A permanent magnet eddy current separator based on Halbach array

Abstract

A permanent magnet eddy current separator based on a Halbach array belongs to the technical field of permanent magnet eddy current separators. This invention addresses the resonance problem existing Halbach array permanent magnet eddy current separators suffer from. It includes: performing modal vibration analysis on the magnetic roller of the eddy current separator to calculate the natural frequency; measuring the frequency of the electromagnetic force wave to obtain the waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave; and determining the ratio range of the main magnetic pole angle to the equivalent magnetic pole number of the magnetic roller based on the waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave; the ratio range of the main magnetic pole angle to the equivalent magnetic angle of the magnetic roller is 0.4 to 0.6. This invention is suitable for use in permanent magnet eddy current separators.

Description

Technical Field

[0001] The invention relates to a permanent magnet eddy current separator based on a Halbach array, belonging to the technical field of permanent magnet eddy current separators. Background Art

[0002] Permanent magnetic eddy current separator is a kind of equipment widely used in sorting electronic waste and metal mines. The traditional permanent magnetic eddy current separator magnet is radially magnetized. Figure 1 and Figure 2 As shown, Figure 1 The cross-sectional structure of the magnetic roller of the traditional uniform permanent magnet eddy current separator is 24 poles, and the size of each magnetic pole is uniform. Figure 2 This is the cross-sectional structure of the magnetic roller of a traditional non-uniform permanent magnet eddy current separator. The magnetic roller has 24 poles and the size of a single magnetic pole is uneven. The magnetic roller has 18 poles and the size of a single magnetic pole is uneven. The size of a single magnetic pole on the right is twice that of the left.

[0003] The magnetization characteristics and advantages of the Halbach Array are that it uses the least amount of permanent magnets under the arrangement of magnetic blocks in different magnetization directions, controls the flow direction of the magnetic circuit, and thus generates a maximum unilateral magnetic field with good sinusoidal distribution, improves the torque output of permanent magnet electromagnetic equipment, is suitable for non-torque output, and effectively reduces vibration and noise. Therefore, it has a very wide range of applications.

[0004] The permanent magnet eddy current separator, operating in cylindrical coordinates, separates objects with both tangential and radial components. Analyzing the objects, the tangential electromagnetic force causes them to be ejected tangentially, while the radial electromagnetic force causes them to detach from the conveyor belt and levitate. Due to the interaction between objects, the permanent magnet eddy current separator experiences both tangential and radial forces. The radial force does not produce work and can damage the bearings.

[0005] During the research on permanent magnet eddy current separators, it was found that the unilaterally magnetized Halbach array magnetization structure can increase the utilization rate of permanent magnet materials and improve the separation effect of the permanent magnet eddy current separator. However, during operation, this type of eddy current separator often encounters eddy current electromagnetic force waves with a frequency higher than the natural frequency of the magnetic roller structure. This will cause a serious decline in the electromagnetic and mechanical properties of the permanent magnet eddy current separator and even lead to resonance, thereby threatening its stable and safe operation and reducing the service life of the magnetic roller. Summary of the Invention

[0006] The purpose of the present invention is to solve the problem of resonance phenomenon in the existing Halbach array permanent magnet eddy current separator, and provide a permanent magnet eddy current separator based on the Halbach array.

[0007] The present invention discloses a permanent magnet eddy current separator based on a Halbach array. The magnetic roller of the eddy current separator adopts a Halbach array. Each magnetic pole includes a main magnetic pole and an auxiliary magnetic pole. The main magnetic pole angle is η, the auxiliary magnetic pole angle is γ, the equivalent magnetic pole angle of a single magnetic roller is η+γ, and the number of magnetic poles is 2π / (η+γ).

[0008] The ratio of the main magnetic pole angle η to the magnetic roller equivalent magnetic angle η+γ ranges from 0.4 to 0.6.

[0009] Preferably, the ratio of the main magnetic pole angle η to the magnetic roller equivalent magnetic angle η+γ is 0.5.

[0010] Preferably, the Halbach array of the magnetic roller includes a non-uniform Halbach array and a uniform Halbach array;

[0011] The main magnetic pole angle η and the auxiliary magnetic pole angle γ of the non-uniform Halbach array are different;

[0012] The main magnetic pole angle η and the auxiliary magnetic pole angle γ of the uniform Halbach array are the same.

[0013] Preferably, the method for determining the ratio range of the main magnetic pole angle to the number of equivalent magnetic poles of the magnetic roller includes:

[0014] S1. Perform modal vibration analysis on the magnetic roller of the eddy current separator and calculate the natural frequency;

[0015] S2. Calculate the frequency of the electromagnetic force wave to obtain the waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave;

[0016] S3. Based on the natural frequency and vibration mode analysis results obtained in S1 and the waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave obtained in S2, the ratio range of the main magnetic pole angle and the equivalent magnetic pole number of the magnetic roller is obtained.

[0017] Preferably, the specific method of performing modal vibration analysis on the magnetic roller of the eddy current separator and calculating the natural frequency in S1 includes:

[0018] S1-1. Simplify the magnetic roller yoke and the rotating shaft into the magnetic roller rotor shaft and draw a three-dimensional model of the magnetic roller;

[0019] S1-2. Based on the fact that the magnetic roller rotor shaft has infinite degrees of freedom and assuming that the mass distribution of the elastic system is continuous, a finite element modal calculation model is established;

[0020] The first six natural frequencies of the magnetic roller of S1-3 and eddy current separator are 275Hz, 275Hz, 295Hz, 780Hz, 781Hz and 1024Hz respectively;

[0021] The results of the S1-4 vibration mode analysis are as follows: with the eddy current force between the magnetic roller and the non-ferrous metal as the external excitation, the 1st, 2nd, 4th and 5th order vibration modes are all radial bending, which are in the same plane as the external excitation received by the magnetic roller; the 3rd and 6th order vibrations are axial vibrations, which are not in the same plane as the external excitation received by the magnetic roller.

[0022] Preferably, the specific method of measuring the frequency of the electromagnetic force wave in S2 to obtain the waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave includes:

[0023] The external excitation is the radial eddy current electromagnetic force wave, and the eddy current electromagnetic force wave is:

[0024] F r =uKA1 2 sin 2 [k(α-ω m t)]

[0025] +2uKA1A3 sin[k(α-ω m t)]sin[3k(α-ω m t)]

[0026] +uKA3 2 sin 2 [3k(α-ω m t)]

[0027] Among them: F r It represents the instantaneous value of the eddy current electromagnetic force wave. The magnetic induction intensity on the surface of the magnetic roller is u times the radial magnetic induction intensity. K represents a coefficient that is independent of frequency. A1 represents the amplitude of the first harmonic. A3 represents the amplitude of the third harmonic. k represents the number of magnetic poles. α represents the magnetic pole offset angle. ω m represents the angular velocity of the magnetic roller, and t represents time;

[0028] Calculation results:

[0029]

[0030] The waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave are:

[0031] The waveform amplitude of the fundamental wave is 0.5uKA1 2 -uKA1A3, waveform frequency is

[0032] The waveform amplitude of the second harmonic is uKA1A3, and the waveform frequency is

[0033] The waveform amplitude of the 3rd harmonic is 0.5uKA3 2 , the waveform frequency is

[0034] Therefore: the amplitude of the second harmonic is A3 / (0.5A1-A3) times of the fundamental wave.

[0035] The amplitude of the third harmonic is A3 of the fundamental wave 2 / (A1 2 -2A1A3) times.

[0036] Advantages of the present invention: Based on the analysis of electromagnetic force wave frequency and natural frequency, combined with modal vibration analysis, the present invention adjusts the magnetization structure of the Halbach array and proposes a permanent magnet eddy current separator based on the Halbach array, which can avoid the low-order harmonic electromagnetic force wave frequency being higher than the natural frequency, and can effectively avoid the low-frequency resonance of the magnetic roller structure, effectively improving the stability of the permanent magnet eddy current separator equipment and extending its service life.

[0037] In addition, the permanent magnet eddy current separator based on the Halbach array proposed in the present invention uses permanent magnet materials of the same size and obtains higher electromagnetic performance and sorting effect of the eddy current separator by rationally designing the magnetization and arrangement sequence of the permanent magnet materials, thereby improving the utilization rate of the permanent magnets. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 It is the cross-sectional structure of the magnetic roller of the traditional uniform permanent magnet eddy current separator;

[0039] Figure 2 It is the cross-sectional structure of the magnetic roller of the traditional non-uniform permanent magnet eddy current separator;

[0040] Figure 3 Schematic diagram of the magnetic pole structure of the permanent magnetic eddy current separator based on the Halbach array of the present invention, wherein 1 represents the main magnetic pole, 2 represents the auxiliary magnetic pole, and the main magnetic pole accounts for 0.5;

[0041] Figure 4 This is a schematic diagram of the magnetic pole structure of the permanent magnet eddy current separator when the main magnetic pole ratio is 0.4;

[0042] Figure 5 This is a schematic diagram of the magnetic pole structure of the permanent magnet eddy current separator when the main magnetic pole ratio is 0.6;

[0043] Figure 6 It is the decomposition diagram of the finite element modal calculation model;

[0044] Figure 7 These are the vibration mode diagrams corresponding to the first six natural frequencies of the magnetic roller of the eddy current separator, where (a) represents the first-order vibration mode, (b) represents the second-order vibration mode, (c) represents the third-order vibration mode, (d) represents the fourth-order vibration mode, (e) represents the fifth-order vibration mode, and (f) represents the sixth-order vibration mode. DETAILED DESCRIPTION

[0045] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making any creative efforts shall fall within the scope of protection of the present invention.

[0046] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0047] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but they are not intended to limit the present invention.

[0048] Example 1:

[0049] The following combination Figure 3 This embodiment describes a permanent magnet eddy current separator based on a Halbach array. The magnetic roller of the eddy current separator adopts a Halbach array. Each magnetic pole includes a main magnetic pole and an auxiliary magnetic pole. The main magnetic pole angle is η, the auxiliary magnetic pole angle is γ, the equivalent magnetic pole angle of a single magnetic roller is η+γ, and the number of magnetic poles is 2π / (η+γ).

[0050] The ratio of the main magnetic pole angle η to the magnetic roller equivalent magnetic angle η+γ ranges from 0.4 to 0.6.

[0051] Furthermore, the ratio of the main magnetic pole angle η to the magnetic roller equivalent magnetic angle η+γ is 0.5.

[0052] Furthermore, the Halbach array of the magnetic roller includes a non-uniform Halbach array and a uniform Halbach array;

[0053] The main magnetic pole angle η and the auxiliary magnetic pole angle γ of the non-uniform Halbach array are different;

[0054] The main magnetic pole angle η and the auxiliary magnetic pole angle γ of the uniform Halbach array are the same.

[0055] In this embodiment, the magnetic pole angle is the angle occupied by the magnetic pole on the inner circle of the stator.

[0056] Example 2:

[0057] The following combination Figure 6 and Figure 7 This embodiment describes a permanent magnet eddy current separator based on a Halbach array. The magnetic roller of the eddy current separator adopts a Halbach array. Each magnetic pole includes a main magnetic pole and an auxiliary magnetic pole. The method for determining the ratio range of the main magnetic pole angle to the equivalent magnetic pole number of the magnetic roller includes:

[0058] S1. Perform modal vibration analysis on the magnetic roller of the eddy current separator and calculate the natural frequency;

[0059] S2. Calculate the frequency of the electromagnetic force wave to obtain the waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave;

[0060] S3. Based on the natural frequency and vibration mode analysis results obtained in S1 and the waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave obtained in S2, the ratio range of the main magnetic pole angle and the equivalent magnetic pole number of the magnetic roller is obtained.

[0061] Furthermore, the specific method of performing modal vibration analysis on the magnetic roller of the eddy current separator and calculating the natural frequency described in S1 includes:

[0062] S1-1. Simplify the magnetic roller yoke and the rotating shaft into the magnetic roller rotor shaft and draw a three-dimensional model of the magnetic roller;

[0063] S1-2. Based on the fact that the magnetic roller rotor shaft has infinite degrees of freedom and assuming that the mass distribution of the elastic system is continuous, a finite element modal calculation model is established;

[0064] The first six natural frequencies of the magnetic roller of S1-3 and eddy current separator are 275Hz, 275Hz, 295Hz, 780Hz, 781Hz and 1024Hz respectively;

[0065] The results of the S1-4 vibration mode analysis are as follows: with the eddy current force between the magnetic roller and the non-ferrous metal as the external excitation, the 1st, 2nd, 4th and 5th order vibration modes are all radial bending, which are in the same plane as the external excitation received by the magnetic roller; the 3rd and 6th order vibrations are axial vibrations, which are not in the same plane as the external excitation received by the magnetic roller.

[0066] In this embodiment, the physical parameters of the rotating shaft, the magnetic yoke, and the permanent magnet poles are similar. Since the various components of the magnetic roller are closely matched, the magnetic roller yoke and the rotating shaft are simplified to the magnetic roller rotor shaft. The structural material parameters of the magnetic roller are shown in Table 1:

[0067] Table 1

[0068] part Material Density / (kg / m3) Young's modulus / (1011Pa) Poisson's ratio / Pa shaft Structural Steel 7500 2.0 0.30 yoke Q235-A 7850 2.1 0.33 magnetic pole Nd-Fe-B N52 7600 1.6 0.24

[0069] The decomposition diagram of the finite element modal calculation model is as follows: Figure 6 As shown. Solid Works software was used to draw the three-dimensional model of the magnetic roller. ANSYS Workbench software was used to establish the finite element calculation model. The vibration mode diagram corresponding to the first 6 natural frequencies of the magnetic roller of the eddy current separator is shown as follows. Figure 7 shown.

[0070] Furthermore, the specific method of measuring the frequency of the electromagnetic force wave in S2 to obtain the waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave includes:

[0071] The external excitation is the radial eddy current electromagnetic force wave, and the eddy current electromagnetic force wave is:

[0072] F r =uKA1 2 sin 2 [k(α-ω m t)]

[0073] +2uKA1A3 sin[k(α-ω m t)]sin[3k(α-ω m t)]

[0074] +uKA3 2 sin 2 [3k(α-ω m t)]

[0075] Among them: F r It represents the instantaneous value of the eddy current electromagnetic force wave. The magnetic induction intensity on the surface of the magnetic roller is u times the radial magnetic induction intensity. K represents a coefficient that is independent of frequency. A1 represents the amplitude of the first harmonic. A3 represents the amplitude of the third harmonic. k represents the number of magnetic poles. α represents the magnetic pole offset angle. ω m represents the angular velocity of the magnetic roller, and t represents time;

[0076] Calculation results:

[0077]

[0078] The waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave are:

[0079] The waveform amplitude of the fundamental wave is 0.5uKA1 2 -uKA1A3, waveform frequency is

[0080] The waveform amplitude of the second harmonic is uKA1A3, and the waveform frequency is

[0081] The waveform amplitude of the 3rd harmonic is 0.5uKA3 2 , the waveform frequency is

[0082] Therefore: the amplitude of the second harmonic is A3 / (0.5A1-A3) times of the fundamental wave.

[0083] The amplitude of the third harmonic is A3 of the fundamental wave 2 / (A1 2 -2A1A3) times.

[0084] In this embodiment, based on the calculation results of the eddy current electromagnetic force wave, the finite element calculation results of the radial component of the magnetic induction intensity of the magnetic roller of the eddy current separator are shown in Table 2:

[0085] Table 2

[0086]

[0087]

[0088] The amplitude and waveform frequency expressions of the fundamental wave and the second and third harmonics of the eddy current electromagnetic force are shown in Table 3:

[0089] Table 3

[0090]

[0091] Therefore, the amplitude of the second harmonic is A3 / (0.5A1-A3) times of the fundamental wave, and the amplitude of the third harmonic is A3 times of the fundamental wave. 2 / (A1 2 -2A1A3) times. The main component of the magnetic induction intensity outside the magnetic roller is the fundamental wave, which accounts for more than 90% of the total magnetic induction intensity amplitude. Its amplitude is A1. The amplitude of the third harmonic A3 is less than 10% of the fundamental wave amplitude, so A3 is used. 2 The eddy current electromagnetic force wave component with a coefficient of can be approximately ignored. The fifth harmonic of magnetic induction intensity is only 7% of the fundamental wave at most and can be approximately ignored. That is, THD% is the percentage of the third harmonic relative to the fundamental wave.

[0092] In this embodiment, according to the waveform amplitude and waveform frequency of each harmonic of the eddy current electromagnetic force wave, the ratio range of the main magnetic pole angle to the equivalent magnetic pole number of the magnetic roller is obtained based on the electromagnetic force frequency and electromagnetic force amplitude respectively. The specific analysis process is as follows:

[0093] According to the finite element calculation results of the radial component of the magnetic induction intensity in Table 2, the calculation results of the second and third harmonics of the radial eddy current electromagnetic force wave are obtained, as shown in Table 4:

[0094] Table 4

[0095]

[0096] As for the electromagnetic force frequency of the Halbach array permanent magnet eddy current separator, it can be seen from Table 4 that as the equivalent magnetic pole angle of the magnetic roller increases, the fundamental frequency of the eddy current electromagnetic force gradually decreases. When the Halbach array main magnetic pole to the entire magnetic pole angle ratio η / (η+γ) is 0.5, the eddy current electromagnetic force wave has no harmonics. When the number of magnetic poles is 24, the fundamental frequency f1 of the eddy current electromagnetic force generated by the magnetic roller is 320Hz, which is higher than the first-order natural frequency of the magnetic roller of 275Hz, which easily leads to resonance of the magnetic roller structure. When the number of magnetic poles is 12 and 16, the fundamental frequency f1 of the eddy current electromagnetic force generated by the magnetic roller is 160Hz and 212Hz, both lower than the natural frequencies of the magnetic roller. The second and third harmonic frequencies are both higher than the first-order natural frequency of the magnetic roller structure. When η / (η+γ) is 0.4~0.6, the amplitude of the second harmonic is small. The maximum amplitude of the second harmonic of the eddy current electromagnetic force generated by the magnetic roller in other proportions exceeds the amplitude of the fundamental component, which can easily lead to more serious resonance of the magnetic roller structure. Therefore, the ratio of the main magnetic pole to the equivalent magnetic pole angle should be avoided to exceed the range of 0.4~0.6, and the best effect of avoiding low-frequency resonance is when the magnetic pole accounts for the equivalent magnetic pole angle equal to 0.5.

[0097] For the electromagnetic force amplitude of a Halbach array permanent magnet eddy current separator, the amplitudes of the second and third harmonics of the eddy current electromagnetic force wave increase as the main magnetic pole angle ratio gradually increases or decreases. To reduce the degree of structural resonance of the magnetic roller, the optimal ratio of the main magnetic pole to the total magnetic pole ratio in a Halbach array non-ferrous metal eddy current separator is 0.4 to 0.6 when the magnetic roller is designed with non-uniform and uniform magnetic poles, which can effectively prevent excessive structural resonance of the magnetic roller.

[0098] Based on the above analysis of the electromagnetic force frequency and amplitude of the Halbach array permanent magnet eddy current separator, the main magnetic pole ratio is 0.4 to 0.6. The schematic diagram of the magnetic pole structure when the main magnetic pole ratio is 0.4 is as follows: Figure 4 As shown, the magnetic pole structure diagram when the main magnetic pole ratio is 0.6 is as follows Figure 5 As shown, the closer the main magnetic pole accounts for the equivalent magnetic pole angle to 0.5, the better the effect. The magnetic pole structure diagram when the main magnetic pole accounts for 0.5 is shown in Figure 3 shown.

[0099] Although the present invention is described herein with reference to specific embodiments, it should be understood that these embodiments are merely illustrative of the principles and applications of the invention. It should be understood that many modifications may be made to the illustrative embodiments, and that other arrangements may be devised, without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that the various dependent claims and features described herein may be combined in ways other than those described in the original claims. It should also be understood that features described in conjunction with individual embodiments may be employed in conjunction with other described embodiments.

Claims

1. A permanent magnet eddy current separator based on Halbach array, characterized in that: The magnetic roller of the eddy current separator adopts the Halbach array. Each magnetic pole includes a main magnetic pole and an auxiliary magnetic pole. The main magnetic pole angle is η, the auxiliary magnetic pole angle is γ, the equivalent magnetic pole angle of a single magnetic roller is η+γ, and the number of magnetic poles is 2π / (η+γ). The ratio of the main magnetic pole angle η to the equivalent magnetic angle η+γ of the magnetic roller is in the range of 0.4 to 0.6; The method for determining the ratio range of the main magnetic pole angle to the number of equivalent magnetic poles of the magnetic roller includes: S1. Perform modal vibration analysis on the magnetic roller of the eddy current separator and calculate the natural frequency; S2. Calculate the frequency of the electromagnetic force wave to obtain the waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave; S3. Based on the natural frequency and vibration mode analysis results obtained in S1 and the waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave obtained in S2, the ratio range of the main magnetic pole angle and the equivalent magnetic pole number of the magnetic roller is obtained.

2. A permanent magnet eddy current separator based on a Halbach array according to claim 1, characterized in that: The specific method for performing modal vibration analysis on the magnetic roller of the eddy current separator and calculating the natural frequency as described in S1 includes: S1-1. Simplify the magnetic roller yoke and the rotating shaft into the magnetic roller rotor shaft and draw a three-dimensional model of the magnetic roller; S1-2. Based on the fact that the magnetic roller rotor shaft has infinite degrees of freedom and assuming that the mass distribution of the elastic system is continuous, a finite element modal calculation model is established; The first six natural frequencies of the magnetic roller of S1-3 and eddy current separator are 275Hz, 275Hz, 295Hz, 780Hz, 781Hz and 1024Hz respectively; S1-4. The results of the vibration mode analysis are as follows: taking the eddy current force between the magnetic roller and the non-ferrous metal as the external excitation, the 1st, 2nd, 4th and 5th order vibration modes are all radial bending, which are in the same plane as the external excitation received by the magnetic roller; the 3rd and 6th order vibrations are axial vibrations, which are not in the same plane as the external excitation received by the magnetic roller.

3. The permanent magnet eddy current separator based on the Halbach array according to claim 2, characterized in that: The specific method for measuring the frequency of the electromagnetic force wave and obtaining the waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave as described in S2 includes: The external excitation is the radial eddy current electromagnetic force wave, and the eddy current electromagnetic force wave is: F r =uKA1 2 sin 2 [k(a-ω m (t)] +2uKA1A3sin[k(α-ω m t)]sin[3k(α-ω) m (t)] +uKA3 2 sin 2 [3k(a-ω m (t)] Among them: F r It represents the instantaneous value of the eddy current electromagnetic force wave. The magnetic induction intensity on the surface of the magnetic roller is u times the radial magnetic induction intensity. K represents a coefficient that is independent of frequency. A1 represents the amplitude of the first harmonic. A3 represents the amplitude of the third harmonic. k represents the number of magnetic poles. α represents the magnetic pole offset angle. ω m represents the angular velocity of the magnetic roller, and t represents time; Calculation results: The waveform amplitude and waveform frequency of each harmonic of the electromagnetic force wave are: The waveform amplitude of the fundamental wave is 0.5uKA1 2 -uKA1A3, waveform frequency is The waveform amplitude of the second harmonic is uKA1A3, and the waveform frequency is The waveform amplitude of the 3rd harmonic is 0.5uKA3 2 , the waveform frequency is Therefore: the amplitude of the second harmonic is A3 / (0.5A1-A3) times of the fundamental wave. The amplitude of the third harmonic is A3 of the fundamental wave 2 / (A1 2 -2A1A3) times.

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