Design Study Reducing Core Volume in Matrix Transformers It is desirable to minimize the volume of a transformer core. It saves weight, space and cost. Some magnetic materials are quite expensive, and reducing the amount of material used may result in substantial savings. One way of reducing the core volume is to increase the flux density and/or the frequency of operation. Another may be to increase the number of primary turns so that fewer elements are used. However, this design study examines ways in which the core volume may be reduced while holding flux density, frequency and the number of elements constant. Figure 1 shows a hypothetical magnetic core in the shape of a cylinder. For comparison purposes, its cross sectional area is 360 mm 2 and its volume is 20,358 mm 3. Often in analyzing matrix transformer designs, the core losses will dominate and the winding losses will be small. If that is the case, winding losses may be allowed to increase to achieve greater reductions in core losses. Figure 2 shows a core having the same area and inside diameter as the core in figure 1 but it is twice as long. The volume is significantly less at 14,431 mm 3, 71 % of the volume of the core in figure 1. Taking this one step further, the core of figure 3 also has the same area and inside diameter as the core in figure 1. The length has been doubled again, and the volume is now 10,179 mm 3, 50 % of the volume of the core in figure 1. For the same operating conditions, since the area is the same, the flux density and frequency will be the same. Therefore, the loss per volume is the same, but the volume is one half, so the losses are one half as great. The winding losses will be four times greater than in the original design, but the total losses may be substantially less. The cost of the magnetic material will also be substantially less. Further, the mean magnetic path is less, so the inductance will be greater. If the core was selected for inductance, the area can be reduced as well. If the original core had its flux density de-rated for thermal considerations, that is very likely no longer necessary, and the core area can be reduced. Figures 4, 5 and 6 show the cores of figures 1, 2 and 3 with their inside diameter reduced by 50%. The volume of the fatter core is reduced somewhat, but the volume of the skinny core is reduced substantially, to 33 % of the volume of the core of figure 1. Thus, to reduce core volume, use longer, skinnier cores and reduce the inside diameter as much as possible. Although the winding losses are greater, the heat is dissipated over a much greater area, and the thermal path is much shorter, so the temperature rise may be much lower. If the temperature rise in the windings was a limiting factor in the original design, it may very well be possible to use a higher current
Fig. 1 V = 20,358 100 % Fig. 2 V = 14,431 71 % Fig. 3 V = 10,179 50 % Fig. 4 V = 16,965 83 % Fig. 5 V = 9,347 46 % Fig. 6 V = 6,786 33 %
density in the windings. That too would increase the winding losses, but the windings would be smaller. A smaller winding requires a smaller core, reducing core losses further, so the total losses and temperature rise may be lower. Figure 7 shows the effect of the penetration depth on a conductor. At dc, all of the conductor is carrying current, so a decrease in the diameter of a conductor results in a decrease in conductivity as the square of the diameter. For example, if the diameter is reduced to 50 %, the conductivity reduces to 25 %. In a conductor that is large compared to the penetration depth, the relationship is different. In the limit, with the diameter very much larger than the penetration depth, the effective area will reduce as the circumference of the conductor, so that if the diameter is reduced to 50%, the effective conductivity will also reduce to 50%. In figure 7, for the conductor shown, a reduction to 50 % of the diameter results in reduction of the conductivity to 44 %. A very good winding arrangement for a matrix transformer is coaxial or triaxial conductors. Figure 8 shows a cross section of a triaxial conductor which could be used in a forward converter with the outer layer as the secondary conductor, the next layer as the primary conductor with an optional center conductor as a reset winding. The penetration depth is shown. Figure 9 shows a cross section of coaxial or triaxial windings that have been optimized for minimum outside diameter. The actual area of the outer conductor has been made equal to the effective area of the inner conductor. Figure 10 shows coaxial or triaxial windings in a magnetic core, with allowance for insulation and clearance for assembly. For comparison purposes, this core has a normalized volume of V = 1. For lower ripple, the double forward converter is popular, and is shown in figure 11. For comparison purposes, its volume is V = 2. (Its power rating will also be twice as great) Figure 12 shows a push-pull transformer. The core volume V = 1.45 is substantially less than the core volume of the double forward converter, though the assembly and terminations are more complex. Figure 13 shows an optimized forward converter core and winding. To optimize the core, the inside diameter was reduced as much as practical. First, the outside conductor was made thinner, so that its cross sectional area equaled the effective area of the inside conductor (Figure 9). Next, the insulation and the clearance were reduced. By letting the core be part of the secondary circuit, only working insulation is required, not safety insulation. This can be very thin with modern dielectrics and dielectric coatings. If the core material has any significant resistivity, no insulation may be needed at all. This results in a relative core volume V=0.70, a 30% savings in core loss and core material cost. The profile is lower and the thermal conduction from the winding to the outside surface of the core is much better without the extra insulation.
Relative area d = 0.5 Fig. 7 A = 0.25 A = 0.444 λ Coaxial, Triaxial Fig. 8 Optimized Coaxial, Triaxial Fig. 9
Forward Converter With reset winding V = 1 Double Forward Converter With reset windings Fig. 10 V = 2 Push-Pull Converter v = 1.45 Fig. 11 Optimized Forward Converter With reset winding Fig. 12 V = 0.70 Fig. 13 Optimized Double Forward Converter With reset windings V = 1.40 Fig. 14
Figure 14 shows a pair of the core and windings of figure 13, for use as a double forward converter. The total relative core volume is 1.40, just about the same as for a push-pull converter. Thus, by optimizing the core and winding, and allowing the core to be part of the secondary circuit, the double forward converter can be substituted for the push pull converter with no penalty in core loss or core material volume. It will have a substantially lower temperature rise, so if the hot spot temperature was the limiting factor in determining the transformer rating, this design can have a substantially higher rating. Figures 15 through 19 show a possible core construction for a forward or double forward converter. The secondary conductor is a thin tube upon which the core is placed. For a metal or amorphous metal core, the magnetic material may be in strip form and can be wound directly on the secondary conductor. A film insulation on the conductor may be needed, but probably is unnecessary. For a ferrite core, a number of small toroids or beads can be slipped over the conductor as in figure 20. Ends can be formed or attached, and may have surface mount terminals as shown or through hole pins. Figure 20 shows a coaxial cable installed as the primary wire with an optional reset winding. If a reset winding is not needed, ordinary hook up wire can be used as shown in figure 21.
Fig.15 Fig.16 Fig.17 Primary Fig. 18 Fig.19 Reset Insulation Secondary Fig.20 Primary Insulation Secondary Fig.21