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Estimation of one-dimensional complex-permittivity profiles: a feasibility study
Journal of
ELECTROSTATICS Journal of Electrostatics 34 (1995) 263-277
ELSEVIER
Estimation of one-dimensional complex-permittivity profiles: a feasibility study P.A. yon Guggenberg*, M.C. Zaretsky 1 Continuum Electromechanics Group, Laboratoryfor Electromagnetic and Electronic Systems, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 13 September 1993; accepted after revision 3 October 1994
Abstract This paper examines the feasibility of using the frequency responses of parallel-plate and interdigital-electrode structures to determine the one-dimensional complex-permittivity distribution of heterogeneous materials. This is achieved by relating the complex capacitance of these structures to the complex-permittivity distribution in a material. For parallel-plate structures, it is shown that the temporal frequency response provides, at most, information about the dispersive nature of the complex permittivity averaged over the thickness of the material. Similarly, the temporal frequency response of interdigital-electrode structures yields a nonunique distribution for piecewise-homogeneous materials in which the spatial variation in the complex permittivity is a perturbation about a uniform value. For these same materials, the spatial frequency response of interdigital-electrode structures, obtained by means of electrode structures with multiple wavelengths, is shown to yield a unique complex-permittivity distribution.
1. Introduction 1.1. Classical dielectrometry
Classical dielectrometry is a technique used to extract information a b o u t the state of a system from the frequency response of the dielectric properties of the material in the system. The information thus obtained is limited by the constitutive relation that describes the dependence of the dielectric properties on the temperature, pressure, and a m o u n t s of constituents in the system. In the simplest of configurations, the dielectric properties of a material are measured by means of a parallel-plate structure, such as the one shown schematically in
*Corresponding author. Present address: Doble Engineering Company, Watertown, MA 02172, USA. 1Present address: Eastman Kodak Company, Rochester, NY 14652, USA. 0304-3886/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved. SSDI 0 3 0 4 - 3 8 8 6 ( 9 4 ) 0 0 0 3 7 - 9
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Fig. 1. Schematicillustrationof a parallel-plate structure containingheterogeneousmaterial betweenthe electrodes.
Fig. 1. Here, the application o f a sinusoidally varying potential of complex magnitude 17 results in the flow of a terminal current [ that is dependent on the dielectric properties of the material under test. If the material is homogeneous, the measured frequency response will reflect the dispersive nature of the dielectric properties. If the material is heterogeneous, the frequency response will exhibit dispersive behavior even if the dielectric properties of the material are not dispersive. In many applications, information about the heterogeneity of the material is not available a priori and assumptions about the structure of the material are made in the process of interpreting measurement results obtained with parallel-plate structures. These assumptions can often be avoided by applying interdigital-dielectrometry techniques to the measurement of the dielectric properties of heterogeneous materials. 1.2. Interdigital dielectrometry
Interdigital dielectrometry is a technique used to measure the dielectric properties of a material, wherein a periodic electric field is imposed by means of a pair of interdigitated electrodes. A typical array of such electrodes is shown in Fig. 2. Here, the electrodes are in contact with the material of interest from above and with an insulating substrate backed by a ground plane from below. Of the two electrodes, one is driven with a sinusoidally varying voltage of complex amplitude 17o and frequency f = ~o/2~, while the other is connected to a high-impedance buffer used to measure the floating potential Vv. The periodicity of the electrode structure is denoted by the spatial wavelength 2 = 2~/k. One inherent benefit of this structure arises from the fact that the coupling of the applied field into the medium can be achieved from a single surface. This feature has been exploited by Senturia et al. in the development of a microdielectrometer used to monitor the curing of epoxy resins in situ [-1-3]. This device has also been used to monitor the moisture content of transformer oil by coating the electrodes with a moisture-sensitive thin film [4]. Similar structures have been used to estimate the thickness, complex permittivity, and surface conductivity of insulating materials [5]. Analogously, a magnetometer has been used to estimate the thickness and conductivity of metal foils [6]. This latter structure consists of a periodic set of intermeandering electrodes that form the primary and secondary coils of a planar transformer. In contrast with remote sensing, where electromagnetic fields are used to resolve the properties of materials at distances large compared with the wavelength of the
P.A. oon Guggenberg,
M.C. Zarets~lJournal
Fig,. 2. Schematic illustration of an interdigital-electrode the electrodes.
qf Electrostatics
34 (1995) 263-277
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structure probing heterogeneous materi la1abc we
electromagnetic wave, intimate sensing makes use of the exponentially decaying fields associated with interdigital-electrode structures to measure the dielectric properties of materials in the immediate neighborhood of the electrodes, namely a distance approximately equal to one-quarter of the spatial wavelength. This feature can be used to measure the heterogeneity of the material above the electrodes by means of electrode structures with multiple wavelengths. Because of the periodicity imposed in one direction (y), only the heterogeneity of the medium in the direction perpendicular to the plane of the electrodes (x) can be discerned [7]. In the process of measuring the dielectric properties of a material using interdigital-electrode structures with various wavelengths, the response of the electrode structure with the shortest wavelength will reflect the dielectric properties of the material in the immediate vicinity of the electrodes. The longer-wavelength electrode structure will not only sample the material adjacent to the electrodes, but that further out as well. As the spatial wavelength is increased further, the dielectric properties of the material at an increasing range will influence the measurement. Thus, the one-dimensional complex-permittivity profile of the material can be recovered from measurement results obtained with multiple-wavelength interdigital-electrode structures. The ability to independently vary the applied frequency and the spatial periodicity of the electrodes in these structures allows one to measure both the temporal and the spatial frequency response of the material. Variations in the applied frequency give rise to the temporal frequency response; variations in the spatial wavelength give rise to the spatial frequency response. Because the temporal (CO)and spatial (k) domains
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are independent, this technique has been referred to as the 'imposed ~o-k' approach to dielectrometry I-8,9]. What differentiates this approach from classical techniques involving parallel-plate structures is the fact that the heterogeneity of the material under test can be deduced independently from the temporal frequency response. This can be achieved by performing variable-spatial-wavelength measurements at the same temporal frequency. The spatial distribution of the dielectric properties can thus be deduced without any assumptions about the nature of the material. This additional freedom allows an unconstrained evaluation of the physical mechanisms that govern the dispersive nature of the dielectric properties. Use of the spatial frequency response to determine the distribution of properties in materials has been exploited by several researchers in the field. Applications thus far have included the monitoring of moisture absorption into transformer pressboard [10], the examination of interfacial structures such as the double layer [10], the measurement of spatial variations in ice/water mixtures 1-11], and the nondestructive evaluation of surface flaws in dielectric and metal objects 1-12]. The approach has also been proposed for the measurement of complex permeability and conductivity profiles in magnetic and conducting media [ 13] and the detection of cartilage degeneration in articular joints [14].
1.3. Design and implementation An important aspect of the design of an experiment, in which dielectrometry techniques are used to measure the dielectric properties of a material, is the selection of the temporal frequency. For an ohmic medium, both the permittivity and the conductivity of the material can be estimated from a single measurement if the frequency is such that the period of the excitation signal is comparable to the electrical relaxation time of the material (coTe ~ 1). Provided that the dielectric properties of the medium do not suffer extreme variations, this condition can be met everywhere in the region of interest with a single temporal frequency. If the experiment involves the use of multiple-wavelength interdigital-electrode structures, the specification of the system is completed by the selection of the spatial wavelengths. Because the electric field associated with interdigital-electrode structures decays exponentially with a characteristic length that is proportional to the spatial wavelength, this selection begins with a wavelength that is short enough to resolve the distribution close to the electrodes and ends with a wavelength long enough to insure that the electric field extends into the material up to the desired depth. Physical constraints on the dimensions of the electrode structure may limit this depth. Thus, with properly designed electrodes, profiles can be sensed to a depth proportional to the longest wavelength. One method of implementing the multiple-wavelength interdigital-electrode structure described above would be to fabricate several interdigital-electrode structures, similar to the one shown in Fig. 2, each having a different spatial wavelength. If the process requires that the measurement be made with all the electrode structures in contact with the material at the same time, then the spatial distribution of the
P.,4. yon Guggenberg, M.C. Zaretsky/Journal of Electrostatics 34 (1995) 263 277
267
^
J
VDo ^
VFo
I I I
;,l, i (
I
I
;
Fig. 3. Apparatus used to generate multiple spatial wavelengths by means of a set of switchable electrodes.
dielectric properties must be the same everywhere. This constraint can be relaxed if the measurements can be made consecutively, allowing each array to make contact with the material at approximately the same location. A convenient method of using just one set of electrodes, which allows the spatial wavelength to be varied through changes in the electrical connections of the electrodes, is illustrated in Fig. 3. In this array, each electrode may be connected to the driven terminal, connected to the floating terminal, or left disconnected. The shortest spatial wavelength can be imposed by connecting every other electrode together, one set connected to the driven terminal and the other connected to the floating terminal. The next longer wavelength may be obtained by connecting pairs of adjacent electrodes together, again half the pairs connected to the driven terminal and the other half connected to the floating terminal. In this latter configuration, one electrode between driven and floating electrode pairs is left disconnected. These latter electrodes will assume a potential somewhere between the driven and floating voltages, as dictated by the constraint that they carry no net charge, thereby preventing the higher spatial harmonics arising from the coupling between the edges of adjacent electrodes from dominating the response.
2. Objectives The objectives for the remainder of this paper are: 1. to demonstrate that the frequency response of parallel-plate structures cannot be used to uniquely determine the spatial distribution of the dielectric properties in a heterogeneous material, 2. to demonstrate that the temporal frequency response of single-wavelength interdigital-electrode structures cannot be used to determine the spatial distribution of the dielectric properties in a material when the spatial variation in the complex permittivity is a perturbation about a uniform value, and
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3. to demonstrate that the spatial frequency response of multiple-wavelength interdigital-electrode structures can be used to determine the spatial distribution of the dielectric properties of a material. The statement contained in the last objective will be proven analytically for the special case of a material in which the spatial variation in the complex permittivity is a perturbation about a uniform value.
3. Governing laws At frequencies sufficiently low that the wavelength of the electromagnetic wave is long compared with the characteristic length of the system, the dynamics described by the wave equation can be approximated by the quasistatic form of Maxwell's equations. In the electroquasistatic regime, these equations can be written as V ' D = p., VxE=0, V - J -4- ~
--- 0.
(1)
If the system is in the sinusoidal steady state, each variable can be expressed as the real part of a complex quantity
A(r, t) = ~R{A(r, o))eJ~'t},
(2)
so that the electroquasistatic laws can be written in the frequency domain as V./~ = ~., Vx~=0, V" 3 + jc@, = 0.
(3)
Note that the form of Eq. (2) implies that the time-domain variables are related by linear differential equations. If the medium is assumed to be isotropic, the displacement flux density and the current density can be related to the electric field by a dielectric constant e' and loss factor ~",
) = coe"/~,
(4)
so that Gauss' law can be written in a form that includes the effects of charge conservation, V" ~*/~ = 0,
(5)
P.A. yon Guggenberg, M C Zaretsky/Journal of Electrostatics 34 (1995) 263-277
269
where the complex permittivity is defined as e*(r, co) = e' (r, co) - je" (r, to),
(6)
with the frequency dependence of the real and imaginary parts related by the Kramers-Kr6nig relations [15]. The electroquasistatic laws can thus be summarized as follows: V" ~*~ = 0, v x ~ = 0.
(7)
Note that, because the electric field is irrotational, it can be expressed as the gradient of a potential = - v,b.
(8)
4. Feasibility Using the electroquasistatic form of Maxwell's equations in the sinusoidal steady state, what follows is an attempt to show the conditions under which the spatial profile of the dielectric properties of a material can be determined from measurement results obtained with parallel-plate and interdigital-electrode structures. 4.1. Parallel-plate structures
A common structure used to measure the dielectric properties of materials consists of two electrodes arranged on either side of the material under test. Such a structure is illustrated in Fig. 1, where the application of a potential v(t) = ~{l?e j'~'}
(9)
results in the flow of a current at the terminals:
i(t)
= ~ {ieJ~'}.
(10)
If the plate separation is much smaller than the dimensions of the plates and the complex permittivity varies only in the x direction, the electric field can be assumed to be one-dimensional and Gauss' law (Eq. (5)) reduces to ~
~*(x, to)~x(x)
-- 0.
(11)
Integration of this equation from x = 0 to an arbitrary position in the medium then yields 7
~*(x, to)/~x(x) = - jtoA'
(12)
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P.A. yon Guggenberg, M.C Zaretsky/Journal of Electrostatics 34 (1995) 263 277
where the integral form of Eq. (5) applied to the upper electrode was used to express e*(0, to)/~x(0) in terms of the terminal current. According to Eq. (8), the electric field can be expressed in terms of the electric potential as
d~
/~x = -- d-~"
(13)
Integrating this expression across the medium gives 12 =
/~xdx = j - - ~
co----~ e*(x, dx,
(14)
where Eq. (12) was used to express the electric field in terms of the complex permittivity of the material. The response of the electrode structure to the applied voltage can be expressed in terms of a complex capacitance per unit area
i (~ --- jmAl~'
(15)
which, with the help of Eq. (14), is related to the complex permittivity of the medium by the following integral:
~(~o)=[
"
1
fo e*(x,~o)dx]
-1
.
(16)
From this expression it is clear that the measurement of the complex capacitance of a parallel-plate structure provides, at most, a spatial average of the complex-permittivity distribution. Furthermore, it is evident that, at a given frequency, there are many different distributions giving rise to the same complex capacitance. Contributions to the integral coming from the neighborhood of x = 0 can be interchanged with those near x = d and the integral will remain the same. If assumptions concerning the frequency dependence of the complex permittivity are allowed, it is possible to use the full temporal frequency response to distinguish between certain attributes of the heterogeneity. However, the necessity for making assumptions of this type strongly limits what additional information can be obtained concerning the physical mechanisms underlying the complex permittivity. 4.2. Interdigital-electrode structures
Intuitively, it makes more sense to determine the heterogeneity of the medium by probing spatially rather than temporally. Because the electroquasistatic fields associated with interdigital-electrode structures are periodic in one direction, they decay in the other direction at a rate that is inversely proportional to the spatial wavelength. Thus, the process of determining the spatial distribution of the complex permittivity at a specified temporal frequency can be achieved by varying the spatial frequency of the applied potential.
P.A. yon Guggenberg, M.C. Zaretsky/Journal of Electrostatics 34 (1995) 263-277
271
4.2.1. General approach Consider the interdigital-electrode structure of Fig. 2. The material under test lies in the infinite half-space x >~ 0, where x = 0 is the plane containing the electrodes. Because the depth of the structure is much greater than the spatial wavelength, the potential can be described as Cb(x,y, t) = 9t{~(x,y)eJ°"},
(17)
where variations in the y direction are assumed to be periodic, q-oo
~(X, y) = ~ ~n(x)e -jt"y,
(18)
-oo
and the wavenumber associated with the nth mode is defined in terms of the spatial wavelength 2 as k.=
2/~n 2
(19)
Assuming that the complex permittivity varies only in the x direction, Gauss' law becomes
ax [~*(x,~o)t~x] + e*(x,~o) ~-y = o
(20)
and, because the electric field is irrotational, the y component of the electric field can be evaluated using Eq. (18):
/~
-
~
Oy
_
+
~ jk. ~.(x)e-jk.r
(21)
-co
Using this result, Eq. (20) can be written in the form Ox [g*(x,o))/~] + e*(x, eo) ~ k2q).(x)e -it"' = O.
(22)
Integrating this equation from x = 0 +, where e*/~x = ~, Ex, ^a to infinity, where the electric field goes to zero and the complex permittivity remains finite, we find an expression for the normal component of the complex displacement flux density just above the surface of the electrodes: =
eaE, e -oo
" ---
~*(x,~o) ~ k2.~,(x)e-Jk"Ydx.
(23)
-oc
The medium above the electrodes can be described in terms of a surface capacitance density. This quantity represents the response of the medium to one Fourier component of the potential at the electrodes. It is defined as the ratio of the normal component of the complex displacement flux density evaluated at the interface ~* E. ^~ to the
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complex amplitude of the interfacial potential ~ [9]: (~,
.
(24)
Note that this quantity has the same units as the complex capacitance per unit area defined in Eq. (15) for parallel-plate structures. Using Eq. (23), the surface capacitance density can be written in terms of the following integral: ~a(o),kn ) = k. 0 ® e*(x,e))~.(x)k, dx.
(25)
~'. Jo
In contrast with the result obtained in Section 4.1, the spatial distribution of the complex permittivity acts as a weighting function to the potential distribution. For a given wavenumber, the potential decays with a characteristic length scale proportional to the imposed spatial wavelength. Thus, it seems possible to extract the spatial distribution of the complex permittivity from measurement results obtained with multiple-wavelength interdigital-electrode structures. 4,2.2. Piecewise-homogeneous media The evaluation of the above integral is, in general, quite complex. Given a complex-permittivity distribution, the potential in the heterogeneous medium has to be solved for in order to carry out the integral. This can be done analytically only for a select number of complex-permittivity distributions. Consider, for example, the piecewise-homogeneous medium shown in Fig. 4. Of the M layers, the top one (j = 1) is of semi-infinite extent. The jth layer has a complex permittivity e*, a thickness d j, and an upper surface designated by (j). In this case, the desired expression for the surface capacitance density can be obtained by solving Laplace's equation in each layer, applying continuity conditions for the potential and the normal component of the complex displacement flux density at each interface, and carrying out the integral in a piecewise fashion over all the layers. Alternatively, the surface capacitance density can be derived by sequentially applying transfer relations to each layer in the medium. These matrix relations summarize the field laws in each layer by relating the normal components of the complex displacement flux density to the corresponding potentials at the upper and lower surfaces of the layers [16]. Because a surface capacitance density can be defined for each interface in terms of the same variables, this iterative process can be cast in terms of the surface capacitance densities of the interfaces. In particular, the surface capacitance density of the (j + 1)st interface is related to that of the jth interface by the following recursive relation [9]: (~*kn) 2
~(j+ 1) = e* k, coth(k~dj) - sinh2(k,dj ) [e*k, coth(k,dj) + C(J)] "
(26)
Thus, in the process of deriving the surface capacitance density at the electrodes, this relation is applied sequentially, starting from the top layer (j = 1) and working down
P.A. yon Guggenberg, M.C. Zaretsky/Journal of Electrostatics 34 (1995) 263 277
[
273
I
I
t di
'
t
dM
k"-S""eee J / £~
Layer j
e~
/
/ h
Fig. 4. Schematic illustration of a piecewise-homogeneous medium composed of M layers.
to the bottom layer (j = M). The desired expression is then given by d~ = d(~+ l).
(27)
Note that, because the surface associated with the top layer is at infinity (dl ~ ~), ~t2) can be evaluated without the need to specify ~ 1 ) If we assume that the complex permittivity of each layer differs from a common uniform value e* by a small perturbation g*, the surface capacitance density above the electrodes can be expressed as a linear superposition of the complex permittivities of the layers in the medium. In particular, for a medium composed of one, two, or three layers, the linearized form of Eq. (26) yields M = 1 (~ = e~'k,,, M = 2
t~." = 8~k,, - (e* - e*)k.e -2k-a2,
M = 3
~." = ~ ' k .
- (~* -
E * ) k . e - 2 ' o " ' -- ( ~ ' - - 4 ) k . e - ' k ° ( " 2 + ' .
(28)
The form of these relations suggests the following expression for the linearized surface capacitance density of a medium with M layers: M-1
C~ = e*k. + ~ j=l
(~* - e*+x)k.e-2k"hj,
(29)
P.A. yon Guggenberg, M.C Zarets~. /Journal o f Electrostatics 34 (1995) 263 277
274
where the height of each layer above the surface of the electrodes is given by M
hi= ~
(30)
di.
i=j+ 1
Thus, the surface capacitance density above the electrodes can be expressed as a linear combination of the complex permittivities of the layers, M
(31)
C~,,l(o~,, k,,,) = ~ Bi.,(dj, hi, k,,,)e*(~oi), j=l
where knz is the wavenumber associated with the nth mode and the lth wavelength: 2rm k.l = - 21
(32)
In a situation where the complex permittivities of the M layers are to be determined, a minimum of M measurement results is required. If this information is obtained from measurements performed at a fixed temporal frequency using M distinct spatial wavelengths, the surface capacitance densities associated with the measurement results can be related to the complex permittivities of the layers by the following matrix relation:
Lc . ,j
=LB-
.,.,
.
B...
(33)
Thus, because the rows of the matrix are linearly independent, the complex permittivities of the layers can be uniquely related to the surface capacitance densities associated with multiple-wavelength measurements. If, on the other hand, the measurement results are obtained from experiments performed at M distinct temporal frequencies using a single-wavelength interdigitalelectrode structure, the surface capacitance densities associated with the multiplefrequency measurements can be written as follows:
•
=
"
"
'
.
(34)
LBM.u With the goal of expressing each of the complex permittivities at the same temporal frequency, the layers are assumed to be composed of ohmic media• The complex permittivity of each layer can then be written in terms of a dielectric constant ej and a conductivity a j: e~
=
ej -
j aj. O9
(35)
P.A. yon Guggenberg, M.C, Zaretsky/Journal of Electrostatics 34 (1995) 263 277
275
Using this constitutive relation, the real and imaginary parts of the surface capacitance densities can be related to the permittivities and conductivities of the layers by the following matrix relations:
¢ol
-- 3
=
LCMntJ
"
~l
".
"
0"1
(36)
L~
The determinant of each of the matrices in these relations is zero .Thus, the complex permittivities of the layers are not uniquely related to the surface capacitance densities and, as a result, cannot be uniquely determined from multiple-frequency measurements. Note that this result applies only when the spatial variations in the complex permittivities are a small perturbation about a uniform value. If the electrical relaxation times eg/0"~ of the layers are sufficiently distinct and decrease with distance away from the electrodes, the complex permittivities of the layers can be uniquely determined from measurement results obtained at multiple frequencies 1-17]. The discussion thus far has centered on the relationship between the complex permittivities of the layers and the surface capacitance densities associated with the measurement results. In practice, however, measurement results obtained with interdigital-electrode structures are expressed in terms of a complex gain I?F = ~ - ( ~ _ ~b)
(37)
which depends on the surface capacitance densities of the medium above and below the electrodes. In this case, the surface capacitance density of the medium below the electrodes (~ is fixed by the properties of the insulating substrate, namely the thickness h and the permittivity ~s- Thus, a unique relationship between the complex permittivities of the layers and the surface capacitance densities associated with the measurement results is a necessary but not sufficient condition to insure that the complex permittivities of the layers can be uniquely determined from the measured complex gains. The relationship between the complex gain and the surface capacitance density above the electrodes is nonlinear. Thus, uniqueness is not guaranteed. However, earlier work has shown that the relationship between the complex permittivity of a uniform medium and the associated complex gain is single-valued [1,9]. This implies that, even though there may be more than one value of the surface capacitance density for a given value of the complex gain, only one of these values is associated with a physical complex permittivity. Thus, if we limit ourselves to physical values for the complex permittivity, a unique relationship between the surface capacitance density and the
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P..4, oon Guggenberg, M.C, Zaretsky/Journal of Electrostatics 34 (1995) 263-277
complex gain does exist and the determination of the complex permittivities of the layers in a piecewise-homogeneous medium from measurement results obtained with multiple-wavelength interdigital-electrode structures is feasible.
5. Conclusions In contrast with classical dielectrometry, where the temporal frequency response of a material is the only measurable response, interdigital dielectrometry allows both the temporal and spatial frequency responses to be measured with multiple-wavelength interdigital-electrode structures. This additional degree of freedom permits the independent examination of the temporal and spatial dispersions in a material, thereby eliminating the need for making assumptions about the uniformity or frequency dependence of the dielectric properties. The temporal frequency response of parallel-plate structures cannot be used to infer the spatial distribution of the dielectric properties in a material. If no assumptions are made about the frequency dependence of the complex permittivity, the temporal frequency response of these structures can only provide information about the dispersive nature of the complex permittivity averaged over the thickness of the material. Except for specific cases, the complex-permittivity profile of a material cannot be determined from the temporal frequency response of single-wavelength interdigitalelectrode structures. Only the spatial frequency response of interdigital-electrode structures yields such information. This was demonstrated for a piecewise-homogeneous medium in which the spatial variation in the complex permittivity is a perturbation about a uniform value.
Acknowledgements This paper is based, in part, on the authors' theses submitted to the Massachusetts Institute of Technology (MIT) in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The research work was carried out in the Continuum Electromechanics Group of the Laboratory for Electromagnetic and Electronic Systems at MIT with financial support from the Electric Power Research Institute and members of MIT's Electric Utilities Program. The companies participating in the program included Allegheny Power Company, American Electric Power Service Corporation, Boston Edison Company, Empire State Electric Energy Research Corporation, Northeast Utilities Service Company, New York Power Authority, Southern California Edison Company, and Tokyo Electric Power Company. The ideas presented in this paper are based on the research of the late Professor James R. Melcher who, with the help of his graduate students, pioneered the application of spectral methods to the field of interdigital dielectrometry. We are grateful to have had the opportunity to study under his tutelage. We would also like to thank Professor Markus Zahn and Mr. Yanko Sheiretov of MIT and Dr. Stuart Inkpen of Instrumar Limited for helpful suggestions and constructive criticisms.
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277
References [1] N.F. Sheppard, D.R. Day, H.L Lee and S.D. Senturia, Microdielectrometry, Sensors and Actuators, 2(3) (1982) 263. [2] S.D. Senturia, N.F. Sheppard, Jr., H.L. Lee and D.R. Day, In-situ measurement of the properties of curing systems with microdielectrometry, J. Adhesion, 15(1) (1982) 69. [3] S.D. Senturia and S.L. Garverick, Methods and apparatus for microdielectrometry, US Patent No. 4,423,371, December 27, 1983. [4] M.C. Zaretsky, J.R. Melcher and C.M. Cooke, Moisture sensing in transformer oil using thin-film microdielectrometry, IEEE Trans. Electrical Insulation, 24(6) (1989) 1167. [5] M.C. Zaretsky, P. Li and J.R. Melcher, Estimation of thickness, complex bulk permittivity and surface conductivity using interdigital dielectrometry, IEEE Trans. Electrical Insulation, 24(6) (1989) 1159. [6] N.J. Goldfine, Uncalibrated, absolute property estimation and measurement optimization for conducting and magnetic media using imposed co k magnetometry, Ph.D. thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, September 1990. [7] J.R. Melcher and M.C. Zaretsky, Apparatus and methods for measuring permittivity in materials, US Patent No. 4,814,690, March 21, 1989. [8] J.R. Melcher, Electrohydrodynamic surface waves. In: R.E. Meyer (Ed.), Waves on Fluid Interfaces, Academic Press, New York, 1983, pp. 167 200. [9] M.C. Zaretsky, L. Mouayad and J.R. Melcher, Continuum properties from interdigital dielectrometry, IEEE Trans. Electrical Insulation, 23(6) (1988) 897. [10] P.A. von Guggenberg, Applications of interdigital dielectrometry to moisture and double layer measurements in transformer insulation, Ph.D. thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, June 1993. [11] S. Inkpen, C. Brobeck and B. Parsons, Measurement of spatial variations in electrical properties of model ice, Cold Regions Science and Technology, 16(2) (1989) 201. [12] M. Gimple and B.A. Auld, Variable geometry capacitive probes for multipurpose sensing, Research in Nondestructive Evaluation, 1(2) (1989) 111. [13] J.R. Melcher, Apparatus and methods for measuring permeability and conductivity in materials using multiple wavenumber magnetic interrogations, US Patent No. 5,015,951, May 14, 1991. [14] E.H. Frank, E.P. Salant and A.J. Grodzinsky, Probe system and method for detecting cartilage degradation, US Patent No. 5,246,013, September 21, 1993. [15] A.K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectrics Press, London, 1983, Sec. 2.8. [16] J.R Melcher, Continuum Electromechanics, MIT Press, Cambridge, MA, 1981, Sec. 2.16. [17] M.C. Zaretsky, Parameter estimation using microdielectrometry with application to transformer monitoring, Ph.D. thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, November 1987, Sec. 3.2.
ELECTROSTATICS Journal of Electrostatics 34 (1995) 263-277
ELSEVIER
Estimation of one-dimensional complex-permittivity profiles: a feasibility study P.A. yon Guggenberg*, M.C. Zaretsky 1 Continuum Electromechanics Group, Laboratoryfor Electromagnetic and Electronic Systems, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 13 September 1993; accepted after revision 3 October 1994
Abstract This paper examines the feasibility of using the frequency responses of parallel-plate and interdigital-electrode structures to determine the one-dimensional complex-permittivity distribution of heterogeneous materials. This is achieved by relating the complex capacitance of these structures to the complex-permittivity distribution in a material. For parallel-plate structures, it is shown that the temporal frequency response provides, at most, information about the dispersive nature of the complex permittivity averaged over the thickness of the material. Similarly, the temporal frequency response of interdigital-electrode structures yields a nonunique distribution for piecewise-homogeneous materials in which the spatial variation in the complex permittivity is a perturbation about a uniform value. For these same materials, the spatial frequency response of interdigital-electrode structures, obtained by means of electrode structures with multiple wavelengths, is shown to yield a unique complex-permittivity distribution.
1. Introduction 1.1. Classical dielectrometry
Classical dielectrometry is a technique used to extract information a b o u t the state of a system from the frequency response of the dielectric properties of the material in the system. The information thus obtained is limited by the constitutive relation that describes the dependence of the dielectric properties on the temperature, pressure, and a m o u n t s of constituents in the system. In the simplest of configurations, the dielectric properties of a material are measured by means of a parallel-plate structure, such as the one shown schematically in
*Corresponding author. Present address: Doble Engineering Company, Watertown, MA 02172, USA. 1Present address: Eastman Kodak Company, Rochester, NY 14652, USA. 0304-3886/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved. SSDI 0 3 0 4 - 3 8 8 6 ( 9 4 ) 0 0 0 3 7 - 9
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Fig. 1. Schematicillustrationof a parallel-plate structure containingheterogeneousmaterial betweenthe electrodes.
Fig. 1. Here, the application o f a sinusoidally varying potential of complex magnitude 17 results in the flow of a terminal current [ that is dependent on the dielectric properties of the material under test. If the material is homogeneous, the measured frequency response will reflect the dispersive nature of the dielectric properties. If the material is heterogeneous, the frequency response will exhibit dispersive behavior even if the dielectric properties of the material are not dispersive. In many applications, information about the heterogeneity of the material is not available a priori and assumptions about the structure of the material are made in the process of interpreting measurement results obtained with parallel-plate structures. These assumptions can often be avoided by applying interdigital-dielectrometry techniques to the measurement of the dielectric properties of heterogeneous materials. 1.2. Interdigital dielectrometry
Interdigital dielectrometry is a technique used to measure the dielectric properties of a material, wherein a periodic electric field is imposed by means of a pair of interdigitated electrodes. A typical array of such electrodes is shown in Fig. 2. Here, the electrodes are in contact with the material of interest from above and with an insulating substrate backed by a ground plane from below. Of the two electrodes, one is driven with a sinusoidally varying voltage of complex amplitude 17o and frequency f = ~o/2~, while the other is connected to a high-impedance buffer used to measure the floating potential Vv. The periodicity of the electrode structure is denoted by the spatial wavelength 2 = 2~/k. One inherent benefit of this structure arises from the fact that the coupling of the applied field into the medium can be achieved from a single surface. This feature has been exploited by Senturia et al. in the development of a microdielectrometer used to monitor the curing of epoxy resins in situ [-1-3]. This device has also been used to monitor the moisture content of transformer oil by coating the electrodes with a moisture-sensitive thin film [4]. Similar structures have been used to estimate the thickness, complex permittivity, and surface conductivity of insulating materials [5]. Analogously, a magnetometer has been used to estimate the thickness and conductivity of metal foils [6]. This latter structure consists of a periodic set of intermeandering electrodes that form the primary and secondary coils of a planar transformer. In contrast with remote sensing, where electromagnetic fields are used to resolve the properties of materials at distances large compared with the wavelength of the
P.A. oon Guggenberg,
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Fig,. 2. Schematic illustration of an interdigital-electrode the electrodes.
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34 (1995) 263-277
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structure probing heterogeneous materi la1abc we
electromagnetic wave, intimate sensing makes use of the exponentially decaying fields associated with interdigital-electrode structures to measure the dielectric properties of materials in the immediate neighborhood of the electrodes, namely a distance approximately equal to one-quarter of the spatial wavelength. This feature can be used to measure the heterogeneity of the material above the electrodes by means of electrode structures with multiple wavelengths. Because of the periodicity imposed in one direction (y), only the heterogeneity of the medium in the direction perpendicular to the plane of the electrodes (x) can be discerned [7]. In the process of measuring the dielectric properties of a material using interdigital-electrode structures with various wavelengths, the response of the electrode structure with the shortest wavelength will reflect the dielectric properties of the material in the immediate vicinity of the electrodes. The longer-wavelength electrode structure will not only sample the material adjacent to the electrodes, but that further out as well. As the spatial wavelength is increased further, the dielectric properties of the material at an increasing range will influence the measurement. Thus, the one-dimensional complex-permittivity profile of the material can be recovered from measurement results obtained with multiple-wavelength interdigital-electrode structures. The ability to independently vary the applied frequency and the spatial periodicity of the electrodes in these structures allows one to measure both the temporal and the spatial frequency response of the material. Variations in the applied frequency give rise to the temporal frequency response; variations in the spatial wavelength give rise to the spatial frequency response. Because the temporal (CO)and spatial (k) domains
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are independent, this technique has been referred to as the 'imposed ~o-k' approach to dielectrometry I-8,9]. What differentiates this approach from classical techniques involving parallel-plate structures is the fact that the heterogeneity of the material under test can be deduced independently from the temporal frequency response. This can be achieved by performing variable-spatial-wavelength measurements at the same temporal frequency. The spatial distribution of the dielectric properties can thus be deduced without any assumptions about the nature of the material. This additional freedom allows an unconstrained evaluation of the physical mechanisms that govern the dispersive nature of the dielectric properties. Use of the spatial frequency response to determine the distribution of properties in materials has been exploited by several researchers in the field. Applications thus far have included the monitoring of moisture absorption into transformer pressboard [10], the examination of interfacial structures such as the double layer [10], the measurement of spatial variations in ice/water mixtures 1-11], and the nondestructive evaluation of surface flaws in dielectric and metal objects 1-12]. The approach has also been proposed for the measurement of complex permeability and conductivity profiles in magnetic and conducting media [ 13] and the detection of cartilage degeneration in articular joints [14].
1.3. Design and implementation An important aspect of the design of an experiment, in which dielectrometry techniques are used to measure the dielectric properties of a material, is the selection of the temporal frequency. For an ohmic medium, both the permittivity and the conductivity of the material can be estimated from a single measurement if the frequency is such that the period of the excitation signal is comparable to the electrical relaxation time of the material (coTe ~ 1). Provided that the dielectric properties of the medium do not suffer extreme variations, this condition can be met everywhere in the region of interest with a single temporal frequency. If the experiment involves the use of multiple-wavelength interdigital-electrode structures, the specification of the system is completed by the selection of the spatial wavelengths. Because the electric field associated with interdigital-electrode structures decays exponentially with a characteristic length that is proportional to the spatial wavelength, this selection begins with a wavelength that is short enough to resolve the distribution close to the electrodes and ends with a wavelength long enough to insure that the electric field extends into the material up to the desired depth. Physical constraints on the dimensions of the electrode structure may limit this depth. Thus, with properly designed electrodes, profiles can be sensed to a depth proportional to the longest wavelength. One method of implementing the multiple-wavelength interdigital-electrode structure described above would be to fabricate several interdigital-electrode structures, similar to the one shown in Fig. 2, each having a different spatial wavelength. If the process requires that the measurement be made with all the electrode structures in contact with the material at the same time, then the spatial distribution of the
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^
J
VDo ^
VFo
I I I
;,l, i (
I
I
;
Fig. 3. Apparatus used to generate multiple spatial wavelengths by means of a set of switchable electrodes.
dielectric properties must be the same everywhere. This constraint can be relaxed if the measurements can be made consecutively, allowing each array to make contact with the material at approximately the same location. A convenient method of using just one set of electrodes, which allows the spatial wavelength to be varied through changes in the electrical connections of the electrodes, is illustrated in Fig. 3. In this array, each electrode may be connected to the driven terminal, connected to the floating terminal, or left disconnected. The shortest spatial wavelength can be imposed by connecting every other electrode together, one set connected to the driven terminal and the other connected to the floating terminal. The next longer wavelength may be obtained by connecting pairs of adjacent electrodes together, again half the pairs connected to the driven terminal and the other half connected to the floating terminal. In this latter configuration, one electrode between driven and floating electrode pairs is left disconnected. These latter electrodes will assume a potential somewhere between the driven and floating voltages, as dictated by the constraint that they carry no net charge, thereby preventing the higher spatial harmonics arising from the coupling between the edges of adjacent electrodes from dominating the response.
2. Objectives The objectives for the remainder of this paper are: 1. to demonstrate that the frequency response of parallel-plate structures cannot be used to uniquely determine the spatial distribution of the dielectric properties in a heterogeneous material, 2. to demonstrate that the temporal frequency response of single-wavelength interdigital-electrode structures cannot be used to determine the spatial distribution of the dielectric properties in a material when the spatial variation in the complex permittivity is a perturbation about a uniform value, and
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3. to demonstrate that the spatial frequency response of multiple-wavelength interdigital-electrode structures can be used to determine the spatial distribution of the dielectric properties of a material. The statement contained in the last objective will be proven analytically for the special case of a material in which the spatial variation in the complex permittivity is a perturbation about a uniform value.
3. Governing laws At frequencies sufficiently low that the wavelength of the electromagnetic wave is long compared with the characteristic length of the system, the dynamics described by the wave equation can be approximated by the quasistatic form of Maxwell's equations. In the electroquasistatic regime, these equations can be written as V ' D = p., VxE=0, V - J -4- ~
--- 0.
(1)
If the system is in the sinusoidal steady state, each variable can be expressed as the real part of a complex quantity
A(r, t) = ~R{A(r, o))eJ~'t},
(2)
so that the electroquasistatic laws can be written in the frequency domain as V./~ = ~., Vx~=0, V" 3 + jc@, = 0.
(3)
Note that the form of Eq. (2) implies that the time-domain variables are related by linear differential equations. If the medium is assumed to be isotropic, the displacement flux density and the current density can be related to the electric field by a dielectric constant e' and loss factor ~",
) = coe"/~,
(4)
so that Gauss' law can be written in a form that includes the effects of charge conservation, V" ~*/~ = 0,
(5)
P.A. yon Guggenberg, M C Zaretsky/Journal of Electrostatics 34 (1995) 263-277
269
where the complex permittivity is defined as e*(r, co) = e' (r, co) - je" (r, to),
(6)
with the frequency dependence of the real and imaginary parts related by the Kramers-Kr6nig relations [15]. The electroquasistatic laws can thus be summarized as follows: V" ~*~ = 0, v x ~ = 0.
(7)
Note that, because the electric field is irrotational, it can be expressed as the gradient of a potential = - v,b.
(8)
4. Feasibility Using the electroquasistatic form of Maxwell's equations in the sinusoidal steady state, what follows is an attempt to show the conditions under which the spatial profile of the dielectric properties of a material can be determined from measurement results obtained with parallel-plate and interdigital-electrode structures. 4.1. Parallel-plate structures
A common structure used to measure the dielectric properties of materials consists of two electrodes arranged on either side of the material under test. Such a structure is illustrated in Fig. 1, where the application of a potential v(t) = ~{l?e j'~'}
(9)
results in the flow of a current at the terminals:
i(t)
= ~ {ieJ~'}.
(10)
If the plate separation is much smaller than the dimensions of the plates and the complex permittivity varies only in the x direction, the electric field can be assumed to be one-dimensional and Gauss' law (Eq. (5)) reduces to ~
~*(x, to)~x(x)
-- 0.
(11)
Integration of this equation from x = 0 to an arbitrary position in the medium then yields 7
~*(x, to)/~x(x) = - jtoA'
(12)
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where the integral form of Eq. (5) applied to the upper electrode was used to express e*(0, to)/~x(0) in terms of the terminal current. According to Eq. (8), the electric field can be expressed in terms of the electric potential as
d~
/~x = -- d-~"
(13)
Integrating this expression across the medium gives 12 =
/~xdx = j - - ~
co----~ e*(x, dx,
(14)
where Eq. (12) was used to express the electric field in terms of the complex permittivity of the material. The response of the electrode structure to the applied voltage can be expressed in terms of a complex capacitance per unit area
i (~ --- jmAl~'
(15)
which, with the help of Eq. (14), is related to the complex permittivity of the medium by the following integral:
~(~o)=[
"
1
fo e*(x,~o)dx]
-1
.
(16)
From this expression it is clear that the measurement of the complex capacitance of a parallel-plate structure provides, at most, a spatial average of the complex-permittivity distribution. Furthermore, it is evident that, at a given frequency, there are many different distributions giving rise to the same complex capacitance. Contributions to the integral coming from the neighborhood of x = 0 can be interchanged with those near x = d and the integral will remain the same. If assumptions concerning the frequency dependence of the complex permittivity are allowed, it is possible to use the full temporal frequency response to distinguish between certain attributes of the heterogeneity. However, the necessity for making assumptions of this type strongly limits what additional information can be obtained concerning the physical mechanisms underlying the complex permittivity. 4.2. Interdigital-electrode structures
Intuitively, it makes more sense to determine the heterogeneity of the medium by probing spatially rather than temporally. Because the electroquasistatic fields associated with interdigital-electrode structures are periodic in one direction, they decay in the other direction at a rate that is inversely proportional to the spatial wavelength. Thus, the process of determining the spatial distribution of the complex permittivity at a specified temporal frequency can be achieved by varying the spatial frequency of the applied potential.
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4.2.1. General approach Consider the interdigital-electrode structure of Fig. 2. The material under test lies in the infinite half-space x >~ 0, where x = 0 is the plane containing the electrodes. Because the depth of the structure is much greater than the spatial wavelength, the potential can be described as Cb(x,y, t) = 9t{~(x,y)eJ°"},
(17)
where variations in the y direction are assumed to be periodic, q-oo
~(X, y) = ~ ~n(x)e -jt"y,
(18)
-oo
and the wavenumber associated with the nth mode is defined in terms of the spatial wavelength 2 as k.=
2/~n 2
(19)
Assuming that the complex permittivity varies only in the x direction, Gauss' law becomes
ax [~*(x,~o)t~x] + e*(x,~o) ~-y = o
(20)
and, because the electric field is irrotational, the y component of the electric field can be evaluated using Eq. (18):
/~
-
~
Oy
_
+
~ jk. ~.(x)e-jk.r
(21)
-co
Using this result, Eq. (20) can be written in the form Ox [g*(x,o))/~] + e*(x, eo) ~ k2q).(x)e -it"' = O.
(22)
Integrating this equation from x = 0 +, where e*/~x = ~, Ex, ^a to infinity, where the electric field goes to zero and the complex permittivity remains finite, we find an expression for the normal component of the complex displacement flux density just above the surface of the electrodes: =
eaE, e -oo
" ---
~*(x,~o) ~ k2.~,(x)e-Jk"Ydx.
(23)
-oc
The medium above the electrodes can be described in terms of a surface capacitance density. This quantity represents the response of the medium to one Fourier component of the potential at the electrodes. It is defined as the ratio of the normal component of the complex displacement flux density evaluated at the interface ~* E. ^~ to the
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complex amplitude of the interfacial potential ~ [9]: (~,
.
(24)
Note that this quantity has the same units as the complex capacitance per unit area defined in Eq. (15) for parallel-plate structures. Using Eq. (23), the surface capacitance density can be written in terms of the following integral: ~a(o),kn ) = k. 0 ® e*(x,e))~.(x)k, dx.
(25)
~'. Jo
In contrast with the result obtained in Section 4.1, the spatial distribution of the complex permittivity acts as a weighting function to the potential distribution. For a given wavenumber, the potential decays with a characteristic length scale proportional to the imposed spatial wavelength. Thus, it seems possible to extract the spatial distribution of the complex permittivity from measurement results obtained with multiple-wavelength interdigital-electrode structures. 4,2.2. Piecewise-homogeneous media The evaluation of the above integral is, in general, quite complex. Given a complex-permittivity distribution, the potential in the heterogeneous medium has to be solved for in order to carry out the integral. This can be done analytically only for a select number of complex-permittivity distributions. Consider, for example, the piecewise-homogeneous medium shown in Fig. 4. Of the M layers, the top one (j = 1) is of semi-infinite extent. The jth layer has a complex permittivity e*, a thickness d j, and an upper surface designated by (j). In this case, the desired expression for the surface capacitance density can be obtained by solving Laplace's equation in each layer, applying continuity conditions for the potential and the normal component of the complex displacement flux density at each interface, and carrying out the integral in a piecewise fashion over all the layers. Alternatively, the surface capacitance density can be derived by sequentially applying transfer relations to each layer in the medium. These matrix relations summarize the field laws in each layer by relating the normal components of the complex displacement flux density to the corresponding potentials at the upper and lower surfaces of the layers [16]. Because a surface capacitance density can be defined for each interface in terms of the same variables, this iterative process can be cast in terms of the surface capacitance densities of the interfaces. In particular, the surface capacitance density of the (j + 1)st interface is related to that of the jth interface by the following recursive relation [9]: (~*kn) 2
~(j+ 1) = e* k, coth(k~dj) - sinh2(k,dj ) [e*k, coth(k,dj) + C(J)] "
(26)
Thus, in the process of deriving the surface capacitance density at the electrodes, this relation is applied sequentially, starting from the top layer (j = 1) and working down
P.A. yon Guggenberg, M.C. Zaretsky/Journal of Electrostatics 34 (1995) 263 277
[
273
I
I
t di
'
t
dM
k"-S""eee J / £~
Layer j
e~
/
/ h
Fig. 4. Schematic illustration of a piecewise-homogeneous medium composed of M layers.
to the bottom layer (j = M). The desired expression is then given by d~ = d(~+ l).
(27)
Note that, because the surface associated with the top layer is at infinity (dl ~ ~), ~t2) can be evaluated without the need to specify ~ 1 ) If we assume that the complex permittivity of each layer differs from a common uniform value e* by a small perturbation g*, the surface capacitance density above the electrodes can be expressed as a linear superposition of the complex permittivities of the layers in the medium. In particular, for a medium composed of one, two, or three layers, the linearized form of Eq. (26) yields M = 1 (~ = e~'k,,, M = 2
t~." = 8~k,, - (e* - e*)k.e -2k-a2,
M = 3
~." = ~ ' k .
- (~* -
E * ) k . e - 2 ' o " ' -- ( ~ ' - - 4 ) k . e - ' k ° ( " 2 + ' .
(28)
The form of these relations suggests the following expression for the linearized surface capacitance density of a medium with M layers: M-1
C~ = e*k. + ~ j=l
(~* - e*+x)k.e-2k"hj,
(29)
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274
where the height of each layer above the surface of the electrodes is given by M
hi= ~
(30)
di.
i=j+ 1
Thus, the surface capacitance density above the electrodes can be expressed as a linear combination of the complex permittivities of the layers, M
(31)
C~,,l(o~,, k,,,) = ~ Bi.,(dj, hi, k,,,)e*(~oi), j=l
where knz is the wavenumber associated with the nth mode and the lth wavelength: 2rm k.l = - 21
(32)
In a situation where the complex permittivities of the M layers are to be determined, a minimum of M measurement results is required. If this information is obtained from measurements performed at a fixed temporal frequency using M distinct spatial wavelengths, the surface capacitance densities associated with the measurement results can be related to the complex permittivities of the layers by the following matrix relation:
Lc . ,j
=LB-
.,.,
.
B...
(33)
Thus, because the rows of the matrix are linearly independent, the complex permittivities of the layers can be uniquely related to the surface capacitance densities associated with multiple-wavelength measurements. If, on the other hand, the measurement results are obtained from experiments performed at M distinct temporal frequencies using a single-wavelength interdigitalelectrode structure, the surface capacitance densities associated with the multiplefrequency measurements can be written as follows:
•
=
"
"
'
.
(34)
LBM.u With the goal of expressing each of the complex permittivities at the same temporal frequency, the layers are assumed to be composed of ohmic media• The complex permittivity of each layer can then be written in terms of a dielectric constant ej and a conductivity a j: e~
=
ej -
j aj. O9
(35)
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275
Using this constitutive relation, the real and imaginary parts of the surface capacitance densities can be related to the permittivities and conductivities of the layers by the following matrix relations:
¢ol
-- 3
=
LCMntJ
"
~l
".
"
0"1
(36)
L~
The determinant of each of the matrices in these relations is zero .Thus, the complex permittivities of the layers are not uniquely related to the surface capacitance densities and, as a result, cannot be uniquely determined from multiple-frequency measurements. Note that this result applies only when the spatial variations in the complex permittivities are a small perturbation about a uniform value. If the electrical relaxation times eg/0"~ of the layers are sufficiently distinct and decrease with distance away from the electrodes, the complex permittivities of the layers can be uniquely determined from measurement results obtained at multiple frequencies 1-17]. The discussion thus far has centered on the relationship between the complex permittivities of the layers and the surface capacitance densities associated with the measurement results. In practice, however, measurement results obtained with interdigital-electrode structures are expressed in terms of a complex gain I?F = ~ - ( ~ _ ~b)
(37)
which depends on the surface capacitance densities of the medium above and below the electrodes. In this case, the surface capacitance density of the medium below the electrodes (~ is fixed by the properties of the insulating substrate, namely the thickness h and the permittivity ~s- Thus, a unique relationship between the complex permittivities of the layers and the surface capacitance densities associated with the measurement results is a necessary but not sufficient condition to insure that the complex permittivities of the layers can be uniquely determined from the measured complex gains. The relationship between the complex gain and the surface capacitance density above the electrodes is nonlinear. Thus, uniqueness is not guaranteed. However, earlier work has shown that the relationship between the complex permittivity of a uniform medium and the associated complex gain is single-valued [1,9]. This implies that, even though there may be more than one value of the surface capacitance density for a given value of the complex gain, only one of these values is associated with a physical complex permittivity. Thus, if we limit ourselves to physical values for the complex permittivity, a unique relationship between the surface capacitance density and the
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complex gain does exist and the determination of the complex permittivities of the layers in a piecewise-homogeneous medium from measurement results obtained with multiple-wavelength interdigital-electrode structures is feasible.
5. Conclusions In contrast with classical dielectrometry, where the temporal frequency response of a material is the only measurable response, interdigital dielectrometry allows both the temporal and spatial frequency responses to be measured with multiple-wavelength interdigital-electrode structures. This additional degree of freedom permits the independent examination of the temporal and spatial dispersions in a material, thereby eliminating the need for making assumptions about the uniformity or frequency dependence of the dielectric properties. The temporal frequency response of parallel-plate structures cannot be used to infer the spatial distribution of the dielectric properties in a material. If no assumptions are made about the frequency dependence of the complex permittivity, the temporal frequency response of these structures can only provide information about the dispersive nature of the complex permittivity averaged over the thickness of the material. Except for specific cases, the complex-permittivity profile of a material cannot be determined from the temporal frequency response of single-wavelength interdigitalelectrode structures. Only the spatial frequency response of interdigital-electrode structures yields such information. This was demonstrated for a piecewise-homogeneous medium in which the spatial variation in the complex permittivity is a perturbation about a uniform value.
Acknowledgements This paper is based, in part, on the authors' theses submitted to the Massachusetts Institute of Technology (MIT) in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The research work was carried out in the Continuum Electromechanics Group of the Laboratory for Electromagnetic and Electronic Systems at MIT with financial support from the Electric Power Research Institute and members of MIT's Electric Utilities Program. The companies participating in the program included Allegheny Power Company, American Electric Power Service Corporation, Boston Edison Company, Empire State Electric Energy Research Corporation, Northeast Utilities Service Company, New York Power Authority, Southern California Edison Company, and Tokyo Electric Power Company. The ideas presented in this paper are based on the research of the late Professor James R. Melcher who, with the help of his graduate students, pioneered the application of spectral methods to the field of interdigital dielectrometry. We are grateful to have had the opportunity to study under his tutelage. We would also like to thank Professor Markus Zahn and Mr. Yanko Sheiretov of MIT and Dr. Stuart Inkpen of Instrumar Limited for helpful suggestions and constructive criticisms.
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