Tao Chen 2023a - Revision history

Rimni at 15:13, 8 November 2023

2023-11-08T15:13:41Z

Rimni at 15:07, 8 November 2023

2023-11-08T15:07:58Z

Rimni at 15:05, 8 November 2023

2023-11-08T15:05:05Z

Rimni: /* 3 Experiment */

2023-11-08T15:01:58Z

‎3 Experiment

Rimni at 15:00, 8 November 2023

2023-11-08T15:00:18Z

Show changes

Rimni at 14:49, 8 November 2023

2023-11-08T14:49:44Z

Rimni: /* 2. Mathematical model of the sensor with a stator frame */

2023-11-08T14:48:03Z

‎2. Mathematical model of the sensor with a stator frame

Rimni at 14:46, 8 November 2023

2023-11-08T14:46:34Z

Rimni at 14:41, 8 November 2023

2023-11-08T14:41:00Z

Rimni at 14:39, 8 November 2023

2023-11-08T14:39:53Z

@@ Line 225: / Line 225: @@
 # The imaginary part of impedance  <math>Z_{123}</math> is different from that of traditional regulation. The last item shows that the gradient of the equivalent inductance is damped, unlike that in the traditional regulation. Fortunately, compared with the equivalent resistance of the sensor coil, it does not reverse the monotonicity of the equivalent inductance with displacement owing to the relation shown in <math>-k_{12}^2-k_{13}^2\leq -2k_{12}k_{13}\leq -k_{12}k_{13}k_{23}</math>,  <math>0\leq k_{12},k_{13},k_{23}\leq 1</math>.
-==3 Experiment==
+==3. Experiment==
 Titanium alloy was used for the stator and rotor frame, considering the LVAD biocompatibility requirements. However, the conductivity of titanium alloys is relatively low. The titanium alloy frame on the stator and rotor must be as thin as possible to increase the sensitivity of the eddy current sensor. In addition, copper was added beneath the rotor titanium frame to improve the conductivity. The parameters are shown in [[#tab-1|Table 1]].

@@ Line 385: / Line 385: @@
 The thickness of the titanium alloy frame is limited by the manufacturing process in actual practice. The proposed model can guide designers in conducting studies and experiments to determine the optimal system frequency. However, the model cannot directly aid designers in determining the theoretical optimal frequency. This is a limitation of the mathematical model, which will be addressed in future studies.
-=5 Conclusion=
+==5. Conclusion==
-<span id='_Hlk129195482'></span>The aim of this study was the development of mathematical models for a stator composed of a titanium alloy frame and analyses of the sensitivity of an eddy current displacement sensor applied in a full-maglev LVAD that contains a titanium alloy frame. The proposed models exhibit good universality, can be adapted to general engineering applications, and can provide better solutions to engineering designs. Experiments using different configurations the stator with titanium alloy frames and copper rings showed different equivalent inductance at different excitation frequencies and presented the relationship between equivalent resistance and rotor displacement. These results provide a theoretical basis for future LVAD upgrades.
+The aim of this study was the development of mathematical models for a stator composed of a titanium alloy frame and analyses of the sensitivity of an eddy current displacement sensor applied in a full-maglev LVAD that contains a titanium alloy frame. The proposed models exhibit good universality, can be adapted to general engineering applications, and can provide better solutions to engineering designs. Experiments using different configurations the stator with titanium alloy frames and copper rings showed different equivalent inductance at different excitation frequencies and presented the relationship between equivalent resistance and rotor displacement. These results provide a theoretical basis for future LVAD upgrades.
 ==References==
-[1] Oberle M., Reutemann R., Hertle R., Huang Q. A 10-mW two-channel fully integrated system-on-chip for eddy-current position sensing [J]. IEEE Journal of Solid-State Circuits, 37(7):916–925, 2002.
+[1] Oberle M., Reutemann R., Hertle R., Huang Q. A 10-mW two-channel fully integrated system-on-chip for eddy-current position sensing. IEEE Journal of Solid-State Circuits, 37(7):916–925, 2002.
-[2] Fang J., Wen T. A wide linear range eddy current displacement sensor equipped with dual-coil probe applied in the magnetic suspension flywheel [J]. Sensors (Basel, Switzerland), 12(8):10693–10706, 2012.
+[2] Fang J., Wen T. A wide linear range eddy current displacement sensor equipped with dual-coil probe applied in the magnetic suspension flywheel. Sensors (Basel, Switzerland), 12(8):10693–10706, 2012.
-[3] Wang K., Zhang L., Han B., Chen S. Analysis and experiment of self-differential eddy current displacement sensor for AMBs used in molecular pump[J]. IEEE Transactions on Instrumentation and Measurement, 67(8): 1815–1824, 2018.
+[3] Wang K., Zhang L., Han B., Chen S. Analysis and experiment of self-differential eddy current displacement sensor for AMBs used in molecular pump. IEEE Transactions on Instrumentation and Measurement, 67(8):1815–1824, 2018.
-[4] Pai C.N., Shinshi T., Asama J., Takatani S., Shimokohbe A. Development of a compact centrifugal rotary blood pump with magnetically levitated impeller using a titanium housing [J]. Journal of Advanced Mechanical Design Systems & Manufacturing, 2(3):345–355, 2008.
+[4] Pai C.N., Shinshi T., Asama J., Takatani S., Shimokohbe A. Development of a compact centrifugal rotary blood pump with magnetically levitated impeller using a titanium housing. Journal of Advanced Mechanical Design Systems & Manufacturing, 2(3):345–355, 2008.
-[5] Ahn C. B., Kim K. H., Moon K. C., Jeong K.S., Kim H.C., Lee J.J., Hwang C.M., Sun K. Development of eddy current sensor systems in artificial heart for noncontact gap sensing [J]. Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 4:3913–3915, 2005.
+[5] Ahn C.B., Kim K.H., Moon K.C., Jeong K.S., Kim H.C., Lee J.J., Hwang C.M., Sun K. Development of eddy current sensor systems in artificial heart for noncontact gap sensing. Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 4:3913–3915, 2005.
-[6] Lee J. J., Chi B. A., Choi J., Park J.W., Song S.-J., Sun K. Development of magnetic bearing system for a new third‐generation blood pump [J]. Artificial Organs, 35(11):1082–1094, 2011.
+[6] Lee J.J., Chi B.A., Choi J., Park J.W., Song S.-J., Sun K. Development of magnetic bearing system for a new third‐generation blood pump. Artificial Organs, 35(11):1082–1094, 2011.
-[7] Sun M., Zhou J., Dong B., Zheng S. Driver circuit improvement of eddy current sensor in displacement measurement of high-speed rotor [J]. IEEE Sensors Journal, 21(6):7776–7783, 2021.
+[7] Sun M., Zhou J., Dong B., Zheng S. Driver circuit improvement of eddy current sensor in displacement measurement of high-speed rotor. IEEE Sensors Journal, 21(6):7776–7783, 2021.
-[8] Nagaoka E., Fujiwara T., Kitao T., Sakota D., Shinshi T., Arai H., Takatani S. MedTech Mag-Lev, single-use, extracorporeal magnetically levitated centrifugal blood pump for mid-term circulatory support [J]. Asaio Journal, 59(3):246–252, 2013.
+[8] Nagaoka E., Fujiwara T., Kitao T., Sakota D., Shinshi T., Arai H., Takatani S. MedTech Mag-Lev, single-use, extracorporeal magnetically levitated centrifugal blood pump for mid-term circulatory support. Asaio Journal, 59(3):246–252, 2013.
-[9] Raghunathan P., Logashanmugam E. Position servo controller design and implementation using low cost eddy current sensor for single axis active magnetic bearing [J]. Journal of ambient intelligence and humanized computing, 10(9):3481–3492, 2019.
+[9] Raghunathan P., Logashanmugam E. Position servo controller design and implementation using low cost eddy current sensor for single axis active magnetic bearing. Journal of Ambient Intelligence and Humanized Computing, 10(9):3481–3492, 2019.
-[10] Zhan H., Wang L., Wang T., Yu J. The influence and compensation method of eccentricity for cylindrical specimens in eddy current displacement measurement [J]. Sensors, 20(22):6608, 2022.
+[10] Zhan H., Wang L., Wang T., Yu J. The influence and compensation method of eccentricity for cylindrical specimens in eddy current displacement measurement. Sensors, 20(22):6608, 2022.
 <br/>

@@ Line 369: / Line 369: @@
 |}
-=4 Discussion=
+==4. Discussion==
 The displacement sensor in a full-maglev LVAD is vital to the system. Sensitivity and robustness directly affect the performance of the magnetic-bearing system. However, the titanium alloy frame of the pump significantly affects the impedance of the eddy current sensor. This paper proposes a mathematical model to demonstrate the relationship between the eddy current sensor and the titanium alloy frame. Several experiments were conducted for analysis and comparison.
-The curves C and D in Figure 4 correspond to change in the imaginary part of Eq. (4) and Eq. (2) respectively. Due to the last term of the imaginary part in Eq. (4) can be guaranteed as <math>R_2R_3<\omega^2L_2L_3</math>, at the same excitation frequency, the change of gradient of equivalent inductance with rotor displacement changing in group C is lower than that in group D.
+The curves C and D in [[#img-4|Figure 4]] correspond to change in the imaginary part of Eq. (4) and Eq. (2), respectively. Due to the last term of the imaginary part in Eq. (4) can be guaranteed as <math>R_2R_3<\omega^2L_2L_3</math>, at the same excitation frequency, the change of gradient of equivalent inductance with rotor displacement changing in group C is lower than that in group D.
-According to Figures 5, 6, and 7, as the excitation frequency goes higher, when the stator is with titanium  alloy frame, the variational trend of the sensor coil's equivalent resistance changing with rotor displacement varying goes to the opposite. It shows that the value of the last term of the real part in Eq. (4) is indeed greater than the sum of the two front terms as the excitation frequency increases. When the excitation frequency exceeds 400 kHz, the variational trend goes to the opposite direction.
+According to [[#img-5|Figures 5]], [[#img-6|6]], and [[#img-7|7]], as the excitation frequency goes higher, when the stator is with titanium  alloy frame, the variational trend of the sensor coil's equivalent resistance changing with rotor displacement varying goes to the opposite. It shows that the value of the last term of the real part in Eq. (4) is indeed greater than the sum of the two front terms as the excitation frequency increases. When the excitation frequency exceeds 400 kHz, the variational trend goes to the opposite direction.
-The curve C in Figure 8 verifies the conclusion that when there is stator titanium alloy frame, as the excitation frequency goes higher, both <math>k_{12}</math> and <math>k_{13}</math>in Eq. (11) go up as well. However, the sensitivity does not monotonically increase with the excitation frequency increment. A peak value exists. That is to say, when a titanium alloy frame is used on the stator, an optimal excitation frequency exists, while higher excitation frequency does not always result in better sensitivity. This result indicates that the sensitivity is limited by the titanium alloy frame on the stator.
+The curve C in [[#img-8|Figure 8]] verifies the conclusion that when there is stator titanium alloy frame, as the excitation frequency goes higher, both <math>k_{12}</math> and <math>k_{13}</math> in Eq. (11) go up as well. However, the sensitivity does not monotonically increase with the excitation frequency increment. A peak value exists. That is to say, when a titanium alloy frame is used on the stator, an optimal excitation frequency exists, while higher excitation frequency does not always result in better sensitivity. This result indicates that the sensitivity is limited by the titanium alloy frame on the stator.
-Comparing C and E in Figure 8 illustrates that when the rotor was embedded with high conductivity metal, <math>k_{13}</math>in Eq. (11) will increase, resulting in higher sensitivity, i.e., embedded copper ring under the rotor titanium alloy frame enhances the conductivity of the rotor. This property increases the sensitivity of the eddy current sensor.
+Comparing C and E in [[#img-8|Figure 8]] illustrates that when the rotor was embedded with high conductivity metal, <math>k_{13}</math> in Eq. (11) will increase, resulting in higher sensitivity, i.e., embedded copper ring under the rotor titanium alloy frame enhances the conductivity of the rotor. This property increases the sensitivity of the eddy current sensor.
 The excitation frequency of the sensor directly influences its bandwidth when a titanium alloy frame is added to the stator. The designer must consider the tradeoff between sensitivity and bandwidth. With respect to the eddy current displacement sensor, whose excitation frequency is normally 500 kHz, its effective bandwidth can ensure the competency of a magnetic bearing system with 20000-rpm capacity. That characteristic is reasonable for an implantable LVAD. Alternatively, the dynamic performance of the sensor can be improved with a trade-off in sensitivity in favor of excitation frequency.

@@ Line 335: / Line 335: @@
 | style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 6'''. Equivalent resistance affected by different excitation frequencies in group C from 100 to 300 kHz
 |}
 <div id='img-7'></div>
@@ Line 358: / Line 359: @@
 In group C, when the excitation frequency ranged from 100 to 475 kHz, the FRIG increased monotonically. The maximum value appeared when the excitation frequency reached 475 kHz; subsequently, the FRIG decreased while the excitation frequency continued to increase. Furthermore, when the excitation frequency exceeds 475 kHz, stator titanium alloy frames cause greater attenuation of the sensor sensitivity, which is not compensated by the sensitivity increase generated by the eddy current in the rotor. This phenomenon causes a steady decrease in sensitivity, which continues to decrease when the excitation frequency increases. In the range of 250 to 750 kHz, the FRIG remained within 5% attenuation of the maximum value. However, the attenuation of the FRIG increased to more than 12.5% of the maximum when the excitation frequency exceeded 1000 kHz. Thus, it is not beneficial for sensitivity to use an excitation frequency that is too high or too low in the presence of a stator titanium alloy frame, although a narrow range of excitation frequencies exists to choose in an actual use environment.
-In group E, the FRIG monotonically increased when the excitation frequency ranged from 100 to 1000 kHz. The maximum gain value appeared at 1800 kHz frequency (not marked on the curve), and it decreased as the excitation frequency continued to increase. The maximum FRIG in group E reached only 60% of that in group C. Thus, the sensitivity is improved by adding copper beneath the titanium alloy frame in the rotor. Figure 9 shows the experimental platform.
+In group E, the FRIG monotonically increased when the excitation frequency ranged from 100 to 1000 kHz. The maximum gain value appeared at 1800 kHz frequency (not marked on the curve), and it decreased as the excitation frequency continued to increase. The maximum FRIG in group E reached only 60% of that in group C. Thus, the sensitivity is improved by adding copper beneath the titanium alloy frame in the rotor. [[#img-9|Figure 9]] shows the experimental platform.
-[[Image:Draft_Tao_267486902-image76.jpeg|294px]]
+<div id='img-9'></div>
+{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
-'''Figure 9'''. Experimental platform
+|-style="background:white;"
 =4 Discussion=

@@ Line 104: / Line 104: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>Z_{123}=\lerft(R_1+\displaystyle\frac{{\omega }^2M_{12}^2}{R_2^2+{\omega }^2L_2^2}R_2+\displaystyle\frac{{\omega }^2M_{13}^2}{R_3^2+{\omega }^2L_3^2}R_3-\displaystyle\frac{{\omega }^4M_{12}M_{13}M_{23}(R_3L_2+R_2L_3)}{\left(R_2^2+{\omega }^2L_2^2)(R_3^2+{\omega }^2L_3^2\right)}\right)+</math><math>j\omega \left(L_1-\displaystyle\frac{{\omega }^2M_{12}^2}{R_2^2+{\omega }^2L_2^2}L_2-\displaystyle\frac{{\omega }^2M_{13}^2}{R_3^2+{\omega }^2L_3^2}L_3-\displaystyle\frac{{\omega }^2M_{12}M_{13}M_{23}(R_2R_3-{\omega }^2L_2L_3)}{\left(R_2^2+{\omega }^2L_2^2)(R_3^2+{\omega }^2L_3^2\right)}\right)</math>.
+| <math>Z_{123}=\left(R_1+\displaystyle\frac{{\omega }^2M_{12}^2}{R_2^2+{\omega }^2L_2^2}R_2+\displaystyle\frac{{\omega }^2M_{13}^2}{R_3^2+{\omega }^2L_3^2}R_3-\displaystyle\frac{{\omega }^4M_{12}M_{13}M_{23}(R_3L_2+R_2L_3)}{\left(R_2^2+{\omega }^2L_2^2)(R_3^2+{\omega }^2L_3^2\right)}\right)+</math><math>j\omega \left(L_1-\displaystyle\frac{{\omega }^2M_{12}^2}{R_2^2+{\omega }^2L_2^2}L_2-\displaystyle\frac{{\omega }^2M_{13}^2}{R_3^2+{\omega }^2L_3^2}L_3-\displaystyle\frac{{\omega }^2M_{12}M_{13}M_{23}(R_2R_3-{\omega }^2L_2L_3)}{\left(R_2^2+{\omega }^2L_2^2)(R_3^2+{\omega }^2L_3^2\right)}\right)</math>.
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (4)
@@ Line 208: / Line 208: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\begin{array}{ll}
+| <math>\begin{align}
-Sy=&\displaystyle\frac{\partial (1-k_{12}^2(\sigma ,\omega )-k_{13}^2(\sigma ,x,\omega )+k_{12}(\sigma ,\omega )k_{13}(\sigma ,x,\omega )k_{23}(\sigma ,x,\omega ))/\partial x}{\partial (1-k_{13}^2(\sigma ,x,\omega ))/\partial x}\\
+Sy & =\displaystyle\frac{\partial (1-k_{12}^2(\sigma ,\omega )-k_{13}^2(\sigma ,x,\omega )+k_{12}(\sigma ,\omega )k_{13}(\sigma ,x,\omega )k_{23}(\sigma ,x,\omega ))/\partial x}{\partial (1-k_{13}^2(\sigma ,x,\omega ))/\partial x}\\
-= & 1-\displaystyle\frac{1}{2}k_{12}(\sigma ,\omega )[\displaystyle\frac{k_{23}(\sigma ,x,\omega )}{k_{13}(\sigma ,x,\omega )}+\displaystyle\frac{\partial k_{23}(\sigma ,x,\omega )/\partial x}{\partial k_{13}(\sigma ,x,\omega )/\partial x}]
+ & = 1-\displaystyle\frac{1}{2}k_{12}(\sigma ,\omega )[\displaystyle\frac{k_{23}(\sigma ,x,\omega )}{k_{13}(\sigma ,x,\omega )}+\displaystyle\frac{\partial k_{23}(\sigma ,x,\omega )/\partial x}{\partial k_{13}(\sigma ,x,\omega )/\partial x}]
-\end{array}</math> .
+\end{align}</math> .
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (11)

@@ Line 74: / Line 74: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>Z_{13}=(R_1+\frac{{\omega }^2M_{13}^2}{R_3^2+{\omega }^2L_3^2}R_3)+</math><math>j\omega (L_1-\frac{{\omega }^2M_{13}^2}{R_3^2+{\omega }^2L_3^2}L_3)</math> .
+|<math>Z_{13}=\left(R_1+\displaystyle\frac{{\omega}^2M_{13}^2}{R_3^2+{\omega }^2L_3^2}R_3\right)+</math> <math> j\omega \left(L_1-\displaystyle\frac{{\omega }^2M_{13}^2}{R_3^2+{\omega }^2L_3^2}L_3\right)</math>.
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (2)
@@ Line 89: / Line 89: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>Z_{12}=(R_1+\frac{{\omega }^2M_{12}^2}{R_2^2+{\omega }^2L_2^2}R_2)+</math><math>j\omega (L_1-\frac{{\omega }^2M_{12}^2}{R_2^2+{\omega }^2L_2^2}L_2)</math> .
+| <math>Z_{12}=\left(R_1+\displaystyle\frac{{\omega }^2M_{12}^2}{R_2^2+{\omega }^2L_2^2}R_2\right)+</math><math>j\omega \left(L_1-\displaystyle\frac{{\omega }^2M_{12}^2}{R_2^2+{\omega }^2L_2^2}L_2\right)</math> .
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3)
@@ Line 104: / Line 104: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>Z_{123}=(R_1+\frac{{\omega }^2M_{12}^2}{R_2^2+{\omega }^2L_2^2}R_2+</math><math>\frac{{\omega }^2M_{13}^2}{R_3^2+{\omega }^2L_3^2}R_3-</math><math>\frac{{\omega }^4M_{12}M_{13}M_{23}(R_3L_2+R_2L_3)}{\left(R_2^2+{\omega }^2L_2^2)(R_3^2+{\omega }^2L_3^2\right)})+</math><math>j\omega (L_1-\frac{{\omega }^2M_{12}^2}{R_2^2+{\omega }^2L_2^2}L_2-</math><math>\frac{{\omega }^2M_{13}^2}{R_3^2+{\omega }^2L_3^2}L_3-</math><math>\frac{{\omega }^2M_{12}M_{13}M_{23}(R_2R_3-{\omega }^2L_2L_3)}{\left(R_2^2+{\omega }^2L_2^2)(R_3^2+{\omega }^2L_3^2\right)})</math> .
+| <math>Z_{123}=\lerft(R_1+\displaystyle\frac{{\omega }^2M_{12}^2}{R_2^2+{\omega }^2L_2^2}R_2+</math><math>\displaystyle\frac{{\omega }^2M_{13}^2}{R_3^2+{\omega }^2L_3^2}R_3-</math><math>\displaystyle\frac{{\omega }^4M_{12}M_{13}M_{23}(R_3L_2+R_2L_3)}{\left(R_2^2+{\omega }^2L_2^2)(R_3^2+{\omega }^2L_3^2\right)}\right)+</math><math>j\omega \left(L_1-\displaystyle\frac{{\omega }^2M_{12}^2}{R_2^2+{\omega }^2L_2^2}L_2-\displaystyle\frac{{\omega }^2M_{13}^2}{R_3^2+{\omega }^2L_3^2}L_3-\displaystyle\frac{{\omega }^2M_{12}M_{13}M_{23}(R_2R_3-{\omega }^2L_2L_3)}{\left(R_2^2+{\omega }^2L_2^2)(R_3^2+{\omega }^2L_3^2\right)}\right)</math>.
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (4)
@@ Line 176: / Line 176: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math display="inline">h=\sqrt{\frac{2}{\omega \sigma \mu }}</math> ,
+| <math display="inline">h=\sqrt{\displaystyle\frac{2}{\omega \sigma \mu }}</math> ,
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (10)
@@ Line 210: / Line 210: @@
 | <math>\begin{array}{ll}
 Sy=&\displaystyle\frac{\partial (1-k_{12}^2(\sigma ,\omega )-k_{13}^2(\sigma ,x,\omega )+k_{12}(\sigma ,\omega )k_{13}(\sigma ,x,\omega )k_{23}(\sigma ,x,\omega ))/\partial x}{\partial (1-k_{13}^2(\sigma ,x,\omega ))/\partial x}\\
-= =1-\displaystyle\frac{1}{2}k_{12}(\sigma ,\omega )[\frac{k_{23}(\sigma ,x,\omega )}{k_{13}(\sigma ,x,\omega )}+\frac{\partial k_{23}(\sigma ,x,\omega )/\partial x}{\partial k_{13}(\sigma ,x,\omega )/\partial x}]
+& = 1-\displaystyle\frac{1}{2}k_{12}(\sigma ,\omega )[\displaystyle\frac{k_{23}(\sigma ,x,\omega )}{k_{13}(\sigma ,x,\omega )}+\displaystyle\frac{\partial k_{23}(\sigma ,x,\omega )/\partial x}{\partial k_{13}(\sigma ,x,\omega )/\partial x}]
 \end{array}</math> .
 |}

@@ Line 208: / Line 208: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\begin{array}{c}
+| <math>\begin{array}{l}
-Sy=\frac{\partial (1-k_{12}^2(\sigma ,\omega )-k_{13}^2(\sigma ,x,\omega )+k_{12}(\sigma ,\omega )k_{13}(\sigma ,x,\omega )k_{23}(\sigma ,x,\omega ))/\partial x}{\partial (1-k_{13}^2(\sigma ,x,\omega ))/\partial x}\\
+Sy=&\frac{\partial (1-k_{12}^2(\sigma ,\omega )-k_{13}^2(\sigma ,x,\omega )+k_{12}(\sigma ,\omega )k_{13}(\sigma ,x,\omega )k_{23}(\sigma ,x,\omega ))/\partial x}{\partial (1-k_{13}^2(\sigma ,x,\omega ))/\partial x}\\
-\mbox{    }=1-\frac{1}{2}k_{12}(\sigma ,\omega )[\frac{k_{23}(\sigma ,x,\omega )}{k_{13}(\sigma ,x,\omega )}+\frac{\partial k_{23}(\sigma ,x,\omega )/\partial x}{\partial k_{13}(\sigma ,x,\omega )/\partial x}]
+= =1-\frac{1}{2}k_{12}(\sigma ,\omega )[\frac{k_{23}(\sigma ,x,\omega )}{k_{13}(\sigma ,x,\omega )}+\frac{\partial k_{23}(\sigma ,x,\omega )/\partial x}{\partial k_{13}(\sigma ,x,\omega )/\partial x}]
 \end{array}</math> .
 |}