HIGH FREQUENCY MODELS FOR AIR-CORE
REACTORS USING 3D EQUIVALENT CIRCUIT THEORY Mathias A. Enohnyaket & Jonas Ekman
EISLAB
Dept. of Computer Science and Electrical Engineering Luleå University of Technology
97187, Luleå Sweden, emc@csee.ltu.se
Outline
¾ Background
¾ Method
¾ Air-core reactor structure
¾ Numerical examples, PEEC model results against measurements
¾ Conclusions
Background
¾ Project goal: introduce, evaluate and develop the PEEC method for combined electric and EM modeling of reactors.
¾ Applications of reactors: current limiting, neutral grounding, filtering, and shunting.
¾ Faster switching operations in PE components used in power distribution systems demand for HF (a few MHz) electromagnetic (EM) models.
¾ HF model for air-core reactor requires detailed discretization of the windings.
¾ Partial Element Equivalent Circuit (PEEC) method, developed at IBM Research
Center, Yorktown (NY), to study cross-talk on PCB’s, is accurate even at high
frequencies, and same model can be used in both time and frequency domain
analysis.
The PEEC method
Starting from the expressioin for the total electric field in free space
Rewritten to the EFIE using definitions of EM potentials:
By mathematical manipulation, interpreted as Kirchoffs voltage law for
basic PEEC cell.
Interpretation of PEEC method
¾ Adjacent nodes connected through self partial inductances.
¾ Mutual partial inductances represents the magnetic field coupling between volume cells.
¾ Each node is connected to infinity through self coefficients of potential.
¾ Mutual coefficients of potential represents the E-field coupling between the surface cells
¾ System is assembled using KVL and KCL.
¾ Software used is developed by Luleå University of Technology (SW), University of L’Aquila (IT), and IBM T.J. Watson Research Center (USA). See further:
http://www.csee.ltu.se/peec
Reactor PEEC model
¾ Each turn is discretized into finite number of rectangular bars in the current direction.
¾ For each volume cell there is a surface cell (charge) discretisation.
¾ Computes partial inductances L
pand resistances R for each volume cell using closed formulas.
¾ Computes coefficients of potential P
from surface cells.
PEEC-model for 4 turn reactor using 14 segements per turn.Numerical example for rectangular reactor model
¾ Lab model: 90 turns wound copper wire, d=2.0 mm, on a sparse wooden support, 49x58 cm, constant pitch of 10.0 mm.
¾ Measurements: Input impedence measurements were made using a vector network analyser for 10 KHz to 5 MHz.
¾ PEEC model:90 turns, with 4 bars per turn. Excited with a unitary current source, gives I
inand V
in, input impedance Z
incalculated.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
10−1 100 101 102
Freq. [MHz]
Input impedance [k Ω]
PEEC model Measurements Lumped model
Input impedance for rectangular reactor (blue PEEC, green- measurements, red- lumped ).
Numerical example for circular reactor model
¾ Lab model: 133 turns wound copper wire, d=0.7 mm, on a plastic cylindrical plastic support, d=0.4m, constant pitch of 2.5 mm.
¾ Measurements: Input impedence measurements for 10 KHz to 5 MHz.
¾ PEEC model: 133 turns, with 20 rectangular bars per turn. Excited with a unitary current source, gives I
inand V
in.Z
incalculated.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
102 103 104 105 106
frequency [MHz]
Input inpedence [Ω]
measurement PEEC QS
Input impedance for circular reactor (blue-measurements, red- PEEC model).
Comment on results
¾ Fairly good agreement in resonance frequencies
¾ Slight mismatch in amplitudes at resonances
¾ Possible explanation :
Î skin-effect and dielectric material NOT modeled
Conculsions
¾ Good agreement between PEEC model simulations and measurements
= Promising.
¾ Easy to include transmission line (effects), driving circuitry, and measurement equipment.
¾ Can handle circular, rectangular, and any other 3D reactor.
¾ Fast, especially for time domain modeling.
¾ Next steps:
¾ Industry reactor modeling.
¾ Include Skin effect.
¾ Include dielectric material.
¾ Future: Develop the theory and code for PEEC + magnetic materials.
Thank You for Your attention
Questions?
Time complexity for 210 turns PEEC model -on regular workstation
Step Time [min] Time [min]
Solver type FD-PEEC TD-PEEC
Parsing & Meshing 0.08 0.08
Calculating partial inductances
0.7 0.7
Calculating coefficients of potentials
6.0 6.0
Solver 1034 (100 frequencies) 15 (1000 time steps)
Total ~ 1040 ~ 22
PEEC solution (considering only 4 bars per turn)
¾ Assemble system using KVL : AV − (R + jωLp)I = Vs
¾ Enforce continuity equation at each node
jωP-1V − ATI = Is
¾ A is connectivity matrix
¾ Is and Vs are current and voltage sources respectively
Lp22
Lp33
Lp44
P133 Ip
P144 Ip P111 Ip
P122 Ip
-
+Lp11
-
+-
+-
+VL
VL
VL
VL
1
4
3 2
Ip1
Ip4
Ip3 Ip2
3
2
1
4
I3
I1 I2
I4
f2
f1
f4
f3
