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Hysteretic Modeling of Output Characteristics of Giant Magnetoresistive Current Sensors Jinchi Han, Student Member, IEEE, Jun Hu, Member, IEEE, Yong Ouyang, Student Member, IEEE, Shan X. Wang, Fellow, IEEE, and Jinliang He, Fellow, IEEE
Abstract—Current sensing based on the giant magnetoresistance (GMR) effect has been gaining attention due to its outstanding merits. In this paper, hysteretic models of the output characteristics of GMR sensors are presented, and corresponding algorithms are successfully applied to practical GMR sensors to compensate for measurement error due to hysteresis, which is particularly important for high-frequency applications. A 91% decrease in nonlinear error is achieved by the proposed advanced hysteretic model, and the applicable frequency range of the GMR sensor can be extended to around the cutoff frequency of sensor hardware, which is nearly 50 times larger than that based on the conventional linear models. This paper provides optimized solutions for GMR current sensors in different frequency ranges. Index Terms—Current sensor, giant magnetoresistance (GMR) effect, hysteretic modeling, output characteristics.
I. I NTRODUCTION
C
URRENT sensing techniques play a significant role in various industrial applications, e.g., real-time control in power electronic modules and motor drives [1]–[6], fault detection and diagnosis of electric motors [7]–[9], and appliance load monitoring [10]. Standard current sensors include resistive shunts, current transformers, and Hall sensors. None of them, however, promise the integrated power electronic applications [11]. The magnetoresistance (MR) effect is the phenomenon in which the resistance of ferromagnetic metal and alloy would undergo regular changes under the influence of an external magnetic field. Various MR effects based on different magnetic materials and structures were proposed and researched, among which the giant MR (GMR) effect discovered by Fert et al. [12] and Grünberg et al. [13] in 1988 was regarded as one of
Manuscript received September 16, 2013; revised January 8, 2014 and March 1, 2014; accepted April 12, 2014. Date of publication May 29, 2014; date of current version December 19, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 51028701 and Grant 51077085 and in part by the National Basic Research Program of China (973 Program) under Grant 2013CB228206. J. Han, J. Hu, Y. Ouyang, and J. He are with the State Key Laboratory of Power Systems, Department of Electrical Engineering, Tsinghua University, Beijing 100084, China (e-mail:
[email protected]; hjun@ tsinghua.edu.cn;
[email protected];
[email protected]). S. X. Wang is with the Center for Magnetic Nanotechnology, Stanford University, Stanford, CA 94305 USA, and also with the State Key Laboratory of Power Systems, Department of Electrical Engineering, Tsinghua University, Beijing 100084, China (e-mail: sxwang@ stanford.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2014.2326989
the most promising mechanisms for sensing techniques. With respect to current sensing, GMR sensors possess a good number of advantages over existing techniques, e.g., wide bands, high sensitivity, high stability, small size, low price, simple structure, low power consumption, and ease of integration. Up until now, GMR current sensors have been already successfully applied to various power electronic circuits [14]–[18]. Presently, the current surge of smart grid technology brings about acute needs for the high-frequency current measurement. Related smart grid applications include monitoring of power quality, fault detection and diagnosis in power system, monitoring of electromagnetic transient process and electromagnetic environment, etc. [19]. Most of the solutions in these fields rely on a precise understanding of high-frequency currents or components (up to 100 MHz), which hence demand good highfrequency performance of the current sensors. However, the highest applicable frequency of GMR current sensors is usually lower than megahertz in consideration of linearity. The most critical problem that causes large nonlinear error and greatly restricts the applicable frequency range is the hysteretic output characteristics of GMR sensors. Generally, there are two categories of methods to achieve a linearized output characteristic, i.e., special hardware designs or signal conditioning with appropriate models. With respect to the first category, efforts and attempts include the closed-loop operation [20], the design of counteracting magnetic field [21], the low-frequency capture [22], and the automatic calibration and adjustment [23]. However, most of these special designs are in effect complicated, expensive, and energy-consuming, which makes them unfavorable for wireless sensor network and lowpower sensing applications. Alternatively, modeling of output characteristics offers ways to improve the high-frequency performance by conditioning the output voltages with appropriate models and algorithms. Still, hysteresis is the most challenging issue. An ideal model should ensure linearized output characteristics in the entire frequency range of application at the lowest possible cost. In order to describe the features of hysteresis, attempts were made in the mathematical modeling of the hysteresis loops. Common mathematical models include the power series models [24] and the rational polynomials [25]–[27]. In addition, there exists a series of more theoretical hysteresis models, the original version of which was presented by Preisach [28] and later expressed mathematically by Krasnosel’skii and Pokrovskii [29]. This model was further developed and numerically implemented by Mayergoyz [30], [31]. The Preisach–Krasnosel’skií model and the Jiles–Atherton model [32] both provided an
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accurate understanding of hysteresis and have been the focus of many researches in recent years. However, it should be noticed that both of these two theoretical models are quite complex, noninvertible, and difficult to implement on digital processers. Although various hysteretic models have been proposed and developed for years, a limited number of them were successfully applied to GMR materials. The scalar Preisach model was verified to describe the behavioral characteristics of GMR structures [33], [34]. Despite its high accuracy, the Preisach model should not be regarded as the appropriate hysteretic model for linearizing the output characteristics considering the difficulty of implementation. Jedlicska et al. utilized a seventhdegree polynomial model to fit the hysteretic relationship between the measured currents and output voltages, achieving a 75% decrease in the nonlinear error [35], [36]. However, since the output voltages were much smaller than the currents (usually below 1 V), the high-degree polynomial coefficients were as high as 1011 for a seventh-degree polynomial coefficient, as reported in [36], which resulted in difficulty in current calculation. In addition, its reliance on the measurement history caused error propagation during measurement. To deal with this problem, a special hardware design and a complicated algorithm were involved to realize the reset process. In view of these, it is quite essential to search for more efficient hysteretic models and algorithms for conditioning the output characteristics of GMR current sensors to achieve competitive performance at acceptable costs. This paper focuses on the hysteretic output characteristics of GMR sensors. Novel hysteretic models and corresponding algorithms for GMR sensors are proposed, based on which the output characteristic of the GMR sensors were conditioned and successfully linearized. As a result, the applicable frequency has been extended to the cutoff frequency of the sensor hardware, which enables various high-frequency applications. Compared with the conventional complicated hysteretic models, the proposed ones improve the accuracy and extend the applicable frequency range at acceptable costs. II. L INEAR M ODEL AND F REQUENCY C OMPENSATION A. CLM Presently, the linear function is most widely utilized to translate the output voltage signals from GMR sensors into currents, but susceptible to hysteresis. With the increase in current frequency, hysteresis of the output characteristics, which results from the phase lag, becomes more and more evident and compounds the nonlinear error. Moreover, the attenuation characteristic of frequency response further increases the nonlinear error. Therefore, the applicable frequency range of the conventional linear model (CLM) is greatly constrained by the hysteretic characteristic and the frequency response of the GMR sensors. B. LMFC Linear model combined with frequency compensation algorithm (LMFC) helps improve the frequency response of GMR sensors. The single-valued output characteristic can be obtained by processing raw voltage data with the method of
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phase compensation, and magnitude compensation ensures a constant slope and a constant intercept of the linear fitting. The compensation algorithm is briefly introduced as follows. The magnitude and phase of each component are obtained by fast Fourier transform (FFT) analysis of the acquired voltage data [A(f0 , . . . , fs /2) , ϕ (f0 , . . . , fs /2)] = FFT(v(t0 , . . . , ts ))
(1)
where A and ϕ are the magnitude array and the phase array of frequency components f0 , . . . , fs /2, respectively, and v is the sampled output voltage of the GMR current sensor. Then, the magnitude gain and the phase lag of each component can be obtained by interpolation of pretested frequency responses, and a new set of magnitude and phase will be generated by Anew (f0 , . . . , fs /2)=A (f0 , . . . , fs /2)/Again (f0 , . . . , fs/2) ϕnew (f0 , . . . , fs /2) = ϕ (f0 , . . . , fs /2)−ϕl (f0 , . . . , fs /2) (2) where Again and ϕl are arrays of magnitude gain and phase lag from the pretested frequency response. The processed components are converted into the compensated voltage data, which can be translated into measured currents with a certain slope and intercept obtained in the dc condition vnew (t0 , . . . , ts ) = IFFT([Anew(f0 , . . . , fs /2) , ϕnew(f0 , . . . , fs /2)]) .
(3)
In spite of the outstanding merits, the digital frequency compensation exerts too much pressure on data processing. Moreover, it is difficult to apply the digital frequency compensation algorithms to sensor nodes to realize real-time data processing and measurement. III. N ONLINEAR H YSTERETIC M ODELS OF O UTPUT C HARACTERISTICS A. BNLM The basic nonlinear model (BNLM) proposed in this paper is based on the rational polynomial functions shown in (4) and does not rely on the integration of the measurement history. Thus, it will not be affected by error propagation. y =A·
a0 + a1 x + a2 x 2 + · · · + an x n . b0 + b 1 x + b2 x 2 + · · · + b n x n
(4)
Since the saturated section is not involved in the applied range, the first-order rational polynomial functions are enough to fit the branches of resistance variation. The model derived from the output expression of the Wheatstone full-bridge is shown in (5) VGMR = Vs · ΔR/R = a0 ·
I + a1 I + a2
(5)
where VGMR is the output voltage of the Wheatstone bridge, I the current to be measured, and ΔR/R the relative change of resistance.
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The expressions of amca1 and amcd1 are complicated, and an approximation is made by analyzing their Taylor series of m = 1, presented as amc1 = mah1 − Ic + = mah1 −Ic + − Fig. 1. Hysteretic characteristics of the GMR current sensor. (a) Low hysteresis. (b) High hysteresis. Major Loop (solid), general minor loop (dashed), and half-range minor loop (dash-dotted) are included.
The coefficients (a0 , a1 , a2 ) of the rational linear functions with the ascending branch and those with the descending branch of a hysteresis loop are different. In addition, the coefficients are related to the range of the currents to be measured. In most cases, a mismatch of resistance causes the imbalance of the Wheatstone bridge, which leads to zero drifts of GMR sensors. To derive the relationship between the coefficients and the current range, the symmetry center of the output characteristic is defined as (Ic , Vc ). The half-range minor loop is introduced for the convenience of deduction, which has half of the range of the major loop, as shown in Fig. 1. The asymmetry minor loops can be transformed into the corresponding symmetry minor loops by means of vertical and horizontal shifts [35], [36]. The coefficients of the half-range symmetry minor loop with a current range of Ih are defined as (aha0 , aha1 , aha2 ) and (ahd0 , ahd1 , ahd2 ). The subscripts h, a, and d represent halfrange, ascending branch, and descending branch, respectively. The output characteristics of the half-range minor loop can be described as ha1 dI aha0 · I+a I+aha2 , dt ≥ 0 VGMR = (6) hd1 dI ahd0 · I+a I+ahd2 , dt < 0. The minor loop with coefficients (amca0 , amca1 , amca2 ), (amcd0 , amcd1 , amcd2 ), and a current range of Ir = mIh can be turned into the half-range minor loop by shift and scaling transformations (VGMR − Vc )/m = ah0 ·
(I − Ic )/m + ah1 (I − Ic )/m + ah2
(7)
which can be equivalently expressed as VGMR = (mah0 + Vc )
I+
mah2 Vc +m2 ah0 ah1 mah0 +Vc
(ah2 −ah1 )Vc (ah2 −ah1 )Vc2 + (m − 1) ah0 +Vc (ah0 + Vc )2
2ah0 (ah2 −ah1 )Vc2 (m−1)2 +O (m−1)3 3 (ah0 +Vc )
(10)
where O((m − 1)3 ) is the third-order infinitesimal. When the imbalance is not noticeable, and the frequency of the measured currents is lower than the cutoff frequency fc of the GMR sensor, the coefficients can be compared as follows:
⎧
2 ⎨
(ah2 −ah1 )V2c
(ah2 −ah1 )Vc
(ah0 +Vc ) ah0 +Vc
(11) ⎩
2ah0 (ah2 −ah13)Vc2
(ah2 −ah1 )Vc
. (ah0 +Vc ) ah0 +Vc Therefore, the expression of amc1 can be simplified as (ah2 − ah1 )Vc amc1 ∼ − Ic . = mah1 + ah0 + Vc
(12)
Substituting the relationship Ir = mIh into (9) and (12), the coefficients are simplified as ⎧ amca0 = aIha0 I r + Vc ⎪ h ⎪ ⎪ −aha1 )Vc ⎪ a ha1 ⎪ amca1 = Ih Ir + (aha2 − Ic ⎪ aha0 +Vc ⎪ ⎨a aha2 mca2 = Ih Ir − Ic (13) amcd0 = aIhd0 I r + Vc ⎪ h ⎪ ⎪ ⎪a (ahd2 −ahd1 )Vc ahd1 ⎪ − Ic ⎪ mcd1 = Ih Ir + ahd0 +Vc ⎪ ⎩a ahd2 mcd2 = Ih Ir − Ic . It can be observed that the expressions of amca0 , amcd0 , amca2 , and amcd2 are approximately in direct proportion to the current range. However, there are offsets in the expressions of amca1 and amcd1 , resulting from the imbalance of the GMR sensor. The algorithm used to calculate the measured currents is summarized in Fig. 2. The turning point of current needs to be detected to split the model into branches [35], [36]. By taking the hysteresis into consideration, the nonlinear error is greatly decreased. B. ANLM
− Ic
I + mah2 − Ic
.
(8)
The coefficients can be concluded as ⎧ amca0 ⎪ ⎪ ⎪ a ⎪ ⎪ ⎨ mca1 amca2 amcd0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ amcd1 amcd2
m(ah2 − ah1 )Vc mah0 + Vc
= maha0 + Vc Vc +m2 aha0 aha1 = maha2ma − Ic ha0 +Vc = maha2 − Ic = mahd0 + Vc Vc +m2 ahd0 ahd1 = mahd2ma − Ic hd0 +Vc = mahd2 − Ic .
(9)
Despite the good performance in low-hysteresis situations, the above algorithm still cannot avoid great nonlinear error at the terminals at higher frequencies (e.g., the cutoff frequency of sensor hardware), for rate of change of the rational function cannot match that of the curvature of the hysteresis loops. To further ensure the feasibility of the nonlinear hysteretic model, a new algorithm is proposed in this paper, which is improved by weighting the current data at both ends. After the coefficients of both branches are obtained, the raw data of the GMR output voltages should be classified into four intervals, i.e., the lower terminal interval, the ascending branch
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boundaries of the intervals are determined. Since it is relatively easier to locate the maximum and the minimum of the output voltages, the two boundary points are determined by the product of the extreme values and the designated coefficient k. The corresponding current values of the turning points (I1 , V1 ) and (I2 , V2 ) are calculated by the following equations: I1 =
Fig. 2. Flowchart of the BNLM and the corresponding algorithm. The main decision is the detection of the turning points. If a turning point is detected, the extremum is stored in the buffer and the coefficients of the new branch are calculated.
ad2 V1 − ad0 ad1 , ad0 − V1
I2 =
aa2 V2 − aa0 aa1 aa0 − V2
(14)
which are transformed from (5). Their symmetry points (I3 , V3 ) and (I4 , V4 ) with respect to the major symmetry axis of hysteresis loop are set as the other two boundaries. The slope and the intercept of the symmetry axis can be extracted from the processed single-valued output characteristics by averaging corresponding points, and the voltages of the symmetry points (I3 , V3 ) and (I4 , V4 ) can be worked out by
I1 +I3 2 V3 −V1 I3 −I1
3 = kp · V1 +V + bp 2 , = −kp
I2 +I4 2 V4 −V2 I4 −I2
4 = kp · V2 +V + bp 2 = −kp (15)
where kp and bp are the slope and the intercept of the major symmetry axis of the hysteresis loop, respectively. This way, the raw voltage data in a single period can be divided into the ascending branch interval V4 , V1 , the descending branch interval V3 , V2 , the upper terminal interval V1 , V3 , and the lower terminal interval V2 , V4 . The measured currents in the terminal intervals are worked out by averaging the results calculated with both the ascending and the descending branch functions, whereas the expressions for the ascending and the descending intervals are the same as those in the basic model. In consideration of the concavities and rates of change in the terminal intervals, the quadratic weights are adopted. The weights of a point indexed as m in an interval with N points can be expressed as
w1 = 1 − (m/N )2 , w2 = (m/N )2
0≤m≤N
(16)
where w1 is the weight of the result based on the currently adopted branch model, and w2 is the weight of the result based on the other branch model. The currents calculated with the advanced nonlinear hysteretic model (ANLM) are expressed as Fig. 3. Flowchart of the ANLM and corresponding algorithm. Two main decisions are the detection of the extremum and the turning point. The extremum determines the coefficients of the branch model. The turning point indicates the change of interval, and the corresponding algorithm will be adopted.
interval, the upper terminal interval, and the descending branch interval. Each interval has a specified algorithm for current calculation. The overall algorithm for classification and current calculation is summarized in Fig. 3. The output characteristic with high hysteresis is presented in Fig. 1(b). In order to classify the raw voltage data, the
⎧ aa2 V −aa0 aa1 , V ∈ V4 , V1 ⎪ aa0 −V ⎪ ⎪ ⎨ ad2 V −ad0 ad1 , V ∈ V , V 3 2 ad0 −V I= ad2 V −ad0 ad1 −aa0 aa1 ⎪ w1 · aa2 Vaa0 + w · , 2 ⎪ −V ad0 −V ⎪ ⎩ aa2 V −aa0 aa1 ad2 V −ad0 ad1 w2 · + w · , 1 aa0 −V ad0 −V
V ∈ V1 , V3 V ∈ V2 , V4 . (17)
The proposed advanced model can deal with the terminal nonlinear errors, and is thus able to increase the linearity of high-frequency (up to the cutoff frequency fc of the sensor hardware) current measurement.
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Fig. 4. Photograph of the two sensors and the experimental platform. Sensor A contains a TMR sensing chip, whereas Sensor B contains a GMR sensing chip.
IV. E XPERIMENTAL S ETUP A. GMR Sensor and TMR Sensor There are four typical structures that have GMR effect, i.e., the multilayer structure, the spin-valve structure, the magnetic tunneling structure, and the granular alloy structure. Among these structures, the tunneling magnetoresistive (TMR) sensors enjoy many merits, such as wide linear range, inhibition of temperature drift, self-compensation, etc. TMR sensors were also chosen as the object of research in this paper, because the proposed models are applicable to both GMR and TMR sensors. Moreover, TMR sensors usually possess much lower cutoff frequencies than GMR sensors, which make TMR sensors even more suitable for studying the performance of the proposed models. A TMR sensor (Sensor A), as shown in Fig. 4, was fabricated based on a ML54P TMR sensing chip provided by Dowaytech Inc. In addition, a GMR sensor (Sensor B) was fabricated based on a NVE AA002-02 GMR sensing chip, the bias field of which was provided by permanent magnets. A ferrite magnetic ring was utilized to enlarge the magnetic field for the sensing chip. Moreover, it can inhibit the external magnetic interference and make the sensor insensitive to the relative position and angle. Signal conditioning circuits were designed to amplify the voltage signal and to add offsets [35]–[37].
B. Experimental Platform The sensors were located on a measurement platform (Fig. 4), which was designed to simulate a power line. The currents to be measured were generated by a DOSF series power amplifier produced by Matsusada Precision Inc. The reference values of the measured currents were converted into voltage signals (0.1 V/A) by a Yokogawa 701932 current probe. Both the reference voltages of the currents and the output voltages of the sensor were sampled by an NI USB-5133 data acquisition card, whose maximum sample rate is 100 MS/s. The voltage signals were transmitted into the customized test and measurement system based on LabVIEW, in which the models and corresponding algorithms were integrated.
Fig. 5. Relationships between the coefficients of the BNLM and the current range. (Solid line) Coefficients of the ascending branch under the frequency of 1 kHz. (Dashed line) Descending branch.
Fig. 6. Comparison on nonlinear error between the linear model and the basic hysteretic nonlinear model. The conventional linear model and the basic nonlinear hysteretic model are abbreviated as CLM and BNLM, respectively.
V. E XPERIMENTAL R ESULTS A. BNLM The basic nonlinear hysteretic model is based on rational functions, whose parameters are obtained by fitting the branches of the output characteristics of a GMR sensor or a TMR sensor. The relationships between the parameters of the BNLMs and the range of the measured currents were studied, and a part of the results of Sensor A is shown in Fig. 5. It can be observed that the linear functions represent the relationships
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Fig. 7.
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Comparison among the conditioned output characteristics based on proposed models. (a) CLM. (b) BNLM. (c) ANLM.
TABLE I L INEARITIES BASED ON D IFFERENT M ODELS AND A LGORITHMS (TMR)
well, which is in good agreement with the theoretical derivation in (13). Under the condition of steady-state currents, the nonlinear error of the two sensors based on the BNLMs and that based on the CLMs are compared in Fig. 6. It shows that the nonlinear error greatly decreases, and that the values are limited to around 1% at the frequencies of 500 Hz for the TMR sensor and 10 kHz for the GMR sensor. B. ANLM The BNLM was verified to be efficient to linearize a shuttleshaped output characteristic of the GMR sensor. However, when the frequency further increases, a large nonlinear error occurs around the extremes of the measured currents, resulting from the transition between adjacent branches and the mismatch of change rates between the rational linear functions and the hysteresis loop. The advanced nonlinear model and the corresponding algorithm are proposed to address these issues. The conditioned output characteristics of Sensor A at the specific frequency of 2 kHz are compared in Fig. 7. The hysteretic characteristic is evident in the linear model, and the measurement error is obvious around the terminals of the output characteristic based on the BNLM. With the special treatment around the extremes of the currents, the output characteristic based on the advanced hysteretic model is successfully linearized. Under the condition of steady-state currents, the respective linearities of the TMR and the GMR sensors based on different models and algorithms are compared in Tables I and II. It indicates that the GMR sensor experiences an 89.5% improvement in the linearity at 50 kHz, and that a 91% decrease in nonlinear error is achieved by the proposed ANLM at the cutoff frequency fc of the TMR sensor (i.e., 5 kHz). C. Dynamic Performance The dynamic performance of the proposed advanced hysteretic nonlinear model was evaluated by applying a 2-kHz
TABLE II L INEARITIES BASED ON D IFFERENT M ODELS AND A LGORITHMS (GMR)
sinusoidal current signal with an oscillating amplitude, as shown in Fig. 8(a). By adopting the appropriate N with relation to the current frequency, the algorithm of the ANLM is also applicable to the measurement of dynamic currents. The conditioned output characteristic of the Sensor A based on the CLM, and that based on the proposed hysteretic nonlinear model are compared in Fig. 8(b). The maximum nonlinear error based on the hysteretic model was 1.48%, whereas that based on the linear model was above 8%. The 80% improvement in the linearity under the condition of dynamic currents confirms the feasibility and merits of the proposed hysteretic model and the corresponding algorithm. VI. C OMPARISON AMONG D IFFERENT M ODELS AND A LGORITHMS Four categories of models are analyzed, i.e., the CLM, the proposed BNLM and ANLM, and LMFC. The slope and the intercept of the CLM need to be extracted in advance from the output characteristics, and their values vary with the frequency of currents. Although high accuracy can be achieved in low-frequency situations, distortion and large nonlinear error occur when the hysteresis is noticeable, which limits the applicable frequency range to around 0.02fc . The two hysteretic nonlinear models are proposed to address the problem of large nonlinear error. Different from the existing hysteretic models that utilize complicated functions, the proposed models make use of corresponding rational linear functions for the ascending and descending branches. By detecting the extreme values of the output voltages, a designated rational function with proper coefficients is employed to calculate the currents, extending the applied frequency range of the GMR sensors to around 0.1fc . When the frequency further increases, the rational linear function cannot match the change rate of the actual hysteretic output characteristics, and large nonlinear error occurs around the extremes of the currents. In view of this,
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TABLE III P ERFORMANCE OF P ROPOSED M ODELS AND O PTIMIZED S ELECTION
VII. C ONCLUSION
Fig. 8. Comparison of dynamic performance between the CLM and the ANLM. (a) Applied currents with an oscillating amplitude. (b) conditioned output characteristics based on CLM and ANLM.
the advanced hysteretic model and algorithm are proposed to control the terminal error by weighting the currents calculated by the functions of both branches. Such modification improves the accuracy of the GMR sensors and extends the applied frequency range to around the cutoff frequency of the sensor hardware. If it is required to measure currents at frequencies much higher than fc , the digital frequency compensation seems to be the only solution. By correcting the lagged phase and the diminished magnitude, the current values can be calculated by a simple linear function with a constant slope and intercept. High accuracy can be achieved in the entire frequency range with such a method, but the cost for data processing is considerable. To summarize, the applied frequency range can be extended at the cost of complicating the models. Despite the limited bandwidth, the linear models are the most popular choices because of their simple algorithms. The proposed nonlinear hysteretic models with the basic and the advanced algorithms largely improve the linearity of the GMR sensor at an acceptable cost for current calculation. Although the linear model combined with digital compensation algorithms shows good applicability in the entire frequency range, the considerable costs and difficulty in implementation decrease its feasibility and popularity in real-time monitoring situations, where huge amounts of data need to be sampled at high rates. The performance of the analyzed models in different applied frequency ranges and the best selection in each area are summarized in Table III.
The hysteresis imposes great restrictions on the applicable frequency range of GMR sensors, and it is regarded as one of the most challenging issues in applying GMR sensors to the measurement of high-frequency currents. Although the linear models are simple and easy to implement, the single-valued feature cannot deal with the hysteretic output characteristics; therefore, large nonlinear error would occur in high-frequency applications. In this paper, hysteretic modeling of output characteristics of GMR current sensors is studied, and the corresponding algorithms are put forward to verify the feasibility and performance of the proposed models. The proposed novel hysteretic models based on rational linear functions are proved capable of successfully linearizing the output characteristics in both steady-state and dynamic-state applications. The basic hysteretic model extended the applicable frequency range to five times as large as the original value based on the linear models, and a 50 times improvement is achieved by the advanced hysteretic model at acceptable costs. The advanced hysteretic algorithm reduces nonlinear error by 91% under the condition of static-state currents compared with the CLM, and the improvement under the condition of dynamic currents is above 80%. Optimized selection of adopted models and algorithms is conducted on the basis of the frequency of currents. R EFERENCES [1] C. Younghoon, T. LaBella, and L. Jih-Sheng, “A three-phase current reconstruction strategy with online current offset compensation using a single current sensor,” IEEE Trans. Ind. Electron., vol. 59, no. 7, pp. 2924–2933, Jul. 2012. [2] B. Metidji, N. Taib, L. Baghli, T. Rekioua, and S. Bacha, “Phase current reconstruction using a single current sensor of three-phase ac motors fed by SVM-controlled direct matrix converters,” IEEE Trans. Ind. Electron., vol. 60, no. 12, pp. 5497–5505, Dec. 2013. [3] W. Fengjiang, S. Bo, Z. Ke, and S. Li, “Analysis and solution of current zero-crossing distortion with unipolar hysteresis current control in gridconnected inverter,” IEEE Trans. Ind. Electron., vol. 60, no. 10, pp. 4450– 4457, Oct. 2013. [4] M. Khazraei and M. Ferdowsi, “Modeling and analysis of projected cross point control—A new current-mode-control approach,” IEEE Trans. Ind. Electron., vol. 60, no. 8, pp. 3272–3282, Aug. 2013. [5] X. Changliang, X. Youwen, C. Wei, and S. Tingna, “Torque ripple reduction in brushless dc drives based on reference current optimization using integral variable structure control,” IEEE Trans. Ind. Electron., vol. 61, no. 2, pp. 738–752, Feb. 2014. [6] Z. Hongzhong and H. Fujimoto, “Overcoming current quantization effects for precise current control by combining dithering techniques and Kalman filter,” in Proc. IEEE IECON, Oct. 2012, pp. 3826–3831.
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Jinchi Han (S’12) was born in Liaoyang, China, in 1990. He received the B.S. degree in electrical engineering from the Department of Electrical Engineering, Tsinghua University, Beijing, China, in July 2013, where he is currently working toward the M.S. degree. His research interests include giant magnetoresistive and tunneling MR current sensors, magnetoelectric materials, and energy harvesting devices.
Jun Hu (M’10) was born in Ningbo City, China, in 1976. He received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from the Department of Electrical Engineering, Tsinghua University, Beijing, China, in July 1998, July 2000, and July 2008, respectively. He is currently an Associate Professor in the Department of Electrical Engineering, Tsinghua University. His research fields include overvoltage analysis in power systems, dielectric materials, and surge arrester technology.
Yong Ouyang (S’13) received the B.S degree from the Department of Electrical Engineering, Tsinghua University, Beijing, China, in July 2011. He is currently working toward the Ph.D. degree in the Institute of High Voltage and Insulation Technology, Tsinghua University. His research interests include electromagnetic compatibility, magnetoresistance materials, and current sensors applied to the smart grid.
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Shan X. Wang (M’88–SM’06–F’09) received the B.S. degree in physics from the University of Science and Technology of China, Hefei, China, in 1986, the M.S. degree in physics from Iowa State University, Ames, IA, USA, in 1988, and the Ph.D. degree in electrical and computer engineering from Carnegie Mellon University, Pittsburgh, PA, USA, in 1993. He currently serves as the Director of the Stanford Center for Magnetic Nanotechnology and a Professor of Materials Science and Engineering, jointly of Electrical Engineering, and by courtesy, a Professor of Radiology with Stanford University, Stanford, CA, USA. He is a Co-Principal Investigator of the Stanford-led Center for Cancer Nanotechnology Excellence and Translation (CCNE-T). His research interests lie in magnetic nanotechnologies and information storage in general and include magnetic biochips, in vitro diagnostics, cell sorting, magnetic nanoparticles, nanopatterning, spin electronic materials and sensors, magnetic inductive heads, and magnetic integrated inductors and transformers.
Jinliang He (M’02–SM’02–F’08) was born in Changsha, China, in 1966. He received the B.Sc. degree from Wuhan University of Hydraulic and Electrical Engineering, Wuhan, China, the M.Sc. degree from Chongqing University, Chongqing, China, and the Ph.D. degree from Tsinghua University, Beijing, China, all in electrical engineering, in 1988, 1991, and 1994, respectively. In 1994, he became a Lecturer and in 1996, an Associate Professor with the Department of Electrical Engineering, Tsinghua University. From April 1997 to April 1998, he was a Visiting Scientist with the Korea Electrotechnology Research Institute, Changwon, Korea, involved in research on metal oxide varistors and high-voltage polymeric metal oxide surge arresters. He is currently the Chair of the High-Voltage Research Institute, Tsinghua University. His research interests include overvoltages and electromagnetic compatibility in power systems and electronic systems, lightning protection, grounding technology, power apparatus, and dielectric material.
