Tesla coil distribution parameter circuit model based on half-wavelength segmentation

Abstract

The invention belongs to the field of Tesla coil distribution parameter circuit model design, and discloses a Tesla coil distribution parameter circuit model based on half-wavelength segmentation. The model construction method comprises the following steps: (1) carrying out segmented modeling on a high-voltage coil according to wavelength; (2) respectively establishing a high-voltage winding circuit model and a low-voltage winding circuit model of the Tesla coil, and combining the complete Tesla coil circuit models; (3) calculating resistance, inductance, mutual inductance and capacitance parameters in the model by using an analytical expression; and (4) constructing a T parameter two-port network based on the established circuit model, and analyzing frequency characteristics such as input impedance and voltage gain. The model established by the invention can accurately analyze the frequency characteristic and the output characteristic of the Tesla coil, is wide in application range, can be applied to all specifications of Tesla coils, is high in model precision, small in error and high in operation speed, and expands the field in which a lumped parameter circuit model cannot be applied.

Description

Technical Field

[0001] The invention belongs to the field of Tesla coil distributed parameter circuit model design, and relates to a Tesla coil distributed parameter circuit model based on half-wavelength segmentation. Background Art

[0002] Tesla Coil is a transformer that operates on the principle of resonance. It was invented by Serbian-American scientist Nikola Tesla in 1891. It is mainly used to produce ultra-high voltage but low current and high frequency AC power. Tesla Coil consists of two (sometimes three) coupled resonant circuits, and its working principle is based on electromagnetic induction and resonant conversion. The structure of the coil consists of a low-impedance main winding coil and a high-impedance secondary coil. When the main winding coil is energized, a high voltage is generated in the secondary coil through electromagnetic induction. The resonant circuit composed of appropriate capacitors and inductors makes the secondary coil and capacitor reach a resonant state, further increasing the output voltage.

[0003] At present, Tesla coils at home and abroad are mainly used in the fields of wireless power transmission and single-line power transmission. In 2008, the Nevada Lightning Laboratory in the United States studied the electric field coupling relationship between two Tesla coils. The results showed that there was obvious electric field coupling between the two Tesla coils, and 800W of power was transmitted within a distance of 5m, with a transmission efficiency of about 22.27%. The experiment also found that even if the two Tesla coils have different sizes and resonant frequencies, energy can be transmitted. In 2023, Professor Chen Xiyou's team at Dalian University of Technology proposed a single-conductor power transmission system based on multi-layer Tesla coils. When the transmission distance is 70m, the transmission power is 1150W and the efficiency is 90%. The power transmission efficiency of 5kw is 87% and the transmission distance is 5km.

[0004] Although the applications of Tesla coils like the above are increasing, the current research on Tesla coils is still mostly in the experimental stage. The few studies on Tesla coil circuit models are also insufficient, and the research in the field of Tesla coil design is almost blank, and the design is mainly based on experience. In the long run, the accurate circuit model and systematic design method of Tesla coils are the basis for the large-scale application of Tesla coils in the future, and it is of great significance to overcome them.

[0005] Professor Chen Xiyou's team at Dalian University of Technology has conducted numerous studies in the field of wireless power transmission and single-line power transmission. There are many systems based on Tesla coils, among which the established circuit models include the lumped parameter circuit model of the Tesla coil, which is equivalent to the mutual inductance model. The primary coil is regarded as a series connection of resistance and inductance, and the secondary coil is regarded as a parallel connection of resistance, inductance and capacitance. However, this type of lumped parameter circuit model is only applicable to small-sized Tesla coils in low-frequency environments. In reality, many applications of Tesla coils are in high-frequency environments, and the length of large Tesla coils can be compared to the wavelength of electromagnetic waves. At this time, the lumped parameter circuit model is no longer applicable, and the distribution of coil parameters must be considered.

[0006] Dr. JV oitkans of Latvia has published a paper on the theoretical model and experimental verification of Tesla coils, considering Tesla coils as transmission lines with distributed parameter effects. The length of large Tesla coils is similar to or greater than the wavelength, and the voltage and current in the circuit are not only functions of time, but also functions of position. In this case, the resistance, inductance, capacitance and other parameters in the circuit can no longer be regarded as concentrated at a certain point, but distributed throughout the circuit. For example, in high-frequency circuits, the resistance, inductance and capacitance of the wire will vary with the length, diameter, material and other factors of the wire. These parameters are called distributed parameters.

[0007] In summary, the distributed parameters of the Tesla coil (such as distributed resistance, distributed inductance, and distributed capacitance) cannot be ignored, and a distributed parameter circuit model needs to be used for analysis and design to ensure the accuracy of the analysis. Summary of the invention

[0008] In view of the problem that the existing lumped parameter circuit model cannot accurately simulate the circuit characteristics of the Tesla coil under high frequency conditions, the present invention designs a Tesla coil circuit model based on distributed parameters. By segmenting the Tesla coil into models and simulating the distributed parameter circuit model, accurate simulation of the Tesla coil in any environment is achieved.

[0009] The technical solution of the present invention:

[0010] A Tesla coil distributed parameter circuit model based on half-wavelength segmentation, the steps are as follows:

[0011] Step 1: Segment the high voltage coil according to wavelength

[0012] Calculate the operating wavelength of the Tesla coil Where c represents the speed of light, f represents the operating frequency, and λ represents the wavelength. The average number of segments m of the Tesla coil distributed parameter circuit model is calculated based on the operating wavelength and length of the Tesla coil. The calculation formula is as follows. m takes the smallest odd number that meets the conditions to ensure that the length of each coil segment is less than one quarter of the wavelength.

[0013]

[0014] In the formula, l w represents the total length of the high voltage coil, f and c = 3×10 8 m / s represents the operating frequency and the speed of light respectively.

[0015] Step 2: Build the Tesla Coil Circuit Model

[0016] Step 2.1: Establish the Tesla coil high-voltage winding model according to the model segment number derived in step 1, and divide the high-voltage coil wound on a cylindrical polyvinyl chloride (PVC) frame into m segments, forming a U-shaped structure. The circuit model of the high-voltage winding consists of the resistance, self-inductance, mutual inductance of each segment of the coil, and stray capacitance between adjacent segments of each segment. Since each segment of the high-voltage coil has the same physical structure and the size of each segment is less than a quarter of a wavelength, the resistance and inductance of each segment of the coil can be used to represent the electrical characteristics of each segment of the coil. The self-capacitance of each segment of the coil is extremely small and can be ignored. Then, the mutual inductance of each segment of the coil and the stray capacitance between adjacent segments are used to represent the correlation and influence of the electrical characteristics between the coils. It should be noted that when the Tesla coil is at the operating frequency, the influence of the surrounding environment on it cannot be ignored. In order to make the circuit model more accurate, the stray capacitance of the high-voltage coil to the ground must also be considered.

[0017] Step 2.2: Add the Tesla coil low-voltage winding circuit model to the Tesla coil high-voltage winding distributed parameter circuit model. Since the length of the low-voltage winding coil is much smaller than the wavelength, there is no need to consider the distributed parameter effect. The low-voltage winding lumped parameter circuit model is established by connecting the resistor and inductor in series. The low-voltage winding lumped parameter circuit model and the high-voltage winding distributed parameter circuit model together constitute a complete Tesla coil distributed parameter circuit model. w Represents the resistance of each section of the high-voltage coil after it is divided into m sections at equal distances. L S Represents the inductance of each section of the high-voltage coil after it is divided into m sections at equal distances. C s Represents the coupling capacitance between two adjacent segments. C t Represents the capacitance of the overall high-voltage coil to ground. Represents power supply, R 0 Represents the equivalent resistance of the low-voltage coil, L 0 Represents the equivalent inductance of the low-voltage coil. M 01 -M 0m M is the mutual inductance between each section of the high voltage coil and the low voltage coil. ij Represents the mutual inductance between segments of the high-voltage coil.

[0018] Step 3: Analyze the resistance, inductance, mutual inductance and capacitance in the circuit model

[0019] Step 3.1: Analyze the resistance of the m-segment high-voltage coil and low-voltage coil in the circuit model respectively:

[0020]

[0021] Where R wdc is the DC resistance of the circular conductor, F R is the AC-DC resistance ratio. d is the wire diameter, p is the coil turn spacing, l w is the total length of the coil winding, ρ w is the resistivity of copper wire, δ w is the skin depth of the winding. Skin depth calculation formula Where f is the operating frequency and μ is the magnetic permeability of vacuum.

[0022] Step 3.2: Analyze the inductance of the m-segment high-voltage coil and low-voltage coil in the circuit model respectively:

[0023]

[0024] The inductance of one layer of the coil is determined by the energy stored in the wire and the energy stored in the coil core. w is the inductance caused by the energy stored in the coil wire, L c The inductance is the energy stored in the magnetic core. w is the total length of the coil winding, N m Indicates the number of turns of the coil, S c and l c are the cross-sectional area and length of the coil respectively. N The inductance of the coil will change according to the shape of the coil. This coefficient is the correction coefficient of the coil. D c Indicates the coil diameter.

[0025] Step 3.3: Analyze the mutual inductance between coils in the circuit model:

[0026] The Tesla coil can be regarded as a system consisting of a thin-walled solenoid and a thick circular coil with a rectangular cross-section. The basic formula for its mutual inductance can be expressed as:

[0027]

[0028] Among them, N 1 、N 2 is the total number of turns of each coil, r and θ represent the radius and angle of the cylindrical coordinate system respectively. 1 and 2 represents the height of the cylindrical coordinate system; R represents the radius of the high-voltage winding, R 1 and R 2 Respectively represent the inner diameter and outer diameter of the low voltage winding, z 1 and z2 Respectively represent the height of the lowest and highest points of the low voltage winding, z 3 、z 4 Respectively represent the height of the lowest point and the highest point of the high-voltage winding; In order to quickly calculate the mutual inductance formula of the quadruple integral, the basic formula of the mutual inductance of the system is simplified;

[0029]

[0030] Simplified formula p n =ρ n -1, α n =ρ n c n sgn(p n )[1-Λ 0 (ε n ,k n )],β n =[1-Λ 0 (θ n ,k n )]+sgn(ρ n -q n )[1-Λ 0 (ζ n ,k n )],

[0031] The simplified formula can be used in any case, and is dimensionless, K(k n ) and E(k n ) are the complete elliptic integrals of the first and second kind, Λ 0 (θ n ,k n ) and Λ 0 (ζ n ,k n ) represents an incomplete elliptic integral;

[0032]

[0033] Step 3.4: Analyze the capacitance in the circuit model:

[0034] The capacitance in the Tesla coil distributed parameter circuit model is the coupling capacitance between adjacent sections of the high-voltage coil. The coupling capacitance is composed of adjacent turn-to-turn capacitance and non-adjacent turn-to-turn capacitance. t(i,j) It is equal to the capacitance associated with the insulating coating and the capacitance associated with the air gap between each turn in series. The capacitance associated with the insulating coating between turns is C c , the capacitance related to the air gap between turns is Cg .

[0035]

[0036] Among them, ε 0 represents the vacuum dielectric constant (permittivity), ε r Represents the relative dielectric constant of the insulating layer, l T represents the average length of a coil turn, d i Indicates the inner diameter of the bare conductor, d o Represents the outer diameter of the insulating layer, and dθ in the capacitance formula is a micro-angle.

[0037] After all the turn-to-turn capacitances are calculated, the capacitances between the segments are further calculated. To find the stray capacitance between adjacent coils with a large number of turns, starting from a two-turn or three-turn network with an equivalent capacitance, the number of turns is gradually increased on one side of the network until the network covers all the turns of a coil. Each additional turn brings an adjacent turn-to-turn capacitance and a turn-to-coil capacitance, which means that the equivalent capacitance of the previous network is connected in series with a turn-to-turn capacitance and then in parallel with a turn-to-coil capacitance. Therefore, for a network with a large number of turns, the total stray capacitance is obtained.

[0038]

[0039] Among them, C t(1,n+i) Indicates the inter-turn capacitance between the 1st turn and the n+ith turn in a layer of coil, C ts (n) represents the stray capacitance of the nth turn of one coil to the other coil in two adjacent coils, and the inter-segment capacitance C s (N) represents the stray capacitance of a coil with N turns to another coil.

[0040] The calculation method of coil-to-ground capacitance is similar to that of coil segment capacitance. First, calculate the capacitance of each turn of a layer of coil to ground. The formula is as follows:

[0041]

[0042] In the formula, C d (n) represents the capacitance of the nth turn of the coil to the ground, h represents the distance between the nth turn of the coil and the ground, θ=atan(1 / (0.1n+1)) is the effective angle, l T Represents the length of one turn of the coil, ε 0 represents the dielectric constant of vacuum.

[0043] and the inter-segment capacitance C s The algorithm is similar to that of the equivalent capacitance network of a single turn and the ground. The number of turns is gradually increased on one side of the network. Each additional turn brings an adjacent turn capacitance and a ground capacitance. The capacitance C of the N-turn coil to the ground is t The specific formula is as follows:

[0044]

[0045] Among them, C t(1,2) Represents the capacitance between adjacent turns, C t (N) represents the capacitance to ground of a coil with N turns.

[0046] Step 4: Construct a T-parameter two-port network

[0047] First, the electricity

[0048] First, simplify the circuit parameters, and the simplified formula is as follows:

[0049]

[0050] In the formula, R 0 Represents the low voltage coil resistance, L 0 Represents the low voltage coil inductance, Z 0 Indicates the low voltage coil impedance. R w Represents the resistance of each high-voltage coil, L s Represents the inductance of each high-voltage coil, Z 1 Indicates the impedance of each high voltage coil. M ij Represents the mutual inductance between coil segments i and j. Here, the low-voltage coil is segment 0, and the high-voltage coil is divided into segments 1 to m. Z mij Represents the mutual inductance impedance between coil segments i and j. s Represents the stray capacitance between two adjacent sections of the high-voltage coil, Z Cs Represents the inter-segment capacitance impedance. C t Represents the capacitance of the high voltage coil to ground, Z t Represents the capacitance impedance to ground; ω=2πf is the angular frequency, Z 0 Indicates the low voltage coil impedance.

[0051] According to the loop current method, write the KVL and KCL equations for the mesh and branches:

[0052]

[0053] Where U b and I b Represent the branch voltage and branch current respectively, U sb Represents the branch source voltage, Z b Represents the branch impedance matrix. Where B(b ij ) is the basic loop matrix of the equation, and the mathematical expression is as follows.

[0054]

[0055] Perform matrix operations on the network to obtain the m+2 mesh current formulas, and then calculate the admittance parameter matrix. The derivation process is as follows:

[0056]

[0057] In the formula, I n represents the current of the nth mesh, Y ij represents the admittance parameter matrix.

[0058] The T parameter two-port matrix of the Tesla coil is derived based on the admittance parameter matrix. The formula is as follows:

[0059]

[0060] According to the T parameter matrix, the input impedance, output impedance, voltage gain and other characteristics of the Tesla coil are calculated.

[0061] The beneficial effects of the present invention are as follows: an accurate Tesla coil distributed parameter circuit model based on wavelength segmentation is established, and the transmission parameter matrix of the circuit model is derived, so that the characteristics of the Tesla coil at any frequency can be analyzed, which is helpful for all subsequent experimental studies based on the Tesla coil. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 This is a simulation diagram of a Tesla coil.

[0063] Figure 2 This is a segmented schematic diagram of Tesla's high-voltage coil.

[0064] Figure 3 Schematic diagram of the segmented U-shaped circuit model of Tesla's high-voltage coil.

[0065] Figure 4 A distributed parameter circuit model of a quarter-wavelength Tesla coil was invented.

[0066] Figure 5 This is the main view of the Tesla coil mutual inductance calculation principle.

[0067] Figure 6 This is a top view of the Tesla coil mutual inductance calculation principle.

[0068] Figure 7 It is the capacitance equivalent circuit of Tesla's high voltage coil (single-layer solenoid air-core inductor).

[0069] Figure 8 The schematic diagram of the T-parameter two-port network is constructed for the circuit model according to the loop current method.

[0070] Fig. 9 Frequency characteristics of the input impedance calculated by the model for an open-circuit Tesla coil.

[0071] Fig.10 Frequency characteristics of the model's experimentally measured input impedance when the Tesla coil is open-circuited.

[0072] Fig.11 Frequency characteristics of the input impedance calculated for the model when the Tesla coil is short-circuited.

[0073] Fig.12 This is the frequency characteristic of the input impedance measured by the model experiment when the Tesla coil is short-circuited.

[0074] Fig.13 The frequency characteristics of the voltage gain calculated by the model for an open-circuit Tesla coil.

[0075] Fig.14 The frequency characteristics of the voltage gain of the model experimentally measured when the Tesla coil is open circuited. DETAILED DESCRIPTION

[0076] The following is a detailed description of the embodiments of the present invention in conjunction with the accompanying drawings and technical solutions.

[0077] Tesla coil structure Figure 1 As shown, the specifications of the Tesla coil selected by the present invention are as follows: the high-voltage coil has 1800 turns, a diameter of 110 mm, and a wire diameter of 0.21 mm; the low-voltage coil has 6 turns, a diameter of 160 mm, and a wire diameter of 1.5 mm; the operating frequency is within 625 kHz. The specific steps of the present invention are as follows:

[0078] Step 1: Segment the high voltage coil according to wavelength

[0079] Calculate the operating wavelength of the Tesla coil Where c represents the speed of light, f represents the operating frequency, and λ represents the wavelength. The average number of segments m of the Tesla coil distributed parameter circuit model is calculated based on the operating wavelength and length of the Tesla coil. The calculation formula is as follows. m takes the smallest odd number that meets the conditions to ensure that the length of each coil segment is less than one quarter of the wavelength.

[0080]

[0081] In the formula, l w Indicates the total length of the high-voltage coil, f and c = 3×10 8 m / s represents the operating frequency and the speed of light respectively.

[0082] like Figure 2 As shown, the length of the example coil is 622m, the minimum wavelength of the frequency range is 480m, and one quarter of the wavelength is 120m. In order to make the length of each section less than or approximately 120m, the high-voltage coil needs to be divided into seven sections.

[0083] Step 2: Build the Tesla Coil Circuit Model

[0084] Step 2.1: According to the model segment number derived in step 1.1, a Tesla coil high-voltage winding model is established. The high-voltage coil wound on a cylindrical polyvinyl chloride (PVC) frame is divided into m segments in a U-shaped structure, such as Figure 3 As shown. The circuit model of the high-voltage winding consists of the resistance, self-inductance, mutual inductance of each coil segment and stray capacitance between adjacent segments. Since each high-voltage coil segment has the same physical structure and the size of each segment is less than a quarter wavelength, the resistance and inductance of each coil segment can be used to represent the electrical characteristics of each coil segment. The self-capacitance of each coil segment is extremely small and can be ignored. Then, the mutual inductance of each coil segment and the stray capacitance between adjacent segments are used to represent the correlation and influence of the electrical characteristics between coils. It should be noted that when the Tesla coil is at the operating frequency, the influence of the surrounding environment on it cannot be ignored. In order to make the circuit model more accurate, the stray capacitance of the high-voltage coil to the ground must also be considered.

[0085] Step 2.2: Add the Tesla coil low-voltage winding circuit model to the Tesla coil high-voltage winding distributed parameter circuit model. Since the length of the low-voltage winding coil is much smaller than the wavelength, there is no need to consider the distributed parameter effect. The low-voltage winding lumped parameter circuit model is established by connecting the resistor and inductor in series. The low-voltage winding lumped parameter circuit model and the high-voltage winding distributed parameter circuit model together constitute a complete Tesla coil distributed parameter circuit model. Figure 4 As shown, R w Represents the resistance of each section of the high-voltage coil after it is divided into m sections at equal distances. L s Represents the inductance of each section of the high-voltage coil after it is divided into m sections at equal distances. C s Represents the coupling capacitance between two adjacent segments. C t Represents the capacitance of the overall high-voltage coil to ground. Represents power supply, R 0 Represents the equivalent resistance of the low-voltage coil, L 0 Represents the equivalent inductance of the low-voltage coil. M 01 -M 0m M is the mutual inductance between each section of the high voltage coil and the low voltage coil. ij Represents the mutual inductance between segments of the high-voltage coil.

[0086] Step 3: Analyze the resistance, inductance, mutual inductance and capacitance in the circuit model

[0087] Step 3.1: Analyze the resistance of the m-segment high-voltage coil and low-voltage coil in the circuit model respectively:

[0088]

[0089] Where R wdc is the DC resistance of the circular conductor, F Ris the AC-DC resistance ratio. d is the wire diameter, p is the coil turn spacing, l w is the total length of the coil winding, ρ w is the resistivity of copper wire, δ w is the skin depth of the winding. Skin depth calculation formula Where f is the operating frequency and μ is the vacuum magnetic permeability. According to calculations, the low-voltage coil AC resistance of the Tesla coil in the example is 0.2445Ω at 550kHz, and the AC resistance of each high-voltage coil is 112Ω.

[0090] Step 3.2: Analyze the inductance of the m-segment high-voltage coil and low-voltage coil in the circuit model respectively:

[0091]

[0092] The inductance of one layer of the coil is determined by the energy stored in the wire and the energy stored in the coil core. w is the inductance caused by the energy stored in the coil wire, L c The inductance is the energy stored in the magnetic core. w is the total length of the coil winding, N m Indicates the number of turns of the coil, S c and l c are the cross-sectional area and length of the coil respectively. N The inductance of the coil will change according to the shape of the coil. This coefficient is the correction coefficient of the coil. D c Indicates the coil diameter. According to calculation, the low-voltage coil inductance of the Tesla coil in the example is 10.45μH, and the inductance of each high-voltage coil is 11.2mH.

[0093] Step 3.3: Analyze the mutual inductance between coils in the circuit model:

[0094] The Tesla coil can be viewed as a system consisting of a thin-walled solenoid and a thick circular coil with a rectangular cross section (e.g. Figure 5 Figure 6 As shown), the basic formula of mutual inductance can be expressed as:

[0095]

[0096] Among them, N 1 、N 2 is the total number of turns of each coil, r and θ represent the radius and angle of the cylindrical coordinate system respectively. 1 and 2 represents the height of the cylindrical coordinate system; R represents the radius of the high-voltage winding, R 1 and R 2 Respectively represent the inner diameter and outer diameter of the low voltage winding, z 1 and z2 Respectively represent the height of the lowest and highest points of the low voltage winding, z 3 、z 4 Respectively represent the height of the lowest point and the highest point of the high-voltage winding; In order to quickly calculate the mutual inductance formula of the quadruple integral, the basic formula of the mutual inductance of the system is simplified;

[0097]

[0098] Simplified formula p n =ρ n -1, α n =ρ n c n sgn(p n )[1-Λ 0 (ε n ,k n )],β n =[1-Λ 0 (θ n ,k n )]+sgn(ρ n -q n )[1-Λ 0 (ζ n ,k n )],

[0099] The simplified formula can be used in any case, and is dimensionless, K(k n ) and E(k n ) are the complete elliptic integrals of the first and second kind, Λ 0 (θ n ,k n ) and Λ 0 (ζ n ,k n ) represents an incomplete elliptic integral;

[0100]

[0101] In this example, by calculating the above formula through MATLAB, the mutual inductance between the coils can be obtained as follows:

[0102] Table 1 Mutual inductance of each section of high voltage coil

[0103]

[0104] Step 3.4: Analyze the capacitance in the circuit model:

[0105] The capacitance equivalent circuit of the Tesla coil distributed parameter circuit model is as follows Figure 7 As shown, the capacitor in the circuit model is the coupling capacitor between adjacent sections of the high-voltage coil, and the coupling capacitor is composed of adjacent turn-to-turn capacitance and non-adjacent turn-to-turn capacitance.

[0106] Inter-turn capacitance C t(i,j) It is equal to the capacitance associated with the insulating coating and the capacitance associated with the air gap between each turn in series. The capacitance associated with the insulating coating between turns is C c , the capacitance related to the air gap between turns is C g .

[0107]

[0108] Among them, ε 0 represents the vacuum dielectric constant (permittivity), ε r Represents the relative dielectric constant of the insulating layer, l T represents the average length of a coil turn, d i Indicates the inner diameter of the bare conductor, d o represents the outer diameter of the insulating layer, and dθ in the capacitance formula is a micro-element angle;

[0109] After all the turn-to-turn capacitances are calculated, the capacitances between the segments are further calculated. To find the stray capacitance between adjacent coils with a large number of turns, starting from a two-turn or three-turn network with an equivalent capacitance, the number of turns is gradually increased on one side of the network until the network covers all the turns of a coil. Each additional turn brings an adjacent turn-to-turn capacitance and a turn-to-coil capacitance, which means that the equivalent capacitance of the previous network is connected in series with a turn-to-turn capacitance and then in parallel with a turn-to-coil capacitance. Therefore, for a network with a large number of turns, the total stray capacitance is obtained.

[0110]

[0111] Among them, C t(1,n+i) Indicates the inter-turn capacitance between the 1st turn and the n+ith turn in a layer of coil, C ts (n) represents the stray capacitance of the nth turn of one coil to the other coil in two adjacent coils, C s (N) represents the stray capacitance of a coil with N turns to another coil.

[0112] The calculation method of coil-to-ground capacitance is similar to that of coil segment capacitance. First, calculate the capacitance of each turn of a layer of coil to ground. The formula is as follows:

[0113]

[0114] In the formula, C d(n) represents the capacitance of the nth turn of the coil to the ground, h represents the distance between the nth turn of the coil and the ground, θ=atan(1 / (0.1n+1)) is the effective angle, l T Represents the length of one turn of the coil, ε 0 represents the dielectric constant of vacuum.

[0115] and the inter-segment capacitance C s The algorithm is similar to that of the equivalent capacitance network of a single turn and the ground. The number of turns is gradually increased on one side of the network. Each additional turn brings an adjacent turn capacitance and a ground capacitance. The capacitance C of the N-turn coil to the ground is t The specific formula is as follows:

[0116]

[0117] Among them, C t(1,2) Represents the capacitance between adjacent turns, C t (N) represents the capacitance of the N-turn coil to ground.

[0118] Calculate the coil in the example according to the above formula, and finally get C s =3.389nF, C t =3.004nF.

[0119] Step 4: Construct a T-parameter two-port network

[0120] First, simplify the circuit parameters, and the simplified formula is as follows:

[0121]

[0122] In the formula, R 0 Represents the low voltage coil resistance, L 0 Represents the low voltage coil inductance, Z 0 Indicates the low voltage coil impedance. R w Represents the resistance of each high-voltage coil, L s Represents the inductance of each high-voltage coil, Z 1 Indicates the impedance of each high voltage coil. M ij Represents the mutual inductance between coil segments i and j. Here, the low-voltage coil is segment 0, and the high-voltage coil is divided into segments 1 to m. Z mij Represents the mutual inductance impedance between coil segments i and j. s Represents the stray capacitance between two adjacent sections of the high-voltage coil, Z Cs Represents the inter-segment capacitance impedance. C t Represents the capacitance of the high voltage coil to ground, Z t Represents the capacitance impedance to ground; ω=2πf is the angular frequency, Z 0 Indicates the low voltage coil impedance.

[0123] According to the loop current method, Figure 8 Write the KVL and KCL equations for the mesh and branches:

[0124]

[0125] Where U b and I b Represent the branch voltage and branch current respectively, U sb Represents the branch source voltage, Z b Represents the branch impedance matrix. Where B(b ij ) is the basic loop matrix of the equation, and the mathematical expression is as follows.

[0126]

[0127] Perform matrix operations on the network to obtain the m+2 mesh current formulas, and then calculate the admittance parameter matrix. The derivation process is as follows:

[0128]

[0129]

[0130] In the formula, I n represents the current of the nth mesh, Y ij represents the admittance parameter matrix.

[0131] The T parameter two-port matrix of the Tesla coil is derived based on the admittance parameter matrix. The formula is as follows:

[0132]

[0133] According to the T parameter matrix, the input impedance, output impedance, voltage gain and other characteristics of the Tesla coil are calculated.

[0134] The input impedance and voltage gain characteristics of the Tesla coil in the example are calculated based on the T parameter matrix. Fig. 9 and Fig.10 As shown in the figure, the amplitude-frequency characteristic and phase angle characteristic of the system input impedance when the high-voltage coil is open circuit are shown respectively. Fig.11 and Fig.12The amplitude-frequency characteristics and phase angle characteristics of the system input impedance when the high-voltage coil is short-circuited are shown respectively. The blue curve is the actual measurement result, and the red curve is the circuit model simulation result. The frequency range is from 1kHz to 550kHz. There are 4 resonance points in the open circuit experiment and 2 resonance points in the short circuit experiment within the frequency range. The phase angle changes dramatically at the resonance point, and the input impedance first increases rapidly and then decreases rapidly. The Tesla coil circuit model is completely consistent with the actual measurement results, and the resonant frequencies of the two are completely consistent. At the frequency corresponding to the extreme points, the errors between the measurement results of the four extreme points of the open circuit experiment and the model calculation results are 0.3%, 0.5%, 0.5% and 0.4% respectively. Regardless of whether the high-voltage coil is short-circuited or open-circuited, the curve trend of the complete model is consistent with the experimental curve, indicating that the circuit model is valid. The T parameter matrix can also analyze the voltage gain frequency characteristics of the Tesla coil, such as Fig.13 and Fig.14 shown.

[0135] In summary, this model can accurately analyze the frequency characteristics of Tesla coils, and the model has a wide range of applications, fast model parameter calculation speed and accurate algorithm. It expands the fields that lumped parameter circuit models cannot be applied to.

[0136] The above shows and describes the basic principles and main features of the present invention and the advantages of the present invention. It should be understood by those skilled in the art that the present invention is not limited to the above embodiments. The above embodiments and descriptions are only for explaining the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention may have various changes and improvements, which fall within the scope of the present invention to be protected. The scope of protection of the present invention is defined by the attached claims and their equivalents.

Claims

1. A Tesla coil distributed parameter circuit model based on half-wavelength segmentation, characterized in that: Here are the steps: Step 1: Model the high-voltage coil segment by segment according to the wavelength; Step 2: Establish the Tesla coil high voltage winding circuit model and the Tesla coil low voltage winding circuit model respectively, and merge the complete Tesla coil distributed parameter circuit model; Step 3: Use analytical expressions to calculate the resistance, inductance, mutual inductance and capacitance parameters in the Tesla coil distributed parameter circuit model; Step 4: Based on the established Tesla coil distributed parameter circuit model, construct a T-parameter two-port network of the Tesla coil distributed parameter circuit model, and analyze the input impedance, output impedance, and voltage gain characteristics.

2. The Tesla coil distributed parameter circuit model based on half-wavelength segmentation according to claim 1, characterized in that: Step 1 The specific process of segmenting the high voltage coil according to the wavelength is as follows: Calculate the operating wavelength of the Tesla coil Where c represents the speed of light, f represents the operating frequency, and λ represents the wavelength. The average number of segments m of the Tesla coil distributed parameter circuit model is calculated based on the operating wavelength and length of the Tesla coil. The calculation formula is as follows. m takes the smallest odd number that meets the conditions to ensure that the length of each coil segment is less than one quarter of the wavelength. In the formula, l w represents the total length of the high voltage coil, f and c = 3×10 8 m / s represents the operating frequency and the speed of light respectively.

3. The Tesla coil distributed parameter circuit model based on half-wavelength segmentation according to claim 1, characterized in that: The specific process of establishing the Tesla coil distributed parameter circuit model in step 2 is as follows: Step 2.1: Establish a Tesla coil high-voltage winding circuit model according to the number of segments of the high-voltage coil derived in step 1, divide the high-voltage coil wound on a cylindrical polyvinyl chloride frame into m segments, and establish a Tesla coil high-voltage winding distributed parameter circuit model with a U-shaped structure; the Tesla coil high-voltage winding distributed parameter circuit model is mainly composed of the resistance, self-inductance, mutual inductance of each segment of the coil, and stray capacitance between adjacent segments of the coil. Each segment of the coil has the same physical structure, and the size of each segment of the coil is less than a quarter of a wavelength. The resistance and inductance of each segment of the coil represent the electrical characteristics of each segment of the coil. The self-capacitance of each segment of the coil is extremely small and is ignored; The mutual inductance of each coil segment and the stray capacitance between adjacent coil segments are used to represent the correlation and influence of the electrical characteristics between coils. When the Tesla coil is at the operating frequency, the influence of the surrounding environment on it cannot be ignored. In order to make the Tesla coil high-voltage winding circuit model more accurate, the stray capacitance of the high-voltage coil to the ground must also be considered. Step 2.2: Add the Tesla coil low-voltage winding circuit model to the Tesla coil high-voltage winding distributed parameter circuit model. Since the length of the low-voltage winding coil is much smaller than the wavelength, there is no need to consider the distributed parameter effect. The low-voltage winding lumped parameter circuit model is established by connecting the resistor and inductor in series. The low-voltage winding lumped parameter circuit model and the Tesla coil high-voltage winding distributed parameter circuit model together constitute a complete Tesla coil distributed parameter circuit model.

4. The Tesla coil distributed parameter circuit model based on half-wavelength segmentation according to claim 1, characterized in that: Step 3 The specific process of parsing the resistance, inductance, mutual inductance and capacitance parameters in the Tesla coil distributed parameter circuit model is as follows: Step 3.1: Analyze the resistance of the m-segment high-voltage coil and low-voltage coil in the Tesla coil distributed parameter circuit model respectively: Among them, R wdc is the DC resistance of the circular conductor, F R is the AC-DC resistance ratio, d is the wire diameter, p is the coil turn spacing, l w is the total length of the coil winding, ρ w is the resistivity of copper wire, δ w is the skin depth of the winding; the skin depth calculation formula Where f is the operating frequency, μ is the vacuum permeability; Step 3.2: Analyze the inductance of the m-segment high-voltage coil and low-voltage coil in the Tesla coil distributed parameter circuit model respectively: The inductance of one layer of the coil is determined by the energy stored in the wire and the energy stored in the coil core; where L w is the inductance caused by the energy stored in the coil wire, L c is the inductance generated by the energy stored in the magnetic core, l w is the total length of the coil winding, N m is the number of turns of the coil, S c and l c are the cross-sectional area and length of the coil, μ0=4π×10 -7 H / m is the vacuum magnetic permeability, K N The inductance of the coil will change according to the shape of the coil. The Nagaoka coefficient is the correction factor of the coil. K N =1 / [1+0.45D c / l c -0.005(D c / l c ) 2 ], D c Indicates the coil diameter; Step 3.3: Analyze the mutual inductance between coils in the Tesla coil distributed parameter circuit model The Tesla coil is considered as a system consisting of a thin-walled solenoid and a thick circular coil with a rectangular cross-section. The basic formula for mutual inductance is expressed as: Among them, N1 and N2 are the total number of turns of each coil; r and θ represent the radius and angle of the cylindrical coordinate system respectively; y1 and y2 represent the height of the cylindrical coordinate system; R represents the radius of the high-voltage winding, R1 and R2 represent the inner diameter and outer diameter of the low-voltage winding respectively; z1 and z2 represent the height of the lowest and highest points of the low-voltage winding respectively, and z3 and z4 represent the height of the lowest and highest points of the high-voltage winding respectively; In order to quickly calculate the mutual inductance formula of the quadruple integral, the basic formula of the mutual inductance of the system is simplified; Simplified formula a n =ρ n c n sgn(p n )[1-Λ0(e n ,k n )], b n =[1-Λ0(θ n ,k n )]+sgn(ρ n -q n )[1-Λ0(ζ n ,k n )], The simplified formula can be used in any case, and is dimensionless, K(k n ) and E(k n ) are the first and second complete elliptic integrals, Λ0(θ n ,k n ) and Λ0(ζ n ,k n ) represents an incomplete elliptic integral; Step 3.4: Resolve the capacitance in the Tesla coil distributed parameter circuit model The capacitance in the Tesla coil distributed parameter circuit model is the coupling capacitance C between adjacent sections of the high voltage coil. s , the coupling capacitance is composed of adjacent turn capacitance and non-adjacent turn capacitance; the turn capacitance C between the i-th turn and the j-th turn in a layer of coil t(i,j) It is equal to the capacitance associated with the insulating coating and the capacitance associated with the air gap between each turn in series; the capacitance associated with the insulating coating between turns is C c , the capacitance related to the air gap between turns is C g ; Where ε0 represents the dielectric constant of vacuum, ε r Represents the relative dielectric constant of the insulating layer, l T represents the average length of a coil turn, d i Indicates the inner diameter of the bare conductor, d o represents the outer diameter of the insulating layer, and dθ in the capacitance formula is a micro-element angle; After all the turn-to-turn capacitances are calculated, the capacitances between the segments are further calculated. To find the stray capacitance between adjacent coils with a large number of turns, starting from a two-turn or three-turn network with an equivalent capacitance, the number of turns is gradually increased on one side of the network until the network covers all the turns of a coil. Each additional turn brings an adjacent turn-to-turn capacitance and a turn-to-coil capacitance, which means that the equivalent capacitance of the previous network is connected in series with a turn-to-turn capacitance and then in parallel with a turn-to-coil capacitance. Therefore, for a network with a large number of turns, the total stray capacitance is obtained. Among them, Ct (1 ,n + i ) Indicates the inter-turn capacitance between the 1st turn and the n+ith turn in a layer of coil, C ts (n) represents the stray capacitance of the nth turn of one coil to the other coil in two adjacent coils, C s (N) represents the stray capacitance of a coil with N turns to another coil; The calculation method of the coil-to-ground capacitance is to first calculate the capacitance of each turn of a layer of coil to ground. The formula is as follows: In the formula, C d (n) represents the capacitance of the nth turn of the coil to the ground, h represents the distance between the nth turn of the coil and the ground, θ=atan(1 / (0.1n+1)) is the effective angle, l T represents the length of one turn of the coil, ε0 represents the dielectric constant of vacuum; Starting from the equivalent capacitance network of a single turn and the ground, the number of turns is gradually increased on one side of the network. Each additional turn brings an adjacent turn capacitance and a ground capacitance. The capacitance C of the N-turn coil to the ground is t The specific formula is as follows: Among them, Ct (1,2) Represents the capacitance between adjacent turns, C t (N) represents the capacitance of the N-turn coil to ground.

5. The Tesla coil distributed parameter circuit model based on half-wavelength segmentation according to claim 1, characterized in that: Step 4: Construct the T parameter two-port network of the Tesla coil distributed parameter circuit model as follows: First, simplify the circuit parameters, and the simplified formula is as follows: In the formula, R0 represents the resistance of the low-voltage coil, L0 represents the inductance of the low-voltage coil, Z0 represents the impedance of the low-voltage coil, R w Represents the resistance of each high-voltage coil, L s Represents the inductance of each high-voltage coil, Z1 represents the impedance of each high-voltage coil, M ij Represents the mutual inductance between coil segments i and j. Here, the low-voltage coil is segment 0, and the high-voltage coil is divided into segments 1 to m. Z mij represents the mutual inductance impedance between coil segments i and j, C s Represents the stray capacitance between two adjacent sections of the high-voltage coil, Z Cs Represents the impedance of the inter-segment capacitance, C t Represents the capacitance of the high voltage coil to ground, Z t represents the capacitance impedance to ground, ω=2πf is the angular frequency, and Z0 represents the impedance of the low-voltage coil; According to the loop current method, write the KVL and KCL equations for the mesh and branches: Where U b and I b Represent the branch voltage and branch current respectively, U sb Represents the branch source voltage, Z b represents the branch impedance matrix; where B(b ij ) is the basic loop matrix of the equation, and the mathematical expression is as follows: Perform matrix operations on the network to obtain the m+2 mesh current formulas, and then calculate the admittance parameter matrix. The derivation process is as follows: In the formula, I n represents the current of the nth mesh, Y ij represents the admittance parameter matrix; The T parameter two-port matrix of the Tesla coil is derived based on the admittance parameter matrix. The formula is as follows: According to the T parameter matrix, the input impedance, output impedance and voltage gain are calculated.

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