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Structural Parameter Optimization of Inductors Used in Inductive Pulse Power Supply Zhen Li, Member, IEEE, Xinjie Yu, Member, IEEE, Shangang Ma, and Yuzhou Sha
Abstract— Structural parameters directly affect the energy density and the total resistance of the inductors used in inductive pulse power supply (IPPS). This paper elaborates the relationship between the structural parameters and electrical performances of the inductors used in IPPS and set up the inductors for the experiment in slow transfer of energy through capacitive hybrid meat grinder circuit using air-core flat spirals of strip coil. Then we compile a program that can obtain the structural parameters of the inductors that have the highest energy density under the condition of given charging and discharging currents as well as inductances in the primary and secondary sides. The calculation results of the program are analyzed and verified by the measured values of the real inductors. At the end, on the condition of a constant current density, we put forward a rule that along with the increase in the system energy, the energy density of the IPPS became higher and the resistance of the inductors became lower. Index Terms— Inductive pulse power supply (IPPS), inductor, optimization, structural parameters.
I. I NTRODUCTION
I
NDUCTIVE pulse power supply (IPPS) has recently become the focus of research due to the facts that inductor has a relatively high energy density and is easy to be cooled, and so on. Some institutes such as Institute for Advanced Technology (IAT) at the University of Texas and German–French Research Institute of Saint Louis have made great progress in the designing, analysis, and implementation of IPPS [1]–[3]. Inductors are the core elements in IPPS, of which energy density and resistance are two key factors in the designing, analysis, and implementation of the IPPS. Energy density is related to the inductance, charging current, and volume of the inductors. When inductance and charging current are given, energy density is related to the volume, which is determined by the structural parameters. Resistance is related to the conductor length and section area, which is also determined by the structural parameters. Therefore, the structural parameters of the inductors affect the two key Manuscript received October 11, 2014; accepted January 11, 2015. Date of publication March 12, 2015; date of current version May 6, 2015. This work was supported in part by the Tsinghua University Initiative Scientific Research Program under Grant 20121087927 and in part by the National Natural Science Foundation of China under Project 51377087. Z. Li, X. Yu, and S. Ma are with the State Key Laboratory of Power System, Department of Electrical Engineering, Tsinghua University, Beijing 100084, China (e-mail:
[email protected];
[email protected];
[email protected]). Y. Sha is with Dalian No.1 Instrument Transformer Company, Ltd., Dalian 116200, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPS.2015.2409062
factors: 1) the energy density and 2) the resistance and need to be studied further. In the slow transfer of energy through capacitive hybrid (STRETCH) meat grinder circuit, which is proposed by IAT, as in [4], the inductors are made up of a Brooks coil. A Brooks coil is a special structure that has the maximum inductance with a given length of a chosen wire [5]. Because the Brooks coil structural parameters have constant proportion, we cannot flexibly modify all the structural parameters that affect the energy density and the resistance. The Brooks coil can be regarded as a special type of air-core flat spirals of strip coil (AFSSC), which has the advantages of high coupling factor and flexible structure. However, in the published literature, there are no discussions on how to design the structural parameters to attain the highest energy density. This paper takes the structural parameters of the inductors in the STRETCH meat grinder circuit as the optimization object. On the condition of given charging current (I1 ) and discharging current (I2 ) as well as given primary side inductance (L 1 ) and secondary side inductance (L 2 ), this paper compiles a program that can use the inner diameter, the outer diameter, and the number of turns as the independent variables to find the inductors’ minimum volume, which, furthermore, can be used to implement the maximum energy density and the minimum total resistance of L 1 and L 2 in series connection. Of course, the optimized inductors can be used not only in the STRETCH meat grinder circuit, but also in other IPPS with a pulse-compression circuit. Section II introduces the structure and parameters of the inductors, Section III describes the design idea of the program, Section IV discusses the key links in the optimization approaches, Section V shows the experimental verification and result analysis of the program, and Section VI gives the optimized structural parameters of the inductors in 1- and 25-kJ IPPS systems, respectively, and discusses the rule of how the energy density and total resistance change along with the increase in the system energy. II. S TRUCTURE AND PARAMETERS OF THE I NDUCTORS A. Structure of the Inductors The basic unit of the inductors is a sheet of AFSSC shown in Fig. 1. The inductors are constructed by sheets of AFSSCs with series or parallel connection. Every sheet of AFSSC is made up of aluminum plate with a thickness of 5.8 mm and is processed by wire cutting. The structural variable definition is also given in Fig. 1.
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LI et al.: STRUCTURAL PARAMETER OPTIMIZATION OF INDUCTORS USED IN IPPS
Fig. 1.
One sheet of AFSSC.
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Fig. 3. Real inductors used in STRETCH meat grinder with ICCOS circuit experiment.
four parallel connections in the basic module of L 2 . Both modules of L 1 and L 2 are connected seven times in series. According to these connections, the r _l and r _i are both four. We need to note that only the outside connections can be seen, and the invisible inside connections are symmetrical with the outside ones. B. Parameters of Inductors Fig. 2.
STRETCH meat grinder circuit [4].
In the STRETCH meat grinder circuit, as shown in Fig. 2, L 1 should be larger than L 2 to get a large multiplication factor of I2 to I1 . That is, L 1 should be implemented with more AFSSC basic units in series than L 2 . Considering the high coupling factor of AFSSCs, with two units connected in series, the inductance becomes four times than the one unit; three units in series will contribute nine times than the one unit, and so on. Therefore, if L 1 and L 2 use AFSSC as the basic unit, the ratio of L 1 to L 2 , i.e., r _l, should be a square number, such as 4, 9, 16, and so on. Because the discharging current I2 is larger than the charging current, we can set the ratio of I2 to I1 to be r _i . Supposing that L 1 and L 2 have the same current density, L 2 should have r _i AFSSCs in parallel to bear I2 , if L 1 uses one AFSSC to bear I1 . In practice, L 1 and L 2 are made up of basic modules separately. The basic module of L 1 is connected in series by sheets of AFSSCs, and the basic module of L 2 is connected in parallel by sheets of AFSSCs. In a basic module of L 1 , the number of the series connection of AFSSCs is the root of r _l; and in a basic module of L 2 , the number of the parallel connection of AFSSCs is r _i . In this way, when a basic module of L 1 is connected in series with a basic module of L 2 , the ratio of the inductance is r _l, and the ratio of maximum current is r _i . If the inductance of one basic module of L 1 and L 2 cannot meet the given L 1 and L 2 , then we can add another two basic modules, L 1 and L 2 , connected in series separately. This procedure can be processed until the given L 1 and L 2 have been met. Fig. 3 shows the structure of the real inductors used in the STRETCH meat grinder with Inverse Current Commutation with Semiconductor (ICCOS) devices circuit experiment [6], [7]. There are 42 sheets of AFSSCs in total. The sheets of inductors are sprayed with insulating paint in Fig. 3. In Fig. 3, there are two series connections in the basic module of L 1 , and correspondingly, there are
The parameters of the inductors can be divided into two groups, i.e., the structural parameters and the electrical parameters. Parts of the electrical parameters can be derived from the structural ones. The structural parameters include the inner diameter d, the outer diameter D, the number of turns N, the height of a sheet of AFSSC h, the width of a turn w, the distance between two adjacent turns d_turns, the distance between two adjacent sheets d_sheets, and the height and the volume of the whole inductors h_total and V , respectively, where the former five parameters have been marked in Fig. 1. The above mentioned parameters are not all independent variables, such as w can be derived from the given d, D, N, and d_turns. Likewise, the section area of one turn in Fig. 1 can be derived from the given current I and a proper current density J , with which we can get h. In general, we can find w and h if d, D, N, d_turn, I , and J are given. h_total can be obtained from h and d_sheets which is decided by the thickness of the insulation between two layers of the AFSSCs. Finally, we can use the following to calculate V : π D2 · h_total. (1) 4 It should be pointed out that V contains the volume of the air region inside the inductors. Later, we will use V to calculate the energy density. Therefore, the calculated energy density is rather conservative. The electrical parameters include the inductance and the resistance of a sheet of AFSSC L_single and R_single, respectively, the inductance of the primary and the secondary inductor L 1 and L 2 , respectively, the total resistance of L 1 and L 2 connected in series R_total, the current density J , and the coupling factor k. After determining the structural parameters of a sheet of AFSSC, we can use the formulas of inductance of flat spirals of strip coil and mutual inductance between two same flat coils in [8] to calculate L_single and the mutual inductance between two coaxial AFSSCs with arbitrary distance. On this basis, the inductance of a module V =
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of L 1 or L 2 was calculated along with and the inductance of the modules in series connection using the method introduced in Section IV. The parameters J and k can be assigned with previous experiences as in Sections IV and VI, respectively. R_single is calculated by the resistance formula, and then R_total can be derived from the series and parallel connections of R_single. The calculations of V and R_total just mentioned are the key parts of the program introduced below. III. D ESIGN I DEA OF THE P ROGRAM The design idea of the inductors aims at calculating d, D, and N of the inductors which have the minimum volume V in terms of the given I1 , I2 , L 1 , and L 2 (other useful parameters J , k, d_turns, and d_sheets are assigned by analyses or experiences). After that, the corresponding energy density and R_total can be obtained. To attain the expected aim, we can compile a program to optimize d, D, and N of the inductors. In the program, the parameters to be optimized, i.e., d, D, and N, are given a range separately first. And multiple evenly distributed samples between the ranges are selected. Second, for a set of the sampling parameters d, D, and N, L 1 and L 2 are calculated using d, D, N, I1 , I2 , J , d_turns, and d_sheets. If the calculated L 1 and L 2 exceed 5% of the given values, the program should be stopped, and V is given a punishing value (a relatively large number); if the calculated L 1 and L 2 drop in the ±5% range of the given values, the specific d, D, and N are saved as feasible numbers, and the corresponding V and R_total are calculated too. The program goes on looping until all the d, D, and N samples in their given ranges are used. After that, we can get two sets of values V and R_total, respectively, with the corresponding d, D, and N. Finally, we find the minimum V as the optimized result and record the corresponding d, D, N, and R_total. Fig. 4 shows the flowchart of program. To simplify the calculations, we assume that the charging current I1 flows through AFSSCs of L 1 in series connection, i.e., there are no parallel AFSSCs in L 1 . At the same time, for the purposes of making the optimized structural parameters with practical significance, we give a number of constraints such as minimum width of a turn and so on. IV. R ELATED K EY D ISCUSSION A. Current Density Suppose the current is given; the current density determines the section area, which further affects two important parameters: the energy density and the resistance. If the current density is set to be low, it will lead to a large section area and a large volume, so the energy density is low and the resistance is small. Otherwise, if the current density is set to be high, it will lead to a small section area and a small volume, so the energy density is high and the resistance is large. There is a contradiction between the energy density and the resistance. Therefore, we need a trade-off current density to resolve. In practice, we need the energy density to be as
Fig. 4.
Flowchart of the program.
high as possible on the promise of the acceptable resistance, i.e., we need to choose a relative high current density. In the STRETCH meat grinder circuit, as shown in Fig. 2, when the opening switch is triggered to open, the pulse of the current will maintain about 10 ms [4]. Supposing that the energy loss in the inductors is consumed in the resistors of the inductors via the heat loss and to get a high safety margin for the energy rise, we assume that the highest allowed temperature rise of the inductors is 10 °C. Because both the time difference and the temperature rise are macroscopic normal values, the following can be used: (2) Q = I 2 · R · t Q = C · M · T (3) M (4) ρ = l·S l (5) R = ρr S where Q is the heat loss in the resistors, C is the specific heat capacity, M is the mass, T is the temperature rise, I is the current, R is the resistance, t is the time difference,
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where
⎛ ⎞ 1 ⎜1⎟ ⎜ ⎟ ⎜ ⎟ B = ⎜1⎟ ⎜ .. ⎟ ⎝.⎠ ⎛
Fig. 5.
Schematic of n parallel AFSSCs.
ρ is the mass density, l is the length, S is the section area, and ρr is the resistivity. For the aluminum materials, C = 905 J/kg · °C, ρ = 2.7 × 103 kg/m3, and ρr = 2.82 × 10−8 · m. Solving (2)–(5) simultaneously, where Q, M, R, and l are eliminated as intermediate variables, we can find that the current density J = I /S is 294 A/mm2 . That means, for the aluminum materials, if J is taken as 294 A/mm2 , the temperature will rise 10 °C after triggering once in the STRETCH meat grinder circuit. Because we take a high safety margin before and it is not possible that all the energy stored in the inductors is used for the heat loss of the resistors, we take J as 300 A/mm2 in the program. If we pay more attention to the resistance than energy density, we can take a smaller J ; otherwise, we can take a larger one. B. Inductance Calculations of n AFSSCs in Series and Parallel The basic modules of L 1 and L 2 contain several AFSSCs in series and parallel connections, respectively. Therefore, after getting L_single and the mutual inductance between two coaxial AFSSCs with arbitrary distance, the induction L_s of n AFSSCs in series connection and the induction L_ p of n AFSSCs in parallel connection need to be calculated. Fig. 5 shows a schematic of n same AFSSCs connected in parallel. Because L 1 –L n are the same in values and are connected coaxially, there are n − 1 mutual inductances Mi (i = 1 − n) in total and the differences of Mi are only determined by the distance between two AFSSCs. In Fig. 5, the relation between voltage u and currents i j ( j = 1 − n) is ⎧ di 1 di 2 di 3 di n ⎪ u=L + M1 + M2 + · · · + Mn−1 ⎪ ⎪ ⎪ dt dt dt dt ⎪ ⎪ ⎪ ⎪ di 2 di 1 di 3 di n ⎪ ⎪ +L + M1 + · · · + Mn−2 u = M1 ⎪ ⎪ dt dt dt dt ⎪ ⎨ di 3 di 1 di 2 di n + M1 +L + · · · + Mn−3 u = M2 ⎪ ⎪ dt dt dt dt ⎪ ⎪ ⎪ ⎪ . ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ di n di 1 di 2 di 3 ⎩u = M + Mn−2 + Mn−3 + ··· + L . n−1 dt dt dt dt Then the upper expression can be written as u·B = A·
di dt
(6)
1
⎜ L ⎜ ⎜ M1 ⎜ A=⎜ ⎜ M2 ⎜ . ⎜ .. ⎝ Mn−1 ⎞ di 1 ⎜ dt ⎟ ⎜ ⎟ ⎜ di 2 ⎟ ⎜ ⎟ ⎜ dt ⎟ ⎜ ⎟ ⎜ di 3 ⎟ di ⎜ ⎟. =⎜ ⎟ dt ⎜ dt ⎟ ⎜ ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎝ di n ⎠ ⎛
..
.
M1
M2
L M1
M1 L
M2 .. .
M1
M2 M1 .. .
M2
M1
⎞ Mn−1 ⎟ .. ⎟ . ⎟ ⎟ M2 ⎟ ⎟ ⎟ M1 ⎟ ⎠ L
dt Also, because there is I = i 1 + i 2 + i 3 + · · · + i n dI di = B · . dt dt Solving (6) and (7) simultaneously, there is
(7)
dI = B · A−1 · B · u dt where B · A−1 · B is a real number. Thus, there is 1 . (8) L p = −1 B ·A ·B The induction L_s of n AFSSCs in series connection is relevant and easy to calculate, which is equal to the sum of all the elements in matrix A. V. E XPERIMENTAL V ERIFICATIONS AND R ESULT A NALYSIS The key section of the program is the calculation of the h_total, L 1 , L 2 , and R_total, which can be tested by the measured values of the real inductors in Fig. 3. The measured values of both the structural parameters and the electrical parameters are shown in Table I. To raise the comparability, the structural parameters d, D, N, d_turns, and d_sheets of real inductors are given as the input variables, and h_total, R_single, L_single, L 1 , and L 2 are the output results calculated by the program, which are shown in Table II. The errors are calculated by comparing the calculation results with the measured value in Table I. It can be seen that the errors are within the acceptable threshold, i.e., the calculation results by the program are credible. The rest of the program is to calculate h_total, L 1 , L 2 , and R_total in the loop cycle indicated in Fig. 4 until all the d,
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TABLE I M EASURED VALUES OF R EAL I NDUCTORS
TABLE IV A SSIGNED VALUES B EFORE O PTIMIZATION
TABLE II C ALCULATION R ESULTS AND E RRORS TABLE V O PTIMIZED R ESULTS
TABLE III T EST S YMBOLS
D, and N samples in their respective given ranges are tried and find out the minimum V and so on, which need not to be tested. To test the progress of the entire program, three symbols n_nan, n_range, and n_normal are set to mark in the results. The meanings of these three symbols and their proportion in the results are shown in Table III. Table III shows that according to the flowchart in Fig. 4, the number of n_range is in the majority. This is because the distances between modules in L 1 are different from those in L 2 , so when the modules in L 1 and L 2 are series connected with same times, the total values of L 1 and L 2 have different amplifications. However, this situation does not affect the correctness of the program. VI. O PTIMIZATION OF 1- AND 25-kJ OF IPPS S YSTEMS On the conditions of the given parameters in Table IV, the structural parameters of 1-kJ and 25-kJ IPPS systems are optimized by the program, the results of which are shown in Table V. It should be pointed out that the coupling factor k between L 1 and L 2 , which is given in Table IV, is used to calculate
the total inductance L along with L 1 or L 2 , and L is further used to calculate the system energy along with I1 . Using the given L 1 , L 2 , k, and I1 , as shown in Table IV, it can be calculated that the system energy is 1 and 25 kJ when I1 values are 1 and 5 kA, respectively. The coupling factor k of inductors made up of AFSSCs is so very high that the measured value k of inductors in Fig. 3 is 0.985, and in the inductors we constructed preciously with 12 sheets of AFSSCs, k is 0.992. Based on the facts just mentioned, k is assigned a value of 0.95. From Table V, it can be seen that along with the higher system energy, the energy density became higher and the total resistance of the inductors became lower, which is further shown in Fig. 6. With the parameters given in Table IV, Fig. 6 shows the change rules of the energy density and the total resistance along with increasing I1 from 2 to 10 kA (system energy increases from 4 to 100 kJ). Specifically, when the charging current I1 is 10 kA, the energy density is 13.23 MJ/m3 and the R_total is 38.37 m. It can be observed from Fig. 6 that the rules just mentioned are clear and correct. This is because when the current density is constant, increase in the charging current leads to increase in the section area, which in further leads to increase in the volume. Whereas, the increasing rate of the volume is less than that of the system energy because J = 0.5L I 2 , so the
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[6] X. Yu and X. Chu, “STRETCH meat grinder with ICCOS,” IEEE Trans. Plasma Sci., vol. 41, no. 5, pp. 1346–1351, May 2013. [7] X. Yu, S. Ma, and Z. Li, “System implementation and testing of STRETCH meat grinder with ICCOS,” in Proc. 17th Int. Symp. Electromagn. Launch Technol. (EML), Jul. 2014, pp. 1–6. [8] P. L. Kalantarov and L. A. Zeitlin, Calculation of Inductance Handbook. Moscow, Russia: Atomic Energy Press, 1986, pp. 251–276.
Fig. 6. Change rules of energy density and total resistance along with the increase in charging current I1 .
energy density increases. At the same time, the total resistance decreases obviously because the section area is increased, but the volume changes little because it is mainly determined by D and the number of AFSSCs, which are determined by L 1 and L 2 . Fig. 6 shows the advantage of IPPS with the increase in the energy density and the decrease in the total resistance, which also bring convenience into the design of IPPS.
Zhen Li (M’12) received the B.S. degree in electrical engineering and automation from Beijing Technology and Business University, Beijing, China, in 2004, and the M.S. degree in electrical engineering from Tsinghua University, Beijing, in 2008. He has been an Engineer of Electrical Engineering with Tsinghua University, since 2010. His current research interests include pulsed-power supply and wireless energy transmission.
Xinjie Yu (M’01) received the B.S. and Ph.D. degrees in electrical engineering from Tsinghua University, Beijing, China, in 1996 and 2001, respectively. He is currently an Associate Professor of Electrical Engineering with Tsinghua University. His current research interests include pulsed-power supply, power electronics, and computational intelligence.
VII. C ONCLUSION This paper compiles a program that can give the minimum volume and the related resistance with the condition of given primary and secondary inductances and charging and discharging currents. The inductors made up of sheets of AFSSCs and the measurement results are used to verify the program. It is pointed that along with increase in the system energy of IPPS, the energy density increases and the total resistance decreases, which highlights the advantages of IPPS. R EFERENCES [1] I. R. McNab, “Developments in pulsed power technology,” IEEE Trans. Magn., vol. 37, no. 1, pp. 375–378, Jan. 2001. [2] A. Sitzman, D. Surls, and J. Mallick, “STRETCH meat grinder: A novel circuit topology for reducing opening switch voltage stress,” in Proc. 13th IEEE Int. Pulsed Power Conf., Monterey, CA, USA, Jun. 2005, pp. 493–496. [3] P. Dedie, V. Brommer, and S. Scharnholz, “Experimental realization of an eight-stage XRAM generator based on ICCOS semiconductor opening switches, fed by a magnetodynamic storage system,” IEEE Trans. Magn., vol. 45, no. 1, pp. 266–271, Jan. 2009. [4] A. Sitzman, D. Surls, and J. Mallick, “Design, construction, and testing of an inductive pulsed-power supply for a small railgun,” IEEE Trans. Magn., vol. 43, no. 1, pp. 270–274, Jan. 2007. [5] F. W. Grover, Inductance Calculations: Working Formulas and Tables. Mineola, NY, USA: Dover, 2009, pp. 97–98.
Shangang Ma received the B.S. degree in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 2005, and the M.S. degree in materials science from Qinghai University, Xining, China, in 2010. He is currently pursuing the Ph.D. degree with the Department of Electrical Engineering, Tsinghua University, Beijing, China. His current research interests include pulsed-power technology and applications.
Yuzhou Sha was born in Dalian, China, in 1969. He received the B.S. degree in electrical engineering and automation from Sichuan University, Chengdu, China, in 2011. He has been a Senior Engineer with Dalian No.1 Instrument Transformer Company, Ltd., Dalian, since 2007. He has been involved in theoretical research, product design, and manufacturing processes of transformer products.
