Output error of converse piezoelectric fiber voltage sensor caused by optical fiber factors

Abstract

In fiber voltage sensor, imperfect fiber splicing angle and length difference between the sensing fiber and the compensating fiber usually influence the system output performance. This article established the mathematical model of the system output error using the Jones matrix and analyzed the relationship between the system output error and the above two error factors, respectively. The results show that the angle error of 90° splice has a significant influence on the system output, and the tolerance of angle error varies with different test voltages. When the test voltage is higher than 6 kV, if the expected system error is less than 0.02%, the splicing angle error should be less than 0.15°. Additionally, the length difference also has a significant influence on the system output accuracy. The length difference between sensing fiber and compensating fiber should be less than 1 µm when the test voltage is 110 kV so that the system can meet the accuracy requirements of IEC 0.2s. This research provides a reference for the development of fiber voltage sensor based on the converse piezoelectric effect and also provides a basis for the design of intelligent power detection robot system.

Introduction

Optical fiber sensing has been applied in robot field due to its advantages of long detection distance, short response time, and high resolution. ¹ With the development of smart grid, intelligent inspection robot is widely used in power system in order to obtain information of power equipment accurately and safely. ^{2 –4} However, the voltage is one of the power systems. In order to obtain standardized and electrically isolated voltage signals and ensure the safe operation of the power system, it is necessary to use voltage sensors to collect the voltage signal. Compared with traditional voltage sensor, optical voltage sensor (OVS) has a series of advantages, such as a relative simple setup, intrinsic safety, excellent scale factor, small size, low weight, and, with closed-loop detection, a linear response over an extended dynamic range. ^{5 –8} Up to date, the two kinds OVS, which the Pockels effect and the converse piezoelectric effect, popular with most researchers. ⁹ The OVS based on the Pockels effect is more mature and has trial operation equipment. However, fluctuations in light intensity affect the accuracy of the measurement. The linear response range depends on the half-wave voltage of the Pockels crystal, which limits dynamic range and sensitivity. ¹⁰ Compared with it, there is no discrete optical component in the converse piezoelectric effect OVS, such as slides, analyzers, and bulk crystal, which avoid interfering of other electro-optic effects on the optical path and simplify the structure and manufacturing process. ¹¹ Meantime, reflective reciprocity structure is also highly appreciated by many researchers in the research of OVSs and optical current sensors. ^{12 –14} However, the converse piezoelectric all-fiber voltage sensor fundamentally improves the temperature resistance and long-term operational stability of the sensor system, but there are few reports on the research of it. In 2010, ABB proposed a converse piezoelectric OVS using fiber optic gyro technology, and the piezoelectric deformation of a transducer by an applied alternating current (AC) voltage is transmitted to a birefringent fiber resulting in a period phase shift of the two orthogonal polarizations of the fundamental fiber mode, which has attracted the attention of the industry. ^15,12 There are two-segment fibers in the sensor head, which makes the second compensating fiber serves to balance the different modal group delay and temperature-induced phase changes in the first sensing fiber. The reflective reciprocal optical path can theoretically offset the phase shift caused by the change of the external environment, but it is not easy to realize in actual production. Meanwhile, it is essential to splice some optical devices at a special angle for control the two orthogonal polarization states of polarized light in the optical fiber, but the perfect splice angle is difficult to achieve in practical applications.

In 2014, Jia et al. proposed two quartz crystals scheme to improve the stability and immunity of OVS to external environmental disturbance. ¹⁶ But it is difficult to obtain two quartz crystals with the same parameters to achieve the reciprocity. At the same time, microstructured fiber splicing has also attracted the attention of researchers in the research of optical sensors. ^{17 –20} More researchers proved that special fiber splices such as 45° splice and 90° splice are crucial for all-fiber voltage sensors. ^21,22 However, the influence of them on converse piezoelectric OVS is still an urgent problem to be solved. In this article, the relationship between the output error of reflective all-fiber optic voltage sensor system and the error factor caused by optical fiber is analyzed and deduced using matrix optics method. The research results lay a foundation for the development of all-fiber voltage sensor based on converse piezoelectric effect and also provide a reference and new ideas for the design of intelligent inspection robot system in electric power.

Theory

Principle of reflective OVS based on converse piezoelectric effect

Figure 1 illustrates that the light from the amplified spontaneous emission (ASE) module (1550 nm) passes through a polarizer to form linearly polarized light which has two orthogonal modes after 45° splice between the polarizer and the phase modulator. Polarized light carries phase-modulated signals along the optical path, and its two orthogonal polarization modes change after passing through special fiber splices and optical devices. After arriving at the reflector, it can be back along the original light path. Polarized light passes through the sensing fiber twice, so the phase shift detected by the sensor is twice the phase shift generated by the test voltage. The compensating fiber and sensing fiber are connected by 90°, which can balance the phase change caused by external environment disturbance in the ideal. ^15,12

Figure 1.

Structure diagram of OVS system. OVS: optical voltage sensor.

Figure 2 shows the transformation process of two orthogonal polarization modes. The system is reciprocity, due to the two orthogonal polarization states transmit along the X-axis and the Y-axis of PMF, respectively. Therefore, there is only the phase modulation signal and phase shift caused by the AC voltage in the ideal output interference signal.

Figure 2.

Conversion of two orthogonal polarization modes of polarized light. OVS: optical voltage sensor.

Voltage sensing mechanism

The sensing fiber is wound around the quartz crystal to perceive the piezoelectric deformation of the crystal, which makes the phase shift generated between two orthogonal polarization modes of polarized light. According to Wildermuth et al., ¹² the change in the optical wave phase depends on the change of length, the core refractive index, and the core diameter of the sensing fiber. And the phase shift caused by core diameter change can be ignored because it is 2–3 orders of magnitude smaller than the first two terms in the longitudinal strain of optical fiber. ¹² As a result, the phase shift caused by the AC voltage is

$\phi =\beta \Delta L+L\Delta \beta =\beta L\frac{\Delta L}{L}+L\frac{\partial \beta }{\partial n}\Delta n$

1

where β is the optical propagation constant, $\beta =\frac{2\pi }{\lambda }$ ; λ is the wavelength of light propagating in the optical fiber, $\lambda =\frac{{\lambda }_{\text{vac}}}{n}$ ; λ _vac is the wavelength of light propagation in vacuum and n is the refractive index of the medium of light propagation; Δn is the variation of the refractive index of the optical fiber; L is the length of the optical fiber and ΔL is the variation of the length of the optical fiber.

The variation of the length of the fiber under the deformation of the quartz crystal is

$\Delta L=N\cdot \Delta l=N\cdot \left(-d_{11}{E}_x\pi R\right)$

2

where Δl is the change of length of single turn sensing fiber; N is the number of turns wound on the quartz crystal by the sensing fiber; E_x is the intensity of the electric field, $E_x=\frac{U}{d}$ , U is the test voltage, and d is the distance between the two electrodes, that the height of the quartz crystal; d ₁₁ is the piezoelectric coefficient of the quartz crystal, $d_{11}=-2.31\times {10}^{-12}\hspace{0.17em}\text{C/N}$ .

According to Ulrich, ²³ the variation of the bending birefringence is

$\begin{array}{l}\Delta \beta \text{r}={\displaystyle \underset{0}{\overset{2\pi r}{\int }}}-5.54\times {10}^5{r}^2\left({\kappa }^2-\frac{1}{R^2}\right)\text{d}R\hfill \\ \hfill \\ =-5.54\times {10}^5{r}^2\frac{d_{11}{E}_x\pi }{R}\hfill \end{array}$

3

therefore

$\Delta \beta L=N\cdot \Delta \beta r=N\cdot \left(-5.54\times {10}^5{r}^2\frac{d_{11}{E}_x\pi }{R}\right)$

4

where κ represents the curvature of fiber bending, 2R is the diameter of the quartz crystal, and 2r is the diameter of the optical fiber.

In summary, the relationship between the piezoelectric phase shift caused by the converse piezoelectric effect and the test voltage is

$\phi =N\cdot \left[-d_{11}{E}_x\pi R\left(\beta +5.54\times {10}^5\frac{r^2}{R^2}\right)\right]$

5

It can be seen that there is a linear relationship between the test voltage and phase shift. Therefore, the measurement accuracy of the sensor can be judged by calculating the phase shift.

Mathematical model for optical polarization in OVS

The polarization characteristics of each optical element can be expressed by the Jones matrix. ^10,13 The expression of light emitted by the ASE (1550 nm) source ${\overrightarrow{E}}_{\text{in}}$ is

${\overrightarrow{E}}_{\text{in}}\hspace{0.17em}=\left(\begin{array}{l}{E}_x\hfill \\ E_y\hfill \end{array}\right)$

6

where E_x is propagating along the x-axis and E_y is propagating along the y-axis.

Then, the light from the source transmits through the polarizer to form linearly polarized light, and the expression of the polarizer P is

$\overrightarrow{P}=\overleftarrow{P}=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)$

7

where $\overrightarrow{P}$ and $\overleftarrow{P}$ are the forward and backward light through the polarizer, respectively.

Special fiber splices can effectively change the direction of the orthogonal polarization modes. $M_{{\theta }_i}$ represents the mathematical model of fiber splice, which can be expressed as

$\begin{array}{l}{\overrightarrow{M}}_{{\theta }_i}=\left(\begin{array}{cc}\text{cos}\left({\theta }_i\right)& \text{sin}\left({\theta }_i\right)\\ -\text{sin}\left({\theta }_i\right)& \text{cos}\left({\theta }_i\right)\end{array}\right)\hfill \\ \hfill \\ {\overleftarrow{M}}_{{\theta }_i}=\left(\begin{array}{cc}\text{cos}\left({\theta }_i\right)& -\text{sin}\left({\theta }_i\right)\\ \text{sin}\left({\theta }_i\right)& \text{cos}\left({\theta }_i\right)\end{array}\right)\hfill \end{array}$

8

where ${\overrightarrow{M}}_{{\theta }_i}$ and ${\overleftarrow{M}}_{{\theta }_i}$ are the forward and backward light through the fiber connection, respectively. And ${\theta }_i$ is the splicing angle of the fiber, perfect splicing ${\theta }_1{=45}^{\circ }$ , ${\theta }_2{=45}^{\circ }$ , and ${\theta }_3{=90}^{\circ }$ .

The mathematical expressions of the phase modulator T are

$\overrightarrow{T}=\left(\begin{array}{cc}1& 0\\ 0& e^{j\phi \left(t-\tau \right)}\end{array}\right),\overleftarrow{T}=\left(\begin{array}{cc}1& 0\\ 0& e^{j\phi \left(t\right)}\end{array}\right)$

9

where $\overrightarrow{T}$ and $\overleftarrow{T}$ are the forward and backward light, respectively, and τ represents the transit time of polarized light passing through the phase modulator twice.

The Faraday rotator M _F that makes the two orthogonal modes is interchanged by changing the polarization plane of the polarized light and offset the phase shift caused by the external environment changes, and the expression is

${\overrightarrow{M}}_F={\overleftarrow{M}}_F=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& -1\\ 1& 1\end{array}\right)$

10

where ${\overrightarrow{M}}_F$ and ${\overleftarrow{M}}_F$ express the forward and backward light, respectively.

The expression of polarization-maintaining delay fiber M is

$\overrightarrow{M}=\overleftarrow{M}=\left(\begin{array}{cc}{e}^{i\frac{2\pi }{\lambda }{n}_{x}L}& 0\\ 0& e^{i\frac{2\pi }{\lambda }{n}_{y}L}\end{array}\right)$

11

where $\overrightarrow{M}$ and $\overleftarrow{M}$ are the forward and backward light, respectively; L is the length of polarization-maintaining delay fiber, which is 200 m in the experiment.

The sensing fiber $S_1$ and the compensating fiber $S_2$ are

${\overrightarrow{S}}_1={\overleftarrow{S}}_1=\left(\begin{array}{cc}{e}^{i\left(\frac{2\pi }{\lambda }{n}_x{L}_1+{\phi }_x\right)}& 0\\ 0& e^{i\left(\frac{2\pi }{\lambda }{n}_y{L}_1+{\phi }_y\right)}\end{array}\right)$

12

where ${\overrightarrow{S}}_1$ and ${\overleftarrow{S}}_1$ are the forward and backward light, respectively. $L_1$ is the length of the sensing fiber, ${\phi }_x$ and ${\phi }_y$ represent the phase shift caused by the AC voltage, $\phi ={\phi }_x-{\phi }_y$

${\overrightarrow{S}}_2={\overleftarrow{S}}_2=\left(\begin{array}{cc}{e}^{i\frac{2\pi }{\lambda }{n}_x\left(L_1+\Delta L_2\right)}& 0\\ 0& e^{i\frac{2\pi }{\lambda }{n}_y\left(L_1+\Delta L_2\right)}\end{array}\right)$

13

where ${\overrightarrow{S}}_2$ and ${\overleftarrow{S}}_2$ are the forward and backward light, respectively. $\Delta L_2$ is the length difference between the sensing fiber and the compensating fiber, and the phase shift caused by $\Delta L_2$ is $\Delta \delta =\frac{2\pi }{\lambda }\left(n_x-n_y\right)\Delta L_2$

The expression of the reflector $\overrightarrow{M_R}$ is

${\overrightarrow{M}}_{\text{R}}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

14

As stated in the working principle of the OVS, the output of the OVS can be obtained

$\begin{array}{c}{E}_{\text{out}}=\hspace{0.17em}\overleftarrow{P}{\overleftarrow{M}}_{{\theta }_1}\overleftarrow{T}\overleftarrow{M}{\overleftarrow{M}}_F{\overleftarrow{M}}_{{\theta }_2}{\overleftarrow{S}}_1{\overleftarrow{M}}_{{\theta }_3}{\overleftarrow{S}}_2{\overrightarrow{M}}_R\\ \cdot {\overrightarrow{S}}_2{\overrightarrow{M}}_{{\theta }_3}{\overrightarrow{S}}_1{\overrightarrow{M}}_{{\theta }_2}{\overrightarrow{M}}_F\overrightarrow{M}\overrightarrow{T}{\overrightarrow{M}}_{{\theta }_1}\overrightarrow{P}{\overrightarrow{E}}_{\text{in}}\end{array}$

15

therefore, the expression of output light intensity is

$I_{\text{out}}\hspace{0.17em}=\hspace{0.17em}|E_{\text{out}}{ }^{*}\cdot E_{\text{out}}|$

16

that is

$I_{\text{out}}=\frac{E_x{ }^2}{2}\left\{1-\text{cos}\left[2\phi +\phi \left(t\right)-\phi \left(t-\tau \right)\right]\right\}$

17

where $2\phi$ is the phase shift generated by the test voltage; $\phi \left(t\right)$ and $\phi \left(t-\tau \right)$ are the modulation signal generated by the phase modulator, which makes the cosine signal of the interference light intensity is converted into a sinusoidal signal; $\phi \left(t\right)\hspace{0.17em}-\hspace{0.17em}\phi \left(t-\tau \right)\hspace{0.17em}\text{=}\hspace{0.17em}\pm \frac{\pi }{2}\hspace{0.17em}+\hspace{0.17em}{\phi }_f$ to make OVS work near the zero points and improve the sensitivity of the system; ${\phi }_f$ is the feedback phase shift. Conforming to the closed-loop signal modulation system, ${\phi }_f$ and $2\phi$ are equal in magnitude and opposite in direction, that is, ${\phi }_f+2\phi =0$ .

Therefore, the output sinusoidal signal is

$I_{\text{out}}=E_x{ }^2\text{sin}\left(2\phi +{\phi }_f\right)$

18

and

${\phi }_f=-2\phi$

19

Mathematical model of output error of OVS

The error caused by external interference can be theoretically offset, but it is difficult to achieve in actual because of the imperfect fiber splice and the length difference between the sensing fiber and the compensating fiber. Therefore, it is crucial to analyze these imperfect factors, because it could introduce system output errors and leads to inaccurate measurements. In this article, ${\eta }_x\left(x=1,2,3,4\right)$ represents the output error caused by unsatisfactory ${\theta }_1$ , ${\theta }_2$ , ${\theta }_3$ , and $\Delta L_2$ , respectively.

${\eta }_x=\frac{\Delta I_{\text{out}}^x-\Delta I_{\text{out}}}{\Delta I_{\text{out}}}\times 100\%$

20

where $\Delta I_{\text{out}}^x\left(x=1,2,3,4\right)$ represents the output sinusoidal signal generated by the imperfect ${\theta }_1$ , ${\theta }_2$ , ${\theta }_3$ , and $\Delta L_2$ , respectively.

Firstly, the output error only caused by the 45° ( ${\theta }_1$ ) splice between the polarizer and the phase modulator was analyzed. The output light intensity signal ( $I_{\text{out}}^1$ ) has been calculated, which can be expressed as

$I_{\text{out}}^{\text{1}}=\frac{E_x{ }^2}{2}{{\displaystyle \text{sin}}}^{2}2{\theta }_1\left[1-\text{cos}\left(2\phi +\Delta \phi \right)\right]$

21

where $2\phi$ is the phase shift caused by the AC voltage and $\Delta \phi$ is the modulation signal generated by the phase modulator, $\Delta \phi =\phi \left(t\right)-\phi \left(t-\tau \right)$ .

When a bias phase $\frac{\pi }{2}$ is introduced, formula (21) becomes

$\Delta I_{\text{out}}^{\text{1}}=E_x{ }^2{{\displaystyle \text{sin}}}^{2}2{\theta }_1\text{sin}\left(2\phi +{\phi }_f\right)$

22

and

${\phi }_f=-2\phi$

23

It can be known that the phase shift caused by the 45° splice between the polarizer and the phase modulator is equal the phase shift in ideal, which cannot introduce an output error ( ${\eta }_1=0$ ) from the formula (23). But the intensity becomes smaller, which can affect the sensitivity.

Then, the influence of another 45° ( ${\theta }_2$ ) splice between the pigtail of Faraday rotator and the sensing fiber was analyzed, when other optical components are ideal. The output light intensity signal ( $I_{\text{out}}^{\text{2}}$ ) can be expressed as

$I_{\text{out}}^{\text{2}}=\frac{E_x{ }^2}{8}\left\{\begin{array}{c}{\left(1-\text{sin}2{\theta }_2\right)}^2\left[1-\text{cos}\left(2\phi -\Delta \phi \right)\right]\\ +2{{\displaystyle \text{cos}}}^{2}2{\theta }_2\left(\text{cos}2\phi -\text{cos}\Delta \phi \right)+\\ {\left(1+\text{sin}2{\theta }_2\right)}^2\left[1-\text{cos}\left(2\phi +\Delta \phi \right)\right]\end{array}\right\}$

24

sinusoidal transformation of the equation (24)

$\Delta I_{\text{out}}^{\text{2}}=\frac{E_x{ }^2}{4}\left[\begin{array}{l}\begin{array}{l}{\left(1+\text{sin}2{\theta }_2\right)}^2\text{sin}\left(2\phi +{\phi }_f\right)\hfill \\ +2{{\displaystyle \text{cos}}}^{2}2{\theta }_2\text{sin}{\phi }_f-\hfill \\ {\left(1-\text{sin}2{\theta }_2\right)}^2\text{sin}\left(2\phi -{\phi }_f\right)\hfill \end{array}\hfill \end{array}\right]$

25

therefore, the feedback phase can be expressed

${\phi }_f=-\text{arctan}\frac{2\text{sin}2{\theta }_2\text{sin}2\phi }{{\text{cos}}^{2}2{\theta }_2+\left(1+{\text{sin}}^{2}2{\theta }_2\right)\text{cos}2\phi }$

26

From formula (26), it can be concluded that the imperfect 45° splice between the pigtail fiber of Faraday rotator and the sensing fiber can lead the output error of OVS.

In addition, the 90° ( ${\theta }_3$ ) splice between the sensing fiber and the compensating fiber is crucial to the OVS system. The output light intensity signal ( $I_{\text{out}}^{\text{3}}$ ) and its sinusoidal expression ( $\Delta I_{\text{out}}^{\text{3}}$ ) as

$I_{\text{out}}^{\text{3}}=\frac{E_x{ }^2}{2}\left\{\begin{array}{l}\begin{array}{l}{{\displaystyle \text{sin}}}^4{\theta }_3\left[1-\text{cos}\left(2\phi +\Delta \phi \right)\right]+\hfill \\ {{\displaystyle \text{cos}}}^4{\theta }_3\left[1-\text{cos}\left(4\delta +2\phi +\Delta \phi \right)\right]+\hfill \\ \frac{1}{2}{{\displaystyle \text{sin}}}^{2}2{\theta }_3\left[\text{cos}2\delta -\text{cos}\left(2\delta +2\phi +\Delta \phi \right)\right]\hfill \end{array}\hfill \end{array}\right\}$

27

and

$\Delta I_{\text{out}}^{\text{3}}=E_x{ }^2\left[\begin{array}{l}\begin{array}{l}{{\displaystyle \text{sin}}}^4{\theta }_3\text{sin}\left(2\phi +{\phi }_f\right)+\hfill \\ {{\displaystyle \text{cos}}}^4{\theta }_3\text{sin}\left(4\delta +2\phi +{\phi }_f\right)+\hfill \\ \frac{1}{2}{{\displaystyle \text{sin}}}^{2}2{\theta }_3\text{sin}\left(2\delta +2\phi +{\phi }_f\right)\hfill \end{array}\hfill \end{array}\right]$

28

where $\delta$ is the phase shift caused by the inherent birefringence of sensing fiber, and $\delta =\frac{2\pi }{\lambda }\left(n_x-n_y\right)L_1$ .

The feedback phase is

${\phi }_f=-\text{arctan}\frac{\begin{array}{l}\begin{array}{l}{{\displaystyle \text{sin}}}^4{\theta }_3\text{sin}2\phi +{{\displaystyle \text{cos}}}^4{\theta }_3\text{sin}\left(4\delta +2\phi \right)\hfill \\ +\frac{1}{2}{{\displaystyle \text{sin}}}^{2}2{\theta }_3\text{sin}\left(2\delta +2\phi \right)\hfill \end{array}\hfill \end{array}}{\begin{array}{l}\begin{array}{l}{{\displaystyle \text{sin}}}^4{\theta }_3\text{cos}2\phi +{{\displaystyle \text{cos}}}^4{\theta }_3\text{cos}\left(4\delta +2\phi \right)\hfill \\ +\frac{1}{2}{{\displaystyle \text{sin}}}^{2}2{\theta }_3\text{cos}\left(2\delta +2\phi \right)\hfill \end{array}\hfill \end{array}}$

29

From formula (29), it can be seen that the phase shift caused by the inherent birefringence of sensing fiber mixed with the phase shift caused by the converse piezoelectric effect, resulting in the output error of sensor.

In the end, the influence of the length difference ( $\Delta L_2$ ) between the sensing fiber and the compensating fiber was analyzed. It assumed the relationship is $L_2=L_1+\Delta L_2$ , and the perfect length difference is zero, that, $\Delta L_2=0$ , which makes the external environment factors lead to the additional phase difference can be offset. However, limited by nonideal factors such as technological level, $\Delta L_2\ne 0$ . The mathematical model of system output ( $I_{\text{out}}^{\text{4}}$ ) and its sinusoidal expression ( $\Delta I_{\text{out}}^{\text{4}}$ ) are

$I_{\text{out}}^{\text{4}}=\frac{E_x{ }^2}{2}\left[1-\text{cos}\left(2\phi -2\Delta \delta +\Delta \phi \right)\right]$

30

and

$\Delta I_{\text{out}}^{\text{4}}=E_x{ }^2\text{sin}\left(2\phi -2\Delta \delta +{\phi }_f\right)$

31

where $\Delta \delta$ is the phase shift caused by the length difference between the sensing fiber and the compensating fiber, and $\Delta \delta =\frac{2\pi }{\lambda }\left(n_x-n_y\right)\Delta L_2$ .

The mathematical model of the feedback phase shift is

${\phi }_f=-\left(2\phi -2\Delta \delta \right)$

32

From formula (32), it can be seen that the phase shift $2\Delta \delta$ makes the feedback in ideal smaller, which introduce the output error of the sensor.

Simulation analysis of system output error

According to the established output error mathematical model, the special fiber splice and the length difference between sensing fiber and compensating fiber can affect the measurement accuracy of the OVS system. Combined with the simulation parameters in Table 1, the quantitative analysis of output errors is as follows.

Table 1.

Simulation parameters of the output error.

Parameters	Value
Operating wavelength	1550 nm
Fiber diameter	125 µm
The length of sensing fiber and compensation fiber	5 m
Quartz crystal diameter	30 mm
Quartz crystal heights	10 cm
Winding turns of sensing fiber	40
Polarization-maintaining delay fiber	200 m
Testing voltage (kV)	6, 10, 20, 35, 66, 110

First of all, from the above analysis that the 45° splice ( ${\theta }_1$ ) between the polarizer and the phase modulator only affects the sensitivity of the system without any error. As shown in Figure 3, the pigtail splicing angle between the polarizer and the phase modulator only affects the sensitivity of the sensor, which is consistent with the ideal state when the splicing angle is 45° and 135°. Furthermore, the sensitivity decreases as the splicing angle gradually deviates from the ideal value, and the sensor cannot recognize the minimum voltage when the splicing angle is 0°, 90°, and 180°.

Figure 3.

The effect of θ ₁ OVS’s sensitivity. OVS: optical voltage sensor.

Next, from formula (26), it can be concluded that the 45° splice ( ${\theta }_2$ ) between the Faraday rotator and the sensing fiber can introduce an additional phase which causes a system output error shown in Figure 4. It can be seen from the results that the output error does not change with the test voltage grade. The output error is within 0.2% when the splicing angle of pigtail fiber between Faraday rotator and sensing fiber at the range of 43.2°–46.8°. However, this is not the only source of the system error. When the splicing angle is 44.4°–45.6°, that is, the angle deviation is less than 0.6°, the system error caused by this factor is less than 0.02%. The system can more easily meets the requirements of IEC 0.2s.

Figure 4.

Output errors caused by 45° splice (θ ₂).

Then, the 90° splice ( ${\theta }_3$ ) between the sensing fiber and the compensating fiber has been quantitative analyzed. As shown in Figure 5, the output error depends on the test voltage. At the same splicing angle, the higher the test voltage grade, the smaller the impact of 90° splice on the system. When the test voltage is 6 kV and the splicing angle is 89.55°–90.45°, the output error caused by ${\theta }_3$ is within 0.2%. As we all known, this is not the only source of the system error. Therefore, when the splicing angle is 89.85°–90.15°, that is, the angle deviation is less than 0.15°, the output error caused by the 90° splice is within 0.02%. At this point, the system can more easily meets the requirements of IEC 0.2s. In addition, it can be concluded that the output error tolerance range caused by the splicing angle depends on the test voltage. And compared with the 45° splice of optical device, the 90° splice is more sensitive to the influence of system output. The relation between output error and splicing angle ( ${\theta }_3$ ) is presented in Table 2.

Figure 5.

The output error of the OVS system caused by imperfect 90° splice ( ${\theta }_3$ ). OVS: optical voltage sensor.

Table 2.

The relation between output error and splicing angle ( ${\theta }_3$ ).

Voltage (kV)	Splicing angle (/°)
	The output error is within 0.2%	The output error is within 0.2%
6	89.55–90.45	89.85–90.15
10	89.42–90.58	89.84–90.18
20	89.20–90.80	89.75–90.25
35	89.00–91.00	89.70–90.30
66	88.65–91.35	89.55–90.40
110	88.30–91.70	89.40–90.60

In the end, the system output error caused by the length difference ( $\Delta L_2$ ) between the sensing fiber and the compensating fiber has been quantitatively analyzed, and it shows the simulation result in Figure 6. The results show that the output error depends on the test voltage and decreases with the increase of the test voltage. Compared with the fiber splicing angle, the system has more strict requirements on the fiber length difference. When the test voltage is 110 kV, the tolerable length difference needs to less than 1 μm, which is difficult to realize in process. However, this is not the only source of the system output error. When the length difference is less than 0.28 µm and the output error is within 0.1%, the system can meet the requirements of IEC 0.2s. It is extremely difficult to meet the standard of length difference in actual production. Therefore, the corresponding error suppression scheme must be put forward to improve this defect and improve the measurement accuracy of the system. The relation between output error and length difference ( $\Delta L_2$ ) is presented in Table 3.

Figure 6.

Output error of system caused by ( $\Delta L_2$ ).

Table 3.

The relation between output error and length difference ( $\Delta L_2$ ).

Voltage (kV)	Splicing angle (the length difference ( $\Delta L_2$ /µm)
	The output error is within 0.2%	The output error is within 0.2%
6	≤0.031	≤0.015
10	≤0.052	≤0.018
20	≤0.104	≤0.050
35	≤0.188	≤0.090
66	≤0.401	≤0.170
110	≤1.000	≤0.280

According to the data in Table 3, the influence of the length difference cannot be ignored. Under the condition of not changing the structure and sensing mechanism, we can suppress the output error from the perspective of fiber itself properties, for example, considering the effects that temperature produces to the PM, we can adopt insensitive polarization-maintaining fiber.

Conclusions

In this article, the relationship among the system output error of optic voltage sensor and the parameters of fiber splicing angle deviation, fiber length difference, and test voltage was studied. The research indicated that different fiber splices have different effects on the system output. Among them, the output error caused by the 45° splice is not affected by the test voltage. And when the splicing angle error is less than 0.6°, the system output error is less than 0.02%. However, the system output error caused by the 90° splice depends on the test voltage. When the test voltage is higher than 6 kV, and its angle deviation is less than 0.15°, the resulting system output error is less than 0.02%. Additionally, the research has also shown that the fiber length difference influences the system output a lot. When the test voltage is 110 kV if the impact of other factors on the system output can be ignored, the length difference between the sensing fiber and the compensating fiber should be less than 1 µm. The research results further deepen the theoretical basis of the all-fiber voltage sensor. And it provides a theoretical basis for the production of optical fiber voltage sensors with different testing voltage requirements and different precision requirements. Especially in the production of the sensor head, it provides a reference for the selection of application parameters of optical fiber. At the same time, it provides reference and new ideas for the design of intelligent inspection robot system for power system under different voltage levels.