Question 1: What is the physical principle of operation of the current sensor (sensing element)? Is there a difference when using a rigid head versus a flexible loop?

Fig. 1 Physical principle of current measurement
The basis for measuring electrical current magnitude with the device manufactured by "Profrotech" lies in the application of the Faraday effect within a quartz waveguide. Fig. 1 shows how the Faraday effect is used to measure electrical current. Two light waves—one with right-handed and the other with left-handed circular polarization (gray and green in the figure)—are introduced into a multi-turn fiber optic loop that encircles the conductor carrying the current. If there is no current in the wire, the light waves propagate at the same speed and arrive at the output with zero relative phase difference. In the presence of an electrical current in the conductor, the waveguide is subjected to the longitudinal magnetic field of the flowing current. This induces circular birefringence in the waveguide, causing the propagation speeds of the light waves along the loop to differ. Consequently, a time delay and a relative phase shift $\phi$ arise between the waves. If the waveguide has uniform magneto-optical sensitivity along its length and the fiber optic loop is closed (the beginning and end of the sensing fiber in the loop coincide), then the relationship between the phase shift and the current is expressed by the simple formula $\phi = 2VNI$, where $V$, $N$, and $I$ are the magneto-optical constant of quartz, the number of turns, and the measured current, respectively. Thus, by measuring the relative phase shift between the light waves, we obtain information about the magnitude of the current. The phase shift is measured using a low-coherence fiber optic interferometer included in the device under discussion.
From the standpoint of the physical principle of operation, there is no difference between a rigid sensing loop (rigid head) and a flexible loop.
Both the rigid head and the flexible loop represent a closed fiber optic loop (single- or multi-turn) into which two light waves with orthogonal circular polarizations are introduced. The difference can be seen in the number of turns used and the nuances of installation on the current busbar. Specifically, the number of fiber optic turns in the sensing loop is determined by the magnitude and range of the measured electrical current and the accuracy class of electronic optical current transformers (EOCT). In practice, a rigid head may have from 1 to 100 fiber optic turns, while a flexible loop may have from 1 to 20 turns.

Read also: All the truth about optical transformers: part 1

Question 2: How are light waves with circular polarizations formed at the input of the sensitive fiber loop? What is a low-coherence fiber interferometer?

To ensure an error in measuring the electric current magnitude at a level of 0.2% or lower, the measurement of the phase shift between light waves induced by the magnetic field of the measured current must be performed with an error of no more than several microradians in a 1 Hz bandwidth. This is very high precision, and it is achieved using the method of low-coherence fiber-optic interferometry, which has two distinctive features: the use of low-coherence optical radiation (the coherence length is several tens of wavelengths) and the application of a mutual fiber interferometer with an optical path difference close to zero for the operating light waves. Hence the term "low-coherence fiber interferometer" arose. The diagram of such a device is shown in Fig. 2.
Fig. 2. Low-coherence fiber interferometer: 1. Low-coherence radiation source. 2. Photodetector. 3. Fiber coupler. 4. Fiber polarizer. 5. Piezoelectric fiber modulator. 6. Fiber delay line. 7. Fiber polarization converter. 8. Magneto-sensitive multi-turn fiber loop. 9. Mirror. 10. Current busbar.
Two light waves with orthogonal linear polarizations, formed by the fiber polarizer (4), arrive at the polarization converter (7). Using the polarization converter (7) at the input of the fiber loop (8), two light waves with right- and left-handed circular polarizations are formed; after passing through the sensing fiber loop, they are specularly reflected at its end (mirror, 9) and then travel in the reverse direction. During specular reflection, the circular polarizations are converted into orthogonal ones (left to right and vice versa), which allows for the compensation of all parasitic mutual interactions within the sensing fiber loop that could affect the accuracy of measuring the phase shift caused by the magnetic field of the electric current. Similarly, parasitic effects in the optical part of the interferometer are minimized from the polarizer (4), where the waves interfere, up to the polarization converter (7)—the only difference being that here, during reverse propagation, orthogonally polarized waves are converted into linearly polarized waves. As a result, the difference in the optical paths of the interferometer's working waves is determined solely by the nonreciprocal Faraday phase shift caused by the measured current, which ensures the required high measurement accuracy.

Question 3: Why does the measuring loop measure the current only within the loop itself and remains insensitive to external fields (including the Earth's magnetic field), and why does the positioning of the current busbar relative to the measuring loop not affect the measurement accuracy?

These properties of the measuring loop are determined by a fundamental physical law of nature, expressed by one of Maxwell's equations—the theorem of circulation of the magnetic field strength vector around an arbitrary closed loop. The essence is as follows. In its original form, the expression for the relative phase shift between the working light waves induced by the magnetic field of the measured current looks like this: The circulation theorem states that the circulation of the magnetic field vector $\mathbf{H}$ (excited by stationary currents) around an arbitrary closed loop (closure is key) is equal to the sum of those currents $I_j$ that cross the surface bounded by this loop (formula (2), Fig. 3):
Fig. 3. Graphical explanation of the circulation theorem
This theorem is absolute; that is, if a current crosses the surface bounded by the loop, it contributes to the circulation (regardless of how the current is positioned within the loop). The loop senses the current identically, regardless of the position of the busbar within the loop. If the current does not cross the surface bounded by the loop, then the circulation is zero and $\Delta\phi$ in formula (1) is zero; therefore, the loop does not sense the current of a busbar located outside the loop. External magnetic fields can be modeled by external currents that do not cross the surface bounded by the loop. Sensitivity to such fields is zero. The contribution to the signal in this case is also zero. The same applies to the Earth's magnetic field: it is created by currents in the Earth's core, which, naturally, do not cross the surface bounded by the fiber loop. If the geometry of the closed loop changes—for example, if a flexible loop is deformed—this does not affect the magnitude of the circulation, provided that the number and strength of the currents crossing the surface bounded by this loop remain unchanged. Circulation is an integral quantity that does not depend on the method used to create the closed loop. It is calculated as the sum of the projections of the magnetic field at the point where the element $d\vec{I}$ is located, multiplied by the length $|dl|$. When a closed loop is deformed, the magnetic fields at each point may change, but the integral quantity—the circulation of the magnetic field—will remain unchanged. From here, we obtain conclusions that answer the following questions:
  • The signal measured by the current sensor is proportional to the sum of currents crossing the surface bounded by the loop. All geometric characteristics remain within the magnetic field circulation.
  • External magnetic fields can be modeled by external currents that do not cross the surface bounded by the loop. Sensitivity to such currents is zero. By modeling the Earth's magnetic field also as an external current, we similarly obtain zero sensitivity to the Earth's magnetic field.
  • Moving the current busbar within the loop does not change the conditions of the circulation theorem and, accordingly, does not affect measurement accuracy: the same current still crosses the same surface bounded by the loop. The field strength at the locations of elements $d\vec{I}$, generally speaking, changes when the busbar is displaced; however, the circulation as an integral remains unchanged.

Question 4: What is the property of optical loop closure (i.e., alignment of marks), and how can it be physically explained? What is the essence of the closure, and what will happen if the marks are not aligned with each other, for example, by 10 cm? How close must the fibers being aligned be to each other in the area of the marks, provided that the marks are aligned in a plane, while the turns are spaced 10 cm apart?

A closed fiber loop refers to the complete alignment of the start and end of the sensitive spun fiber along all three coordinates (i.e., axis-axis). Regarding two coordinates, there are no fundamental restrictions on alignment in loops from "Profrotech". For the third coordinate, the limit is the fiber diameter (250 $\mu$m) or the cable diameter (1 cm). In this case, the closure error can be estimated by the value $d_{vol}/L_{cont}$ (rigid head) or $D_{kab}/L_{cont}$ (flexible loop), where $d_{vol}$ is the diameter of the fiber in its protective cladding, $D_{kab}$ is the diameter of the flexible loop cable, and $L_{cont}$ is the length of the loop. In the case of a rigid head with $L_{cont} = 100\text{ m}$ and $d_{wave} = 250\text{ \mu m}$, the closure error is $2.5 \times 10^{-6}$. This value can be used to estimate the residual influence of current bus positioning within the rigid head loop on measurement accuracy. In this case, for a fixed temperature, the residual influence of bus positioning will be $2.5 \times 10^{-6}$ of the measured current value. Within an operating temperature range of $100^\circ\text{C}$, with a loop length of $100\text{ m}$, the gap will change by $1\text{ cm}$. The residual influence of bus positioning will increase to $0.0001$ of the measured current. Such a value is small ($0.01\%$), i.e., it is the level of a good reference transformer. The contribution from external currents will be $0.0001$ of the value of their magnetic fields within the loop area. Using the formula for a flexible loop, one can see that a displacement of $10\text{ cm}$ along one coordinate relative to each other results in a closure error of $0.01$ (at a full loop length of $10\text{ m}$), which can change current measurement readings by up to $1\%$ when positioning the bus within the loop. It is clear that this is unacceptable. The desirable minimum acceptable displacement is $0.001$ of the flexible loop length and does not exceed $0.0001$ for a rigid measurement loop. In reality, the closure of the sensing fiber loop occurs by aligning a fiber phase plate, located at the input of the sensing fiber, with a mirror located at the other end of the same fiber. In the case of a rigid head, such alignment is performed once during manufacturing assembly. Measures are taken to ensure that mechanical impacts do not disrupt the loop's closure. In the case of a flexible loop, loop closure is required during each new installation of the loop at the site. In this case, the criterion for closure is the alignment of marks applied to the outer side of the fiber cable loop. The alignment of these marks is secured by a special and very reliable lock.

Read also: The whole truth about optical transformers: part 2

Question 5: What is the physical essence of closing an optical loop? Why is it that specifically upon aligning the marks, the influence of currents from the outside of the fiber disappears, yet sensitivity to currents located inside the closed optical loop remains?

A strict answer to these questions for a closed loop of arbitrary shape is provided by one of Maxwell's equations (the theorem of circulation of the magnetic field vector $\mathbf{H}$ excited by stationary currents). A qualitative explanation can also be provided using the example of a simple circular loop. The phase difference between right and left circularly polarized waves (Faraday shift) when waves pass through an elementary directed section of a waveguide (element) under the influence of a current's magnetic field is determined by the scalar product of the magnetic field vector created by the current and the direction of this fiber element. Hence, the sign of the Faraday shift is determined by the projection of the magnetic field vector created by the current onto the direction of this fiber element. A closed circular loop can be represented as a sum of similar directed elements; in this case, if the busbar is inside the loop, then at any given moment in time, the projection of the magnetic field vector created by the current in the busbar onto a directed element has the same sign for every element of the loop. Consequently, the sign of the Faraday phase shift is the same for any element of the loop. The Faraday shift across the entire loop will be equal to the simple arithmetic sum of the elementary Faraday shifts (the current is sensed inside the loop).
For a busbar outside the loop, the situation is different.
For half of the elements forming the loop, the projection of the external current's magnetic field vector onto the element is negative, and for the other half of the elements, it is positive. Consequently, one half of the Faraday shifts will have a negative sign, and the other half will be positive. In this case, the total shift around the loop will be equal to the algebraic sum of the elementary shifts, half of which are "plus" and half of which are "minus". A circular loop can be divided into elements such that elementary shifts with opposite signs will be equal in magnitude. As a result, all positive shifts are compensated by negative ones (the external current is not sensed by the closed loop). If the loop is not fully closed, the zero balance is disrupted—and the loop senses the residual external current. If you have any questions about optical transformers, ask them in the comments on the website, in our social networks (vk, fb), or in the Telegram chat; we will certainly find an answer.