Three-dimensional modeling of multilayer busbar ... › bitstream › 2268 › 222058 › 1 ›...
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Three-dimensional modeling of multilayer busbar incorporating domains of lesser dimension J. Velasco 1 , M. Spirlet 1 , C. Geuzaine 1 and F. Henrotte 1,2 1 University of Liège, Department of Electrical Engineering and Computer Science, Liège 4000, Belgium 2 EPL-iMMC-MEMA, Université Catholique de Louvain, Louvain-la-Neuve 1348, Belgium Introduction For the accurate modeling of a busbar, it is needed to solve the full-wave Maxwell s equations, to include inductive and capacitive effects, in the calculation of the input impedance. Their opera- tion at higher frequencies is costly as the skin depth becomes smaller than the thickness of the conducting domain. The chal- lenge therein lies in finding a trade-off between computational cost and the accuracy of the solution. T HIN S HELL M ODEL Let us consider a wide structure with infinitesimal thicknes d. Un- der these conditions we may assume that the current distribution across the thin structure will occur tangentially to the larger sur- face, and negligible across the thickness. d n - t n + t Γ + t Γ - t Γ = t Performing the analytic integration of the 1D problem we may ex- press the volume integral of the curlh as Z V curlh dV = σ tanh(γ d/2) γ ( e t | Γ + + e t | Γ - ) Where γ = q iωσμ(1 + iω σ ) The physical model can be coupled to a finite element model where it can be used as a type of impedance boundary condi- tion, tied to the traces of the magnetic field and the relationship to the magnetic vector potential. ˆ n t × h t | Γ + + ˆ n t × h t | Γ - = - 1 μβ 0 a c,t | Γ + ˆ n t × h t | Γ + - ˆ n t × h t | Γ - = -σβ 0 iω (2a c,t | Γ + +2a d,t | Γ + ) The results at low frequency can be compared to Problem 6 of the Testing Electromagnetic Analysis Methods (TEAM). 0 50 100 150 200 250 300 0.5 1 1.5 x (mm) b z (T) Volume @ 50Hz Shell @ 50Hz [1]C. Geuzaine, P. Dular, and W. Legros, “Dual formulations for the modeling of thin electromagnetic shells using edge elements,” IEEE Transactions on Magnetics, vol. 36, no. 4, pp. 799–803, Jul 2000. [2] L. Krahenbuhl and D. Muller, “Thin layers in electrical engineering-example of shell models in analysing eddy-currents by boundary and finite element methods,” IEEE Transactions on Magnetics, vol. 29, no. 2, pp. 1450–1455, Mar 1993. [3]G. A. P.Magnusson and V. Tripathi, Transmission lines and Wave Propagation. CRC Press, 2000. [4] Miscellaneous, The Finite Element Method for Electromagnetic Modeling. Wiley and Sons, Inc., 2008. Acknowledgments: This work was supported under grand PIT7508 (ATAC-HP). L INE R EGION AND B ESSEL F UNCTIONS The behavior of the current density in the time-harmonic case can be described in terms of modified Bessel functions. j = Re i 3/2 k 2π R J 0 (i 3/2 k r ) J 1 (i 3/2 kR) I exp i(ωt-β z) ˆ z The assumption is that in an infinitesimal long wire, the current density distribution will be symmetrical to the central axis of the conductor. -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0 2 4 6 ·10 6 Radius (mm) Re j z (A/m) Analytical @ 1MHz Numerical @ 1MHz Analytical @ 1e-3Hz Numerical @ 1e-3Hz Z A ˆ n · j dA = -σ iω a s Re 2π R i 3/2 k J 1 (i 3/2 kR) J 0 (i 3/2 kR) 1 exp i(ωt-β z) B USBAR M ODEL AND I NPUT I MPEDANCE Busbars are commonly used for power distribution in a variety of applications including circuit coupling and load control. Due to the diversity in functionality, their size and shape vary substan- tially from design to design, although their main concept resem- bles that of a parallel-plate transmission line, consisting of parallel conducting plates separated by a dielectric slab. This model aims to replicate the behavior of a no-load three- phase inverter, where we have an input terminal and three vias representing the IGBTs of the inverter. The results below show the behavior of the impedance at a wide range of frequencies. Although the phase shows that the busbar is mainly inductive, it is possible to see the effect of the skin effect on the real part of the impedance 10 0 10 3 10 6 10 9 0 20 40 Frequency (Hz) |Z |(Ω) Volume Shell - Constant Shell - Skin Effect 10 0 10 3 10 6 10 9 0 50 Frequency (Hz) θ (deg) Volume Shell - Constant Shell - Skin Effect 10 0 10 3 10 6 10 9 10 -3 10 -2 10 -1 Frequency (Hz) Re Z (Ω) Volume Shell - Constant Shell - Skin Effect 10 0 10 3 10 6 10 9 0 20 40 Frequency (Hz) Im Z (Ω) Volume Shell - Constant Shell - Skin Effect
Transcript of Three-dimensional modeling of multilayer busbar ... › bitstream › 2268 › 222058 › 1 ›...
Three-dimensional modeling of multilayer busbar incorporatingdomains of lesser dimension
J. Velasco 1, M. Spirlet 1 , C. Geuzaine 1 and F. Henrotte 1,2
1University of Liège, Department of Electrical Engineering and Computer Science, Liège 4000, Belgium2EPL-iMMC-MEMA, Université Catholique de Louvain, Louvain-la-Neuve 1348, Belgium
Introduction
For the accurate modeling of a busbar, it is needed to solve thefull-wave Maxwell′s equations, to include inductive and capacitiveeffects, in the calculation of the input impedance. Their opera-tion at higher frequencies is costly as the skin depth becomessmaller than the thickness of the conducting domain. The chal-lenge therein lies in finding a trade-off between computationalcost and the accuracy of the solution.
THIN SHELL MODEL
Let us consider a wide structure with infinitesimal thicknes d. Un-der these conditions we may assume that the current distributionacross the thin structure will occur tangentially to the larger sur-face, and negligible across the thickness.
dn
t
n+t
+t
t
=t
Performing the analytic integration of the 1D problem we may ex-press the volume integral of the curlh as∫
V
curlh dV = σtanh(γd/2)
γ( et|Γ+ + et|Γ−)
Where γ =√
iωσµ(1 + iωεσ )
The physical model can be coupled to a finite element modelwhere it can be used as a type of impedance boundary condi-tion, tied to the traces of the magnetic field and the relationshipto the magnetic vector potential.
nt × ht|Γ+ + nt × ht|Γ− = − 1µβ0
ac,t|Γ+
nt × ht|Γ+ − nt × ht|Γ− = −σβ0iω(2ac,t|Γ+ + 2ad,t|Γ+)
The results at low frequency can be compared to Problem 6 ofthe Testing Electromagnetic Analysis Methods (TEAM).
0 50 100 150 200 250 300
0.5
1
1.5
x (mm)
b z(T
)
Volume @ 50HzShell @ 50Hz
[1] C. Geuzaine, P. Dular, and W. Legros, “Dual formulations for the modeling of thin electromagnetic shells using edge elements,” IEEETransactions on Magnetics, vol. 36, no. 4, pp. 799–803, Jul 2000.
[2] L. Krahenbuhl and D. Muller, “Thin layers in electrical engineering-example of shell models in analysing eddy-currents by boundary andfinite element methods,” IEEE Transactions on Magnetics, vol. 29, no. 2, pp. 1450–1455, Mar 1993.
[3] G. A. P.Magnusson and V. Tripathi, Transmission lines and Wave Propagation. CRC Press, 2000.
[4] Miscellaneous, The Finite Element Method for Electromagnetic Modeling. Wiley and Sons, Inc., 2008.
Acknowledgments: This work was supported under grand PIT7508 (ATAC-HP).
LINE REGION AND BESSEL FUNCTIONS
The behavior of the current density in the time-harmonic casecan be described in terms of modified Bessel functions.
j = Re[
i 3/2k2πR
J0(i 3/2k r)J1(i 3/2kR)
I expi(ωt−βz)]z
The assumption is that in an infinitesimal long wire, the currentdensity distribution will be symmetrical to the central axis of theconductor.
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
0
2
4
6
·106
Radius (mm)
Rej z(A
/m)
Analytical @ 1MHzNumerical @ 1MHzAnalytical @ 1e-3HzNumerical @ 1e-3Hz
∫A
n · jdA = −σiωasRe[
2πRi 3/2k
J1(i 3/2kR)J0(i 3/2kR)
1expi(ωt−βz)
]
BUSBAR MODEL AND INPUT IMPEDANCE
Busbars are commonly used for power distribution in a variety ofapplications including circuit coupling and load control. Due tothe diversity in functionality, their size and shape vary substan-tially from design to design, although their main concept resem-bles that of a parallel-plate transmission line, consisting of parallelconducting plates separated by a dielectric slab.
This model aims to replicate the behavior of a no-load three-phase inverter, where we have an input terminal and three viasrepresenting the IGBTs of the inverter.
The results below show the behavior of the impedance at a widerange of frequencies. Although the phase shows that the busbaris mainly inductive, it is possible to see the effect of the skin effecton the real part of the impedance
100 103 106 109
0
20
40
Frequency (Hz)
|Z|(Ω
)
Volume
Shell - Constant
Shell - Skin Effect
100 103 106 109
0
50
Frequency (Hz)
θ(d
eg)
Volume
Shell - Constant
Shell - Skin Effect
100 103 106 109
10−3
10−2
10−1
Frequency (Hz)
ReZ
(Ω)
Volume
Shell - Constant
Shell - Skin Effect
100 103 106 109
0
20
40
Frequency (Hz)
ImZ
(Ω)
Volume
Shell - Constant
Shell - Skin Effect
