CN117129012A - Coil constant and magnetic field non-orthogonal angle calibration method for atomic inertial measurement - Google Patents
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Abstract
The invention discloses a coil constant and magnetic field non-orthogonal angle calibration method for atomic inertial measurement. The method takes a magnetic field coil in an atomic spin inertia measurement system as a research object, and provides an innovative scheme for simultaneously measuring the constant of the magnetic field coil and the non-orthogonal angle of the magnetic field aiming at the problem that the measurement system is easily influenced by the residual magnetic field in the device. The method realizes the decoupling of electrons and nuclei by applying a larger magnetic field in the direction of the pumping laser path. The invention can realize the simultaneous calibration of the magnetic field coil constant and the non-orthogonal angle under the condition of not damaging the spin polarization state of the atoms, thereby obviously improving the compensation precision of the residual magnetic field in the atomic spin inertia measurement system and providing reliable guarantee for improving the stability of the system.
Description
Technical Field
The invention relates to a coil constant and magnetic field non-orthogonal angle calibration method for atomic inertial measurement, belonging to the technical field of high-precision active compensation of residual magnetic field of an atomic inertial measurement device.
Background
Inertial navigation is a key technology, and is widely applied to the fields of aerospace, underwater exploration and the like, and is used for sensing carrier motion information. In inertial navigation systems, gyroscopes are a key core component in measuring angular velocity, playing an important role. With the breakthrough progress of optical detection and quantum control technologies, an atomic spin inertial measurement device based on a spin-free relaxation (SERF) principle has attracted a great deal of attention, and is considered to be a development direction of ultra-high precision and small-volume gyroscopes in the future.
Atomic spin inertial measurements are based on the SERF state, operating under extremely weak magnetic strips. The core air chamber part is easily affected by the residual magnetic field in the device. Therefore, compensation of the residual magnetic field within the device is very important. High-precision compensation of the residual magnetic field requires that the magnetic field coil constants and the non-orthogonal angles produced when the magnetic field coils are placed be obtained as accurately as possible. The current measuring method for the non-orthogonal angle of the magnetic field coil is complex, and can influence the polarization of electrons and nuclei to destroy the normal working state of the inertial measurement device.
Disclosure of Invention
The technical problems to be solved by the invention are as follows: the method for calibrating the coil constant and the non-orthogonal angle of the magnetic field for atomic inertial measurement is beneficial to improving the compensation precision of the magnetic field coil to the residual magnetic field in the device.
The technical scheme of the invention is as follows:
a method for calibrating a coil constant and a magnetic field non-orthogonal angle for atomic inertial measurement, comprising the steps of:
step A, an atomic air chamber in an atomic inertial measurement device is arranged in a non-magnetic heating device, the non-magnetic heating device is arranged in a magnetic field coil, the magnetic field coil comprises a magnetic field coil in the z-axis direction, a magnetic field coil in the y-axis direction and a magnetic field coil in the x-axis direction, a pumping laser light path passes through the atomic air chamber along the z-axis direction, a detection laser light path passes through the atomic air chamber along the x-axis direction, the non-magnetic heating device is started to heat the atomic air chamber, atomic polarization is enabled to reach a working state by pumping light, and residual magnetic fields in the device are compensated to be near zero by the magnetic field coil;
step B, for the magnetic field coil constant in the z-axis directionMeasuring;
step C, for the magnetic field coil constant in the y-axis directionAnd measuring a non-orthogonal angle beta of the magnetic field in the directions of the y axis and the z axis;
step D, for the magnetic field coil constant in the x-axis directionAnd measuring a magnetic field non-orthogonal angle of the x-axis and the z-axis.
The step B comprises the following steps:
step B1, applying increasing voltage to the magnetic field coil in the z-axis directionFrom 6000mV to 9000mV, adjacent voltage points are separated by 1000mV, and nuclear precession signals are recorded in sequence>Wherein i=1, 2,3,4;
step B2, obtaining the first nucleon precession frequency by using the following fitting formula
Wherein k is 1 Is a coefficient, t is time, t 0 For initial time, b 1 Is a constant value;
step B3, using the fittedCalculating a first total magnetic field vector module value B according to the following formula 1 ||：
Wherein gamma is n Is nuclear gyromagnetic ratio;
step B4, fitting the obtained number by using the following formulaThe units are nT/mV:
B′ x ＝0，
B′ y ＝0，
wherein the method comprises the steps ofFor the residual weak magnetic field in the z-axis direction, B' x Is the x-axis magnetic field B x X-axis projection component of B' y Is the y-axis magnetic field B y Is the y-axis projection component of B' z Is an intermediate quantity.
The step C comprises the following steps:
step C1, applying voltage to the magnetic coil in the z-axis direction Applying increasing voltage U to magnetic coil in y-axis direction y ，U y From 1000mV to 9000mV, adjacent voltage points are separated by 1000mV, and nuclear precession signals are recorded in sequence>Where j=1, 2, …,9;
step C2, obtaining the second nucleon precession frequency by using the following fitting formula
Wherein k is 2 As coefficients, b 2 Is a constant value;
step C3, using the fittedCalculating a second total magnetic field vector module value B according to the following formula 2 ||：
Step C4, fitting the obtained number by using the following formulaUnits are nT/mV, and beta:
B’ x ＝0，
wherein B' 2 Is of intermediate quantity, B z Is the magnetic field of the z-axis,for the remaining weak magnetic field in the z-axis direction, +.>A weak magnetic field remains for the y-axis direction.
The step D comprises the following steps:
step D1, applying voltage to the magnetic field coil in the z-axis direction Applying increasing voltage U to magnetic field coil in x-axis direction x ，U x From 1000mV to 9000mV, adjacent voltage points are separated by 1000mV, and nuclear precession signals are recorded in sequence>Where k=1, 2, …,9;
step D2, obtaining a third nucleon precession frequency by using the following fitting formula
Wherein k is 3 As coefficients, b 3 Is a constant value;
step D3, using the fittedCalculating a third total magnetic field vector module value B according to the following formula 3 ||：
Step D4, fitting the obtained number by using the following formulaUnits are nT/mV and α:
B’ y ＝0,
wherein B' z Is an intermediate amount of the total number of the components,for the remaining weak magnetic field in the z-axis direction, +.>The weak magnetic field remaining in the x-axis direction.
Said alpha is B x And B' x Included angle between, said beta is B y And B' y An included angle between the two.
The invention has the following technical effects: the invention provides a coil constant and magnetic field non-orthogonal angle calibration method for atomic inertial measurement, which takes a magnetic field coil in an atomic spin inertial measurement system as a research object, and provides an innovative scheme for simultaneously measuring the magnetic field coil constant and the magnetic field non-orthogonal angle aiming at the problem that the measurement system is easily influenced by a residual magnetic field in a device. The method realizes the decoupling of electrons and nuclei by applying a larger magnetic field in the direction of the pumping laser path. The invention can realize the simultaneous calibration of the magnetic field coil constant and the non-orthogonal angle under the condition of not damaging the spin polarization state of the atoms, thereby obviously improving the compensation precision of the residual magnetic field in the atomic spin inertia measurement system and providing reliable guarantee for improving the stability of the system.
Compared with the prior art, the invention has the advantages that: (1) According to the calibration method for the coil constant and the magnetic field non-orthogonal angle for atomic inertia measurement, provided by the invention, the magnetic coil in the z-axis direction is applied with a larger voltage, so that the z-axis direction is always in a larger magnetic field, at the moment, electrons and nuclei are in a decoupling state, and the normal working state of the atomic spin inertia measurement device is not damaged. (2) The calibration method for the coil constant and the non-orthogonal angle of the magnetic field for atomic inertia measurement can simultaneously realize the accurate calibration of the coil constant and the non-orthogonal angle of the magnetic field.
Drawings
Fig. 1 is a schematic structural diagram of an atomic inertial measurement unit according to the method for calibrating coil constants and non-orthogonal angles of magnetic fields for atomic inertial measurement.
Fig. 2 is a schematic diagram of the magnetic field coordinate system and the non-orthogonal angle of the magnetic field in the present invention.
FIG. 3 is a flow chart of a method of calibrating coil constants and magnetic field non-orthogonal angles for atomic inertial measurement embodying the present invention. Step 1 is included in fig. 3 between the beginning and the end, turning on heating and pumping light to polarize atoms; step 2, compensating the residual magnetic field in the device; step 3, applying a variable voltage in the z-axis direction; step 4, fitting by using sine functionAnd get ||B 1 ||，/>Is the first nuclear precession frequency, ||B 1 I is the first total magnetic field vector modulus; step 5, fitting to obtain->Is the z-direction magnetic field coil constant; step 6, applying a fixed voltage in the z-axis direction; step 7, applying a variable voltage in the y-axis direction; step 8, fitting out +_with sine function>And get ||B 2 ||，/>Is the second nuclear precession frequency, ||B 2 I is the second total magnetic field vector modulus; step 9, fitting to obtain->And angle beta->Is the magnetic field coil constant in the y direction, and the beta angle is the non-orthogonal angle of the magnetic field in the y axis and the z axis; step 10, applying a fixed voltage in the z-axis direction; step 11, applying a variable voltage in the x-axis direction; step 12, fitting with sine functionAnd get ||B 3 ||，/>Is the third nuclear precession frequency, ||B 3 I is the third total magnetic field vector modulus; step 13, fitting to obtain->And angle alpha>Is the x-direction magnetic field coil constant, and the alpha angle is the non-orthogonal angle of the magnetic field in the x-axis and z-axis directions.
The reference numerals are explained as follows: 1-pumping a laser light path; 2-detecting a laser light path; a 3-mirror; a 4-atom gas cell; 5-a non-magnetic heating device; 6-an optical detector; 7-z-axis direction magnetic field coils; 8-y axis direction magnetic field coils; 9-x axis direction magnetic field coils; a 10-z axis magnetic field function generator; an 11-y axis magnetic field function generator; a 12-x axis magnetic field function generator; 13-pumping laser; 14-a detection laser; xyz-rectangular coordinate system three axes (i.e., x-axis, y-axis, and z-axis); b (B) x X-axis magnetic field (i.e. x-axis direction magnetic)A magnetic field vector applied by the field coil); b (B) y -y-axis magnetic field (i.e. magnetic field vector applied by y-axis direction magnetic field coil); b'. x -B x An x-axis projection component of (2); b'. y -B y A y-axis projection component of (2); non-orthogonal angles of the magnetic field in the alpha-x and z directions (i.e. B x And B' x An included angle therebetween); the non-orthogonal angle of the magnetic field in the beta-y and z directions (i.e., B y And B' y Included angle between them).
Detailed Description
The invention is described below with reference to the figures (fig. 1-3) and examples.
Fig. 1 is a schematic structural diagram of an atomic inertial measurement unit according to the method for calibrating coil constants and non-orthogonal angles of magnetic fields for atomic inertial measurement. Fig. 2 is a schematic diagram of the magnetic field coordinate system and the non-orthogonal angle of the magnetic field in the present invention. FIG. 3 is a flow chart of a method of calibrating coil constants and magnetic field non-orthogonal angles for atomic inertial measurement embodying the present invention. Referring to fig. 1 to 3, a coil constant and magnetic field non-orthogonal angle calibration method for atomic inertial measurement includes the steps of: step A, an atomic air chamber 4 in an atomic inertial measurement device is arranged in a non-magnetic heating device 5, the non-magnetic heating device 5 is arranged in a magnetic field coil, the magnetic field coil comprises a z-axis direction magnetic field coil 7 (which is connected with a z-axis magnetic field function generator 10), a y-axis direction magnetic field coil 8 (which is connected with a y-axis magnetic field function generator 11) and an x-axis direction magnetic field coil 9 (which is connected with an x-axis magnetic field function generator 12), a pumping laser light path 1 (pumping light of which is sourced from a pumping laser 13) passes through the atomic air chamber 4 along the z-axis direction, a detection laser light path 2 (detection light of which is sourced from a detection laser 14, the detection laser 14 passes through the atomic air chamber 4 along the x-axis direction sequentially through a reflecting mirror 3 and the atomic air chamber 4, the non-magnetic heating device 5 is started to heat the atomic air chamber 4, atomic polarization is enabled to reach a working state by pumping light, and residual magnetic fields in the device are compensated to a near zero state by the magnetic field coil; step B, for the magnetic field coil constant in the z-axis directionMeasuring;step C, magnetic field coil constant in y-axis direction>And measuring a non-orthogonal angle beta of the magnetic field in the directions of the y axis and the z axis; step D, magnetic field coil constant in x-axis direction +.>And measuring a magnetic field non-orthogonal angle of the x-axis and the z-axis.
The technical problems to be solved by the invention are as follows: the method for calibrating the coil constant and the non-orthogonal angle of the magnetic field for atomic inertial measurement is provided to improve the compensation precision of the magnetic field coil to the residual magnetic field in the device.
The coil constant and magnetic field non-orthogonal angle calibration method for atomic inertia measurement is characterized in that an atomic air chamber is in a high temperature state through a non-magnetic heating device, atomic polarization is enabled to reach a working state through pumping light, and residual magnetic field in the device is compensated to be close to zero value through the magnetic field coil, so that residual magnetism in a barrel does not influence subsequent measurement. First, the magnetic field coil constants in the z-axis direction are calibrated, and the magnetic field coil constants in the y-axis direction and the x-axis direction and the non-orthogonal angles of the magnetic field formed by the magnetic field coil constants in the y-axis direction and the x-axis direction are respectively calibrated.
The measuring step of the magnetic field coil constant in the z-axis direction comprises the following steps:
step 1, voltage is respectively applied to magnetic field coils in the z-axis directionFrom 6000mV to 9000mV (adjacent voltage points are separated by 1000 mV), while corresponding precession signals generated by nuclei are recorded +.>Where i=1, 2,3,4.
Step 2, fitting the recorded 4 points by standard sine functionObtaining the frequency of nuclear precession->
Step 3, according to the fittedCalculating the module value B of the total magnetic field vector in the device 1 ||。
Step 4, according to B 1 And appliedFitting to obtain the magnetic field coil constant in the z directionThe unit is nT/mV.
The measuring steps of the magnetic field coil constant and the non-orthogonal angle in the y-axis direction are as follows:
step 1, applying voltage to a magnetic field coil in the z-axis direction Applying voltages U to the magnetic field coils in the y-axis direction y ，U y From 1000mV to 9000mV (adjacent voltage points are separated by 1000 mV), while recording the precession signal generated by the nuclei +.>Where j=1, 2, …,9.
Step 2, fitting the precession frequency of the nuclei by using a standard sine function
Step 3, according to the fitted omega n Calculate B 2 ||。
Step 4, according to B 2 I and applied U y Fitting to obtain the magnetic field coil constant in the y directionAnd a magnetic field non-orthogonal angle beta between the y-axis and the z-axis directions.
The measuring steps of the magnetic field coil constant and the non-orthogonal angle in the x-axis direction are as follows:
step 1, applying voltage to a magnetic field coil in the z-axis direction Applying voltages U to the magnetic field coils in the x-axis direction x ，U x From 1000mV to 9000mV (adjacent voltage points are separated by 1000 mV), while recording the precession signal generated by the nuclei +.>Where k=1, 2, …,9.
Step 2, fitting out by using standard sine function
Step 3, according to the fittedCalculate B 3 ||。
Step 4, according to B 3 I and applied U x Fitting to obtain magnetic field coil constant in x directionAnd a non-orthogonal angle alpha of the magnetic field in the x-axis and z-axis directions.
As shown in fig. 1, the device according to fig. 1 schematically shows that the arranged atomic spin inertia measuring device comprises a pump laser (13) for generating a pump laser (1). The laser detector comprises a detection laser (14) for generating detection laser (2), a reflector (3) for changing the optical path of the detection laser, a core sensitive unit atomic gas chamber (4), a non-magnetic heating device (5) for keeping the atomic gas chamber in a high-temperature working condition, an optical detector (6) for detecting the change of the detection laser, magnetic field coils (7) (8) (9) respectively positioned in three directions of a z axis, a y axis and an x axis for generating a magnetic field in a specific direction, and a function generator (10) (11) (12).
As shown in FIG. 2, wherein B x (1) And B y (3) Magnetic field vectors applied by magnetic field coils on the x axis and the y axis respectively, B' x (2) And B' y (4) Respectively B x And B y The projection components on the x axis and the y axis are respectively the non-orthogonal angle (6) of the magnetic field in the direction of the x axis and the z axis to be measured and the non-orthogonal angle (5) of the magnetic field in the direction of the y axis and the z axis.
As shown in fig. 3, a specific calibration procedure for fitting the calculated data is shown.
The method specifically comprises the following steps:
1. the non-magnetic heating device (5) is used for enabling the atomic air chamber to be in a high-temperature state, pumping laser (1) is used for enabling atomic polarization to be in a working state, and the magnetic coils (7) (8) (9) are used for compensating the residual magnetic field in the device to be close to zero, so that the residual magnetic field in the barrel does not influence subsequent measurement.
2. The measuring process of the magnetic field coil constant in the z-axis direction comprises the following 4 steps:
step 1, voltage is respectively applied to magnetic field coils in the z-axis directionFrom 6000mV to 9000mV (adjacent voltage points are separated by 1000 mV), while corresponding precession signals generated by nuclei are recorded +.>Where i=1, 2,3,4.
Step 2, fitting the recorded 4 points by standard sine functionObtaining the frequency of nuclear precession->The fitting formula is:
wherein k is 1 Is a coefficient, t is time, t 0 For initial time, b 1 Is a constant value.
Step 3, according to the fittedCalculating the module value B of the total magnetic field vector in the device 1 I. The calculation formula is as follows:
wherein gamma is n Is nuclear gyromagnetic ratio.
Step 4, according to B 1 And appliedFitting to obtain the magnetic field coil constant in the z directionThe unit is nT/mV. The fitting formula is as follows:
wherein B' x And B' y Respectively B x And B y Projection components on the x-axis and the y-axis, B 'is due to the fact that no magnetic field is applied in the x-axis direction and the y-axis direction at this time' x ＝0，B′ y ＝0， A weak magnetic field remaining in the z-axis direction.
3. The measuring process of the magnetic field coil constant and the magnetic field non-orthogonal angle in the y-axis direction comprises the following 4 steps:
step 1, applying voltage to a magnetic field coil in the z-axis direction Applying voltages U to the magnetic field coils in the y-axis direction y ，U y From 1000mV to 9000mV (adjacent voltage points are separated by 1000 mV), while recording the precession signal generated by the nuclei +.>Where j=1, 2, …,9.
Step 2, fitting the precession frequency of the nuclei by using a standard sine functionThe fitting formula is:
wherein k is 2 Is a coefficient, t is time, t 0 For initial time, b 2 Is a constant value.
Step 3, according to the fittedCalculate B 2 The calculation formula is as follows:
step 4, according to ||B 2 I and applied U y Fitting to obtain the magnetic field coil constant in the y directionAnd a magnetic field non-orthogonal angle beta between the y-axis and the z-axis directions. The fitting formula is as follows:
wherein B' x And B' y Respectively B x And B y Projection components in x-axis and y-axis, B 'since no magnetic field is applied in the x-axis direction at this time' x ＝0，
For the remaining weak magnetic field in the y-axis direction, +.>A weak magnetic field remaining in the z-axis direction.
4. The measuring process of the magnetic field coil constant and the magnetic field non-orthogonal angle in the x-axis direction comprises the following 4 steps:
step 1, applying voltage to a magnetic field coil in the z-axis direction Applying voltages U to the magnetic field coils in the x-axis direction x ，U x From 1000mV to 9000mV (adjacent voltage point spacing 1000)mV) and recording the precession signal generated by the nuclei simultaneously>Where k=1, 2, …,9.
Step 2, fitting out by using standard sine functionThe fitting formula is:
wherein k is 3 Is a coefficient, t is time, t 0 For initial time, b 3 Is a constant value.
Step 3, according to the fittedCalculate B 3 The calculation formula is as follows:
step 4, according to B 3 I and applied U x Fitting to obtain magnetic field coil constant in x directionAnd a non-orthogonal angle alpha of the magnetic field in the x-axis and z-axis directions. The fitting formula is as follows:
wherein B' x And B' y Respectively B x And B y The projection components in the x-axis and y-axis, since no magnetic field is applied in the y-axis direction at this time,B′ y ＝0,/> for the remaining weak magnetic field in the x-axis direction, +.>A weak magnetic field remaining in the z-axis direction.
What is not described in detail in the present specification belongs to the prior art known to those skilled in the art. It is noted that the above description is helpful for a person skilled in the art to understand the present invention, but does not limit the scope of the present invention. Any and all such equivalent substitutions, modifications and/or deletions as may be made without departing from the spirit and scope of the invention.
Claims (5)
1. A method for calibrating a coil constant and a magnetic field non-orthogonal angle for atomic inertial measurement, comprising the steps of:
step A, an atomic air chamber in an atomic inertial measurement device is arranged in a non-magnetic heating device, the non-magnetic heating device is arranged in a magnetic field coil, the magnetic field coil comprises a magnetic field coil in the z-axis direction, a magnetic field coil in the y-axis direction and a magnetic field coil in the x-axis direction, a pumping laser light path passes through the atomic air chamber along the z-axis direction, a detection laser light path passes through the atomic air chamber along the x-axis direction, the non-magnetic heating device is started to heat the atomic air chamber, atomic polarization is enabled to reach a working state by pumping light, and residual magnetic fields in the device are compensated to be near zero by the magnetic field coil;
step B, for the magnetic field coil constant in the z-axis directionMeasuring;
step C, for the magnetic field coil in the y-axis directionNumber of digitsAnd measuring a non-orthogonal angle beta of the magnetic field in the directions of the y axis and the z axis;
step D, for the magnetic field coil constant in the x-axis directionAnd measuring a magnetic field non-orthogonal angle of the x-axis and the z-axis.
2. The method for calibrating a coil constant and a magnetic field non-orthogonal angle for atomic inertial measurement according to claim 1, wherein the step B comprises:
step B1, applying increasing voltage to the magnetic field coil in the z-axis directionFrom 6000mV to 9000mV, adjacent voltage points are separated by 1000mV, and nuclear precession signals are recorded in sequence>Wherein i=1, 2,3,4;
step B2, obtaining the first nucleon precession frequency by using the following fitting formula
Wherein k is 1 Is a coefficient, t is time, t 0 For initial time, b 1 Is a constant value;
step B3, using the fittedThe first total magnetic field is calculated by the following formulaField vector modulus value B 1 ||：
Wherein gamma is n Is nuclear gyromagnetic ratio;
step B4, fitting the obtained number by using the following formulaThe units are nT/mV:
B′ x ＝0，
B′ y ＝0，
wherein the method comprises the steps ofFor the residual weak magnetic field in the z-axis direction, B' x Is the x-axis magnetic field B x X-axis projection component of B' y Is the y-axis magnetic field B y Is the y-axis projection component of B' b Is an intermediate quantity.
3. The method for calibrating a coil constant and a magnetic field non-orthogonal angle for atomic inertial measurement according to claim 1, wherein the step C comprises:
step C1, applying voltage to the magnetic coil in the z-axis direction Applying increasing voltage U to magnetic coil in y-axis direction y ，U y From 1000mV to 9000mV, adjacent voltage points are separated by 1000mV, and nuclear precession signals are recorded in sequence>Where j=1, 2, …,9;
step C2, obtaining the second nucleon precession frequency by using the following fitting formula
Wherein k is 2 As coefficients, b 2 Is a constant value;
step C3, using the fittedCalculating a second total magnetic field vector module value B according to the following formula 2 ||：
Step C4, fitting the obtained number by using the following formulaUnits are nT/mV, and beta:
B′ x ＝0，
wherein B' z Is of intermediate quantity, B z Is the magnetic field of the z-axis,for the remaining weak magnetic field in the z-axis direction, +.>A weak magnetic field remains for the y-axis direction.
4. The method for calibrating a coil constant and a magnetic field non-orthogonal angle for atomic inertial measurement according to claim 1, wherein the step D comprises:
step D1, applying voltage to the magnetic field coil in the z-axis direction Applying increasing voltage U to magnetic field coil in x-axis direction x ，U x From 1000mV to 9000mV, adjacent voltage points are separated by 1000mV, and nuclear precession signals are recorded in sequence>Where k=1, 2, …,9;
step D2, obtaining a third nucleon precession frequency by using the following fitting formula
Wherein k is 3 As coefficients, b 3 Is a constant value;
step D3, using the fittedCalculating a third total magnetic field vector module value B according to the following formula 3 ||：
Step D4, fitting the obtained number by using the following formulaUnits are nT/mV and α:
B′ y ＝0,
wherein B' z Is an intermediate amount of the total number of the components,for the remaining weak magnetic field in the z-axis direction, +.>The weak magnetic field remaining in the x-axis direction.
5. The method of calibrating a coil constant and a magnetic field non-orthogonal angle for atomic inertial measurement according to claim 1, wherein α is B x And B' x Included angle between, said beta is B y And B' y An included angle between the two.
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