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Skin and Proximity Effects in Two Parallel Plates View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CORE Wright State University CORE Scholar Browse all Theses and Dissertations Theses and Dissertations 2007 Skin and Proximity Effects in Two Parallel Plates Hamdi Eltayib Abdelbagi Wright State University Follow this and additional works at: https://corescholar.libraries.wright.edu/etd_all Part of the Electrical and Computer Engineering Commons Repository Citation Abdelbagi, Hamdi Eltayib, "Skin and Proximity Effects in Two Parallel Plates" (2007). Browse all Theses and Dissertations. 184. https://corescholar.libraries.wright.edu/etd_all/184 This Thesis is brought to you for free and open access by the Theses and Dissertations at CORE Scholar. It has been accepted for inclusion in Browse all Theses and Dissertations by an authorized administrator of CORE Scholar. For more information, please contact [email protected]. SKIN AND PROXIMITY EFFECTS IN TWO PARALLEL PLATES A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Engineering By Hamdi Altayib Abdelbagi B.S.EE, 2004 Wright State University 2007 Wright State University WRIGHT STATE UNIVERSITY SCHOOL OF GRADUATE STUDIES August 14, 2007 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVISION BY Hamdi Eltayib Abdelbagi ENTITLED Skin and Proximity Effects in Two Parallel Plates BE ACCEPTED IN PAR- TIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in Engineering Marian K. Kazimierczuk, Ph.D. Thesis Director Fred D. Garber, Ph.D. Department Chair Committee on Final Examination Marian K. Kazimierczuk, Ph.D. Gregory Kozlowski, Ph.D. Ronald Riechers, Ph.D. Dr. Josph F. Thomas, Jr, Ph.D. Dean, School of Graduate Studies Abstract Abdelbagi, Hamdi. M.S., Department of Electrical Engineering, Wright State University, 2007. SKIN AND PROXIMITY EFFECTS IN TWO PARALLEL PLATES Time varying currents within winding and core conductors induce magnetic fields. When more than one conductor is present the resultant magnetic field can be found by adding the individual magnetic fields by superposition. The resultant magnetic field in turn induces eddy currents within each electrical component within the vicin- ity of the resultant magnetic field. Eddy currents flow in the opposite direction of the primary current and increase the resistance by reducing the area in which the primary current has to travel. Eddy currents also reduce the effectiveness of the conductors to conductor high frequency currents. Skin and proximity effects were numerically investigated for two parallel plate conductors while a laminated core was designed to reduce the power losses. Maxwell’s equations were solved to obtain analytical equa- tions for magnetic fields eddy current distribution and power losses. These equations were illustrated in MATLAB for various frequencies to validate the theoretical analy- sis. Results demonstrate current within an isolated conductor flows near the surface. However, when the same conductor is placed near another conductor the flow path is affected. For the case when the current is flowing in the opposite direction, the magnetic fields are added in the area between the conductors and subtracted on the outer side of the conductor. This causes an increase of the current density within the conductor areas, where the conductors are close to each other. This is the proximity effect. The anti-proximity effect occurs when two conductors carry current in the same direction. In this case the magnetic fields are subtracted from each other in the area between the conductors and are added to each other in the area outside the conductors resulting in a higher current density in these areas. The eddy currents iii can be reduced in two ways. Using a highly resistant material for the core increases the skin depth making the distribution of the magnetic flux more uniform. Laminat- ing the core with an oxide film can be used to reduce the eddy current loss as well. The study shows that the eddy current power loss in a sold core is greater than loss in a laminated core by a factor of K2, where K is the number of the sheets in the laminated core. iv Contents 1 Introduction 2 1.1 Research Background and Motivation of Study . ... 2 1.2 TheObjective............................... 3 1.3 ThesisOutline............................... 4 2 Skin and Proximity Effects 5 2.1 SkinEffect................................. 5 2.2 ProximityEffect.............................. 6 3 Skin Effect in Single Rectangular Plate 9 4 Proximity and Skin Effects in Two Parallel Plates 23 5 Anti-proximity and Skin Effects in Two Parallel Plates 41 6 Laminated Cores 48 6.1 Low-FrequencySolution . 50 6.2 GeneralSolution ............................. 54 7 Summary 72 7.1 FutureWork: ............................... 74 v List of Figures 1 Skin depth δw as a function of frequency f ............... 6 2 Single isolated plate carrying current . .. 9 3 Plotof2a H(x) /I as a function of x/w ................. 10 | | 4 Plotoftherealpartos2a H(x) /I as a function of x/w ........ 10 | | 5 Plot of the imaginary part of 2a H(x) /I as a function of x/w .... 11 | | 6 Plotof2a J(x) /I as a function of x/w ................. 12 | | 7 Plotoftherealpartof2a J(x) /I as a function of x/w ........ 13 | | 8 Plot of the imaginary part of 2a J(x) /I as a function of x/w ..... 13 | | 9 Plotof J(x)/Jdc as a function of x/w at selected values for w/δw ... 14 10 Plot ofthe real partof J(x)/Jdc as a function of x/w at selected values for w/δw .................................. 14 11 Plot of of the imaginary part of J(x)/Jdc as a function of x/w at selected values for w/δw ......................... 15 2 12 Time-average skin-effect power loss 4awPD/bρwI as a function of w/δw at fixed δw ................................. 15 2 13 Time-average skin-effect power loss 4aδwPD/bρwI as a function of w/δw at fixed w .............................. 16 2 14 Time-average skin-effect energy stored in the plate 4aWm/bδwµwI as a function of w/δw ............................ 16 15 Ratioof Rw/Rwdc as a function of w/δw ................. 18 16 Ratioof XL/Rwdc as a function of w/δw ................. 19 17 Plotof Z /R as a function of w/δ ................. 20 | | wdc w 18 Ratioof φZ as a function of w/δw .................... 21 19 Ratioof XL/Rw as a function of w/δw ................. 21 20 Two plates carrying currents in opposite directions. ....... 23 vi 21 Plotof a H(x) /I as a function of x/w for selected values of w/δ in | | w the left plate due to the proximity effect. 24 22 Plot of the real part of a H(x) /I as a function of x/w for selected | | values of w/δw. The current density is given by in the left plate due to theproximityeffect. ........................... 24 23 Plotoftheimaginarypartof a H(x) /I as a functed of x/w for selection | | values of w/δw. The current density is given by in the left plate due to theproximityeffect. ........................... 25 24 Plot of a J(x) /I as a function of x/w for selected values of w/δ in | | w the left plate due to the proximity effect. 27 25 Plot of the real part of a J(x) /I as a function of x/w for selected | | values of w/δw in the left plate due to the proximity effect. 27 26 Plot of theimaginary part of a J(x) /I as a function of x/w for selected | | values of w/δw in the left plate due to the proximity effect. 28 27 Plotof J(x) /J as a function of x/w for selected values of w/δ in | | dc w the left plate due to the proximity effect. 29 28 Plot of the real part of J(x) /J as a function of x/w for selected | | dc values of w/δw in the left plate due to the proximity effect. 30 29 Plotoftheimaginarypartof J(x) /J as a function of x/w for selected | | dc values of w/δw in the left plate due to the proximity effect. 30 30 Plots of P (x)/ρw as a function of x/w for selected values of w/δw in the left plate due to the proximity effect. 31 31 Plotsof J ( w/2)/J ( w/2) as a function of w/δ . ........ 32 | sp − s − | w 32 Plotof J (w/2)/J (w/2) as a function of w/δ ............ 32 | sp s | w 2 33 Plot of the power loss due to skin and proximity effects awPsp/bρwI as a function of w/δw at fixed w. .................... 33 vii 2 34 Plot of the power loss due to skin and proximity effects aδwPsp/bρwI as a function of w/δw at fixed δw ..................... 33 2 35 Plot of the power loss due to skin effect awPs/bρwI as a function of w . 34 2 36 Plot of the power loss due to skin effect aδwPs/bρwI as a function of w/δw at fixed δw. ............................. 34 2 37 Plot of the power loss due to proximity effect awPp/bρwI as a function of w/δw at fixed w............................. 35 2 38 Plot of the power loss due to proximity effect aδwPp/bρwI as a function of w/δw at fixed δw............................. 35 39 Plot of the ratio Pp/Psp as a function of w/δw at fixed w........ 36 40 Plot of the ratio Ps/Psp as a function of w/δw . ............ 36 41 Plot of the ratio Pp/Ps as a function of w/δw .............. 37 42 Plot of the ratio Rw/Rwdc as a function of w/δw . ........... 37 43 Plotof XL/Rwdc as a function of w/δw .
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