T. Magn. Soc. Jpn. (Special Issues)., 1, 61-65 (2017)<Paper>Estimation of eddy current loss
for transformer windings based on RNA
T. Hiwatashi, K. Tajima, and Y. Yoshida*

Department of Cooperative Major in Life Cycle Design Engineering, Akita Univ., 1-1, Tegata Gakuen-machi, Akita 010‒5802,

Japan*Department of Electrical and Electronic Engineering,

Akita Univ., 1-1, Tegata Gakuen-machi, Akita 010‒5802, JapanAn estimation method of eddy current loss for transformer windings based on reluctance network analysis(RNA) is proposed. As the initial stage of study, the authors focus on eddy current loss occurring in the transformer
windings and configured a simple evaluation model to estimate eddy current loss in prism copper wires simulatingtransformer windings. The validity of the proposed method is demonstrated by comparing 3D-FEA and experimentalresults.
Keywords: DC-DC converter, reluctance network analysis (RNA), eddy current loss, transformer winding
1. Introduction
DC-DC converters have been used mainly as theDC-power sources of electronic equipment. With thegrowing demand for compact, high-efficiency electricequipment, DC-DC converters are now also required tobe compact and high-efficiency. Magnetic devices occupy
a large volume in DC-DC converters, and the size ofthese devices has been reduced by higher switchingfrequency. However, higher switching frequency
increases the switching loss. A common method forswitching loss reduction is zero voltage switching (ZVS).
Further, it is well known that transformer cores have agap in some cases, and leakage fluxes have increased
due to gaps in the transformer core1). The leakage ofmagnetic fluxes interlinking with the transformerwinding is what causes the eddy current flow and loss
inside the windings. For the transformer, in order todesign based on complex high-frequency magnetic flux
distribution and current density distribution, asimulation technique having high accuracy andrequiring a short period of time for calculation isneeded.
Reluctance network analysis (RNA) has proveneffective for the design of magnetic devices such as
motors and inductors2),3). In addition, since it is possible
to perform calculations by general circuit simulation
software such as SPICE, coupled analysis of the electric
circuit and magnetic devices such as a DC-DC converter
is possible and can be expected to serve as a highly
practical design technique.
The authors previously studied how to estimate
eddy current loss caused by the leakage magnetic flux
for transformer windings based on simple assumptions4).
In this paper, as the initial stage of study, the authors
focus on eddy current loss occurring in the transformer
windings and configured a simple evaluation model toestimate eddy current loss in prism copper wires

simulating the transformer windings. The validity ofthe proposed model is demonstrated by comparing
3D-FEA and experimental results.
2. Analysis model
2.1 Specifications of the analysis object
Figure 1 shows the shape and specifications of theanalytical and experimental object used in theconsideration. The exciting coils are wound aroundU-shape cores. A total of six prism copper wires sized
2.0 mm  2.0 mm 20 mm are sandwiched by theferrite cores. The number of winding turns of theexciting coils per leg and the depth of the cores andprism copper wires are five turns and 20 mm,respectively.
2.2 Derivation of the RNA model
The schematics of the three dimensional (3D) RNA
model are shown in Fig. 2. Figure 2(a) gives anoverview of the magnetic circuit. Ni are magnetomotiveforces (MMF), Rmcore are the magnetic reluctances of thecore, and RmLcoil and RmLgap are the reluctances ofleakage magnetic flux path in the exciting coils and thegaps. The gaps are divided, as shown in Fig. 2(b). The
air gaps between the core and a prism copper wire aredivided into fine elements with a size of 0.2 mm  0.2mm and the other area around the air gap is divided
with a size of 2.0 mm  0.2 mm or 2.0 mm  0.25 mm. In
order to accurately represent the flow of the magnetic
flux of the gaps, prism copper wires are divided into fineelements with a size of 0.2 mm  0.25 mm. The skineffect is considered in this division. Generally, skindepth  of the conductor is obtained from2 ,(1)where  is resistivity of the conductor,  is angularfrequency, and  is permeability of the conductor. Skin

Transaction of the Magnetics Society of Japan (Special Issues) Vol.1, No.1, 20176159Ferrite
core30lz

20

Exciting
coil30ly50

Rm

2.8Fig. 3 Unit magnetic circuit.

Prism
copper wireyyzx( unit : mm )(a) General view of analysis modelxz

2.0
0.4
0.4

2.0

depth of a prism copper wire is about 0.65 mm infrequency at 10 kHz and  = 1.67310-8 m (at 20degrees Celsius), so prism copper wires are assumed tobe affected by the skin effect. Therefore, the prism
copper wires are divided by 0.2 mm in the x-axis
direction and by 0.25 mm in the y-axis direction.
Each divided element is expressed in a 3D unit
magnetic circuit. Figure 3 shows the unit magnetic
circuit of the prism copper wire and the air region. Inthe figure, lx, ly, and lz are the lengths of the element.
The reluctance Rm in the unit magnetic circuit isobtained fromRm (b) Enlarged view of the gap
Fig. 1 Specifications of analysis object.

RmLcoilNi

Ni

RmLcoil

RmLcoil

RmLgapRmLgapRmLcoil

RmLcoil

RmcoreRmLgapRmLcoil

RmcoreNi

Ni(a) Overview of magnetic circuit(b) Division of gap
Fig. 2 Schematics of 3D-RNA model.

62lxly2 0  r l x  l z ,(2)where 0 is permeability of the vacuum and r isrelative permeability. By connecting all unit magnetic
circuits together, the RNA model is obtained.
In the proposed model, the relative permeability of thecore, the prism copper wires, and the air region are set
to 2300, 1, and 1, respectively.
Figure 4 shows the magnetic circuit of a prism
copper wire and MMFs. Figure 4(a) shows the front
view of a prism copper wire and (b) shows the dottedframe part of Fig. 4(a). When magnetic flux  flows toone element of a prism copper wire, the electromotive
force by Faraday’s law causes eddy current i11 occur.
Eddy current i11 produces an MMF in the magnetic
circuit, as shown in Fig. 4(b). Similarly, affected by
the MMF flows to the element below. Thus, the skineffect can be taken into account by placing the MMFs in
the y-axis direction.
Figure 5 shows the division of the z-axis direction ofa prism copper wire. In order to consider the eddy
current distribution accurately, a prism copper wire isdivided into 10 splits in the z-axis direction. The tip of
the wire is divided by 0.2 mm.
Figure 6(a) shows the electric circuits of a prism
copper wire. Eddy currents calculated by the four-layer
independent electric circuit can be converted into theMMFs in the magnetic circuit, as shown in Fig. 4. Whenthe magnetic flux in the electrical circuit of a certain
element flows, the electromotive force d/dt occurs due
to Faraday’s law. Eddy current i is calculated by theelectromotive force and the electric resistances of theelements. Each divided element is expressed in a unit

Transaction of the Magnetics Society of Japan (Special Issues) Vol.1, No.1, 2017

0.20



0.25

i11i12y(a) Electric circuit of a prism copper wirexz(a) Front view of a prism copper wire i11

z
z

i12rz
y i21(b) Part of the enlarged viewFig. 4 Magnetic circuit of a prism copper wire andmagnetomotive force.

rx
2rz
2rx

rz
2rz
2rx
x

i22rx
2rz

rx
2rx
2rz
2x(b) Electric resistances of circuitFig. 6 Electric circuit of a prism copper wire and theelectric resistances.

120.2

RmLcoil

9.2

0.2

430

520.0

RmLgap9.2

RmLcoilzxxzFig. 7 Setting method of the reluctance of leakage magnetic flux
path in exciting coil and the gap.

zyyy

x

2.0

Fig. 5 Division of z-axis direction of a prism copperwire.
electric circuit, as shown in Fig. 6(b). The resistances rx
and rz in the unit electric circuit are obtained from2x ,(3)
r x

yz
2z ,rz  xy(4)where x is the element length in the x-axis direction,
y is the element length in the z-axis direction, and z
is the element length in the z-axis direction,respectively. If there is an adjacent element, theresistance in the circuit becomes halved because thetwo adjacent resistances are connected in parallel.
Figure 7 shows the setting method of thereluctances of leakage flux paths in the exciting coils
and the gaps. At high frequency, the quantity of leakageof the magnetic flux is expected to be greater than at byTransaction of the Magnetics Society of Japan (Special Issues) Vol.1, No.1, 2017

63

Prism copperwires

U-shaped cores(PC40:TDK)

Ferritecore

Supporter
Bipolar
power
supplyKIKUSUIPBZ40-10vePowermeterieN

Excitingcoil
Prism
copper wires

YOKOGAWA
2533EN

10 mmNN

Fig. 9 Configuration of experimental set-up.
Core

Exciting coils(5 turn)(a) Exposed view

Prism
copper wires

Supporter

Fig. 10 Extraction methods of eddy current loss
occurring in the prism copper wires: (a) case of prism
copper wires being sandwiched and (b) case of only thesupporter being sandwiched.

yzx(b) Assembled set-up
Fig. 8 Photographs of experimental apparatus.
low frequency. In this paper, the leakage magnetic
paths in the x-axis direction and the z-axis direction areapproximated straight lines (gap length) and arcs torepresent the leakage magnetic flux. The leakage
magnetic flux paths around the exciting coils areapproximated by only arcs. Further, reluctances areconnected in parallel with the magnetic flux path flow
to the main magnetic flux such that the leakage
magnetic flux paths are combined into a single path, asshown in Fig. 2(a).
3. Comparison with experimental results
To verify the calculation accuracy of the RNA model,
the calculated values were compared to the measured
ones. Figure 8 shows a photograph of the experimental
apparatus, where (a) is an exposed view. U-shapedcores are made of ferrite (PC40, TDK). Supporters (PLAresin) are sandwiched between the cores to fix thepositions of the cores and the prism copper wires.
Figure 8(b) shows the assembled experimental set-up
and Fig. 9 shows the configuration of experimental
set-up. Bipolar power supply (PBZ40-10, KIKUSUI) wasused for the power supply. Excited voltage ve andcurrent ie are measured with a power meter (2533E,YOKOGAWA). The sum of iron loss in the core, copper
loss in the exiting coils, and eddy current loss in theprism copper wires Wtotal is given by
Wtotal 1 Tve ie dt .
T 1



(5)

To extract eddy current loss occurring in the prism
copper wires, the method shown in Fig. 10 was adopted.

64

Figure 10(a) shows the case of prism copper wires being
sandwiched. In this case, occurring total loss WA isgiven by(6)W A  Wi  Wc  We ,where Wi is iron loss in the core, Wc is copper loss of theexciting coil, and We is eddy current loss of the prism
copper wires. Figure 10(b) shows a case without theprism copper wires, where occurring loss WB is given by(7)
WB  Wi 'Wc ' .

For both loss measurements, the exciting current wassine wave at 4.0 A root-mean-square value constant andsame frequency. Therefore, copper loss Wc and Wc’ wereequal. Iron loss Wi was not equal to Wi’ because theprism copper wires were sandwiched. However, since
the ferrite core was used, the iron loss itself wassmaller than the eddy current loss of the prism copperwires, so the difference between Wi and Wi’ was ignoredin this paper. Thus, eddy current loss We in the prism
copper wires is given by(8)We  W A  WB .

Figure 11 shows the experimental results of loss
measurement and the calculation results in 3D-FEA.
The exciting current was sine wave at 4.0 A
root-mean-square value constant, and the frequencies ofthe exciting current were 1, 3, 5, 8, and 10 kHz. Copper
loss of the exciting coil in the experiment was obtained
from the product of the square of effective value of
excited current and DC resistance (RDC = 0.086 ) in
the exciting coil. Since the excited current was constantfor each frequency, copper loss was almost constant.
Iron loss in the core obtained by 3D-FEA tended toincrease with frequency. Since there was almost nodifference in iron loss with and without the prism

Transaction of the Magnetics Society of Japan (Special Issues) Vol.1, No.1, 2017

Exciting current : 4Arms

876Loss (W)Current density
J (A/m2)

Iron loss (Calc., without prism copper wires)
Iron loss (Calc.)
Eddy currrent loss (Calc.)
Total loss (Meas.)
Eddy current loss (Meas.)
Iron loss (Meas.)
Copper loss (Meas.)543

4.00E+07
3.56E+07
3.11E+07
2.67E+07

2

2.22E+07

1

002

4

6810

Frequency (kHz)Fig. 11 Experimental results of loss measurement andcalculation results in 3D-FEA.

Eddy current loss (W)Calc. (3D-FEA)

3Meas.

2.5

4.40E+06

xRNA

3D-FEA

0The prism copper wire was chosen to be sandwiched in
the middle of the right leg in Fig. 1. Both current
density distributions show a similar tendency, which
demonstrates that skin effect can be considered in theproposed model.

2
1.5
1
0.5
0

8.89E+06

zFig. 13 Comparison of current density distribution in a
prism copper wire in exciting current at maximum andfrequency at 10 kHz.

Calc. (RNA)

3.5

1.33E+07

yExciting current : 4Arms

4

1.78E+07

120246Frequency (kHz)81012Fig. 12 Comparison of eddy current loss in prism copper
wires obtained from proposed model, experiment, and3D-FEA.
copper wires, the validity of the experimental method
was confirmed. Measured values and calculation valuesof iron loss showed similar tendencies, but there weresomediscrepancies.
These
discrepancies
wereconsidered to be due to experimental error. Since theeddy current loss occurring in the prism copper wires issubtracted the sum of iron loss and copper loss fromtotal loss as determined from equation (8), it isconsidered that the measurement error of iron loss
directly affected the difference in eddy current loss.
Figure 12 shows the relationship between eddy
current loss of the prism copper wires and frequency.
Calculating conditions were the same as those in Fig.
11. As seen in the figure, the calculated values obtained
from the proposed RNA model were in almost perfectagreement with the ones obtained from the experimentand 3D-FEA.
The validity of the division of a prism copper wire
was confirmed by current density distribution. Figure13 shows the current distributions of RNA and 3D-FEA
in the exciting current at maximum and frequency at 10kHz.

4. Conclusion
An estimation method of eddy current loss in prism
copper wires based on RNA was proposed. The
usefulness of the proposed method was demonstrated bycomparisons with the FEA and an experiment. Furtherstudies will attempt to estimate eddy current loss fortransformer windings in a DC-DC converter by using
the proposed method.
References
1) T. Ninomiya, IEEJ Trans. IA, 117, 2, 120 (1997).
2) K. Tajima and T. Sato, J. Magn. Soc. Jpn., 34, 3, 367 (2010).
3) K. Nakamura, K, Honma, T. Ohinata, K. Arimatsu, T.
Shirasaki, and O. Ichinokura, J. Magn. Soc. Jpn., 38, 4, 174(2014).
4) T. Hiwatashi, K. Tajima, and Y. Yoshida, The papers ofTechnical Meeting on Magnetics, IEE Jpn, MAG-16-035(2016).
Received Oct. 10, 2016; Revised Dec. 9, 2016; Accepted Mar.
4, 2017

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