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Systems of planar electrodes arranged on dielectric or piezoelectric layers are applied in numerous sensors and transducers. In this paper electrostatics of such electrode systems is presented and exploited in the analysis of distributed piezoelectric transducer dedicated to surface elastometry of biological tissues characterized by large Poisson modulus. The fundamental Matlab^{®} code for analyzing complex planar multiperiodic electrode systems is also presented.

In many sensors, electric field is applied to investigate body by multiple electrodes which frequently can be considered periodic. This for example is the case of surface wave sensors of gas, actuators and linear ultrasonic motors utilizing planar metal strips as electrodes. Analysis of the field distribution and the electric property of electrode systems is usually necessary for the design and evaluation of the sensor parameters. The analysis of conducting strip is the general subject of this paper, and its application for elastometry is proposed as an example of its usefulness.

Typically, elastic properties can be evaluated by measuring the ultrasonic wave time of flight over certain distance. In the case of tissue however, small dimension of tissue sample and considerable wave damping makes the task difficult. In this paper we show how to measure the wave velocity within one ultrasonic transducer applying multiple strip electrodes on a piezoelectric layer applied to the tissue sample, in which case the detailed analysis of the strip system is indispensable because the frequency characteristic of the system, instead of the time of flight, is measured.

The known identity [

Two equations can be casted from the above identity which can be easily interpreted in electrostatic terms (the second equation results from the first one after substitutions

The electrostatic interpretation of the above complementary set of functions is the following. The first equation is the Fourier expansion of the normal electric induction on planar periodic perfectly conducing strips arranged along axis

In order to make use of the above set of expansion, we need certain characterization of the electrostatic medium the strips to be embedded in. Consider the dielectric halfspace _{x}_{x}φ_{n}_{n}P_{n}_{z}_{x}

The application of different electric potentials to subsequent strips breaks the electric field periodicity, hence the above Fourier expansion must be generalized into Bloch expansion of the planar electric field components by corresponding multiplication by _{m}_{n}_{m}_{e} =

Usually, the most interesting for applications are the strip potentials (which frequently are given) and the resulting strip charges or currents _{l}

The ^{th} strip potential _{l}

Analogous integral of the normal induction ^{th} strip). The value of the ^{th} strip charge is the difference of the integral values at these two points: at the space centers after and before the ^{th} strip:

Note that both _{m}_{l}_{l}

If strips are placed on a layered substrate, the

The fundamental feature of such system is that for large wavenumber value (say, for |_{e}

We apply the field expansion _{m}_{n}

These equations result from _{n}, E_{n}_{n}x_{n}_{n}_{e}_{n}S_{n}_{−}_{m}_{n}_{n}_{n}S_{n}_{−}_{m}_{m}

Note that the number of unknowns _{m}

Evaluation of the strip voltages and currents (charges) requires integration of _{lk}_{k}

This is the equation which, appended to _{m}_{k}^{th} strip potential _{l}

It is evident that the strip admittance depends on the piezoelectric and elastic property of the layered media on which the strips reside. In the case of biological tissue of large Poisson module (_{t}_{l}_{t}_{t}

In typical cases, the spectral function _{R}

The fundamental properties of the considered layered system are described by its effective dielectric permittivity _{o}_{o}_{o}_{t}_{l}

It is convenient to present first the results for infinite periodic system of 5-strip cells presented in

Due to the system exact periodicity and different voltages within periodic cells, the summation over the repeating strips (over indices _{m}_{0} + _{2}. Naturally, the measured current in real system would be the sum over all cells multiplied by the system aperture-width. For convenience of further discussion, we define the observed signal as _{0} + _{2})/

In the case of finite number of cells, the ^{±} of upper and lower bus-bars, respectively; ^{+} − ^{−} = ^{±} = ±^{o}^{o}^{o}

The resulting signal _{o}_{t}_{t}_{o}_{t}

In more general case, there are different strips within periodic cells. In the above-discussed sensor, for example, strips number 3 and 4 can be joined without any space between them. Such system is no longer simple-periodic; it is called [^{®} code for direct evaluation of the spatial Fourier expansion coefficients of the corrected template function

The electric fields is constructed using these Fourier coefficients analogously to

This can be avoided well by applying small ^{−4} is a good choice). Taking the output of FFT{[_{n}_{−}_{m}F_{n}_{−}_{m}_{n}_{−}_{m}_{m}_{n}_{−}_{m}_{n}_{−}_{m}

Modern electronic technology allows one to easily fabricate planar system of strips, which contributes to their wide applications in many electronic devices including sensors (SAW gas sensors, for instance) and actuators (piezoelectric linear motors, for instance). Rigorous electric field analysis is usually necessary for the design and evaluation of electric properties of such devices. In this paper, a method of analysis of periodic strips or periodic groups of strips (with arbitrary width and spacing within cells) is presented in interesting application for surface elastographic sensor, which may contribute to better accuracy of measurements of tissue properties required in medical investigation and diagnosis of skin [

This work was supported by the Polish Ministry of Science and Higher Education (Grant NO. N515 500540).

Consider an isotropic elastic halfspace _{1,2} and normal traction _{3}_{i}, i_{i}z_{1} = _{3} =

Much more complicated is the characterization [_{3}) on the layer upper side, which is considered traction-free on normal displacement (_{3}) and traction _{33} applied to its bottom side which is considered metalized:

Substitution of

Consider even number _{n}

Periodic system of strips with external cross-less connections within the strip cells including five strips; the arrangement used in the discussed sensor.

Typical frequency dependence (

Example _{t}