CN107273563A - A kind of flat surface transformer PCB windings parasitic capacitance Equivalent calculation method - Google Patents

A kind of flat surface transformer PCB windings parasitic capacitance Equivalent calculation method Download PDF

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CN107273563A
CN107273563A CN201710270652.0A CN201710270652A CN107273563A CN 107273563 A CN107273563 A CN 107273563A CN 201710270652 A CN201710270652 A CN 201710270652A CN 107273563 A CN107273563 A CN 107273563A
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mrow
msub
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theta
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CN107273563B (en
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钱钦松
张乐
刘琦
孙伟锋
陆生礼
时龙兴
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Southeast University
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    • H—ELECTRICITY
    • H01—ELECTRIC ELEMENTS
    • H01F—MAGNETS; INDUCTANCES; TRANSFORMERS; SELECTION OF MATERIALS FOR THEIR MAGNETIC PROPERTIES
    • H01F27/00—Details of transformers or inductances, in general
    • H01F27/34—Special means for preventing or reducing unwanted electric or magnetic effects, e.g. no-load losses, reactive currents, harmonics, oscillations, leakage fields
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/30—Circuit design
    • G06F30/32—Circuit design at the digital level
    • G06F30/327—Logic synthesis; Behaviour synthesis, e.g. mapping logic, HDL to netlist, high-level language to RTL or netlist
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/30—Circuit design
    • G06F30/39—Circuit design at the physical level
    • G06F30/398—Design verification or optimisation, e.g. using design rule check [DRC], layout versus schematics [LVS] or finite element methods [FEM]
    • H—ELECTRICITY
    • H01—ELECTRIC ELEMENTS
    • H01F—MAGNETS; INDUCTANCES; TRANSFORMERS; SELECTION OF MATERIALS FOR THEIR MAGNETIC PROPERTIES
    • H01F27/00—Details of transformers or inductances, in general
    • H01F27/28—Coils; Windings; Conductive connections
    • H01F27/2804—Printed windings
    • H01F2027/2809—Printed windings on stacked layers
    • H—ELECTRICITY
    • H01—ELECTRIC ELEMENTS
    • H01F—MAGNETS; INDUCTANCES; TRANSFORMERS; SELECTION OF MATERIALS FOR THEIR MAGNETIC PROPERTIES
    • H01F27/00—Details of transformers or inductances, in general
    • H01F27/28—Coils; Windings; Conductive connections
    • H01F27/2804—Printed windings
    • H01F2027/2819—Planar transformers with printed windings, e.g. surrounded by two cores and to be mounted on printed circuit

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Abstract

The invention discloses a kind of flat surface transformer PCB windings parasitic capacitance Equivalent calculation method, pass through the modeling and analysis to flat surface transformer PCB windings, the distribution situation of analog PCB winding parasitic capacitance, there is between each adjacent two layers of parasitic capacitance the size of parasitic capacitance storage energy by calculating and be overlapped, utilize the relation and law of conservation of energy between energy and parasitic capacitance, the overall parasitic capacitance of PCB windings is calculated, then it is equivalent to the calculating of the electric capacity of three parts, electric capacity C i.e. between primary side winding Same Name of Ends and vice-side winding Same Name of Ends1, electric capacity C between primary side winding Same Name of Ends and vice-side winding different name end2And the electric capacity C between primary side winding different name end and vice-side winding Same Name of Ends3。

Description

Equivalent calculation method for parasitic capacitance of PCB winding of planar transformer
Technical Field
The invention relates to a PCB type planar transformer, in particular to an equivalent calculation method for PCB winding parasitic capacitance of a planar transformer, which can be applied to the parasitic parameter analysis of the planar transformer and the establishment of an EMI conduction model in the field of electromagnetic compatibility.
Background
The planar transformer has the advantages of low height, small volume, good consistency, convenience for batch production, small leakage inductance, good heat dissipation, good electromagnetic compatibility and the like, and is a novel transformer which is widely applied at present. There are various planar transformers, among which the PCB planar transformer is most widely used due to its advantages of low cost, stable structure, convenient fabrication, etc. The structure is simple, and the through holes of the windings of all layers are welded. The parasitic capacitance of the planar transformer is an unavoidable practical problem in the design of the switching power supply, and often causes the working state of the switching power supply to deviate from the design result, even seriously affects the technical index of the power supply. The effect of these parasitic capacitances must therefore be taken into account during the design of the switching power supply. Therefore, it is very important to calculate the magnitude of the parasitic capacitance of the PCB winding of the planar transformer by using any method.
The planar transformer can be divided into 4 types of Printed Circuit Board (PCB), thick film, thin film and submicron according to different design and manufacturing processes. The PCB type planar transformer mainly comprises a planar magnetic core, a PCB winding and an insulating layer. Planar transformers require a planar structure of magnetic cores and windings, so multilayer PCB windings should be used, generally eight to ten layers, and more nearly twenty layers. In the PCB type planar transformer, the PCB winding is a round flat conducting wire made on a printed circuit board or is directly copper foil, the distance between the windings of each layer is small, and a larger facing area exists, so that parasitic capacitance inevitably exists. It can be considered that the parasitic capacitance of the PCB type planar transformer exists mainly between the PCB windings. The parasitic capacitance can make the working state of the switching power supply deviate from the design result, and even seriously affect the technical index of the power supply. The effects of parasitic capacitances cannot be eliminated, but we can make an estimate to take these parasitic capacitances into account during the design of the switching power supply.
The planar transformer PCB winding is divided into two parts, namely a primary winding (P) and a secondary winding (S). The number of turns of the primary winding is increased in a series connection mode; in order to increase the current carrying capacity of the winding, the secondary winding is generally in a parallel structure. The primary winding of the planar transformer is generally in a single-turn form or a multi-turn form and adopts a series connection form; the secondary side windings are all in a single-turn form and adopt a parallel structure. There are various ways for the winding structure of the PCB type planar transformer, which can be divided into a simple structure, a sandwich structure and a staggered structure. Although the three winding forms have different structures, parasitic capacitance is inevitably generated. 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 the size of the parasitic capacitance is calculated, the capacitance of the non-adjacent two layers of windings is much smaller than the capacitance between the adjacent two layers of windings and can be ignored; although the single-layer winding also has a capacitor, each layer of winding of the common planar transformer is very thin and can be ignored; the secondary windings are in parallel connection, when two adjacent layers of windings are the secondary windings, potential difference does not exist, and parasitic capacitance of the secondary windings can be considered to be zero.
Capacitance exists between any metal parts, and the existing planar transformer PCB winding parasitic capacitance calculation method is to regard each layer of winding as an equipotential surface, namely no voltage drop exists on the winding, and calculate the capacitance between the windings by using a static capacitance formula C-Q/U. However, in a transformer, the potential varies between turns of the same winding, between different windings, between windings and shields, and along a certain line length, and the capacitance thus formed is different from the static capacitance, which is called distributed capacitance. Therefore, the distributed capacitance cannot be directly calculated by the method of calculating the static capacitance. The invention provides a more accurate and strong-adaptability calculation method for the parasitic capacitance of the PCB winding of the planar transformer, which is used for solving the existing problem.
Disclosure of Invention
The invention provides an equivalent calculation method for the parasitic capacitance of the PCB winding of the planar transformer aiming at the inapplicability of the conventional calculation method for the parasitic capacitance of the planar transformer.
The purpose of the invention is realized by the following technical scheme: a planar transformer PCB winding parasitic capacitance equivalent calculation method, the said planar transformer includes the plane magnetic core, PCB winding and insulating layer, the PCB winding is a multilayer structure, set up a winding on each layer, primary winding and secondary winding are set up on different layers, the primary winding adopts one of single turn or multiturn, the secondary winding adopts the round shape of single turn, the multiturn adopts the spiral, the primary winding on different layers is connected in series, the secondary winding on different layers is connected in parallel, under the condition that two adjacent layers are primary winding and secondary winding separately, there is parasitic capacitance not neglected between two adjacent layers at this moment;
the method is characterized in that: simulating the distribution condition of the parasitic capacitance of the PCB winding by modeling and analyzing the PCB winding of the planar transformer, calculating and superposing the energy storage capacity of the parasitic capacitance between each two adjacent layers with the parasitic capacitance, calculating the overall parasitic capacitance of the PCB winding by utilizing the relation between the energy and the parasitic capacitance and the energy conservation law, and then equating the overall parasitic capacitance into three capacitors, namely a capacitor C between the homonymous end of the primary winding and the homonymous end of the secondary winding1Capacitor C between homonymous terminal of primary winding and heteronymous terminal of secondary winding2And a capacitor C between the different name terminal of the primary winding and the same name terminal of the secondary winding3(ii) a The method comprises the following steps:
(1) single layer winding potential calculation
1) Single layer single turn primary side winding potential representation
Assuming that the current of a primary winding flows in a clockwise direction, the voltage of the winding uniformly drops along the current direction, a polar coordinate system is established by taking a ray from a polar core to a current inflow point as a polar diameter, the angle parameter of the position polar coordinate is theta, a section of circular arc on the winding is taken as a infinitesimal d theta, and the infinitesimal voltage is expressed as:
V1is the current inflow point voltage, V2Is the current sink voltage;
2) single layer single turn secondary side winding potential representation
The entering positions of the primary current and the secondary current are mirror images, the flowing directions of the currents are opposite, and the voltage of a infinitesimal d theta on the secondary side winding is expressed as follows:
3) single layer multi-turn primary side winding potential representation
When the primary side winding is in a multi-turn condition, the difference with a primary single-turn winding is larger, the coil is a section of spiral line, parasitic capacitance exists between every two turns of spiral line, but each layer of winding of a general planar transformer is very thin, the capacitance can be ignored, an Archimedes spiral equation is applied to fit the shape of the winding, and the Archimedes spiral equation is expressed as:
r*＝h·θ (3)
r denotes the pole diameter, h is a constant to denote the spiral ratio;
a polar coordinate system is established by taking a ray from a polar core to a current inflow point as a polar diameter, a length dL of a section on the multi-turn coil is taken as a infinitesimal, and the voltage of the infinitesimal is expressed as:
a represents a linear distance from a current flow point to the pole center, θ1Represents the angle from the point of current entry along the helix to the pole center, with the magnitude:
l represents the length from the current entry point to the current exit point, and has the magnitude:
(r)' represents the first derivative of r, B represents the linear distance from the point of current entry to the pole center, and θ2Representing the angle from the current exit point around the helix to the pole center, with the magnitude:
(2) calculation of capacitance energy of adjacent two layers of windings
1) Capacitance energy calculation of adjacent two layers of single-turn primary winding and single-turn primary winding
Two adjacent layers of single-turn windings are primary windings, the current directions of the two adjacent layers of single-turn windings are the same, and the capacitance infinitesimal is expressed as follows:
r is the outer diameter of the winding, R is the inner diameter of the winding and is the dielectric constant between the plates, and d is the vertical distance between the two layers of windings;
the relationship of capacitance C to energy W is expressed as:
U1，U2respectively representing the potential of the upper and lower electrode plates of the capacitor;
the capacitance energy of the single-turn primary winding and the single-turn primary winding of two adjacent layers is expressed by the formulas (1), (8) and (9),
V3voltage, V, representing the current inflow point of the other layer of winding4Representing the voltage of the current flowing point of the other layer of winding;
2) capacitance energy calculation of adjacent two layers of single-turn primary winding and single-turn secondary winding
One layer of the adjacent two layers of windings is a single-turn primary winding, the other layer of the adjacent two layers of windings is a single-turn secondary winding, the current directions of the two layers of windings are opposite at the moment, and the capacitance energy of the adjacent two layers of single-turn primary windings and the single-turn secondary winding is expressed by the following formulas (2), (8) and (9):
3) capacitance energy calculation of adjacent two-layer multi-turn primary winding and multi-turn primary winding
Two adjacent layers of multi-turn windings are primary windings, the current directions of the two adjacent layers of multi-turn windings are the same, and the capacitance micro element is expressed as follows:
w represents the width of each turn of the multi-turn coil winding,
by the formulas (1), (9) and (12), the capacitance energy of the adjacent two layers of multi-turn primary winding and multi-turn primary winding is expressed as:
wherein n represents the total number of turns of the single-layer multi-turn coil, the number of turns is calculated from a current inflow point, one turn is calculated when each turn passes 2 pi, less than one turn is calculated according to one turn, and i represents the ith turn;
4) capacitance energy calculation of adjacent two-layer multi-turn primary winding and single-turn secondary winding
One layer of the adjacent two layers of windings is a multi-turn primary winding, the other layer of the adjacent two layers of windings is a single-turn secondary winding, the current directions of the two layers of windings are opposite, and the two adjacent layers of multi-turn primary windings and the single-turn secondary winding are expressed by formulas (2), (9) and (12):
(3) equivalent parasitic capacitance calculation
Is provided with Ck,k+1Represents the parasitic capacitance between two adjacent windings of the k layer and the k +1 layer, Wk,k+1Is represented by Ck,k+1According to the adjacent winding condition of the PCB type planar transformer, the sum W of the capacitance energy between all adjacent two layers of windings is calculated by using the formulas (10), (11), (13) and (14)ps：
Wps＝W12+W23...+Wk,k+1+...(15)
The primary and secondary total pressure drops are respectively VpAnd VsThe total voltage drop on each layer of the primary side is the same, namely the input voltage and the output voltage of the primary winding of each layer can be respectively VpThe secondary side adopts a parallel structure, and the pressure drop of each layer is VsBy E, an itemF represents the term Vp·VsG represents the term Vs2, the sum of coefficients (15) is:
the total energy contained in the parasitic capacitance of the PCB winding is equal to three equivalent capacitances C1，C2，C3The sum of the energies contained, the equivalent capacitance C1，C2，C3The energy stored in the three capacitors is calculated,
due to Wps＝W1+W2+W3Comparing equations (16), (17), we obtain:
three equivalent capacitances C are obtained by calculation of (18)1，C2，C3The capacitance value of (2).
Compared with the prior art, the method has the following advantages and remarkable effects:
1. the voltage changes of different positions of the single-layer winding are fully considered, the single-layer winding is not regarded as an equipotential body, and the calculation result is more accurate.
2. The parasitic capacitance of the PCB winding of the planar transformer is solved by utilizing the relation between the energy and the capacitance, the whole parasitic capacitance is equivalent to the capacitance between the primary winding and the secondary winding, and the method has higher practical value.
3. The method has wide adaptability to the calculation of the parasitic capacitance of the existing PCB type planar transformer, and provides an idea for solving the problems of the analysis of the parasitic parameters of the planar transformer and the establishment of an EMI conduction model in the field of electromagnetic compatibility. The calculation is rapid and accurate, the adaptability is strong, and the engineering value is high.
Drawings
FIG. 1 is a schematic diagram of an equivalent capacitance;
FIGS. 2a, 2b and 2c are schematic diagrams of three structures of a planar transformer PCB winding respectively;
FIG. 3 is a schematic diagram of a single turn primary side winding voltage gradient;
FIG. 4 is a schematic diagram of a single turn secondary side winding voltage gradient;
FIG. 5 is a schematic diagram of a multi-turn primary side winding voltage gradient;
FIG. 6 is a schematic diagram of a single-turn primary side winding and a single-turn primary side winding capacitance energy calculation;
FIG. 7 is a schematic diagram of a single-turn primary side winding and single-turn secondary side winding capacitance energy calculation;
FIG. 8 is a schematic diagram of the calculation of the capacitance energy of the primary-side winding and the 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 Description
The size of the parasitic capacitance in the planar transformer is equal to the superposition of countless capacitances, which cannot be represented one by one. The capacitance is essentially generated by the creation of an electric field between two conductors having different potential differences, and the presence of the electric field means energy. Thus, capacitance effectively characterizes the ability to store energy. In other words, the equivalent overall capacitance is effectively equivalent to the capacity of one transformer as a whole to store energy. Therefore, the magnitude of the equivalent capacitance can be calculated by calculating the energy. In the design analysis process of the switching power supply, an equivalent model is established to represent the parasitic capacitance between the planar transformer PCB windings. As shown in fig. 1, the present invention equates the parasitic capacitance between the planar transformer PCB windings to three parts: (1) capacitor C between homonymous terminal of primary winding and homonymous terminal of secondary winding1(ii) a (2) Capacitance C between homonymous terminal of primary winding and heteronymous terminal of secondary winding2(ii) a (3) Capacitance C between different-name end of primary winding and same-name end of secondary winding3。
The size of the capacitor is closely related to the winding pattern of the planar transformer distributed on the PCB. As shown in fig. 2, the winding structure of the PCB type planar transformer may be divided into a simple structure (fig. 2a), a sandwich structure (fig. 2b), and a staggered structure (fig. 2 c). Taking eight layers of PCB windings as an example, the solid part represents a primary winding (P) which is formed by connecting four layers of windings in series, the windings are all in a single-turn form or a multi-turn form, and the current direction is assumed to be clockwise; the hollow part represents a secondary winding (S) which is formed by connecting four layers of windings in parallel, the windings are all in a single-turn structure, and the current direction of the windings is opposite to the primary side and is in a counterclockwise direction. The parasitic capacitance is calculated in two cases: the first condition is that two adjacent layers of windings are positioned at the same side and are both primary side windings, and the current directions of the two windings are the same; in the second case, two adjacent layers of windings are respectively positioned at two sides, one layer is a primary winding, the other layer is a secondary winding, and the current directions of the two windings are opposite. Now, the energy contained in the parasitic capacitance of the primary side winding is calculated when the primary side winding is single-turn and multi-turn respectively.
(one) Single layer winding potential calculation
1. Single layer single turn primary side winding potential representation
As shown in fig. 3, assuming that the voltage at the current flow point a is V1The voltage at the current outflow point b is V2. It is readily seen that the voltage across the winding decreases with the distance from the point of current flow and that the current flows in a clockwise direction. Using the polar coordinate system shown in fig. 3, assume that the inner diameter of the winding is R and the outer diameter is R. Taking a section of circular arc on the circular ring as a infinitesimal d theta, wherein the angle parameter of the position polar coordinate is theta, and assuming that voltage drops from an inflow point to an outflow point are uniformly distributed on the coil, the voltage on the section of infinitesimal circular arc is as follows:
2. single layer single turn secondary side winding potential representation
As shown in fig. 4, since the inflow positions of the primary current and the secondary current are mirror images and the current flowing directions are opposite, the voltage on a certain section of infinitesimal arc on the secondary side winding is:
3. single layer multi-turn primary side winding potential representation
When the primary side winding is in a multi-turn condition, the difference from a single-turn winding is larger, the coil is a section of spiral line, parasitic capacitance exists between every two turns of spiral line, but each layer of winding of a general planar transformer is very thin, and the capacitance can be ignored. Fitting the winding shape using an Archimedes' spiral equation, expressed as:
r*＝h·θ (3)
wherein r denotes the pole diameter, and h is a constant to denote the spiral ratio;
as shown in FIG. 5, assume that the voltage at the current inflow point is V1Voltage at current outflow point is V2. It is readily observed that the voltage on the spiral decreases linearly with the distance from the point of current flow. A polar coordinate system shown in the figure is adopted, a length of a segment on the multi-turn coil is taken as a infinitesimal, the length of the infinitesimal is dL, an angle parameter of a position polar coordinate is theta, and an angle which the infinitesimal bypasses from a spiral line to a polar center o is represented; theta1Represents the angle from the point of current entry along the helix to the pole center o; theta2Representing the angle at which the coil passes from the current exit point along the helix to the pole center o. Thus, the voltage over the length of the section of infinitesimal is:
wherein,
l represents the length from the current entry point to the current exit point, and has the magnitude:
wherein,
(II) calculating capacitance energy of two adjacent layers of windings
1. Capacitance energy calculation of adjacent two 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 condition, that is, the two adjacent layers of windings are both primary windings, and the current directions of the two adjacent layers of windings are the same. As shown in fig. 6, two layers of PCB windings are adjacent to each other, and it is assumed that the upper layer winding is denoted as winding 1 and the lower layer winding is denoted as winding 2. a. The terminals c are current input terminals, and the terminals b and d are current output terminals. In winding 1, the voltage at a is V1B terminal voltage is V2，V1>V2Current flows in a clockwise direction; in winding 2, the voltage at c-terminal is V3D terminal voltage is V4，V3>V4The current also flows in a clockwise direction. R is the outer diameter of the ring, R is the inner diameter of the ring, the dielectric constant between the plates, and d is the linear distance between the two layers of windings. The capacitive micro-elements may be represented as,
the relationship of the capacitance C to the energy W is expressed as,
U1，U2individual watchShowing the potential of the upper and lower plates of the capacitor.
By the formulas (1), (8) and (9), the capacitance energy of the adjacent two layers of single-turn primary winding and single-turn primary winding is expressed as:
2. capacitance energy calculation of adjacent two 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 condition, that is, one layer of the two adjacent layers of windings is the primary winding, and the other layer of the two adjacent layers of windings is the secondary winding, and the current directions of the two adjacent layers of windings are opposite, as shown in fig. 7. By the formulas (2), (8) and (9), the capacitance energy of the adjacent two layers of single-turn primary winding and single-turn secondary winding is expressed as:
3. capacitance energy calculation of adjacent two-layer multi-turn primary winding and multi-turn primary winding
The capacitive infinitesimal can be represented as:
w represents the width of each turn of the multi-turn coil winding,
if two adjacent layers of multi-turn windings are in the first condition, as shown in fig. 8, only length integration is needed, and at this time, the capacitance energy of the two adjacent layers of multi-turn primary windings and the multi-turn primary winding is expressed by equations (1), (9) and (12):
wherein n represents the total number of turns of the single-layer multi-turn coil (the number of turns is calculated from the voltage inflow end, one turn is calculated after every 2 pi, and the number of turns is calculated according to one turn when the number of turns is less than one turn), and i represents the ith 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 condition, the primary winding has multiple turns and the secondary winding has a single turn, as shown in fig. 9. At this time, the energy of the adjacent two layers of windings is expressed by the equations (2), (9) and (12):
(III) calculation of equivalent parasitic capacitance
Take eight layers of PCB windings as an example, C12，C23。。。C78Representing the parasitic capacitance, W, between adjacent layers of windings12，W23...W78Respectively represent C12，C23。。。C78According to the adjacent winding condition of the PCB type planar transformer, calculating the sum of capacitance energy between all adjacent two layers of windings by using the formulas (10), (11), (13) and (14):
Wps＝W12+W23+W34+W45+W56+W67+W78(15)
the primary and secondary total pressure drops are respectively VpAnd VsThe total voltage drop on each layer at the primary side is the same, i.e. the input voltage and the output voltage of each layer of winding can be respectively VpThe secondary side adopts a parallel structure, and the pressure drop of each layer is VsBy E, an itemF represents the term Vp·VsG represents the term Vs 2The sum of coefficients of (15) is arranged as:
the total energy contained by the parasitic capacitance of the PCB winding is equal to three equivalent capacitances C1，C2，C3The sum of the energies involved. Equivalent capacitance C1，C2，C3The energy stored in the three capacitors is calculated,
due to Wps＝W1+W2+W3By comparing the formulas (16) and (17),
three equivalent capacitances C are obtained by calculation of (18)1，C2，C3The capacitance value of (2). Thereby obtaining the specific value of the parasitic capacitance in the equivalent model of the planar transformer.
The calculation result of the invention has accuracy, and can be used for calculating the parasitic capacitance of the PCB winding of the planar transformer, thereby being applied to the parasitic parameter analysis of the planar transformer and the establishment of an EMI conduction model in the field of electromagnetic compatibility.
The practical application result shows that the purpose of the invention is realized and the effect is achieved.
While the present invention has been described in detail with reference to specific embodiments thereof, it will be apparent to one skilled in the art that various changes and modifications can be made therein without departing from the spirit and scope thereof as defined in the appended claims.

Claims (1)

1. A planar transformer PCB winding parasitic capacitance equivalent calculation method, the said planar transformer includes the plane magnetic core, PCB winding and insulating layer, the PCB winding is a multilayer structure, set up a winding on each layer, primary winding and secondary winding are set up on different layers, the primary winding adopts one of single turn or multiturn, the secondary winding adopts the round shape of single turn, the multiturn adopts the spiral, the primary winding on different layers is connected in series, the secondary winding on different layers is connected in parallel, under the condition that two adjacent layers are primary winding and secondary winding separately, there is parasitic capacitance not neglected between two adjacent layers at this moment;
the method is characterized in that: simulating the distribution condition of the parasitic capacitance of the PCB winding by modeling and analyzing the PCB winding of the planar transformer, calculating and superposing the energy storage capacity of the parasitic capacitance between each two adjacent layers with the parasitic capacitance, calculating the overall parasitic capacitance of the PCB winding by utilizing the relation between the energy and the parasitic capacitance and the energy conservation law, and then equating the overall parasitic capacitance into three capacitors, namely a capacitor C between the homonymous end of the primary winding and the homonymous end of the secondary winding1Capacitor C between homonymous terminal of primary winding and heteronymous terminal of secondary winding2And a capacitor C between the different name terminal of the primary winding and the same name terminal of the secondary winding3(ii) a The method comprises the following steps:
(1) single layer winding potential calculation
1) Single layer single turn primary side winding potential representation
Assuming that the current of a primary winding flows in a clockwise direction, the voltage of the winding uniformly drops along the current direction, a polar coordinate system is established by taking a ray from a polar core to a current inflow point as a polar diameter, the angle parameter of the position polar coordinate is theta, a section of circular arc on the winding is taken as a infinitesimal d theta, and the infinitesimal voltage is expressed as:
<mrow> <msub> <mi>V</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <mfrac> <mi>&amp;theta;</mi> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
V1is the voltage of the current inflow point，V2Is the current sink voltage;
2) single layer single turn secondary side winding potential representation
The entering positions of the primary current and the secondary current are mirror images, the flowing directions of the currents are opposite, and the voltage of a infinitesimal d theta on the secondary side winding is expressed as follows:
<mrow> <msub> <mi>V</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mfrac> <mi>&amp;theta;</mi> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>&amp;theta;</mi> <mo>&amp;le;</mo> <mi>&amp;pi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mi>&amp;theta;</mi> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mo>(</mo> <mn>3</mn> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mrow> <mi>&amp;pi;</mi> <mo>&lt;</mo> <mi>&amp;theta;</mi> <mo>&amp;le;</mo> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
3) single layer multi-turn primary side winding potential representation
When the primary side winding is in a multi-turn condition, the difference with a primary single-turn winding is larger, the coil is a section of spiral line, parasitic capacitance exists between every two turns of spiral line, but each layer of winding of a general planar transformer is very thin, the capacitance can be ignored, an Archimedes spiral equation is applied to fit the shape of the winding, and the Archimedes spiral equation is expressed as:
r*＝h·θ (3)
r denotes the pole diameter, h is a constant to denote the spiral ratio;
a polar coordinate system is established by taking a ray from a polar core to a current inflow point as a polar diameter, a length dL of a section on the multi-turn coil is taken as a infinitesimal, and the voltage of the infinitesimal is expressed as:
<mrow> <msub> <mi>V</mi> <mi>&amp;theta;</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msubsup> <mo>&amp;Integral;</mo> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <mi>&amp;theta;</mi> </mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> </msubsup> <mn>1</mn> <mi>d</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
a represents a linear distance from a current flow point to the pole center, θ1Represents the angle from the point of current entry along the helix to the pole center, with the magnitude:
<mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mi>A</mi> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> <mi>h</mi> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mo>=</mo> <mfrac> <mi>A</mi> <mi>h</mi> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
l represents the length from the current entry point to the current exit point, and has the magnitude:
<mrow> <mi>L</mi> <mo>=</mo> <msubsup> <mo>&amp;Integral;</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </msubsup> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <mi>r</mi> <mo>*</mo> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mo>*</mo> </mrow> <mo>)</mo> </mrow> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mi>d</mi> <mi>&amp;theta;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
(r)' represents the first derivative of r, B represents the linear distance from the point of current entry to the pole center, and θ2Representing the angle from the current exit point around the helix to the pole center, with the magnitude:
<mrow> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mi>B</mi> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> <mi>h</mi> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mo>=</mo> <mfrac> <mi>B</mi> <mi>h</mi> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
(2) calculation of capacitance energy of adjacent two layers of windings
1) Capacitance energy calculation of adjacent two layers of single-turn primary winding and single-turn primary winding
Two adjacent layers of single-turn windings are primary windings, the current directions of the two adjacent layers of single-turn windings are the same, and the capacitance infinitesimal is expressed as follows:
<mrow> <mi>d</mi> <mi>C</mi> <mo>=</mo> <mfrac> <mrow> <mi>&amp;epsiv;</mi> <mrow> <mo>(</mo> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mi>d</mi> <mi>&amp;theta;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
r is the outer diameter of the winding, R is the inner diameter of the winding and is the dielectric constant between the plates, and d is the vertical distance between the two layers of windings;
the relationship of capacitance C to energy W is expressed as:
<mrow> <mi>W</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>U</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;CenterDot;</mo> <mi>C</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
U1，U2respectively representing the potential of the upper and lower electrode plates of the capacitor;
the capacitance energy of the single-turn primary winding and the single-turn primary winding of two adjacent layers is expressed by the formulas (1), (8) and (9),
<mrow> <mi>W</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <munderover> <mo>&amp;Integral;</mo> <mn>0</mn> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </munderover> <msubsup> <mi>V</mi> <mi>&amp;theta;</mi> <mn>2</mn> </msubsup> <mo>&amp;CenterDot;</mo> <mi>d</mi> <mi>C</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <munderover> <mo>&amp;Integral;</mo> <mn>0</mn> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mfrac> <mi>&amp;theta;</mi> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>3</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>4</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>4</mn> </msub> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;CenterDot;</mo> <mfrac> <mrow> <mi>&amp;epsiv;</mi> <mrow> <mo>(</mo> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mi>d</mi> <mi>&amp;theta;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
V3voltage, V, representing the current inflow point of the other layer of winding4Representing the voltage of the current flowing point of the other layer of winding;
2) capacitance energy calculation of adjacent two layers of single-turn primary winding and single-turn secondary winding
One layer of the adjacent two layers of windings is a single-turn primary winding, the other layer of the adjacent two layers of windings is a single-turn secondary winding, the current directions of the two layers of windings are opposite at the moment, and the capacitance energy of the adjacent two layers of single-turn primary windings and the single-turn secondary winding is expressed by the following formulas (2), (8) and (9):
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>W</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <munderover> <mo>&amp;Integral;</mo> <mn>0</mn> <mi>&amp;pi;</mi> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>4</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>&amp;theta;</mi> </mrow> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <mo>+</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>V</mi> <mn>4</mn> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;CenterDot;</mo> <mfrac> <mrow> <mi>&amp;epsiv;</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mi>d</mi> <mi>&amp;theta;</mi> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <munderover> <mo>&amp;Integral;</mo> <mi>&amp;pi;</mi> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>4</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>&amp;theta;</mi> </mrow> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <mo>+</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>-</mo> <mfrac> <mrow> <mn>3</mn> <msub> <mi>V</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>3</mn> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;CenterDot;</mo> <mfrac> <mrow> <mi>&amp;epsiv;</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mi>d</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
3) capacitance energy calculation of adjacent two-layer multi-turn primary winding and multi-turn primary winding
Two adjacent layers of multi-turn windings are primary windings, the current directions of the two adjacent layers of multi-turn windings are the same, and the capacitance micro element is expressed as follows:
<mrow> <mi>d</mi> <mi>C</mi> <mo>=</mo> <mfrac> <mrow> <mi>&amp;epsiv;</mi> <mo>&amp;CenterDot;</mo> <mi>w</mi> </mrow> <mi>d</mi> </mfrac> <mi>d</mi> <mi>L</mi> <mo>=</mo> <mfrac> <mrow> <mi>&amp;epsiv;</mi> <mo>&amp;CenterDot;</mo> <mi>w</mi> </mrow> <mi>d</mi> </mfrac> <mi>h</mi> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>&amp;theta;</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mi>d</mi> <mi>&amp;theta;</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
w represents the width of each turn of the multi-turn coil winding,
by the formulas (1), (9) and (12), the capacitance energy of the adjacent two layers of multi-turn primary winding and multi-turn primary winding is expressed as:
<mrow> <mi>W</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> <mi>i</mi> </mrow> </msubsup> <msup> <mrow> <mo>&amp;lsqb;</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mi>&amp;theta;</mi> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;theta;</mi> </mrow> <mrow> <mi>&amp;theta;</mi> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msubsup> <mn>1</mn> <mi>d</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mi>&amp;theta;</mi> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;theta;</mi> </mrow> <mrow> <mi>&amp;theta;</mi> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msubsup> <mn>1</mn> <mi>d</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>3</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>V</mi> <mn>3</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>C</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
wherein n represents the total number of turns of the single-layer multi-turn coil, the number of turns is calculated from a current inflow point, one turn is calculated when each turn passes 2 pi, less than one turn is calculated according to one turn, and i represents the ith turn;
4) capacitance energy calculation of adjacent two-layer multi-turn primary winding and single-turn secondary winding
One layer of the adjacent two layers of windings is a multi-turn primary winding, the other layer of the adjacent two layers of windings is a single-turn secondary winding, the current directions of the two layers of windings are opposite, and the two adjacent layers of multi-turn primary windings and the single-turn secondary winding are expressed by formulas (2), (9) and (12):
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>W</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>&amp;pi;</mi> </mrow> </msubsup> <msup> <mrow> <mo>&amp;lsqb;</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mi>&amp;theta;</mi> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;theta;</mi> </mrow> <mrow> <mi>&amp;theta;</mi> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </msubsup> <mn>1</mn> <mi>d</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>&amp;theta;</mi> <mo>-</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>4</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>V</mi> <mn>4</mn> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>C</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>&amp;pi;</mi> </mrow> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> <mi>i</mi> </mrow> </msubsup> <msup> <mrow> <mo>&amp;lsqb;</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msubsup> <mo>&amp;Integral;</mo> <mrow> <mi>&amp;theta;</mi> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;pi;</mi> <mo>-</mo> <mi>&amp;theta;</mi> </mrow> <mrow> <mi>&amp;theta;</mi> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;pi;</mi> </mrow> </msubsup> <mn>1</mn> <mi>d</mi> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>&amp;theta;</mi> <mo>-</mo> <mn>2</mn> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>4</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <msub> <mi>V</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>V</mi> <mn>3</mn> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&amp;rsqb;</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>C</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>
(3) equivalent parasitic capacitance calculation
Is provided with Ck,k+1Represents the parasitic capacitance between two adjacent windings of the k layer and the k +1 layer, Wk,k+1Is represented by Ck,k+1According to the adjacent winding condition of the PCB type planar transformer, the sum W of the capacitance energy between all adjacent two layers of windings is calculated by using the formulas (10), (11), (13) and (14)ps：
Wps＝W12+W23...+Wk,k+1+...(15)
The primary and secondary total pressure drops are respectively VpAnd VsThe total voltage drop on each layer of the primary side is the same, namely the input voltage and the output voltage of the primary winding of each layer can be respectively VpThe secondary side adopts a parallel structure, and the pressure drop of each layer is VsBy E, an itemF represents the term Vp·VsSum of coefficients, G represents termThe sum of coefficients of (15) is arranged as:
<mrow> <msub> <mi>W</mi> <mrow> <mi>p</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mi>E</mi> <mo>&amp;CenterDot;</mo> <msubsup> <mi>V</mi> <mi>p</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mi>F</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>V</mi> <mi>p</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>V</mi> <mi>s</mi> </msub> <mo>+</mo> <mi>G</mi> <mo>&amp;CenterDot;</mo> <msubsup> <mi>V</mi> <mi>s</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>
the total energy contained in the parasitic capacitance of the PCB winding is equal to three equivalent capacitances C1，C2，C3The sum of the energies contained, the equivalent capacitance C1，C2，C3The energy stored in the three capacitors is calculated,
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>W</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>&amp;CenterDot;</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mi>p</mi> </msub> <mo>-</mo> <msub> <mi>V</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>&amp;CenterDot;</mo> <msubsup> <mi>V</mi> <mi>p</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>V</mi> <mi>p</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>V</mi> <mi>s</mi> </msub> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>&amp;CenterDot;</mo> <msubsup> <mi>V</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>W</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>&amp;CenterDot;</mo> <msubsup> <mi>V</mi> <mi>p</mi> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>W</mi> <mn>3</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>C</mi> <mn>3</mn> </msub> <mo>&amp;CenterDot;</mo> <msubsup> <mi>V</mi> <mi>s</mi> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>
due to Wps＝W1+W2+W3Comparing equations (16), (17), we obtain:
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>2</mn> <mi>F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mi>E</mi> <mo>-</mo> <mi>F</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mi>G</mi> <mo>-</mo> <mi>F</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>
three equivalent capacitances C are obtained by calculation of (18)1，C2，C3The capacitance value of (2).
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