APPLICATION OF A THREE-REFERENCE CALIBRATION ALGORITHM FOR AN ELECTRICAL IMPEDANCE SPECTROMETER YUXIANG YANG, JUE WANG, FEILONG NIU AND HONGWU WANG

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1 INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume, Number 4, Pages Institute for Scientific Computing and Information APPLICATION OF A THREE-REFERENCE CALIBRATION ALGORITHM FOR AN ELECTRICAL IMPEDANCE SPECTROMETER YUXIANG YANG, JUE WANG, FEILONG NIU AND HONGWU WANG Abstract Accurate measurement of complex impedance in a wide frequency range is the foundation of electrical impedance spectroscopy technique. In order to improve the accuracy of a portable electrical impedance spectrometer (EIS) by correcting the systematic errors mathematically, a low cost and accurate three-reference calibration (TRC) algorithm based on quadratic Lagrange interpolation is applied. The principle of the TRC algorithm and the selection of the three RC calibration references are expounded and experimental measurements on a RC measurand are carried out to evaluate the performance of the TRC algorithm. The results find a good agreement between the calibrated data based on the TRC algorithm using the original data measured by the EIS and the standard data measured by impedance analyzer HP4194, with a maximum magnitude error of 0.34% and a maximum phase error of 0.6 over the frequencies from 0 khz to 1 MHz. The selection strategy of the calibration references is discussed. Key Words, Electrical impedance spectrometer, Systematic error, Three-reference calibration algorithm, Quadratic Lagrange interpolation, Accuracy improvement 1. Introduction As a quick, affordable, portable and harmless technique, electrical impedance spectroscopy is rapidly gaining popularity in a wide field of bio-research applications such as tissue ischemia and organ inflammation[1, ]. In the project for early detection of pressure ulcer, we have developed a portable, low-cost electrical impedance spectrometer (EIS) which can make a non-invasive measurement of tissue impedance over frequencies from 0 khz to 1 MHz[3]. However, the measurement accuracy is not so satisfactory, and large offsets exist for the original results both in magnitudes and in phases, especially at high frequencies. In fact, errors are inevitable in real world measurements between the original measured results and the true value of a measurand for any impedance measurement system. The uncertainty of measurements may result mostly from the systematic errors, because the random errors can be reduced to be negligible compared with the systematic errors with the help of averaging and smoothing procedure at the stage of sampling[4]. There are mainly two sources that contribute to the systematic errors: i) stray fields and parasitic elements of the measurement fixtures and connecting cables, and ii) instrumentation artifacts[5], which pose complicated influences on measurement outputs. Hence, system calibration techniques play an important role in achieving accurate measurements. In order to improve the measurement accuracy of the EIS system, a three-reference calibration (TRC) algorithm based on quadratic Lagrange-interpolation, which mathematically corrects the systematic errors, is Received August 10,

2 THREE REFERENCE CALIBRATION ALGORITHM 53 introduced and its performance is evaluated in this paper. The selection strategy of the calibration references is discussed in the last part of the paper.. Three references calibration algorithm.1. Calibration Theory The calibration method for the EIS is based on an assumption that there is a quadratic nonlinear relationship between the original measured values Z m (hereafter, bold symbols represent complex quantities) and the true values Z t. The input/output relation of the nonlinear system can be depicted by a quadratic polynomial in equation (1) and a curve in Figure 1. Z = k Z + k Z + k (1) t m 1 m where k is a calibration factor with the same unit as admittance, k 1 as a coefficient without unit, and k 0 as a constant with the same unit as impedance denoting the offset error of the original measuring system. Z t is a complex function of Z m so that the measurement error will affect both the magnitude and phase of the result. 0 Figure 1. Characteristic curve of calibration method In our application, a three-reference calibration (TRC) algorithm proposed by Liu[4] was adopted. With the use of three calibration references (RC circuits), their true values (Z r0, Z r1, Z r ) and original measured values (Z mr0, Z mr1, Z mr ) by the EIS can be obtained, which fix three points on the quadratic curve with known coordinate values (Z r0, Z mr0 ), (Z r1, Z mr1 ) and (Z r, Z mr ). These three known points can then uniquely determine the relation between Z t and Z m by a quadratic Lagrange-interpolation[4]: ( Z Z m mr1)( Z Z m mr) Zt = Z + r0 Z Z Z Z Z Z r1 r ( mr0 mr1)( mr0 mr ) ( Z Z )( Z Z m mr m mr0) ( Z Z 1 )( Z Z mr mr mr1 mr0) ( Z Z m mr0)( Z Z m mr1) ( Z Z )( Z Z ) mr mr0 mr mr1 + ()

3 54 Y YANG, J WANG, F NIU AND H WANG Before calibration, the true values (Z r0, Z r1, Z r ) can be obtained by a precise impedance analyzer (section.), while the measured values (Z mr0, Z mr1, Z mr ) can be measured by the EIS (section.3). Since the six symbols (Z r0, Z r1, Z r ) and (Z mr0, Z mr1, Z mr ) in equation () are constants, the calibrated result Z t can be calculated by this equation with its corresponding input, the original impedance Z m measured by the EIS. Note that values of all symbols in equation () are in complex form... RC Calibration References We use the three-element RC circuits shown in Figure as reference impedances for calibration. The reference circuit consists of a series/parallel association of two resistors and a capacitor, with the hypothesis that the extra-cellular liquid and intra-cellular liquid act like resistors and the cell membrane like an imperfect capacitor in biological tissues[6]. The complex impedance between terminal A and B can be calculated using Z r ( ) R R + R + R X R X = + + X 1 1 C i R X R C where X C is the impedance of the capacitor C and X C 1 = π f C denoting that Z X is a function of frequency ƒ. The RC circuit has a characteristic frequencies ƒ C where phase θ reaches its peak θ peak. The ƒ C and θ peak can be calculated by 1 R + R 1 f = C π C RR 1 1 R θ = arctan peak R1( R + R 1 ) C C (3). (4) Three such circuits with the same structure as shown in Figure were deliberately chosen as the calibration references Z r0, Z r1, Z r, which give the same characteristic frequency ƒ C at 100 khz. Table 1 shows the three groups of R 1, R and C parameters, which are all market purchasable. Figure. Circuit modality of RC calibration reference Three practical reference circuits are constructed in the form shown in Figure. Each circuit is soldered together using two four 1/4 watt metal-film resistors with ±1% tolerance and a low-loss metalized polyester film capacitor with ±5% tolerance, in which the parameters of R 1, R and C conform to table 1. The reference circuits are measured one by one on impedance analyzer HP4194A, whose outputs will be used as

4 THREE REFERENCE CALIBRATION ALGORITHM 55 the standard impedance values of Z r0, Z r1, Z r. The device used to connect the reference circuits with HP4194A is Hewlett-Packard Model 16047C test fixture. In order to acquire the measurement results at 10 well-chosen frequency points ranging from 0 khz to 1 MHz, the START and STOP frequencies of the HP4194 must be deliberately set. The measured impedance data at interested frequency points are recorded in table and stored in the on-board FLASH memory. It has been found that there is about % difference both in impedance magnitude and phase between theoretic calculation and actual measurement, which may derive from the synthetical effects of physical characteristics of the resistors and capacitors, such as parameter errors of the resistors and capacitors, the parasitical inductance and capacitance components in resistors and the parasitical inductance and resistance components in capacitors. Table 1 Parameters of the three calibration references R 1 ( Ω ) R ( Ω ) C ( nf ) ƒ C ( khz ) θ peak ( ) Z r Z r Z r Table Standard values of the three calibration references measured by HP4194 ƒ Z r0 Z r1 Z r (khz) Z r0 ( Ω ) θ r0 ( ) Z r1 ( Ω ) θ r1 ( ) Z r ( Ω ) θ r ( )

5 56 Y YANG, J WANG, F NIU AND H WANG Impedance magnitude (ohm) Impedance phase (degree) Zr0 Zr1 Zr Frequency (khz) (a) Zr0 Zr1 Zr Frequency (khz) (b) Figure 3. The magnitude-frequency and phase-frequency characteristics of the three RC calibration references at 10 well-chosen frequency points measured by HP4194: (a) the magnitude-frequency characteristics and (b) the phase-frequency characteristics..3. Calibration Measurements Calibration measurements must be performed by the EIS, whose measurement results on the three reference circuits will act as Z mr0, Z mr1, Z mr respectively in interpolation equation (). Three successive sweep-frequency measurements on the calibration circuits are performed and each sweep-frequency measurement rolls up all of the 10 frequency points as those chosen in table. In order to improve repeatability, every result is averaged by sampling 18 periods, which makes the random errors negligible compared to system errors. The calibration measurement results Z mr0, Z mr1, Z mr are recorded in

6 THREE REFERENCE CALIBRATION ALGORITHM 57 table 3 and stored in the on-board FLASH memory. Every time when computing the value of Z t using interpolation equation () at a certain frequency, both the original measurement values Z mr0, Z mr1, Z mr and the standard values Z r0, Z r1, Z r at relevant frequency will be read out. Table 3 Original values of the three calibration references measured by the EIS ƒ Z mr0 Z mr1 Z mr (khz) Z mr0 ( Ω ) θ mr0 ( ) Z mr1 ( Ω ) θ mr1 ( ) Z mr ( Ω ) θ mr ( ) System Performance In order to evaluate the performance of the EIS, we prepare another RC circuit as the measurand which is constructed with the same form shown in Figure and different parameters presented in table 4. The measurand is firstly tested on HP4194, whose outputs at the same 10 frequency points are regarded as their standard values of Z X. Then sweep-frequency measurements using the EIS on the RC circuit is performed and the original results are recorded as Z m. The ultimate calibrated values Z t is computed using the interpolation equation () at each frequency. The standard results measured by HP4194, the original results measured by the EIS and the calibrated results by the TRC algorithm are presented in table 5. The errors analysis between the calibrated results and the standard values of the measurand is shown in table 6, in which the magnitude absolute errorδ Z t = Z t - Z X and the phase absolute errorδθ t = θ t -θ X. The magnitude relative errors and the phase absolute errors are depicted in Figure 4 (a) and (b) respectively. From table 5, we can see that there are large offsets both in magnitudes and in phases between the original measurement values Z m, θ m and the standard values Z X, θ X. But after calibration based on the TRC algorithm, the differences between the calibrated values Z t, θ t and the standard values Z X, θ X are very small as shown in table 6 and figure 4, where the relative errors of impedance magnitudes are less than 0.34%, while the absolute errors of impedance phases are below 0.6 among the frequency range from 0 khz to 1 MHz. The maximum absolute error of impedance magnitude is about 0.58Ω, while the maximum relative error of phase reaches 3.99% because of smaller comparing base. Table 4 Parameters of the measurand

7 58 Y YANG, J WANG, F NIU AND H WANG R 1 ( Ω ) R ( Ω ) C ( nf ) ƒ C ( khz ) θ peak ( ) Z X Table 5 The standard results measured by HP4194, the original results measured by the EIS and the calibrated results by the TRC algorithm ƒ Standard results Z X Original results Z m Calibrated results Z t (khz) Z X ( Ω ) θ X ( ) Z m ( Ω ) θ m ( ) Z t ( Ω ) θ t ( ) Table 6 Error analysis between the calibrated results and the standard values of the measurand ƒ Magnitude Errors Phase Errors (khz) Δ Z t ( Ω ) Δ Z t / Z X (%) Δθ t ( ) Δθ t /θ X (%)

8 THREE REFERENCE CALIBRATION ALGORITHM Magnitude Errors Magnitude relative errors (%) Frequency (khz) (a) Phase Errors Phase absolute errors (degree) Frequency (khz) (b) Figure 4. Errors between the calibrated results based on the TRC algorithm and the standard values of the measure and: (a) the relative errors of the impedance magnitudes, (b) the absolute errors of the phases. 4. Discussion The selection strategy of the three calibration reference circuits is an importance factor that could greatly influence the measurement accuracy. In order to improve the measuring accuracy, it is important that the DUT must lie in the range the three references span through the whole frequency spectrum, and the distances between references had better be well-proportioned. This feature is crucial to ensure the validity of the calibration owing to the requirements of feasible interpolation according to

9 530 Y YANG, J WANG, F NIU AND H WANG equation (). Hence, parameters of R 1, R and C must be deliberately selected to avoid intercrossing among magnitude-frequency and phase-frequency characteristic curves in the measuring frequency range. For the measurement examples used in the paper, it is a perfect configuration as presented in table 1 for Z r0, Z r1 and Z r, where at each frequency point the three magnitudes and the three phases change in accord with an increasing trend from the calibration reference Z r0 via Z r1 to Z r as shown in figure 3 (a) and (b). However, this configuration may be not suitable for pure resistor measurement because the resistor s phase (less than 1 anyway) is not located in the range the three-reference span. Obviously, using three pure resistors with different magnitudes as calibration references may be better for this case. For biological uses where phases may range from very small to dozens degrees, it is reasonable to combine a pure resistor with two RC circuits as the three calibration references. 5. Conclusion In order to improve measurement accuracy, a TRC algorithm based on quadratic Lagrange interpolation is applied to the EIS. The experimental results show that the TRC algorithm corrects the systematic errors effectively and the measurement accuracy is greatly improved in the whole frequency range from 0 khz to 1 MHz, which demonstrates the applicability of the proposed algorithm. Acknowledgments This study was supported by a grant of the National Natural Science Foundation of China (Grant No ) and a grant of Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China (Grant No. 380). References [1] C. A. Gonzalez-Correa, B. H. Brown, R. H. Smallwood, T. J. Stephenson, C. J. Stoddard, and K. D. Bardhan (003), "Low frequency electrical bioimpedance for the detection of inflammation and dysplasia in Barrett's oesophagus", Physiological Measurement, 4(),pp [] C. e. A. Gonz alez, C. Villanueva, S. Othman, R. u. Narv aez, and E. Sacristan (003), "Impedance spectroscopy for monitoring ischemic injury in the intestinal mucosa", Phsiological Measurement, 4(3),pp [3] Y. Yang and J. Wang (005), "A Portable Bioimpedance Spectrometer for Early Detection of Pressure Ulcer", Proceeding of 14th International Conference of Medical Physics of the International Organization for Medical Physics (IOMP), Biomedizinische Technik/Biomedical Engineering, 50(Supplementary vol 1, part),pp [4] J.-G. Liu, U. Frühauf, and A. Schönecker (1999), "Accuracy improvement of impedance measurements by using the self-calibration", Measurement, [5] J.-Z. Bao, C. C. Davis, and R. E. Schmukler (1993), "Impedance system spectroscopy of human erythrocytes calibration and nonlinear modeling", IEEE Transactions on Biomedical Engineering, 40(4),pp [6] C. E. B. Neves and M. N. Souza (000), "A method for bio-electrical impedance analysis based on a step-voltage response", Physiological Measurement, 1(3),pp

THREE REFERENCE CALIBRATION ALGORITHM 531 Black/white Photograph Yuxiang Yang, Key Laboratory of Biomedical Information Engineering of Education Ministry, Xi an Jiaotong University, Xi an, China.

10 THREE REFERENCE CALIBRATION ALGORITHM 531 Black/white Photograph Yuxiang Yang, Key Laboratory of Biomedical Information Engineering of Education Ministry, Xi an Jiaotong University, Xi an, China. Mr. Yang is now a doctorial candidate of Xi an Jiaotong University. He received his MSc degree in Instrument & Measurement from Hunan University in 00. His research interests are in the areas of Medical Jue Wang, Key Laboratory of Biomedical Information Engineering of Education Ministry, School of Life Science and Technology, Xi an Jiaotong University, Xi an , People s Republic of China. juewang@mail.xjtu.edu.cn. Feilong Niu, Key Laboratory of Biomedical Information Engineering of Education Ministry, School of Life Science and Technology, Xi an Jiaotong University, Xi an , People s Republic of China. feilong.niu@gmail.com. Hongwu Wang, Key Laboratory of Biomedical Information Engineering of Education Ministry, School of Life Science and Technology, Xi an Jiaotong University, Xi an , People s Republic of China. hwwang0301@gmail.com.

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