Issue #7/2018

Fiber-Optic Transformers of Electric Current: Physical Principles & Technical Realizations. Part I

**V. P. Gubin, N. I. Starostin, Y. V. Przhiyalkovsky, S. K. Morshnev, A. I. Sazonov, S. Y. Otrokhov**Fiber-Optic Transformers of Electric Current: Physical Principles & Technical Realizations. Part I

Physical principles of an optical method for an electrical current measurement based on the Faraday effect are considered. Its application for a creation of fiber-optic transformers of electric current (FOTC) is discussed. The properties of a special fiber for a sensing element of up-to-date FOTC are analyzed. The description of FOTC technical realizations is presented.

DOI: 10.22184/1993-7296.2018.12.7.704.715

DOI: 10.22184/1993-7296.2018.12.7.704.715

Теги: faraday effect optical current transformer spun fiber spun-волокно оптический трансформатор тока эффект фарадея

INTRODUCTION

Measurement of electric current for metering of electricity consumed, control and protection of electrical networks is one of the main tasks of power electric power engineering. Until recently, similar problems were solved based on traditional transformers using the principle of electromagnetic induction. Electromagnetic transformers (ET), which have been intensively used for about 70 years, are distinguished by relative reliability, in particular, in the voltage range of 6–35 kV. ET able to work in fairly severe climatic conditions. At the same time, one of the unrecoverable drawbacks of traditional transformers is the saturation of the magnetic circuit of the aperiodic component of the short circuit current and, as a result, the lack of transmission of current information in the line during the first time of the emergency process. The problem aspects also include the lack of reliability of high-voltage insulation in transformers for higher voltages (110–750 kV) and the explosion and fire hazard of such devices [1].

Among the alternative methods for measuring current a fiber-optic method for measuring current using the Faraday effect in an optical fiber has been actively developing for the last two decades [2–4]. Based on this method, fiber-optic current transformers (FOCT) [5, 6] were created (in Russia, in the State Register of Measuring Instruments, these devices are referred to as the electro-optical current transformers). These devices are free from the aforementioned disadvantages inherent in traditional current transformers. They provide increased operational safety for service personnel, require less installation and maintenance costs. According to the accuracy characteristics, FOCT is not inferior and in some cases even surpasses traditional current transformers. An important advantage is that the output data on the current are issued directly in digital form.

Currently, the main part of the market of fiber-optic current transformers is represented by such leading global companies as General Electric (GE), ABB, Arteche. GE, in particular, supplied about 12,000 optical transformer phases to consumers. In Russia, the only manufacturer of such devices is "Profotech".

The purpose of the article is a brief overview of the physical basis of FOCT.

1.

THE PHYSICAL BASIS OF THE OPTICAL METHOD OF MEASURING ELECTRIC CURRENT

1.1. Faraday effect in a quartz fiber

The magneto-optical Faraday effect in a dielectric medium located in a magnetic field was discovered by M. Faraday in 1845. Figure 1 shows the scheme of the experiment, which usually demonstrates the Faraday effect. Let’s consider the propagation of a light wave along a segment of a quartz fiber (QF) in the presence of an external magnetic field H directed along the central axis of the QF z (H ↑↑ z). Let the wave propagating in the positive direction (wave vector k ↑↑ z) be in the state of linear polarization with the electric field vector Ein oriented along the y axis (Fig. 1a). Then the plane of polarization of the light at the QF output (field vector Eout) will be rotated due to the Faraday effect by the angle θF counterclockwise [2]. With the opposite direction of propagation of the same light wave (wave vector –k ↓ z), i. e. against the direction of the magnetic field vector H, the plane of polarization after passing the QF will be rotated to the same absolute angle –θF clockwise (Fig. 1b). Thus, the sign of the angle of rotation of the polarization plane depends on the reciprocal orientation of the direction of the magnetic field vector H and the direction of light propagation ξ = k / |k|, which can be characterized by the scalar product of these vectors. As a result, in the case of a direct fiber and a uniform longitudinal magnetic field, the angle of rotation of the polarization plane (Faraday angle) θF is determined by the expression

θF = VL (Н∙ξ), (1)

where V is the Verde constant of the core material of the QF, L is the length of the segment of the QF. This simple relationship (1) between the angle of rotation of the plane of polarization of light and the magnetic field vector is the basis of measuring the electric current by the optical method.

Note that this property of the optical Faraday effect, the dependence on the reciprocal orientation of the direction of propagation of light and the magnetic field vector, is characteristic of nonreciprocal optical effects. For comparison, Fig. 1 shows the effect on linearly polarized light of a dielectric medium where a similar rotation of the polarization plane takes place, however, due to the reciprocal optical effect (the electric field vector E and the angle of rotation of the polarization plane θc are represented by a dotted line). In this case, the directions of rotation of the polarization plane with the forward and backward passage of radiation through the fiber have the same sign (in the figure – clockwise). Reciprocal optical effects are manifested, for example, in the medium with various types of deformations, in particular, in the QF in presence of compression, bending or axial torsion. Another example of the manifestation of a reciprocal effect is the evolution of polarization in the medium with a spiral anisotropy of the refractive index (spun fiber in the case of the QF), which is discussed in Section 1.4.

Due to the non-reciprocal nature of the Faraday effect with a double pass of a segment of the QF with mirror reflection of light at its end, the angle of rotation of the polarization plane is doubled. Indeed, after a straight pass, for example, at the exit of the QF we have (Fig. 1 (a)) a rotation angle of θF>0. Taking into account the observation system chosen above, after reflection from the mirror, the plane of polarization of the wave will be rotated through an angle of –θF. After the backward pass through the QF (Fig. 1), the polarization plane will rotate according to (1) through an angle of – θF, so the resulting angle of rotation will be – 2θF, which is twice as large in absolute value as in one pass of the QF. It should be noted that in the case of reciprocal effects, the angle of rotation of the polarization plane after a similar propagation of light in the opposite direction is completely compensated. The noted features of the above effects are key in the optical scheme of the fiber-optic current transformer.

The strict mathematical description of the Faraday effect is quite complex. The effect is based on the interaction of the electromagnetic field of a light wave with electrons of atoms of a dielectric medium located in an external magnetic field. It can be simplified to assume that the orbital axes of electrons are oriented along a magnetic field, and due to the Lorentz force, the orbit axes rotate (precess) around the field direction with a frequency proportional to the field, and the direction of rotation (precession) is determined by the direction of the magnetic field vector. The precession of the orbits is similar to the precession of a mechanical whipping top, the axis of which does not coincide with the direction of gravity. The theory shows that the precession of electron orbits in the atoms of the medium affects the speed of propagation of a light wave that is in a state of circular polarization.

1.2.

Phenomenological description of the Faraday effect

The electric field of a linearly polarized light wave can be represented as a sum of fields of orthogonal circularly polarized waves. With circular polarization, the end of the electric field vector E of the light wave rotates around a circle with an optical frequency clockwise (right, ER) or counterclockwise (left, EL). For definiteness, we assume that the direction of rotation of the vector E is determined by the same rule as the sign of the rotation of the polarization plane of linearly polarized light, i. e. when observing a wave going towards the observer (see footnote above).

In a dielectric environment in a magnetic field, orthogonal circularly polarized waves have different phase velocities. The reason for this is, as noted above, the influence of an external magnetic field on the electrons of the atoms of the environment, which leads to different coefficients of the effective refractive index n for orthogonal circularly polarized waves ER and EL. The physical mechanism of this effect manifests itself differently for the ER and EL waves, since for one of them the direction of rotation of the vector E coincides with the direction of precession, and for the other it is opposite to it. If there is no field, then nR = nL = n0 and the waves, when propagating in the environment, will acquire the same phase shifts; n0 is the part of the refractive index of the environment that is independent of the magnetic field. In the presence of an external magnetic field, the refractive indices of the left and right waves become different:

nR = n0 + δn (slow wave), (2a)

nL = n0 – δn (fast wave), (2b)

where δn << n0 for real magnetic fields. The sign of the amendment δn, and therefore, which of the waves will be slow and which is fast (in the case of a specific medium) depends on the ratio of the directions of propagation of light and the magnetic field vector.

In the presence of a magnetic field, the ER and EL waves after passing through a direct fiber with a length L along the direction of the magnetic field will acquire, according to (2a, 2b), different phase shifts due to the Faraday effect:

ϕR = 2π L nR / λ = ϕ0 + ϕF, (slow wave) (3a)

ϕL = 2π L nL / λ = ϕ0 – ϕF, (fast wave) (3b)

where ϕ0 = 2π L n0 / λ, and ϕF = θF = VLHz according to relation (1), Hz is the projection of the magnetic vector field in the direction of wave propagation. When waves propagate in the opposite direction, the signs in front of ϕF in (3) are reversed.

With a direct pass, the phase difference of the ER and EL waves due to the Faraday effect (Faraday phase shift) ΔϕF = ΔϕFfor will be equal to

ΔϕF = ϕR – ϕL = ΔϕFfor = 2ϕF = 2 VLHz . (3с)

Let’s now consider the total Faraday phase shift with a double passage of radiation through a fiber with a mirror at its end.

During the reverse propagation of a circularly polarized wave after its specular reflection at the end of the QF, there are two factors, each of which changes the sign of the Faraday phase shift. First, during reflection, the circular polarization changes to orthogonal. This is due to the change in the direction of wave propagation, and hence the direction of the observer’s look on the wave, to the opposite (this effect can be illustrated as a change in the direction of rotation of the clock when looking at the dial on the reverse side). Secondly, after specular reflection, the reciprocal orientation of the direction of propagation of light and the magnetic field vector changes. As a result of both factors, the sign of the phase shift during the back pass is preserved, so that ΔϕFback = 2VLHz, and the total phase shift is equal to

ΔϕF = ΔϕFfor + ΔϕFback = 4ϕF = 4 VLHz, (3d)

The relations (3a‑3d) represent the Faraday effect in terms of phase shifts of circularly polarized light waves.

The Faraday phase shift ΔϕF completely determines the angle θF of the rotation of the polarization plane of a linearly polarized wave, as follows from relations (3a‑3c). Let us explain this in the vector diagram in fig. 2. This shows the position of the field vectors ER and EL of the right and left circularly polarized waves at the input (a) and output (b) of the segment of the QF that is in a longitudinal magnetic field. Note that a change in the angle of rotation of these vectors corresponds to a change in the current phase of the waves; in particular, the full circle corresponds to a phase change of 2π. Let ЕR is a slow wave, ЕL is a fast wave. Suppose that, at the QF input, the waves have equal initial phases – the ЕR and ЕL vectors are oriented along the A – A line (Fig. 2a). At the QF output, the orientation of the wave vectors will have the form shown in Figure 2b. The ЕR and ЕL vectors, according to (3a, 3b), rotate at different angles ϕ0 + ϕF > ϕ0 – ϕF, respectively (vectors denoted by dotted lines indicate the position of ЕR and ЕL in the absence of a magnetic field ϕF=0). A phase difference ΔϕF = 2ϕF (Faraday phase shift) will arise between the field vectors in accordance with (3c). The B-B bisector between the ЕR and ЕL vectors, which is the sum of circularly polarized waves, corresponds to the orientation of the polarization plane of a linearly polarized wave at the QF output. The angle of rotation of the polarization plane is θF = ΔϕF / 2.

With the opposite direction of the magnetic field vector, when ЕR is a fast wave, ЕL is a slow wave, the polarization plane will turn through the angle θF = –ΔϕF / 2. Note that the phase of the waves and the angle of rotation of the polarization plane are expressed in the same units (radians) and the measurement of any of these parameters allows you to get information about the magnetic field of the current and, as a consequence, the current itself. Below, we will present further material in terms of the Faraday phase shifts of orthogonal circular waves, since in practice, it is easier to implement a highly accurate measurement of the electric current through the measurement of the relative phase shift between orthogonal light waves.

1.3.

Theorem on the circulation of a magnetic field in a closed loop

To obtain information on the magnitude of the measured current, the optical fiber is wound around a current bus. In practice, it is the fiber coil that is the basis of the sensing element of a real optical transformer. For high-precision measurement, the fiber loop must be closed (the beginning and end of the loop must be spatially aligned). This requirement is based on the fundamental law of nature, expressed by one of the Maxwell equations, namely, circuital law. This law is formulated as follows: the circulation (linear integral) of the magnetic field strength vector, excited by slowly varying electric currents along a closed loop L of an arbitrary shape, is equal to the algebraic sum of the currents I circuital law by this loop:

. (4)

If a closed loop consists of N windings of arbitrary shape covering the currents, then bypassing each turn of the loop will contribute I to the result of integration, therefore (4) will be expressed as follows:

. (5)

The Faraday phase shift during the passage of circular polarized light waves of the elementary part of the fiber dl, according to (3c), will have the form dϕF = 2VHdl. Performing integration over a closed contour L with regard to (5) (assuming a uniform magneto-optical sensitivity along the entire length of the contour), we obtain the relation connecting Faraday phase shift with the measured current:

.

In the case of specular reflection at the end of the fiber (the so-called reflective type of the optical scheme), the radiation passes the loop twice, in the forward and reverse directions. Therefore, similarly to (3d), the total Faraday shift also doubles:

. (6)

In conclusion, we emphasize that, as follows directly from the circuital law, with a closed loop of an arbitrary shape, the currents of adjacent buses do not contribute to integral (5); the position of the bar inside the contour does not affect the result either. Therefore, the condition of closed loop is one of the key requirements for high-precision measurement of electric current.

1.4.

Magnetic sensors for optical current transformers

In order for radiation to effectively accumulate Faraday’s phase shift due to the magnetic field of an electric current, an optical fiber must have the ability to maintain a circular or similar state of polarization of radiation (such fibers are called magnetooptically sensitive). Conventional single-mode ("isotropic") fibers used in communication lines do not fully possess this property due to induced birefringence due to external mechanical effects, as well as the presence of residual internal birefringence, in particular, because of the non-ideal initial preform fiber.

Currently, one of the most attractive fibers for current measurement is the spun fiber [2–4]. Spun fibers are obtained from a preform with a strong built in linear birefringence, which is rapidly rotated when drawing. We emphasize that the drawing technology assumes the absence of any elastic torsional stresses in the fiber. The key advantage of such fibers is the combination of high sensitivity to the Faraday effect with its relative resistance to external mechanical influences. These properties have largely determined the widespread use of spun fibers in industrial fiber-optic current transformers.

This type of optical fibers has a number of features, which will be discussed below. Due to the rotation of the preform when drawing, the axis of the embedded linear birefringence of the spun fiber have a spiral structure, therefore the polarization properties of the spun fibers are determined by two main parameters: the beat length of the embedded linear birefringence Lb and the pitch structure Ls. Such a structure has the ability to maintain the original direction of rotation of the vector of the electric field of the wave, to maintain in the fiber (on average) an elliptical polarization state, which ensures the accumulation of the Faraday phase shift when radiation propagates through the fiber. On the other hand, the spiral structure of the birefringence provides, to a certain extent, the suppression of linear birefringence induced by external influences on the fiber.

The main polarization properties of a spun fiber are characterized by a σ parameter [7]: σ = Ls / (2Lb). Thus, for σ << 1, the fiber has a small birefringence and / or a small pitch of the spiral and the ellipticity of the polarization state (PS) is close to unity (quasi-circular polarization state), and for σ ≥ 1 – a strong birefringence and / or large pitch of the helix, ellipticity of polarization averaged over the spectrum is substantially less than unity.

The small ellipticity (less than unity) of the polarization state leads to a decrease in the magneto-optical sensitivity of the spun fiber. This can be qualitatively understood by representing a wave with an elliptical PS as a sum of orthogonal circularly polarized components with different weights: a component with lower weight accumulates a Faraday phase shift with the opposite sign, thereby reducing the contribution of the main component to the resulting shift Δϕ F. These polarization properties are considered in the formula (6) for the Faraday phase shift by multiplying by the coefficient S = 1 / (1+σ2)1 / 2 [4, 7]:

ΔϕF = 4SVNI. (6а)

For typical spun fibers used in optical transformers current, Lb ~ 8 … 10 mm (λ = 1550 nm) and Ls≈3 mm, which means σ << 1, i. e. the state of polarization of radiation is close to circular one and, according to (6a), the sensitivity decreases slightly.

It should be noted that the phase difference of orthogonal circularly (elliptically) polarized light waves propagating through the spun fiber is influenced not only by the magnetic field of the measured current (Faraday effect), but also by the spiral the structure of the axes of the linear birefringence [8]. In the case of σ << 1, the phase difference Δϕc acquired by quasi-circularly polarized light waves when they propagate in the spun fiber is determined by the ratio

Δϕc ≈ πσL / Lb, (7)

where L is the fiber length. The physical mechanism of the influence of the spiral structure on the phase difference of light waves is associated with the presence of different phase velocities for orthogonal circularly polarized waves: for one wave, the direction of rotation of the electric field vector coincides with the direction of birefringence axis rotation, for orthogonal waves these directions are opposite.

It is important to note that the spiral structure of the birefringence is symmetric about the direction of light propagation (as opposed to the longitudinal component of the magnetic field vector), and its effect on the light wave manifests itself as a reciprocal effect. Therefore, when the circularly polarizated light waves pass through a spun fiber with mirror at its end, where after reflection waves exchange their polarization states, the total phase shift Δφc between them due to the birefringence axis spiral structure becomes equal to zero (compensated). This fact is undoubtedly an advantage of a sensitive fiber coil with a double radiation path (reflective version), since in this case the phase difference between the working wave is determined only by the magnetic field of the current being measured.

Furthermore, if a wide-band light source is used the phase shift (7), appeared between the quasi-circular light waves due to the birefringence axis spiral structure of the spun fiber, leads to a loss of coherence between the waves (depolarization) during their propogation through the fiber. The criterion for this process is the depolarization length Ld, which is determined by the parameters of the spun fiber and broadband radiation. For a rectangular spectrum, it is equal to [7]:

Ld = (Ls / 2) [(σ2+1)1 / 2 / σ2] (λ / Δλ), (8)

where λ is the central wavelength of the source, Δλ is the width of the spectrum. The phenomenon of the depolarization of light plays an important role in the operation of FOCT and will be discussed in the second part of the review.

The technical implementation of the method will be discussed in the second part of the article.

[2] Note: the state of polarization (in particular, the sign of the angle of rotation of the polarization plane), is considered in this review when observing a wave towards. A positive rotation angle corresponds to a counterclockwise rotation, and a negative one – to a clockwise rotation.

Measurement of electric current for metering of electricity consumed, control and protection of electrical networks is one of the main tasks of power electric power engineering. Until recently, similar problems were solved based on traditional transformers using the principle of electromagnetic induction. Electromagnetic transformers (ET), which have been intensively used for about 70 years, are distinguished by relative reliability, in particular, in the voltage range of 6–35 kV. ET able to work in fairly severe climatic conditions. At the same time, one of the unrecoverable drawbacks of traditional transformers is the saturation of the magnetic circuit of the aperiodic component of the short circuit current and, as a result, the lack of transmission of current information in the line during the first time of the emergency process. The problem aspects also include the lack of reliability of high-voltage insulation in transformers for higher voltages (110–750 kV) and the explosion and fire hazard of such devices [1].

Among the alternative methods for measuring current a fiber-optic method for measuring current using the Faraday effect in an optical fiber has been actively developing for the last two decades [2–4]. Based on this method, fiber-optic current transformers (FOCT) [5, 6] were created (in Russia, in the State Register of Measuring Instruments, these devices are referred to as the electro-optical current transformers). These devices are free from the aforementioned disadvantages inherent in traditional current transformers. They provide increased operational safety for service personnel, require less installation and maintenance costs. According to the accuracy characteristics, FOCT is not inferior and in some cases even surpasses traditional current transformers. An important advantage is that the output data on the current are issued directly in digital form.

Currently, the main part of the market of fiber-optic current transformers is represented by such leading global companies as General Electric (GE), ABB, Arteche. GE, in particular, supplied about 12,000 optical transformer phases to consumers. In Russia, the only manufacturer of such devices is "Profotech".

The purpose of the article is a brief overview of the physical basis of FOCT.

1.

THE PHYSICAL BASIS OF THE OPTICAL METHOD OF MEASURING ELECTRIC CURRENT

1.1. Faraday effect in a quartz fiber

The magneto-optical Faraday effect in a dielectric medium located in a magnetic field was discovered by M. Faraday in 1845. Figure 1 shows the scheme of the experiment, which usually demonstrates the Faraday effect. Let’s consider the propagation of a light wave along a segment of a quartz fiber (QF) in the presence of an external magnetic field H directed along the central axis of the QF z (H ↑↑ z). Let the wave propagating in the positive direction (wave vector k ↑↑ z) be in the state of linear polarization with the electric field vector Ein oriented along the y axis (Fig. 1a). Then the plane of polarization of the light at the QF output (field vector Eout) will be rotated due to the Faraday effect by the angle θF counterclockwise [2]. With the opposite direction of propagation of the same light wave (wave vector –k ↓ z), i. e. against the direction of the magnetic field vector H, the plane of polarization after passing the QF will be rotated to the same absolute angle –θF clockwise (Fig. 1b). Thus, the sign of the angle of rotation of the polarization plane depends on the reciprocal orientation of the direction of the magnetic field vector H and the direction of light propagation ξ = k / |k|, which can be characterized by the scalar product of these vectors. As a result, in the case of a direct fiber and a uniform longitudinal magnetic field, the angle of rotation of the polarization plane (Faraday angle) θF is determined by the expression

θF = VL (Н∙ξ), (1)

where V is the Verde constant of the core material of the QF, L is the length of the segment of the QF. This simple relationship (1) between the angle of rotation of the plane of polarization of light and the magnetic field vector is the basis of measuring the electric current by the optical method.

Note that this property of the optical Faraday effect, the dependence on the reciprocal orientation of the direction of propagation of light and the magnetic field vector, is characteristic of nonreciprocal optical effects. For comparison, Fig. 1 shows the effect on linearly polarized light of a dielectric medium where a similar rotation of the polarization plane takes place, however, due to the reciprocal optical effect (the electric field vector E and the angle of rotation of the polarization plane θc are represented by a dotted line). In this case, the directions of rotation of the polarization plane with the forward and backward passage of radiation through the fiber have the same sign (in the figure – clockwise). Reciprocal optical effects are manifested, for example, in the medium with various types of deformations, in particular, in the QF in presence of compression, bending or axial torsion. Another example of the manifestation of a reciprocal effect is the evolution of polarization in the medium with a spiral anisotropy of the refractive index (spun fiber in the case of the QF), which is discussed in Section 1.4.

Due to the non-reciprocal nature of the Faraday effect with a double pass of a segment of the QF with mirror reflection of light at its end, the angle of rotation of the polarization plane is doubled. Indeed, after a straight pass, for example, at the exit of the QF we have (Fig. 1 (a)) a rotation angle of θF>0. Taking into account the observation system chosen above, after reflection from the mirror, the plane of polarization of the wave will be rotated through an angle of –θF. After the backward pass through the QF (Fig. 1), the polarization plane will rotate according to (1) through an angle of – θF, so the resulting angle of rotation will be – 2θF, which is twice as large in absolute value as in one pass of the QF. It should be noted that in the case of reciprocal effects, the angle of rotation of the polarization plane after a similar propagation of light in the opposite direction is completely compensated. The noted features of the above effects are key in the optical scheme of the fiber-optic current transformer.

The strict mathematical description of the Faraday effect is quite complex. The effect is based on the interaction of the electromagnetic field of a light wave with electrons of atoms of a dielectric medium located in an external magnetic field. It can be simplified to assume that the orbital axes of electrons are oriented along a magnetic field, and due to the Lorentz force, the orbit axes rotate (precess) around the field direction with a frequency proportional to the field, and the direction of rotation (precession) is determined by the direction of the magnetic field vector. The precession of the orbits is similar to the precession of a mechanical whipping top, the axis of which does not coincide with the direction of gravity. The theory shows that the precession of electron orbits in the atoms of the medium affects the speed of propagation of a light wave that is in a state of circular polarization.

1.2.

Phenomenological description of the Faraday effect

The electric field of a linearly polarized light wave can be represented as a sum of fields of orthogonal circularly polarized waves. With circular polarization, the end of the electric field vector E of the light wave rotates around a circle with an optical frequency clockwise (right, ER) or counterclockwise (left, EL). For definiteness, we assume that the direction of rotation of the vector E is determined by the same rule as the sign of the rotation of the polarization plane of linearly polarized light, i. e. when observing a wave going towards the observer (see footnote above).

In a dielectric environment in a magnetic field, orthogonal circularly polarized waves have different phase velocities. The reason for this is, as noted above, the influence of an external magnetic field on the electrons of the atoms of the environment, which leads to different coefficients of the effective refractive index n for orthogonal circularly polarized waves ER and EL. The physical mechanism of this effect manifests itself differently for the ER and EL waves, since for one of them the direction of rotation of the vector E coincides with the direction of precession, and for the other it is opposite to it. If there is no field, then nR = nL = n0 and the waves, when propagating in the environment, will acquire the same phase shifts; n0 is the part of the refractive index of the environment that is independent of the magnetic field. In the presence of an external magnetic field, the refractive indices of the left and right waves become different:

nR = n0 + δn (slow wave), (2a)

nL = n0 – δn (fast wave), (2b)

where δn << n0 for real magnetic fields. The sign of the amendment δn, and therefore, which of the waves will be slow and which is fast (in the case of a specific medium) depends on the ratio of the directions of propagation of light and the magnetic field vector.

In the presence of a magnetic field, the ER and EL waves after passing through a direct fiber with a length L along the direction of the magnetic field will acquire, according to (2a, 2b), different phase shifts due to the Faraday effect:

ϕR = 2π L nR / λ = ϕ0 + ϕF, (slow wave) (3a)

ϕL = 2π L nL / λ = ϕ0 – ϕF, (fast wave) (3b)

where ϕ0 = 2π L n0 / λ, and ϕF = θF = VLHz according to relation (1), Hz is the projection of the magnetic vector field in the direction of wave propagation. When waves propagate in the opposite direction, the signs in front of ϕF in (3) are reversed.

With a direct pass, the phase difference of the ER and EL waves due to the Faraday effect (Faraday phase shift) ΔϕF = ΔϕFfor will be equal to

ΔϕF = ϕR – ϕL = ΔϕFfor = 2ϕF = 2 VLHz . (3с)

Let’s now consider the total Faraday phase shift with a double passage of radiation through a fiber with a mirror at its end.

During the reverse propagation of a circularly polarized wave after its specular reflection at the end of the QF, there are two factors, each of which changes the sign of the Faraday phase shift. First, during reflection, the circular polarization changes to orthogonal. This is due to the change in the direction of wave propagation, and hence the direction of the observer’s look on the wave, to the opposite (this effect can be illustrated as a change in the direction of rotation of the clock when looking at the dial on the reverse side). Secondly, after specular reflection, the reciprocal orientation of the direction of propagation of light and the magnetic field vector changes. As a result of both factors, the sign of the phase shift during the back pass is preserved, so that ΔϕFback = 2VLHz, and the total phase shift is equal to

ΔϕF = ΔϕFfor + ΔϕFback = 4ϕF = 4 VLHz, (3d)

The relations (3a‑3d) represent the Faraday effect in terms of phase shifts of circularly polarized light waves.

The Faraday phase shift ΔϕF completely determines the angle θF of the rotation of the polarization plane of a linearly polarized wave, as follows from relations (3a‑3c). Let us explain this in the vector diagram in fig. 2. This shows the position of the field vectors ER and EL of the right and left circularly polarized waves at the input (a) and output (b) of the segment of the QF that is in a longitudinal magnetic field. Note that a change in the angle of rotation of these vectors corresponds to a change in the current phase of the waves; in particular, the full circle corresponds to a phase change of 2π. Let ЕR is a slow wave, ЕL is a fast wave. Suppose that, at the QF input, the waves have equal initial phases – the ЕR and ЕL vectors are oriented along the A – A line (Fig. 2a). At the QF output, the orientation of the wave vectors will have the form shown in Figure 2b. The ЕR and ЕL vectors, according to (3a, 3b), rotate at different angles ϕ0 + ϕF > ϕ0 – ϕF, respectively (vectors denoted by dotted lines indicate the position of ЕR and ЕL in the absence of a magnetic field ϕF=0). A phase difference ΔϕF = 2ϕF (Faraday phase shift) will arise between the field vectors in accordance with (3c). The B-B bisector between the ЕR and ЕL vectors, which is the sum of circularly polarized waves, corresponds to the orientation of the polarization plane of a linearly polarized wave at the QF output. The angle of rotation of the polarization plane is θF = ΔϕF / 2.

With the opposite direction of the magnetic field vector, when ЕR is a fast wave, ЕL is a slow wave, the polarization plane will turn through the angle θF = –ΔϕF / 2. Note that the phase of the waves and the angle of rotation of the polarization plane are expressed in the same units (radians) and the measurement of any of these parameters allows you to get information about the magnetic field of the current and, as a consequence, the current itself. Below, we will present further material in terms of the Faraday phase shifts of orthogonal circular waves, since in practice, it is easier to implement a highly accurate measurement of the electric current through the measurement of the relative phase shift between orthogonal light waves.

1.3.

Theorem on the circulation of a magnetic field in a closed loop

To obtain information on the magnitude of the measured current, the optical fiber is wound around a current bus. In practice, it is the fiber coil that is the basis of the sensing element of a real optical transformer. For high-precision measurement, the fiber loop must be closed (the beginning and end of the loop must be spatially aligned). This requirement is based on the fundamental law of nature, expressed by one of the Maxwell equations, namely, circuital law. This law is formulated as follows: the circulation (linear integral) of the magnetic field strength vector, excited by slowly varying electric currents along a closed loop L of an arbitrary shape, is equal to the algebraic sum of the currents I circuital law by this loop:

. (4)

If a closed loop consists of N windings of arbitrary shape covering the currents, then bypassing each turn of the loop will contribute I to the result of integration, therefore (4) will be expressed as follows:

. (5)

The Faraday phase shift during the passage of circular polarized light waves of the elementary part of the fiber dl, according to (3c), will have the form dϕF = 2VHdl. Performing integration over a closed contour L with regard to (5) (assuming a uniform magneto-optical sensitivity along the entire length of the contour), we obtain the relation connecting Faraday phase shift with the measured current:

.

In the case of specular reflection at the end of the fiber (the so-called reflective type of the optical scheme), the radiation passes the loop twice, in the forward and reverse directions. Therefore, similarly to (3d), the total Faraday shift also doubles:

. (6)

In conclusion, we emphasize that, as follows directly from the circuital law, with a closed loop of an arbitrary shape, the currents of adjacent buses do not contribute to integral (5); the position of the bar inside the contour does not affect the result either. Therefore, the condition of closed loop is one of the key requirements for high-precision measurement of electric current.

1.4.

Magnetic sensors for optical current transformers

In order for radiation to effectively accumulate Faraday’s phase shift due to the magnetic field of an electric current, an optical fiber must have the ability to maintain a circular or similar state of polarization of radiation (such fibers are called magnetooptically sensitive). Conventional single-mode ("isotropic") fibers used in communication lines do not fully possess this property due to induced birefringence due to external mechanical effects, as well as the presence of residual internal birefringence, in particular, because of the non-ideal initial preform fiber.

Currently, one of the most attractive fibers for current measurement is the spun fiber [2–4]. Spun fibers are obtained from a preform with a strong built in linear birefringence, which is rapidly rotated when drawing. We emphasize that the drawing technology assumes the absence of any elastic torsional stresses in the fiber. The key advantage of such fibers is the combination of high sensitivity to the Faraday effect with its relative resistance to external mechanical influences. These properties have largely determined the widespread use of spun fibers in industrial fiber-optic current transformers.

This type of optical fibers has a number of features, which will be discussed below. Due to the rotation of the preform when drawing, the axis of the embedded linear birefringence of the spun fiber have a spiral structure, therefore the polarization properties of the spun fibers are determined by two main parameters: the beat length of the embedded linear birefringence Lb and the pitch structure Ls. Such a structure has the ability to maintain the original direction of rotation of the vector of the electric field of the wave, to maintain in the fiber (on average) an elliptical polarization state, which ensures the accumulation of the Faraday phase shift when radiation propagates through the fiber. On the other hand, the spiral structure of the birefringence provides, to a certain extent, the suppression of linear birefringence induced by external influences on the fiber.

The main polarization properties of a spun fiber are characterized by a σ parameter [7]: σ = Ls / (2Lb). Thus, for σ << 1, the fiber has a small birefringence and / or a small pitch of the spiral and the ellipticity of the polarization state (PS) is close to unity (quasi-circular polarization state), and for σ ≥ 1 – a strong birefringence and / or large pitch of the helix, ellipticity of polarization averaged over the spectrum is substantially less than unity.

The small ellipticity (less than unity) of the polarization state leads to a decrease in the magneto-optical sensitivity of the spun fiber. This can be qualitatively understood by representing a wave with an elliptical PS as a sum of orthogonal circularly polarized components with different weights: a component with lower weight accumulates a Faraday phase shift with the opposite sign, thereby reducing the contribution of the main component to the resulting shift Δϕ F. These polarization properties are considered in the formula (6) for the Faraday phase shift by multiplying by the coefficient S = 1 / (1+σ2)1 / 2 [4, 7]:

ΔϕF = 4SVNI. (6а)

For typical spun fibers used in optical transformers current, Lb ~ 8 … 10 mm (λ = 1550 nm) and Ls≈3 mm, which means σ << 1, i. e. the state of polarization of radiation is close to circular one and, according to (6a), the sensitivity decreases slightly.

It should be noted that the phase difference of orthogonal circularly (elliptically) polarized light waves propagating through the spun fiber is influenced not only by the magnetic field of the measured current (Faraday effect), but also by the spiral the structure of the axes of the linear birefringence [8]. In the case of σ << 1, the phase difference Δϕc acquired by quasi-circularly polarized light waves when they propagate in the spun fiber is determined by the ratio

Δϕc ≈ πσL / Lb, (7)

where L is the fiber length. The physical mechanism of the influence of the spiral structure on the phase difference of light waves is associated with the presence of different phase velocities for orthogonal circularly polarized waves: for one wave, the direction of rotation of the electric field vector coincides with the direction of birefringence axis rotation, for orthogonal waves these directions are opposite.

It is important to note that the spiral structure of the birefringence is symmetric about the direction of light propagation (as opposed to the longitudinal component of the magnetic field vector), and its effect on the light wave manifests itself as a reciprocal effect. Therefore, when the circularly polarizated light waves pass through a spun fiber with mirror at its end, where after reflection waves exchange their polarization states, the total phase shift Δφc between them due to the birefringence axis spiral structure becomes equal to zero (compensated). This fact is undoubtedly an advantage of a sensitive fiber coil with a double radiation path (reflective version), since in this case the phase difference between the working wave is determined only by the magnetic field of the current being measured.

Furthermore, if a wide-band light source is used the phase shift (7), appeared between the quasi-circular light waves due to the birefringence axis spiral structure of the spun fiber, leads to a loss of coherence between the waves (depolarization) during their propogation through the fiber. The criterion for this process is the depolarization length Ld, which is determined by the parameters of the spun fiber and broadband radiation. For a rectangular spectrum, it is equal to [7]:

Ld = (Ls / 2) [(σ2+1)1 / 2 / σ2] (λ / Δλ), (8)

where λ is the central wavelength of the source, Δλ is the width of the spectrum. The phenomenon of the depolarization of light plays an important role in the operation of FOCT and will be discussed in the second part of the review.

The technical implementation of the method will be discussed in the second part of the article.

[2] Note: the state of polarization (in particular, the sign of the angle of rotation of the polarization plane), is considered in this review when observing a wave towards. A positive rotation angle corresponds to a counterclockwise rotation, and a negative one – to a clockwise rotation.

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