1. Introduction

Fiber-optic current sensors (FOCS) based on the Faraday effect in a coil of fused silica fiber have unique advantages over conventional inductive current transformers in electric power transmission systems. In particular, the weight and size of FOCS are dramatically reduced so that the sensors can be easily integrated into other primary equipment of high voltage substations such as circuit breakers. Furthermore, the current is measured with higher fidelity and larger bandwidth due to the absence of magnetic saturation (e.g., at fault currents). The digital output of FOCS is directly compatible with modern digital substation control and protection systems. Other benefits include inherent galvanic separation of electronics from high voltage and high safety of operation (e.g., no electrical safety hazards or risk of explosive failures). Fiber-optic current sensors have been under investigation since the 1970s [1,2]. A particularly robust sensor configuration which has paved the way towards commercial applications employs a reflective fiber-coil and a non-reciprocal phase modulation interrogation scheme adopted from fiber-optic gyroscopes [3–5]. First commercial applications of fiber-optic current sensors have included the measurement of high direct current in the electro-winning industry (for example aluminum smelters) [6] where the sensors have gained significant market share since their introduction in 2005.

The application of fiber-optic current sensors in electrical power transmission as well as industrial process control demands high measurement accuracy (often within ± 0.2% or even ± 0.1%) over a wide temperature range, e.g., from −40 °C to + 85 °C. For comparison, the Faraday effect (Verdet constant) in silica fiber varies by almost 1% over this range [5]. Since conventional temperature measurements are not easily implemented on high voltage potential, methods of inherent temperature compensation are preferred whereby different temperature-dependent contributions to the sensor signal cancel each other. We have shown earlier for coils from low-birefringent fiber that the fiber retarder at the coil entrance which generates the left and right circularly polarized light waves in the sensing fiber coil can be prepared such that it compensates the temperature dependence of the Faraday effect. The method was demonstrated for both, coils with negligible bend-induced birefringence (thermally annealed coils and fiber loops of large diameter [5,6]) and coils with moderate amounts of bend-induced birefringence [7,8]. The fiber coils were operated in an interferometric sensor configuration (Sagnac or in-line interferometers) with interrogation based on non-reciprocal phase modulation.

In this work, we extend the method of inherent temperature compensation to spun highly birefringent fiber coils [9–17]. Spun birefringent fiber is produced by spinning the preform of a linearly birefringent fiber in the drawing process so that the principal fiber axes rotate along the fiber. Such a fiber is elliptically birefringent, i.e., the fiber has elliptically polarized eigenmodes. Depending on the parameters of the spun fiber, these modes can be very close to circularly polarized modes. The elliptical birefringence quenches to a large extent disturbing linear birefringence from external sources, while maintaining the fiber’s sensitivity to the Faraday effect. As a result, the fiber coils are largely insensitive to bend-induced birefringence or birefringence due to stress from the fiber coating and other packaging related stress. This is a significant advantage over coils from low birefringent fiber where sophisticated packaging methods are needed in order to attain stable operation over extended temperature ranges and bend-induced birefringence typically limits the number of fiber loops to a few [5, 6]. However, the birefringence of a spun highly birefringent fiber commonly varies with temperature, which can result in a significant extra temperature dependence of the sensor, in addition to the temperature dependence of the Faraday effect.

In the following, we theoretically and experimentally investigate the temperature dependence of reflective spun fiber coils in an interferometric current sensor with non-reciprocal phase modulation. We determine how the spun fiber parameters, in particular, the embedded linear birefringence (linear beat length), the spin pitch, and the temperature dependence of the beat length, affect the sensor scale factor as a function of temperature. We show how the parameters of the fiber retarder at the fiber coil entrance (retardation and orientation with respect to the spun fiber axes) must be chosen for a temperature-insensitive sensor signal. In particular, we show that there are sets of retarder parameters for which the temperature dependence of the beat length does not enter into the sensor signal. In the regime of small magneto-optic phase shifts the sensing coil then behaves like an ideal coil free of any linear birefringence. It turns out that the apparent magneto-optic phase shift of a temperature-compensated sensor increases slightly nonlinearly with the applied current. We also determine sensor parameters that result in a perfectly linear relationship, however, at the expense of some remaining temperature dependence.

2. Sensor configuration

The sensor configuration is depicted in Fig. 1(a) [3,5]. The broadband output from a superluminescent light emitting diode (SLED, center wavelength λ ≈1310 nm, full width at half maximum Δλ ≈40 nm) is polarized and then excites, at a 45°-splice, the two orthogonal polarization states of a polarization maintaining (PM) fiber with equal amplitudes. In the basic configuration, a quarter-wave retarder (QWR) consisting of a piece of elliptical-core fiber (beat length around 6 mm) at the entrance to the sensing fiber coil converts the orthogonal linear polarization states into left and right circularly polarized light waves. The two light waves are reflected by a metallic coating on the far end of the fiber coil, retrace the optical path with swapped polarization states (the polarization swapping is a combined effect of reflection and double passage through the QWR), and finally interfere at the fiber polarizer. During their round trip through the fiber coil, the circularly polarized light waves accumulate a magneto-optic phase shift Δϕ given by

 

Fig. 1 (a) Sensor configuration, SLED: superluminescent light emitting diode, PD: photodiode. (b) Relative orientation of slow axes of PM lead fiber, retarder, and spun fiber.

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The evolution of the polarization states of the two light waves during their roundtrip through the ideal sensor can be described by

The temperature dependence of the retardation ρ of the fiber retarder can be employed to compensate for the temperature dependence of the Verdet constant, (1/V)·dV/dT = 0.7 × 10⁻⁴ °C⁻¹ [5]. To this end, the retarder is detuned by an amount ε from quarter-wave retardation so that the light waves in the sensing fiber assume elliptical polarization states. The elliptical states can be described as superposition of left and right circularly polarized light waves with amplitudes that are modified, compared to the case of a perfect QWR, by terms cos (ε/2) and sin (ε/2):

3. Sensor employing spun highly birefringent fiber

A spun highly birefringent fiber is characterized by the embedded linear beat length L_LB and the spin pitch length p. The spin ratio is defined as x = 2 L_LB/p. For spun fibers that are appropriate for current sensing, the linear beat length exceeds the spin pitch [9]. The optical circuit including the spun fiber can be theoretically investigated using the Jones matrix formalism. Neglecting bend-induced birefringence (which is justified for the coil diameter of 170 mm used below), the analytical expression of the Jones matrix of a spun fiber is obtained by integration of the corresponding coupled mode equation [19].

In the absence of a magnetic field, the eigenmodes of a spun fiber, defined in the reference frame of local birefringence axes rotating along the spun fiber, are denoted as RCP* and LCP*, and are here represented as a (not normalized) superposition of left and rightly circular polarized waves:

During their propagation through the spun sensing fiber coil, the eigenmodes of the spun fiber experience a mode dispersion per unit length of λ/(cL_EB) due to the birefringence of the spun fiber [10]. On the return path, the light states reflected by the end mirror with swapped polarizations do no longer exactly match the eigenmodes of the Jones matrix of the spun fiber [because of the sign change in Eq. (5)]. Accordingly, secondary light states are created upon reflection for which the spun fiber’s polarization mode dispersion is not compensated on the return path, but doubles. Hence, for short sensing fiber lengths (modal group delay smaller or comparable to the coherence length of the light source), the sensor signal is overlaid by signal oscillations as a function of temperature (due to the temperature dependence of the birefringence) or the fiber length; the normalized amplitude of these unwanted signal oscillations calculates to

A retarder orientated at α₀ = ± 45° with a retardation deviating by ε = $\mp$cot⁻¹ x from perfect quarter-wave retardation maps the incoming orthogonal linear polarization modes of the PM fiber one-to-one onto the elliptical eigenmodes of the spun fiber since in this case $i\tan \frac{\epsilon }{2}=\pm L_{LB}/L_{EB}{e}^{\mp 2i{\alpha }_0}$ [see Eqs. (3) and (5)]:

We calculate the Jones matrix for the entire sensor system with non-reciprocal phase modulation (using the Jones matrices for spun fiber from [19]) to derive an expression for the modulated detector signal. The apparent magneto-optic phase shift Δϕ can be retrieved from the detector signal as shown in [5, 18]. We find:

Here, it is assumed that the Faraday phase shift is small (4NVI << 1) and the fiber is sufficiently long to prevent the mentioned sensitivity oscillations (l >> L_LBλ/Δλ). Apart from Section 5 where also high magneto-optic phase shifts are considered, all calculations and experiments are conducted with parameters such that these two assumptions are valid.

For the special retarder orientations α₀ = ± 45°, Eq. (8) reduces to:

Note that, in this case, the detected magneto-optic phase shift is the same as in case of a sensor with ideal non-birefringent sensing fiber [Eq. (4)]. In particular, it is interesting to note that for α₀ = 0°, 90° the sensor signal is not affected by the variation of the linear beat length L_LB with temperature. Still, in this case, in contrast to the case of a low-birefringent sensing fiber described by Eq. (4), there are also secondary states excited for ε = 0° and the interferometer visibility is reduced with respect to the ideal case.

Figure 2 shows the normalized signal, i.e., the apparent magneto-optic phase shift Δϕ relative to the phase shift of an ideal sensor according to Eq. (1) as a function of the retarder length for retarder orientations of α₀ = 0° and ± 45°. The assumed fiber parameters are given in Table 1 (fiber A). For α₀ = ± 45°, the minima of the parabola are reduced by a factor of $\sqrt{1+x^{-2}}$ and are shifted in the horizontal direction by ± cot⁻¹x compared to the cases of α₀ = 0°, 90° [or the case of a non-birefringent fiber, Eq. (4)]. A retardation of 90° ± cot⁻¹x yields injection of pure eigenmodes into the fiber as discussed above [see Eqs. (3), (5), and (7)].

 

Fig. 2 Calculated normalized magneto-optic phase shift vs. deviation from quarter-wave retardation for different choices of the spun fiber’s angular orientation (fiber A, see Table 1).

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Table 1. Linear Beat Length and Spin Pitch of the Spun Highly Birefringent Fibers Used for This Work

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4. Inherent temperature compensation for current sensors employing spun fiber

From Eq. (8), it is obvious that the retarder’s contribution to the sensor’s overall temperature dependence can be controlled by appropriately adjusting α₀ and ε. Further temperature dependent contributions stem from the Verdet constant and from the spun fiber’s birefringence. The contribution of the birefringence itself depends on the choice of α₀ and ε [see Eq. (8)]. Figure 3(a) shows for a temperature compensated sensor the calculated individual temperature contributions from the retarder, the Verdet constant, and the spun fiber’s birefringence as well as their sum. With α₀ = 45°, beat length and pitch according to fiber A (see Table 1), and the measured temperature coefficients of the retarder and the fiber birefringence (dρ/(ρdT) = −3.2 × 10⁻⁴ °C⁻¹ and dL_LB/(L_LBdT) = 1.0 × 10⁻³ °C⁻¹), the room temperature retardation ρ_RT must be set to ρ_RT = 107.2° (ε = 17.2°) in order to achieve temperature compensation. Figure 3(b) depicts the proper choice of ε for temperature compensation as a function of the spinning ratio x for the cases of α₀ = 0°, ± 45°, and 90°. Note that the solution for α₀ = 0°, 90° (black line) is independent of the spinning ratio since the birefringence of the spun fiber does not contribute to the overall temperature dependence.

 

Fig. 3 (a) Calculated normalized magneto-optic phase shift vs. temperature for a temperature-compensated sensor (black). The other lines depict the individual contributions from the Verdet constant (green), the spun fiber’s birefringence (orange), and the fiber retarder (blue), see text for parameters. (b) Retarder detuning ε required for temperature compensation vs. spin ratio [p = 2.9 mm].

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The experimental data of Fig. 4 confirms the relationship of Eq. (8). Figure 4(a) depicts for spun fiber A the contribution of the retarder to the temperature dependence of the sensor signal, i.e., the term d(lnΔϕ)/dT_retarder as a function of the orientation angle α₀ of a straight spun fiber (red squares). Qualitatively, we observe that the orientation of the spun fiber with respect to the retarder is an important sensor parameter. The spun fiber orientation has a significant influence of the sensor’s temperature dependence and we can even demonstrate a sign change of d(lnΔϕ)/dT_retarder. By contrast, in case of low-birefringent sensing fiber, the retarder orientation has no influence for a straight fiber (and only must be taken into account if the fiber is bent to a coil [6]). In addition, the data is in good agreement with numerical results (black line). The theoretical curve was calculated with the fiber parameters given in Table 1 (fiber A), a retarder detuning angle ε(20 °C) = −2.5°, and a retarder temperature dependence of dρ/(ρdT) = −1.6 × 10⁻⁴ °C⁻¹. Here, ε and dρ/(ρdT) were treated as adjustable parameters and, as only relative changes of α₀ were experimentally accessible, the absolute value of α₀ was adjusted to the theoretical expectations by adding an offset. (In this particular experiment, there was a short section of low-birefringent fiber spliced to the retarder to manufacture the retarder as described in [5]. The same retarder could then be used for different spun fiber orientations. However, this hindered knowledge of absolute values for α₀ and also experimental determination of ε with an accuracy better than within a few degrees.) The retrieved value for dρ/(ρdT) is a factor of 2 smaller than measured for long undisturbed sections of the same elliptical-core fiber (−3.2 × 10⁻⁴ °C⁻¹). In part, the deviation may be explained by modifications of the retarder fiber properties caused by the splice procedure.

 

Fig. 4 (a) Contribution of the retarder, d(lnΔϕ) /dT_retarder, to the temperature dependence of the sensor signal as a function of the spun fiber orientation angle α₀, experimental: red squares, calculated: black line. (b) Normalized sensor signal as a function of the temperature of the retarder (sensing fiber at room temperature, blue squares) and of the temperature of the entire fiber coil (retarder and sensing fiber, black data points); the red line is a linear fit (sensor parameters: fiber type B, α₀ = 0°, ε = 0°). (c) Same as in (b), but with ε ≈7.5° so that sensor signal is temperature compensated to within < ± 0.2%.

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We have fabricated temperature-insensitive current sensors with various types of spun fiber and different choices of the fiber orientation α₀. Note that there is a small uncertainty of α₀ introduced by aligning the spun fiber with respect to the retarder: While the PM fiber orientation angle and thus the retarder orientation can be precisely determined before splicing by means of polarizers, the angular orientation of the spun fiber is set by eye using an end view of the fiber’s facet. The estimated error is within ± 2.5° for elliptical-core spun fiber and less for spun fibers with stress bodies such as bow-tie spun fiber. This uncertainty of α₀ results in turn in an uncertainty of the retarder contribution to the overall temperature-dependence of the sensor and, accordingly, in an uncertainty of the optimum retarder length for temperature compensation. It is obvious from Fig. 4(a) that errors in α₀ are least critical at α₀ = ± 45°. In practice, accuracy of angular orientation within ± 2.5° is sufficient for the targeted sensor accuracy at all angles α₀, though. In the following, typical results are shown for a spun elliptical-core fiber (fiber B in Table 1) and α₀ = 0°. According to Eq. (10), the temperature sensitivity for this sensor originates from the Verdet constant and the retarder, but not from the spun fiber’s birefringence. Figure 4(b) depicts the signal versus temperature of an uncompensated sensor with an exact quarter-wave retarder (ε = 0, black data points), showing a slope of dΔϕ /(Δϕ dT) ≈0.7 × 10⁻⁴ °C⁻¹ (fit, red line) in excellent agreement with values reported for the temperature dependence of the Verdet constant in fused silica fiber [6]. The blue squares in Fig. 4(b) show the sensor signal vs. the retarder temperature (with the spun fiber kept at room temperature). It is obvious that, as expected for ε = 0, the retarder does not contribute to the temperature dependence. Figure 4(c) shows corresponding results for a temperature compensated sensor (ε ≈7.5°). The retarder contribution (experiment: blue squares, fit: red line) is now opposite to that of the Verdet constant and as a result the sensor signal becomes insensitive to temperature within < ± 0.2% from −40 to + 85 °C (black data points).

5. Sensor response at large magneto-optic phase shifts

It is known that the apparent magneto-optic phase shift of a sensor using low-birefringent sensing fiber and a retarder that deviates from perfect quarter-wave retardation (e.g., for the purpose of temperature compensation) exhibits some nonlinearity as a function of current [6–8]. In addition to the retarder, bend-induced birefringence in the fiber coil also influences the signal vs. current relationship. The nonlinearity is again the result of the interference of primary and secondary pairs of orthogonal light waves as described in Section 2. Similar effects also occur in sensors employing spun fiber.

In the remainder of the paper, we theoretically and experimentally examine the nonlinearity of sensors with spun fiber coils and show that the retarder can be prepared in a way that non-linearity can be avoided. To experimentally examine magneto-optic phase shifts up to ± π, a sensor was fabricated with 385 spun fiber windings with a diameter of 165 mm (fiber C in Table 1). The total fiber length was thus about 200 m. The retarder was set so that the signal was temperature compensated at low magneto-optic phase shifts (Δϕ << 1, α₀ = 0°). Figure 5(a) shows the measured signal as a function of current, i.e., vs. the terms 4NVI and NI, for coil temperatures of −20 °C (black squares) and 70 °C (red squares). The solid lines are calculated assuming a retardation of the fiber retarder of ρ = 96° [ε = 6°, dρ/(ρdT) = −3.2 × 10⁻⁴ °C⁻¹], the fiber parameters from Table 1 [dL_LB/(L_LBdT) = 1.0 × 10⁻³ °C⁻¹], cross coupling of 26 dB at splices in the PM fiber lead (cross-coupling at PM fiber splices also leads to the generation of secondary light states in the sensing fiber), and a Verdet constant of V = 1.055 μradA⁻¹ in good agreement with literature values [20] with temperature dependence of dV /(VdT) = 0.7 × 10⁻⁴ °C⁻¹. At a magneto-optic phase shift of 180°, the signal deviates from a linear relationship by several percent. Note also that the temperature compensation works only in the regime of small magneto-optic phase shifts. This is however not a limitation for most practical applications since for high-voltage substation applications the sensors are prepared such that the rated currents correspond to the low current regime of the sensor. The high current range is reserved for short-time fault currents. Fault currents can be 1-2 orders of magnitude higher than the rated currents but need to be measured only with relatively moderate accuracy.

 

Fig. 5 (a) Normalized sensor signal as a function of applied current for a temperature-compensated sensor with 385 fiber windings at temperatures of −20 °C (black squares) and + 70 °C (red squares), the lines represent corresponding theoretical data. (b) Normalized sensor signal as a function of applied current for ε = 0° (black line) and α₀ = ± 45° and ε = ± cot⁻¹ x (red line).

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The non-linearity in the signal can be eliminated with a retarder that maps the incoming pair of orthogonal linearly polarized waves one-to-one on the left and right elliptical eigen-modes of the spun fiber. As argued in Section 3, this can be achieved by setting the spun fiber’s angular orientation to α₀ = ± 45° and the retarder to 90° $\mp$ cot⁻¹ x. Figure 5(b) depicts for this case the calculated normalized sensor signal Δϕ/(4NVI) (red line, parameters of fiber A, Table 1) showing negligibly small non-linearity. For comparison, the calculation was also conducted for a perfect quarter-wave retarder (ε = 0°, black line) which does result in a nonlinear signal due to the generation of secondary light states, contrary to the case of a non-birefringent sensing fiber with such a retarder. It should be noted that a linear sensor output excludes inherent temperature compensation and requires an extra temperature measurement if compensation is necessary. Note that, at an apparent magneto-optic phase shift of 180° × $\sqrt{1+x^{-2}}$ = 1.04 × 180° where the black and red lines cross in Fig. 5(b), the coherent primary and secondary light states of each polarization are in phase. The signal is then the same for all choices of α₀ and ε.

6. Conclusion

In conclusion, we have experimentally and theoretically demonstrated that the various contributions to the temperature dependence of an interferometric current sensor with spun highly birefringence fiber can be balanced such that the sensor signal becomes independent of temperature. A residual variation of the sensor signal as a function of fiber coil temperature of less than ± 0.2% was achieved between −40 and + 85 °C in the regime of small magneto-optic phase shifts (this range is the normal operating range of the sensor and relevant for metering applications whereas the range of large phase shifts is reserved for fault currents; typically, measurement of the latter requires only accuracy to within ± 5%). Hence, the sensor can meet the stringent accuracy requirements of electric power transmission systems without the need of an extra temperature sensor on high voltage. Furthermore, we have determined design rules for sensors with spun fiber that result in a perfectly linear signal as a function of current.