CN107273563B - Equivalent calculation method for parasitic capacitance of PCB winding of planar transformer - Google Patents

Abstract

The invention discloses an equivalent calculation method for the parasitic capacitance of the PCB winding of a planar transformer. By modeling and analyzing the PCB winding of the planar transformer, the distribution of the parasitic capacitance of the PCB winding is simulated. The parasitic capacitance stores the energy and superimposes them. Using the relationship between energy and parasitic capacitance and the law of energy conservation, the parasitic capacitance of the PCB winding as a whole is calculated, and then it is equivalent to the calculation of the capacitance of the three parts, namely Capacitance C ₁ between the same-named end of the primary winding and the same-named end of the secondary winding, capacitance C2 between the same _- named end of the primary winding and the same-named end of the secondary winding, and the difference between the same-named end of the primary winding and the same-named end of the secondary winding capacitor _C3 between.

Description

An Equivalent Calculation Method for Parasitic Capacitance of PCB Windings of Planar Transformers

technical field

The invention relates to a PCB-type planar transformer, in particular to an equivalent calculation method for the parasitic capacitance of the PCB winding of the planar transformer, which can be applied to the analysis of the parasitic parameters of the planar transformer and the establishment of an EMI conduction model in the field of electromagnetic compatibility.

Background technique

Planar transformers have the advantages of low height, small size, good consistency, easy mass production, small leakage inductance, good heat dissipation, and good electromagnetic compatibility. They are new types of transformers that are widely used. There are many kinds of planar transformers. Among them, PCB planar transformers are the most widely used due to their advantages of low cost, stable structure, and convenient production. Its structure is relatively simple, and the via holes of each layer of windings can be welded. The parasitic capacitance of the planar transformer is an unavoidable practical problem in the design of switching power supplies, which often makes the working state of the switching power supply deviate from the design results, and even seriously affects the technical indicators of the power supply. Therefore, the influence of these parasitic capacitances must be considered in the design process of the switching power supply. Therefore, it is very important to use which method to calculate the parasitic capacitance of the PCB winding of the planar transformer.

Planar transformers can be divided into four types: printed circuit (PCB) type, thick-film type, thin-film type, and sub-micron type according to different design and manufacturing processes. Among them, the PCB-type planar transformer is mainly composed of three parts: a planar magnetic core, a PCB winding and an insulating layer. Planar transformers require the magnetic core and winding to be of a planar structure, so multi-layer PCB windings should be used, generally eight to ten layers, and more than twenty layers. In the PCB-type planar transformer, the PCB windings are round flat conductive wires made on the printed circuit board or directly made of copper. exist. It can be considered that the parasitic capacitance of the PCB-type planar transformer mainly exists between the PCB windings. The parasitic capacitance will make the working state of the switching power supply deviate from the design results, and even seriously affect the technical indicators of the power supply. The effects of parasitic capacitance cannot be eliminated, but we can estimate to account for these parasitic capacitance effects in the design of switching power supplies.

The planar transformer PCB winding is divided into two parts, the primary winding (P) and the secondary winding (S). The primary winding is connected in series to increase the number of turns; in order to increase the current carrying capacity of the winding, the secondary winding generally adopts a parallel structure. The primary windings of planar transformers are generally single-turn or multi-turn, and are in series; the secondary windings are single-turn and parallel. There are many ways of winding structure of PCB type planar transformer, which can be divided into simple structure, sandwich structure and staggered structure. Although the three types of windings have different structures, parasitic capacitance will inevitably occur. The parasitic capacitance of the simple structure is small; the parasitic capacitance of the staggered structure is large; the parasitic capacitance of the sandwich structure is between the two. When calculating the size of the parasitic capacitance, the capacitance of the non-adjacent two-layer windings is much smaller than the capacitance between the adjacent two-layer windings and can be ignored; although the single-layer winding itself also has capacitance, the windings of each layer of the general planar transformer are very small. Thin, can be ignored; the secondary winding adopts a parallel structure, when the two adjacent layers of windings are both secondary windings, there is no potential difference, and the parasitic capacitance can be considered to be zero.

There is capacitance between any metal parts. The existing method for calculating the parasitic capacitance of PCB windings of planar transformers is to treat each layer of windings as an equipotential surface, that is, there is no voltage drop on the windings, and use the static capacitance formula C=Q/U to calculate. Calculate the size of the capacitance between the windings. However, in the transformer, the potential between turns of the same winding, between different windings, between the windings to the shielding layer, and between the lengths of a line varies, so the capacitance formed in this way is different from the electrostatic capacitance, which is called distributed capacitance. Therefore, the distributed capacitance cannot be directly calculated by the method of calculating the electrostatic capacitance. The present invention proposes a more accurate and highly adaptable method for calculating the parasitic capacitance of the PCB winding of a planar transformer to solve the existing problems.

SUMMARY OF THE INVENTION

Aiming at the inapplicability of the existing plane transformer parasitic capacitance calculation method, the invention provides an equivalent calculation method for the parasitic capacitance of the PCB winding of the plane transformer, which is fast, accurate, adaptable and has high practical application value.

The object of the present invention is achieved by the following technical solutions: an equivalent calculation method for the parasitic capacitance of the PCB winding of a planar transformer, the planar transformer includes a planar magnetic core, a PCB winding and an insulating layer, and the PCB winding is a multi-layer structure, and each A winding is arranged on the layer, the primary winding and the secondary winding are arranged on different layers, the primary winding adopts one of single-turn or multi-turn, the secondary winding adopts a single-turn circular ring, and the multi-turn adopts a spiral shape. The primary windings are connected in series, and the secondary windings on different layers are connected in parallel. When two adjacent layers are both primary windings and two adjacent layers are primary windings and secondary windings, respectively, At this time, there is a non-negligible parasitic capacitance between two adjacent layers;

It is characterized in that: through the modeling and analysis of the PCB winding of the planar transformer, the distribution of the parasitic capacitance of the PCB winding is simulated, and the energy stored in the parasitic capacitance between the two adjacent layers where the parasitic capacitance exists is calculated and superimposed, and the energy is utilized. The relationship between the parasitic capacitance and the energy conservation law, the parasitic capacitance of the PCB winding as a whole is calculated, and then it is equivalent to the capacitance of three parts, that is, the capacitance between the same-named end of the primary winding and the same-named end of the secondary winding _C1 . Capacitor C2 between the same _- named end of the primary winding and the synonymous end of the secondary winding, and capacitance C3 between the synonymous end of the primary winding and the same _- named end of the secondary winding; including the following steps:

(1) Calculation of single-layer winding potential

1) Single-layer single-turn primary side winding potential representation

Assuming that the primary winding current flows in a clockwise direction, and the winding voltage drops uniformly along the current direction, a polar coordinate system is established with the ray from the pole center to the current inflow point as the pole diameter, and a segment of the arc on the winding is taken as the micro-element dθ, The angle parameter of the polar coordinate of the position of the micro-element dθ is θ, and the voltage of the micro-element is expressed as:

V _1P is the current inflow point voltage of single-turn primary side winding, V _2P is the current outflow point voltage of single-turn primary side winding;

2) Representation of single-layer single-turn secondary winding potential

The primary side current and the secondary side current entry position are mirror images of each other and the current flows in opposite directions. The micro-element dθ voltage on the secondary side winding is expressed as:

V _1S is the current inflow point voltage of the single-turn secondary side winding, V _2S is the current outflow point voltage of the single-turn secondary side winding;

3) Single-layer multi-turn primary side winding potential representation

When the primary side winding is multi-turn, different from the primary side single-turn winding, the coil itself is a helix, and there will also be parasitic capacitance between each turn of the helix, but the thickness of each layer of the planar transformer is small enough to Ignoring the parasitic capacitance between the spirals, the Archimedes spiral equation is used to fit the winding shape. The Archimedes spiral equation is expressed as:

r*=h·θ _x (3)

r* represents the polar diameter, h is a constant representing the helical ratio,

The polar coordinate system is established with the ray from the pole center to the current inflow point as the pole diameter, and a length dL on the multi-turn coil is taken as the micro-element, and θ _x represents the angle parameter of the micro-element dL along the spiral line to the pole center; the micro-element The element voltage is expressed as:

A represents the straight-line distance from the current inflow point to the pole center, θ ₁ represents the angle from the current inflow point along the spiral line to the pole center, and the magnitude is:

L represents the length from the current inflow point to the current outflow point, and its size is:

(r*)' represents the first derivative with respect to r*, B represents the straight-line distance from the current outflow point to the pole center, θ ₂ represents the angle from the current outflow point along the spiral line to the pole center, the size is:

(2) Calculation of capacitance energy of adjacent two-layer windings

1) Calculation of capacitance energy between two adjacent layers of single-turn primary winding and single-turn primary winding

The two adjacent layers of single-turn windings are both primary windings. At this time, the current direction of the two layers is the same, and the capacitance element is expressed as:

R is the outer diameter of the winding, r is the inner diameter of the winding, ε is the dielectric constant between the plates, and d is the vertical distance between the two layers of windings;

The relationship between capacitance C and energy W is expressed as:

U ₁ , U ₂ respectively represent the electric potential of the upper and lower plates of the capacitor;

From equations (1), (8) and (9), the capacitance energy of two adjacent layers of single-turn primary winding and single-turn primary winding is expressed as

2) Capacitance energy calculation of two adjacent layers of single-turn primary winding and single-turn secondary winding

Two adjacent layers of windings: One layer is a single-turn primary winding, and the other layer is a single-turn secondary winding. At this time, the current directions of the two are opposite. The single-turn primary winding and single-turn secondary winding capacitance energy is expressed as:

3) Calculation of capacitance energy between two adjacent layers of multi-turn primary winding and multi-turn primary winding

The two adjacent layers of multi-turn windings are all primary windings. At this time, the two current directions are the same, and the capacitance element is expressed as:

w represents the width of each turn of the multi-turn coil winding,

From equations (1), (9) and (12), the capacitance energy of two adjacent layers of multi-turn primary windings and multi-turn primary windings is expressed as:

Among them, n represents the total number of turns of the single-layer multi-turn coil, the number of turns is calculated from the current inflow point, one turn is counted for every 2π revolution, and less than one turn is counted as one turn, and i represents the i-th turn;

4) Calculation of capacitance energy of adjacent two-layer multi-turn primary winding and single-turn secondary winding

Two adjacent layers of windings: One layer is a multi-turn primary winding, and the other is a single-turn secondary winding. At this time, the current directions of the two are opposite. The capacitance energy of multi-turn primary winding and single-turn secondary winding is expressed as:

(3) Calculation of equivalent parasitic capacitance

Let C _k,k+1 represent the parasitic capacitance between the two adjacent windings of the kth layer and the k+1th layer, and W _k,k+1 represent the capacitance energy of C _k,k+1 . According to the adjacent PCB-type planar transformer For the winding condition, use equations (10), (11), (13), (14) to calculate the sum of capacitance energy W _ps between all adjacent two-layer windings:

W _ps =W ₁₂ +W ₂₃ ...+W _k,k+1 +...(15)

的系数和、F表示项V_p·V_s的系数和、G表示项V_s ²的系数和，将(15)整理为：The total voltage drops on the primary and secondary sides are respectively V _p and V _s . The total voltage drop on each layer on the primary side is the same, that is, the input voltage and output voltage of the primary winding of each layer can be represented by V _p respectively, while the secondary side adopts a parallel connection. structure, each pressure drop is _Vs , term E

, F represents the coefficient sum of the term V _p ·V _s , G represents the coefficient sum of the term V _s ² , and (15) is organized as:

The total energy contained in the parasitic capacitance of the PCB winding is equal to the sum of the energy contained in the three equivalent capacitances C ₁ , C ₂ , and C ₃ , and the energy stored in the three equivalent capacitances C ₁ , C ₂ , and C ₃ calculated,

Since W _ps =W ₁ +W ₂ +W ₃ , comparing formulas (16) and (17), we get:

The capacitance values of the three equivalent capacitances C ₁ , C ₂ , and C ₃ are calculated from (18).

Compared with the existing method, the present invention has the following advantages and remarkable effects:

1. Fully consider the voltage changes at different positions of the single-layer winding instead of treating the single-layer winding as an equipotential body, and the calculation results are more accurate.

2. Using the relationship between energy and capacitance to solve the parasitic capacitance of the PCB winding of the planar transformer, the overall parasitic capacitance is equivalent to the capacitance between the primary winding and the secondary winding, which has higher practical value.

3. The present invention has wide adaptability to the calculation of the parasitic capacitance of the existing PCB-type planar transformer, and provides an idea for solving the parasitic parameter analysis of the planar transformer and the establishment of an EMI conduction model in the field of electromagnetic compatibility. The calculation is fast, accurate, adaptable, and has high engineering value.

Description of drawings

Figure 1 is a schematic diagram of equivalent capacitance;

Figures 2a, 2b, and 2c are schematic diagrams of three structures of planar transformer PCB windings;

Figure 3 is a schematic diagram of a single-turn primary side winding voltage gradient;

Figure 4 is a schematic diagram of a single-turn secondary winding voltage gradient;

FIG. 5 is a schematic diagram of a multi-turn primary side winding voltage gradient;

Figure 6 is a schematic diagram of calculating the capacitance energy of a single-turn primary side winding and a single-turn primary side winding;

Fig. 7 is a schematic diagram of calculating the capacitance energy of a single-turn primary side winding and a single-turn secondary side winding;

FIG. 8 is a schematic diagram of calculating the capacitance energy of a multi-turn primary side winding and a multi-turn primary side winding;

FIG. 9 is a schematic diagram of the calculation of the capacitance energy of the multi-turn primary side winding and the single-turn secondary side winding.

Detailed ways

The size of the parasitic capacitance in the planar transformer is equal to the superposition of countless capacitances, and it is impossible to express them one by one. Capacitance is essentially due to an electric field between two conductors with different potential differences, and the presence of an electric field means energy. Therefore, capacitance actually characterizes the ability to store energy. In other words, the equivalent bulk capacitance is actually equivalent to the overall energy storage capacity of a transformer. Therefore, the size of the equivalent capacitance can be calculated by calculating the energy. In the design and analysis process of the switching power supply, an equivalent model is established to represent the parasitic capacitance between the PCB windings of the planar transformer. As shown in FIG. 1 , the present invention equivalently divides the parasitic capacitance between the PCB windings of the planar transformer into three parts: (1) the capacitance C ₁ between the same-named end of the primary winding and the same-named end of the secondary winding; (2) the primary winding Capacitance C ₂ between the synonymous end of the secondary winding and the synonymous end of the secondary winding; ( ₃ ) Capacitance C3 between the synonymous end of the primary winding and the synonymous end of the secondary winding.

The size of the capacitance is closely related to the way the windings of the planar transformer are distributed on the PCB. As shown in Fig. 2, the winding structure of the PCB-type planar transformer can be divided into a simple structure (Fig. 2a), a sandwich structure (Fig. 2b), and a staggered structure (Fig. 2c). Taking the eight-layer PCB winding as an example, the solid part represents the primary winding (P), which is formed by connecting four layers of windings in series. The windings are either single-turn or multi-turn, assuming that the current direction is clockwise; the hollow part Indicates the secondary winding (S), which is composed of four layers of windings in parallel. The windings are all single-turn structures, and the current direction is opposite to the primary side, which is counterclockwise. The calculation of parasitic capacitance is divided into two cases: the first case is that two adjacent layers of windings are located on the same side, both of which are primary side windings, and the current directions of the two are the same; the second case is that the two adjacent layers of windings are respectively On both sides, one layer is the primary winding, the other is the secondary winding, and the current directions of the two are opposite. Now calculate the energy contained in the respective parasitic capacitances of the two cases when the primary side windings are single-turn and multi-turn respectively.

(1) Calculation of single-layer winding potential

1. Single-layer single-turn primary side winding potential representation

As shown in Figure 3, the single-turn primary side winding is assumed to be V _1P at the current inflow point and V _2P at the current outflow point. It is not difficult to find that the voltage on the winding decreases with the distance from the point where the current flows, and the current flows in a clockwise direction. Using the polar coordinate system shown in Figure 3, it is assumed that the inner diameter of the winding is r and the outer diameter is R. Take an arc on the ring as the micro-element dθ, and the angle parameter of its position polar coordinate is θ. Assuming that the voltage drop from the inflow point to the outflow point is evenly distributed on the coil, the voltage on this micro-element arc is :

2. Single-layer single-turn secondary winding potential representation

As shown in Figure 4, for the single-turn secondary side winding, since the inflow positions of the primary side current and the secondary side current are mirror images of each other and the current flow directions are opposite, the voltage on a certain segment of the micro-element arc on the secondary side winding is:

3. Single-layer multi-turn primary side winding potential representation

When the primary side winding is multi-turn, different from the primary side single-turn winding, the coil itself is a helix, and there will also be parasitic capacitance between each turn of the helix, but the thickness of each layer of the planar transformer is small enough to Ignoring the parasitic capacitance between the spirals, the Archimedes spiral equation is used to fit the winding shape, and the equation is expressed as:

r*=h·θ _x (3)

Among them, r* represents the polar diameter, and h is a constant representing the helical ratio.

As shown in FIG. 5 , it is assumed that the current inflow point voltage is V _1PM , and the current outflow point voltage is V _2PM . It is not difficult to find that the voltage on the helix decreases linearly with the distance from the point where the current flows. Using the polar coordinate system shown in the figure, take a length of a multi-turn coil as a micro-element, its length is dL, and the angle parameter of the position polar coordinate is θx, indicating that the micro-element bypasses along the spiral line to the polar center o Angle; _θ1 represents the angle from the current inflow point along the spiral to the pole center o _; θ2 represents the coil from the current outflow point along the spiral to the pole center o around the angle. Therefore, the voltage on the length of the micro-element is:

A represents the straight-line distance from the current inflow point to the pole center, θ ₁ represents the angle from the current inflow point along the spiral line to the pole center, and the magnitude is:

L represents the length from the current inflow point to the current outflow point, and its size is:

(r*)' represents the first derivative with respect to r*, B represents the straight-line distance from the current outflow point to the pole center, θ ₂ represents the angle from the current outflow point along the spiral line to the pole center, the size is:

(2) Calculation of capacitance energy of adjacent two-layer windings

1. Calculation of capacitance energy between two adjacent layers of single-turn primary winding and single-turn primary winding

At this time, the two adjacent layers of single-turn windings are in the first situation, that is, the two adjacent layers of windings are both primary windings, and the current directions of the two are the same. As shown in Figure 6, there are two adjacent layers of PCB windings up and down, assuming that the upper layer winding is recorded as winding 1, and the lower layer winding is recorded as winding 2. The a and c terminals are the current inflow terminals respectively, and the b and d terminals are the current outflow terminals respectively. In winding 1, the voltage of terminal a is V _1P , the voltage of terminal b is V _2P , V _1P > V _2P , and the current flows clockwise; in winding 2, the voltage of terminal c is V _3P , the voltage of terminal d is V _4P , V _3P > V _4P , the current also flows clockwise. R is the outer diameter of the ring, r is the inner diameter of the ring, ε is the dielectric constant between the plates, and d is the straight-line distance between the two layers of windings. The capacitance element can be expressed as,

The relationship between capacitance C and energy W is expressed as,

U ₁ and U ₂ respectively represent the potential of the upper and lower plates of the capacitor.

By formulas (1), (8), (9), the capacitance energy of two adjacent layers of single-turn primary winding and single-turn primary winding is expressed as,

2. Calculation of capacitance energy between two adjacent layers of single-turn primary winding and single-turn secondary winding

At this time, the two adjacent layers of single-turn windings are in the second situation, that is, one layer of the adjacent two layers of windings is the primary winding, and the other layer is the secondary winding. At this time, the current directions of the two are opposite, as shown in Figure 7 . V _1P and V _1S represent the current inflow point voltages of the primary and secondary windings of a single turn, respectively, and V _2P and V _2S represent the current outflow point voltages of the primary and secondary windings of a single turn, respectively. By formulas (2), (8), (9), the capacitance energy of two adjacent layers of single-turn primary winding and single-turn secondary winding is expressed as:

3. Calculation of capacitance energy between adjacent two-layer multi-turn primary windings and multi-turn primary windings

The capacitance element can be expressed as:

w represents the width of each turn of the multi-turn coil winding. If the adjacent two-layer multi-turn windings are in the first case, as shown in Figure 8, it is only necessary to integrate the length. At this time, by formulas (1) and (9) ), (12), the capacitance energy of two adjacent layers of multi-turn primary winding and multi-turn primary winding is expressed as:

Among them, n represents the total number of turns of the single-layer multi-turn coil (the number of turns is calculated from the inflow end of the voltage, one turn is counted for every 2π revolution, and the less than one turn is counted as one turn), and i is the i-th turn.

4. Capacitance energy calculation of adjacent two-layer multi-turn primary winding and single-turn secondary winding

If two adjacent layers of windings are in the second case, and the primary winding is multi-turn, the secondary winding is single-turn, as shown in Figure 9. At this time, according to equations (2), (9) and (12), the energy of two adjacent layers of windings is expressed as:

(3) Calculation of equivalent parasitic capacitance

Take the eight-layer PCB winding as an example, C ₁₂ , C ₂₃ . . . C ₇₈ represents the parasitic capacitance between two adjacent layers of windings, W ₁₂ , W ₂₃ . . . W ₇₈ represent C ₁₂ , C ₂₃ , respectively. . . The energy of C ₇₈ , according to the situation of the adjacent windings of the PCB-type planar transformer, use the formulas (10), (11), (13), (14) to calculate the sum of the capacitance energy between all the adjacent two-layer windings:

W _ps = W ₁₂ +W ₂₃ +W ₃₄ +W ₄₅ +W ₅₆ +W ₆₇ +W ₇₈ (15)

的系数和、F表示项V_p·V_s的系数和、G表示项V_s ²的系数和，将(15)整理为：The total voltage drops on the primary and secondary sides are respectively V _p and V _s . The total voltage drop on each layer on the primary side is the same, that is, the input voltage and output voltage of the primary winding of each layer can be represented by V _p respectively, while the secondary side adopts a parallel connection. structure, each pressure drop is _Vs , term E

, F represents the coefficient sum of the term V _p ·V _s , G represents the coefficient sum of the term V _s ² , and (15) is organized as:

The total energy contained in the parasitic capacitance of the PCB winding is equal to the sum of the energy contained in the three equivalent capacitances C ₁ , C ₂ , and C ₃ , and the energy stored in the three equivalent capacitances C ₁ , C ₂ , and C ₃ calculated,

Since W _ps =W ₁ +W ₂ +W ₃ , comparing formulas (16) and (17), we get:

The capacitance values of the three equivalent capacitances C ₁ , C ₂ , and C ₃ are calculated from (18). From this, the specific value of the parasitic capacitance in the equivalent model of the planar transformer is obtained.

The calculation result of the invention is accurate, can be used to calculate the parasitic capacitance of the PCB winding of the planar transformer, and can be applied to the parasitic parameter analysis of the planar transformer and the establishment of the EMI conduction model in the field of electromagnetic compatibility.

The practical application of the method for calculating the parasitic capacitance of the PCB winding of the planar transformer of the present invention shows that the purpose of the present invention is achieved and the stated effect is achieved.

The specific embodiment of the present invention has made a detailed description to the content of the present invention, but is not limited to this embodiment, and any obvious changes made by those skilled in the art according to the inspiration of the present invention belong to the scope of protection of the present invention.

Claims (1)

1. An equivalent calculation method for the parasitic capacitance of a PCB winding of a planar transformer, the planar transformer includes a planar magnetic core, a PCB winding and an insulating layer, the PCB winding is a multi-layer structure, and each layer is provided with a winding, the primary winding and the secondary winding; The side windings are arranged on different layers, the primary winding is one of single-turn or multi-turn, the secondary winding is a single-turn circular ring, and the multi-turn is spiral. The primary windings on different layers are connected in series, different The secondary windings on the layers are connected in parallel. When the two adjacent layers are both primary windings and the adjacent two layers are primary windings and secondary windings respectively, there is an inability between the two adjacent layers at this time. Negligible parasitic capacitance; It is characterized in that: through the modeling and analysis of the PCB winding of the planar transformer, the distribution of the parasitic capacitance of the PCB winding is simulated, and the energy stored in the parasitic capacitance between the two adjacent layers where the parasitic capacitance exists is calculated and superimposed, and the energy is utilized. The relationship between the parasitic capacitance and the energy conservation law, the parasitic capacitance of the PCB winding as a whole is calculated, and then it is equivalent to the capacitance of three parts, that is, the capacitance between the same-named end of the primary winding and the same-named end of the secondary winding _C1 . Capacitor C2 between the same _- named end of the primary winding and the synonymous end of the secondary winding, and capacitance C3 between the synonymous end of the primary winding and the same _- named end of the secondary winding; including the following steps: (1) Calculation of single-layer winding potential 1) Single-layer single-turn primary side winding potential representation Assuming that the primary winding current flows in a clockwise direction, and the winding voltage drops uniformly along the current direction, a polar coordinate system is established with the ray from the pole center to the current inflow point as the pole diameter, and a segment of the arc on the winding is taken as the micro-element dθ, The angle parameter of the polar coordinate of the position of the micro-element dθ is θ, and the voltage of the micro-element is expressed as:

V _1P is the current inflow point voltage of single-turn primary side winding, V _2P is the current outflow point voltage of single-turn primary side winding; 2) Representation of single-layer single-turn secondary winding potential The primary side current and the secondary side current entry position are mirror images of each other and the current flows in opposite directions. The micro-element dθ voltage on the secondary side winding is expressed as:

V _1S is the current inflow point voltage of the single-turn secondary side winding, V _2S is the current outflow point voltage of the single-turn secondary side winding; 3) Single-layer multi-turn primary side winding potential representation When the primary side winding is multi-turn, different from the primary side single-turn winding, the coil itself is a helix, and there will also be parasitic capacitance between each turn of the helix, but the thickness of each layer of the planar transformer is small enough to Ignoring the parasitic capacitance between the spirals, the Archimedes spiral equation is used to fit the winding shape. The Archimedes spiral equation is expressed as: r*=h·θ _x (3) r* represents the polar diameter, h is a constant representing the helical ratio, The polar coordinate system is established with the ray from the pole center to the current inflow point as the pole diameter, and a length dL on the multi-turn coil is taken as the micro-element, and θ _x represents the angle parameter of the micro-element dL along the spiral line to the pole center; the micro-element The element voltage is expressed as:

A represents the straight-line distance from the current inflow point to the pole center, θ ₁ represents the angle from the current inflow point along the spiral line to the pole center, and the magnitude is:

L represents the length from the current inflow point to the current outflow point, and its size is:

(r*)' represents the first derivative with respect to r*, B represents the straight-line distance from the current outflow point to the pole center, θ ₂ represents the angle from the current outflow point along the spiral line to the pole center, the size is:

(2) Calculation of capacitance energy of adjacent two-layer windings 1) Calculation of capacitance energy between two adjacent layers of single-turn primary winding and single-turn primary winding The two adjacent layers of single-turn windings are both primary windings. At this time, the current direction of the two layers is the same, and the capacitance element is expressed as:

R is the outer diameter of the winding, r is the inner diameter of the winding, ε is the dielectric constant between the plates, and d is the vertical distance between the two layers of windings; The relationship between capacitance C and energy W is expressed as:

U ₁ , U ₂ respectively represent the electric potential of the upper and lower plates of the capacitor; From equations (1), (8) and (9), the capacitance energy of two adjacent layers of single-turn primary winding and single-turn primary winding is expressed as

2) Capacitance energy calculation of two adjacent layers of single-turn primary winding and single-turn secondary winding Two adjacent layers of windings: One layer is a single-turn primary winding, and the other layer is a single-turn secondary winding. At this time, the current directions of the two are opposite. The single-turn primary winding and single-turn secondary winding capacitance energy is expressed as:

3) Calculation of capacitance energy between two adjacent layers of multi-turn primary winding and multi-turn primary winding The two adjacent layers of multi-turn windings are all primary windings. At this time, the two current directions are the same, and the capacitance element is expressed as:

w represents the width of each turn of the multi-turn coil winding, From equations (1), (9) and (12), the capacitance energy of two adjacent layers of multi-turn primary windings and multi-turn primary windings is expressed as:

Among them, n represents the total number of turns of the single-layer multi-turn coil, the number of turns is calculated from the current inflow point, one turn is counted for every 2π revolution, and less than one turn is counted as one turn, and i represents the i-th turn; 4) Calculation of capacitance energy of adjacent two-layer multi-turn primary winding and single-turn secondary winding Two adjacent layers of windings: One layer is a multi-turn primary winding, and the other is a single-turn secondary winding. At this time, the current directions of the two are opposite. The capacitance energy of multi-turn primary winding and single-turn secondary winding is expressed as:

(3) Calculation of equivalent parasitic capacitance Let C _k,k+1 represent the parasitic capacitance between the two adjacent windings of the kth layer and the k+1th layer, and W _k,k+1 represent the capacitance energy of C _k,k+1 . According to the adjacent PCB-type planar transformer For the winding condition, use equations (10), (11), (13), (14) to calculate the sum of capacitance energy W _ps between all adjacent two-layer windings: W _ps =W ₁₂ +W ₂₃ ...+W _k,k+1 +...(15) The total voltage drops on the primary and secondary sides are respectively V _p and V _s . The total voltage drop on each layer on the primary side is the same, that is, the input voltage and output voltage of the primary winding of each layer can be represented by V _p respectively, while the secondary side adopts a parallel connection. structure, each pressure drop is _Vs , term E

, F represents the coefficient sum of the term V _p ·V _s , G represents the coefficient sum of the term V _s ² , and (15) is organized as:

The total energy contained in the parasitic capacitance of the PCB winding is equal to the sum of the energy contained in the three equivalent capacitances C ₁ , C ₂ , and C ₃ , and the energy stored in the three equivalent capacitances C ₁ , C ₂ , and C ₃ calculated,

Since W _ps =W ₁ +W ₂ +W ₃ , comparing formulas (16) and (17), we get:

The capacitance values of the three equivalent capacitances C ₁ , C ₂ , and C ₃ are calculated from (18).

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