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D. Riedinger – Efficient computation of general eddy current problems for arbitrary conductor geometries 24.09.2017

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Efficient computation of general eddy current problems for arbitrary conduc-tor geometries Dirk Riedinger It is well known that the computation of general current displacement (eddy current) problems is extraor-dinarily complicated. Especially if the aim is to compute the current density distribution of large dimen-sion conductors of arbitrary geometry and arbitrary magnetic coupling like found at electric arc furnace high current systems. Usual field computation methods are not accurate and not stable enough for such large problems. Presented here is a new method which solves the eddy current problem in general for the first time. Finite Network Method (FNM)

Badische Stahl-Engineering GmbH (BSE), Kehl, Germany, has developed a program system with graph-ical modelling (Delphi, C) that is based on the theoretical description of the Finite Network Method (FNM) created by Prof. Dr.-Ing. habil Abbas Farschtschi [1-3]. This method is a very refined innovation because it allows to compute current density distributions and eddy currents respectively very efficiently and accurately for arbitrary conductor geometries. Therefore the geometry is transformed into a resistive-inductively coupled electrical network. Our FNM program system has got the following features:

Not the fields or the vector potential are the primary quantities but straightforward the current densities of all volume elements. These are directly computed from the properties of the electrical network that represents the simulated conductors.

Only conductive areas are to be considered. Thus definitions of boundary conditions and discreti-sation of empty space is superfluous.

FNM is a semi-analytic method. The network parameters of the discretised geometry are comput-ed exactly applying analytic formulas. The accuracy of the solution only depends on the discreti-sation depth (and thus on the computation capacity).

No convergence problems or numerical instabilities do occur. The equation system is solved exactly, the computation runs always stable. Already rough discretisation results in useful solu-tions.

Thanks to parallelisation the computation time is very efficient also for fine discretisation and de-pends on the number of available processors (cores) and clock rate.

Peripheral magnetically coupled conductive areas (not part of the electrical circuit) where eddy currents are induced are considered automatically. Moving conductors are not considered.

Frequency range is from DC to several kHz (no capacitive effects considered). Permeability is fix with µr = 1.

Because the primary quantity of FNM is the current density per volume element also the important sec-ondary quantities can be computed easily and accurately. These are:

Magnetic induction/flux (B-field exposition, screening). This is computed for each volume el-ement when the acc. forces are wanted or for free points in space.

Magnetic forces and torques ( dimensioning). The total force on a volume with a current densi-ty distribution is the sum of the forces on all contained volume elements.

Electric power losses ( heating, efficiency). The total power loss of a volume with a current density distribution is the sum of the power losses of the contained volume elements.


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The magnetic forces with their dynamic and static components can be computed accurately for arbitrary geometry for the first time. Further lumped impedances of a configuration based on a proper equivalent circuit or the current distribu-tion in parallel conductors can be accurately calculated considering all current displacement (skin, prox-imity) effects. Future extensions of the FNM program system are:

Presently FNM utilizes complex arithmetic with sinusoidal source voltages. A computation with arbitrary source voltage time functions can be implemented easily. Then the time step evolution of the current density distribution, of the B-field and of the forces on a system can be simulated.

Presently the consideration of ferromagnetic matter like steel with µr = f (H) is not implemented but the FNM program system can be extended in a straightforward way.

Example electric arc furnace (EAF)

The FNM program system was designed by BSE especially for computation of arc furnace high current systems. EAF have got very extensive high current systems with large conductor cross sections that are subject to significant current displacement effects at mains frequency (50 / 60 Hz). The two arc furnaces of Badische Stahlwerke GmbH (BSW), Kehl, Germany, melt steel scrap very efficiently. The temperature of the liquid steel is about 1600 °C. The electrical energy is supplied to arc furnaces via furnace trans-formers that transform a medium voltage of typically 33 kV to typically 700 to 1200 V. The currents con-ducted by the high current system are typically about 50 to 80 kA (average). The average active power per furnace at BSW is 75 MW. The scrap is molten by the arcs which have core temperatures of about 10000 K under normal athmospheric conditions. Only electrodes made from graphite can withstand these temperatures. However these are consumed over time and need to be replaced regularly. From the point of view of electrical engineering the arc furnace is a very demanding aggregate. The elec-trical quantities severely fluctuate depending on the melting process. Frequently the supply network is loaded unsymmetrically. The arcs are active power „consumers“ though they have got non-linear charac-teristics and create harmonics. The three phases of the high current system are inductively coupled by the strong magnetic fields and thus influence each other. To be able to compute the electrical properties of high current systems of arc furnaces very refined methods are required. These are now available with the FNM program system. BSE mainly applies FNM for optimal design of high current systems (form, impedance, unsymmetry, forces, torques), thus for optimal power input in the furnace, loss minimization and for solution of prob-lems arising from eddy currents or locally excessive current densities. In the following some examples are explained that indicate the performance of the FNM system. Figure 1 depicts the high current system of furnace No. 2 of BSW in short circuit modus without arcs. On the right are the graphite electrodes where the arcs are burning at the tips, on the left are the flexible high current cables and the external delta closure of the furnace transformer. The electrodes are held by means of horizontal conductive electrode arms made of aluminium. The vertical movement happens via hydrau-lically actuated steel masts that are insulated from the electrical circuit. The upper part of the masts is subject to induced eddy currents. The current density distribution is extremely inhomogenous, local dif-ferences of greater then 1:100 do appear. The dimensions of the whole system are approx. L x W x H = 15 x 4 x 8 m. The graphite electrodes have a diameter of 610 mm, the electrode arm bodies have dimen-sions of approx. H x W = 800 x 450 mm, their wall thickness is approx. 40 mm.


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Figure 1: Inhomogeneous current density distribution in the high current system of furnace No. 2 of BSW.

Logarithmic Scale. The simulated model depicted in figure 1 consists of about 10000 volume elements. Figure 2 depicts a cross section of the graphite electrodes. Here skin and proximity effect are clearly visi-ble. The current displacement in the electrodes is the cause of dynamic torques acting on the electrodes. If the rotating field of the three phase system is in correct sequence then the electrode joints are tightened, otherwise loosened. A well known phenomenon at arc furnaces.

Bild 2: Inhomogenous current density distribution (skin+proximity effect)

in graphite electrodes. Linear scale.


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Figure 2a depicts the magnetic forces on the electrodes qualitatively. The static components are directed outwards, the dynamic components rotate with double mains frequency.

Figure 2a: Magnetic forces on the electrodes of an arc furnace. A strength of FNM is the efficient computation of the lumped impedances of simulated circuits. No new complete solutions of the linear equation system are necessary. The lumped (global) impedances of the proper equivalent circuit result from all local parameters of the volume elements of the system and thus contain all current displacement effects, figure 3.

Figure 3: Equivalent circuit of the high current system of figure 1 with lumped impedances.

The impedance values of the circuit of figure 1, 3 are: Phases: Delta connection: Z1 = 0,290 +j 2,12 mOhm Za = 0,027 +j 0,52 mOhm Z2 = 0,280 +j 1,68 mOhm Zb = 0,027 +j 0,52 mOhm Z3 = 0,288 +j 2,11 mOhm Zc = 0,019 +j 0,33 mOhm


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The significant unsymmetry of the phase reactances results from the almost coplanar configuration of the electrode arms. Application example: induction in a plate

The Finite Network Method is universally applicable and of course not limited to arc furnace problems. Arbitrary three dimensional geometries (objects) and circuits can be simulated. The following example shall demonstrate the capabilities of FNM. The model has got the following parameters:

Dimensions of the plate: 2000 x 2000 mm, 10 mm thick Dimensions of the ring: diameters corner-corner 1 m, pipe diameter 40 mm, wall thickness

10 mm Current in the ring: 8386 ARMS, sinusoidal Frequency: 1 kHz Materials: Copper

The computation of the model requires the solution of a dense (full) and complex linear equation system with 25641 independent loops for 11048 volume elements. The parallel Gauss algorithm is applied. The computation time on a standard workstation with two 10 core Xeon CPU and 64 GB RAM is 70 min in total. The solution of the linear equation system requires 50 min. Figure 4 depicts the course of the calcu-lation time.

Figure 4: Computation time as a function of the computation progress

The following figures 5-9 depict the solution in form of the current density distribution in logarithmic scaling. Though the discretisation is relatively rough the solution is outstandingly clear and accurate.


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Figure 5: Configuration of the model with current density distribution

Figure 6: Cross section of the plate with 4 layers a 2.5 mm


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Figure 7: Current density distribution in the copper plate, front side, connection wires: bottom

Figure 8: Current density distribution in the copper plate, back side, connection wires: top


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The computation of the magnetic field of the current density distribution as the basis for the force calcula-tion requires extensive computation and is also parallelized. Once the magnetic induction in every volume element is known, the forces and torques can easily be calculated. Figure 9 depicts the calculated total force on the plate as well as the torques along two middle axes on the front side of the plate. The force is directed downward perpendicularly, the torques want to bent the edges of the plate upward. The total force also acts on the ring in opposite direction but with a slight y-component caused by the connection wires. Ring and plate repell each other.

Figure 9: Total force (red) and torques (orange) along two centre lines acting on the plate

Force and torques are composed of a static and a dynamic part. The static part of the force / torque vector is represented by the red / orange lines, the dynamic parts by the ellipses at the end of the lines. The static force on the plate (red line) is Fz-stat = 190.6 N, the dynamic part is Fz-dyn = 6 N. The torque in the cen-tre of the plate is zero and maximal at the edges with M = 190.6 6 Nm. Conclusion

The Finite Network Method (FNM) is specialized to solve eddy current problems of high power electrical engineering at mains frequency. The solutions are stable and accurate, the computation time is efficient and allows to compute different scenarios or discretisations in short time using a normal workstation. Badische Stahl-Engineering GmbH (BSE) applies the FNM program system as explained for the analysis, optimization and dimensioning of arc furnace high current systems. For other application fields BSE can provide a computation service.

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References

[1] A. Farschtschi: „Neuartiges Berechnungssystem löst elektromagnetische Probleme an Elektrolichtbo-genöfen“, stahl & eisen 131 (2011) Nr. 6/7

[2] A. Farschtschi: „An advanced computation system to solve electromagnetic problems in arc furnaces”, Steel Times International, September 2011

[3] D. Riedinger, A. Vogel, A. Farschtschi: „A new dimension of designing arc furnace high current sy-stems“, stahl & eisen 135 (2015) Nr. 8

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