Thermodynamic Modeling and Analysis of an Optical Electric-Field Sensor

Transcription

1 Sensors 15, 15, ; doi:1.9/s Article OPEN ACCESS sensors ISSN Thermodynamic Modeling and Analysis of an Otical Electric-Field Sensor Xia Xiao *, Yan Xu and Zexing Dong State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and Technology, 17 Luoyu Road, Wuhan 474, China; s: (Y.X.); (Z.D.) * Author to whom corresondence should be addressed; xiaoxiahust@16.com; Tel./Fax: Academic Editor: Lorenzo Pavesi Received: 4 December 14 / Acceted: 18 March 15 / Published: 4 March 15 Abstract: The stability of the otical electric field sensor (OEFS) in actual oeration is affected by environmental factors such as temerature and SF6 (sulfur hexafluoride). To analyze the oerational environment arameters affecting the otical roerties of crystals, a thermodynamic model of the OEFS in which the otical roerties of the crystal are changed by the first-order effects and the second-order effects was established. The intensity arameters such as electric, stress and temerature fields were introduced. The theoretical analysis results show that under temerature, stress and electric field conditions, the otical roerties of the sensing crystals are no longer changed only by the electro-otic effect, but also by the temerature and the stress fields. Further synthesis suggests the exected otical roerty changes under the effect of the environment fields. OEFS tests show that the accuracy of OEFS is deendent on temerature with a ratio error of.8%~1.5% in the temerature range from 5 C to +4 C. Keywords: otical electric-field sensor; thermodynamic model; sensor crystal; otical roerties; stability

2 Sensors 15, Introduction Passive electronic voltage transformers (PEVTs) based on the rincile of the electro-otic effect have bright rosects in high voltage grade alications because of a series of advantages they dislay such as their wide measuring range, good frequency characteristics, a simle insulation structure and higher safety [1 ]. However, at resent PEVTs can t be widely alied in ower systems, mainly because the long term running stability of the otical electric field sensor (OEFS) cannot meet the ower grid requirements. There have been many studies aimed at imroving the stability of OEFS, such as emloying software comensation or using a double otical ath structure to imrove the temerature erformance of OEFS [4 9]. The temerature effect on OEFS shows that macroscoically there is a certain relationshi between the sensor outut and the temerature. In [9], Filiov identified in the exeriments the temerature characteristics of each OEFS in actual oeration, and obtained the oerational temerature of the sensor. Thus the deendence of OEFS on temerature could be comensated to some extent. In the 199s, Lee imroved the temerature stability of OEFS from ±7.% to ±.75% within C~65 C with dual light ath comensation [1]. In recent literature, the accuracy of OEFS was reorted to be imroved to ±.5% within 4 C~+6 C by introducing a reference voltage for comarison [11]. In actual oeration the electro-otic sensing crystal of the OEFS interacts not only with the electric field to be tested, but also with the temerature field and stress field. The electric, temerature and stress fields can change the otical roerties of electro-otic crystals. The relationshis between the various arameters of the crystal system are shown in Figure 1. Therefore, in any further study of the stability of PEVT, the sensor must be seen as a system that can resond in many ways to a series of environmental fields. Figure 1. The relationshi between various arameters of a crystal system.. The Princile and Scheme of OEFS OEFS based on bismuth germinate (BGO) crystals are a common research subject [1,1], and their basic rincile of oerations is known as the linear electro-otic effect. The transverse modulation OEFS rincile diagram is shown in Figure. The crystal refractive index changes with the electric field E,

3 Sensors 15, which is called the linear electro-otic effect. The variable Δ β = β β of the reverse dielectric tensor 1 β ( β = ) is usually used to describe the changes of the otical roerties of crystals, where, β and n β are the reverse dielectric tensor with electric field and without electric field, resectively, and n is the refractive index. α Figure. The rincile of OEFS. Without an electric field, the otical roerties of a BGO crystal could be exressed with the refractive index ellisoid equation as follows [14]: β + β + β = (1) x1 x x 1 With an electric field, the inverse dielectric tensor β would be changed as: Δ β = γ E ij ijk () In Equation (), E is the electric field, γ ijk is the linear electro-otic coefficient matrix, and: γ ijk = γ 41 γ 41 γ 41 In the ideal case, when the light direction is along the 1 1 direction of the BGO crystal and the electric field E k is alied erendicular to the 1 surface of the crystal, the birefringence hase retardation caused by electro-otic effect is: δ πl n γ E λ = 41 k () where λ is the wave length of inut light, γ 41 is the electro-otic coefficient of the crystal, l is the length of the light ath through the crystal. From Equation (), the corresonding alied electric field can be obtained by measuring the birefringence hase retardation. In OEFS based on the transverse modulation structure, the term E k in Equation () is the electric field where the light asses through the crystal, which is generated by the aroriate electrode structure. As

4 Sensors 15, shown in Figure, a measured voltage is alied between the uer electrode and the ground electrode, the middle of which is suorted by the casing. The main insulation, sensing crystals and the air ga make u the medium to roduce the electric field E k to be tested. To ensure a sufficient insulation intensity, the sensor is laced in a sulfur hexafluoride (SF6) gas environment. Uer electrode Main insulation casing Air ga BGO Ground LED PIN1 PIN. Thermodynamic Modeling of OEFS Figure. Structural diagram of a sensor. The hysical roerties of crystals describe the relationshi between the indeendent variable and the deendent variable. The hysical basis of OEFS to measure the electric field is the linear electro-otic effect, that is, the change in the otical roerties of the sensing crystal with an electric field. However, in ractical cases, what affects the crystal otical roerties is not only the electric field, but also the temerature field and stress field. OEFS oerates under a gas ressure and temerature environment, and OEFS works with the surrounding environment under a thermal equilibrium state. The relationshi between the environmental arameters and the hysical roerties of the crystal indicated in the thermodynamic model is shown in Figure 1. The intensity arameters such as temerature θ, stress T and electric field intensity E show an ' effect on the crystal; extensive arameters such as the entroy S, the electric olarization P and strain S show the deendent variables caused by the effect of the crystal. The BGO crystals used for the OEFS sensor belong to the cubic crystal system with no yroelectric effect. In Figure 1, the effects marked with a thick arrow include the main effects, which describe the relationshi between the same kind of intensity arameters and the extensive arameters, such as thermo main effect, electrical rimary effect and mechanical effect. The relationshi between the indeendent variables and other extensive arameters marked with a thin arrow are the cross effects, such as mechanical effect, electrothermo effect and thermoelastic effect. As discussed above, various hysical roerties of the crystal are intrinsically linked. Under different conditions, the same hysical roerties tend to be different. Therefore, in the study of the crystal roerties, all kinds of ossible effects must be taken into consideration, in articular, the thermodynamic relationshi between the various rocesses. The resent study is focused on dielectric olarization, because the otical roerties of the crystal and its olarization or dielectric roerties have a direct relationshi. In Figure 1, the intensity arameters

5 Sensors 15, marked by the arrow ointing directly to the crystal electric olarization are the electric field, stress field and temerature field. The relationshi among them are first-order effects including the electro-otic effect, elastic-otic effect and thermo-otic effect. There are still some second-order effects. One is the electrostrictive effect generated by the electric field, in which the inverse iezoelectric effect and electrostriction strain caused by the electric field roduce a strain, while the elastic-otic effect by the strain changes the crystal olarizability; the other is the hotoelastic effect induced by the stress with the change of temerature. Therefore, the analysis of crystal otical roerties in the thermodynamics system should be done taking into consideration the interactions of electric field, temerature field and stress field. The electric field causes the first-order electro-otic effect. The temerature field on the one hand causes the thermo-otic effect, and roduces the elastic-otic effect created by the unequal heat within the crystal or temerature mutation as the effect of stress would. Gas ressure would also cause elastic-otic effects in the crystal. Changes of suscetibility state can be exressed with differential relationshi between the deendent variable and indeendent variable as: Pi Pi Pi dpi = ( ) E, θdtkl + ( ) T, θde j + ( ) T, Edθ T E θ (4) kl In Equation (4), the right first item is for the iezoelectric modulus, the second item is for the electric suscetibility and the third item is for the yroelectric coefficient. Integrating the intensity arameter, the function of the crystal olarization intensity as the intensity arameters for a certain value is: ij j θ T, θ T Pi = ijkltkl + εχij Ej + i Δ θ (5) T, θ where χ is the crystal olarizability. θ ijkl is the elastic otic effect coefficient. T i is the yroelectric coefficient. From Equation (5), under the effects of temerature, stress and electric field on the OEFS, the sensing crystal otical roerties change not only with the electro-otic effect, but also with the temerature field and the stress field. 4. Analysis of OEFS Otical Proerties in the Thermodynamic System 4.1. Effect of Temerature on the Otical Proerties of Crystals From Figure 1, the effect of temerature on the otical roerties of BGO includes two arts. One is the olarization caused by the thermo-otic effect, and the other is the olarization caused by the elastic-otic effect The Thermo-Otic Effect on BGO Crystals BGO crystals belong to the cubic system, in which the thermo-otic effect is exressed as [14]: Δβ = b Δθ (6)

6 Sensors 15, In Equation (6), Δ θ is the temerature variable, b is the thermo-otic effect coefficient. Suosing that the coordinate axis is the sindle of the refractive index ellisoid, the thermo-otic coefficient matrix is: b ij b = 11 b 11 b The Elastic-Otic Effect Caused by Thermo Stress in BGO Crystals When the environmental temerature changes in a crystal, the otical roerties of BGO would change with the thermo stress due to the elastic-otic effect. The elastic-otic effect can be exressed as: Δ β = P T (i,j,k,l = 1,,) (7) ijkl kl In Equation (7), P ijkl is the elastic-otic coefficient. Equation (7) could be reresented with comonents as: Δ β = P T ij ijkl kl where P ijkl is a four order tensor. Δ βij is a symmetric tensor. T kl is the symmetric tensor too, so both of the first two subscrits i, j and the latter two subscrits kl, have a symmetric ermutation. The elastic-otic coefficient matrix P ijkl could be simlified as [14]: As shown in Figure, the crystal together with two triangular risms is bonded on the ground electrode. When the environmental temerature changes, the BGO crystal will be deformed because of heat exansion and cold contraction. In the coordinate system as shown in Figure, the BGO crystal could exand freely because of the lack of constraints in the direction arallel to the electric field, while ' the deformation in the light assing direction X will be subject to constraints because the crystal and the triangular rism have different exansion coefficients. According to the thermal stress theory, thermal stress will be roduced. Selecting the coordinate system X 1, X, X as the rincile stress axes, the shear stress comonent is zero. T 1, T and T are the main stresses arallel to X 1, X and X resectively. T 4, T 5 and T 6 are the shear stresses in lane XX 1, XX and XX 1 resectively. Because of the smaller size of BGO crystals and the slower temerature change, the temerature gradient can be regarded as zero. Then: T1 = T, T = T, T = T4 = T5 = T6 =

7 Sensors 15, The Elastic Otic Effect Caused by SF6 Pressure OVS oerates under a SF6 environment and the otical roerties of BGO crystals are changed by hotoelastic effect due to comressive stress of SF6. When SF6 is in equilibrium, in the coordinate system as shown in Figure, P 1, P and P are the main stresses arallel to X 1, X and X resectively. P 4, P 5 and P 6 are the shear stresses in lane XX, 1 XX and XX, 1 resectively. The comressive stress is: P1 = P = P = P, P = P = P = 4.. Effect of All Environment Fields on the Otical Proerties of Crystals In the sensing system shown in Figure 1, an electric field is alied erendicular to the 1 direction of the crystal, then E1 = E =, E = E, and the light ass through the crystal along the 1 1 direction. With the effect of thermo stress caused by the temerature variation Δ θ, and with the effect of SF6 gas comressive stress, the change of crystal otical roerties is synthetical for each effect. That is: Δ β = γijk Ek + bδ θ + mntn + mnpn (8) Substituting the matrix γ ijk, b and mn into Equation (8), then: ( 11 1) T ( ) P+ b11δθ A ( ) T ( + ) P+ b Δθ B ( + ) P+ b Δθ C γ E γ E Δ β = = (9) Combining Equation (1), the refractive index ellisoid equation indicating the otical roerties of crystal is: ' ' ' ( β + A+ γ41esin α) x1 ( β + B γ41ecos α) x + ( β + C) x (1) + [( A+ B)sin α + γ Ecos α] x ' x ' = where x ', x ', x ' 1 is the new coordinate system of the refractive index ellisoid. The coefficient of the item x1' x' should be zero, namely: and so: ( A+ B)sinα + γ Ecosα = (11) γ41e γ41e tan α = = B A ( ) T α is the rotation angle that the new index ellisoid coordinate system rotates around the axis x, and is deendent on the alied electric field and temerature. α =± π /4 is the ideal rotation angle without stress. From Equation (1), under the combination of environmental fields, the refractive index ellisoid rincile axis of the crystal is rotated around the x axis with a rotation angle α, and the change of the three main shafts length is not equal. The birefringence hase retardation due to environmental fields is: (1)

8 Sensors 15, π λ Δ ϕ = n l[ γ41esin α+ ( 11 1) T] (1) For the summation terms in the bracket in Equation (1), the former is the hase retardation caused by the electric field, whereas the latter is the hase retardation caused by thermal stress due to the change of temerature. The hase retardation caused by the thermo-otic effect is zero because of the otical symmetry of BGO crystals. 5. Tests of OEFS The OEFS is fixed inside the insulation sleeve as a test device which is filled with SF6 gas. The test rincile diagram of the sensor system is shown in Figure 4. The ste-u transformer oututs the measured voltage. The voltage is connected to the high voltage electrode of the test device through a wire, making the electric field distribution inside the BGO crystal aroximately equal to the actual runtime distribution. The light outut of the OEFS is converted to an electrical signal which is amlified and then transferred to the merging unit (MU) where the outut of MU is connected to the calibrated channel of the calibrator. In addition, the voltage to be measured is transformed by the standard voltage transformer to small voltage signals which are connected to the standard channel of the calibrator. The ratio error and the hase error of OEFS are calculated in the calibrator. Figure 4. OEFS Tests Schematic Diagram.

9 Sensors 15, The set-u transformer The temerature control box MU the calibrator the standard voltage transformer Figure 5. The Temerature Test Photo of OEFS. The insulating sleeve is laced in the temerature control box. The measured voltage is alied to the high-voltage electrode. A hotograh of the test system is shown in Figure 5. The temerature characteristics of OEFS were obtained. Figure 6 shows the ratio error curve and the hase error curve of OEFS at room temerature within 4 h at the alied voltage u to 1 kv. Figure 7 shows the ratio error curve and the hase error curve of OEFS with temerature at an alied voltage u to 1 kv. From Figure 6, the accuracy of OEFS within 4 h at room temerature is within ±.%. When the range of temerature is much smaller and with slow changes, the stress roduced in the crystal will be less. From Equation (1), the second term with the stress T will be less, so the accuracy of the OEFS can be ket within ±.%. However, when the temerature changes greatly and the rate of change is larger, the thermal stress roduced in the crystal will change with temerature and the accuracy of OEFS will be worse. From Figure 7, the ratio error of OEFS is within.8%~1.5%, and the hase error is within 6 ~+15 with the temerature range from C to +4 C..4 Ratio error (%) Phase error (min) Time (hours) Time (hours) (a) (b) Figure 6. (a) The Ratio Error Curve within 4 h; (b) The Phase Error Curve within 4 h.

10 Sensors 15, the ratio error(%) temerature( ) (a) the hase error temerature( ) (b) Figure 7. (a) The Ratio Error Curve with Temerature; (b) The Phase Error Curve with Temerature. 6. Conclusions The nature of OEFS is that an external electric field changes their otical roerties. The analysis of a thermodynamic system model of OEFS in actual oeration suggests that the external temerature field and ressure field will also change the otical roerties of BGO crystals, thus resulting in instability of OEFS. The tests results show that the accuracy of OEFS at room temerature is within ±.%, but deendent on temerature. Acknowledgments We would like to exress our gratitude for financial from the National Natural Science Foundation of China under Grant Author Contributions Xia Xiao and Yan Xu conceived and designed the study. Zexing Dong erformed the exeriments. Xia Xiao wrote the aer. Yan Xu and Zexing Dong reviewed and edited the manuscrit. All authors read and aroved the manuscrit. Conflicts of Interest The authors declare no conflict of interest. References 1. Laurensse, I.J.; Koreman, C.G.A.; Rutgers, W.R.; van der Wey, A.H. Alications for Otical Current and Voltage Sensors. Sens. Actuators 1989, 17, Bohnert, K.; Gabus, P.; Kostovic, J.; Brandle, H. Otical fiber sensors for the electric ower industry. Ot. Laser. Eng. 5, 4, Kumada, A.; Hidaka, K. Directly High-Voltage Measuring System Based on Pockels Effect. IEEE Trans. Power Deliv. 1, 8, Mitsui, T.; Hosoe, K.; Usami, H.; Miyamoto, S. Develoment of Fiber-otic Voltage and Magnetic-field Sensors. IEEE Trans. Power Deliv. 1987,, 87 9.

11 Sensors 15, Luo, S.; Ye, M.; Xu, Y. Stability Research on Otical Fiber Voltage Transformer. Proc. Chin. Soc. Electr. Eng.,, Gu, P.; Hu, G.; Hua, S. A New Inorganic Scintillation Crystal BGO. J. Funct. Mater. 1994, 5, Williams, P.A.; Rose, A.H.; Lee, K.S.; Conrad, D.C.; Day, G.W.; Hale, P.D. Otical, Thermo-otic, Electro-otic, and Photoelastic Proerties of Bismuth Germanate (BGO). Al. Ot. 1996, 519, Monsef, H.; Ghomian, T. Modified quadrature method for accurate voltage measurement in otical voltage transducer. IEE Proc. Gener. Trans. Distrib. 6, 15, Filiov, V.N.; Starodumov, A.N.; Minkovich, V.P.; Lecona, F.G.P. Fiber Sensor for Simultaneous Measurement of Voltage and Temerature. IEEE Photon. Technol. Lett., 1, Lee, K.S. Electrootic Voltage Sensor: Birefringence effects and Comensation Methods. Al. Ot. 199, 9, Wang, H. Research of Caacitor Divider Otical Voltage. Ph.D. Thesis, Harbin Institute of Technology, Harbin, China, Araia, P.; Daonte, P.; Grimaldi, D.; Michaeli, L. ANN-Based Error Reduction for Exerimentally Modeled Sensors. IEEE Trans. Instrum. Meas., 51,. 1. Xiao, X.; Xu, Y.; Xu, K.; Ye, M. Influence of temerature and stress field on otical voltage sensor. In Proceedings of the 9 Symosium on Photonics and Otoelectronics, Wuhan, China, August 9; Chen, G.; Liao, L.; Hao, W. Fundermentials of Crystal Physics, nd ed.; Science Press: Beijing, China, by the authors; licensee MDPI, Basel, Switzerland. This article is an oen access article distributed under the terms and conditions of the Creative Commons Attribution license (htt://creativecommons.org/licenses/by/4./).