Design method for realizing planar transformer with any non-integer turn ratio and planar transformer

Abstract

The invention discloses a design method for achieving a planar transformer with any non-integer turn ratio and the planar transformer. The design method comprises the steps that the to-be-designed turn ratio of the planar transformer is determined; np1 turns of primary windings are wound on the magnetic column 1, Np2 turns of primary windings are wound on the magnetic column 2, Ns turns of secondary windings are wound on the magnetic column 3, the magnetic column 1, the magnetic column 2 and the magnetic column 3 are sequentially distributed, the magnetic resistance of the magnetic column 1 is Rg, and the magnetic resistance of the magnetic column 2 and the magnetic resistance of the magnetic column 3 are xRg and yRg respectively; determining the magnetic resistance proportion x of the magnetic column 2 and the magnetic column 1 and the magnetic resistance proportion y of the magnetic column 3 and the magnetic column 1 according to the to-be-designed turn ratio of the planar transformer; and designing the planar transformer according to the magnetic resistance ratio x of the magnetic column 2 to the magnetic column 1 and the magnetic resistance ratio y of the magnetic column 3 to the magnetic column 1. The method and the planar transformer can meet the requirement of any non-integer turn ratio.

Description

Technical Field

[0001] The invention belongs to the field of power electronic technology DC / DC converters and relates to a design method for realizing a planar transformer with arbitrary non-integer turns ratio and a planar transformer. Background Art

[0002] The continuous development of aerospace, information technology, electric vehicles and other fields has put forward higher requirements for power converters in terms of efficiency, power density, current capability, etc. Planar transformers based on PCB windings have the advantages of high power density, good consistency, and easy winding interleaving, and have been widely used. However, under high current applications, due to the limitations of the number of PCB layers and the number of winding turns, planar transformers cannot refer to traditional transformers to convert any non-integer turns ratio into an integer turns ratio with a high number of turns, which also brings challenges to subsequent planar transformers.

[0003] In order to meet the requirements of achieving non-integer turns ratio at low winding turns, the existing fractional turns solution can achieve the effect of fractional turns by dividing the secondary winding into integer segments and connecting them in parallel with the output capacitor. However, the existing fractional turns solution can only complete a few fixed groups of turns such as 1:1 / 2, 1:1 / 3, 1:1 / 4, etc., which has great limitations and is difficult to meet the requirements of any non-integer turns ratio. At the same time, the fractional turn transformer will cause the excitation voltage of the magnetic core to increase exponentially, and the volume of the magnetic core will increase, which will ultimately affect the overall power density, efficiency, heat dissipation and reliability of the converter. Summary of the invention

[0004] The purpose of the present invention is to overcome the shortcomings of the above-mentioned prior art and provide a design method for a planar transformer with an arbitrary non-integer turns ratio and a planar transformer. The method and the planar transformer can meet the requirements of any non-integer turns ratio.

[0005] In order to achieve the above object, the present invention adopts the following technical scheme:

[0006] In one aspect, the present invention provides a design method for realizing an arbitrary non-integer turns ratio planar transformer, comprising:

[0007] Determine the turns ratio of the planar transformer to be designed;

[0008] Wind N around magnetic column 1 p1 The primary winding is wound around the magnetic column 2. p2 The primary winding is wound around the magnetic column 3. s The secondary winding has magnetic columns 1, 2 and 3, and the magnetic resistance of magnetic column 1 is R. g , the magnetic resistance of magnetic column 2 and magnetic column 3 are xR g and yR g ;

[0009] Determine the magnetic resistance ratio x of magnetic column 2 to magnetic column 1 and the magnetic resistance ratio y of magnetic column 3 to magnetic column 1 according to the turns ratio to be designed of the planar transformer;

[0010] The planar transformer is designed based on the magnetic resistance ratio x of the magnetic column 2 to the magnetic column 1 and the magnetic resistance ratio y of the magnetic column 3 to the magnetic column 1.

[0011] The design method of the planar transformer for realizing arbitrary non-integer turns ratio according to the present invention is further improved in that:

[0012] Furthermore, the turns ratio n to be designed of the planar transformer is:

[0013]

[0014] Among them, φ1 is the magnetic flux of magnetic column 1, φ2 is the magnetic flux of magnetic column 2, and φ3 is the magnetic flux of magnetic column 3.

[0015] Furthermore, the magnetic flux φ1 of the magnetic column 1 is:

[0016]

[0017] Furthermore, the magnetic flux φ2 of the magnetic column 2 is:

[0018]

[0019] Furthermore, the magnetic flux φ3 of the magnetic column 3 is:

[0020]

[0021] Furthermore, the magnetic resistance of the magnetic column 1 is adjusted by adjusting the area of ​​the magnetic column 1 and the air gap length;

[0022] Furthermore, the magnetic resistance of the magnetic column 2 is adjusted by adjusting the area of ​​the magnetic column 2 and the air gap length;

[0023] Furthermore, the magnetic resistance of the magnetic column 3 is adjusted by adjusting the area of ​​the magnetic column 3 and the air gap length.

[0024] In a second aspect of the present invention, the present invention provides a planar transformer designed based on the design method for realizing a planar transformer with arbitrary non-integer turns ratio.

[0025] The planar transformer of the present invention is further improved in that:

[0026] Furthermore, the turns ratio n to be designed of the planar transformer is:

[0027]

[0028] Among them, φ1 is the magnetic flux of magnetic column 1, φ2 is the magnetic flux of magnetic column 2, and φ3 is the magnetic flux of magnetic column 3.

[0029] The present invention has the following beneficial effects:

[0030] In the design method for realizing a planar transformer with an arbitrary non-integer turns ratio and the specific operation of the planar transformer described in the present invention, magnetic pillars 1, 2 and 3 are distributed in sequence, and the magnetic resistance ratio x between magnetic pillars 2 and 1 and the magnetic resistance ratio y between magnetic pillars 3 and 1 are adjusted, and then the designed turns ratio of the planar transformer is adjusted to meet the requirements of any non-integer turns ratio. At the same time, the core excitation voltage will not be increased exponentially, and the problems of large core area, high magnetic flux density, and heat dissipation difficulty existing in fractional turn transformers are solved, the winding loss is significantly reduced, and the final efficiency of the converter is greatly improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] The accompanying drawings constituting a part of the present invention are used to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the accompanying drawings:

[0032] Figure 1a It is the structural diagram of the transformer with arbitrary turns ratio;

[0033] Figure 1b is the equivalent magnetic circuit diagram of any turns ratio transformer;

[0034] Figure 2a The structural diagram of the transformer with the air gap on the right eliminated;

[0035] Figure 2b The equivalent magnetic circuit diagram of the transformer with the air gap on the right eliminated;

[0036] Figure 3 is a schematic diagram of an arbitrary non-integer turns ratio transformer using an EI-like core;

[0037] Figure 4 A schematic diagram of a 3D model of a transformer constructed in the present invention;

[0038] Figure 5 The simulation result diagram of the transformer with an arbitrary non-integer turns ratio designed by the present invention;

[0039] Figure 6 This is the simulation result diagram of the transformer designed by the traditional method. DETAILED DESCRIPTION

[0040] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0041] In the description of the present invention, it should be understood that the terms “include” and “comprises” indicate the presence of described features, wholes, steps, operations, elements and / or components, but do not exclude the presence or addition of one or more other features, wholes, steps, operations, elements, components and / or collections thereof.

[0042] It should also be understood that the terms used in the present specification are only for the purpose of describing specific embodiments and are not intended to limit the present invention. As used in the present specification and the appended claims, unless the context clearly indicates otherwise, the singular forms "a", "an" and "the" are intended to include plural forms.

[0043] It should be further understood that the term "and / or" used in the present specification and the appended claims refers to any combination of one or more of the associated listed items and all possible combinations, and includes these combinations. For example, A and / or B can represent: A exists alone, A and B exist at the same time, and B exists alone. In addition, the character " / " in the present invention generally indicates that the associated objects are in an "or" relationship.

[0044] It should be understood that, although the terms first, second, third, etc. may be used to describe preset ranges, etc. in the embodiments of the present invention, these preset ranges should not be limited to these terms. These terms are only used to distinguish preset ranges from each other. For example, without departing from the scope of the embodiments of the present invention, the first preset range may also be referred to as the second preset range, and similarly, the second preset range may also be referred to as the first preset range.

[0045] The word "if" as used herein may be interpreted as "at the time of" or "when" or "in response to determining" or "in response to detecting", depending on the context. Similarly, the phrases "if it is determined" or "if (stated condition or event) is detected" may be interpreted as "when it is determined" or "in response to determining" or "when detecting (stated condition or event)" or "in response to detecting (stated condition or event)", depending on the context.

[0046] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all of the embodiments. The components of the embodiments of the present invention described and shown in the drawings here can usually be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the drawings is not intended to limit the scope of the claimed invention, but merely represents selected embodiments of the present invention. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0047] Various structural schematic diagrams of the embodiments disclosed in the present invention are shown in the accompanying drawings. These figures are not drawn to scale, and some details are magnified and some details may be omitted for the purpose of clear expression. The shapes of various regions and layers shown in the figures and the relative sizes and positional relationships therebetween are only exemplary, and may deviate in practice due to manufacturing tolerances or technical limitations, and those skilled in the art may additionally design regions / layers with different shapes, sizes, and relative positions according to actual needs.

[0048] As is known, a planar transformer is a transformer with high frequency, low profile, small height and high operating frequency. The following is a detailed introduction to the planar transformer: A planar transformer is an ultra-thin, low-height flat transformer, mainly used for voltage conversion. Its winding is generally composed of copper foil on a printed circuit board or copper wires directly deposited on a magnetic film.

[0049] Embodiment 1

[0050] The design method of the planar transformer for realizing arbitrary non-integer turns ratio according to the present invention comprises:

[0051] Determine the turns ratio of the planar transformer to be designed;

[0052] Wind N around magnetic column 1 p1 The primary winding is wound around the magnetic column 2. p2 The primary winding is wound around the magnetic column 3. s The secondary winding has magnetic columns 1, 2 and 3, and the magnetic resistance of magnetic column 1 is R. g , the magnetic resistance of magnetic column 2 and magnetic column 3 are xR g and yR g ;

[0053] Determine the magnetic resistance ratio x of magnetic column 2 to magnetic column 1 and the magnetic resistance ratio y of magnetic column 3 to magnetic column 1 according to the turns ratio to be designed of the planar transformer;

[0054] The planar transformer is designed based on the magnetic resistance ratio x of the magnetic column 2 to the magnetic column 1 and the magnetic resistance ratio y of the magnetic column 3 to the magnetic column 1.

[0055] Embodiment 2

[0056] In the embodiment, V in =30V, V o =12V, P=120W, L m =2.4μH The turns ratio is 2.5:1:1 as a parameter to design any non-integer turns ratio transformer.

[0057] refer to Figure 1a , wind N around magnetic column 1 p1 The primary winding is wound around the magnetic column 2. p2 turns of the primary winding, and N s The secondary winding is wound around the magnetic column 3. Assume that the magnetic resistance of magnetic column 1 is R g , the magnetic resistance of column 2 is xR g , the magnetic resistance of column 3 is yR g The equivalent magnetic circuit of the transformer is as follows: Figure 1b As shown, by the superposition theorem, the magnetic flux φ1 of magnetic column 1, the magnetic flux φ2 of magnetic column 2 and the magnetic flux φ3 of magnetic column 3 are calculated as follows:

[0058]

[0059]

[0060]

[0061] The turns ratio of the transformer is

[0062] Since the turns ratio is 2.5:1:1, let N p1 =3,N p2 =1,N s =1, equivalent turns ratio Therefore, by adjusting the parameters x and y, x>0, y>0, that is, the magnetic resistance of the middle magnetic column and the right magnetic column, any ratio greater than or equal to 2 can be achieved. Similarly, for the remaining N p1 and N p2 A combination of can achieve a transformation ratio greater than or equal to N.

[0063] refer to Figure 2a and Figure 2b ,because Figure 1aThe middle magnetic column 1, magnetic column 2 and magnetic column 3 all have air gaps, which results in some defects in manufacturing. Therefore, the air gap of the right magnetic column is eliminated, and the secondary winding is transferred to the two primary windings, resulting in the transformer structure and equivalent magnetic circuit shown in Figure 2. This is equivalent to: Let y = 0, so the turns ratio is For better EMI performance and mechanical performance, Figure 2a The transformer core side column is divided into two parts, using an EI-like structure. Figure 3 As shown, since the equivalent magnetic circuit does not change, the turns ratio remains unchanged.

[0064] Since the transformer turns ratio to be designed is 2.5:1:1, n=2.5 is set, and x=1 is obtained, that is, when the magnetic resistance of the magnetic column 1 is equal to that of the magnetic column 2, the transformer turns ratio is 2.5:1:1.

[0065] The ratio of the lengths of the air gaps of the two magnetic columns and the specific lengths are obtained by the magnetic flux of each magnetic column and the value of the excitation inductance required by the transformer. In this embodiment, the air gap length of magnetic column 1 is gap1 = 0.57 mm, and the air gap length of magnetic column 2 is gap2 = 0.19 mm, that is, At the same time, the model parameters are optimized according to the loss, and the 3D model of the required magnetic core and winding is obtained. Figure 4 .

[0066] The transformer is simulated in the finite element software Ansys, and the following is obtained: Figure 5 The magnetic flux density and current density shown in the figure meet the design requirements. At the same time, a transformer with the same turns ratio is designed using the traditional method. The results are as follows Figure 6 . Referring to Table 1, the results of the two are compared. It can be concluded from Table 1 that when the magnetic flux density and the board area are similar, the design loss of the present invention is about 50% of that of the traditional transformer, while reducing the number of PCB board layers and reducing costs. At the same time, the turns ratio of the traditional transformer is controlled by the relationship of the number of turns. However, this relationship is affected by non-ideal factors when designing the winding distribution, which will cause the turns ratio to fluctuate and be difficult to adjust. The present invention can fine-tune the turns ratio by adjusting the air gap and then adjusting the magnetic resistance ratio to achieve an ideal state.

[0067] Table 1

[0068] Any turns ratio Traditional methods PCB Layers 4 6 Original secondary turns 4+1+1＝6 5+2+2＝9 Final turns ratio 2.5 2.543 Exhibition board area 33*19*4mm 31.5*18*4mm Winding loss 1.1142W 2.6203W Core loss 0.30267W 0.32349W Total loss 1.41687W 2.94379W

[0069] Embodiment 3

[0070] The planar transformer described in this embodiment is designed based on the design method for realizing a planar transformer with arbitrary non-integer turns ratio.

[0071] The planar transformer of the present invention is further improved in that:

[0072] Furthermore, the turns ratio n to be designed of the planar transformer is:

[0073]

[0074] Among them, φ1 is the magnetic flux of magnetic column 1, φ2 is the magnetic flux of magnetic column 2, and φ3 is the magnetic flux of magnetic column 3.

[0075] The present invention has the following characteristics:

[0076] When traditional planar transformers achieve the same turns ratio by increasing the number of turns exponentially, they will bring about great winding losses under high current applications and greatly increase the transformer footprint. To this end, the present invention reduces winding losses by reducing the total number of winding turns, reduces the volume of the magnetic core by reducing the magnetic flux density, and enhances the heat dissipation performance. Taking a 1.8:1 transformer as an example, the minimum required design of a traditional transformer is 9:5, that is, 9 turns on the primary side, 5 turns on the secondary side, and the final total number of turns is 14 turns. At the same time, the current density will greatly increase the transformer's footprint and the number of PCB layers. Using the design method of the present invention, the total number of turns is 4 turns, the winding length is greatly reduced, and the winding loss is reduced. At the same time, since the primary magnetic flux flows through two magnetic columns, the magnetic flux density is relatively reduced, the size of the magnetic core can be reduced, and the transformer footprint can be reduced.

[0077] In addition, compared with the traditional method that cannot fine-tune the turns ratio after board making, the design method of the present invention can fine-tune the turns ratio by changing the air gap ratio, further reducing the impact of simulation errors and improving circuit efficiency.

[0078] Embodiment 4

[0079] A computer device comprises a memory, a processor and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, the steps of the method for designing a planar transformer with an arbitrary non-integer turns ratio are implemented, for example, including: determining the turns ratio to be designed of the planar transformer; winding N on a magnetic column 1; p1 The primary winding is wound around the magnetic column 2. p2 The primary winding is wound around the magnetic column 3. s The secondary winding has magnetic columns 1, 2 and 3, and the magnetic resistance of magnetic column 1 is R. g , the magnetic resistance of magnetic column 2 and magnetic column 3 are xR g and yR g; Determine the magnetic resistance ratio x of magnetic column 2 to magnetic column 1 and the magnetic resistance ratio y of magnetic column 3 to magnetic column 1 according to the turn ratio to be designed of the planar transformer; design the planar transformer according to the magnetic resistance ratio x of magnetic column 2 to magnetic column 1 and the magnetic resistance ratio y of magnetic column 3 to magnetic column 1. Wherein, the memory may include a memory, such as a high-speed random access memory, and may also include a non-volatile memory, such as at least one disk memory, etc.; the processor, the network interface, and the memory are interconnected through an internal bus, and the internal bus may be an industrial standard architecture bus, a peripheral component interconnection standard bus, an extended industrial standard architecture bus, etc. The bus can be divided into an address bus, a data bus, a control bus, etc. The memory is used to store programs, and specifically, the program may include a program code, and the program code includes computer operation instructions. The memory may include a memory and a non-volatile memory, and provide instructions and data to the processor.

[0080] Embodiment 5

[0081] A computer-readable storage medium stores a computer program, and when the computer program is executed by a processor, the steps of the method for designing a planar transformer with an arbitrary non-integer turns ratio are implemented, for example, including: determining the turns ratio to be designed of the planar transformer; winding N on a magnetic column 1; p1 The primary winding is wound around the magnetic column 2. p2 The primary winding is wound around the magnetic column 3. s The secondary winding has magnetic columns 1, 2 and 3, and the magnetic resistance of magnetic column 1 is R. g , the magnetic resistance of magnetic column 2 and magnetic column 3 are xR g and yR g ; Determine the magnetic resistance ratio x of magnetic column 2 to magnetic column 1 and the magnetic resistance ratio y of magnetic column 3 to magnetic column 1 according to the turns ratio to be designed of the planar transformer; Design the planar transformer according to the magnetic resistance ratio x of magnetic column 2 to magnetic column 1 and the magnetic resistance ratio y of magnetic column 3 to magnetic column 1. Specifically, the computer-readable storage medium includes, but is not limited to, for example, volatile memory and / or non-volatile memory. The volatile memory may include random access memory (RAM) and / or cache memory (cache), etc. The non-volatile memory may include read-only memory (ROM), hard disk, flash memory, optical disk, magnetic disk, etc.

[0082] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment in combination with software and hardware. Moreover, the present application may adopt the form of a computer program product implemented in one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that include computer-usable program code.

[0083] The present application is described with reference to the flowchart and / or block diagram of the method, device (system) and computer program product according to the embodiment of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, and the combination of the process and / or box in the flowchart and / or block diagram can be realized by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device produce a device for realizing the function specified in one process or multiple processes in the flowchart and / or one box or multiple boxes in the block diagram.

[0084] These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to work in a specific manner, so that the instructions stored in the computer-readable memory produce a manufactured product including an instruction device that implements the functions specified in one or more processes in the flowchart and / or one or more boxes in the block diagram.

[0085] These computer program instructions may also be loaded onto a computer or other programmable data processing device so that a series of operational steps are executed on the computer or other programmable device to produce a computer-implemented process, whereby the instructions executed on the computer or other programmable device provide steps for implementing the functions specified in one or more processes in the flowchart and / or one or more boxes in the block diagram.

[0086] Those skilled in the art will readily appreciate other embodiments of the present invention after considering the specification and disclosure of the invention. This application is intended to cover any variations, uses or adaptations of the present invention that follow the general principles of the present invention and include common knowledge or customary techniques in the art that are not disclosed by the present invention. The specification and examples are to be considered exemplary only, and the true scope and spirit of the present invention are indicated by the following claims.

[0087] It should be understood that the present invention is not limited to the exact construction that has been described above and shown in the drawings and that various modifications and changes may be made without departing from the scope thereof. The scope of the present invention is limited only by the appended claims.

[0088] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any way. Any simple modification, change and equivalent structural change made to the above embodiment based on the technical essence of the present invention still falls within the protection scope of the technical solution of the present invention.

Claims

1. A design method for realizing a planar transformer with arbitrary non-integer turns ratio, characterized in that: include: Determine the turns ratio of the planar transformer to be designed; Wind N around magnetic column 1 p1 The primary winding is wound around the magnetic column 2. p2 The primary winding is wound around the magnetic column 3. s The secondary winding has magnetic columns 1, 2 and 3, and the magnetic resistance of magnetic column 1 is R. g , the magnetic resistance of magnetic column 2 and magnetic column 3 are xR g and yR g ; Determine the magnetic resistance ratio x of magnetic column 2 to magnetic column 1 and the magnetic resistance ratio y of magnetic column 3 to magnetic column 1 according to the turns ratio to be designed of the planar transformer; The planar transformer is designed based on the magnetic resistance ratio x of the magnetic column 2 to the magnetic column 1 and the magnetic resistance ratio y of the magnetic column 3 to the magnetic column 1.

2. The design method for realizing an arbitrary non-integer turns ratio planar transformer according to claim 1, characterized in that: The turn ratio n to be designed for the planar transformer is: Among them, φ1 is the magnetic flux of magnetic column 1, φ2 is the magnetic flux of magnetic column 2, and φ3 is the magnetic flux of magnetic column 3.

3. The design method for realizing an arbitrary non-integer turns ratio planar transformer according to claim 2, characterized in that: The magnetic flux φ1 of magnetic column 1 is:

4. The design method for realizing an arbitrary non-integer turns ratio planar transformer according to claim 2, characterized in that: The magnetic flux φ2 of magnetic column 2 is:

5. The design method for realizing an arbitrary non-integer turns ratio planar transformer according to claim 2, characterized in that: The magnetic flux φ3 of magnetic column 3 is:

6. The design method for realizing an arbitrary non-integer turns ratio planar transformer according to claim 1, characterized in that: The magnetic resistance of the magnetic column 1 is adjusted by adjusting the area of ​​the magnetic column 1 and the air gap length.

7. The design method for realizing an arbitrary non-integer turns ratio planar transformer according to claim 1, characterized in that: The magnetic resistance of the magnetic column 2 can be adjusted by adjusting the area of ​​the magnetic column 2 and the air gap length.

8. The design method for realizing an arbitrary non-integer turns ratio planar transformer according to claim 1, characterized in that: The magnetic resistance of the magnetic column 3 is adjusted by adjusting the area of ​​the magnetic column 3 and the air gap length.

9. A planar transformer, characterized in that: The design is based on the design method for realizing arbitrary non-integer turns ratio planar transformer as described in claim 1.

10. The planar transformer according to claim 9, characterized in that: The turn ratio n to be designed for the planar transformer is: Among them, φ1 is the magnetic flux of magnetic column 1, φ2 is the magnetic flux of magnetic column 2, and φ3 is the magnetic flux of magnetic column 3.

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