Analytical calculation method for self-inductance value of air-core reactor
1. An analytical calculation method for a self-inductance value of an air-core reactor comprises the following steps: the method is characterized in that: calculating the self-inductance parameter of the air reactor by adopting the following calculation steps:
（1）lis the axial length of the air-core reactor,ris the radius of the air-core reactor,
calculated by the following formulaA i ：
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In the formula:
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preferablyTaking 0.005-0.01; sinh is a hyperbolic sine function, arcsin is an inverse hyperbolic sine function;
(2) respectively takei=1、2、3、4……N-1、NThen respectively calculating according to the formula in the step (1)A 1、A 2、A 3、A 4…A N-1、A N WhereinNThe number of turns of the air-core reactor;
(3) calculating the self-inductance of the air-core reactor according to the following formulaL：
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Background
From the electrical model, the hollow solenoid can be regarded as an electrified solenoid with limited axial length, the magnetic field of the electrified solenoid is a symmetrical magnetic field of a central axis, the analytical analysis is carried out on the magnetic field by applying the biot-savart law, an elliptic integral can be obtained, but the elliptic integral can not be represented by an elementary function, so that the magnetic field distribution of the hollow solenoid and the parameter calculation can not be represented by the elementary function for a long time, the self-inductance calculation value of the hollow reactor can only be realized by numerical calculation of a finite element method, the problems of time and labor are solved, and particularly, when the electrical design is carried out in a transformer substation, how to develop an elementary function representation formula of the hollow solenoid is adopted, so that a designer can calculate the magnetic field intensity quickly, and the problem which needs to be solved on site is provided.
Disclosure of Invention
The invention provides an analytical calculation method for a self-inductance value of an air reactor, and aims to develop an elementary function expression formula of an air solenoid and solve the problem of calculation of a corresponding magnetic field of a transformer substation.
The invention solves the technical problems by the following technical scheme:
the general concept of the invention is: the invention develops a new method, and gets around direct magnetic density analysis through ingenious integral transformation to obtain an analytic expression of vector magnetic potential and further obtain an analytic calculation method of a self-inductance value of a vector air reactor to replace a finite element numerical solution.
The vector magnetic bit generated by a conductive object placed in a single medium can be calculated by equation (1) as follows:
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considering a magnetic field generated by a conductive cylinder with radius r placed in the air, neglecting the radial thickness for simplicity, the vector magnetic potential element generated by the current element can be obtained according to the above formula as a calculation formula (2):
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it is easy to know that the directions of the generated vector magnetic bits in the whole space are annular tangential directions concentric with the circular ring, and because of the central symmetry, it is known that the magnetic field distribution on any plane passing through the center of the circular ring is the same, therefore, one of the planes is taken, a rz coordinate system is established in the plane, the vector magnetic bits in the plane are all normal directions, the vector magnetic bit infinitesimal generated by any current infinitesimal on any point P on any section perpendicular to the central axis is considered, as shown in fig. 1, the point P' in the figure is the projection of the point P on the section, it is easy to know that the sum of the radial components of the vector magnetic bits generated by the whole current circumference at the point P is zero, so only the tangential component is considered, and the tangential component of the vector magnetic bit infinitesimal generated by the current infinitesimal at the point P shown in fig. 1 can be calculated by the following formula (3):
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whereinThe physical meaning of (c) is shown in fig. 1, and the following formula (4) can be obtained from fig. 1 and the cosine theorem:
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the calculation formula (5) of the vector magnetic potential is:
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the formula can not obtain the analysis result, however, the integrated quantity is skillfully converted into the integrated quantity, and the analysis result can be obtained; as can be seen from fig. 2, in the right triangle, the following formula (6) is calculated:
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FIG. 2 is to be integratedConverted to an integrated quantityThe following calculation formula (7) is obtained;
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thus, the calculation formula (8) is obtained,
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thus, the calculated formula (9):
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when in useDue toSince the higher order of yes is infinitesimal but the same order is infinitesimal, the calculation formula (9) becomes the following calculation formula (10):
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calculation formula (11) can be obtained by substituting calculation formula (10) into calculation formula (3) and by combining calculation formula (4):
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for the formula (11), the sum is calculatedlBy performing integration and performing a complicated integration operation, formula (12) can be obtained:
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in the formula:
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because the vector magnetic potential is in the tangential direction and the vector magnetic potential module values of all points on any concentric ring are equal, the double integral calculation of the magnetic flux can be converted into simple product calculation by applying the Stokes theorem, and the calculation difficulty is greatly simplified. According to equation (12), the flux linked with a certain turn of coil in the reactor is calculated by using stokes law as equation (13):
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wherein, the ordinate of the position of the turn coil is used to calculate the inductance value of the reactor as the calculation formula (14):
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the foregoing is the principle of the invention. The specific technical scheme provided by the invention is as follows:
calculating the self-inductance parameter of the air reactor by adopting the following calculation steps:
lis the axial length of the air-core reactor,ris the radius of the center line of the air core reactor coil as shown in figure 1. Calculated by the following formulaA i ：
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In the formula:
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preferablyTaking 0.005-0.01; sinh is a hyperbolic sine function, arcsin is an inverse hyperbolic sine function;
respectively takei=1、2、3、4……N-1、NThen respectively calculating according to the formula in the step (1)A 1、A 2、A 3、A 4…A N-1、A N WhereinNThe number of turns of the air-core reactor;
calculating the self-inductance of the air-core reactor according to the following formulaL：
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The above formulas all adopt the international system of units.
And determining the inductance parameter of the air solenoid through an analytical expression according to the size parameter and the winding parameter of the air solenoid. The invention has the advantages of saving calculation time and calculation resources.
Drawings
FIG. 1 is a vector magnetic potential infinitesimal generated by any current infinitesimal at any point P on any cross section of an electrical cylinder perpendicular to a central axis;
FIG. 2 is a vector magnetic bitmap generated at point P for the entire current circle of the present invention;
fig. 3 is a sectional view of an air-core reactor of the present invention.
Detailed Description
The invention is described in detail below with reference to the accompanying drawings:
for an air-core reactor with the radius of 1.108m, the axial length of 2.47m and the number of turns of 560, the self-inductance is calculated by the calculation method provided by the invention, and thenr=1.108，l=2.47，N=560, substitute for calculation step of the invention, calculate self-inductanceL=0.436H。
