CN103728463B - Ultrasonic Wind Meter and Measuring Method - Google Patents

Abstract

The invention provides an ultrasonic anemometer and a measurement method. The anemometer includes a casing, a wind speed and direction measurement module, a direction correction module and a main control module. The invention adopts one ultrasonic transmitter and three ultrasonic receivers to make the circuit structure simpler, and the wind speed and direction can be measured only by sending ultrasonic waves once. The invention has a direction correction module, which eliminates the need to consider the installation direction of the equipment during the installation of the equipment. During the measurement process, the measurement direction can be corrected to a pre-established direction, which simplifies the installation process and improves the measurement accuracy. . The ultrasonic anemometer circuit provided by the invention is simple in circuit design, easy in installation, simple in measurement method and accurate in measurement result.

Description

Ultrasonic Wind Meter and Measuring Method

Technical field:

The invention relates to the field of wind speed and direction measurement, in particular to an ultrasonic wind measuring instrument and a method for measuring wind speed and direction.

Background technique:

Anemometers play an important role in the fields of meteorology, civil aviation, highways, agriculture and new energy. At present, the ultrasonic anemometer has become the mainstream of anemometer application and development.

The current ultrasonic anemometer is mainly based on the measurement principles of time difference method and phase difference method. In the design of the device, it is often necessary to place the ultrasonic sensor in the same direction as the geographical north-south and east-west. The pre-set direction and the geographical north-south and east-west directions are corrected for installation. If there is an error in the installation direction, it will cause errors in the measurement results. At the same time, most of the existing ultrasonic anemometers adopt the time-sharing principle, that is, a pair of ultrasonic transducers send and receive signals in turn, requiring many measurements, and the circuit structure is complex, often resulting in errors in measurement results due to multiple measurements.

Invention content:

The object of the present invention is to provide an ultrasonic wind measuring instrument with simple circuit structure, easy installation and accurate measurement results.

Another object of the present invention is to provide a method for measuring wind speed and direction, which has a simple and convenient measurement process and accurate results.

Technical scheme of the present invention is as follows:

An ultrasonic anemometer includes a casing, a wind speed and direction measurement module, a direction correction module and a main control module,

The casing includes an upper installation box and a lower installation box, the upper installation box and the lower installation box are connected through a bracket, and a mounting column is provided below the lower installation box for supporting and fixing;

The wind speed and direction measurement module includes an ultrasonic transmitting sensor and three ultrasonic receiving sensors, which are respectively connected with the main control module; the direction correction module is also connected with the main control module;

The main control module and the direction correction module are placed in the lower installation box, three ultrasonic receiving sensors are embedded in the lower installation box and distributed in an equilateral triangle, the ultrasonic emission sensors are embedded in the upper installation box, and the ultrasonic emission sensors are located in the three directly above the three ultrasonic receiving sensors, and correspondingly located in the center of the equilateral triangle formed by the three ultrasonic receiving sensors.

The main control module includes a CPLD/FPGA controller, a power module, an RS485 interface module, an ultrasonic drive circuit, a temperature compensation module, a signal conditioning circuit and a threshold comparison circuit, and the power module, the RS485 interface module, a temperature compensation module, an ultrasonic Both the drive circuit and the threshold comparison circuit are connected with the CPLD/FPGA controller, the power supply module is also connected with the ultrasonic drive circuit module, the ultrasonic transmitting sensor is connected with the ultrasonic drive circuit module, and the ultrasonic receiving sensor is connected with the signal conditioning circuit .

The direction correction module includes a magnetic sensor. The ultrasonic transmitting sensor and the ultrasonic receiving sensor have a certain radiation direction opening angle.

The method for measuring wind speed and direction by using the above-mentioned anemometer comprises the following steps:

1) The main control module obtains the ambient temperature data, and determines the temperature compensation according to the calculation;

2) The main control module generates a signal to make the ultrasonic transmitting sensor send a pulse signal, and at the same time start the counter corresponding to the ultrasonic receiving sensor in the main control module;

3) When the ultrasonic pulse signals are respectively received by the three ultrasonic receiving sensors, the received signals are amplified and processed by their respective follow-up signal conditioning circuits, and the corresponding counter interrupt stop signals are generated by the threshold comparison circuit, and the ultrasonic pulses are respectively obtained from The time elapsed from sending to being received by each ultrasonic receiver;

4) The main control module corrects the wind speed and direction by obtaining data from the magnetic sensor, and obtains the measured wind speed and direction;

5) The main control module outputs the data through the RS485 interface module.

The present invention has following beneficial effect:

1. The ultrasonic anemometer of the present invention adopts an ultrasonic transmitter and three ultrasonic receivers, which makes the circuit structure simpler. During the measurement process, the wind speed and direction can be measured by sending ultrasonic waves only once.

2. The ultrasonic anemometer of the present invention has a direction correction module, which does not need to consider the installation direction of the equipment during the installation of the equipment. During the measurement process, the measurement direction can be corrected to a pre-established direction, which simplifies the installation process. Improved measurement accuracy.

3. The circuit design of the ultrasonic anemometer of the present invention is simple and low in cost, and the circuit adopts a modular design.

4. The present invention has strong environmental adaptability. The upper installation box can block rain, snow and dust, and reduce the influence of foreign objects on wind speed measurement. At the same time, the upper installation box can also prevent the interference of wind in the vertical direction, improving the measurement accuracy of the entire system.

Description of the drawings:

Figure 1 is a schematic diagram of the structure of the ultrasonic wind measuring instrument.

Fig. 2 Schematic diagram of the top view structure of the ultrasonic wind measuring instrument.

Figure 3 is a block diagram of the main control module circuit.

Fig. 4 is a schematic diagram of the principle of the method for the wind speed and direction measurement device of the present invention to correct the measured parameters by using the magnetic sensor to obtain the wind speed and direction parameters under actual environmental conditions.

Fig. 5 is the transmitting driving circuit of the ultrasonic transmitting sensor of the present invention.

Fig. 6 is a schematic diagram of the signal conditioning circuit and the threshold comparison circuit of the ultrasonic receiving sensor of the present invention.

FIG. 7 is a circuit schematic diagram of the direction correction module of the present invention.

Figure 8 is a schematic diagram of the temperature compensation circuit.

In the figure: 1-upper installation box; 21-bracket; 22-bracket; 23-bracket; 31-ultrasonic receiving sensor A; 32-ultrasonic receiving sensor B; 33-ultrasonic receiving sensor C; 4-ultrasonic transmitting sensor; 5. Lower the installation box; 6. Install the column.

Detailed ways:

The present invention will be described below in conjunction with the accompanying drawings and specific embodiments.

As shown in Fig. 1 and Fig. 2, the ultrasonic anemometer of the present invention includes a casing, a wind speed and direction measurement module, a direction correction module and a main control module.

The casing includes an upper installation box 1 and a lower installation box 5, the upper installation box 1 and the lower installation box 5 are connected by a bracket 3, and the bottom of the lower installation box 5 is connected to the installation column 6, and the installation column is used for supporting and fixing. The upper installation box 1 and the lower installation box 5 are arranged oppositely, and are connected as a whole by three brackets.

The wind speed and direction measurement module includes an ultrasonic transmitting sensor 4 and three ultrasonic receiving sensors A, B, and C, which are respectively connected to the main control module; the direction correction module is also connected to the main control module;

The main control module and the direction correction module are placed in the lower installation box 5, three ultrasonic receiving sensors are embedded in the lower installation box 5 and distributed in an equilateral triangle, the ultrasonic emission sensor 4 is embedded in the upper installation box 1, and the ultrasonic emission sensor 4 It is located directly above the three ultrasonic receiving sensors, and is correspondingly located at the center of an equilateral triangle formed by the three ultrasonic receiving sensors. Its orthographic projection is at the center of the triangle.

As shown in Figure 3, the main control module includes: CPLD/FPGA controller, power module, RS485 interface module, ultrasonic drive circuit, temperature compensation module, signal conditioning circuit and threshold comparison circuit, described power module, RS485 interface module, The temperature compensation module, the ultrasonic driving circuit and the threshold comparison circuit are all connected with the CPLD/FPGA controller.

In this embodiment, the selection of the ultrasonic transmitter/ultrasonic receiver needs to have a certain radiation direction opening angle, so as to ensure that the ultrasonic pulse signal sent by the ultrasonic transmitter can be stably and reliably received by the ultrasonic receiver.

The ultrasonic transmitting/receiving sensor selected in the present invention has higher sensitivity and a certain opening angle, and adopts a model DYA-125-02A transceiver integrated sensor to form a wind speed and direction measurement module.

The CPLD/FPGA controller in the main control module of the present invention adopts the CPLD chip EPM240T100C5 of Altera Company, which has stable performance, low power consumption, high cost performance, and has the characteristics of parallel processing, which can realize simultaneous processing of signals of 3-way ultrasonic conditioning circuits.

Figure 5 shows the above-mentioned ultrasonic driving circuit (that is, the ultrasonic transmitting sensor transmitting driving circuit). In the figure, QU1 is connected to the I/O pin of the controller CPLD, and the controller CPLD generates a signal to control the on-off of the switch transistor Q1, so that the secondary side of the transformer T1 generates a DC pulse voltage of about 120V to drive the ultrasonic transmitter FS1 Generate an ultrasonic pulse signal.

Figure 6 shows the schematic diagram of the above-mentioned signal conditioning circuit (that is, the signal conditioning circuit of the ultrasonic receiving sensor) and the threshold comparison circuit. In the figure, the ultrasonic receiver FS2 receives the ultrasonic signal sent by the ultrasonic transmitter, and the signal is amplified by the two-stage amplifier circuit composed of the operational amplifier U1 to generate the voltage signal required by the subsequent circuit, and sent to the threshold comparison circuit composed of the comparator U2 to generate timing Interrupt signal, the interrupt signal is sent to the port connected to CPLD and INT2 for processing.

As shown in Fig. 7, the magnetic sensor of model LSM303DLM is used in the present invention to form a direction correction module, and the chip uses I2C protocol for data transmission, which has a high data transmission rate. Its circuit schematic diagram is shown in Fig. 7. The chip control port and data port pins are directly connected to CPLD. CPLD generates interrupt control signals through INT1 and INT2 pins to read the current installed azimuth angle data. ^The direction sensor sends data to CPLD through I2C interface for processing. Correct the orientation.

During the measurement process of the anemometer, the main controller needs to obtain the current ambient temperature data to perform temperature compensation on the measurement unit in real time to ensure measurement accuracy. The temperature compensation module in the present invention mainly includes a temperature detection circuit and a low temperature compensation circuit. The temperature detection circuit adopts a digital temperature sensor DS18B20 chip to form a detection module. The low temperature measurement range can reach -55 degrees Celsius, and the temperature measurement error is less than 0.5 degrees Celsius. High, no external components are needed in use, the temperature measurement circuit is simple, and it is directly connected with the CPLD controller. Figure 8 shows the low temperature compensation circuit. The heating wire H1 is respectively placed in the upper installation box and the lower installation box to heat the area between the upper and lower installation boxes. The control terminal Con1 of the compensation circuit is directly connected to the CPLD. Control the on-off of the switch transistor Q5 to control the work of the heating wire.

Ultrasonic anemometer of the present invention carries out the measuring method of wind speed and wind direction, specifically comprises the following steps:

Figure 4 shows two coordinate systems, the dotted line is the coordinate system during actual installation, and the solid line is the coordinate system to be corrected. In the following steps, step (4) and step (5) are to calculate the wind speed and direction values for the solid line coordinate system, and the others are performed according to the dotted line coordinate system.

(1) The main controller CPLD/FPGA obtains the data of the temperature compensation module, judges whether temperature compensation is required, and makes the sound velocity always maintain a constant value of 340m/s during the measurement process through compensation. The calculation formula of temperature compensation is as follows:

in is the speed of sound, is the current ring temperature value;

(2) The main controller CPLD/FPGA generates a signal to make the ultrasonic drive circuit module drive the ultrasonic transmitter to send a pulse signal, and simultaneously start the counters corresponding to the three ultrasonic receivers inside the main controller CPLD/FPGA, when the ultrasonic pulse The signals are respectively received by the ultrasonic receiver in the lower installation box;

(3) The signals received by the ultrasonic receiver are respectively amplified and processed by their respective follow-up signal conditioning circuits, and the corresponding counter interrupt stop signals are generated by the threshold comparison circuit, and the ultrasonic pulses are respectively obtained from being sent to being received by each ultrasonic receiver. To the elapsed time t, the wind velocity value components on each side of the equilateral triangle structure formed by the three ultrasonic receivers can be obtained by calculation:

in is the wind speed value on the upper side of the equilateral triangle structure formed by three ultrasonic receivers, c is the speed of sound, and t is the propagation time of the ultrasonic pulse signal from the ultrasonic transmitter to the time it is received by the ultrasonic receiver, is the maximum cross-sectional angle of the spatial structure formed by the ultrasonic transmitter and three ultrasonic receivers.

(4) The main controller obtains the corresponding data through the magnetic sensor to correct the wind speed and direction and obtain the measured wind speed and direction;

(5) Finally, the main controller sends the data through the RS485 interface module.

Fig. 4 is a schematic diagram of the principle of the method for the wind speed and direction measurement device of the present invention to correct the measured parameters by using the magnetic sensor to obtain the wind speed and direction parameters under actual environmental conditions. Wherein, the coordinates of X and Y dotted lines are the geographic orientation to which the entire wind speed and direction measuring device is installed, and the coordinates of X and Y dotted lines are the coordinate system established with the sensor A as the designated direction. The solid line coordinates are the coordinates in the actual geographical north-south and east-west directions, where the angle r is the deviation angle between the actual installation direction measured by the magnetic sensor and the geographical north-south and east-west directions. A, B, and C in Figure 4 are the schematic diagrams in Figure 1 The 3 ultrasonic receivers on the lower installation box. Angle a is the maximum cross-sectional angle of the spatial structure formed by the ultrasonic transmitter and the three ultrasonic receivers.

As shown in Figure 4, it is assumed that the propagation time of the ultrasonic pulse signal sent by the ultrasonic transmitter in the upper installation box to the ultrasonic receivers A, B, and C in the lower installation box is: .

According to the formula , the wind speed component values on each side of the equilateral triangle formed by A, B, and C can be obtained:

According to the basic theorem of equilateral triangles, the wind speed component values of the wind direction shown in the figure in the installation direction X and Y directions can be obtained :

According to the data read by the magnetic sensor, the deflection angle r between the actual geographical north-south, east-west direction and the installation direction can be obtained, and the wind speed value component of the actual wind can be obtained according to the deflection angle , and the wind direction can be obtained according to the wind speed value:

According to the principle of vector synthesis, the values of actual wind speed and wind direction can be obtained:

.

Claims (4)

1. An ultrasonic anemometer, is characterized in that: comprise casing, wind speed and wind direction measurement module, direction correction module and main control module, The casing includes an upper installation box and a lower installation box, the upper installation box and the lower installation box are connected through a bracket, and a mounting column is provided below the lower installation box for supporting and fixing; The wind speed and direction measurement module includes an ultrasonic transmitting sensor and three ultrasonic receiving sensors, which are respectively connected with the main control module; the direction correction module is also connected with the main control module; The main control module and the direction correction module are placed in the lower installation box, three ultrasonic receiving sensors are embedded in the lower installation box and distributed in an equilateral triangle, the ultrasonic emission sensors are embedded in the upper installation box, and the ultrasonic emission sensors are located in the three directly above the first ultrasonic receiving sensor, and correspondingly located at the center of the equilateral triangle formed by the three ultrasonic receiving sensors; The main control module includes a CPLD/FPGA controller, a power module, an RS485 interface module, an ultrasonic drive circuit, a temperature compensation module, a signal conditioning circuit and a threshold comparison circuit, and the power module, the RS485 interface module, a temperature compensation module, an ultrasonic Both the driving circuit and the threshold comparison circuit are connected with the CPLD/FPGA controller, the power module is also connected with the ultrasonic driving circuit, the ultrasonic transmitting sensor is connected with the ultrasonic driving circuit, and the ultrasonic receiving sensor is connected with the signal conditioning circuit. 2. A method utilizing the anemometer according to claim 1 to measure wind speed and direction, comprising the following steps: (1) The main control module obtains the ambient temperature data, and determines the temperature compensation according to the calculation; (2) the main control module generates a signal to make the ultrasonic transmitting sensor send a pulse signal, and starts the counter corresponding to the ultrasonic receiving sensor in the main control module simultaneously; (3) When the ultrasonic pulse signals are respectively received by the three ultrasonic receiving sensors, the received signals are respectively amplified and processed by the respective follow-up signal conditioning circuits, and the corresponding counter interrupt stop signals are generated by the threshold comparison circuit to obtain the ultrasonic pulses respectively The time elapsed from sending to being received by each ultrasonic receiver; (4) The main control module obtains data through the magnetic sensor to perform wind speed and wind direction correction, and obtains the measured wind speed and wind direction; (5) The main control module outputs the data through the RS485 interface module. 3. The method according to claim 2, characterized in that: in (1), the main control module obtains the ambient temperature data of the current temperature compensation module, determines whether to perform corresponding temperature compensation, and makes the sound velocity in the measurement process through compensation Always maintain a constant value of 340m/s; the calculation formula of temperature compensation is as follows: ν _c ＝331.3+0.606·T _c Among them, ν _c is the speed of sound, and T _c is the current ambient temperature value. 4. method as claimed in claim 2, it is characterized in that: in described (4), the time that the ultrasonic pulse signal that the ultrasonic transmitting sensor that obtains by measurement sends arrives three-way ultrasonic receiving sensor A, B, C propagates is: t _A , t _B , t _C , the angle r is the deflection angle between the actual installation direction measured by the magnetic sensor and the geographical north-south and east-west directions, c is the speed of sound, and it can be obtained that the wind speed value is the wind speed component value on each side of the equilateral triangle formed by the ultrasonic receiving sensors A, B, and C, and α is the maximum cross-sectional angle of the spatial structure formed by the ultrasonic transmitter and three ultrasonic receivers : ${\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; C</span>}}<span\; class="notranslate">\; =</span>\sqrt{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; a</span>}$ ${\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; C</span>}\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; =</span>\sqrt{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; C</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; a</span>}$ ${\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; =</span>\sqrt{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; c</span>}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; a</span>}$ According to the basic theorem of equilateral triangles, the wind speed component values V _X , V _Y of the wind direction in the installation direction X and Y directions can be obtained: ${\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; C</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; C</span>}\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; B</span>}}$ ${\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; Y</span>}}<span\; class="notranslate">\; =</span>\frac{\sqrt{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; C</span>}}<span\; class="notranslate">\; +</span>\frac{\sqrt{\mathrm{<span\; class="notranslate">\; 3</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&CenterDot;</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; C</span>}\mathrm{<span\; class="notranslate">\; B</span>}}$ According to the data read by the magnetic sensor, the deflection angle r between the actual geographical north-south, east-west direction and the installation direction can be obtained. According to the deflection angle, the wind speed value components V' _X and V _Y ' of the actual wind can be obtained, and The wind direction can be obtained from the value wind volume: V′ _X ＝V _X ·cosr V′ _Y ＝V _Y ·cosr According to the principle of vector synthesis, the values of actual wind speed and wind direction can be obtained: $\mathrm{<span\; class="notranslate">\; V</span>}<span\; class="notranslate">\; =</span>\sqrt{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; Y</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}$ $\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>\frac{{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; x</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}{{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; Y</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span>$