2019-10-29 08:57

*Abstract***—High frequency transformers are an integral part ofpower electronics devices and their parasitic parameters influence theperformance and efficiency of the overall system. In this paper, transformerleakage inductances and parasitic capacitances are analyzed using finiteelement method (FEM) for different structures and windings arrangements of highfrequency transformers. Also, magnetic field, electric field, and voltagedistribution within the transformer is simulated and analyzed. Six differenthigh frequency transformers with toroidal, EE, and UU cores with differentwindings are investigated for a 400(V)/400(V), 8 kVA transformer operating at10 kHz. Additionally, interleaved windings for EE core are simulated andresults compared with previous outcomes. Analysis results will help categorizeeach structure, based on its balance between leakage inductances and seriesparasitic capacitance. This information can later be used for optimal selectionof transformers as a function of their operating frequency and enable designersto compromise between various parameters in different applications, especiallynew fast switches such as SiC and GaN.**

*Index Terms***—High Frequency Transformers, LeakageInductance, Magnetic andElectric Field Distribution, Parasitic Capacitance, Winding Arrangements**

_{}

_{P}ARASITIC parametersin power electronics devices play animportantrole in device performance and efficiency. Many applications in powerelectronics use magnetic components. Because of advantages such as compactness,active/reactive power flow control, and protection, [1,2], HF galvanic isolation-basedtopologies have been widely incorporated in

power electronics applications.

The leakageinductance in transformers results from imperfect coupling between the primaryand the secondary of transformer windings which leads to voltage spikes, higherrectification losses, and efficiency reduction [3]. Additionally, parasiticcapacitances lead to the injection of high frequency currents which increasethe electromagnetic interferences (EMI). Moreover, parasitic capacitances mayform electrostatic coupling to other parts of the circuit. Decreasing these twoparameters not only improve switching waveforms, but also improve systems’efficiency.

Estimating the value of theseparasitic parameters, can be very helpful during the design process. In someapplications such as soft-switching converters, it is possible to use parasiticparameters of transformers as the additional inductors needed to form LC tanks.In [4], using leakage inductance of

transformer as the inductance oftank circuit has reduced the total volume of the system by 15%.

As the frequencyincreases, the size of passive components, decreases. Hence, increasing thefrequency can help the miniaturization of the converter. However, at higherfrequencies, the importance of the parasitic, intensifies. For instance, theutilization of resonant soft-switching converters at higher frequencies rely onan accurate estimation of the parasitic in the final product. But unwantedleakage inductances and parasitic capacitances may interrupt resonant modes andproduce unwanted resonant frequencies, which can distort voltage and currentwaveforms and decrease the efficiency. Also, this may lead to the interruptionin control system [5].

In the designprocess of DC-DC converters, high frequency transformer design is an importantstep. Depending on the application and the design method, designers shoulddecide to choose a suitable core shape and winding formation to deal with theparasitic parameters. Different transformer core shapes and windingarrangements, lead to different parasitic parameters. Therefore, it isnecessary to understand the parasitic behavior of various transformer cores andways of reducing and changing those parameters.

In this paper, sixdifferent transformer structures are investigated for high frequency and highpower applications. The transformers are 1:1 and working at 400V for bothprimary and secondary at 10 kHz for power rating of 8 kVA. The windings arewound with AWG 8 (with diameter of 3.2639

By using FEM magnetic field, electric field, andvoltage distributions are computed for each structure then leakage inductanceand parasitic capacitance for each of them are obtained. Then, by usinginterleaved windings and changing the voltage distribution along the cores, theparasitic parameters changed and the results compared with previouscomputations.

II. TRANSFORMERDESIGNAND CORE SELECTION

Selecting magneticmaterial is a main step in designing high frequency transformers for powerelectronics applications. Cost, size, and performance are three major factors,which determine type of magnetic material in each design. These factors areinfluenced by relative permeability, temperature stability, core loss, cost,etc. Many magnetic materials can be used for the power electronics devices; anddesigners need to make a trade-off in choosing materials for each specificapplication. In TABLE I five magnetic materials are compared. It is noted thateach feature is totally dependent to specific

TABLE I

MagneticMaterials key features

Molyperm | Iron | Ferrite | Ferrite | |||

alloy | Sendust | |||||

Powder | MnZn | NiZn | ||||

(MPP) | ||||||

Temperature | Very Good | Very | Very | Fair | Fair | |

Stability | Good | Good | ||||

Relative | 14-550 | 26-125 | 4-100 | 750- | 15- | |

Permeability | 15000 | 1500 | ||||

Core Loss | Very Low | Low | Moderate | Very | Low | |

Low | ||||||

Relative Cost | High | Low | Very | Very | Very | |

Low | Low | Low | ||||

manufacturer. Inthis table ferrites are soft iron and the others are powder materials.

For the high frequency transformer in thispaper, iron powder is chosen, mainly due to the lower frequency of application.Iron powder has a very low cost which make it suitable for mass production andshows very good temperature stability. Fig. 1 represent the B-H curve of theIron powder used in this paper.

By using (1), andfollowing the design steps in [6], number of turn for both primary andsecondary in all the transformers is set on 80 turns and the cross sectionalarea in all of them is 1600 mm^{2}. The turn ratio and cross sectional for all transformerstructures set on the same values in order to make a better comparison, betweenthe structures and transformers parasitic behavior.

= (1)

In (1), , , ,, , , and are apparent power, waveformcoefficient, window utilization factor, flux density,

current density, and area product respectively.

To find core lossdensity the Steinmetz equation is widely used. However, [6] represents a moreaccurate formula for micro metals, which can be used for Iron powder-Mix-08. In

(2), is the average of power lossdensity (watts/kg), is the amplitude of the flux density, is the systemfrequency, D is the duty cycle in square waveforms (D is 0.5 for sinusoidalwaveforms) and , , , are constants, which related to each material. For aspecific material, at higher frequency, medium frequency and lower frequencyvalues of these constants are different. These values for the ironpowder-Mix-08 used here, are shown in TABLE IV.

= |
| + | |||||||||||

+681( ) | |||||||||||||

100( ) | 1 | + | 1 | ). | 1 | (2) | |||||||

4 | |||||||||||||

1− | TABLE II | ||||||||||||

Constant values of the iron powder-Mix-08 | |||||||||||||

0.01235 | 0.8202 | 1.4694 | 3.85×10 |

III. TRANSFORMERSSTRUCTURES

Toroidal, UU, andEE cores are widely used, in the power electronics applications for the followingreasons. The geometry of toroidal cores forces the magnetic field lines to formin closed circular loops and constrain the magnetic flux inside the core andreduce the EMI. Also, toroidal cores offer less weight and volume, and sincethey have a lower length of copper, the winding resistance decreases and due tolarge surface area of these core heat removal via convection and radiation isbetter in comparison. But winding copper wires on toroidal cores is moreexpensive than bobbin-based cores. Also, placing airgaps on the toroidal coresis quite difficult [7]. UU cores are also popular because of their simplicity,and high window utilization factor [7]. In some applications by using CC cores,which have rounded corners, high density of electric field can be removed.Lastly, EE cores are widely used in the power electronics applications. Ifairgaps are needed, EE cores allow forgapping of only the center leg which willreduce the EMI and fringing issues that arise when all legs are gapped [7].Cooling of EE and UU cores are easier than toroidal cores due to theirstructure, which allows for a flow of coolant through the windings if windingspacers are used. Since, the windings in the

EE cores are notcompletely covered by magnetic material, there is more EMI in these cores,which renders them not suitable for higher frequency applications. In suchscenarios, pot cores can replace EE cores.

Fig. 2 shows the 3Dtransformer cores used for the simulations to obtain the parasitic parametersin this paper. Fig. 3 and TABLE III show the toroidal cores dimensions. Twotoroidal cores types are investigated in this paper with different dimensions.

As it is shown inFig. 2(a) and (b) both UU and EE cores are made with eight block sets. Fig. 4and TABLE IV represent the block set and front and side views of the UU and EEcores. Moreover, in fig. 5 the winding arrangements used for the cores aredepicted. The toroidal core with winding arrangements of same turns on thesecondary of the toroidal core are wound in

(a) (b) (c)

Fig. 2. The 3D transformers cores. a) Toroidal core. b) UUcore. c) EE core

(a) (b)

Fig. 3. a)Front view of the toroidal core. b) Side view of the toroidal core (shows thedepth of the core).

TABLEIII

Dimensionsof the toroidal cores

A | 60 (mm) | ||

Case 1 | B | 40 (mm) | |

H | 80 (mm) | ||

A | 70 (mm) | ||

Case 2 | B | 30 (mm) | |

H | 40 (mm) |

(a) (b) (c) (d)

(e) (f)

Fig. 4. Side andfront views of the UU and the EE cores. a) Front view of the block set. b) Sideview of the block set. c) Front view of the UU core. d) Side view of the

UU core.e) Front view of the EE core. f) Side view of the EE core. Side views show thedepth of the cores.

(a) (b)

(c) (d)

Fig. 5. Winding arrangementsof the transformers. a) Toroidal core with 3 layers for

each winding (non-overlaid windings). b) Toroidal core with 1 layer for each

winding (overlaid windings).c) UU core winding arrangement with 4 layers for each

winding. d) EE core windingarrangement with 4 layers for each winding.three layers which is clear in Fig.5(a). But for the toroidal core of Fig. 5(b) there is just one layer of 80turns for the primary and the same for the secondary. This winding arrangementsand layers are expected to completely influence the parasitic parameters.Furthermore, for the UU and the EE cores, 80 turns of the primary and the sameturns of the secondary are wound in four layers for each of them.

IV. PARASITICPARAMETERSCALCULATIONS

In this sectionmagnetic field, electric field, and voltage distributions for all thetransformer cores, depicted in the section III, are displayed. Then, leakageinductance and parasitic capacitance for each structure calculated. Leakageinductance in transformers calculated by measuring energy and magneto motiveforce (MMF) stored in the space, while the secondary is short circuited and theparasitic capacitance calculated is between primary and secondary winding.

In high frequency,skin effect is the tendency of currents in conductors to concentrate on surfaceand by considering this phenomenon the leakage inductance values decrease. Foraccuracy purpose the skin effect is observed in the FEM simulations.

*A. Magnetic Field Distribution and LeakageInductance Calculations*

Fig. 6 showsmagnetic flux density distributions, for all the cores. The flux density inthis figure are obtained when the transformers work at the full load. As it isclear in this figure, the maximum flux density, in case 2 with 3 layers andcase 2 with 1 layer are slightly greater than case 1 of both of 3 and 1 layersas well as flux density distribution linearity. In toroidal cores there is nosaturation but in both UU and EE cores some points are in saturation. The windingarrangements of the layers for the EE core here, is the same as Fig. 5(d) witharrangement of P-P-P-P-S-S-S-S (P for primary and S for secondary) from left toright in the right window and a symmetry arrangement for the left window.

(a) (b)

(c) (d)

(e) (f)

Fig. 6. Magnetic flux density(T). a) Toroidal core with non-overlaid windings, case

1. b) Toroidal corewith non-overlaid windings, case 2. c) Toroidal core with overlaid windings,case 1. d) Toroidal core with overlaid windings, case 2. e) UU core with 4layers on each winding. f) EE core with 4 layers on each winding.

Byshort circuiting the secondary, MMF distribution at the space, especially thewindow area, is the key factor to find and analysis of leakage inductance.Hence, Leakage inductance is directly proportional to the energy stored in thespace and with equations (4) and (5) magnetic energy in the space is related tothe magnetic field strength (H).

= ∮ . ℓ | 1 | (3) | ||||||

| ∰ . | (4) | ||||||

= | 2 | (5) | ||||||

1 | ||||||||

= | . | . | (6) | |||||

2 | ||||||||

In the equations(3-6), ℓ is thelength, is the volume of the space that energy is calculated on, is thepermeability of the medium, is the leakage inductance from primary side, and isthe current of primary winding.

Fig.7 represents the magnetic field strength (H) distribution in the transformercores, winding arrangements, and window area while the secondary windings areshort circuited. It is noted that besides the cores, the relative permeabilityof other parts such as insulations, copper of windings, and air are considered

1. But the topology and structures of the transformersinfluence the leakage inductance values. In these cores, the windings in alllayers are wound homogeneously and the insulation between bare areas of thewinding conductors is fixed on 0.15mm for all the topologies. Also, insulationand bobbin thickness between

(a) (b)

(c) (d)

(e) (f)

Fig. 7. Magnetic field strength(A/mm). a) Toroidal core with non-overlaid windings, case 1. b) Toroidal corewith non-overlaid windings, case 2. c) Toroidal core with overlaid windings,case 1. d) Toroidal core with overlaid windings, case 2. e) UU core with 4layers on each winding. f) EE core with 4 layers on each winding with the samewinding arrangements of Fig. 6(f).

TABLE V

Magnetizinginductance, leakage inductances, and windings AC resistance of the

transformers

Transformer | Magnetizing | Leakage | AC | |

Type | Inductance( H) | Inductance( H) | Resistance(m ) | |

a | 2061.5 | 736.25 | 34.055 | |

b | 1944.0 | 324.89 | 18.343 | |

c | 1774.5 | 15.21 | 50.817 | |

d | 1820.0 | 9.23 | 25.496 | |

e | 2160.8 | 450.42 | 36.306 | |

f | 2005.1 | 79.39 | 25.038 |

the windings and the core in thetoroidal, UU and the EE cores is 2mm.

The leakageinductance and the magnetizing inductance for all the cores are achieved by FEMand displayed in TABLE V. Results obtained by considering magnetic fieldstrength not only in the represented areas of Fig. 7, but also in the wholespace. Also, in this table AC resistance of the windings from primary side areshown. The magnetizing inductance of the transformer types purposely designedto be almost near to each other, to make a better comparison between theparasitic parameters and choose the suitable type for each application.a,b,c,d,e,f types are the same as Fig. 6 and Fig. 7.

It is clear inTABLE V, leakage inductance can vary in a wide range by changing core andarrangements. By comparing leakage inductances of types a, b, c, and d andconsidering magnetic field strength of Fig. 7, it is apparent that the windowarea volume and the winding arrangements, which lead to better magneticcoupling are remarkably important in the results.

*B. Insulation Designs, VoltageDistributions, and Capacitance Calculations*

Parasiticcapacitance between primary and secondary windings can be found by calculatingtotal energy stored in the electric field. Then, by having total amount ofstored energy between the primary and the secondary, capacitance can becalculated. Equations (7-9) explain these relations.

| 1 | ∭ . | (7) | |||

2 | ||||||

= | 1 | (8) | ||||

= | . | (9) | ||||

2 | ||||||

In equations (7-9), is the volume ofthe space that energy is calculated on, is the permittivity of medium, is therelated voltage, and is the capacitance.

The capacitancebetween the primary and the secondary is influenced by structure oftransformers, winding arrangements,and permittivity of used materials.The insulation dimensions between layers, windings, cores, and bobbinthickness, for this study are the same as mentioned before. Insulation materialbetween bare conductors is silicon vanish with relative permittivity of 3.1 andthe bobbin between windings and core are made from plastic with relativepermittivity of 2.2.

(a) (b)

(c)

(d)

Fig. 8. Electric field distribution(kV/mm) in the transformers at highest insulation test voltage of 10kV. a) EEcore, case 1, and case 2 of the toroidal cores with overlaid windings (thesecores follow a similar maximum electric field pattern). Maximum electric fieldof 67(kV/mm) occurs between primary and secondary layers of these cores. b)Case 1 of the toroidal core with non-overlaid windings. Maximum electric fieldof 13.8(kV/mm) occurs between primary layers and the core. c) Case 2 of thetoroidal core with non-overlaid windings. Maximum electric field of 14.7(kV/mm)occurs between primary layers and the core. d) UU core. Maximum electric fieldof 7(kV/mm) occurs between primary layers and the core.

From insulationdesign point of view, in addition to dielectric constant (relativepermittivity) and distances, dielectric strength of the insulators are alsoimportant. The dielectric strength for silicon varnish and plastic of bobbinsare considered 120(kV/mm) and 25(kV/mm) respectively, also air dielectric

(a) (b)

(c) (d)

(e) (f)

Fig. 9. Voltage distribuiton in thetransforemrs by applying 1V to a winding and geounding the other one. a)Toroidal core with non-overlaid windings, case 1. b) Toroidal core withnon-overlaid windings, case 2. c) Toroidal core with overlaid windings, case 1.d) Toroidal core with overlaid windings, case 2. e) UU core with 4 layers oneach winding. f) EE core with 4 layers on each winding.

TABLE VI

Parasiticcapacitacne(pF) between primary and secondary windings by considering

twocore types

Transformer Type | Iron Powder-Mix-08 | Ferrite-NiZn |

a | 76.258 | 1.869 |

b | 30.603 | 1.346 |

c | 6444.137 | 6444.278 |

d | 3222.014 | 3222.269 |

e | 60.577 | 7.3117 |

f | 960.103 | 958.872 |

strength is considered 3(kV/mm). Itis important to attention that the dielectric constants and strength for thesematerials can vary, because of different manufacturers and this may change thecapacitance. According to the with IEEE Std. C57.12.01 dry-type transformersstandard, which used in [8], the insulation designs of this study transformers,should be based on 10kV. To examine if the insulation materials and dimensionssatisfy the standard or not; electric field distribution for the transformerscomputed with FEM. In this distribution, the primary is connected to the highinsulation test voltage of 10kV and both secondary and cores are grounded. Asit is shown in Fig. 8 maximum electric field in insulations is much lower thanthe dielectric strength of the used materials.

In order to findcapacitance between windings by following procedure used in [9], 1V applied tothe primary windings and 0V to the other ones. Fig. 9 shows voltagedistribution in the transformer structures. Also, TABLE VI displays capacitancebetween primary and secondary windings the transformers. Since the conductivityof the core influence capacitance values, in this table the values byconsidering ferrite-NiZn as the core (with a very low conductivity) are alsomentioned. a,b,c,d,e,f types are the same as Fig. 9.

As it is clear inTABLE VI parasitic capacitacne can change hugely for different structures.Additionally, by substituting a conductive core (Iron Powder) with anonconductive core (Ferrite-NiZn with 10^{5}(Ω.) resistivity) the capacitacne can dramatically change for some cases.

*C.* *InterleavedWindings*

By usinginterleaved windings, instead of the winding arrangements explained in theprevious part, leakage inductance significantly decrease. For the EE core here,using interleaved windings reduce leakage energy stored in the window area anddecrease MMF. Thus, without changing magnetizing inductance, leakage inductancedecreases. As well, by using interleaved windings, parasitic capacitancechanges too. Fig. 10 shows magnetic field strength distribution for three

interleaved winding arrangements(while the secondary is short circuited) for the EE core. TABLE VII display newleakage inductance and parasitic capacitance for these windings. a,b,c, typesare the same as Fig. 10.

(a) (b)

(c)

Fig. 10. Magnetic field strength(A/mm) for EE core by interleaved windings. These Color diagrams are scaled tobe compared with Fig. 7(f). a) Interleaved winding arrangements ofP-S-P-S-P-S-P-S from left to right in the right window and a symmetryarrangement for the left window. b) Interleaved winding arrangements ofP-P-S-S-P-P-SS from left to right in the right window and a symmetryarrangement for the left window. c) Interleaved winding arrangements ofP-S-S-P-P-S-S-P from left to right in the right window and a symmetryarrangement for the left window.

TABLEVII

Leakageinductance and parasitic capacitance of the EE core with interleaved

windings

Interleaved EE | Leakage | Parasitic |

Transformer Type | Inductance( H) | Capacitance(pF) |

a | 5.12 | 6704.1 |

b | 19.90 | 2875.0 |

c | 5.51 | 3832.3 |

The maximum ofmagnetic field strength in both Fig. 10(a) and (c) reaches to about 6 (A/mm)and in Fig. 10(b) it reaches to almost 11.8 (A/mm), while in the Fig. 7(f) withP-P-P-P-S-S-S-S arrangement the maximum is 23.3 (A/mm). However, magnetic field,leakage stored energy in the window area and consequently leakage inductancefor this transformer decreases significantly by interleaved windings, theparasitic capacitance increases significantly. Additionally, in TABLE VII typea and b have almost the same leakage inductance but a completely differentcapacitance.

V. CONCLUSION

By comparing the results of the structures andthe winding arrangements, it can be concluded that by changing designparameters a wide range of leakage inductance and parasitic capacitance can beobtained. Depend on the application and utilization, all of the studiedstructures can be used for the high frequency transformers. To change theleakage inductance changing the magnetic field strength distribution isnecessary because it leads to changing magnetic stored energy. By this methodor using different structures, the leakage inductance for high frequencytransformers can change and by reduction of that, the efficiency can increase.But this may lead to a different parasitic capacitance between primary andsecondary windings which in someapplications with fast switches (SiC and GaN) with high result toinjecting high frequency current that may create many troublesfor the system. In some structures like EE cores using interleavedwindings, may be a good solution to reduce leakageinductance, although this leads to a higher parasiticcapacitance. Finally, to choose a structure and a specific design strategy,each application and system should be considered and analyzed separately, andbetween leakage inductance and parasitic capacitance a trade-off is vital.Also, manufacturing factors in each application are important and may imposesome design compromises.

ACKNOWLEDGEMENT

This material is based upon worksupported by the U.S. Department of Energy, "Enabling Extreme FastCharging with Energy Storage", DE-EE0008449.

REFERENCES

[1] H. Qin and J.Kimball, “Ac-ac dual active bridge converter for solid state transformer,” *2009IEEE Energy Conversion Congress and Exposition*, 2009.

[2] B. Zhao, Q. Song, W.Liu, and Y. Sun, “Overview of Dual-Active-Bridge Isolated Bidirectional DC–DCConverter for High-Frequency-Link Power-

ConversionSystem,” *IEEE Transactions on Power Electronics*, vol. 29, no. 8, pp.4091–4106, 2014.

[3] Z. Ouyang, J. Zhang,and W. G. Hurley, “Calculation of Leakage Inductance for High-FrequencyTransformers,” *IEEE Transactions on Power**Electronics*, vol. 30,no. 10, pp. 5769–5775, 2015.

[4] J.-M. Choi, B.-J.Byen, Y.-J. Lee, D.-H. Han, H.-S. Kho, and G.-H. Choe,

“Design of Leakage Inductance in Resonant DC-DC Converterfor Electric

VehicleCharger,” *IEEE Transactions on Magnetics*, vol. 48, no. 11, pp. 4417–4420, 2012.

[5] S. Yazdani and M.Ferdowsi, “Robust Backstepping Control of Synchronverters under Unbalanced GridCondition,” *2019 IEEE Power and**Energy Conference at Illinois (PECI)*,2019.

[6] C. W. T. McLyman, *Transformerand inductor design handbook*. CRC Press, 2017.

[7] M.K. Kazimierczuk, *High-frequency magnetic components*. Chichester: JohnWiley & Sons, 2014.

[8] S. Zhao, Q. Li, F. C. Lee, andB. Li, “High-Frequency Transformer Design for Modular Power Conversion FromMedium-Voltage AC to 400 VDC,” *IEEE**Transactions on Power Electronics*,vol. 33, no. 9, pp. 7545–7557, 2018.

[9] M. B. Shadmand and R. S. Balog,“A finite-element analysis approach to determine the parasitic capacitances ofhigh-frequency multiwinding transformers for photovoltaic inverters,” *2013IEEE Power and Energy**Conference at Illinois (PECI)*, 2013.