HIGH FREQUENCY INTEGRATED PLANAR MAGNETICS FOR A BIDIRECTIONAL AC TO DC CLLC RESONANT CONVERTER

Abstract

A transformer for a power converter, comprising: a first auxiliary subcore, a central subcore, and a second auxiliary subcore, each respective subcore comprising a lower plate, at least one pair of central spacers, and an upper plate, the lower plate, at least one pair of central spacers, and the upper plate of each subcore, being respectively separated by a gap; the first auxiliary subcore and the central subcore being separated by a gap; the second auxiliary subcore and the central subcore being separated by a gap; a primary coil, encircling a first spacer of the first auxiliary subcore and a first spacer of the central subcore; and a secondary coil, encircling a second spacer of the second auxiliary subcore and a second spacer of the central subcore.

Claims

1. A transformer for a power converter, comprising: a central subcore comprising: a central subcore lower plate separated by a gap from at least one gap-separated set of central subcore spacers, and a central subcore upper plate separated by a gap from the at least one gap-separated set of central subcore spacers; a first auxiliary subcore comprising a first auxiliary subcore lower plate separated by a gap from at least one gap-separated set of first auxiliary subcore spacers separated by at least one gap, and a first auxiliary subcore upper plate separated by a gap from the at least one gap-separated set of first auxiliary subcore spacers; a second auxiliary subcore comprising a second auxiliary subcore lower plate separated by a gap from at least one gap-separated set of second auxiliary subcore spacers separated by at least one gap, and a second auxiliary subcore upper plate separated by a gap from the at least one gap-separated set of second auxiliary subcore spacers; the first auxiliary subcore and the central subcore being separated by a gap; the second auxiliary subcore and the central subcore being separated by a gap; a primary coil, encircling a first spacer of the first auxiliary subcore and a first spacer of the central subcore; and a secondary coil, encircling a second spacer of the second auxiliary subcore and a second spacer of the central subcore.

2. The transformer according to claim 1, wherein the primary coil and the secondary coil are each configured to produce a central magnetic field having an axis intersecting the lower plate and the upper plate of the central subcore.

3. The transformer according to claim 2, wherein the central subcore, the first auxiliary subcore, and second first auxiliary subcore are each gapped along the axis.

4. The transformer according to claim 2, wherein at least one of the central subcore, the first auxiliary subcore, and second first auxiliary subcore is split along the axis by at least two gaps.

5. The transformer according to claim 2, wherein each of the at least one gap-separated set of central subcore spacers is split along the axis by at least one gap.

6. The transformer according to claim 2, wherein each of the central subcore, the first auxiliary subcore, and second first auxiliary subcore is split along the axis by a common number of gaps.

7. The transformer according to claim 1, wherein: the gap separating the first auxiliary subcore and the central subcore is configured to decouple a magnetic flux therebetween; the gap separating the second auxiliary subcore and the central subcore is configured to decouple a magnetic flux therebetween; the first auxiliary subcore is configured to provide a reluctance path to a flow of leakage flux to achieve a value of a first side inductance; the second auxiliary subcore is configured to provide a value of a secondary-side inductance; and the center subcore carries a flux to meet a magnetizing inductance to transfer power between the first side to the second side.

8. The transformer according to claim 1, wherein: ϕ.sub.1 is a flux covering a reluctance path provided by the first auxiliary subcore comprising R.sub.1 and R.sub.2; ϕ.sub.2 is a mutual flux covering a reluctance path shared by a portion of the first auxiliary subcore and a first portion of the central subcore comprising R.sub.1, R.sub.1′, and R.sub.4; ϕ.sub.3 is a flux covering a reluctance path provided by the central subcore comprising R.sub.3, R.sub.4, and R.sub.5; ϕ.sub.4 is a mutual flux covering a reluctance path shared by a portion of the second auxiliary subcore and a second portion of the central subcore comprising R.sub.5, R.sub.1′, and R.sub.6; ϕ.sub.5 is the flux covering the reluctance path provided by the second auxiliary subcore comprising R.sub.6 and R.sub.7; L.sub.11 is a primary coil resonant inductance; L.sub.12 is a secondary coil resonant inductance; L.sub.m1 is a magnetizing inductance of the transformer; V.sub.1 is a voltage excitation of the primary coil; V.sub.2 is a voltage excitation of the secondary coil; i.sub.1 is a current entering through the primary coil; i.sub.2 is a current entering through the secondary coil; N.sub.1 is a number of turns of the primary coil; N.sub.2 is a number of turns of the secondary coil; R.sub.equ1═ is an equivalent reluctance of the primary coil; R.sub.equ3═ is an equivalent reluctance of the secondary coil; such that: $\begin{array}{cc}{\Phi }_1=\frac{N1*i1}{\mathrm{Requ}1}-\frac{\left(N1*i1+N2*i2\right)*K}{\mathrm{Requ}3},& \left(1\right)\end{array}$ $\begin{array}{cc}{\Phi }_3=\frac{\left(N1*i1+N2*i2\right)}{\mathrm{Requ}3},& \left(2\right)\end{array}$ $\begin{array}{cc}{\Phi }_5=\frac{N1*i1}{\mathrm{Requ}1}-\frac{\left(N1*i1+N2*i2\right)*K}{\mathrm{Requ}3},& \left(3\right)\end{array}$ $\begin{array}{cc}{\Phi }_2=\frac{N1*i1}{R1}-\left\{\frac{2N1*i1\left(R1+R2+R^{\prime }\right)-2*\mathrm{\Phi 3}*R1*R2}{R1*\left(R1+2R2+2R^{\prime }\right)}\right\},& \left(4\right)\end{array}$ $\begin{array}{cc}{\Phi }_4=\frac{N2*i2}{R1}-\left\{\frac{2N2*i2\left(R1+R2+R^{\prime }\right)-2*\mathrm{\Phi 3}*R1*R2}{R1*\left(R1+2R2+2R^{\prime }\right)}\right\},& \left(5\right)\end{array}$ $\begin{array}{cc}K=\frac{R2}{R1+2R2+2R^{\prime }},& \left(6\right)\end{array}$ $\begin{array}{cc}{R}_{\mathrm{equ}1}=\frac{R1*\left(R1+2R2+2R^{\prime }\right)}{R1+R2+2R^{\prime }},& \left(7\right)\end{array}$ $\begin{array}{cc}{R}_{\mathrm{equ}3}=\frac{\left(2R2+R3\right)*R1*\left(R1+2R2+2R^{\prime }\right)-2R1*R{2}^2}{R1*\left(R1+R^{\prime }\right)+R1*\left(R2+R^{\prime }\right)},& \left(8\right)\end{array}$ $\begin{array}{cc}{\lambda }_1=N_1*\left({\Phi }_{1+}{\Phi }_3\right),& \left(9\right)\end{array}$ $\begin{array}{cc}{\lambda }_2=N_2*\left({\Phi }_{3+}{\Phi }_5\right),& \left(10\right)\end{array}$ $\begin{array}{cc}{\lambda }_1=\frac{N{1}^2*i1}{\mathrm{Requ}1}+\frac{N{1}^2*i1}{\mathrm{Requ}3}*\left(1-K\right)+\frac{N1*N2}{\mathrm{Requ}3}*i2*\left(1-K\right),& \left(11\right)\end{array}$ $\begin{array}{cc}{\lambda }_2=\frac{N{2}^2*i2}{\mathrm{Requ}1}+\frac{N{2}^2*i2}{\mathrm{Requ}3}*\left(1-K\right)+\frac{N1*N2}{\mathrm{Requ}3}*i1*\left(1-K\right),& \left(12\right)\end{array}$ $\begin{array}{cc}{V}_1=\frac{d\lambda 1}{dt},& \left(13\right)\end{array}$ $\begin{array}{cc}{V}_2=\frac{d\lambda 2}{dt},& \left(14\right)\end{array}$ $\begin{array}{cc}\left[\begin{array}{c}V1\\ V2\end{array}\right]=\left[\begin{array}{cc}\mathrm{Lm}1+\mathrm{Ll}1& \mathrm{Lm}1*\left(\frac{N2}{N1}\right)\\ \mathrm{Lm}2*\left(\frac{N1}{N2}\right)& \mathrm{Lm}2+\mathrm{Ll}2\end{array}\right]\left[\begin{array}{c}\frac{\mathrm{di}1}{\mathrm{dt}}\\ \frac{\mathrm{di}2}{\mathrm{dt}}\end{array}\right],& \left(15\right)\end{array}$

9. A multicore transformer, comprising: a first auxiliary subcore, a central subcore, and a second auxiliary subcore, each respective subcore comprising a magnetically permeable material; the first auxiliary subcore and the central subcore being separated by a respective gap to decouple respective a first auxiliary subcore flux and the central subcore flux; the second auxiliary subcore and the central subcore being separated by a respective gap to decouple respective a second auxiliary subcore flux and the central subcore flux; the first auxiliary subcore and the second auxiliary subcore being separated by the central subcore; a primary coil, encircling a portion of the first auxiliary subcore and a first portion of the central subcore; and a secondary coil, encircling a portion of the second auxiliary subcore and a second portion of the central subcore, wherein the first auxiliary subcore provides a reluctance path to a flow of leakage flux to achieve a primary side resonant inductance, the secondary auxiliary subcore provides a reluctance path to a flow of leakage flux to achieve a secondary side resonant inductance, and the center core carries a main flux to meet a magnetizing inductance of the transformer, such that the primary side resonant inductance is defined independently of the secondary side resonant inductance.

10. A method of inductively transferring power, comprising: providing a first auxiliary subcore, a central subcore, and a second auxiliary subcore, each respective subcore comprising a magnetically permeable material, each respective subcore being separated by a respective flux-decoupling gap, having a primary coil encircling a portion of the first auxiliary subcore and a first portion of the central subcore and a secondary coil encircling a portion of the second auxiliary subcore and a second portion of the central subcore, to thereby define a multicore transformer, exciting the primary coil to supply a magnetizing flux of the multicore transformer at an excitation frequency; and transferring the magnetizing flux to the secondary coil, to thereby generate an electrical current in the secondary coil at the excitation frequency, wherein the first auxiliary subcore provides a reluctance path to a flow of leakage flux to achieve a primary side resonant inductance, the secondary auxiliary subcore provides a reluctance path to a flow of leakage flux to achieve a secondary side resonant inductance, and the center core carries a main flux to meet the magnetizing inductance of the transformer, such that the primary side resonant inductance is defined independently of the secondary side resonant inductance.

11. The method according to claim 10, wherein the primary coil and the secondary coil are each configured to produce a central magnetic field having an axis intersecting the lower plate and the upper plate of the central subcore.

12. The method according to claim 11, wherein each subcore is split along the axis by at least one gap.

13. The method according to claim 11, wherein at least one subcore is split along the axis by at least two gaps.

14. The method according to claim 11, wherein each central spacer is split along the axis by at least one gap.

15. The method according to claim 11, wherein each subcore is split along the by at least one gap, and each subcore has the same number of vertical gaps.

16. The method according to claim 10, wherein: the first auxiliary subcore, central subcore, and second auxiliary subcore are configured with a respective gap between the first auxiliary subcore and the central subcore, and the central core and the second auxiliary subcore, so that magnetic flux linking adjacent subcores is decoupled; the first auxiliary subcore is configured to provide a reluctance path to a flow of leakage flux to achieve a desired value of a first side inductance; the second auxiliary subcore is configured to provide a desired value of a secondary-side inductance; and the center subcore mainly carries a main flux to meet a magnetizing inductance to transfer power between the first side to the second side.

17. The method according to claim 10, further comprising defining an electromagnetic model of a transformer comprising the first auxiliary subcore, the central subcore, the second auxiliary subcore, the gaps, the primary coil, and the secondary coil; defining a magnetizing inductance required for the transformer, and constraining the central subcore to supply the magnetizing inductance from the primary coil; defining the primary side resonant inductance, and constraining the first auxiliary subcore to provide the reluctance path to the flow of leakage flux to achieve the primary side resonant inductance from the primary coil; defining the secondary side resonant inductance, and constraining the second auxiliary subcore to provide the reluctance path to the flow of leakage flux to achieve the secondary side resonant inductance from the from the secondary primary coil; and optimizing at least a size, shape, and gap configuration of the first auxiliary subcore, the central subcore, and the second auxiliary subcore in accordance with the electromagnetic model and constraints to meet at least a defined magnetizing inductance, a defined primary side resonant inductance, a defined secondary side resonant inductance, and a required power transfer capability.

18. The method according to claim 17, wherein: ϕ.sub.1 is a flux covering a reluctance path provided by the first auxiliary subcore comprising R.sub.1 and R.sub.2; ϕ.sub.2 is a mutual flux covering a reluctance path shared by a portion of the first auxiliary subcore and a first portion of the central subcore comprising R.sub.1, R.sub.1′, and R.sub.4; ϕ.sub.3 is a flux covering a reluctance path provided by the central subcore comprising R.sub.3, R.sub.4, and R.sub.5; ϕ.sub.4 is a mutual flux covering a reluctance path shared by a portion of the second auxiliary subcore and a second portion of the central subcore comprising R.sub.5, R.sub.1′, and R.sub.6; ϕ.sub.5 is the flux covering the reluctance path provided by the second auxiliary subcore comprising R.sub.6 and R.sub.7; L.sub.11 is a primary coil resonant inductance; L.sub.12 is a secondary coil resonant inductance; L.sub.m1 is a magnetizing inductance of the transformer; V.sub.1 is a voltage excitation of the primary coil; V.sub.2 is a voltage excitation of the secondary coil; i.sub.1 is a current entering through the primary coil; i.sub.2 is a current entering through the secondary coil; N.sub.1 is a number of turns of the primary coil; N.sub.2 is a number of turns of the secondary coil; R.sub.equ1═ is an equivalent reluctance of the primary coil; R.sub.equ3═ is an equivalent reluctance of the secondary coil; and the electromagnetic model comprises: $\begin{array}{cc}{\Phi }_1=\frac{N1*i1}{\mathrm{Requ}1}-\frac{\left(N1*i1+N2*i2\right)*K}{\mathrm{Requ}3},& \left(1\right)\end{array}$ $\begin{array}{cc}{\Phi }_3=\frac{\left(N1*i1+N2*i2\right)}{\mathrm{Requ}3},& \left(2\right)\end{array}$ $\begin{array}{cc}{\Phi }_5=\frac{N1*i1}{\mathrm{Requ}1}-\frac{\left(N1*i1+N2*i2\right)*K}{\mathrm{Requ}3},& \left(3\right)\end{array}$ $\begin{array}{cc}{\Phi }_2=\frac{N1*i1}{R1}-\left\{\frac{2N1*i1\left(R1+R2+R^{\prime }\right)-2*\mathrm{\Phi 3}*R1*R2}{R1*\left(R1+2R2+2R^{\prime }\right)}\right\},& \left(4\right)\end{array}$ $\begin{array}{cc}{\Phi }_4=\frac{N2*i2}{R1}-\left\{\frac{2N2*i2\left(R1+R2+R^{\prime }\right)-2*\mathrm{\Phi 3}*R1*R2}{R1*\left(R1+2R2+2R^{\prime }\right)}\right\},& \left(5\right)\end{array}$ $\begin{array}{cc}K=\frac{R2}{R1+2R2+2R^{\prime }},& \left(6\right)\end{array}$ $\begin{array}{cc}{R}_{\mathrm{equ}1}=\frac{R1*\left(R1+2R2+2R^{\prime }\right)}{R1+R2+2R^{\prime }},& \left(7\right)\end{array}$

19. The method according to claim 17, wherein said optimizing comprises receiving quantitative functional criteria for the transformer, and solving the equations of the electromagnetic model to achieve the quantitative functional criteria.

20. The method according to claim 10, wherein: the first auxiliary subcore comprises a first pair of magnetically permeable plates, separated by a first set of segmented magnetically permeable spacers defining a first magnetically coupled path; the second auxiliary subcore comprises a second pair of magnetically permeable plates, separated by second set of segmented magnetically permeable spacers defining a second magnetically coupled path; the central subcore comprises a third pair of magnetically permeable plates, separated by third and fourth sets of segmented magnetically permeable spacers defining a third magnetically coupled path; the first magnetically coupled path, the second magnetically coupled path, and the third magnetically coupled path being sufficiently spaced to be respectively magnetically decoupled; the primary coil is a planar primary coil, surrounding the first set of segmented magnetically permeable spacers and the third set of magnetically permeable segmented spacers, such that a magnetizing flux is induced in the third set of magnetically permeable segmented spacers to meet a magnetizing inductance of the multicore transformer and a first leakage flux is induced in the first set of magnetically permeable segmented spacers to provide a reluctance path to a flow of leakage flux to achieve a primary side resonant inductance; and the secondary coil is a planar secondary coil, surrounding the second set of segmented magnetically permeable spacers and the fourth set of magnetically permeable segmented spacers, such that the magnetizing flux in the fourth set of magnetically permeable segmented spacers is coupled to the planar secondary coil, and a second leakage flux is induced in the second set of magnetically permeable segmented spacers to provide a reluctance path to a flow of leakage flux to achieve a secondary side resonant inductance.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0129] FIG. 1 shows the architecture of a bidirectional AC to DC resonant converter with integrated magnetics, as known in the prior art.

[0130] FIGS. 2A and 2B show a magnetic core arrangements with air gaps according to the present invention.

[0131] FIG. 3 shows an integrated magnetic design with multiple air gaps across core heights.

[0132] FIGS. 4A and 4B show magnetic design arrangements with their dimensions for a 1300V/45V, 12 kW bidirectional AC to DC resonant converter with integrated magnetics. FIG. 4A shows a front view and FIG. 4B shows a trimetric view with the planar windings.

[0133] FIG. 5A shows the primary winding planar PCB winding layout.

[0134] FIG. 5B shows the secondary winding planar PCB winding layout.

[0135] FIG. 6 shows a reluctance network model of the integrated magnetic core structure.

[0136] FIG. 7 shows a simplified reluctance network model of the integrated magnetic core structure.

[0137] FIG. 8 shows an equivalent electrical circuit diagram of proposed magnetic structure showing different elements.

[0138] FIG. 9 shows a graph of the variation of flux linkages per turn ϕ.sub.1 (filled markers, rising) and 42 (open markers, declining) with the air gap between middle cores to left cores in meters.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0139] For high frequency CLLC resonant converters, gapped planar transformers are being used to avoid saturation and obtain the precise inductance value to achieve soft switching in the converter. However, the gap mainly creates the fringing fluxes, inducing extra magnetic losses in the core surfaces near to the air gap. Because of the air gap, some fluxes fringe and complete their path through the air. These fringing fluxes strike the core and cause excess magnetic losses in the core, called fringing losses. Also, the fringing field that exists around the air gap results in extra winding losses due to the induced eddy current in the winding.

[0140] A large air gap causes a larger reluctance, lower magnetizing inductance, wider space for the windings, and better performance characteristics for the CLLC resonant converters. However, it may also produce higher losses. Essentially, fringing losses are directly proportional to the air gap length and the core strip width. While the air gap is a necessary evil in the magnetic design, the resent technology provides a solution to reduce the fringing losses.

[0141] As shown in FIG. 3, by distributing the air gap across the core height primarily, the fringing fluxes can be reduced in the surrounding the gaps. Furthermore, by having multiple smaller length cores and combining them while maintaining certain clearance (e.g., 1 mm) the core strip width can be effectively reduced as shown in FIG. 2B. Therefore, the fringing losses may be reduced.

[0142] The three magnetic cores shown in FIGS. 2A and 2B, namely left, center and right, are arranged with a calculated air gap between each of the cores, so that the flux linking through each of these cores is very lightly coupled with one another. The left core provides a reluctance path to the flow of leakage flux to achieve a desired value of primary side resonant inductance. Similarly, the right core is optimally designed to achieve a desired value of secondary side resonant inductance, while the center core mainly carries the flux to meet the magnetizing inductance of the integrated transformer.

[0143] Magnetic Structure and Winding Layout for a 1300V Input Voltage to 30-60 VV Output Voltage, 12 kW CLLC Converter

[0144] The magnetic core dimensions and arrangements for a 1300V DC input and 30V-60V DC output with a peak power of 12 kW is shown in FIGS. 4A and 4B (dimensions in mm).

[0145] FIG. 3 shows three multiple small length magnetic elements assembled for the left side core, and two small length magnetic elements each for middle and right-side cores, to have significant reduction in fringing loss. However, if the core elements are smaller beyond a certain length, it puts a limit on the manufacturing process.

[0146] Therefore, relatively larger magnetic elements are placed for each of the sides (i.e., left, middle and right), with air gaps on top and bottom that provide fringing loss close to the arrangements shown in FIG. 3. The modified magnetic arrangements of the three cores are shown in FIGS. 4A and 4B, with the planar windings placed as shown in FIG. 4B. Furthermore, the windings on primary and secondary sides are kept at a distance of 4 mm and 2 mm respectively from the air gap of the core, so that the AC loss due to fringing flux can be minimized as shown in FIG. 4A. Also, to reduce proximity effects, both the windings are placed at a distance of 3 mm from each other.

[0147] Primary and secondary winding layers are kept on a separate layout over two different limbs of the core. The designed turn's ratio of the planar transformer is 13:1 due to high DC link voltage (1300V) and low DC output voltage (30V-60V). To have a reduced turn on the primary side, only one turn is provided on the secondary side layer. Primary and secondary sides have 6 layer and 10 layer Printed Circuit Boards (PCBs) respectively. The layout of for primary winding is shown in FIG. 5A, and the layout of the secondary winding is shown in FIG. 5B. To mitigate the high insulation requirement at the primary by dint of high voltage, windings are laid on the 2.sup.nd to 5.sup.th layers, keeping layer 1 and layer 6 copper-free). Moreover, the winding power loss is proportional to the number of layers in a high-frequency operation. Bearing this in mind, 13 turns are set out in four layers at the primary side with series-connected copper windings. However, as the secondary side has only one turn, to supply the hundreds of ampere current, ten layers are connected in parallel. To reduce the DC resistance, the secondary layers are widened to 68 mm as shown in FIG. 4B.

[0148] Modelling of Electrical Circuit and Air Gap Determination for Light Coupling of Three Magnetic Cores

[0149] The reluctance network for the proposed integrated magnetic structure is shown in FIG. 6. As shown in FIG. 6, R.sub.4=R.sub.5 as the two arms of the middle core have the same length and air gap. Similarly, R′.sub.1═R′.sub.2, R.sub.1═R.sub.2 and R.sub.6═R.sub.7. Also, from FIG. 2A, when N.sub.1=N.sub.2, the two side cores (i.e., left and right) are of equal size and air gaps are the same. The modified reluctance network for FIG. 2A can be drawn as shown in FIG. 7.

[0150] Having solved the above reluctance model, the following expressions for the different fluxes are obtained,

[0151] ϕ.sub.1 is the flux covering the reluctance path provided by the first auxiliary subcore including airgaps;

[0152] ϕ.sub.2 is the mutual flux covering the reluctance path shared by a portion of the first auxiliary subcore and a first portion of the central subcore including airgaps;

[0153] ϕ.sub.3 is the flux covering the reluctance path provided by the central subcore including airgaps;

[0154] ϕ.sub.4 is the mutual flux covering the reluctance path shared by a portion of the second auxiliary subcore and a second portion of the central subcore including airgaps;

[0155] ϕ.sub.5 is the flux covering the reluctance path provided by the second auxiliary Zo subcore including airgaps.

[0156] L.sub.11 and L.sub.12 are the primary and secondary side resonant inductances, respectively;

[0157] L.sub.m1 is the magnetizing inductance of the transformer;

[0158] V.sub.1 and V.sub.2 are the voltage excitations at the primary and secondary terminals, respectively;

[0159] i.sub.1 and i.sub.2 are the current entering through the primary and secondary terminals, respectively;

[0160] N.sub.1 and N.sub.2 are the number of primary and secondary turns, respectively,

[00004]$\begin{array}{cc}{\Phi }_1=\frac{N1*i1}{\mathrm{Requ}1}-\frac{\left(N1*i1+N2*i2\right)*K}{\mathrm{Requ}3}& \left(1\right)\end{array}$$\begin{array}{cc}{\Phi }_3=\frac{\left(N1*i1+N2*i2\right)}{\mathrm{Requ}3}& \left(2\right)\end{array}$$\begin{array}{cc}{\Phi }_5=\frac{N1*i1}{\mathrm{Requ}1}-\frac{\left(N1*i1+N2*i2\right)*K}{\mathrm{Requ}3}& \left(3\right)\end{array}$$\begin{array}{cc}{\Phi }_2=\frac{N1*i1}{R1}-\left\{\frac{2N1*i1\left(R1+R2+R^{\prime }\right)-2*\mathrm{\Phi 3}*R1*R2}{R1*\left(R1+2R2+2R^{\prime }\right)}\right\}& \left(4\right)\end{array}$$\begin{array}{cc}{\Phi }_4=\frac{N2*i2}{R1}-\left\{\frac{2N2*i2\left(R1+R2+R^{\prime }\right)-2*\mathrm{\Phi 3}*R1*R2}{R1*\left(R1+2R2+2R^{\prime }\right)}\right\}& \left(5\right)\end{array}$$\begin{array}{cc}K=\frac{R2}{R1+2R2+2R^{\prime }}& \left(6\right)\end{array}$

[0161] R.sub.equ1 and R.sub.equ3 are the equivalent reluctances

[0162] K is a constant,

λ.sub.1=N.sub.1*(ϕ.sub.1+ϕ.sub.3)  (9)

λ.sub.2=N.sub.2*(ϕ.sub.3+ϕ.sub.5)  (10)

[0163] Solving equations (1)-(10) yields,

[00005]$\begin{array}{cc}{\lambda }_1=\frac{N{1}^2*i1}{\mathrm{Requ}1}+\frac{N{1}^2*i1}{\mathrm{Requ}3}*\left(1-K\right)+\frac{N1*N2}{\mathrm{Requ}3}*i2*\left(1-K\right)& \left(11\right)\end{array}$$\begin{array}{cc}{\lambda }_2=\frac{N{2}^2*i2}{\mathrm{Requ}1}+\frac{N{2}^2*i2}{\mathrm{Requ}3}*\left(1-K\right)+\frac{N1*N2}{\mathrm{Requ}3}*i1*\left(1-K\right)& \left(12\right)\end{array}$

[0164] The voltages applied across the two windings can be calculated as,

[00006]$\begin{array}{cc}{V}_1=\frac{d\lambda 1}{dt}& \left(13\right)\end{array}$$\begin{array}{cc}{V}_2=\frac{d\lambda 2}{dt}& \left(14\right)\end{array}$

[0165] Solving equations (11)-(14) the voltages, currents and inductances forms a matrix as follows,

[00007]$\begin{array}{cc}\left[\begin{array}{c}V1\\ V2\end{array}\right]=\left[\begin{array}{cc}\mathrm{Lm}1+\mathrm{Ll}1& \mathrm{Lm}1*\left(\frac{N2}{N1}\right)\\ \mathrm{Lm}2*\left(\frac{N1}{N2}\right)& \mathrm{Lm}2+\mathrm{Ll}2\end{array}\right]\left[\begin{array}{c}\frac{\mathrm{di}1}{\mathrm{dt}}\\ \frac{\mathrm{di}2}{\mathrm{dt}}\end{array}\right]& \left(15\right)\end{array}$$\begin{array}{cc}{L}_{11}=\frac{N{1}^2}{\mathrm{Requ}1}& \left(16\right)\end{array}$$\begin{array}{cc}{L}_{12}=\frac{N{2}^2}{\mathrm{Requ}1}& \left(17\right)\end{array}$$\begin{array}{cc}{L}_{m1}=\frac{N{1}^2}{\mathrm{Requ}3}*\left(1-K\right)& \left(18\right)\end{array}$$\begin{array}{cc}{L}_{m2}=\frac{N{2}^2}{\mathrm{Requ}3}*\left(1-K\right)& \left(19\right)\end{array}$$\begin{array}{cc}\mathrm{Generally},& \left(20\right)\end{array}$$\begin{array}{cc}{\Phi }_1>>{\Phi }_2& \left(21\right)\end{array}$

[0166] In some cases, these limitations do not apply.

[0167] Therefore, the equivalent electrical circuit diagram of transformer can be drawn as shown in FIG. 8.

[0168] Now, to determine the air gap that needs to be placed among the three magnetic structures, the impact of the air gap clearance on the flux linkages is needed. Hence, to understand it more clearly, the variation of flux linkages per turn ϕ.sub.1 and ϕ.sub.2 with the air gap between middle cores to left cores is shown in FIG. 9 for the design shown in FIG. 2A and FIG. 2B. ϕ.sub.3 doesn't have any variations with the air gap clearance, whereas ϕ.sub.1 increases and ϕ.sub.2 decreases with the air gap. To achieve a light coupling, a condition where ϕ.sub.1>>ϕ.sub.2 should exist. The same holds true for ϕ.sub.4 and ϕ.sub.5 (i.e., ϕ.sub.5>>ϕ.sub.4). From FIG. 9, one can notice for an air gap clearance of 7-8 mm is an optimal to achieve the light coupling.

[0169] By following the above approach, for the design shown in FIG. 4A and FIG. 4B, where a high step-down voltage (i.e., 1300V/45V) is required, an air gap (i.e., 5 mm) between left and middle cores, and an air gap (i.e., 2 mm) between right and middle cores are set to achieve a light coupling.

[0170] Although the invention(s) have been described with reference to specific embodiments, it will be understood by those skilled in the art that various changes may be made, and equivalents may be substituted, for elements thereof without departing from the true spirit and scope of the invention. In addition, modifications may be made without departing from the essential teachings of the invention. The invention is described by way of various embodiments and features. This disclosure is intended to encompass all consistent combinations, subcombinations, and permutations of the different options and features, as if expressly set forth herein individually.

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