CN104965175B - A kind of detection method in the static fault of eccentricity orientation of power generator air gap and fault degree - Google Patents

A kind of detection method in the static fault of eccentricity orientation of power generator air gap and fault degree Download PDF

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CN104965175B
CN104965175B CN201510419276.8A CN201510419276A CN104965175B CN 104965175 B CN104965175 B CN 104965175B CN 201510419276 A CN201510419276 A CN 201510419276A CN 104965175 B CN104965175 B CN 104965175B
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CN104965175A (en
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何玉灵
邓玮琪
刘会兰
唐贵基
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Shenzhen union power (Heyuan) Co., Ltd.
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North China Electric Power University
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Abstract

The invention discloses a kind of detection method in the static fault of eccentricity orientation of power generator air gap, comprise the steps:1) voltage signal of induction coil on four vertical distribution measuring points of stator circumference is gathered by Hall voltage sensor, and outputs this to Acquisition Instrument;2) compare four voltage signals, determine quiet fault of eccentricity orientation and fault degree.Technical scheme disclosed by the invention can simply, it is quick, reliably determine the quiet fault of eccentricity of generator air gap, make up the deficiency of the conventional diagnostic technology based on rotor oscillation characteristic, be easy to the maintenance and maintenance of generator.

Description

A kind of detection method in the static fault of eccentricity orientation of power generator air gap and fault degree
Technical field
The static fault of eccentricity orientation of generator air gap and the method for fault degree, category inspection are determined the present invention relates to a kind of Survey technology field.
Background technology
All there is the phenomenon of rotor air gap eccentric centre in most of generators, when eccentric degree is more than a fixed limit to some extent Think that generator has air-gap eccentric fault when spending, such failure can cause dynamo bearing working condition to deteriorate, while plus Acute rotor vibration, causes stator core deformation, winding abrasion and insulation breakdown.
Air gap eccentric centre can be divided into three kinds of quiet eccentric, dynamic eccentric and sound mixing bias etc., wherein, the static state of air gap is eccentric Refer to that the pivot of generator unit stator center and rotor is misaligned, but rotor center is overlapped with pivot, equivalent to guide shaft Hold center to offset from stator center to some direction, make rotor eccentric relative to stator in this direction, rotor air gap becomes Change, as shown in figure 1, relatively quiet eccentricity value withRepresent.
The research for air gap eccentric centre is concentrated on according to generator stator-rotator mechanical oscillation feature, stator parallel branch at present Circulation electric characteristic, event is combined by the air-gap eccentric fault of generator from rotor short-circuit failure and air gap eccentric centre and rotor short-circuit Distinguished in barrier.But for the quiet fault of eccentricity of air gap, the degree size of generation position and bias due to its failure can not Simply obtained in the electric characteristics such as vibration performance, parallel branch circulation from its rotor, maintenance and maintenance band to generator Carry out very big difficulty.
In summary, the generation orientation of the quiet fault of eccentricity of power generator air gap and the degree of failure how are determined, is remained This area urgent problem to be solved.
The content of the invention
In view of the shortcomings of the prior art, the invention discloses a kind of quiet fault of eccentricity orientation of power generator air gap and fault degree Detection method, this method is simple and easy to do, can fast, reliably determine the quiet fault of eccentricity orientation of generator air gap and therefore Barrier degree, data support is provided for the exclusion of the quiet fault of eccentricity of generator.
Specifically, the present invention is achieved through the following technical solutions:
A kind of detection method in the static fault of eccentricity orientation of power generator air gap, comprises the steps:1) Hall voltage is passed through The voltage signal of induction coil on sensor collection four vertical distribution measuring points of stator circumference, and output this to Acquisition Instrument;2) Compare four measuring point voltage signals, determine quiet fault of eccentricity orientation and fault degree.
In order to ensure that voltage signal acquisition is accurate, in step 1) in, the voltage signal collected is passed through into operational amplifier By output voltage stabilization in the collection range of Acquisition Instrument.
In method disclosed by the invention, quiet fault of eccentricity orientation is judged by the voltage signal of measuring point, is specifically sentenced Disconnected method is:
(1) four measuring point voltage is equal and equal to voltage under normal circumstances, then generator operation is normal;
(2) four measuring point any twos are equal with respect to measuring point voltage and are equal to surveyed voltage under normal circumstances, two other Measuring point voltage is respectively maximum and minimum value, then is 90 ° of quiet fault of eccentricity of air gap, and the minimal air gap of failure is located at maximum voltage At the measuring point of value;
(3) four measuring point any two adjacent measuring point voltages are equal, two other adjacent measuring point voltage it is also equal but In the voltage of the first two measuring point, the size of surveyed voltage is then 45 ° of air gaps between this two groups of measuring point voltages under normal circumstances Quiet fault of eccentricity, the minimal air gap of failure is located in the middle of the measuring point of two larger equivalent voltages of magnitude of voltage;
It is unequal that (4) four measuring points survey voltage, then be the quiet fault of eccentricity of θ angles air gap, the minimal air gap of failure is located at The maximum measuring point of magnitude of voltage turns over a θ angular direction, 0 ° of 45 ° of < θ < to the second largest measuring point of magnitude of voltage.
For accurate evaluation fault degree, provide data for maintenance and support that on the basis of the above, the invention also discloses right The detection method of the quiet fault of eccentricity degree of power generator air gap, each measuring point voltage and survey piezoelectric voltage under normal circumstances during using failure The relatively quiet eccentricity value δ s of ratio calculation so as to assessing fault degree.
Technical scheme disclosed in this invention, according to the voltage signal of four induction coils of vertical distribution in stator circumference Diagnose the quiet fault of eccentricity of generator air gap, can not only by direction where the minimal air gap of the quiet fault of eccentricity of air gap, and Relatively quiet eccentric fault degree can accurately be calculated.The detection method of the present invention, it is simple and easy to apply, it can make up with rotor oscillation The deficiency of conventional diagnostic technology based on characteristic, is easy to the maintenance and maintenance of generator.
Brief description of the drawings
Fig. 1 is the quiet fault of eccentricity state coordinate system schematic diagram of power generator air gap;
Fig. 2 measures method schematic diagram for the induction coil voltage signal of the present invention;
Fig. 3 is induction coil layout schematic diagram in air gap;
Fig. 4 is that generator normally runs criterion schematic diagram;
Fig. 5 is the quiet fault of eccentricity criterion schematic diagram of 90 ° of air gaps of generator;
Fig. 6 is the quiet fault of eccentricity criterion schematic diagram of 45 ° of air gaps of generator;
Fig. 7 is the quiet fault of eccentricity criterion schematic diagram of generator θ angles air gap;
Fig. 8 is the quiet fault of eccentricity degree identity process figure of generator;
It is quiet partially that a, b, c, d in Fig. 9 do not occur the quiet fault of eccentricity of air gap, 90 ° of quiet eccentric, 45 ° of quiet eccentric, θ angles respectively The engine condition coordinate system schematic diagram of the heart.
In provided accompanying drawing, label correspondence implication is respectively 1- generators, 2,3,4,5- Hall voltage sensors, 6- axles Bearing, 7- exciters, 8- host computers, 9- slave computers;
Symbol correspondence implication is respectively U in each formula1、U2、U3、U4For each measuring point magnitude of voltage (U0For voltage under normal circumstances Value);(α, (the quiet eccentric only influence magnetic conductance that generator occurs, does not influence magnetic potential to magnetic potential of air gap when t) for normal operation to f, therefore partially After the heart magnetic potential with it is normal equal);FrFor main magnetic potential;FsFor armature reaction magnetic potential;F1For composite magnetic power;ωr=2 π frFor rotor machine Tool angular frequency;frFor the mechanical frequency of rotor;G is average airgap;Λ is air-gap permeance;μ0For space permeability;α is stator Tool angle (is defined using at air gap minimum as starting point, counterclockwise for just);ψ is generator rotor angle in generator;B is magnetic induction intensity;L is The length of induction coil cutting magnetic field;V be magnetic field relative to the cutting speed of induction coil, be numerically ωr(R is in stator to R Footpath);δsFor relatively quiet eccentricity value;θ is eccentric angle.
Embodiment
In order to more accurately, clearly illustrate the specific operation process of detection method, applicant is with reference to accompanying drawing to this Principle, implementation process, the computational methods of invention detection method are illustrated.Specific implementation process as disclosed below is only to show Meaning property, especially limitation is not constituted to the present invention.
With reference to Fig. 2, detection method of the invention is that (generator 1 on bearing block 6 leads in traditional generator set structure Cross the exciter 7 being coaxial therewith and obtain DC current from exciter), generator 1 stator core vertically and horizontally on The Hall voltage sensor 2,3,4,5 of four connection induction coils is installed, the signal of collection is transmitted under by signal transmssion line Position machine 9.Signal after noise suppression preprocessing after filtering is reached host computer 8 and preserved by slave computer, and host computer 8 is by comparing four surveys The size of the voltage signal of point may determine that the direction that quiet fault of eccentricity occurs;, will using the fault degree calculating formula derived The voltage signal collected is updated in formula with voltage signal under normal circumstances, can calculate quiet fault of eccentricity degree.
With reference to Fig. 3, it is shown that install four measuring points 20,30,40,50 of four Hall sensors, these measuring points are in circumference Upper vertical distribution
With reference to Fig. 4-7, it is shown that detection method deterministic process of the invention:The voltage signal collected, may respectively be Following several situations:
①U1=U2=U3=U4=U0
②U1> U2=U3> U4;U4> U2=U3> U1;U2> U1=U4> U3;U3> U1=U4> U2
③U1=U2> U3=U4;U4=U2> U3=U1;U2=U1> U4=U3;U3=U1> U4=U2
④U1> U2> U3> U4;U1> U3> U2> U4;U2> U1> U4> U3;U2> U4> U1> U3;U3> U1> U4> U2;U3> U4> U1> U2;U4> U2> U3> U1;U4> U3> U2> U1
In above-mentioned, the average voltage U of four induction coils when each measuring point average voltage and normal operation0By following formula Calculate:
Wherein, UiFor the rms voltage (virtual value) of i-th of measuring point, m is sampling number, U0Total for four measuring points Average voltage.
In order to improve the accuracy of retrieval gained voltage data, the acquisition methods of measuring point voltage data sample are:Generating electricity After the debugging of machine kludge is finished and when indices tend to normal condition, stator circumference four is measured using Hall voltage sensor Each ten groups of the voltage signal data of induction coil, operational amplifier is passed through by the voltage signal collected on individual vertical distribution measuring point Input solves the average value of every ten groups of voltage data virtual values to Acquisition Instrument, then by the flat of the voltage effective value of four measuring points Average is compared to each other, direction where the minimal air gap of quiet fault of eccentricity is judged according to the magnitude relationship between them, and utilizes event The radiometer of each measuring point voltage and measuring point voltage under normal circumstances calculates relatively quiet eccentricity value δ during barriers
Above-mentioned several situations represent following four kinds of states respectively:
(1) four measuring point voltage is equal and equal to surveyed voltage under normal circumstances, then generator operation is normal;
(2) four measuring point any twos are equal with respect to measuring point voltage and are equal to surveyed voltage under normal circumstances, two other Measuring point voltage is respectively maximum and minimum value, then is 90 ° of quiet fault of eccentricity of air gap, and the minimal air gap of failure is located at maximum voltage At the measuring point of value;
(3) four measuring point any two adjacent measuring point voltages are equal, two other adjacent measuring point voltage it is also equal but In the voltage of the first two measuring point, the size of surveyed voltage is then 45 ° of air gaps between this two groups of measuring point voltages under normal circumstances Quiet fault of eccentricity, the minimal air gap of failure is located in the middle of the measuring point of two larger equivalent voltages of magnitude of voltage;
It is unequal that (4) four measuring points survey voltage, then be the quiet fault of eccentricity of θ angles air gap, the minimal air gap of failure is located at The maximum measuring point of magnitude of voltage turns over a θ angular direction (0 ° of 45 ° of < θ <) to the second largest measuring point of magnitude of voltage.
Based on measured obtained magnitude of voltage U under normal circumstances0With exist it is quiet it is eccentric in the case of magnitude of voltage Ux, utilize survey The ratio of voltage is put to calculate relatively quiet eccentricity value δs
Specifically calculating process is:First with motional electromotive force calculating formula U0=Blv, magnetic induction intensity calculating formula B=f (α, t) Λ (α) and magnetic conduction calculation formula Λ=μ0/ g (α), derives voltage U under normal circumstances0Identity U0G=f (α, t) lv μ0.Motional electromotive force calculating formula is recycled to calculate the measuring point voltage under fault of eccentricity, what is now changed is only that air gap is long Spending g (α) will voltage U under normal circumstances0Identity U0G=f (α, t) lv μ0And it is long on the air gap that stator mechanical angle α changes Spend calculating formula g (α)=g [1- δsCOS α] substitute into after, voltage ratio before and after bias can be utilized and solve relatively quiet eccentricity value δs Equation.Finally substitute into the stator mechanical angle α corresponding to each measuring point, it is possible to obtain relatively quiet eccentricity value δs
Above-mentioned calculating process is applied to the four kinds of quiet eccentric situations of the invention for detecting and obtaining, described in detail respectively such as Under:
Magnetic potential of air gap of institute's survey generator in normal operation is calculated first:
Wherein, FrFor main magnetic potential, FsFor armature reaction magnetic potential, F1For composite magnetic power, ωr=2 π frFor the mechanical angular frequency of rotor Rate, frFor the mechanical frequency of rotor, α is stator mechanical angle, and ψ is generator rotor angle in generator, and is had:
Consider the quiet bias of air gap, select coordinate system as shown in Figure 1, using at rotor minimal air gap as origin, then correspond to Air gap radical length and unit area magnetic conductance be:
Wherein:G is average airgap, δsFor relatively quiet eccentricity value, Λ is unit area air-gap permeance, μ0For space permeability.
Under normal circumstances (Fig. 9-a), each measuring point voltage U1=U2=U3=U4=U0, then have:
By formula (5) four minors can simultaneous obtain following formula:
U0G=f (α, t) lv μ0 (6)
In the case of 90 ° of quiet bias (Fig. 9-b), with measuring point voltage U1> U2=U3> U4Exemplified by, the corresponding stator of each measuring point Tool angle is respectively α1=0 °, α2=270 °, α3=90 °, α4=180 °, the voltage equation of each measuring point is as follows:
On the basis of the 1st minor of formula (7) and the 4th minor, the quiet fault of eccentricity degree δ of correspondence can be obtainedsCalculate public Formula is as follows:
In the case of 45 ° of quiet bias (Fig. 9-c), with measuring point voltage U1=U2> U3=U4Exemplified by, the corresponding stator of each measuring point Tool angle is respectively α1=45 °, α2=315 °, α3=135 °, α4=225 °, the voltage equation of each measuring point is as follows:
On the basis of formula (9), the fault degree δ of corresponding quiet fault of eccentricitysCalculated by following formula:
In the case of the quiet bias in θ angles (Fig. 9-d), with measuring point voltage U1> U2> U3> U4Exemplified by, the corresponding stator of each measuring point Tool angle is respectively α1=θ, α2=3 pi/2+θ, α3=pi/2+θ, α4=π+θ, the corresponding voltage of each measuring point is as follows:
One in formula (11) the 1st, the 4th minor and one in the 2nd, the 3rd minor are carried out into combination of two to be divided by, divided out δs, four corresponding θ calculation formula are can obtain, the θ values that this four formula are obtained are averaged after being added, and obtain θ most Whole expression formula is as follows:
Each minor of formula (11) can correspond to and seek out a δsValue, can obtain four δ altogethersValue, makes even to it , the fault degree δ of corresponding quiet fault of eccentricity is obtainedsCalculated by following formula:

Claims (6)

1. the detection method in a kind of static fault of eccentricity orientation of power generator air gap and fault degree, it is characterised in that including following steps Suddenly:1) voltage signal of induction coil on four vertical distribution measuring points of stator circumference is gathered by Hall voltage sensor, and will It is exported to Acquisition Instrument;2) compare four measuring point voltage signals, determine quiet fault of eccentricity orientation;Wherein, in step 2) in, it is quiet inclined The Rule of judgment in heart failure orientation is that (1) four measuring point voltage is equal and equal to voltage under normal circumstances, then generator operation is being just Often;(2) four measuring point any twos are equal with respect to measuring point voltage and equal to surveyed voltage under normal circumstances, two other measuring point electricity Pressure is respectively maximum and minimum value, then is 90 ° of quiet fault of eccentricity of air gap, and the minimal air gap of failure is located at the survey of maximum voltage value At point;(3) four measuring point any two adjacent measuring point voltages are equal, and two other adjacent measuring point voltage is also equal but before being not equal to The voltage of two measuring points, the size of surveyed voltage is between this two groups of measuring point voltages, then quiet partially for 45 ° of air gaps under normal circumstances Heart failure, the minimal air gap of failure is located in the middle of the measuring point of two larger equivalent voltages of magnitude of voltage;(4) four measuring points survey electricity Pressure is unequal, then is the quiet fault of eccentricity of θ angles air gap, and the minimal air gap of failure is located at the maximum measuring point of magnitude of voltage to magnitude of voltage the Two big measuring points turn over a θ angular direction, 0 ° of 45 ° of < θ <;The average voltage of four induction coils is U when definition is normally run0, The voltage of lower four measuring points of malfunction is followed successively by U from high to low1> U2> U3> U4, then θ angles calculated using following formula
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2. detection method according to claim 1, it is characterised in that in step 1) in, the voltage signal collected is passed through into computing Amplifier is by output voltage stabilization in the collection range of Acquisition Instrument.
3. the detection method of the static fault of eccentricity degree of a kind of power generator air gap, it is characterised in that determine that power generator air gap is quiet first State fault of eccentricity orientation, specifically includes following step:Step:1) stator circumference four is gathered vertically by Hall voltage sensor The voltage signal of induction coil on measuring point is distributed, and outputs this to Acquisition Instrument;2) compare four measuring point voltage signals, determine quiet Fault of eccentricity orientation;It is determined that on the basis of power generator air gap static state fault of eccentricity orientation, during using failure each measuring point voltage with The relatively quiet eccentricity value δ of the ratio calculation of measuring point voltage under normal circumstancessSo as to assess fault degree.
4. detection method according to claim 3, it is characterised in that in the case of 90 ° of quiet bias, measuring point voltage U1> U2=U3> U4Sort, then relatively quiet eccentricity value δsCalculated using following formula:
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5. detection method according to claim 3, it is characterised in that in the case of 45 ° of quiet bias, measuring point voltage U1=U2> U3= U4Sort, then relatively quiet eccentricity value δsCalculated using following formula:
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>s</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>4</mn> </mrow> </msub> </mrow> <mn>4</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msqrt> <mn>2</mn> </msqrt> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mfrac> <msub> <mi>U</mi> <mn>0</mn> </msub> <msub> <mi>U</mi> <mn>1</mn> </msub> </mfrac> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msqrt> <mn>2</mn> </msqrt> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mfrac> <msub> <mi>U</mi> <mn>0</mn> </msub> <msub> <mi>U</mi> <mn>2</mn> </msub> </mfrac> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msqrt> <mn>2</mn> </msqrt> <mrow> <mo>(</mo> <mfrac> <msub> <mi>U</mi> <mn>0</mn> </msub> <msub> <mi>U</mi> <mn>3</mn> </msub> </mfrac> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>4</mn> </mrow> </msub> <mo>=</mo> <msqrt> <mn>2</mn> </msqrt> <mrow> <mo>(</mo> <mfrac> <msub> <mi>U</mi> <mn>0</mn> </msub> <msub> <mi>U</mi> <mn>4</mn> </msub> </mfrac> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> </mrow>
6. detection method according to claim 3, it is characterised in that in the case of the quiet bias in θ angles, measuring point voltage U1> U2> U3> U4 Sort, then relatively quiet eccentricity value δsCalculated using following formula:
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mi>s</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>4</mn> </mrow> </msub> </mrow> <mn>4</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>/</mo> <msub> <mi>U</mi> <mn>1</mn> </msub> </mrow> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mrow> <mo>(</mo> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>/</mo> <msub> <mi>U</mi> <mn>2</mn> </msub> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>/</mo> <msub> <mi>U</mi> <mn>1</mn> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>/</mo> <msub> <mi>U</mi> <mn>2</mn> </msub> </mrow> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mfrac> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>/</mo> <msub> <mi>U</mi> <mn>3</mn> </msub> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>/</mo> <msub> <mi>U</mi> <mn>1</mn> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>/</mo> <msub> <mi>U</mi> <mn>3</mn> </msub> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>/</mo> <msub> <mi>U</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>/</mo> <msub> <mi>U</mi> <mn>4</mn> </msub> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mrow> <mi>s</mi> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>/</mo> <msub> <mi>U</mi> <mn>4</mn> </msub> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mi>cos</mi> <mrow> <mo>(</mo> <mi>arctan</mi> <mfrac> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>/</mo> <msub> <mi>U</mi> <mn>3</mn> </msub> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <msub> <mi>U</mi> <mn>0</mn> </msub> <mo>/</mo> <msub> <mi>U</mi> <mn>4</mn> </msub> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> </mrow> 2
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