Abstract

The design principle exploiting the geometric rotation effect for the sensing coil of the fiber-optic current sensor (FOCS) on the basis of the polarization-rotated reflection interferometer is investigated. The sensing coil is formed by winding the low birefringence single-mode optical fiber in a toroidal spiral. The effects of the linear birefringence on the scale factor of the sensor can be suppressed with the reciprocal circular birefringence by appropriately designing the geometric parameters of the sensing coil. When the rated current is 1200Arms, the designed sensing coil can ensure the scale factor error of the sensor to satisfy the requirements of the 0.2 S class specified in IEC60044-8 over a temperature range from 40°C to 60 °C.

© 2012 Optical Society of America

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  4. K. Bohnert, P. Gabus, J. Kostovic, H. Brandle, and M. G. Brunzel, “Fiber-optic current sensor for electrowinning of metals,” J. Lightwave Technol. 25, 3602–3609 (2007).
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  8. J. D. P. Hrabliuk, “Optical current sensors eliminate CT saturation,” in Proceedings of IEEE Conference on Power Engineering Society Winter Meeting (IEEE, 2002), pp. 1478–1481.
  9. S. X. Short, P. Tantaswadi, R. T. de Carvalho, B. D. Russell, and J. N. Blake, “An experiment study of acoustic vibration effects in optical fiber current sensors,” IEEE Trans. Power Deliv. 11, 1702–1706 (1996).
    [CrossRef]
  10. J. N. Blake and A. H. Rose, “Fiber-optic current transducer optimized for power metering applications,” in Proceedings of IEEE Conference on Transmission and Distribution Conference and Exposition (IEEE, 2003), pp. 405–408.
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    [CrossRef]
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  28. Z. B. Ren, Y. Wang, and Ph. Robert, “Faraday rotation and its temperature dependence measurements in low-birefringence fibers,” J. Lightwave Technol. 7, 1275–1278 (1989).
    [CrossRef]
  29. P. A. Williams, A. H. Rose, G. W. Day, T. E. Milner, and M. N. Deeter, “Temperature dependence of the Verdet constant in several diamagnetic glass,” Appl. Opt. 30, 1176–1178 (1991).
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  30. P. Polynkin and J. N. Blake, “Polarization evolution in bent spun fiber,” J. Lightwave Technol. 23, 3815–3820 (2005).
    [CrossRef]
  31. Z. B. Ren, Ph. Robert, and P. A. Paratte, “Temperature dependence of bend- and twist-induced birefringence in a low-birefringence fiber,” Opt. Lett. 13, 62–64 (1988).
    [CrossRef]
  32. H. C. Lefevre, P. Martin, J. Nehring, P. Simonpietri, P. Vivenot, and H. J. Arditty, “High dynamic range fiber gyro with all-digital signal processing,” Proc. SPIE 1367, 72–80 (1990).
    [CrossRef]
  33. J. L. Cruz, M. V. Andres, and M. A. Hernandez, “Faraday effect in standard optical fibers: dispersion of effective Verdet constant,” Appl. Opt. 35, 922–927 (1996).
    [CrossRef]
  34. A. H. Rose, S. M. Etzel, and C. M. Wang, “Verdet constant dispersion in annealed optical fiber current sensors,” J. Lightwave Technol. 15, 803–807 (1997).
    [CrossRef]
  35. S. X. Short, A. A. Tselikov, J. U. de Arruda, and J. N. Blake, “Imperfect quarter-waveplate compensation in Sagnac interferometer-type current sensors,” J. Lightwave Technol. 16, 1212–1219 (1998).
    [CrossRef]

2007 (1)

2006 (2)

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Yu. K. Chamorovsky, A. I. Oussov, and S. Yu. Otrokhov, “All-fiber optical sensor of electrical current with a spun sensing element,” Proc. SPIE 6251, 62510P (2006).
[CrossRef]

C. D. Perciante and J. A. Ferrari, “Cancellation of bending-induced birefringence in single-mode fibers: applications to Faraday sensors,” Appl. Opt. 45, 1951–1956 (2006).
[CrossRef]

2005 (2)

K. Bohnert, P. Gabus, J. Kostovic, and H. Brandle, “Optical fiber sensors for the electric power industry,” Opt. Lasers Eng. 43, 511–526 (2005).
[CrossRef]

P. Polynkin and J. N. Blake, “Polarization evolution in bent spun fiber,” J. Lightwave Technol. 23, 3815–3820 (2005).
[CrossRef]

2002 (1)

2000 (1)

1998 (3)

1997 (3)

A. H. Rose, S. M. Etzel, and C. M. Wang, “Verdet constant dispersion in annealed optical fiber current sensors,” J. Lightwave Technol. 15, 803–807 (1997).
[CrossRef]

E. M. Frins and W. Dultz, “Rotation of the polarization plane in optical fibers,” J. Lightwave Technol. 15, 144–147 (1997).
[CrossRef]

K. Kurosawa, “Optical current transducers using flint glass fiber as the Faraday sensor element,” Opt. Rev. 4, 38–44 (1997).
[CrossRef]

1996 (4)

A. H. Rose, Z. B. Ren, and G. W. Day, “Twisting and annealing optical fiber for current sensors,” J. Lightwave Technol. 14, 2492–2498 (1996).
[CrossRef]

S. X. Short, P. Tantaswadi, R. T. de Carvalho, B. D. Russell, and J. N. Blake, “An experiment study of acoustic vibration effects in optical fiber current sensors,” IEEE Trans. Power Deliv. 11, 1702–1706 (1996).
[CrossRef]

J. N. Blake, P. Tantaswadi, and R. T. de Carvalho, “In-line Sagnac interferometer current sensor,” IEEE Trans. Power Deliv. 11, 116–121 (1996).
[CrossRef]

J. L. Cruz, M. V. Andres, and M. A. Hernandez, “Faraday effect in standard optical fibers: dispersion of effective Verdet constant,” Appl. Opt. 35, 922–927 (1996).
[CrossRef]

1994 (1)

1991 (2)

D. Tang, A. H. Rose, G. W. Day, and S. M. Etzel, “Annealing of linear birefringence in single-mode fibers coils: application to optical fiber current sensors,” J. Lightwave Technol. 9, 1031–1037 (1991).
[CrossRef]

P. A. Williams, A. H. Rose, G. W. Day, T. E. Milner, and M. N. Deeter, “Temperature dependence of the Verdet constant in several diamagnetic glass,” Appl. Opt. 30, 1176–1178 (1991).
[CrossRef]

1990 (1)

H. C. Lefevre, P. Martin, J. Nehring, P. Simonpietri, P. Vivenot, and H. J. Arditty, “High dynamic range fiber gyro with all-digital signal processing,” Proc. SPIE 1367, 72–80 (1990).
[CrossRef]

1989 (3)

F. Maystre and A. Bertholds, “Magneto-optic current sensor using a helical-fiber Fabry-Perot resonator,” Opt. Lett. 14, 587–589 (1989).
[CrossRef]

R. I. Laming and D. N. Payne, “Electric current sensors employing spun highly birefringence optical fibers,” J. Lightwave Technol. 7, 2084–2094 (1989).
[CrossRef]

Z. B. Ren, Y. Wang, and Ph. Robert, “Faraday rotation and its temperature dependence measurements in low-birefringence fibers,” J. Lightwave Technol. 7, 1275–1278 (1989).
[CrossRef]

1988 (1)

1984 (1)

J. N. Ross, “The rotation of the polarization in low birefringence monomode optical fibers due to geometric effects,” Opt. Quantum Electron. 16, 455–461 (1984).
[CrossRef]

1980 (1)

Andres, M. V.

Ankiewicz, A.

Arditty, H. J.

H. C. Lefevre, P. Martin, J. Nehring, P. Simonpietri, P. Vivenot, and H. J. Arditty, “High dynamic range fiber gyro with all-digital signal processing,” Proc. SPIE 1367, 72–80 (1990).
[CrossRef]

Bertholds, A.

Blake, J. N.

P. Polynkin and J. N. Blake, “Polarization evolution in bent spun fiber,” J. Lightwave Technol. 23, 3815–3820 (2005).
[CrossRef]

S. X. Short, A. A. Tselikov, J. U. de Arruda, and J. N. Blake, “Imperfect quarter-waveplate compensation in Sagnac interferometer-type current sensors,” J. Lightwave Technol. 16, 1212–1219 (1998).
[CrossRef]

S. X. Short, J. U. de Arruda, A. A. Tselikov, and J. N. Blake, “Elimination of birefringence induced scale factor errors in the in-line Sagnac interferometer current sensor,” J. Lightwave Technol. 16, 1844–1850 (1998).
[CrossRef]

S. X. Short, P. Tantaswadi, R. T. de Carvalho, B. D. Russell, and J. N. Blake, “An experiment study of acoustic vibration effects in optical fiber current sensors,” IEEE Trans. Power Deliv. 11, 1702–1706 (1996).
[CrossRef]

J. N. Blake, P. Tantaswadi, and R. T. de Carvalho, “In-line Sagnac interferometer current sensor,” IEEE Trans. Power Deliv. 11, 116–121 (1996).
[CrossRef]

J. N. Blake and A. H. Rose, “Fiber-optic current transducer optimized for power metering applications,” in Proceedings of IEEE Conference on Transmission and Distribution Conference and Exposition (IEEE, 2003), pp. 405–408.

J. N. Blake and A. H. Rose, “Interfacing optical CTs and VTs to relays and meters,” in Proceedings of IEEE Conference on Transmission and Distribution Conference and Exhibition (IEEE, 2006), pp. 1280–1284.

F. Rahmatian and J. N. Blake, “Application of high-voltage fiber-optic current sensors,” presented at IEEE Conference on Power Engineering Society General Meeting, Montreal, Quebec, Canada,19–22 June 2006.

J. N. Blake and A. H. Rose, “Precision fiber-optic current sensor as a check-standard,” in Proceedings of IEEE Conference on Power Engineering Society Summer Meeting (IEEE, 2002), pp. 904–908.

Bohnert, K.

Brandle, H.

Brunzel, M. G.

Chamorovsky, Yu. K.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Yu. K. Chamorovsky, A. I. Oussov, and S. Yu. Otrokhov, “All-fiber optical sensor of electrical current with a spun sensing element,” Proc. SPIE 6251, 62510P (2006).
[CrossRef]

Cruz, J. L.

Culshaw, B.

Dandliker, R.

Day, G. W.

A. H. Rose, Z. B. Ren, and G. W. Day, “Twisting and annealing optical fiber for current sensors,” J. Lightwave Technol. 14, 2492–2498 (1996).
[CrossRef]

D. Tang, A. H. Rose, G. W. Day, and S. M. Etzel, “Annealing of linear birefringence in single-mode fibers coils: application to optical fiber current sensors,” J. Lightwave Technol. 9, 1031–1037 (1991).
[CrossRef]

P. A. Williams, A. H. Rose, G. W. Day, T. E. Milner, and M. N. Deeter, “Temperature dependence of the Verdet constant in several diamagnetic glass,” Appl. Opt. 30, 1176–1178 (1991).
[CrossRef]

de Arruda, J. U.

de Carvalho, R. T.

S. X. Short, P. Tantaswadi, R. T. de Carvalho, B. D. Russell, and J. N. Blake, “An experiment study of acoustic vibration effects in optical fiber current sensors,” IEEE Trans. Power Deliv. 11, 1702–1706 (1996).
[CrossRef]

J. N. Blake, P. Tantaswadi, and R. T. de Carvalho, “In-line Sagnac interferometer current sensor,” IEEE Trans. Power Deliv. 11, 116–121 (1996).
[CrossRef]

Deeter, M. N.

Dultz, W.

E. M. Frins and W. Dultz, “Rotation of the polarization plane in optical fibers,” J. Lightwave Technol. 15, 144–147 (1997).
[CrossRef]

Eickhoff, W.

Etzel, S. M.

A. H. Rose, S. M. Etzel, and C. M. Wang, “Verdet constant dispersion in annealed optical fiber current sensors,” J. Lightwave Technol. 15, 803–807 (1997).
[CrossRef]

D. Tang, A. H. Rose, G. W. Day, and S. M. Etzel, “Annealing of linear birefringence in single-mode fibers coils: application to optical fiber current sensors,” J. Lightwave Technol. 9, 1031–1037 (1991).
[CrossRef]

Ferrari, J. A.

Frins, E. M.

E. M. Frins and W. Dultz, “Rotation of the polarization plane in optical fibers,” J. Lightwave Technol. 15, 144–147 (1997).
[CrossRef]

Frosio, G.

Gabus, P.

Gubin, V. P.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Yu. K. Chamorovsky, A. I. Oussov, and S. Yu. Otrokhov, “All-fiber optical sensor of electrical current with a spun sensing element,” Proc. SPIE 6251, 62510P (2006).
[CrossRef]

Hernandez, M. A.

Hrabliuk, J. D. P.

J. D. P. Hrabliuk, “Optical current sensors eliminate CT saturation,” in Proceedings of IEEE Conference on Power Engineering Society Winter Meeting (IEEE, 2002), pp. 1478–1481.

Isaev, V. A.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Yu. K. Chamorovsky, A. I. Oussov, and S. Yu. Otrokhov, “All-fiber optical sensor of electrical current with a spun sensing element,” Proc. SPIE 6251, 62510P (2006).
[CrossRef]

Kostovic, J.

K. Bohnert, P. Gabus, J. Kostovic, H. Brandle, and M. G. Brunzel, “Fiber-optic current sensor for electrowinning of metals,” J. Lightwave Technol. 25, 3602–3609 (2007).
[CrossRef]

K. Bohnert, P. Gabus, J. Kostovic, and H. Brandle, “Optical fiber sensors for the electric power industry,” Opt. Lasers Eng. 43, 511–526 (2005).
[CrossRef]

Kurosawa, K.

K. Kurosawa, “Optical current transducers using flint glass fiber as the Faraday sensor element,” Opt. Rev. 4, 38–44 (1997).
[CrossRef]

Laming, R. I.

R. I. Laming and D. N. Payne, “Electric current sensors employing spun highly birefringence optical fibers,” J. Lightwave Technol. 7, 2084–2094 (1989).
[CrossRef]

Lefevre, H. C.

H. C. Lefevre, P. Martin, J. Nehring, P. Simonpietri, P. Vivenot, and H. J. Arditty, “High dynamic range fiber gyro with all-digital signal processing,” Proc. SPIE 1367, 72–80 (1990).
[CrossRef]

Martin, P.

H. C. Lefevre, P. Martin, J. Nehring, P. Simonpietri, P. Vivenot, and H. J. Arditty, “High dynamic range fiber gyro with all-digital signal processing,” Proc. SPIE 1367, 72–80 (1990).
[CrossRef]

Maystre, F.

Milner, T. E.

Morshnev, S. K.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Yu. K. Chamorovsky, A. I. Oussov, and S. Yu. Otrokhov, “All-fiber optical sensor of electrical current with a spun sensing element,” Proc. SPIE 6251, 62510P (2006).
[CrossRef]

Nehring, J.

K. Bohnert, P. Gabus, J. Nehring, and H. Brandle, “Temperature and vibration insensitive fiber-optic current sensor,” J. Lightwave Technol. 20, 267–276 (2002).
[CrossRef]

H. C. Lefevre, P. Martin, J. Nehring, P. Simonpietri, P. Vivenot, and H. J. Arditty, “High dynamic range fiber gyro with all-digital signal processing,” Proc. SPIE 1367, 72–80 (1990).
[CrossRef]

Ortega, A.

F. Rahmatian and A. Ortega, “Application of optical current and voltage sensors in high-voltage system,” in Proceedings of IEEE Conference on Transmission & Distribution Conference and Exposition (IEEE, 2006), paper PT1-21.

Otrokhov, S. Yu.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Yu. K. Chamorovsky, A. I. Oussov, and S. Yu. Otrokhov, “All-fiber optical sensor of electrical current with a spun sensing element,” Proc. SPIE 6251, 62510P (2006).
[CrossRef]

Oussov, A. I.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Yu. K. Chamorovsky, A. I. Oussov, and S. Yu. Otrokhov, “All-fiber optical sensor of electrical current with a spun sensing element,” Proc. SPIE 6251, 62510P (2006).
[CrossRef]

Paratte, P. A.

Payne, D. N.

R. I. Laming and D. N. Payne, “Electric current sensors employing spun highly birefringence optical fibers,” J. Lightwave Technol. 7, 2084–2094 (1989).
[CrossRef]

Perciante, C. D.

Polynkin, P.

Rahmatian, F.

F. Rahmatian and J. N. Blake, “Application of high-voltage fiber-optic current sensors,” presented at IEEE Conference on Power Engineering Society General Meeting, Montreal, Quebec, Canada,19–22 June 2006.

F. Rahmatian, “Design and application of optical voltage and current sensors for relaying,” in Proceedings of IEEE Conference on Power Systems Conference and Exposition (IEEE, 2006), pp. 532–537.

F. Rahmatian and A. Ortega, “Application of optical current and voltage sensors in high-voltage system,” in Proceedings of IEEE Conference on Transmission & Distribution Conference and Exposition (IEEE, 2006), paper PT1-21.

Rashleigh, S. C.

Ren, Z. B.

A. H. Rose, Z. B. Ren, and G. W. Day, “Twisting and annealing optical fiber for current sensors,” J. Lightwave Technol. 14, 2492–2498 (1996).
[CrossRef]

Z. B. Ren, Y. Wang, and Ph. Robert, “Faraday rotation and its temperature dependence measurements in low-birefringence fibers,” J. Lightwave Technol. 7, 1275–1278 (1989).
[CrossRef]

Z. B. Ren, Ph. Robert, and P. A. Paratte, “Temperature dependence of bend- and twist-induced birefringence in a low-birefringence fiber,” Opt. Lett. 13, 62–64 (1988).
[CrossRef]

Robert, Ph.

Z. B. Ren, Y. Wang, and Ph. Robert, “Faraday rotation and its temperature dependence measurements in low-birefringence fibers,” J. Lightwave Technol. 7, 1275–1278 (1989).
[CrossRef]

Z. B. Ren, Ph. Robert, and P. A. Paratte, “Temperature dependence of bend- and twist-induced birefringence in a low-birefringence fiber,” Opt. Lett. 13, 62–64 (1988).
[CrossRef]

Rose, A. H.

A. H. Rose, S. M. Etzel, and C. M. Wang, “Verdet constant dispersion in annealed optical fiber current sensors,” J. Lightwave Technol. 15, 803–807 (1997).
[CrossRef]

A. H. Rose, Z. B. Ren, and G. W. Day, “Twisting and annealing optical fiber for current sensors,” J. Lightwave Technol. 14, 2492–2498 (1996).
[CrossRef]

D. Tang, A. H. Rose, G. W. Day, and S. M. Etzel, “Annealing of linear birefringence in single-mode fibers coils: application to optical fiber current sensors,” J. Lightwave Technol. 9, 1031–1037 (1991).
[CrossRef]

P. A. Williams, A. H. Rose, G. W. Day, T. E. Milner, and M. N. Deeter, “Temperature dependence of the Verdet constant in several diamagnetic glass,” Appl. Opt. 30, 1176–1178 (1991).
[CrossRef]

J. N. Blake and A. H. Rose, “Interfacing optical CTs and VTs to relays and meters,” in Proceedings of IEEE Conference on Transmission and Distribution Conference and Exhibition (IEEE, 2006), pp. 1280–1284.

J. N. Blake and A. H. Rose, “Fiber-optic current transducer optimized for power metering applications,” in Proceedings of IEEE Conference on Transmission and Distribution Conference and Exposition (IEEE, 2003), pp. 405–408.

J. N. Blake and A. H. Rose, “Precision fiber-optic current sensor as a check-standard,” in Proceedings of IEEE Conference on Power Engineering Society Summer Meeting (IEEE, 2002), pp. 904–908.

Ross, J. N.

J. N. Ross, “The rotation of the polarization in low birefringence monomode optical fibers due to geometric effects,” Opt. Quantum Electron. 16, 455–461 (1984).
[CrossRef]

Russell, B. D.

S. X. Short, P. Tantaswadi, R. T. de Carvalho, B. D. Russell, and J. N. Blake, “An experiment study of acoustic vibration effects in optical fiber current sensors,” IEEE Trans. Power Deliv. 11, 1702–1706 (1996).
[CrossRef]

Sazonov, A. I.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Yu. K. Chamorovsky, A. I. Oussov, and S. Yu. Otrokhov, “All-fiber optical sensor of electrical current with a spun sensing element,” Proc. SPIE 6251, 62510P (2006).
[CrossRef]

Senthilkumaran, P.

Short, S. X.

Simonpietri, P.

H. C. Lefevre, P. Martin, J. Nehring, P. Simonpietri, P. Vivenot, and H. J. Arditty, “High dynamic range fiber gyro with all-digital signal processing,” Proc. SPIE 1367, 72–80 (1990).
[CrossRef]

Starostin, N. I.

V. P. Gubin, V. A. Isaev, S. K. Morshnev, A. I. Sazonov, N. I. Starostin, Yu. K. Chamorovsky, A. I. Oussov, and S. Yu. Otrokhov, “All-fiber optical sensor of electrical current with a spun sensing element,” Proc. SPIE 6251, 62510P (2006).
[CrossRef]

Tang, D.

D. Tang, A. H. Rose, G. W. Day, and S. M. Etzel, “Annealing of linear birefringence in single-mode fibers coils: application to optical fiber current sensors,” J. Lightwave Technol. 9, 1031–1037 (1991).
[CrossRef]

Tantaswadi, P.

S. X. Short, P. Tantaswadi, R. T. de Carvalho, B. D. Russell, and J. N. Blake, “An experiment study of acoustic vibration effects in optical fiber current sensors,” IEEE Trans. Power Deliv. 11, 1702–1706 (1996).
[CrossRef]

J. N. Blake, P. Tantaswadi, and R. T. de Carvalho, “In-line Sagnac interferometer current sensor,” IEEE Trans. Power Deliv. 11, 116–121 (1996).
[CrossRef]

Thursby, G.

Tselikov, A. A.

Ulrich, R.

Vivenot, P.

H. C. Lefevre, P. Martin, J. Nehring, P. Simonpietri, P. Vivenot, and H. J. Arditty, “High dynamic range fiber gyro with all-digital signal processing,” Proc. SPIE 1367, 72–80 (1990).
[CrossRef]

Wang, C. M.

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Figures (13)

Fig. 1.
Fig. 1.

Polarization orientation of the light in the Frenet–Serret coordinates.

Fig. 2.
Fig. 2.

Toroidal spiral sensing coil.

Fig. 3.
Fig. 3.

Evolution of the circular polarized light in the sensing coil.

Fig. 8.
Fig. 8.

Normalized scale factor versus current.

Fig. 4.
Fig. 4.

Normalized scale factor versus number of the fiber loops.

Fig. 5.
Fig. 5.

(a) Normalized scale factor and (b) bend-induced LB and geometric CB versus APR.

Fig. 6.
Fig. 6.

(a) Normalized scale factor and (b) bend-induced LB and geometric CB versus minor radius.

Fig. 7.
Fig. 7.

(a) Normalized scale factor and (b) bend-induced LB and geometric CB versus major radius.

Fig. 9.
Fig. 9.

Normalized scale factor versus sensing coil temperature.

Fig. 10.
Fig. 10.

FOCS configuration based on a polarization-rotated reflection interferometer.

Fig. 11.
Fig. 11.

Normalized scale factor versus wavelength for different (a) currents and (b) temperatures.

Fig. 12.
Fig. 12.

Scale factor error test result for currents from 12 to 3600Arms.

Fig. 13.
Fig. 13.

Scale factor error test result for different sensing coil temperatures.

Tables (1)

Tables Icon

Table 1. Relative Variation of Scale Factor from 40°C to 60 °C

Equations (31)

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E=Enn+Ebb,
dEds=dEndsn+Endnds+dEbdsb+Ebdbds.
dEds=κ(E·n)t+δ(E×t),
dEds=dEndsn+dEbdsbEbτn+En(τbκt),
dds(EbEn)=(0(τ+δ)(τ+δ)0)(EbEn).
dds(EbEn)=(0ττ0)(EbEn).
dEds=κ(E·n)tδ(E×t),
dEds=dEndsn+dEbdsb+EbτnEn(τbκt).
dds(EbEn)=(0ττ0)(EbEn).
ρ(θ)=(R+rcosnθ)cosθ·i+(R+rcosnθ)sinθ·j+rsinnθ·k,
κ=|ρ×ρ||ρ|3,
τ=(ρ,ρ,ρ)(ρ×ρ)2,
H(θ)=I2π·ρyρx2+ρy2·i+I2π·ρxρx2+ρy2·j,
β(θ)=0.858λ·(rfiberκ(θ))2,
ds=ρxdθ·i+ρydθ·j+ρzdθ·k,
ds=l(θ)dθ=(R+rcosnθ)2+r2n2dθ.
Jin=(ejβ2ds00ejβ2ds)·(cos(τds+VH·ds)sin(τds+VH·ds)sin(τds+VH·ds)cos(τds+VH·ds))(1jβ2ldθ(τl+Vf)dθ(τl+Vf)dθ1+jβ2ldθ),
(Einb(θ+dθ)Einn(θ+dθ))=Jin·(Einb(θ)Einn(θ)),
ddθ(EinbEinn)=(jβ2l(τl+Vf)(τl+Vf)jβ2l)(EinbEinn).
Einl(θ)=Einb(θ)+jEinn(θ)2,
Einr(θ)=Einb(θ)jEinn(θ)2,
ddθ(EinlEinr)=(j(τl+Vf)jβ2ljβ2lj(τl+Vf))(EinlEinr).
Jout(1jβ2ldθ(τlVf)dθ(τlVf)dθ1+jβ2ldθ).
ddθ(EoutlEoutr)=(j(τlVf)jβ2ljβ2lj(τlVf))(EoutlEoutr),
Elin|θ=0=(Elinl,Elinr)T|θ=0=(1,0)T,
Erin|θ=0=(Erinl,Erinr)T|θ=0=(0,1)T,
KSF=Δφ4F,
T=202πτ(θ)l(θ)dθ,
Γ=02πβ(θ)l(θ)dθ.
L=02πl(θ)dθ.
KSF¯=λ1λ2Δφ(λ)dλ4NIλ1λ2V(λ)dλ.

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