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1. Introduction
Fringing electric and magnetic field sensors are widely used for non-destructive measurements of material properties. A simple conventional fringing field sensor consists of a sensor head, a current or voltage source, an impedance measurement circuit, and data processing/data acquisition capability. The sensor head is usually a patterned array of electrodes or windings on an electrically insulating substrate. The voltage or current signal is applied to these electrodes and creates, respectively, electric or magnetic fields in the space around the electrodes or windings. These fields penetrate into the materials near the electrodes or windings. Changes of dielectric and magnetic properties of materials affect the distribution of these fields. The impedance measured between the electrical terminals of the sensor head is a function of the material electrical and geometric properties and the electric and magnetic fields. This general principle allows design of sensors for a very broad spectrum of applications including relating changes in electrical properties to physical properties such as temperature, density, defects, moisture, etc.
2. Advantages of Fringing Field Sensors
2.1.1. One-side access
Several inherent advantages of the planar interdigital geometry attract device designers. One of the most important ones, especially for non-destructive testing (NDT) sensors and piezoacoustic transducers, is that only a single-side access to the test material is required. One can penetrate the sample with electric, magnetic, acoustic, or optical fields from one side of the sample, leaving the other side open to the environment, which can allow absorption of gas, moisture, or chemicals that change electrical properties of the material under test (MUT). A sensitive layer of chemical or biological nature deposited over the electrodes and windings can also interact with a gas or liquid environment, allowing monitoring of concentration of chemicals in such materials as air, transformer oil, or the human body. In some situations, the other side of the material sample may be too far away, or inaccessible due to design limitations for an electrode so that one-sided access is essential.
2.1.2. Control of signal strength
By changing the area of the sensor, the number of fingers, and the spacing between them, one can control the strength of the output signal. A trade-off between the signal-to-noise ratio and the minimum sensing area is selected based on the application requirements. In microchip sensors, the size of the sensitive area is usually of little consequence, whereas in imaging devices it plays a major role.
2.1.3. Imaging capability
Either by moving sensor heads or by forming sensor arrays, one can interrogate different regions of material under test. A simple interdigital structure can be moved up and down to measure the depth profile or float above the material surface to measure variation of changes at a specified depth. A combination of both potentially provides a 3D image of the material under test, but simple interdigital sensors are rarely used in this manner because it can take a long time to scan a sample.
3. Theoretical background
3.1. Maxwell's Equations
All classic electromagnetic phenomena are described by Maxwell's equations. The differential form of Maxwell's equations in linear dielectric and magnetic media is
Law of Conservation of Charge:
(5)
where
is the electric field,
is the magnetic field strength,
is the current density,
is the charge density, m is the magnetic permeability, e is the dielectric permittivity, and
is time. The magnetic permeability
and dielectric permittivity
can be time and space varying in (1) – (5). Under most sensor application conditions the electromagnetic wave radiation wavelength,
where
is the speed and f is the frequency of electromagnetic waves, is much longer than sensor periodicity l of the sensor geometrical structure. For example, a 600 MHz electromagnetic wave has the free-space wavelength of 50 cm. The spatial wavelength of the periodic sensor structures is the distance between the centerlines of the adjacent electrode or winding finger belonging to the same electrode or winding, typically of millimeter order. For quasistatic approximations to be valid, the sensor spatial wavelength has to be much smaller than the radiation wavelength, i.e., l<<lem.
3.2. Electroquasistatics
The operation of fringing electric field sensors is governed by the electroquasistatic approximation to Maxwell's equations. In this approximation, the electric field energy stored in the system is much larger than the magnetic field energy, and the electric field
. This occurs when the system is capacitive and the time variations are sufficiently slow that the time variation on the right hand side of (3) is neglected. Under the electroquasistatic conditions, Maxwell's equations reduce to:
Figure 1 shows a schematic diagram of a generic model-based fringing electric field sensor. The sensor is comprised of a set of four coplanar electrodes: the driving electrode, the sensing electrode, the guard electrode, and the ground electrode. As the name suggests, the driving electrode is used to excite the sensor. Typically, the driving electrode is connected to an AC voltage source, represented by VD in Figure 1. The sensing electrodes are either connected to a voltage measurement circuit or current measurement circuit depending on the detection technique used for the application. When the material under test is present in the near vicinity of the sensor electrodes, the electric fields originating from the driving electrodes penetrate through the bulk of the material under test and then terminate on the sensing electrodes. The dielectric properties of the material under test alter the distribution of the field lines. Hence, the potential or current at the sensing electrodes is also a function of the material's dielectric properties. Thus by studying the variation of sensing current IS or sensing voltage in the time or frequency domain, material properties can be estimated. To prevent interference from external stray electromagnetic fields, a metal layer in the plane just below the driving and sensing electrodes serves as a ground plane. If there are multiple sensor heads on the same substrate, then guard electrodes are used to prevent “cross-talk” between sensors. These electrodes primarily suppress the interaction between adjacent sensor heads. The guard electrodes are either grounded or can be maintained at the same potential as the sensing electrodes.
Figure 1. Generic fringing electric field sensor with interdigital pattern of electrodes [1,2].
Electroquasistatic interdigital dielectrometry sensors use spatially periodic electrode structures to generate spatially periodic electric fields that penetrate into adjacent materials under test with dielectric permittivity e and conductivity s. The electric field distribution in the Z-X plane of a generic interdigital fringing field sensor in air (
,
) is shown in Figure 2. The outer driving electrodes in Figure 2 were excited at 1 V and the center sensing electrode was grounded. The shape of the electric field is independent of frequency while the amplitude depends on the ratio
. Figure 3 shows the variation of the electric field along the normal passing through
. The electric field along the Z axis decays approximately exponentially. Hence, the variations in the permittivity of the media closest to the surface of the electrodes have greater impact on the field distribution than those farther away. The fields die off as 1/r3 in the far field as a point electric dipole, where r is the distance between the point dipole and the field position.
Figure 2. Field distribution of a generic fringing electric field sensor in air. The outer driving electrodes were excited at 1 V and the center sensing electrode was grounded. Each electrode was 0.1l wide with 0.4l wide gap between electrodes. The shading represents the variation of the potential in the X-Z plane
Figure 3. Variation of the non-dimensional magnitude of the electric field as a function of z along the x=l/4 line midway between sense and drive electrodes. The drive electrodes were maintained at VD=1V and the sense electrode was grounded.
Figure 4 shows the electric potential along the same axis as that for Figure 3. The electric scalar potential obeys Laplace's equation with the electric field penetration depth of the order of l/2p, where l is the spatial periodicity of the electrode structure.
Because the electric field has zero curl in (8),
where
is the electric scalar potential. When
and
(10)
The potential as a function of position for z>0 can be written as a Fourier series
The coefficients
and
can be found by evaluating (11) at z=0, where the potentials are constant on each drive and sense electrode, and must be solved by numerical techniques in the space between the electrodes.
Figure 4. Variation of the non-dimensional potential as a function of z along the x=l/4 line midway between sense and drive electrodes. The drive electrodes were maintained at VD=1V and the sense electrode was grounded.
3.3. Magnetoquasistatics
The operation of fringing magnetic field sensors is governed by the magnetoquasistatic approximation. In this approximation, the magnetic field energy stored in the system is much larger than the electric field energy, the system is inductive, and the time variations are sufficiently slow that the displacement current density on the right side of (4) is negligible. Under magnetoquasistatic conditions, Maxwell's equations reduce to:
Figure 5 shows a schematic diagram of a generic fringing magnetic field sensor. The sensor consists of two sets of windings, the primary and secondary, generally on a substrate under the material under test. When electric drive current ID passes through the windings, it induces eddy currents in the conducting material under test. Magnetic permeability of the material also results in an increase of mutual inductance between secondary and primary windings. The open-circuit secondary winding voltage is given by the time-rate change of magnetic flux passing through the secondary winding due to the current in the primary winding.
Figure 5. Generic fringing magnetic field sensor with meandering pattern of windings [3-5].
Quasistatic magnetometry sensors often have spatially periodic windings to generate spatially periodic magnetic fields that penetrate into adjacent materials under test. Because the divergence of
in (16) is zero, a magnetic vector potential can be defined as
For ohmic conducting material, so that
, substituting (17) into (15) with
held constant yields a diffusion equation for
where we use the gauge condition
. For the sensor in Figure 5 in the sinusoidal steady state at radian frequency
, we neglect variation with the y coordinate so that the vector potential is of the form
(19)
so that (18) reduces to
where
is the space varying complex amplitude of
.
Then the solution of (20) is the Fourier series
(21)
where the complex spatial decay coefficient is
(22)
with
known as the skin-depth.
Such sensors are used for measurements of conductivity, complex magnetic permeability, and material thickness for single and multiple layered magnetic and/or conducting media. Typical non-destructive testing and evaluation applications include gas turbine component quality assessment [6]; cold work quality assessment and fatigue characterization [7,8]; and quality control measurements of aircraft propeller blades [9].
3.4. Spectroscopic Systems
All dielectric and magnetic materials consist of electric or magnetic dipoles. When subjected to an external electric or magnetic field, these dipoles re-align so as to partially neutralize the effect of the external field. This re-alignment of dipoles occurs to a varying extent for different materials. Thus, the dielectric or magnetic response of each material across the frequency spectrum is different, and in most cases, unique. The study of this response variation is known as dielectric or magnetic spectroscopy.
The dielectric or magnetic response of a material is generally quantified in terms of its complex dielectric permittivity or magnetic permeability. The complex dielectric permittivity
or magnetic permeability
is usually represented as,
(23)
where
and
are the real parts representing capacitive or inductive energy storage and
and
are the loss.
For all materials, the loss factor is a function of excitation frequency. The loss factor mechanisms are schematically shown in Figure 6. For a few low-loss materials and non-polar materials the variation in the loss factor with frequency is predominantly due to distortion in the electron clouds. Hence, the magnitude of variation of loss factor is negligibly small.
Figure 6. Mechanisms influencing the loss factor of a moist material over a wide range of frequencies f (Hz). (A) DC conductivity, (B) Maxwell-Wagner polarization, (C) dipolar polarization of water bound to the matrix of the material, (D) dipolar polarization of free water [10].
The polarization of molecules arising from their reorientation with the imposed electric field is the most important phenomenon contributing to the loss factor in the radio and microwave frequency ranges (107 to 3x1010 Hz). This includes the dipolar polarization due to bound and free water relaxation. At infrared and visible light frequencies, the loss mechanisms due to atomic and electronic polarization (collectively known as distortion polarization) are the dominating loss mechanisms [10].
The description for the process for pure polar materials was developed by Debye in 1929 [11]. The Debye dielectric relaxation model is the simplest way to analyze polarization in purely polar materials. The complex dielectric permittivity for the Debye relaxation model with a single relaxation time t and DC magnetic susceptibility
is
so that
(26)
The model assumes that the relaxation process is governed by first order dynamics, and hence can be characterized with a single time constant. The model can be derived using basic laws of polarization and conduction [12]. Magnetic materials have analogous frequency dispersion in the complex magnetic permeability.
4. References
[1] A. V. Mamishev, K. Sundara-Rajan, F. Yang, Y. Du, and M. Zahn, "Interdigital Sensors and Transducers," Proceedings of the IEEE, vol. 92, no. 5, pp. 808-845, May 2004.
[2] M. Zahn and H. A. Haus, "Contributions of Prof. James R. Melcher to Engineering-Education," Journal of Electrostatics, vol. 34, no. 2-3, pp. 109-162, Mar. 1995.
[3] A. V. Mamishev, K. Sundara-Rajan, F. Yang, Y. Du, and M. Zahn, "Interdigital Sensors and Transducers," Proceedings of the IEEE, vol. 92, no. 5, pp. 808-845, May 2004.
[4] M. Zahn and H. A. Haus, "Contributions of Prof. James R. Melcher to Engineering-Education," Journal of Electrostatics, vol. 34, no. 2-3, pp. 109-162, Mar. 1995.
[5] Y. Sheiretov, "Deep Penetration Magnetoquasistatic Sensors," Ph.D., MIT, Department of Electrical Engineering and Computer Science, 2001.
[6] N. Goldfine, D. Schlicker, Y. Sheiretov, A. Washabaugh, and V. Zilberstein, "Conformable Eddy-Current Sensors and Arrays for Fleet-Wide Gas Turbine Component Quality Assessment," ASME Turbo Expo, Land, Sea, & Air, New Orleans, LA, 2001.
[7] J. Fischer, N. Goldfine, and V. Zilberstein, "Cold Work Quality Assesment and Fatigue Characterization Using Conformable MWM Eddy-Current Sensors," 49th Defense Working Group on NDT, Biloxi, Mississippi, 2000.
[8] N. Goldfine, A. Washabaugh, V. Zilberstein, D. Schlicker, J. Fischer, T. Lovelett, T. Yentzer, and R. Pittman, "Scanning and Permanently Mounted Conformable MWM Eddy Current Arrays for Fatigue/Corrosion Imaging and Fatigue Monitoring," USAF ASIP Conference, San Antonio , TX, 2000.
[9] T. Yentzer, V. Zilberstein, N. Goldfine, D. Schlicker, and D. Clark, "Anisotropic Conductivity Measurements for Quality Control of C-130/P-3 Propeller Blades Using MWM Sensors With Grid Methods," Fourth DoD/FAA/NASA Conference on Aging Aircraft, St. Louis, MO, 2000.
[10] A. W. Kraszewski, Microwave Aquametry, IEEE Press, 1996.
[11] P. Debye, Polar Molecules, Chemical Catalog Co., 1929.
[12] J. R. MacDonald, Impedance Spectroscopy : Emphasizing Solid Materials and Systems, Wiley New York, 1987.
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(2005 © Kishore Sundara-Rajan