded shield so that it is charged with plate 2 by induction of the side 1 and an increase of the sampled charge is expected, as the observed positive offset signals.
E. Cataldo et al. / Journal of Electrostatics 87 (2017) 26e31
5. Conclusions The experiments reported in the paper demonstrate the physical mechanism underlying the capacitance-to-digital converters operating through the sampling of the charges accumulated on one of a pair of conductor. We have shown that the returned data when two parallel plate are connected to the CDC does not coincide with what we intend usually as capacitance C. Really, the value returned is the coefficient of self-capacitance C11 or the induction capacitance C12 . We have explained also that, due to the convergence of C11 and C12 to the same value when the distance between the plates tends to vanish both the mode of operation of the CDCs return, in this condition, the value of relative capacitance C and this is the result when a small monolithic capacitor is tested. The results show that the instruments, basically applied for the ability to sense a change in sensor capacitance, are able to perform with a little effort accurate measure of the basic properties of an electrostatic system. We checked with a very simple apparatus the dependence of the coefficients of capacitance of two square plates up to a value of the distance several times the side of the plates and we made a strict comparison with the calculated numerical results finding a very good agreement both on the global range of the recorded data, when the effect of the connections, consisting primarily in an overall offset of the measured data, are taken into account properly. This quantitative experimental determination of the fringe effects of a parallel plate capacitor up to a very high value of the aspect ratio d=L is not common in the technical literature. On the other hand, it should be easy to investigate with the same apparatus other systems such as a couple of circular disks or two rectangular plates in the strip configuration. Moreover a very good agreement is found with the asymptotic behaviour of the coefficient of capacitance. The fitting of the well known minimal model for the expected trend give accurate value of the intrinsic capacitance of the plates used, and as the model resulting from eq. (3) is general the procedure should apply with minor modifications to conductors of generic shapes and relative disposition. The fact that sub-leading terms in the higher power of 1=d involve, as demonstrated in Ref. [16], other interesting parameters such as the polarizability of the conducting bodies makes interesting the possibility to refine
31
the apparatus and the data analysis in order to get deeper information on less symmetric systems of two conductors.
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