200 Electronics, Robotics and Automotive Mechanics Conference Control of the Magnetizing Characteristics of a Toroidal Core Using Virtual Ga S. Magdaleno, C. Pérez Rojas 2 INDUSTRIAS IEM S.A. DE C.V., Vía Dr. Gustavo Baz Prada No. 340, Col. Industrial Barrientos, C.P. 5405 Tlalneantla, Edo. de Mexico, Mexico smagdalenoa@hotmail.com 2 U.M.S.N.H, Av. Fco. J. Múgica S/N, C.P. 58030, Morelia, Michoacán, Mexico crojas@umich.mx Abstract A study of the technology of the virtual ga in a toroidal core is analyzed and showed in this aer. We made a study about of the effects of a virtual ga in a toroidal core. Through nonlinear finite element simulations we studied the effects that a virtual ga has on a toroidal core. We realized several analysis magnetic static (using only DC currents). The finite element simulations show the function of a virtual ga in a toroidal core and how this technology can hel to control the magnetizing characteristics of a toroidal core. Furthermore, we roosed a method to estimate the equivalent air ga using the results obtained with a virtual ga in the same toroidal core. Moreover, we show how a virtual ga can control the characteristics of the magnetizing curve of a toroidal core varying the DC current intensity in the windings of the holes and how we can vary the saturation degree and the size of the virtual ga of the same manner. Index Terms Virtual ga, Toroidal core, Air ga, Magnetizing Curve. 2. FINITE ELEMENT SIMULATIONS We realized several simulations using finite elements to rove the function of a virtual ga in a toroidal core with nonlinear characteristics [7], [8]. Fig. shows the toroidal core in 2D used in this aer where OD is the external diameter, ID is the internal diameter and HT is the height. Fig. shows the magnetizing curve of the toroidal core. Note that the toroidal core has a nonlinear region of 0T to.7t; in this region the core is not saturated. But from.7t to 2T is saturated.. INTRODUCTION The technology called virtual ga or virtual air ga was roosed in []. The rincial idea of a virtual ga is to make holes on an iron core and to wind a winding through them. Through the windings we ass DC currents to roduce a localized saturation region. Whit this, one can control the DC current in the windings of the holes; therefore, the size and degree of saturation can be controlled too. In [] virtual gas have been used to limit the inrush current that a transformer draws from the line. In [2] the virtual ga technology has been used to roduce an enhanced soldering machine. The new machine allows the use of AC ower to roduce the high quality soldering results tyical of DC soldering. A rocedure for equivalent air ga length comutation is established by erforming finite element analysis to obtain the change in transformer core reluctance, in [3] and [4]. In [5] and [6] the virtual ga is used for the control of a variable reactor to obtain a continuously variable reactance without introducing the harmonics created by the thyristor switching. In this aer we have used toroidal iron cores because by design they do not have air gas. They are built from a continuous stri of grain oriented silicon steel. Toroidal cores offer simultaneously the best magnetizing roerties, the least tolerance to DC excitation offsets and the largest inrush currents, in [7] and [8]. Fig.. Characteristics of the toroidal core. We considered that the toroidal core has a stacking factor k=. Therefore, the toroidal core has a cross sectional area S=0.025m 2 and a geometric medium length l =0.23569m. We build a virtual ga using four holes. The distances used to localize each one of the four holes of the virtual ga in the toroidal core are show in the Fig.2. Note that the holes are distributed in radial form in the toroidal core. 978-0-7695-4204-/0 $26.00 200 IEEE DOI 0.09/CERMA.200.2 540
shows the number of finite elements used in each art of the model. Fig.5 shows the mesh of the comlete model. TABLE I NUMBER OF FINITE ELEMENTS OF THE MODEL Fig.2. Distances to localize the four holes of the virtual ga. The distances used to localize the virtual ga in a core are: X = X () 3 X = 2 X = 2 X (2) 2 3 If we ut () and (2) in function of the diameters of the toroidal core, we obtain this: ( OD ID) X = X = = 0.25 ( OD ID) (3) 3 8 ( OD ID) X 2 = = 0.25 ( OD ID) (4) 4 We calculated these distances using (3) and (4) and we obtained that X =X 3 =6.25x0-3 m and X 2 =0.025m. Each hole has a diameter of 2mm. In addition, we considered that each hole has a winding with N a = turn and with the same diameter of 2mm. Fig.3 shows the direction of the magnetomotive forces Fa in each one of the windings of the holes. In Out F a =(N a )( ) Fig. 3. Producing a virtual ga in a toroidal core. We collocated a main winding for magnetizing the toroidal core. We considered that the winding has N = turn and a thickness ξ=2.5mm. Fig.4 shows this main winding. F a =(N a )( ) Devanado rincial Main Winding Región externa (Aire) External region (Air) Aire Parts of the Model Number of Finite Elements Saces of air 2062 Exciting winding 06 Windings of the holes 056 Toroidal core 7466 Total=690 y z x Fig. 5. Mesh of the comlete model. A. Study of a Virtual Ga in a Toroidal Core Toroidal Core We magnetized the toroidal core injecting a DC current in the main winding. We took as reference a single oint of the magnetizing curve for magnetizing the toroidal core where H=30.9A t/m and B=.7T. We alied Amere s Law in the geometric medium length l of the toroidal core to calculate the DC current intensity in the main winding. We calculated a DC current I =30.84A and we alied this current in the main winding. In this case, we did not inject DC current in the windings of the holes ( =0A). Fig.6 shows the inactive virtual ga. P X Núcleo ANSS 0.0 NOV 26 2005 4:46:45 STEP= SUB = TIME= RSS=0 EFACET= SMN =.05E-07 S =2.04.05E-07.223755 B[T].4475.67265.89502.07E-07.223752.9.447504.343.67256.566.895008.79.9 2.04.343.566.79 2.04 Length l Devanados de los agujeros Windings of the holes Windings of the holes Devanados de los agujeros Fig. 6. Magnetic flux density in toroidal core with inactive virtual ga. Fig. 4. Main winding. The model was closed in a circular sace of air from radius equal to 00mm. After, the model was meshed using triangular finite elements. Each finite element has six nodes. Table I. For this case we measured the values of H and B out of the region of the virtual ga in the oint P. We measured a value of H=3.845A t/m and B=.63378T in this oint. These values do not corresond to value of H=30.9A t/m and B=.7T. This is caused by the resence of the four holes on 54
the toroidal core. Because of that the holes and their windings have a relative ermeability equal to their reluctance is high comares to ermeability of the core. Note that the magnetic flux does not ass through the holes. This magnetic flux surrounds the holes and accumulates around of them. This accumulation of magnetic flux roduced small regions saturated. For this reason the value of H and B decrease out of the toroidal core. For this new oint (H, B) measured out of the virtual ga corresonds a ermeability µ=.435x0-2 H/m. For roducing a virtual ga in the region of the holes we injected a DC current = 00A in the windings of the holes. Fig.7 shows the active virtual ga. P Fig. 7. Magnetic flux density in toroidal core with active virtual ga. In the region of the virtual ga exists high value of H sat and B sat. These values were measured using the maximum values of H and B localized in the virtual ga. The values measured were H sat =682A t/m and B sat =2.257T. This oint is localized in the saturated region of the magnetizing curve. For this oint corresonds an average ermeability µ sat =2.0209x0-5 H/m. This average ermeability is very close to ermeability of the vacuum. Furthermore, we measured values of H and B out of the virtual ga in the oint P. The new values measured were H ev =25.4334A t/m and B ev =0.25884T. This oint (H ev, B ev ) is localized in the region no saturated of the magnetizing curve. For this oint (H ev, B ev ) corresonds a ermeability value µ ev =.077x0-2 H/m. Fig. 8 shows the oints of oerations when the virtual ga is inactive and active. Table II shows the values of the oints of oeration of the virtual ga. TABLE I I VALUES OF POINTS OF OPERATION OF THE VIRTUAL GAP Points Variable Values Inactive H 3.845A t/m virtual ga (H, B) B.63378T µ.4350x0-2 H/m Active virtual ga (H sat, B sat) Active virtual ga (H ev, B ev) H sat 682A t/m B sat 2.257T µ sat 2.0209x0-5 H/m µ 0 H ev X 25.4334A t/m B ev 0.25884T ANSS 0.0 OCT 9 2005 2:05:8 STEP= SUB = TIME= RSS=0 EFACET= SMN =.35E-05 S =2.257.35E-05.250753 B[T].50502.75225.003.337E-07.254.25088.504.50632.755.752446 2.006.003 2.257.254.505.756 2.007 2.257 Length l µ ev.077x0-2 H/m Fig. 8. Points of oeration of the virtual ga. One can observe that when the virtual ga is active roduce a new oint of oeration (H ev, B ev ). This oint of oeration is different to oint of oeration original (H, B). The virtual ga functions equal that an air ga modifying the oint of oeration original to another oint different on the same magnetizing curve. We can conclude that a virtual ga ermits to modify a oint of oeration in the magnetizing curve to another new oint. This new oint deends of the DC current intensity injected to windings of the holes. B. Comutation of Equivalent Air Ga We urose a method to calculate the equivalent air ga using the results obtained in the case of the active virtual ga with =00A. We suosed that we would have the same values of H ev =25.4334A t/m and B ev =0.25884T in the toroidal core with the air ga. We used these values for calculating the value of the length of the air ga l g. Fig.9 shows a toroidal core with its air ga. Fig. 9. Toroidal core with its equivalent air ga. We suosed that N = turn for the main winding and that the toroidal core has the same magnetic material and the same geometric characteristics used in the ast model. We alied Amere s Law to model of the Fig. 9 through the geometric medium length l of the toroidal core. + F = Hidl H idl = N I (5) g g l l The relation between H g and B g in the air ga is: g B = µ H (6) g 0 g 542
We suosed that the length of the air ga is very small; therefore, we did not take in account the effect of contour in the air ga. Therefore: B = Bg C. Control of the Characteristics of the Magnetizing Curve We made several simulations of finite element to analyze the control that we have on the characteristics of the magnetizing curve of a toroidal core using a virtual ga. Extensive finite element simulations were carried out varying the DC current injected in the windings of the holes for a comlete set of different excitation levels. We varied the DC current in the main winding to imress flux densities in the toroidal core from 0.T to.7t in 0.T intervals. Fig. shows the results obtained on the control of the characteristics of the magnetizing curve of the toroidal core. The results show that we can obtain a comlete control on the characteristics of the magnetizing curve of a toroidal core using a virtual ga. This control deends of the DC current intensity in the windings of the holes. The DC current intensity defines the range of oeration on the magnetizing curve. If the DC current intensity is low then the range of oeration is wide. An examle is showed in Fig. where Ia=0A. For this case we oerate in a nonlinear region in the magnetizing curve in a range of oeration from 0T to.399t. On the other hand, if the DC current intensity is large then the range of oeration is narrow. An examle is showed in Fig.(c) where Ia=50A. For this case we oerate in a linear region in the magnetizing curve in a range of oeration from 0T to 0.299T. (7) Substituting H and Hg in (6) we obtained: Bg B µ i dl + µ i dl g l lg 0 = N I (8) Then, Bl µ + Bg l g µ0 = N I (9) The variables B and µ in (0) corresond to values Bev and µev for the case when the virtual ga is active with Ia=00A. If N= turn and we resolving the equation for lg in (9). I µ0 l µ0 lg = (0) Bev µev Substituting the values of I=30.84A, Bev=0.25884T, µev=.077x0-2 H/m and l=0.23569m in (), we obtain lg=0.2mm. With this methodology we obtained an air ga that is equivalent to an active virtual ga. We also simulated this toroidal core with its air ga of lg=0.2mm. Fig.0 shows the results obtained in the toroidal core with its equivalent air ga. ANSS 0.0 2 2006 Length l FEB 8:00:20 SUB = TIME= BSUM (AVG) RSS=0 EFACET= g SMN =.292E-05 S =.594202.292E-05.8843.237682.356522.475362.594202.292E-05 Air ga l =0.2mm P 27 X 26 25 24 23 22 2 20 9 8 7 B[T] Ia=0A [B=0T to.399t] y [H=0A t/m to 90.04A t/m] (c) Ia=20A (d) [B=0T to 0.7T] y [H=0A t/m to 52.69A t/m] (e) Ia=50A (f) [B=0T to 0.299T] y [H=0A t/m to 28.34A t/m].8843.237682.356522.475362 Fig. 0.Magnetic flux density in toroidal core with equivalent air ga. After we measured the value of H and B out of the air ga in the oint P. We obtained H=24.854A t/m and B=0.25T. These values are a few different to values obtained with the active virtual ga with Ia=00A but they are good and close results. For these values we calculated a ermeability µ=.0098x0-2. Table III shows the results obtained with the simulations of finite elements for both cases. TABLE III COMPARATIVE RESULTS OF A VIRTUAL GAP VERSUS AN AIR GAP Variable B(T) H(A t/m) I(A) µ(h/m) Virtual ga Ia=00A 0.25884 25.4334 30.84.077x0-2 Air ga lg=0.2mm 0.25 24.854 30.84.0098x0-2 Fig.. Results of the control on the characteristics of the magnetizing curve. After we analyzed the behavior of the flux density in the toroidal core when the DC current intensity in the windings of the holes is constant and we varied the DC current in the main winding. 543
ga only works in regions no saturated of the magnetizing curve and that one can control H, B and µ values in and out of the virtual ga varying the DC current intensity in the windings of the holes. Finally, one can observe that a virtual ga deends of the control of the DC current intensity in the windings of the holes and of the current in the main winding. (e) -6 I=0A, Ia=0A, B=0T, Bev=4.3028x0 T -6 I=0A, Ia=00A, B=0T, Bev=8.6450x0 T I=4.9A, Ia=0A, B=0.2T, Bev=0.68T (f) I=4.9A, Ia=00A, B=0.2T, Bev=0.T (c) I=4.57A, Ia=0A, B=0.9T, Bev=0.373T (g) I=4.57A, Ia=00A, B=0.9T, Bev=0.20T D. Control of the Saturation degree of the Virtual Ga To obtain a control on the saturation degree of the virtual ga is necessary to vary the DC current intensity injected in the winding of the holes. For this, we varied the DC current intensity from Ia=0A to 00A in intervals 0A. Fig.3 shows the maximum values obtained for Hsat and Bsat for the different variations of DC current intensity in the windings of the holes. The relation between the Fig.3 and results in the average ermeability of the virtual ga. Fig.4 shows the variations of this ermeability in the virtual ga versus the variations of the DC current intensity in the windings of the holes. (d) I=394.9A, Ia=0A, B=.9T, Bev=.87T (h) I=394.9A, Ia=00A, B=.9T, Bev=.86T Fig.2. Flux densities for different DC current intensities in the main winding. Fig.2 illustrates the sequence of the magnetic field for different main winding excitation levels for the case when the DC current in the windings of the holes is Ia=0A and 00A. As we can see from the figure, the saturation region is a function of the DC current intensity in the windings of the holes and the instantaneous magnetic field strength alied to toroidal core. Fig. 2 and (e) corresond to case where the DC current in the main winding is zero (I=0A) and Ia=0A and 00A. Once can areciate that the DC current in the windings of the holes roduces a local magnetic field in the neighborhood of the holes. There is no net flux circulating around the toroidal core, however when the DC currents in the main winding and in the windings of the holes are large enough the saturation region around the holes disaears, see Fig. 2(d) and (h). The effect of the virtual ga disaears because of that in this saturation condition the toroidal core has the same ermeability in and out of the virtual ga. In this condition the toroidal core is saturated. One can conclude that the virtual Fig. 3. Results obtained of Hsat and Bsat in the virtual ga. Fig. 4. Control of the average ermeability in the virtual ga. E. Control of the Size of the Virtual Ga The size of the virtual ga lev deends of the DC current intensity alied to windings of the holes. We alied a DC 544
current intensity in the windings holes for two cases, for =A and =00A. Fig.5 shows the results of the size of the virtual ga for both cases. MN MN X X ANSS 0.0 NOV 23 2005 22:07:42 STEP= SUB = TIME= RSS=0 EFACET= SMN =.02E-07 S =.762.02E-07 l ev.95746.39492.587238.782984.97873 B[T].74.37.566.58E-0.762.95663.39326.586989.782652.97835.74.37.565.76 Fig. 5. Size for different virtual gas. Fig.5 illustrates how the size of the virtual ga l ev, (the saturated zone around the holes) can be controlled by the DC current intensity alied to them. Fig. 5 shows the magnetic flux density when the current in the holes is A and Fig. 5 shows the magnetic flux density for a current in the holes of 00A. Not only the saturated zone is larger when a larger current is alied to holes, but also the saturation is higher. In Fig. 5 the magnetic flux density varies from 0.9T in the region far from the holes to.76t right at the holes. In Fig. 5 the magnetic flux density goes from 0.25T in the region far from the holes to 2.28T for the internal region. The magnetic flux density of 2.28T corresonds to comlete saturation of the region. Therefore a ga has truly formed since the ermeability of that region has reduced to the same of the air. ANSS 0.0 NOV 23 2005 22:2:47 STEP= SUB = TIME= RSS=0 EFACET= SMN =.285E-06 S =2.288.285E-06.254264 l ev.508528.76279.07 B[T].27.526.78.28E-08 2.034.254267 2.288.508534.76280.07.27.526.78 2.034 2.288 in the windings of the holes. One can work in the nonlinear or linear region on the magnetizing curve; it deends of the DC current intensity in the windings of the holes. Furthermore, we rove that a virtual ga works equal that an air ga in a toroidal core. Both gas control the magnetizing characteristics of the toroidal core. Therefore, we rove that the saturation degree and the size of the virtual ga deend of the DC current intensity in the windings of the holes. REFERENCES [] V. Molcrette, J.L Kotny, J.P. Swan and J.F. Brudny, Reduction of Inrush Current in Single-Phase Transformer using Virtual Air Ga Technique, IEEE Transactions on Magnetics, Vol. 34, No. 4, July 998,. 92-94. [2] Ewa Naieralska Juszczak, Jean Philie Lecointe, The active control of the leakage reluctance in welding transformers, IEEE RVP-AI/O2-AI-08- Acaulco, Mexico, July 2002. [3] A. Konrad, J.F. Brudny, An Imroved Method for Virtual Air Ga Length Comutation, IEEE Transactions on Magnetics, Vol. 4, No. 0,.405 4053, October 2005. [4] A. Konrad, J.F. Brudny, Virtual Air Ga Length Comutation with the Finite Element Method, IEEE Transactions on Magnetics, Vol. 43, No. 4,.829 832, Aril 2007. [5] D.S.L. Dolan, and P.W. Lehn, Analysis of a Virtual Air Ga Variable Reactor, IEEE PESC07 Power Electronics Secialist Conference, Orlando, Florida, June 2007. [6] D.S.L. Dolan, and P.W. Lehn, Harmonics and Dynamic Resonse of a Virtual Air Ga Variable Reactor, IEEE Transactions on Power Delivery, aer TPWRD-00699-2007, submitted November 2007. [7] F. de León, S. Magdaleno, Finite Element Analysis of the Virtual Ga Technology: Controlling the Magnetizing Curve, IEEE RVP-AI/2005-TRO- 0-Acaulco, Mexico, July 2005. [8] S. Magdaleno, The Finite Element in the Control of the Magnetic Saturation of a Toroidal Core Using Virtual Ga, B.Sc. thesis, U.M.S.N.H, Morelia, Michoacán, Mexico, 2008. ACKNOWLEDGMENT The authors would like to recognize the assistance and contribution of Professor Francisco de Leon from the Polytechnic Institute of New ork University. 3. CONCLUSIONS A virtual ga has been created injecting DC current intensities in the windings of the holes in a toroidal core. A virtual ga has been created by magnetic saturation in the region of the holes. We observed that a virtual ga can control the magnetizing characteristics of a toroidal core using a DC current intensity 545