CN109193969B - Microwave energy transmission system and simulation method suitable for moving target - Google Patents

Microwave energy transmission system and simulation method suitable for moving target Download PDF

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CN109193969B
CN109193969B CN201810791190.1A CN201810791190A CN109193969B CN 109193969 B CN109193969 B CN 109193969B CN 201810791190 A CN201810791190 A CN 201810791190A CN 109193969 B CN109193969 B CN 109193969B
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CN109193969A (en
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程晓洁
吴侹
彭革新
卢毅
佘广益
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Seventh Research Institute Of China Electronics Technology Group Corp
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
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    • H02J50/00—Circuit arrangements or systems for wireless supply or distribution of electric power
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Abstract

The invention discloses a microwave energy transmission system and a simulation method suitable for a moving target, which are used for realizing non-Gaussian distribution transmitted waveform-flat wave beam as a transmitting caliber level of a microwave wireless energy transmission system. In order to achieve the purpose, the microwave energy transmission system adopts a transmitting module and a receiving module which comprise distributed antenna arrays, so that multi-frequency multi-point-to-multi-point microwave wireless energy transmission is achieved, high power is not limited, and the application is more flexible. The microwave energy transmission system adopts a flat wave beam with a non-Gaussian distribution waveform as a transmitting caliber level, so that uniformly distributed power density can be obtained on a receiving array surface, the power capacities of the rectifier diodes are the same, the rectifier antenna can be uniformly designed, and the design difficulty is reduced; meanwhile, the uniformly distributed power density can improve the conversion efficiency of the rectifying antenna, so that the efficiency of the whole system is improved; in addition, the design can also solve the problem that the traditional Gaussian distribution waveform burns out the rectifying antenna due to the deviation of the focus central point caused by the movement of the target.

Description

Microwave energy transmission system and simulation method suitable for moving target
Technical Field
The invention relates to the field of microwave transmission, in particular to a microwave energy transmission system and a simulation method suitable for a moving target.
Background
At present, a microwave wireless energy transmission system mainly comprises three parts, namely emission, space propagation and receiving rectification, as shown in fig. 1. The transmitting end microwave power generator converts direct current energy into microwave energy, then the antenna is used for transmitting the microwave energy, the microwave energy is transmitted through free space, and the receiving end rectifying antenna receives and rectifies the microwave energy and converts the microwave energy into direct current energy.
In the prior art, the aperture level forms of a transmitting antenna and a receiving rectifying antenna need to be designed specially, the aperture field level distribution of the transmitting antenna needs to be designed into a Gaussian distribution form, the central transmitting power is high, the peripheral transmitting power is low, the power size accords with the Gaussian distribution, the phased array antenna is generally adopted to realize the distribution, and the cost is high; the receiving rectification antenna is also designed into a Gaussian distribution form correspondingly, the central antenna has low gain and high density, the peripheral antenna has high gain and low density, and the design difficulty is higher.
The power capacity of a rectifying diode in the conventional rectifying antenna can only be designed according to a theoretical calculation result of power density, and the conversion efficiency of the rectifying antenna is reduced due to the inconsistency between the calculation result and the actual power density in actual application.
The transmitting waveform and the receiving rectification antenna in the prior art are designed according to Gaussian distribution, are suitable for microwave wireless energy transmission of a static target, are easily over-high in power density of the rectification units with high gain and low density at the periphery due to deviation of a central focus point when applied to a moving target, burn out a rectification circuit, threaten the use safety of a system, and are not suitable for microwave wireless energy transmission of the moving target.
In the prior art, a microwave wireless energy transmission and emission system of a single-frequency single point to a single point or a single-frequency multi-point to a single point is adopted, because the frequencies are the same, the same-source design is needed, a microwave source is limited during high-power application, distributed array can not be realized, a common-frequency focusing algorithm is complex, and the Gaussian distribution design can not be accurately realized.
Disclosure of Invention
The invention provides a microwave energy transmission system and a simulation method suitable for a moving target, aiming at overcoming the defects in the prior art, ensuring higher capture efficiency and rectification antenna conversion efficiency while meeting the application requirement of the moving target.
In order to solve the technical problems, the technical scheme of the invention is as follows:
a microwave energy delivery system adapted for use with a moving target, comprising: the device comprises a transmitting module, a receiving module and a control module;
the transmitting module comprises an excitation source, a power amplifier and a transmitting antenna, wherein the excitation source generates a microwave signal and transmits the microwave signal to the power amplifier, and the power amplifier amplifies the microwave signal and transmits the microwave signal to the receiving module through the antenna;
the array mode of the transmitting module is multi-frequency multi-area arrangement, transmitting antennas in areas are homologous and coherent, different frequencies in the areas are incoherent, and a plurality of focusing points with different frequencies are formed on a receiving array surface through a self-focusing control algorithm, so that the received power density of the whole plane of the receiving array surface is relatively uniform, and a flat beam is realized;
the receiving module comprises a rectifying antenna, a load and a parameter detection module, the rectifying antenna is a uniformly distributed array surface, adapts and receives the flat wave beam of the transmitting system, converts the microwave signal into a direct current signal and transmits the direct current signal to the load;
the parameter detection module is connected with the rectifying antenna and used for detecting the working state of the rectifying antenna and the parameters of the load power and sending the parameters to the control module, and the control module carries out switching and power adjustment on the transmitting module according to the received parameters;
the transmitting module is in wireless connection with the receiving module, and the control module is in signal connection with the transmitting module and the receiving module respectively.
Preferably, the transmitting antenna and the rectifying antenna are both antenna arrays consisting of a plurality of antennas, and are used for realizing a multi-frequency multi-point-to-multi-point microwave energy aggregation mode.
Preferably, the parameter detection module comprises a communication module, a processor and a microwave sensor, and is used for collecting the state and parameter information of the rectenna and sending the information to the control module.
Preferably, the control module comprises a communication module and a processor, and the communication module is respectively in signal connection with the parameter detection module and the excitation source, analyzes the state of the rectifying antenna, and adjusts the corresponding output power and the switch of the excitation source.
Preferably, the transmitting antenna is a hexagonal antenna array consisting of 1 central transmitting antenna and 6 edge transmitting antennas.
A simulation method of a microwave energy transmission system suitable for a moving target comprises the following steps:
s1, Matlab software is adopted for calculation, antenna array information of a transmitting module A and antenna array information of a receiving module B are obtained, a three-dimensional normalized power density graph is calculated and drawn, an optimized power density distribution is obtained by searching for a variable adjustment rule, the power density fluctuation on an energy aggregation effective surface is required to be less than 3dB, a beam roll-off coefficient approaches to 1, and a power value P point is obtained;
s2, obtaining working frequency f, transceiving interval L, antenna aperture D and antenna interval D, marking 7 transmitting antennas and 7 receiving antennas on a three-dimensional coordinate to obtain A1, A2, A3, A4, A5, A6, A7, B1, B2, B3, B4, B5, B6 and B7, wherein a7 of a central transmitting point is used as an origin,
Figure GDA0003104537520000031
is an angle relative to the bore normal;
s3, obtaining the three-dimensional coordinates through the step S1
Figure GDA0003104537520000032
And
Figure GDA0003104537520000033
the following vector cosine function is firstly obtained:
Figure GDA0003104537520000034
Figure GDA0003104537520000035
also obtained is:
Figure GDA0003104537520000036
Figure GDA0003104537520000037
s4, carrying out the step S2
Figure GDA0003104537520000038
Substituting equation 2 yields:
Figure GDA0003104537520000039
wherein the content of the first and second substances,
Figure GDA00031045375200000310
is the angle of the transmitting antenna A1 relative to the aperture normal of the receiving module B;
s5, mixing
Figure GDA00031045375200000311
Substitution normalized field strength function:
Figure GDA00031045375200000312
wherein the content of the first and second substances,
Figure GDA00031045375200000313
for normalized field intensity E of transmitting antenna A1 at point PA1p(φA1) (ii) a D is the diameter of the transmitting antenna aperture, and the unit is m; λ is the wavelength, in m;
Figure GDA00031045375200000314
is an angle relative to the bore normal; j1 is a first order bezier function;
s6, calculating the normalized field intensity of the A2, the A3, the A4, the A5, the A6, the A7, the B1, the B2, the B3, the B4, the B5, the B6 and the B7 at the P point according to the steps S2-S5 and substituting the normalized field intensity into the following formula:
Figure GDA00031045375200000315
wherein S ispIs the power density value at point P, EiMAX(φi) The value of the maximum field intensity point of the transmitting antenna on the receiving surface;
s7, adopting Matlab to carry out S of the step S6pAnd (4) carrying out simulation solution on the values, and carrying out three-dimensional graph drawing by using a drawing tool to obtain a power density distribution oscillogram of the flat beam.
Compared with the prior art, the technical scheme of the invention has the beneficial effects that:
the scheme of the invention adopts a multi-frequency multi-point to multi-point energy aggregation mode to pursue and realize the flat wave beam with the wave beam roll-off coefficient approaching to 1, and can achieve the following effects:
meanwhile, higher capture efficiency and higher conversion efficiency of the rectifying antenna are ensured, and a beam with the beam energy accounting for 80% of the total energy in a 3dB fluctuation range can be obtained, which means that high capture efficiency and high conversion efficiency can be obtained at the same time;
the method is suitable for the application requirement of a moving target, when the Gaussian distribution mode is applied to the moving target, the defect that the rectifying antenna in the non-uniform distribution mode is instantaneously burnt due to overhigh local unit power caused by central focusing offset exists, and the method is only suitable for wireless energy transmission of a static target; when the center of the flat beam is designed to be deviated, only partial energy is lost, but the rectifying antenna is not burnt, so that the flat beam is suitable for application to a moving target; in addition, the uniformly distributed rectifying antenna can be designed in a special shape, so that the antenna is more easily conformal with a moving target;
the transmitting system can be distributed and arrayed, the subarrays in all areas are designed for different frequencies, the subarrays can be distributed and arrayed at certain intervals, the subarrays are selectively grouped according to the requirements of the moving target, common energy transmission or continuous energy transmission of the moving target is achieved, and the application is more flexible.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.
Fig. 1 is a schematic diagram of a microwave energy delivery system suitable for use with a moving target.
Fig. 2 is a schematic diagram of a calculation model of a transmit antenna array algorithm.
Fig. 3 is a first simulation diagram of a flat beam obtained by multi-frequency multi-point to multi-point energy aggregation.
Fig. 4 is a second simulation of a flat beam obtained by multi-frequency multi-point to multi-point energy aggregation.
Detailed Description
The drawings are for illustrative purposes only and are not to be construed as limiting the patent;
for the purpose of better illustrating the embodiments, certain features of the drawings may be omitted, enlarged or reduced, and do not represent the size of an actual product;
it will be understood by those skilled in the art that certain well-known structures in the drawings and descriptions thereof may be omitted.
The technical solution of the present invention is further described below with reference to the accompanying drawings and examples.
A microwave energy delivery system suitable for use with a moving target, as shown in fig. 1, comprising: the device comprises a transmitting module, a receiving module and a control module;
the transmitting module comprises an excitation source, a power amplifier and a transmitting antenna, wherein the excitation source generates a microwave signal and transmits the microwave signal to the power amplifier, and the power amplifier amplifies the microwave signal and transmits the microwave signal to the receiving module through the antenna;
the array mode of the transmitting module is multi-frequency multi-area arrangement, transmitting antennas in areas are homologous and coherent, different frequencies in the areas are incoherent, and a plurality of focusing points with different frequencies are formed on a receiving array surface through a self-focusing control algorithm, so that the received power density of the whole plane of the receiving array surface is relatively uniform, and a flat beam is realized;
the receiving module comprises a rectifying antenna, a load and a parameter detection module, the rectifying antenna is a uniformly distributed array surface, adapts and receives the flat wave beam of the transmitting system, converts the microwave signal into a direct current signal and transmits the direct current signal to the load;
the parameter detection module is connected with the rectifying antenna and used for detecting the working state of the rectifying antenna and the parameters of the load power and sending the parameters to the control module, and the control module carries out switching and power adjustment on the transmitting module according to the received parameters;
the transmitting module is in wireless connection with the receiving module, and the control module is in signal connection with the transmitting module and the receiving module respectively.
In the embodiment, in order to reduce the size of the antenna and facilitate array, the frequency of the signal generated by the excitation source is preferably an X-band, the center frequency is 11.2GHz, and the frequency of each frequency source is separated by 100 MHz; wherein, the frequency source interval of 100MHz is the optimal scheme of adapting the filter bandwidth of the focusing point arranged on the multi-frequency multi-point receiving surface. Based on symmetry and system implementation considerations, the transmit antenna array is determined as a hexagonal array of 7 transmit elements. Each antenna unit adopts a circularly polarized parabolic antenna with the caliber of 0.4m, the half-power beam angle of the antenna is about 5 degrees, the gain Gt is 30dBi, and the radiation efficiency is more than 50 percent.
According to the far field formula:
L＝2D2/λ
wherein D is the aperture of the antenna, and lambda is the wavelength, the nearest far field distance of the antenna with the aperture of 0.4m and the working frequency of 11.2GHz is 12 m.
After 7 transmitting antennas are arrayed, the power density on the receiving surface can be calculated:
Figure GDA0003104537520000061
wherein Pt is the transmitting power, Gt is the transmitting antenna gain, and L is the transmitting-receiving antenna spacing. Determining the transmitting power of the antenna to be 20W, and calculating the power density of the receiving rectifying antenna surface to be 7.7mW/cm2。
After the basic parameters of the transmitting system are calculated and determined, modeling and simulation are carried out on the array algorithm of the transmitting system. The algorithm adopts a vector matrix method to calculate the field intensity of any point on a receiving surface, adopts Matlab software to calculate and draw a three-dimensional normalized power density graph, obtains the optimized power density distribution by searching a variable adjustment rule, and requires that the power density fluctuation on an energy aggregation effective surface is less than 3dB and the beam roll-off coefficient approaches to 1. Calculating variable parameters in the model, namely working frequency f, transmitting-receiving distance L, antenna aperture D and antenna distance D, respectively marking coordinates of each point on three-dimensional coordinates of 7 transmitting points and 7 receiving points,
Figure GDA00031045375200000612
at an angle relative to the bore normal. The calculation model of the transmit antenna array algorithm is shown in fig. 2.
The following vector cosine function is obtained:
Figure GDA0003104537520000062
Figure GDA0003104537520000063
wherein the content of the first and second substances,
Figure GDA0003104537520000064
the vector coordinates are (-Xo-d/2,
Figure GDA0003104537520000065
-L)，
Figure GDA0003104537520000066
the vector coordinates are (0, 0, -L), resulting in:
Figure GDA0003104537520000067
Figure GDA0003104537520000068
substituting the above value into the above formula 2, we get:
Figure GDA0003104537520000069
will phiA1Value substitution normalized field strength function:
Figure GDA00031045375200000610
wherein: d- - -the diameter of the transmitting antenna aperture, with the unit of m;
λ - - - -wavelength in m;
Figure GDA00031045375200000611
-angle to bore normal;
j1- -first order Bessel function;
obtaining the normalized field intensity E of the transmitting antenna A1 at the point PA1p(φA1) And by analogy, calculating the normalized field intensity of each transmitting antenna A2-A7 at the point P. Because each transmitting antenna has different frequencies, the total power at the point P is the sum of the powers of each transmitting antenna, so that the power density value Sp at the point can be obtained, and the value is an equation of the frequency, the transceiving distance, the antenna aperture and the antenna spacing.
Figure GDA0003104537520000071
Wherein EiMAX(φi) The value of the point of maximum field strength on the receiving surface for a certain transmitting antenna.
Carrying out simulation solving on the power density formula 4 by using Matlab, and simultaneously carrying out three-dimensional graphic description by using a tool to obtain the power density distribution waveform of the flat beam shown in the figures 3 and 4; the influence of each parameter on the beam shape can be determined through simulation, and a size reference required by a receiving front is provided, so that the scheme of the antenna array is further refined and designed on the basis of the basic parameters of the transmitting system.
In this embodiment, the transmitting antenna is a parabolic antenna array composed of a plurality of antennas, so as to realize multi-frequency multi-point to multi-point microwave transmission; the transmitting antenna array is determined to be a hexagonal array consisting of 7 transmitting units. In a modified embodiment, the transmitting antenna may be a phased array antenna array, achieving an effect similar to a parabolic antenna array, and also achieving transmission of a flat beam.
In this embodiment, the parameter detection module includes a communication module, a processor, and a microwave sensor. The parameter detection module detects the working state of the rectifying antenna and the data of the microwave parameters by adopting a microwave sensor, and the data are collected and sorted by the processor and sent to the control module through the wired or wireless module.
In this embodiment, the control module includes a communication module and a processor. The control module can be arranged at any end of the transmitting module or the receiving module, is connected with the locally corresponding excitation source or parameter detection module, and adopts corresponding wireless connection for data transmission when being connected with the other end. The control module firstly receives the working state and the microwave parameters of the rectifying antenna sent by the parameter detection module, and carries out analysis and judgment according to the data, thereby carrying out the adaptive adjustment of the switch and the power of the excitation source.
A simulation method of a microwave energy transmission system suitable for a moving target comprises the following steps:
s1, Matlab software is adopted for calculation, antenna array information of a transmitting module A and antenna array information of a receiving module B are obtained, a three-dimensional normalized power density graph is calculated and drawn, an optimized power density distribution is obtained by searching for a variable adjustment rule, the power density fluctuation on an energy aggregation effective surface is required to be less than 3dB, a beam roll-off coefficient approaches to 1, and a power value P point is obtained;
s2, obtaining working frequency f, transceiving interval L, antenna aperture D and antenna interval D, marking 7 transmitting antennas and 7 receiving antennas on a three-dimensional coordinate to obtain A1, A2, A3, A4, A5, A6, A7, B1, B2, B3, B4, B5, B6 and B7, wherein a7 of a central transmitting point is used as an origin,
Figure GDA0003104537520000081
is an angle relative to the bore normal;
s3, obtaining the three-dimensional coordinates through the step S1
Figure GDA0003104537520000082
And
Figure GDA0003104537520000083
the following vector cosine function is firstly obtained:
Figure GDA0003104537520000084
Figure GDA0003104537520000085
also obtained is:
Figure GDA0003104537520000086
Figure GDA0003104537520000087
s4, carrying out the step S2
Figure GDA0003104537520000088
Substituting equation 2 yields:
Figure GDA0003104537520000089
wherein the content of the first and second substances,
Figure GDA00031045375200000810
is the angle of the transmitting antenna A1 relative to the aperture normal of the receiving module B;
s5, mixing
Figure GDA00031045375200000811
Substitution normalized field strength function:
Figure GDA00031045375200000812
wherein the content of the first and second substances,
Figure GDA00031045375200000813
for normalized field intensity E of transmitting antenna A1 at point PA1p(φA1) (ii) a D is the diameter of the transmitting antenna aperture, and the unit is m; λ is the wavelength, in m;
Figure GDA00031045375200000814
is an angle relative to the bore normal; j1 is a first order bezier function;
s6, calculating the normalized field intensity of the A2, the A3, the A4, the A5, the A6, the A7, the B1, the B2, the B3, the B4, the B5, the B6 and the B7 at the P point according to the steps S2-S5 and substituting the normalized field intensity into the following formula:
Figure GDA0003104537520000091
wherein S ispIs the power density value at point P, EiMAX(φi) The value of the maximum field intensity point of the transmitting antenna on the receiving surface;
s7, adopting Matlab to carry out S of the step S6pAnd (4) carrying out simulation solution on the values, and carrying out three-dimensional graph drawing by using a drawing tool to obtain a power density distribution oscillogram of the flat beam.
The number of area sub-arrays, frequency selection, power amplification power, antenna size, array form and other specific parameter values and implementation manners adopted in the present invention are used in the illustrative embodiments, and are not intended to limit the content of the present application.
It should be understood that the above-described embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention. Other variations and modifications will be apparent to persons skilled in the art in light of the above description. And are neither required nor exhaustive of all embodiments. Any modification, equivalent replacement, and improvement made within the spirit and principle of the present invention should be included in the protection scope of the claims of the present invention.

Claims (5)

1. A microwave energy delivery system adapted for use with a moving target, comprising: the device comprises a transmitting module, a receiving module and a control module;
the transmitting module comprises an excitation source, a power amplifier and a transmitting antenna, wherein the excitation source generates a microwave signal and transmits the microwave signal to the power amplifier, and the power amplifier amplifies the microwave signal and transmits the microwave signal to the receiving module through the antenna;
the array mode of the transmitting module is multi-frequency multi-area arrangement, transmitting antennas in areas are homologous and coherent, different frequencies in the areas are incoherent, and a plurality of focusing points with different frequencies are formed on a receiving array surface through a self-focusing control algorithm to realize flat beams;
the receiving module comprises a rectifying antenna, a load and a parameter detection module, the rectifying antenna is a uniformly distributed array surface, adapts and receives the flat wave beam of the transmitting system, converts the microwave signal into a direct current signal and transmits the direct current signal to the load;
the parameter detection module is connected with the rectifying antenna and used for detecting the working state of the rectifying antenna and the parameters of the load power and sending the parameters to the control module, and the control module carries out switching and power adjustment on the transmitting module according to the received parameters;
the transmitting module is wirelessly connected with the receiving module, and the control module is respectively in signal connection with the transmitting module and the receiving module;
the transmitting antenna is a hexagonal antenna array consisting of 1 central transmitting antenna and 6 edge transmitting antennas.
2. The microwave energy delivery system for mobile objects of claim 1, wherein the rectenna is an antenna array consisting of a plurality of antennas.
3. The microwave energy delivery system for a moving object of claim 1, wherein the parameter detection module comprises a communication module, a processor, and a microwave sensor.
4. The microwave energy delivery system for a moving object of claim 1, wherein the control module comprises a communication module and a processor.
5. A simulation method of a microwave energy transmission system suitable for a moving target based on any one of claims 1 to 4, characterized by comprising the following steps:
s1, Matlab software is adopted for calculation, antenna array information of a transmitting module A and antenna array information of a receiving module B are obtained, a three-dimensional normalized power density graph is calculated and drawn, an optimized power density distribution is obtained by searching for a variable adjustment rule, the power density fluctuation on an energy aggregation effective surface is required to be less than 3dB, a beam roll-off coefficient approaches to 1, and a power value P point is obtained;
s2, obtaining working frequency f, transmitting-receiving antenna distance L, transmitting antenna aperture D and distance D between a center transmitting antenna and 6 edge transmitting antennas, marking 7 transmitting antennas and 7 receiving antennas on a three-dimensional coordinate to obtain A1, A2, A3, A4, A5, A6, A7, B1, B2, B3, B4, B5, B6 and B7, wherein a7 of the center transmitting point is used as an origin,
s3, obtaining the three-dimensional coordinates through the step S2
Figure FDA0003161920390000021
And
Figure FDA0003161920390000022
the following vector cosine function is firstly obtained:
Figure FDA0003161920390000023
Figure FDA0003161920390000024
also obtained is:
Figure FDA0003161920390000025
Figure FDA0003161920390000026
s4, carrying out the step S2
Figure FDA0003161920390000027
Substituting equation 2 yields:
Figure FDA0003161920390000028
wherein phi isA1Is the angle of the transmitting antenna A1 relative to the aperture normal of the receiving module B;
s5. will phiA1Substitution normalized field strength function:
Figure FDA0003161920390000029
wherein E (phi) is the normalized field intensity of the transmitting antenna at the point P, and the normalized field intensity E of the transmitting antenna A1 at the point P is obtainedA1P(φA1) (ii) a D is the diameter of the transmitting antenna aperture, and the unit is m; λ is the wavelength, in m; phi is the angle of the transmitting antenna relative to the caliber normal of the receiving module B; j. the design is a square1Is a first order Bessel function;
s6, calculating the normalized field intensity of the A2, the A3, the A4, the A5, the A6, the A7, the B1, the B2, the B3, the B4, the B5, the B6 and the B7 at the P point according to the steps S2-S5 and substituting the normalized field intensity into the following formula:
Figure FDA00031619203900000210
wherein the content of the first and second substances,
Figure FDA0003161920390000031
is the power density value at point P, EiMAX(φi) The value of the maximum field intensity point of the ith transmitting antenna on the receiving surface is obtained;
s7, adopting Matlab to carry out comparison on the step S6
Figure FDA0003161920390000032
And (4) carrying out simulation solution on the values, and carrying out three-dimensional graph drawing by using a drawing tool to obtain a power density distribution oscillogram of the flat beam.
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