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## Signal Processing for Impedance-Sensors

Prof. Dr. rer. nat. W.-J. Becker
Dipl.-Ing. Torsten Gerhold
Universität Gesamthochschule Kassel

### Abstract

In 1953 F. Förster [1] presented a method to separate the influence of two parameters on the impedance of a coil. He put the coil, applied as a sensor in non-destructive testing, on the specimen´s surface and was able to eliminate the influence of the distance a on the impedance. By that the various distance a between the sensor and the surface did not influence the measurement of the electrical conductivity s of the specimen for example. In 1995 Y. Wang and W.-J. Becker [2] represented a "function Sa". They used this function to measure the distance a independently of the specimen material. This "function S" connects the real and the imaginary part of the impedance . That way you eliminate the influence of two materials on the impedance and reduce the influence of other materials at any distance.They were not forced to put the sensor on the surface of the specimen. From that Flaschke and Tränkler [3] developed a procedure for determination of the soil water content. They defined a "function SY ". This function connects the both parts of the admittance and minimizes the influence of three or more materials on the determination of the soil water content. The coefficients of the "function S Y " were determined using a least square algorithm. In this paper the method of Wang and Becker is enlarged. A new "function T" connects two functions S at different frequencies. That way you can separate the influence of two and more parameters on the impedance of the sensor. You can generalize this procedure and connect the impedances at three and more frequencies for separation of four and more parameters. The conditions result out of the context between the "function Sa" and a method for determining of the concrete cover thickness and the bar diameter [4], [5].

### 1. Introduction

In this paper the following task is considered. The sensor, here a coil with ferrite core, is centered above the metallic bar and is moved down to the top of the bar. This happens step by step and at any step the impedance of the coil is measured (see figure 1). You get curves for both parts of the sensor´s impedance. In figure 2 the real part of the impedance is shown,and in figure 3 the imaginary part is represented. Both are standardized on the value at the distance a = 120 mm.You can describe the curves by the equations (1) and (2), see also [4],[5]. These equations are the result of a linear regression. A method shall be developed, which calculates the distance m between the sensor´s bottom and the top of the bar, the bar diameter d and the type m of bar material. For that the influences of the parameters distance a, diameter d and type of material m on the impedance must be separated.

 Fig 1: structure of experiment Fig 2: standardized Re{} versus distance a Fig 3: standardized Im{} versus distance a

### 2. Separation of two parameters by use of the "function S"

Now we consider metallic bars of one material m1 and various diameter d Î [dd , du] only (see also [4]). The -curves, you get by measurement at one frequency w1, are able to be described by the following equations.

Then the "function S",which Y.Wang and W.-J.Becker defined, goes:

You will be able to eliminate the influence of the bar diameter, if you choose the coeffizient n as follows.

Using equation (7) the "function S" at the distance a = a1 can be expressed as

You see, that the value of the coeffizient n is determined at one distance a = a1 and the influence of the bar diameter d is compensated. The influence at any other distance a will be compensated, if the following equation is valid.

In this experiment the equation (9) will be valid, if the distance a is an element of the intervall A = [ad , au]. Thus, the "function S" is

The equation (10) shows, that the "function S" and the coefficient C* respectively depends on the distance a1. This means, you have to calibrate at the distance a1 before a new measurement starts. You will be able to calibrate at a distance a, which is an element of the interval A = [ad , au], if the coefficients k12 and k22 fulfill the condition:

In that case the "function S" is given by equation (12).

Now you can connect the real and the imaginary part of the sensor´s impedance at the frequency w1 to the "function S" and can calculate the distance a (a Î [ ad , au]) without knowing the bar diameter d. The value of the bar diameter d is able to be determined (equation (3),(4)) by the being calculated value of the distance a (equation (12)).This way the influence of the parameters a and d on the impedance is separated. The figure 4 shows the "function S" for metallic bars of the material brass.

 Fig 4: function S for metallic bars of brass

### 3. Separation of three parameters by using "function T"

In the following metallic bars of several materials m and various diameter d are considered. The curves of the impedance getting by measurement at two frequencies w1 , w2 are able to be described as follows.

On the condition, that the distance a is an element of the intervall [ad , au], the "functions S" are

The "function T" is a combination of these two "functions S" of different frequency w.

The influence of two materials m1 and m2 on the impedance will be compensated, if the coefficient r is computed by:

The influence of other materials is reduced at least. Substituting the expression for r (equation (21)) into (equation (20)) gives the following equation for the "function T":

Like the "function S" you connect the real and the imaginary parts of the sensor´s impedance at two frequencies w1 , w2 to the "function T" and compute the distance a (a Î [ad , au])without knowing the bar diameter d and the bar material m .The values of the diameter d and the material m are able to be determined (equations (13) upto (16)) by the being calculated value of the distance a (equation (22)). That way the influence of the three parameters on the impedance is separated. The figure 5 shows the "function T" for metallic bars of the materials aluminium and brass.

 Fig 5: function T for bars of brass and aluminium

### 4. Generalization of the "functions"

You can generalize the procedures having been described. If you want to separate one more parameter, then you will have to increase the number of frequencies by double of amount. For example, four frequencies are necessary for separating four parameters (see equations (23) and (24)) and so eight are necessary to separate five parameters.

Considering the equations (13) up to (16) it is obvious that here we have to solve a system of equations, which contains a greater number of equations than unknowns. In this case there are four equations (Re{ ( T1,a,d,m)}, Im{ ( T1,a,d,m)}, Re{ ( T2,a,d,m)}, Im{ ( T2,a,d,m)}) and three unknowns (a,d,m). The requirement to solve the system in the way having described is,that the parameters, which you want to separate, appear in the real and the imaginary part of the impedance in same way. For example, if the parameter p appears in the real part as k 13 × f(p), it also must appear in the imaginary part as k 23 × f(p). For instance following functions f(p) are possible: p, p 2 , p n, e p, sin(p),... But no sum of these functions is possible, because then you can not reduce the expression of the coefficient n to a constant (see equations (6) up to (12)). If you are able to describe your system by a mathematical model, which fulfils this requirement, you will be able to separate the influences of the parameters on the impedance having measured.

### References

1. Förster, Fritz: Theoretische und experimentelle Grundlagen der zerstörungsfreien Werkstoffprüfung mit Wirbelstromverfahren, I. Das Tastspulverfahren, Zeitschrift fürMetallkunde, 1953/54, S.163ff.
2. Wang, Y.; Becker, W.-J.: Trennung des Messabstands von den Materialeigenschaften eines Messobjektes mittels eines Wirbelstromsensors, Kongressband zur Sensor 95 AMA, Vortrag A10.1 (1995), S. 231-236.
3. Flaschke, T.; Tränkler, H.-R.: Potential der Impedanz-Sensorik am Beispiel der Bodenfeuchtemessung, Technisches Messen 66 (1999), Heft 4, S. 146-150.
4. Ricken, Werner: Verfahren zur zerstörungsfreien Materialprüfung an Stahlbeton-bewehrungen mit Methoden der Wirbelstromanalyse, Fortschritt Berichte VDI Reihe 8 Nr. 427, VDI Verlag, Düsseldorf, 1994.
5. Gerhold, T.; Becker, W.-J.: Signalverarbeitungen zur Parametertrennung bei Impedanz-Sensorik, Sensoren und Messsysteme 2000, Tagungsband Ludwigsburg, 13. und 14. März 2000, VDI/VDE- Gesellschaft Mess und Automatisierungstechnik (VDI Berichte Nr. 1530), VDI Verlag, Düsseldorf, 2000.

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