Abstract

An optical fiber sensor is presented that allows current and voltage to be measured simultaneously by use of only one block of bismuth germanate crystal. The polarized light from the sensing crystal is split into two light beams: One beam is utilized for current measurement based on the Faraday effect, and the other one is utilized for voltage measurement based on the Pockels effect. Compared with the existing optical sensors that can measure current and voltage simultaneously, this sensor is simple and inexpensive and allows measurement of electric power. The simultaneous measurements of ac electric current from 0.05 to 10 A, voltage from 1 to 235 V, and power from 2 to 1000 W have been achieved with good linear-response characteristics. The input characteristics and measurement uncertainties that are due to the nonlinear error of the sensing system are also discussed.

© 2002 Optical Society of America

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References

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  1. A. J. Rogers, “Method for simultaneous measurement of current and voltage on high-voltage lines using optical techniques,” Proc. IEE 123, 957–960 (1976).
  2. A. J. Rogers, “Optical methods for measurement of voltage and current on power system,” Opt. Laser Technol. 9, 273–283 (1977).
    [CrossRef]
  3. N. Rajkumar, V. Jagadeesh Kumar, P. Sankaran, “Fiber sensor for the simultaneous measurement of current and voltage in a high-voltage system,” Appl. Opt. 32, 1225–1228 (1993).
    [CrossRef] [PubMed]
  4. R. T. de Carvalho, J. Blake, “Simultaneous measurement of electric and magnetic fields using a Sagnac interferometer,” in Tenth International Conference on Optical Fiber Sensors, B. Culshaw, J. D. Jones, eds., Proc. SPIE2360, 411–414 (1994).
    [CrossRef]
  5. Z. Wang, Y. Liao, S. Lai, H. Zhao, X. Chen, “A novel method for simultaneous measurement of current and voltage using one low-birefringence fiber,” Opt. Laser Technol. 30, 257–262 (1998).
    [CrossRef]
  6. C. Li, X. Cui, “Pulse-controlled polarization converter and its application,” Acta Photonica Sin. 26, 929–934 (1997).
  7. Z. Yang, C. Li, W. Zhang, X. Cui, “Novel multifunction optical fiber sensing system for simultaneous measurement of current and voltage,” in Optical and Fiber Optic Sensor System, S. Huang, K. D. Bennett, D. A. Jackson, eds., Proc. SPIE3555, 57–62 (1998).
    [CrossRef]
  8. A. H. Rose, S. M. Etzel, K. B. Rochford, “Optical fiber current sensors in high electric field environments,” J. Lightwave Technol. 17, 1042–1048 (1999).
    [CrossRef]
  9. C. Li, X. Cui, “Coupled-wave analysis on the electro-optic and magneto-optic interaction in BSO crystal and its application to sensors,” Acta Photonica Sin. 27, 122–126 (1998).
  10. R. Nitsche, “Crystal growth and electro-optic effect of bismuth germanate, Bi4(GeO4)3,” J. Appl. Phys. 36, 2358–2360 (1965).
    [CrossRef]
  11. H. Wang, W. Jia, J. Shen, “Magneto-optical Faraday rotation in Bi4Ge3O12 crystal,” Acta Phys. Sin. 34, 126–128 (1985).
  12. P. A. Williams, A. H. Rose, K. S. Lee, D. C. Conrad, G. W. Day, P. D. Hale, “Optical, thermo-optic, electro-optic, and photo-elastic properties of bismuth germinate (Bi4Ge3O12),” Appl. Opt. 35, 3562–3569 (1996).
    [CrossRef] [PubMed]
  13. M. Kanoi, G. Takahashi, T. Sato, M. Higaki, E. Mori, K. Okumura, “Optical voltage and current measuring system for electric power systems,” IEEE Trans. Power Deliv. 1, 91–97 (1986).
    [CrossRef]
  14. J. C. Santos, K. Hidaka, “Optical high voltage measurement technique using Pockels device,” Jpn. J. Appl. Phys. Part I 36, 2394–2398 (1997).
    [CrossRef]
  15. A. Kumada, M. Chiba, K. Hidaka, “Potential distribution measurement of surface discharge by Pockels sensing technique,” J. Appl. Phys. 84, 3059–3065 (1998).
    [CrossRef]
  16. T. A. Maldonado, T. K. Gaylord, “Accurate method to determine the eigenstates of polarization in gyrotropic media,” Appl. Opt. 28, 2075–2086 (1989).
    [CrossRef] [PubMed]
  17. W. J. Tabor, F. S. Chen, “Electromagnetic propagation through materials possessing both Faraday rotation and birefringence: experiments with ytterbium orthoferrite,” J. Appl. Phys. 40, 2760–2765 (1969).
    [CrossRef]
  18. S. Huard, Polarization of Light (Wiley, Chichester, UK, 1997), Chaps. 1 and 3.5.
  19. X. Cui, C. Li, “Optical propagation property of BSO crystal under simultaneously applied electric and magnetic fields and its application in electric power sensor,” Chin. J. Lasers A24, 347–351 (1997).
  20. P. A. Williams, G. W. Day, A. H. Rose, “Compensation for temperature dependence of Faraday effect in diamagnetic materials: application to optical fiber sensors,” Electron. Lett. 27, 1131–1132 (1991).
    [CrossRef]
  21. S. H. Zaidi, R. P. Tatam, “Faraday-effect magnetometry: compensation for the temperature-dependent Verdet constant,” Meas. Sci. Technol. 5, 1471–1479 (1994).
    [CrossRef]
  22. K. S. Lee, “New compensation method for bulk optical sensors with multiple birefringences,” Appl. Opt. 28, 2001–2011 (1989).
    [CrossRef] [PubMed]

1999

1998

C. Li, X. Cui, “Coupled-wave analysis on the electro-optic and magneto-optic interaction in BSO crystal and its application to sensors,” Acta Photonica Sin. 27, 122–126 (1998).

Z. Wang, Y. Liao, S. Lai, H. Zhao, X. Chen, “A novel method for simultaneous measurement of current and voltage using one low-birefringence fiber,” Opt. Laser Technol. 30, 257–262 (1998).
[CrossRef]

A. Kumada, M. Chiba, K. Hidaka, “Potential distribution measurement of surface discharge by Pockels sensing technique,” J. Appl. Phys. 84, 3059–3065 (1998).
[CrossRef]

1997

X. Cui, C. Li, “Optical propagation property of BSO crystal under simultaneously applied electric and magnetic fields and its application in electric power sensor,” Chin. J. Lasers A24, 347–351 (1997).

C. Li, X. Cui, “Pulse-controlled polarization converter and its application,” Acta Photonica Sin. 26, 929–934 (1997).

J. C. Santos, K. Hidaka, “Optical high voltage measurement technique using Pockels device,” Jpn. J. Appl. Phys. Part I 36, 2394–2398 (1997).
[CrossRef]

1996

1994

S. H. Zaidi, R. P. Tatam, “Faraday-effect magnetometry: compensation for the temperature-dependent Verdet constant,” Meas. Sci. Technol. 5, 1471–1479 (1994).
[CrossRef]

1993

1991

P. A. Williams, G. W. Day, A. H. Rose, “Compensation for temperature dependence of Faraday effect in diamagnetic materials: application to optical fiber sensors,” Electron. Lett. 27, 1131–1132 (1991).
[CrossRef]

1989

1986

M. Kanoi, G. Takahashi, T. Sato, M. Higaki, E. Mori, K. Okumura, “Optical voltage and current measuring system for electric power systems,” IEEE Trans. Power Deliv. 1, 91–97 (1986).
[CrossRef]

1985

H. Wang, W. Jia, J. Shen, “Magneto-optical Faraday rotation in Bi4Ge3O12 crystal,” Acta Phys. Sin. 34, 126–128 (1985).

1977

A. J. Rogers, “Optical methods for measurement of voltage and current on power system,” Opt. Laser Technol. 9, 273–283 (1977).
[CrossRef]

1976

A. J. Rogers, “Method for simultaneous measurement of current and voltage on high-voltage lines using optical techniques,” Proc. IEE 123, 957–960 (1976).

1969

W. J. Tabor, F. S. Chen, “Electromagnetic propagation through materials possessing both Faraday rotation and birefringence: experiments with ytterbium orthoferrite,” J. Appl. Phys. 40, 2760–2765 (1969).
[CrossRef]

1965

R. Nitsche, “Crystal growth and electro-optic effect of bismuth germanate, Bi4(GeO4)3,” J. Appl. Phys. 36, 2358–2360 (1965).
[CrossRef]

Blake, J.

R. T. de Carvalho, J. Blake, “Simultaneous measurement of electric and magnetic fields using a Sagnac interferometer,” in Tenth International Conference on Optical Fiber Sensors, B. Culshaw, J. D. Jones, eds., Proc. SPIE2360, 411–414 (1994).
[CrossRef]

Chen, F. S.

W. J. Tabor, F. S. Chen, “Electromagnetic propagation through materials possessing both Faraday rotation and birefringence: experiments with ytterbium orthoferrite,” J. Appl. Phys. 40, 2760–2765 (1969).
[CrossRef]

Chen, X.

Z. Wang, Y. Liao, S. Lai, H. Zhao, X. Chen, “A novel method for simultaneous measurement of current and voltage using one low-birefringence fiber,” Opt. Laser Technol. 30, 257–262 (1998).
[CrossRef]

Chiba, M.

A. Kumada, M. Chiba, K. Hidaka, “Potential distribution measurement of surface discharge by Pockels sensing technique,” J. Appl. Phys. 84, 3059–3065 (1998).
[CrossRef]

Conrad, D. C.

Cui, X.

C. Li, X. Cui, “Coupled-wave analysis on the electro-optic and magneto-optic interaction in BSO crystal and its application to sensors,” Acta Photonica Sin. 27, 122–126 (1998).

C. Li, X. Cui, “Pulse-controlled polarization converter and its application,” Acta Photonica Sin. 26, 929–934 (1997).

X. Cui, C. Li, “Optical propagation property of BSO crystal under simultaneously applied electric and magnetic fields and its application in electric power sensor,” Chin. J. Lasers A24, 347–351 (1997).

Z. Yang, C. Li, W. Zhang, X. Cui, “Novel multifunction optical fiber sensing system for simultaneous measurement of current and voltage,” in Optical and Fiber Optic Sensor System, S. Huang, K. D. Bennett, D. A. Jackson, eds., Proc. SPIE3555, 57–62 (1998).
[CrossRef]

Day, G. W.

P. A. Williams, A. H. Rose, K. S. Lee, D. C. Conrad, G. W. Day, P. D. Hale, “Optical, thermo-optic, electro-optic, and photo-elastic properties of bismuth germinate (Bi4Ge3O12),” Appl. Opt. 35, 3562–3569 (1996).
[CrossRef] [PubMed]

P. A. Williams, G. W. Day, A. H. Rose, “Compensation for temperature dependence of Faraday effect in diamagnetic materials: application to optical fiber sensors,” Electron. Lett. 27, 1131–1132 (1991).
[CrossRef]

de Carvalho, R. T.

R. T. de Carvalho, J. Blake, “Simultaneous measurement of electric and magnetic fields using a Sagnac interferometer,” in Tenth International Conference on Optical Fiber Sensors, B. Culshaw, J. D. Jones, eds., Proc. SPIE2360, 411–414 (1994).
[CrossRef]

Etzel, S. M.

Gaylord, T. K.

Hale, P. D.

Hidaka, K.

A. Kumada, M. Chiba, K. Hidaka, “Potential distribution measurement of surface discharge by Pockels sensing technique,” J. Appl. Phys. 84, 3059–3065 (1998).
[CrossRef]

J. C. Santos, K. Hidaka, “Optical high voltage measurement technique using Pockels device,” Jpn. J. Appl. Phys. Part I 36, 2394–2398 (1997).
[CrossRef]

Higaki, M.

M. Kanoi, G. Takahashi, T. Sato, M. Higaki, E. Mori, K. Okumura, “Optical voltage and current measuring system for electric power systems,” IEEE Trans. Power Deliv. 1, 91–97 (1986).
[CrossRef]

Huard, S.

S. Huard, Polarization of Light (Wiley, Chichester, UK, 1997), Chaps. 1 and 3.5.

Jagadeesh Kumar, V.

Jia, W.

H. Wang, W. Jia, J. Shen, “Magneto-optical Faraday rotation in Bi4Ge3O12 crystal,” Acta Phys. Sin. 34, 126–128 (1985).

Kanoi, M.

M. Kanoi, G. Takahashi, T. Sato, M. Higaki, E. Mori, K. Okumura, “Optical voltage and current measuring system for electric power systems,” IEEE Trans. Power Deliv. 1, 91–97 (1986).
[CrossRef]

Kumada, A.

A. Kumada, M. Chiba, K. Hidaka, “Potential distribution measurement of surface discharge by Pockels sensing technique,” J. Appl. Phys. 84, 3059–3065 (1998).
[CrossRef]

Lai, S.

Z. Wang, Y. Liao, S. Lai, H. Zhao, X. Chen, “A novel method for simultaneous measurement of current and voltage using one low-birefringence fiber,” Opt. Laser Technol. 30, 257–262 (1998).
[CrossRef]

Lee, K. S.

Li, C.

C. Li, X. Cui, “Coupled-wave analysis on the electro-optic and magneto-optic interaction in BSO crystal and its application to sensors,” Acta Photonica Sin. 27, 122–126 (1998).

C. Li, X. Cui, “Pulse-controlled polarization converter and its application,” Acta Photonica Sin. 26, 929–934 (1997).

X. Cui, C. Li, “Optical propagation property of BSO crystal under simultaneously applied electric and magnetic fields and its application in electric power sensor,” Chin. J. Lasers A24, 347–351 (1997).

Z. Yang, C. Li, W. Zhang, X. Cui, “Novel multifunction optical fiber sensing system for simultaneous measurement of current and voltage,” in Optical and Fiber Optic Sensor System, S. Huang, K. D. Bennett, D. A. Jackson, eds., Proc. SPIE3555, 57–62 (1998).
[CrossRef]

Liao, Y.

Z. Wang, Y. Liao, S. Lai, H. Zhao, X. Chen, “A novel method for simultaneous measurement of current and voltage using one low-birefringence fiber,” Opt. Laser Technol. 30, 257–262 (1998).
[CrossRef]

Maldonado, T. A.

Mori, E.

M. Kanoi, G. Takahashi, T. Sato, M. Higaki, E. Mori, K. Okumura, “Optical voltage and current measuring system for electric power systems,” IEEE Trans. Power Deliv. 1, 91–97 (1986).
[CrossRef]

Nitsche, R.

R. Nitsche, “Crystal growth and electro-optic effect of bismuth germanate, Bi4(GeO4)3,” J. Appl. Phys. 36, 2358–2360 (1965).
[CrossRef]

Okumura, K.

M. Kanoi, G. Takahashi, T. Sato, M. Higaki, E. Mori, K. Okumura, “Optical voltage and current measuring system for electric power systems,” IEEE Trans. Power Deliv. 1, 91–97 (1986).
[CrossRef]

Rajkumar, N.

Rochford, K. B.

Rogers, A. J.

A. J. Rogers, “Optical methods for measurement of voltage and current on power system,” Opt. Laser Technol. 9, 273–283 (1977).
[CrossRef]

A. J. Rogers, “Method for simultaneous measurement of current and voltage on high-voltage lines using optical techniques,” Proc. IEE 123, 957–960 (1976).

Rose, A. H.

Sankaran, P.

Santos, J. C.

J. C. Santos, K. Hidaka, “Optical high voltage measurement technique using Pockels device,” Jpn. J. Appl. Phys. Part I 36, 2394–2398 (1997).
[CrossRef]

Sato, T.

M. Kanoi, G. Takahashi, T. Sato, M. Higaki, E. Mori, K. Okumura, “Optical voltage and current measuring system for electric power systems,” IEEE Trans. Power Deliv. 1, 91–97 (1986).
[CrossRef]

Shen, J.

H. Wang, W. Jia, J. Shen, “Magneto-optical Faraday rotation in Bi4Ge3O12 crystal,” Acta Phys. Sin. 34, 126–128 (1985).

Tabor, W. J.

W. J. Tabor, F. S. Chen, “Electromagnetic propagation through materials possessing both Faraday rotation and birefringence: experiments with ytterbium orthoferrite,” J. Appl. Phys. 40, 2760–2765 (1969).
[CrossRef]

Takahashi, G.

M. Kanoi, G. Takahashi, T. Sato, M. Higaki, E. Mori, K. Okumura, “Optical voltage and current measuring system for electric power systems,” IEEE Trans. Power Deliv. 1, 91–97 (1986).
[CrossRef]

Tatam, R. P.

S. H. Zaidi, R. P. Tatam, “Faraday-effect magnetometry: compensation for the temperature-dependent Verdet constant,” Meas. Sci. Technol. 5, 1471–1479 (1994).
[CrossRef]

Wang, H.

H. Wang, W. Jia, J. Shen, “Magneto-optical Faraday rotation in Bi4Ge3O12 crystal,” Acta Phys. Sin. 34, 126–128 (1985).

Wang, Z.

Z. Wang, Y. Liao, S. Lai, H. Zhao, X. Chen, “A novel method for simultaneous measurement of current and voltage using one low-birefringence fiber,” Opt. Laser Technol. 30, 257–262 (1998).
[CrossRef]

Williams, P. A.

P. A. Williams, A. H. Rose, K. S. Lee, D. C. Conrad, G. W. Day, P. D. Hale, “Optical, thermo-optic, electro-optic, and photo-elastic properties of bismuth germinate (Bi4Ge3O12),” Appl. Opt. 35, 3562–3569 (1996).
[CrossRef] [PubMed]

P. A. Williams, G. W. Day, A. H. Rose, “Compensation for temperature dependence of Faraday effect in diamagnetic materials: application to optical fiber sensors,” Electron. Lett. 27, 1131–1132 (1991).
[CrossRef]

Yang, Z.

Z. Yang, C. Li, W. Zhang, X. Cui, “Novel multifunction optical fiber sensing system for simultaneous measurement of current and voltage,” in Optical and Fiber Optic Sensor System, S. Huang, K. D. Bennett, D. A. Jackson, eds., Proc. SPIE3555, 57–62 (1998).
[CrossRef]

Zaidi, S. H.

S. H. Zaidi, R. P. Tatam, “Faraday-effect magnetometry: compensation for the temperature-dependent Verdet constant,” Meas. Sci. Technol. 5, 1471–1479 (1994).
[CrossRef]

Zhang, W.

Z. Yang, C. Li, W. Zhang, X. Cui, “Novel multifunction optical fiber sensing system for simultaneous measurement of current and voltage,” in Optical and Fiber Optic Sensor System, S. Huang, K. D. Bennett, D. A. Jackson, eds., Proc. SPIE3555, 57–62 (1998).
[CrossRef]

Zhao, H.

Z. Wang, Y. Liao, S. Lai, H. Zhao, X. Chen, “A novel method for simultaneous measurement of current and voltage using one low-birefringence fiber,” Opt. Laser Technol. 30, 257–262 (1998).
[CrossRef]

Acta Photonica Sin.

C. Li, X. Cui, “Pulse-controlled polarization converter and its application,” Acta Photonica Sin. 26, 929–934 (1997).

C. Li, X. Cui, “Coupled-wave analysis on the electro-optic and magneto-optic interaction in BSO crystal and its application to sensors,” Acta Photonica Sin. 27, 122–126 (1998).

Acta Phys. Sin.

H. Wang, W. Jia, J. Shen, “Magneto-optical Faraday rotation in Bi4Ge3O12 crystal,” Acta Phys. Sin. 34, 126–128 (1985).

Appl. Opt.

Chin. J. Lasers

X. Cui, C. Li, “Optical propagation property of BSO crystal under simultaneously applied electric and magnetic fields and its application in electric power sensor,” Chin. J. Lasers A24, 347–351 (1997).

Electron. Lett.

P. A. Williams, G. W. Day, A. H. Rose, “Compensation for temperature dependence of Faraday effect in diamagnetic materials: application to optical fiber sensors,” Electron. Lett. 27, 1131–1132 (1991).
[CrossRef]

IEEE Trans. Power Deliv.

M. Kanoi, G. Takahashi, T. Sato, M. Higaki, E. Mori, K. Okumura, “Optical voltage and current measuring system for electric power systems,” IEEE Trans. Power Deliv. 1, 91–97 (1986).
[CrossRef]

J. Appl. Phys.

W. J. Tabor, F. S. Chen, “Electromagnetic propagation through materials possessing both Faraday rotation and birefringence: experiments with ytterbium orthoferrite,” J. Appl. Phys. 40, 2760–2765 (1969).
[CrossRef]

R. Nitsche, “Crystal growth and electro-optic effect of bismuth germanate, Bi4(GeO4)3,” J. Appl. Phys. 36, 2358–2360 (1965).
[CrossRef]

A. Kumada, M. Chiba, K. Hidaka, “Potential distribution measurement of surface discharge by Pockels sensing technique,” J. Appl. Phys. 84, 3059–3065 (1998).
[CrossRef]

J. Lightwave Technol.

Jpn. J. Appl. Phys. Part I

J. C. Santos, K. Hidaka, “Optical high voltage measurement technique using Pockels device,” Jpn. J. Appl. Phys. Part I 36, 2394–2398 (1997).
[CrossRef]

Meas. Sci. Technol.

S. H. Zaidi, R. P. Tatam, “Faraday-effect magnetometry: compensation for the temperature-dependent Verdet constant,” Meas. Sci. Technol. 5, 1471–1479 (1994).
[CrossRef]

Opt. Laser Technol.

Z. Wang, Y. Liao, S. Lai, H. Zhao, X. Chen, “A novel method for simultaneous measurement of current and voltage using one low-birefringence fiber,” Opt. Laser Technol. 30, 257–262 (1998).
[CrossRef]

A. J. Rogers, “Optical methods for measurement of voltage and current on power system,” Opt. Laser Technol. 9, 273–283 (1977).
[CrossRef]

Proc. IEE

A. J. Rogers, “Method for simultaneous measurement of current and voltage on high-voltage lines using optical techniques,” Proc. IEE 123, 957–960 (1976).

Other

R. T. de Carvalho, J. Blake, “Simultaneous measurement of electric and magnetic fields using a Sagnac interferometer,” in Tenth International Conference on Optical Fiber Sensors, B. Culshaw, J. D. Jones, eds., Proc. SPIE2360, 411–414 (1994).
[CrossRef]

Z. Yang, C. Li, W. Zhang, X. Cui, “Novel multifunction optical fiber sensing system for simultaneous measurement of current and voltage,” in Optical and Fiber Optic Sensor System, S. Huang, K. D. Bennett, D. A. Jackson, eds., Proc. SPIE3555, 57–62 (1998).
[CrossRef]

S. Huard, Polarization of Light (Wiley, Chichester, UK, 1997), Chaps. 1 and 3.5.

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Figures (8)

Fig. 1
Fig. 1

Crystallographic orientation and dimensions of the Bi4Ge3O12 crystal used.

Fig. 2
Fig. 2

Experimental setup for the simultaneous measurement of current and voltage by use of the Bi4Ge3O12 (BGO) crystal.

Fig. 3
Fig. 3

Schematic diagram of the signal-processing circuit.

Fig. 4
Fig. 4

Oscilloscope traces of current u o1, voltage u o2, and instantaneous electric power u o3 sensing signals corresponding to a capacitive load.

Fig. 5
Fig. 5

Relative errors of (a) current measurement and (b) voltage measurement.

Fig. 6
Fig. 6

Oscilloscope traces of apparent power signal u o4 and active power signal U o5.

Fig. 7
Fig. 7

Relative errors of (a) active power measurement and (b) apparent power measurement.

Fig. 8
Fig. 8

Nonlinear error δ as a function of measurand current I and voltage U.

Tables (1)

Tables Icon

Table 1 Optical Fiber Sensors for Simultaneous Measurement of Current and Voltage

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

T=cos Θ+j Γ2Θsin Θ-ΦΘsin ΘΦΘsin Θcos Θ-j Γ2Θsin Θ,
Θ=Γ/22+Φ21/2
Ji=Ii21/211,
Jo=TJi=JoxJoy=Ii21/2cos Θ-ΦΘsin Θ+j Γ2Θsin Θcos Θ+ΦΘsin Θ-j Γ2Θsin Θ.
Iox=Jox2=1-ΦΘsin2ΘIi/2, Ioy=Joy2=1+ΦΘsin2ΘIi/2.
Iox 1-2ΦIi/2=1-2k1itIi/2, Ioy 1+2ΦIi/2=1+2k1itIi/2.
Joc=TQWJo=121jj1JoxJoy=12Jox+jJoyjJox+Joy=JocxJocy,
Iocx=Jocx2=1+Γ2Θsin2ΘIi/2, Iocy=Jocy2=1-Γ2Θsin2ΘIi/2.
Iocx 1+ΓIi/2=1+k2utIi/2, Iocy 1-ΓIi/2=1-k2utIi/2,
uo1=raA1k1Iiit,
uo2=0.5rbA2k2Iiut,
uo3=A3uo1uo2=k3utit,
ut=2U sin ωt, it=2I sinωt-ϕ,
uo3=k3UI cos ϕ-UI cos2ωt-ϕ=k3P-S cos2ωt-ϕ,
δ=sin2Θ-y/ymax,

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