1. INTRODUCTION

With the expansion of the modern power grid, the measurement and control of power systems require current transformers with better performance. A conventional electromagnetic current transformer is known to have some drawbacks, such as insulation difficulty and poor dynamic performance [1–3]. Thus, it is difficult to meet the demands for further development of power grids. Based on the Faraday effect [4,5], the fiber optic current transformer (FOCT) has outstanding advantages, including small size, good insulation properties, and no magnetic saturation [6–9]. With further performance improvements, FOCTs can be gradually adopted in large-scale practical applications.

Many researchers have extensively studied various aspects of the FOCT. According to [10], the effective Verdet constant is linear with the square of the optical frequency. An adjustment of the input polarization to the optimum state can minimize the negative effects of linear birefringence. A distributed parameter model describing the FOCT sensing unit in combination with the photoelectric conversion model was developed and was used to describe the open loop mechanism [11]. The noise characteristics of FOCT were also analyzed, in particular, the influence of noise on the sensing output [12]. In [13], the temperature-induced linear birefringence was explained considering the fiber bending properties. In this previous report, temperature changes caused an asymmetric stress on the fiber cross section. At present, the actual operation precision is the main obstacle that restricts further applications of FOCTs. Therefore, based on existing studies, it is very important to analyze the characteristics of the FOCT error and to propose appropriate solutions for improving the accuracy of measurements and optimizing performance.

In this paper, the Jones matrix method is used to analyze the optical path. Error models that take into account the extinction ratio and the polarization direction deviation are established to study the effects of polarization on the FOCT accuracy. The model of the temperature error is combined with changes in the Verdet constant and linear birefringence. Considering the class 0.2 standard, the effects of fiber parameters are analyzed in the temperature range required for high precision, including intrinsic birefringence, bending radius, and the number of winding turns. An experimental platform is constructed to confirm the above analysis.

2. ANALYSIS OF THE POLARIZATION ERROR CHARACTERISTICS

A. Analysis of the Error Caused by the Polarizer and Analyzer

The FOCT configuration based on polarization detection is illustrated in Fig. 1. The subsequent analysis is based on this structure.

 

Fig. 1. FOCT configuration based on polarization detection.

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In practical applications, the performance of optical devices is not ideal, which will lead to an output error. The extinction ratio $e$ can be used to characterize the polarization performance of the polarizer and the analyzer,

To simplify the calculation, it is assumed that the polarization direction of the polarizer coincides with the $x$ axis. Let the optical vector from the polarizer to the sensing fiber be equal to $E_{\text{in}}$. The polarizer extinction ratio $e_1$ is not infinite, and therefore, the component $E_y$ of $E_{\text{in}}$ is not zero. $E_{\text{in}}$ can be expressed as

According to the above analysis, when $e_2$ is $+\infty$, the relationships between $e_1$ and $ϵ$ for linear birefringence $\delta =0.00$, 0.05, and 0.10 rad are shown in Fig. 2.

 

Fig. 2. Effects of the polarizer extinction ratio on the FOCT error.

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For each curve in Fig. 2, the ratio error decreases as the polarizer extinction ratio increases. Improved polarization properties lead to higher precision. The curve slope gradually decreases, and at $e_1>45\mathrm{dB}$ it tends to zero. This indicates that when the polarizer extinction ratio reaches 45 dB, it has a poor effect on accuracy. By comparing several curves, we can infer that a greater linear birefringence leads to a longer distance between the curve and the $e_1$ axis, inducing lower overall accuracy of the FOCT.

The relationships between $e_2$ and $ϵ$ for linear birefringence $\delta =0.00$, 0.05, and 0.10 rad are shown in Fig. 3, when $e_1$ is $+\infty$.

 

Fig. 3. Effects of the analyzer extinction ratio on the FOCT error.

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Figure 3 shows that the error decreases as the analyzer extinction ratio $e_2$ increases, and the curve slope gradually decreases. When $e_2>30\mathrm{dB}$, the curve is nearly parallel to the $e_2$ axis. Thus, the analyzer polarization properties have a negligible effect on the output precision when $e_2$ reaches 30 dB.

The total slope of the curve in Fig. 2 is larger than that in Fig. 3, suggesting that the polarizer extinction ratio has a greater influence on the FOCT ratio error. For example, a comparison of the results at $\delta$ of 0.05 rad shows that for $e_1=20\mathrm{dB}$, the error exceeds 2%; however, for $e_2=20\mathrm{dB}$, the error is less than 0.1%. For further contrast, the error is shown in Fig. 4 as a function of $\delta$ for different values of $e_1$ and $e_2$.

 

Fig. 4. Relationship between $ϵ$ and $\delta$ for different $e_1$ and $e_2$.

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Regarding the class 0.2 standard of the power transformer, when $e_1=30\mathrm{dB}$ and $e_2=20\mathrm{dB}$, the error still exceeds 0.2% in the absence of linear birefringence (i.e., $\delta =0$). The error is slightly reduced when $e_1$ remains at 30 dB and $e_2$ increases to 30 dB. Furthermore, when $e_2$ remains at 20 dB and $e_1$ increases to 40 dB, $ϵ$ decreases noticeably and is within 2% when the linear birefringence ranges from 0 to 0.05 rad. Consequently, the overall performance of the FOCT is limited by the polarizer and analyzer, while the polarizer has a greater effect on the output accuracy.

B. Error Analysis of the Polarization Direction Deviation

In a practical installation, the polarization direction angle between the polarizer and analyzer may deviate from 45°, thereby inducing the signal error. Assuming that the polarizer and analyzer are ideal, the optical vector $E_{\text{in}}$ incident on optical fiber can be written as

 

Fig. 5. Effects of the polarization direction deviation on the FOCT error.

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It can be deduced from Fig. 5 that a greater absolute value of $\mathrm{\Delta }$ results in a greater output error, thus manifesting a lower FOCT accuracy. Each curve is symmetric about the line $\mathrm{\Delta }=0$, which indicates that the effects of deviations along different directions on the error are the same, provided that their deviation magnitudes are equal. In the case of low linear birefringence, the FOCT precision can meet the standard of class 0.2 when $\mathrm{\Delta }$ is within 3°. Nevertheless, it is still difficult to reduce the error to the point where the system satisfies the class 0.2 precision for high linear birefringence, even if the direction deviation is very small. Accordingly, it is necessary to reduce the polarization direction deviation based on the reduction of the linear birefringence in order to achieve the desired accuracy.

C. Optimization Methods for Reducing the Polarization Error

The extinction ratio of linear polarization devices on the market can reach about 40 dB, which basically meets the performance requirements of the analyzer. However, the polarizer properties have a greater influence on the FOCT performance, so other measures must be taken to improve the degree of polarization of the incident light, including the application of a laser with a high degree of polarization and the use of multistage polarizers with a corresponding collimator.

According to the error analysis of the FOCT, a slight deviation of the polarization direction is allowed. However, taking into account the comprehensive effects of various factors, an analyzer with high adjustment accuracy should be applied to bring the polarization angle between the polarizer and analyzer close to 45° and reduce the actual deviation.

3. ERROR CHARACTERISTIC ANALYSIS CAUSED BY TEMPERATURE DRIFT

A. Model of the FOCT Temperature Characteristics

The influence of environmental temperature changes on the sensing output primarily includes two aspects. First, the Verdet constant varies with temperature [15],

Second, the installation of the sensing ring inevitably causes the fiber to bend. A temperature change leads to a stress difference in the fiber cross section [16], resulting in linear birefringence. The temperature-induced linear birefringence coefficient $M$ is defined as the linear birefringence per unit length due to temperature changes,

Considering the Verdet constant and the linear birefringence induced by temperature, the FOCT output signal can be written as

The FOCT is calibrated at temperature $T_0$. The coefficient correction is applied such that the ratio error is zero at $T=T_0$. When the temperature changes from $T_0$ to $T$, the FOCT ratio error can be written as

B. Simulation of the FOCT Temperature Characteristics

When $N=30$ and $R=10\mathrm{cm}$, the relationships between $ϵ$ and $T$ can be obtained at ${\delta }_s$ at 0.000, 0.002, and 0.004 rad/m, as shown in Fig. 6(a). At ${\delta }_s=0.002\mathrm{rad}/\mathrm{m}$ and $N=30$, the relationships between the ratio error $ϵ$ and temperature $T$ are shown in Fig. 6(b) for the bending radius $R=10$, 15, and 20 cm. When ${\delta }_s=0.002\mathrm{rad}/\mathrm{m}$ and $R=10\mathrm{cm}$, the relationships between $ϵ$ and $T$ can be analyzed for different numbers $N$ of winding turns, namely, $N$ equal to 25, 30, and 35, as shown in Fig. 6(c).

 

Fig. 6. (a) Effects of the intrinsic linear birefringence on the relationship between $ϵ$ and $T$. (b) Effects of the bending radius on the relationship between $ϵ$ and $T$. (c) Effects of the number of turns on the relationship between $ϵ$ and $T$.

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Figure 6(a) shows that a larger intrinsic linear birefringence per unit length results in a greater curve slope and causes the temperature to have a greater effect on the FOCT accuracy. Overall, the intrinsic linear birefringence has little effect on the FOCT temperature characteristics. When ${\delta }_s$ is 0.000, 0.002, and 0.004 rad/m, the measurement accuracy at temperatures ranging from 5°C to 40°C can satisfy the standard of class 0.2 current transformers.

Figure 6(b) shows that when $R$ is 10 cm, the temperature range satisfying the accuracy standard of class 0.2 is from 10°C to 40°C. As the bending radius of the sensing fiber increases, the curve slope decreases and the temperature range meeting the accuracy requirements gradually increases. At $R=20\mathrm{cm}$, the FOCT ratio error is within 0.2% at temperatures between $-40°\mathrm{C}$ and 60°C. In comparison, the bending radius has a greater influence on the FOCT temperature characteristics, mainly due to the following two aspects. First, the increased bending radius directly reduces the temperature-induced linear birefringence coefficient, thereby decreasing the temperature-induced birefringence. In contrast, the bending-induced linear birefringence is the main part of the residual linear birefringence in practice. As a result, a larger bending radius results in a smaller bending-induced birefringence, thereby effectively reducing the residual linear birefringence.

According to Fig. 6(c), the FOCT accuracy can meet the class 0.2 standard at temperatures between $-15°\mathrm{C}$ and 55°C at $N=25$. As the number of winding turns increases, the sensing length in the optical path becomes longer, which inevitably increases the cumulative effect of linear birefringence. The cumulative effect leads to an increase in the slope and a decrease in the temperature range that satisfies the measurement accuracy. When $N$ reaches 35, the temperature range required for errors to stay within 0.2% is between 15 °C and 35 °C.

C. Optimization Methods for Reducing the Temperature Drift Error

To reduce the FOCT error and increase the usable temperature range, the following optimization methods can be applied:

4. EXPERIMENTS

A. Experimental Platform

The experimental platform of the FOCT was constructed according to the schematic in Fig. 1. A photograph of this system is shown in Fig. 7. Multi-turn wires passing through the sensing ring were used to simulate high current in the power system. The number of turns of the wire across the fiber was set to 300, constituting a double solenoid structure with the multi-turn optical fiber. When the wire current was 3 A, the equivalent measured current was 900 A. The number of wire turns across the standard current sensor was set to 30, so the equivalent standard current of the FOCT was obtained by amplifying the current value by a factor of 10.

 

Fig. 7. Experimental FOCT platform: 1) Compact laser diode controller and laser diode; 2) polarizer; 3) standard current sensor with high precision; 4) sensing optical fiber (between two splints); 5) analyzer; 6) photodetector.

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An LPS-PM1310-FC 1310 nm laser diode with the extinction ratio of 28.2 dB was driven by a compact laser diode controller. The extinction ratio of the polarizer used in the experiment reached 45 dB. Spun Hi-Bi Fiber (SHB1250) was used as the sensing fiber whose internal birefringence could effectively resist linear birefringence caused by some external factors [17]. The bending radius of the sensing fiber was 12 cm, and the total number of turns was set to 30. A manual analyzer with the extinction ratio of 40 dB was used for polarization detection during rotation from 0° to 360°. The electrical signal through the photodetector contains various noises. A linear Kalman filter was used to filter the noise and perform the subsequent signal processing, thus yielding the real-time data for the current.

B. Experimental Tests

1. Current Test for Different Extinction Ratios of the Input Light

To analyze the effects of the extinction ratio on the output signal, two groups of output data were measured at different currents. In the first group, the laser was connected to the polarizer and the sensing fiber. According to the device parameters, the extinction ratio of the incident light reached 45 dB. In the second group, the laser was directly connected to the sensing fiber, and the corresponding extinction ratio was 28.2 dB. The AC frequency was adjusted to 50 Hz, and the current was increased from small to large values. We obtained standard and measured values of the current in two groups of experiments at different measurement points, as shown in Fig. 8.

 

Fig. 8. Standard and measured current values for different extinction ratios.

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It can be deduced from Fig. 8 that in the current range of 0–1500 A the linearity of the measured current is consistent with the linearity of the standard current, and the measured value of the current is slightly lower than the standard value of the current. These results indicate that the FOCT experimental system is affected by the devices themselves as well as various external factors, but the experimental platform basically meets the measurement requirements. Comparing the measured current values with extinction ratios of 45 dB and 28.2 dB, one can see that the measurement results have smaller errors, higher accuracy, and better linearity when the extinction ratio is 45 dB. Since there was no specific numerical value for the fiber linear birefringence in the experiment, the first group of data was divided by the second group of data to compare the theoretical and experimental results,

2. Test for Different Deviations in the Polarization Direction

The FOCT platform in Fig. 7 was used for experiments because of its high accuracy at an extinction ratio of 45 dB. Considering the good linearity of the experimental system and the uncertainty in the numerical value of the linear birefringence of the fiber, the measured current is directly corrected by the standard current to set the experimental error to zero when the polarization direction deviation is zero. The polarization direction of the analyzer was rotated clockwise (“+”) and counterclockwise (“−”). The output data were recorded each time after adjustment by 1°, and the maximum deviation of the direction reached 6°. A comparison between the corresponding theoretical and experimental ratio errors is shown in Fig. 9.

 

Fig. 9. Comparison of theoretical and experimental ratio errors induced by deviations of the polarization direction.

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Figure 9 shows that a larger deviation of the polarization direction increases the experimental ratio error. When the deviation of the polarization direction is 6°, the absolute value of ratio error reaches 1.35%. When the absolute values of the direction deviation are the same, the effects of the clockwise and counterclockwise deviation on the error are basically the same. The changing trend of the experimental error curve is consistent with the theoretical analysis.

3. Test for Different Temperature Conditions

The sensing system was placed in a temperature control box, and the output test was performed at different temperatures. First, the AC power supply was adjusted to set the equivalent standard current of 900 A. The measured current was corrected to make the ratio error equal to zero at 25°C, and then the current tests were performed at temperatures from $-10°\mathrm{C}$ to 60°C. The sensing system was kept in the temperature control box for 1 h every 5°C, ensuring that the sensing ring can fully contact the environment. The output currents and their correction values were recorded. The theoretical and experimental ratio error curves induced by temperature were obtained, as shown in Fig. 10.

 

Fig. 10. Curves of theoretical and experimental ratio errors induced by temperature.

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According to Fig. 10, when the temperature fluctuates between $-10°\mathrm{C}$ and 60°C, the experimental ratio error varies from $-0.648\%$ to 0.673%. A greater temperature fluctuation exacerbates the FOCT error drift. When the temperature drift reaches a certain level, the precision cannot meet the class 0.2 standard. There is a difference between the theoretical and experimental results due to uneven heating caused by the difference in heat transfer with the environment when the fiber is wound from the inside to the outside. However, the overall trend agrees with the theoretical results, confirming the reliability of the analysis of temperature errors.

5. CONCLUSIONS

In this paper, theoretical analysis and experimental verification of the characteristics of FOCT polarization error and temperature error were investigated. Based on the Jones matrix, FOCT error models were established, taking into account the extinction ratio of polarization devices and deviations in the polarization direction. A greater extinction ratio of the polarizer and analyzer results in a lower output error, and the polarizer performance has a greater effect on FOCT accuracy. A larger absolute value of the deviation of the polarization direction leads to a larger FOCT error. The accuracy can satisfy the standard of class 0.2 when the deviation of the polarization direction is within 3° at low linear birefringence.

The FOCT temperature error model was built considering changes in the Verdet constant and linear birefringence induced by temperature fluctuations. A greater intrinsic linear birefringence, a smaller bending radius, and more winding turns of the sensing optical fiber will cause the temperature-induced error curve to have a larger slope, thus reducing the operating temperature range meeting the accuracy standard of class 0.2.

In experiments, the measurement results have a lower error and better linearity when the extinction ratio of the polarized light incident on the sensing fiber is higher. Tests for different polarization direction deviations indicate that a larger deviation of the polarization direction (its absolute value) leads to a greater output error. Deviations in different directions have the same effect on the measurement precision. When the temperature fluctuates between $-10°\mathrm{C}$ and 60°C, the experimental ratio error varies from $-0.648\%$ to 0.673%. Larger temperature fluctuations exacerbate the FOCT error drift.