IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014
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Numerical and Experimental Validation of Discharge Current With Generalized Energy Method and Integral Ohm’s Law in Transformer Oil Ho-Young Lee1 , Jae-Seung Jung1, Hong-Kyu Kim2 , Il-Han Park3, and Se-Hee Lee1 1 Department
of Electrical Engineering, Kyungpook National University, Daegu 702-701, Korea Electrotechnology Research Institute, Changwon 642-120, Korea 3 College of Information and Communication Engineering, Sungkyunkwan University, Suwon 440-746, Korea 2 Korea
The discharge currents were evaluated and validated using the generalized energy method and the integral Ohm’s law combined with the recently developed discharge analysis technique for dielectric liquid media, such as transformer oil. The terminal current in voltage-driven systems was found to play an essential role in characterizing the pattern of electric discharge, such as corona, breakdown, etc. Until now, the generalized energy method and integral Ohm’s law were normally adopted to evaluate this terminal current, but no experimental validation was reported and no test was performed in a dielectric liquid media. The generalized energy method can be derived directly using Poynting’s theorem and is well suited for the finite element method. As an alternative approach, the integral Ohm’s law can be applied to multiport systems. To confirm the numerical results, an experimental setup was proposed with a multiport system composed of a tip and separated conducting ring shells. The numerical results were compared with those from experiments, which would be the first trial in a dielectric liquid with a multiport system. The calculated current profile was similar to that from the experimental result, but the breakdown voltage from the calculated results was relatively high. Index Terms— Discharge current, energy method, FEM, integral Ohm’s law, Poynting’s theorem.
I. I NTRODUCTION ALCULATIONS of a current flowing between two electrodes during a discharge simulation is important because the terminal current is often one of the only parameters that can be measured during an actual breakdown test [1]–[7]. In a high voltage discharge and plasma analysis with gaseous media, the expanded Sato’s equation has been used widely to calculate the terminal current [8], [9]. The final Sato’s equation can be applied successfully to most discharge problems [9]. In an arc simulation, when the magnetic field effect is significant, the calculation method for a terminal current should be modified and generalized. With the energy balance equation, the generalized energy method can be derived to calculate the terminal current by adopting Poynting’s theorem incorporating the FEM, which has been applied successfully to discharge analysis. In this approach, global quantities, such as energy and Ohmic dissipation, were employed directly in the expression of Poynting’s theorem with the terminal quantities, current, and voltage. This generalized energy method naturally covers the time-varying voltage sources, magnetic field effect, and any dielectric media, such as gas, liquid, and solid. As an alternative method, this paper tested the integral Ohm’s law, which can be applied to multiport systems. Sato’s equation and the generalized energy method, however, can be applied to only a two-terminal electromagnetic system because it is difficult to separate the energy contributions from each port. Although the integral Ohm’s law can calculate the terminal current at each port, the result has some numerical distortions caused by an abrupt change in electric field intensity and space charge density. To test the two proposed
C
Manuscript received June 29, 2013; revised July 30, 2013; accepted August 16, 2013. Date of current version February 21, 2014. Corresponding author: S.-H. Lee (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/TMAG.2013.2279181
methods, the results were compared with those from the Sato’s equation and an analytic solution with a simple geometry, a plane–plane electrode [10]. The numerical results from the generalized energy method and the integral Ohm’s law were then compared with those from the experiments. II. G OVERNING E QUATIONS FOR L IQUID D ISCHARGE The general expression of the governing equations for space charge propagation can be analyzed using the hydrodynamic diffusion–drift model for electrons, positive ions, and negative ions as follows [6], [11], [12]: −∇ · (ε∇V ) = ρ+ + ρ− + ρe ∂ρ+ ρ+ ρ− R+− + ∇ · J + = G I (|E|) + G D (|E| , T ) + ∂t e ρ+ ρe R+e + e ∂ρ− ρe ρ+ ρ− R+− + ∇ · J − = −G D (|E| , T ) − + ∂t e τa ρe ∂ρe ρ+ ρe R+e + ∇ · J e = −G I (|E|) − + ∂t e τa 1 ∂T 2 + v · ∇T = (k T ∇ T + E · J) ∂t ρl cv
(1)
(2) (3) (4) (5)
where the subscripts +, −, and e indicate the positive, negative ions, and electrons, respectively, ε is the dielectric permittivity, V is the electric scalar potential, ρ is the charge density, t is the time, G I (|E|) is the electric field-dependent molecular ionization source term, G D (|E|, T ) is the electric field and the temperature dependent ionic dissociation source term, e is the electric charge, Rx y is the recombination rate of x and y carriers, τa is the electron attachment time constant, T is the temperature, ρl and cv are the mass density and the specific heat capacity, respectively, k T is the thermal conductivity. E·J denotes the electrical power dissipation term in the fluidic medium.
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The setup for G I (|E|) and G D (|E|, T ) was developed in a previous study [11]. The present study adopted the same setup for discharge analysis. To solve these coupled equations, COMSOL Multiphysics software was employed and the detailed numerical setup was implemented in that program. III. R EVISITED T ERMINAL C URRENT C ALCULATING M ETHODS The expanded Sato’s equation considering the three carriers and a time-dependent applied voltage was expressed as [9] ε0 1 ∂E L Jc · E L dv + (6) ·E L dv I = Va Va ∂t
Fig. 1. Lower electrode at x = 0 as a source of injected positive charge with a mobility μ in a medium with permittivity ε. Here, ρ is the charge density and A is the linear charge injection coefficient [10]. C1 and C2 represent the anode and cathode, respectively.
with J c = (N p V p − Ne Ve − Nn Vn − D p ∇ N p + De ∇ Ne +Dn ∇ Nn ) where I is the external circuit current, Va is the applied voltage, Jc is the conduction current, E L is the electric field intensity from the Laplace equation, εo is the dielectric permittivity in air, and dv is a volume integral over the discharge space. In general, the terminal current is due to the conduction current in a dielectric medium as well as the displacement current due to the time rate of the change in the surface charge on the electrodes as follows: ∂E I = · da (7) Jc + ε ∂t S
where E is the total electric field intensity due to the applied voltage and space charge distributions, and (7) is known as the integral’s Ohm’s law. By considering any general field within a volume, one realizes that the energy contained by that field must be distributed throughout space with a local energy density of W at every point in the volume. With field generalization and quasi-static approximations, the power flowing into a volume , enclosed by the surface, Sa , can be expressed as n d Vi Ii = W dv + Pd dv (8) dt v i=1
with W = (1/2)εE · E + (1/2)μE · E and Pd = E · J c where Vi is the applied voltage at a terminal, Ii is the corresponding current at that terminal, μ is the magnetic permeability, and H is the magnetic field intensity. Generally, the FEM provides more accurate global quantities, such as energy and total power dissipation, because the procedure of the FEM follows the global energy minimization condition. This is based on the principle of virtual work, which is incorporated with a variation of the Ritz method for deriving the FEM formulation in static and transient cases. Therefore, this generalized energy method is in harmony with the FEM. Note that (6) and (8) can only be used to establish the current flowing into a volume when the number of terminal pairs n = 1, i.e., when the volume only has two terminals. On the other hand, the integral Ohm’s law, (7) can be applied to a multiport system when n > 1.
Fig. 2. Terminal current profiles from the various methods. The analytic solution was evaluated using Zahn’s approach in [10]. Here, the direct approach represents the integral Ohm’s law.
IV. E XPERIMENTAL AND N UMERICAL R ESULTS A. Verification of Numerical Setup with Parallel Plane Model To verify the numerical setup and compare with each other, first, the governing equations were simplified to a single carrier system in a plane-plane 2-D XY geometry, as shown in Fig. 1. The governing equations are the Poisson’s equation and the charge conservation equations as follows: ∂ρ +∇ ·J = 0 ∂t ∇ · (−ε∇V ) = ρ.
(9) (10)
As shown in Fig. 2, the energy approaches, Sato’s equation and generalized energy method, produced similar results to the analytic solutions [10]. Although the integral Ohm’s law can measure the terminal current on each electrode, it contained some numerical distortions, where the value of the space charge was high. B. Experimental and Numerical Results for a Multiport System With Tip-Ring Electrodes in Transformer Oil Fig. 3 shows a schematic diagram of the experimental setup of the tip-ring electrodes for evaluating the discharge current. The plate electrode was split into eight ring shell electrodes to check the space contributions of the current in each channel, as shown in Fig. 4. When the breakdown voltage was applied between the anode and cathode, the bright light can be viewed in the discharge column, as shown in Fig. 5,
LEE et al.: NUMERICAL AND EXPERIMENTAL VALIDATION OF DISCHARGE CURRENT
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(a)
(b)
Fig. 6. Total current profiles in the experiments. The applied breakdown voltage and peak current were approximately (a) 17 kV and 4.7 A and (b) 35 kV and 4.0 A, respectively. The peak current was decreased by approximately 14.9% with the longer gap.
Fig. 3. Schematic diagram of the set-up for experimental and numerical validation with the tip-plates model for measurements of the discharge current.
Fig. 4. Plate electrodes for current measurements. The electrodes were split and insulated.
Fig. 7. Calculated current profiles with a high voltage input by the integral Ohm’s law (direct approach) and generalized energy method for the tip-plates model. The total current by the direct approach was the same as that by the generalized energy method. TABLE I N UMERICAL R ESULTS IN D IELECTRIC L IQUID
Fig. 5. Snapshot of the breakdown phenomenon between the electrodes with 10 mm gap. The breakdown voltage reached 35 kV.
where the discharge current was measured in the cases of 2.5 and 10 mm gap electrodes, as shown in Fig. 6. The peak current was decreased by approximately 14.9% with a longer gap electrode from 4.7 to 4 A measured in the first period. Using the above numerical setup, the terminal current was obtained with different calculating methods, resulting in the same total terminal current profile, as shown in Fig. 7. The summation of each shell current by the integral Ohm’s law agreed well with that determined using the generalized
energy method. The peak currents determined by the numerical approach were 3.92 and 3.39 A with 2.5 and 10 mm gap electrodes, respectively. The numerical results were lower than those from the experimental approach, as shown in Table I. Although the peak current was not matched precisely, the decreasing rate was 13.6%, which is similar to that of the experimentally determined current between the 2.5 and 10 mm gap electrodes. To enhance the numerical results, the following can be considered: the surface roughness of the electrodes, purity of the transformer oil, and the electro hydrodynamic (EHD) effect [13]. Those issues will be the next important research topics in this area.
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014
was evaluated on the dummy middle line, and was increased and decreased along with the space charge propagation rate, as shown in Fig. 9.
Fig. 8. Calculated current profiles of the displacement and conduction current at the dummy middle surface located at z = 1.25 mm in the 2.5 mm gap electrode model.
V. C ONCLUSION The discharge current was analyzed using the generalized energy method and integral Ohm’s law for unipolar and three charge carrier systems with the FEM. To verify the generalized energy method and integral Ohm’s law, the parallel plane electrodes was tested with the unipolar case and their results were compared with those from the Sato’s approach and Zahn’s analytic solution. After theoretical verification, the generalized energy method and integral Ohm’s law were applied to the three charge carrier system with the tip-ring electrodes, where the experiments were conducted. The experimental results showed that the numerical results were sufficiently valid considering the surface roughness of the electrodes, purity of the dielectric liquid and electro-hydrodynamic (EHD) effect. Those will be critical issues in future discharge simulations with dielectric liquid materials. ACKNOWLEDGMENT This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2013R1A1A2013111). R EFERENCES
(a)
(c)
(b)
(d)
Fig. 9. Temporal electric field distributions with time. (a) Time = 100 ns. (b) Time = 200 ns. (c) Time = 350 ns. (d) Time = 470 ns.
Fig. 8 shows the current components, which can be divided into the displacement and conduction currents in the integral Ohm’s law in (7). Before the breakdown initiation, the displacement current was the main current source and was approximately 100 times larger than the conduction current, which was calculated on the dummy middle line for the 2-Daxial symmetry FE analysis region, as shown in Fig. 3. After 300 ns from the breakdown initiation, the conduction current
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