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An analytical circuit based nonlinear thermal model for capacitor banks
Microelectronics Reliability 88–90 (2018) 524–527
Contents lists available at ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
An analytical circuit based nonlinear thermal model for capacitor banks ⁎
T
Haoran Wang , Huai Wang Center of Reliable Power Electronics (CORPE), Aalborg University, Denmark
A R T I C LE I N FO
A B S T R A C T
Keywords: Capacitor Reliability Thermal model Lifetime
The thermal couplings among capacitors in a bank could significantly alter the reliability performance compared to a single capacitor. The impact of thermal coupling is becoming stronger for high power density systems due to more stringent constraint in volume. Prior-art studies take into account the thermal coupling effects of a capacitor bank by either Finite Element Method (FEM) or experimental characterization, which are case dependent and time-consuming. This paper proposes a nonlinear mathematical model for capacitor banks based on physics of thermal conduction, convection, and radiation. A simplified version of the model is also obtained and represented by an RC circuit network, which enables computational-efficient thermal stress modeling. The proposed models are convenient to use to support model based sizing of capacitor banks and is scalable for multicell rectangle layout. A case study with experimental testing is discussed to verify the accuracy of the models.
1. Introduction Capacitors are widely used in electrolytic systems to buffer the pulsation power, filter the harmonics and support voltage for stable operation. For these applications where single capacitor cannot fulfil the voltage rating or capacitance requirements, capacitor bank is always used as the energy buffer by connecting several capacitors in parallel for larger capacitance, or in series for higher voltage rating [1]. With more stringent reliability constrains brought by automotive, aerospace and energy industries, capacitor bank is one kind of the stand-out components in terms of failure rate in field operation electronic systems. The efforts to overcome the above challenges can be divided into three categories: a) obtain the accurate temperature and lifetime distribution in the design phase [2]; b) optimal capacitor bank design solutions to achieve proper robustness margin and cost-effectiveness [3]; c) adopt novel longer lifetime capacitor technologies [4]. For the three categories, accurate and time efficient temperature estimation is the basic. The existing methods to estimate the temperature distribution of the capacitor banks is by using experiment or FEM simulation, however, it always take long time to acquire the results in the original design phase. Although constant thermal resistance based mathematical thermal models for single capacitor have been explored, which offer a fast way to estimate the status of the capacitors [2], it cannot be used for a capacitor bank directly to acquire the temperature distribution. Several issues need to be overcome: a) one issue is the uneven temperature distribution among the capacitors inside the bank due to
⁎
Corresponding author. E-mail address: [email protected] (H. Wang).
https://doi.org/10.1016/j.microrel.2018.06.112 Received 31 May 2018; Accepted 30 June 2018 0026-2714/ © 2018 Elsevier Ltd. All rights reserved.
thermal coupling effect [5]. The temperature estimation based on a single capacitor becomes useless; b) thermal resistance extraction for temperature estimation consumes much time in either FEM simulation or experiment. An accurate, fast and efficient mathematical model to acquire the thermal resistance as well as the 3-D temperature distribution is highly valuable. Although researchers have developed a mathematical model for capacitor bank [3], it cannot estimate the detailed hot-spot temperature of each cell, because it packages the capacitor bank as a cuboid module for a rough estimation and ignores the nonlinear thermal behavior inside the module. This paper aims to propose a 3-D nonlinear thermal model for capacitor bank, where the conduction, convection and radiation heat transfer among the cells are considered to acquire the hotspot temperature of each cell. Without the time-consuming FEM simulation or experiment, thermal resistance as well as the critical temperature distribution can be fast characterized from circuit simulators (e.g., Simulink, PLECS, Pspice). Scalable circuit design guideline is also studied. The rest of this paper is as follows: Section 2 introduces the general structure and fundamental theory of nonlinear thermal model; Section 3 studied the circuit based nonlinear thermal model and the scalable design guideline; Section 4 demonstrates a case study of a capacitor bank to verify the accuracy of the proposed thermal model. 2. A general nonlinear thermal model A diagram of the proposed nonlinear thermal model is shown in
Microelectronics Reliability 88–90 (2018) 524–527
H. Wang, H. Wang
Fig. 2. Diagram to illustrate the view factor calculation for the efficient surface area.
Fig. 1. Thermal model of two capacitors in a bank.
Fig. 1. Ploss,i and Ploss,j are the power loss of the capacitor i and j, respectively. Th,i, Th,j, Tc,i, Tc,j and Ta are the hot-spot temperature, case temperature and ambient temperature of the capacitor i and j, respectively. Rhc,i and Rhc,j are the thermal resistance of the capacitor i and j, respectively, from hot-spot to case which are determined by the physical structure and assumed constant in different operating conditions (e.g., electrical loading and thermal loading), and locations. Rconv,i and Rconv,j are the thermal resistance of the capacitor i and j, respectively, for convection heat transfer. Rr,i and Rr,j are the thermal resistance of the capacitor i and j, respectively, for radiation heat transfer. The thermal model for convection and radiation heat transfer cannot be obtained by linear thermal resistor and capacitor, while they follow the nonlinear formulas as shown in Eqs. (A.1) and (A.2) which are terminal temperature dependent and effective surface area related. In the multicell capacitor bank, since the impact from neighboring cells is significant, additional aspects need to be considered in the thermal modeling:
Fig. 3. Circuit based thermal model diagram for two capacitors.
C) Effective surface area Effective surface area is a key coefficient to acquire the thermal resistances, which are adaptively changed with the capacitor dimension and layout. For example, as a single capacitor i, all the surface exposes in the ambient air, so that the effective surface area in heat convection and heat radiation are the whole surface. Different from that, for the capacitor i in a bank with two cells, only a part of the surface exposes to the ambient air, other parts are faced to the capacitor j. The effective surface area is defined in Fig. 2, which can be written as
A) Thermal coupling resistance The case temperature of the capacitor is not only determined by power loss and thermal resistance of the capacitor itself, but is also related to the neighboring cells in the bank. A new path for heat transfer among cells which contains heat conduction and radiation thermal resistances connected in parallel from capacitor i to capacitor j should be considered. Rcond,ij and Rr,ij are the thermal coupling resistance between capacitor i and j for conduction and radiation heat transfer through the air. For simplicity, assuming the distance between cells is very small, the complicated heat convection process among cells is ignored and considered as heat conduction through air.
Seffect = 2πri 2 +
⏟
top / bottom
θ 2πri H = 2πri 2 + θri H 2π lateral area
(1)
where H is the height of the capacitor. A view factor θ is used to obtain the effective surface area from the lateral area, which is shown as
rj ⎞ θ = 2 arcsin ⎛⎜ ⎟ r + d i ij + r j ⎠ ⎝
B) Variance of self-heating thermal resistance Another difference between single capacitor and capacitor bank thermal model is the convection and radiation heat transfer coefficient of self-heating thermal resistance for each cell. Taking capacitor i as an example, these two thermal resistances are the nonlinear function of Tc,i, Ta and the effective surface area to ambient air. When capacitor i locates in a bank, the thermal resistance for self-heating is no longer constant, because the effective surface area to ambient air is no longer the whole surface, which should be the whole surface area subtracts the area faced to the neighboring cells. In the thermal model of a capacitor bank, except for the linear thermal resistance from hot-spot to case, the other thermal resistances are always adaptively changed with the layout as well as the effective surface area.
(2)
3. Circuit based nonlinear thermal model Basic modules for building up the thermal model in circuit simulator is discussed firstly in this section, followed by the guideline to illustrate how to implement the thermal model for different application by using the scalable module in circuit simulator. A) Basic circuit modules of the thermal model As an example, the circuit based thermal model for a capacitor bank 525
Microelectronics Reliability 88–90 (2018) 524–527
H. Wang, H. Wang
Fig. 4. Scalable circuit based thermal model for rectangle layout capacitor bank.
Fig. 5. Temperature distribution of the simulation and experimental results. Fig. 6. Comparison of temperature estimation among experiment, simulation and proposed analytical model.
with two cells is shown in Fig. 3. Current sources in the circuit represent the Ploss,i and Ploss,j. Rhc,i and Rhc,j are assumed to be constant with different loading conditions, which are represented as linear resistors in the circuit. Rca,i and Rca,j are temperature, power loss and effective surface area depended, which cannot use linear resistor to stand for. Current-control voltage sources are used to represent the nonlinear function Rca,i and Rca,j, where the “current” is the power loss and the “voltage” is the terminal temperature in the thermal model. The terminal temperature can be obtained with instantaneous power loss. Rcc,ij is temperature, power loss and surface area depended, which should also use a nonlinear resistor to present. The possible choices to represent the nonlinear resistors are the voltage-controlled current source and current-controlled voltage source. Because Tc,i and Tc,j are controlled by the current-controlled voltage source, the voltage control cannot be applied on the same node to represent the thermal coupling resistor. Rcc,i and Rcc,j should be a voltage-controlled current source.
example, the scalable design guideline is discussed in this section. Depending on the locations as well as the effective surface area, three types thermal model can be obtained in Fig. 4 in colors. For the type I corner capacitor, the total view factor is 2 times of the view factor in one neighboring cell case, because there are two neighboring cells close to the corner capacitor. In the same way, 3 times and 4 times view factor for type II and type III can be obtained. The thermal coupling resistance circuit module is capacitor dimension and layout dimension depended. If the dimensions among cells are the same, same thermal coupling resistance module can be applied to connect different capacitor cells. Based on these basic circuit modules, the circuit based thermal model for capacitor banks with rectangle layout can be implemented in circuit simulators. The temperature distribution can be obtained timeefficiently with increased accuracy compared to the models without considering the thermal coupling effects.
B) Scalable design guideline
4. Case study
Taking typical rectangle layout capacitor bank shown in Fig. 4 as an
The temperature distribution of a capacitor bank with nine cells is 526
Microelectronics Reliability 88–90 (2018) 524–527
H. Wang, H. Wang
The deviation comparison among the three results is shown in Fig. 6(b). The errors of the proposed analytical model and the FEM simulation with respect to the experimental results are within 5–10%.
investigated as case study. The voltage rating is 450 V. Capacitance of this bank is 5040 μF (560 μF ∗ 9). The ambient temperature in simulation is 25 °C. For each cell, the diameter is 40 mm and height is 45 mm. The distance between two capacitor cells is 2 mm. The current is shared among the nine capacitors, and the power dispassion for each cell is 1 W. The simulation and experimental results are shown in Fig. 5. The circuit based thermal model is implemented in Simulink, Matlab R2014a. The thermal models for three types are packaged as modules as well as the thermal coupling connection module. By running the circuit simulator, the temperature distribution can be obtained in 5 s, while the FEM simulation takes 5 min. The results based on experiment, simulation, and the proposed model are compared in Fig. 6(a). Type 1, 2 and 3 are the corner, border and middle capacitors in the 3 ∗ 3 capacitor bank. It can be seen that the estimation results of the proposed analytical model and the experimental results have 2 degree maximum difference.
5. Conclusions This paper proposes a nonlinear analytical thermal model for capacitor banks by taking into account the thermal coupling effects. The obtained circuit based model enables a time-efficient thermal stress analysis without the need of FEM simulation and experimental testing. A case study is presented with nine capacitors physically aligned in a rectangular capacitor bank. It reveals a more than 10 times faster thermal modeling compared to FEM simulation, and 5–10% estimation error benchmarked with experimental measurements.
Appendix A A. Convection heat transfer The thermal resistance of convection heat transfer can be obtained from following equation
R conv =
1 hconv A
hconv is usually given as function of ΔT, hconv = 1.42(ΔT/H) area for heat spread [6].
(A.1) , where ΔT is the temperature rise; H is the height of the capacitor; A is the surface
0.25
B. Radiation heat transfer The radiant energy exchange between a hog body with absolute temperature T1 and an enclosing body with absolute temperature T2 is proportional to the difference in the absolute temperatures to the fourth power
q = εσA (T14 − T2 4 )
(A.2)
where ε is the emissivity of the radiating surface and σ is the Stefan-Boltzmann constant 5.67e−8 [6]. C. Conduction heat transfer The heat conduction can be described as
R cond =
d hcond A
(A.3)
hcond is the conduction heat transfer coefficient of the air. A is the surface of the heated body. d is the length of the conduction heat transfer [6].
[4] Nippon-Chemi-con, Aluminum capacitors group chart, Available: https://www. chemicon.co.jp/e/catalog/pdf/ale/alsepae/001guide/algroupcharte171001.pdf. [5] Z. Wang, F. Yan, M. Xu, Z. Wang, X. Wang, Z. Xu, Influence of external factors on selfhealing capacitor temperature field distribution and its validation, IEEE Trans. Plasma Sci. 45 (7) (Jul. 2017) 1680–1688. [6] I. Villar, Multiphysical Characterization of Medium-frequency Power Electronic Transformers, Ecole Polytechnique Fédérale de Lausanne, 2010.
References [1] P. Pelletier, J.M. Guichon, J.L. Schanen, D. Frey, Optimization of a dc capacitor tank, IEEE Trans. Ind. Appl. 45 (2) (Mar. 2009) 880–886. [2] Y. Yang, K. Ma, H. Wang, F. Blaabjerg, Instantaneous thermal modeling of the dc-link capacitor in photovoltaic systems, Proc. IEEE APEC, Mar. 2015, pp. 2733–2739. [3] M.L. Gasperi, N. Gollhardt, Heat transfer model for capacitor banks, Proc. IEEE Industry Applications Conference, vol. 2, Oct. 1998, pp. 1199–1204.
527
Contents lists available at ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
An analytical circuit based nonlinear thermal model for capacitor banks ⁎
T
Haoran Wang , Huai Wang Center of Reliable Power Electronics (CORPE), Aalborg University, Denmark
A R T I C LE I N FO
A B S T R A C T
Keywords: Capacitor Reliability Thermal model Lifetime
The thermal couplings among capacitors in a bank could significantly alter the reliability performance compared to a single capacitor. The impact of thermal coupling is becoming stronger for high power density systems due to more stringent constraint in volume. Prior-art studies take into account the thermal coupling effects of a capacitor bank by either Finite Element Method (FEM) or experimental characterization, which are case dependent and time-consuming. This paper proposes a nonlinear mathematical model for capacitor banks based on physics of thermal conduction, convection, and radiation. A simplified version of the model is also obtained and represented by an RC circuit network, which enables computational-efficient thermal stress modeling. The proposed models are convenient to use to support model based sizing of capacitor banks and is scalable for multicell rectangle layout. A case study with experimental testing is discussed to verify the accuracy of the models.
1. Introduction Capacitors are widely used in electrolytic systems to buffer the pulsation power, filter the harmonics and support voltage for stable operation. For these applications where single capacitor cannot fulfil the voltage rating or capacitance requirements, capacitor bank is always used as the energy buffer by connecting several capacitors in parallel for larger capacitance, or in series for higher voltage rating [1]. With more stringent reliability constrains brought by automotive, aerospace and energy industries, capacitor bank is one kind of the stand-out components in terms of failure rate in field operation electronic systems. The efforts to overcome the above challenges can be divided into three categories: a) obtain the accurate temperature and lifetime distribution in the design phase [2]; b) optimal capacitor bank design solutions to achieve proper robustness margin and cost-effectiveness [3]; c) adopt novel longer lifetime capacitor technologies [4]. For the three categories, accurate and time efficient temperature estimation is the basic. The existing methods to estimate the temperature distribution of the capacitor banks is by using experiment or FEM simulation, however, it always take long time to acquire the results in the original design phase. Although constant thermal resistance based mathematical thermal models for single capacitor have been explored, which offer a fast way to estimate the status of the capacitors [2], it cannot be used for a capacitor bank directly to acquire the temperature distribution. Several issues need to be overcome: a) one issue is the uneven temperature distribution among the capacitors inside the bank due to
⁎
Corresponding author. E-mail address: [email protected] (H. Wang).
https://doi.org/10.1016/j.microrel.2018.06.112 Received 31 May 2018; Accepted 30 June 2018 0026-2714/ © 2018 Elsevier Ltd. All rights reserved.
thermal coupling effect [5]. The temperature estimation based on a single capacitor becomes useless; b) thermal resistance extraction for temperature estimation consumes much time in either FEM simulation or experiment. An accurate, fast and efficient mathematical model to acquire the thermal resistance as well as the 3-D temperature distribution is highly valuable. Although researchers have developed a mathematical model for capacitor bank [3], it cannot estimate the detailed hot-spot temperature of each cell, because it packages the capacitor bank as a cuboid module for a rough estimation and ignores the nonlinear thermal behavior inside the module. This paper aims to propose a 3-D nonlinear thermal model for capacitor bank, where the conduction, convection and radiation heat transfer among the cells are considered to acquire the hotspot temperature of each cell. Without the time-consuming FEM simulation or experiment, thermal resistance as well as the critical temperature distribution can be fast characterized from circuit simulators (e.g., Simulink, PLECS, Pspice). Scalable circuit design guideline is also studied. The rest of this paper is as follows: Section 2 introduces the general structure and fundamental theory of nonlinear thermal model; Section 3 studied the circuit based nonlinear thermal model and the scalable design guideline; Section 4 demonstrates a case study of a capacitor bank to verify the accuracy of the proposed thermal model. 2. A general nonlinear thermal model A diagram of the proposed nonlinear thermal model is shown in
Microelectronics Reliability 88–90 (2018) 524–527
H. Wang, H. Wang
Fig. 2. Diagram to illustrate the view factor calculation for the efficient surface area.
Fig. 1. Thermal model of two capacitors in a bank.
Fig. 1. Ploss,i and Ploss,j are the power loss of the capacitor i and j, respectively. Th,i, Th,j, Tc,i, Tc,j and Ta are the hot-spot temperature, case temperature and ambient temperature of the capacitor i and j, respectively. Rhc,i and Rhc,j are the thermal resistance of the capacitor i and j, respectively, from hot-spot to case which are determined by the physical structure and assumed constant in different operating conditions (e.g., electrical loading and thermal loading), and locations. Rconv,i and Rconv,j are the thermal resistance of the capacitor i and j, respectively, for convection heat transfer. Rr,i and Rr,j are the thermal resistance of the capacitor i and j, respectively, for radiation heat transfer. The thermal model for convection and radiation heat transfer cannot be obtained by linear thermal resistor and capacitor, while they follow the nonlinear formulas as shown in Eqs. (A.1) and (A.2) which are terminal temperature dependent and effective surface area related. In the multicell capacitor bank, since the impact from neighboring cells is significant, additional aspects need to be considered in the thermal modeling:
Fig. 3. Circuit based thermal model diagram for two capacitors.
C) Effective surface area Effective surface area is a key coefficient to acquire the thermal resistances, which are adaptively changed with the capacitor dimension and layout. For example, as a single capacitor i, all the surface exposes in the ambient air, so that the effective surface area in heat convection and heat radiation are the whole surface. Different from that, for the capacitor i in a bank with two cells, only a part of the surface exposes to the ambient air, other parts are faced to the capacitor j. The effective surface area is defined in Fig. 2, which can be written as
A) Thermal coupling resistance The case temperature of the capacitor is not only determined by power loss and thermal resistance of the capacitor itself, but is also related to the neighboring cells in the bank. A new path for heat transfer among cells which contains heat conduction and radiation thermal resistances connected in parallel from capacitor i to capacitor j should be considered. Rcond,ij and Rr,ij are the thermal coupling resistance between capacitor i and j for conduction and radiation heat transfer through the air. For simplicity, assuming the distance between cells is very small, the complicated heat convection process among cells is ignored and considered as heat conduction through air.
Seffect = 2πri 2 +
⏟
top / bottom
θ 2πri H = 2πri 2 + θri H 2π lateral area
(1)
where H is the height of the capacitor. A view factor θ is used to obtain the effective surface area from the lateral area, which is shown as
rj ⎞ θ = 2 arcsin ⎛⎜ ⎟ r + d i ij + r j ⎠ ⎝
B) Variance of self-heating thermal resistance Another difference between single capacitor and capacitor bank thermal model is the convection and radiation heat transfer coefficient of self-heating thermal resistance for each cell. Taking capacitor i as an example, these two thermal resistances are the nonlinear function of Tc,i, Ta and the effective surface area to ambient air. When capacitor i locates in a bank, the thermal resistance for self-heating is no longer constant, because the effective surface area to ambient air is no longer the whole surface, which should be the whole surface area subtracts the area faced to the neighboring cells. In the thermal model of a capacitor bank, except for the linear thermal resistance from hot-spot to case, the other thermal resistances are always adaptively changed with the layout as well as the effective surface area.
(2)
3. Circuit based nonlinear thermal model Basic modules for building up the thermal model in circuit simulator is discussed firstly in this section, followed by the guideline to illustrate how to implement the thermal model for different application by using the scalable module in circuit simulator. A) Basic circuit modules of the thermal model As an example, the circuit based thermal model for a capacitor bank 525
Microelectronics Reliability 88–90 (2018) 524–527
H. Wang, H. Wang
Fig. 4. Scalable circuit based thermal model for rectangle layout capacitor bank.
Fig. 5. Temperature distribution of the simulation and experimental results. Fig. 6. Comparison of temperature estimation among experiment, simulation and proposed analytical model.
with two cells is shown in Fig. 3. Current sources in the circuit represent the Ploss,i and Ploss,j. Rhc,i and Rhc,j are assumed to be constant with different loading conditions, which are represented as linear resistors in the circuit. Rca,i and Rca,j are temperature, power loss and effective surface area depended, which cannot use linear resistor to stand for. Current-control voltage sources are used to represent the nonlinear function Rca,i and Rca,j, where the “current” is the power loss and the “voltage” is the terminal temperature in the thermal model. The terminal temperature can be obtained with instantaneous power loss. Rcc,ij is temperature, power loss and surface area depended, which should also use a nonlinear resistor to present. The possible choices to represent the nonlinear resistors are the voltage-controlled current source and current-controlled voltage source. Because Tc,i and Tc,j are controlled by the current-controlled voltage source, the voltage control cannot be applied on the same node to represent the thermal coupling resistor. Rcc,i and Rcc,j should be a voltage-controlled current source.
example, the scalable design guideline is discussed in this section. Depending on the locations as well as the effective surface area, three types thermal model can be obtained in Fig. 4 in colors. For the type I corner capacitor, the total view factor is 2 times of the view factor in one neighboring cell case, because there are two neighboring cells close to the corner capacitor. In the same way, 3 times and 4 times view factor for type II and type III can be obtained. The thermal coupling resistance circuit module is capacitor dimension and layout dimension depended. If the dimensions among cells are the same, same thermal coupling resistance module can be applied to connect different capacitor cells. Based on these basic circuit modules, the circuit based thermal model for capacitor banks with rectangle layout can be implemented in circuit simulators. The temperature distribution can be obtained timeefficiently with increased accuracy compared to the models without considering the thermal coupling effects.
B) Scalable design guideline
4. Case study
Taking typical rectangle layout capacitor bank shown in Fig. 4 as an
The temperature distribution of a capacitor bank with nine cells is 526
Microelectronics Reliability 88–90 (2018) 524–527
H. Wang, H. Wang
The deviation comparison among the three results is shown in Fig. 6(b). The errors of the proposed analytical model and the FEM simulation with respect to the experimental results are within 5–10%.
investigated as case study. The voltage rating is 450 V. Capacitance of this bank is 5040 μF (560 μF ∗ 9). The ambient temperature in simulation is 25 °C. For each cell, the diameter is 40 mm and height is 45 mm. The distance between two capacitor cells is 2 mm. The current is shared among the nine capacitors, and the power dispassion for each cell is 1 W. The simulation and experimental results are shown in Fig. 5. The circuit based thermal model is implemented in Simulink, Matlab R2014a. The thermal models for three types are packaged as modules as well as the thermal coupling connection module. By running the circuit simulator, the temperature distribution can be obtained in 5 s, while the FEM simulation takes 5 min. The results based on experiment, simulation, and the proposed model are compared in Fig. 6(a). Type 1, 2 and 3 are the corner, border and middle capacitors in the 3 ∗ 3 capacitor bank. It can be seen that the estimation results of the proposed analytical model and the experimental results have 2 degree maximum difference.
5. Conclusions This paper proposes a nonlinear analytical thermal model for capacitor banks by taking into account the thermal coupling effects. The obtained circuit based model enables a time-efficient thermal stress analysis without the need of FEM simulation and experimental testing. A case study is presented with nine capacitors physically aligned in a rectangular capacitor bank. It reveals a more than 10 times faster thermal modeling compared to FEM simulation, and 5–10% estimation error benchmarked with experimental measurements.
Appendix A A. Convection heat transfer The thermal resistance of convection heat transfer can be obtained from following equation
R conv =
1 hconv A
hconv is usually given as function of ΔT, hconv = 1.42(ΔT/H) area for heat spread [6].
(A.1) , where ΔT is the temperature rise; H is the height of the capacitor; A is the surface
0.25
B. Radiation heat transfer The radiant energy exchange between a hog body with absolute temperature T1 and an enclosing body with absolute temperature T2 is proportional to the difference in the absolute temperatures to the fourth power
q = εσA (T14 − T2 4 )
(A.2)
where ε is the emissivity of the radiating surface and σ is the Stefan-Boltzmann constant 5.67e−8 [6]. C. Conduction heat transfer The heat conduction can be described as
R cond =
d hcond A
(A.3)
hcond is the conduction heat transfer coefficient of the air. A is the surface of the heated body. d is the length of the conduction heat transfer [6].
[4] Nippon-Chemi-con, Aluminum capacitors group chart, Available: https://www. chemicon.co.jp/e/catalog/pdf/ale/alsepae/001guide/algroupcharte171001.pdf. [5] Z. Wang, F. Yan, M. Xu, Z. Wang, X. Wang, Z. Xu, Influence of external factors on selfhealing capacitor temperature field distribution and its validation, IEEE Trans. Plasma Sci. 45 (7) (Jul. 2017) 1680–1688. [6] I. Villar, Multiphysical Characterization of Medium-frequency Power Electronic Transformers, Ecole Polytechnique Fédérale de Lausanne, 2010.
References [1] P. Pelletier, J.M. Guichon, J.L. Schanen, D. Frey, Optimization of a dc capacitor tank, IEEE Trans. Ind. Appl. 45 (2) (Mar. 2009) 880–886. [2] Y. Yang, K. Ma, H. Wang, F. Blaabjerg, Instantaneous thermal modeling of the dc-link capacitor in photovoltaic systems, Proc. IEEE APEC, Mar. 2015, pp. 2733–2739. [3] M.L. Gasperi, N. Gollhardt, Heat transfer model for capacitor banks, Proc. IEEE Industry Applications Conference, vol. 2, Oct. 1998, pp. 1199–1204.
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