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Energy and Power Engineering, 2013, 5, 1101-1104 doi:10.4236/epe.2013.54B210 Published Online July 2013 (http://www.scirp.org/journal/epe) Calculation and Design of Dry-type Air-core Reactor Yan Li1, Zhenh ai Zhang1, Longnv Li1,Guoli Li1, Manhua Jiang2 1Research Institute of Special Electrical Machines, Shenyang University of Technology, Shenyang, China 2Tebian Electric Apparatus Shenyang Transformer Group Co., Ltd., Shenyang, China Email: hailong198759@126.com Received March, 2013 ABSTRACT Based on the method of compoun d and additional cond itions under the conditio ns of the equal temperature rise and the equal potential drop (P.D.) of resistance, the application of design software of dry-type air-core reactor is introduced in this thesis. The analytical methods of th e inductance are also given. This approach is proved entirely feasible in theory through the simplification with Bartky transformation, and is able to quickly and accurately calculate reactor inductance. This pape r p r esents the analytica l methods of the loss of dry-type air-core reactor as well. Keywords: Dry-type Air-core Reactor; Bartky Transformation; Compound and Additional Conditions; Software Design 1. Introduction With the advantages of light weight, good linearity, high mechanical strength, low noise and easy maintenance, dry-type air-core reactor is a equipment widely used for the current limit, voltage regulator and reactive power compensation in power system [1]. The factors, such as inductance, temperature rise, height and others, should be considered comprehensively in the design of dry-type air-core reactor. The mutual affection and restraint of various factors make the design of dry-type air-core reactor more complicated. In order to simplify the design and improve the efficiency, it is nec- essary to take appropriate simplification and process. The design adopts the simplified processing mathematical model which is based on the additional constraints. And this paper developed the design software of dry-type air-core reactor. 2. Calculation Methods In the premise of known voltage, capacity, radius and height cond itions, th en the desig n of the dry-typ e air-cor e reactor could obtain parameters of turns, inductance, temperature rise, current, etc. by using relevant theorems and formulas. It is difficult to directly calculate the out- come. Because of the shortage of given conditions. Thus it’s necessary to simplify it without affecting its accuracy. The encapsulating width of the reactor is so far smaller than the coil height that the reactor is considered as the ideal solenoid in the calculation of the inductor. The re- sistance is much smaller than the inductance of the reac- tor, so that wire could be considered as an ideal conduc- tor, which means we can ignore the affection of resis- tance. To make the design to meet certain characteristics, such as making the reactor have a minimum temperature rise, minimum supplies, minimum loss, simple structure and convenient manufacturing, we can use the additional constraints of equivalent temperature rise, height and resistance drop method, which could not only make the design more reasonable and achieve structure optimiza- tion, but also reduce the computing time and improve efficiency. This article uses additional constraints of en- trapment between temperature rise, layer winding resis- tance drop to achieve minimal supplies, losses and tem- perature rise of reactor. 2.1. The Calculation of the Self and Mutual Inductances The inductance of the reactor is the most important pa- rameter in the dry-type air-core reactor and its calcula- tion accuracy directly affects the accuracy of the other parameters of the reactor, and then affects the perform- ance of the reactor. According to the reference [2], the mutual inductance of any two coils is : 32 01212121 121 123 124 M 2 (R R)n n[(,,) (, ,)(, ,)(, ,) i iii CRR Z CRRZ CRRZCRRZ (1) 12 12 22 012 12 222 2 12 12 1 (, ,)22cos 2cossin i RR CRR ZRR RR RRZ RRd (2) Copyright © 2013 SciRes. EPE Y. LI ET AL. 1102 12 1 12 2 12 3 12 4 22 22 22 22 H H ZS H H ZS H H ZS H H ZS (3) where 1、2 nis the number of turns of the two windings per unit length，1 R、2 R, respectively are the radius、 the distance、the height of the two windings and nS presents the horizontal angle. From the above equation, 12 is only related with the radius, height, and relative position of two windings, but regardless of the number of turns of the windings. After analysis, we can know that 12 is a elliptic integral and the computation of its direct in- tegral is complex and not easy to achieve. But it could be simplified and then be solved with the Applied Numeri- cal Solution according to the Bartky transformation me- thod proposed in reference [3]. After using simplifica- tion and the Bartky Transformation, the expression is: (, ,) i CRRZ (, ,) i CRRZ 222 0 2 222 22 2 2 (,,,,)[sincos sin ] cos sincos sin B CABCqkA B d Cqk (4) where 2 12 12 12 2 2 22 22 12 22 12 , 0<1; ; 4 1 1; 3(1 )31 () ; 1; () [] RR z qq RR RR k Ak B kk RR Z kCk RR Z ; when 1, namely q = 0, the parameters need to be adjusted to 2 RR A AC; ; . BBC 0C By the Bartky transformation method, the complicated elliptic integral could be simplified to be easily solved with a more accurate calculation. 2.2. The Calculation of the Encapsulation 1112 2111 112 2 112 2 .................... ..... .... ................... mm N ii iiimmii mmmm mmN Suppose the number of encapsulations in reactor is m, and regard each encapsulation as a coil. Set as the self-inductance of the ith encapsulation, ij i L M as the mutual inductance of the ith and jth encapsulation, i as the resistance of the ith encapsulation and i R I as AC current of the ith encapsulation and N U as the voltage of both ends of reactor. With the basic principle of the circuit, it is possible to calculate the equivalent height, turns number and current of the encapsulation, and then obtain other parameters. Under general conditions, the resistance is much smaller than the inductor. In the case of ignoring encap- sulation resistor, the equation can be equivalent to the following. 1(1,2,3,) mN ijijj j U WW f Iim (6) when solving the number of turns, equation 6 is a nonli- near equation. It is difficult to directly calculate, but we can take advantage of some simplifications under certain constraints. Under the conditions of the equivalent tem- perature rise, we can get the following equation [4]. 1 11 () m ijij j i mm kjkj kj aaf N I I aaf (7) 11 2 1 11 1 () () 1 () mm kjkj kj N im Njij j mm N kjkj mkj jij j aaf U WIaf Laaf af (8) 2 11 max 0.36 ()()( mm pi N kjkj kj ii kJ S aaf kH ) (9) where N N N U L I is the rated inductance, N S is the rated capacity and max is the maximum allowable temperature rise. N j LIj MIj MIRIU jMI jMI j LIj MIRIU j LIjM IjLIRIU From the above equation, we can see that in some cas- es of ignoring the resistor, nonlinear equations could be converted into linear equations to solve through the addi- tional conditions of the equivalent temperature rise, which can greatly simplify the solution process. Under the condition of knowing the height of dry hollow reactor, encapsulation thickness, the calculated inductance and the maximum temperature of encapsulation, the mini- mum encapsulation number and the turns number of the respective encapsulating and current, which satisfy the (5) Copyright © 2013 SciRes. EPE Y. LI ET AL. 1103 requirement, could be calculated. The temperature rise and loss of the encapsulation can be determined after determining the encapsulation turns and current numbers, and then some parameters, such as AWG and the height and width of encapsulation, could be confirmed. 2.3. Calculation of the Layer Current and the Yurns Number The calculation of the layer current an d the turns number is similar to the calculation of the encapsulation. If we regard each layer as a coil, its layer equations are nonlinear known from the circuit principle. The resistance is so much smaller than the inductance that the pressure drop of the resistance could be ignored. Thus it’s unnecessary to solve nonlinear equations, but directly obtain the num- ber of turns, the current, inductance, thickness and loss parameters of each layer under the constraints of resis- tance drop method. Under the constraints of resistance drop method, each layer current and turns number can be obtained as fol- lowing: 1 11 nci cjij jij iN nn ck cjkj kj kj ss f rr I I ss f rr (10) 11 1 nn ck cj N kj kj kj incj ij jj ss Lf rr Wsf r (11) Where ci s is the conductor sectional area of the ith layer of the coil and i is the radius distance between the ith layer and the axis. r From the above equation, we can see that by the way of ignoring the resistor and through the additional condi- tions of the temperature rise, nonlinear equations could be converted into linear equations to solve, which can greatly simplify the solution process. Under the condition of knowing AWG, radius, calculate the inductance and rated current, the turns number and current of each layer, which satisfy the requirement, could be calculated. Then other parameters can be obtained, such as the resistor, the current density, inductance of each layer. 3. Software Design The idea for design is as following: According to the known basic parameters of known, Bartky transformation is applied to solve the mutual inductance coefficient ij f . Under the constraint condition of the encapsulation equivalent temperature rise, we can get the encapsulation number meeting the minimum encapsulating condition according to the equation 8.Then the turn number and current of each encapsulation could be obtained on the basis of the equation 7 and 8. Under the comprehensive constraint condition such as AWG, height, ampere den- sity, encapsulation thickness and equal resistance drop method, the turn number and current of every layer could be obtained by the formula 10, 11. And then the wire gauge, axial and radial parallel conductors, encapsulation height and current density can be confirmed. In the computation of the wire gauge, the current and the turn number of each layer as well as current density, the correct parameters must be chosen to iterate compu- tation and the two results are also supposed to be com- pared to confirm a reasonable error range for a accurate calculation result. If using the wire gauge of the conduc- tor as iteration parameters, the two results would mutate because of the inconsecutive change of the wire gauge of the conductor, thus there is no guarantee for the conver- gence of the wire gauge. Therefore, the wire gauge should not be chosen as iteration parameter. If choose the height or thickness of each layer whose change is not obvious in Iteration calculations, the change of the pa- rameters could not be accurately determined, and so the precision of calculation was poor. Consequently, the height or thickness of each layer should not be also cho- sen as iteration parameter. The current density of the conductor not only changes continuously but also con- verges well, and its variation in the process of iteration is apparent, thus various parameters can be accurately de- termined. According to the mentioned above, this paper chooses the current density as the iterative parameter. The program flowchart is shown in Figures 1 (a) and (b). 4. Example From the software interface, we can see that the basic parameters such as rated voltage, rated capacity, rated current, winding height and winding diameter need to be input to the interface. The other options can be default values or modified values, which makes the design more flexible, convenient, and applicab le. For example, A dry-type air-core reactor with the fol- lowing parameters: rated voltage is 10 kV, rated current is 400 A, rated capacity is 185 kvar, winding height is 0.9 m, winding diameter is 0.8 m, inductance is 3.7674 mH. The design results are shown in Figure 3. The design results are presented in Figure 3. in detail, which include encapsulation number, encapsulation loss, temperature rise, the number of axial and radial parallel conductors, wire gauge, the thickness, current density, the number of turns, inductance, the turn number of each layer, inductance, current and height. These pa- rameters can fully meet the requirements of a dry-type air-core reactor. Copyright © 2013 SciRes. EPE Y. LI ET AL. Copyright © 2013 SciRes. EPE 1104 B Assume the rad ial p arallel conductor numbe r curent density is too large Yes Adjust the radial number of parallel conductors Curent density is reasonable No: Adjust the initial parameters A Yes Calculate and resul t out p ut end The AWG is too small A Yes: Adjust the height No Figure 3. Output result. 5. Conclusions In this paper, the analytical method is used to calculate the inductance of the dry-type air-core reactor, the Bartky transformation is used to simplify it, and the constraints of the temperature rise method and the resistance drop method are used to optimize it. Users only need to enter the basic parameters the design scheme with high accu- racy and fast calculation can be obtained. The developed design software interface is simple and easy to use. REFERENCES [1] Z. G. Liu, J. H. Wang and W. P. Wang, “Development and Application of Dry-type Air-core Reactor Design Software,” Electric Machines and Control, Vol. 6, No. 7, 2003, pp. 103-106. Figure 1. Program flow chart (b). [2] X. L. Wei and S. Ma, “Analytic Calculation Method for the Magnetic Field of Air-core Power Reactor with Mul- ti-Layer in Parallel,” Transformer, Vol. 2, 1993, pp. 12-15. [3] T. H. Fawzi and P. E. Burke, “The Accurate Computation of Self and Mutual Inductances of Circular Coils”. IEEE Trans Power Syst. Vol. 97, No. 2, 1978, pp. 464-468. [4] Z. Min, “Relevant Issues and Computer Aided Design of Hollow Electric Reactor,” Harbin: Harbin University of Science and Technology, Mar 2003. Figure 2. Software interface diagram. |