Splitting Conductors of Coils on PCB
for AC-resistance Reduction
Shunsaku Nomoto, Shinjiro Shimura, Keisuke Kusaka
Dept. of Electrical, Electronics, and Information Engineering
Nagaoka University of Technology
Nagaoka, Japan
Email: s201067@stn.nagaokaut.ac.jp, kusaka@vos.nagaokaut.ac.jp

Takashi Takada
Mobility Business Development Office Automotive Company
AGC Inc.
Tokyo, Japan
Email: takashi.takada@agc.com

Abstract— Coils built on printed circuit boards (PCBs) havebeen widely used for transformers and inductors because they
have the advantages of downsizing and consistency for mounting
components on the PCBs. However, copper loss caused by theskin and proximity effect takes a large proportion of the total lossof the inductors at high-frequency regions. Thus, reducing
copper loss is one of the significant issues for the inductors to
reduce the power loss on the coils. This paper proposes a method
to suppress copper loss (AC-resistance) due to the skin andproximity effect by splitting patterns of a coil and swapping themat the corner of the coil. The proposed structure for the coil ischaracterized as having no via, which results in an increase incost and copper loss. The simulation results demonstrate that a
12-corner coil shape with a splitting pattern into three sections
suppresses the resistance by 16.7% compared to a spiral coil andimproves the quality factor by 10.9%. Practical experiments haverevealed that the optimized 12-corner coil improves resistance by
24.6% and the quality factor by 18.7%.

In order to overcome the problem mentioned above, thispaper proposes new conductor pattern configurations thatsuppress AC-resistance without vias and the increase in lengthof the winding. The rest of this paper is organized as follows.
First, the theory to reduce the resistance with the proposed
structures is analyzed using the theoretical formula for ACresistance due to the skin effect. Then the optimum shape toreduce the AC-resistance is discussed with a finite element
method (FEM). The suppression effect of AC-resistance in the
optimized coil is verified through a measurement of the
prototype.

Keywords— Skin effect, Proximity effect, Spiral coil, Printed
circuit board (PCB)

I. INTRODUCTION
In recent years, magnetic components, such as inductors
and transformers, formed using copper patterns on printed
circuit boards (PCBs), have been used in certain power
conversion circuits [1–7]. These magnetic components offer
advantages in terms of miniaturization and ease of
implementation compared to conventional windings. However,there is a problem of increased AC-resistance due to the skinand proximity effect when these magnetic components are
utilized at high frequencies. In order to address this issue, somemethods have been proposed to mitigate AC-resistance byarranging the patterns in a twisted configuration, similar to alitz wire [8–10].
In [8], AC-resistance is suppressed by arranging patterns of
the coil such as a litz wire. However, the realization of such astructure requires a lot of vias, which may lead to increased DC
resistance and complexity in implementation. In order toreduce the number of vias, a method to minimize the vias has
been proposed [9]. However, in order to achieve the structureof a litz wire, the winding becomes longer, resulting inincreased losses. While the structure has been proposed tosuppress AC-resistance using a configuration different fromthat of a litz wire, there are drawbacks associated with the useof vias [10].

II. PROPOSED STRUCTURE OF A COIL WITHOUT VIAS
A. Skin and proximity effect of conductors
Fig. 1 illustrates the proposed pattern configurations of thecoil without vias, where w and t are the width and thickness ofeach conductor, and c is the clearance between adjacentconductors. In order to reduce the DC resistance of the patterns,
it is necessary to increase their cross-sectional area, as shownin (1)
RDC 

1
A

(1),

where RDC is the DC resistance of a conductor [/m], is the
conductivity of copper [S/m], and A is the cross-sectional areaof a conductor [m2]. However, at the higher frequencies, theskin depth  [m] becomes smaller, as shown in (2)
 

1
πf 

(2),

where f is the frequency [Hz], and  is the permeability ofcopper [H/m], leading to an uneven distribution of current
density in the conductor, increasing AC-resistance. Thus, it isnecessary to split patterns to reduce AC-resistance. The total
resistance for each pattern configuration is calculated using thetheoretical formula for resistance due to the skin and proximity
effect and verified by the effect of AC-resistance suppression.
The total resistance of a conductor is composed of the
conduction and proximity resistance. Each resistance iscalculated by
RAC  Rcond  Rprox,x  Rprox,y

(3),

1
cond
A

(4),

4π
prox,x H x2
(5),

Rcond Rprox,x 

4π
Rprox,y prox,y H y2(6),where RAC is the total resistance of a conductor, Rcond is the
conduction resistance including DC resistance and resistancedue to the skin effect, cond is the parameter representing the
conduction losses depending on frequency and geometry of the
conductor, Hx and Hy is the x and y direction component of theexternal magnetic field, prox,x and prox,y are the parameter
representing the proximity losses due to the Hx and Hy, andRprox,x and Rprox,y is the proximity resistance due to theproximity effect [11]. The external magnetic field is calculated
as follows
H xij yi  y jI
2πn  x  x 2   y  y  2(7),xi  x jI
2πn  x  x 2   y  y 2

(8),

i

H yij  

j

i

i

j

j

i

j

assuming that the conductors are infinitely long, where I is thecurrent flowing through the conductor, n is the number ofconductors in parallel, xi and yi are the center coordinates of the
conductor i, and xj and yj are the center coordinates of the
conductor j [12].
Fig. 2 shows the calculated frequency characteristics of thetotal resistance for each pattern, as shown in Fig. 1 with w =0.4 mm, t = 0.5 mm, and c = 0.1 mm. From Fig. 2, it isobserved that arranging the patterns diagonally, as shown in
Fig. 1 (c), effectively suppresses the AC-resistance. Therefore,
a structure in which patterns interchanged while maintaining
the configuration was considered.
B. Proposed structure
Fig. 3 shows the proposed coil structure. Fig. 3 (b)provides an enlarged view of the pattern. When the patternreaches the corners of the polygon, the positions of the pattern
are interchanged from the inside to the outside of the coil.
Importantly, since the split patterns are arranged on separate
layers, no interference occurs during the interchange.
Fig. 4 and tableⅠ show the geometric parameters of the
proposed coil. This figure represents a coil with the number ofsplitting a pattern of three (n = 3) and five turns (nt = 5).
III. OPTIMIZATION OF THE PROPOSED STRUCTUREn

H x   H xij(9),i 1, i  jn

H y   H yij(10),

A. Consideration of the number of splitting
The parameters of the coil under consideration arepresented in TableⅡ. In order to determine the optimal numberi 1, i  j(a) Lateral(b) Vertical(c) Diagonal

Fig. 1. Configurations of patterns of a coil.
(a) Top view

LateralVerticalDiagonal(b) Enlarged viewFig. 2. AC-resistance of each configuration.

Fig. 3. Proposed coil structure.

Center ofthe coil

wt

w
t

c

din /2ds
dout/2
Fig. 4. Geometric parameters of the proposed coil.

of splitting that effectively suppresses the skin and proximity
effect in the coil on these parameters at 100 kHz, the change in
AC-resistance is analyzed through (3) while keeping the coilshape as a 12-corner polygon and varying the number ofpattern divisions. In order to simplify the calculation, thepattern is assumed to be straight.
Fig. 5 shows the change in AC-resistance as the number ofpatterns splitting varied based on (3). From Fig. 5, the ACresistance is most effectively suppressed when the pattern issplit into three sections. Even with an increase in the numberof splits of more than three sections, there is not much
reduction in AC-resistance.
Based on Fig. 5, the optimal number of splits is three.
However, the performance of a coil is not only determined byresistance. The coil should be evaluated considering both theresistance and inductance. Thus, the quality factor of the coilcalculated by (11) is used for the comparison, where L is theinductance and R is the equivalent series resistance of the coil.
Q

2πfL
R

(11)

An investigation is conducted to determine the optimal numberof splits with the Finite Element Method (FEM). The
simulation conditions are set at the dodecagonal coil structure,
as shown in Fig. 3, with a frequency of 100 kHz, and a currentof 28 A. In order to simplify the FEM model, the coil was
assumed to have a single winding.
Fig. 6 shows the change in AC-resistance as the number ofpatterns splitting varied based on the FEM. From Fig. 6, theTABLE I.

TABLE II.

PARAMETER VALUES OF COILS
Symbol

Value

dout

60 mm

ds

0.5 mmc

0.01 mmwt

1.2 mmt

0.5 mmnt

5AC-resistance is most effectively suppressed when the patternis split into four sections. However, as the number of splits
increases, the inductance decreases. Therefore, the quality
factor is maximized with the patterns divided into three. Also,
when the pattern is split into more than three sections, theresistance does not decrease much compared to the pattern
divided into three. This is explained by the current density.
Fig. 7 shows the current distribution of the coil cross-section.
As the number of splitting increases, the overall thickness
increases due to the need to arrange patterns on different
conductor layers, leading to the influence of the proximity
effect in the thickness direction.
B. AC-resistance suppression effect of the proposed coil
In order to validate the AC-resistance reduction effect ofthe proposed coil, the proposed coil is compared with a spiralcoil, which has the same parameters listed in TableⅡ.

PARAMETERS OF COILS

Parameter

SymbolOuter diameter

dout

Inner diameter

dinSpace of windings

ds

Clearance of each splitting patternc

Total width of the windingwt

Thickness of each splitting patterntWidth of each splitting patternw

Number of splittingnNumber of turns

nt

Fig. 5. Relationships between AC-resistance and split.

(a) Resistance(b) Inductance(a) Spiral(b) 6-corner (6c)(c) 12-corner (12c)(c) Quality factor
Fig. 6. Relationships between resistance, inductance, and quality
factor of the coil.

(a) 3 splits(b) 5 splits

Fig. 7. Current density distributions of the single-turn coils.

1) Current density distributions
Fig. 8 illustrates the current density distributions of the
proposed coil and the spiral coil. FEM analysis is conductedwith a current of 28 A and a frequency of 100 kHz. The pattern
split is three based on the analysis results of the previous
section. Additionally, three different shapes of the proposed(d) 24-corner (24c)Fig. 8. Current density distributions of the five-turn coil.

coil are considered (6-corners, 12-corners, and 24-corners), forthe comparison of AC-resistance.
Fig. 8 represents the current density distribution of thequarter of the coil. Since magnetic flux concentrates at thecenter of the coil, the current tends to bias towards the inside of
the spiral coil. However, it is observed from Fig. 8 that the
proposed coil exhibits less variation in current densityeffectively suppress AC-resistance. At a frequency of 100 kHz,the 12c coil proposed structure achieves a reduction inresistance of at least 16.7% compared to the spiral coil.
Spiral

24c
12c

6c

(a) ResistanceFig. 9 (b) shows that the inductance values of all coils
remain nearly constant regardless of frequency. Additionally,as the coil shape approaches a circle, the inductance increases.
This is because when the outer diameter of the coil is fixed, theinner diameter of the coil increases as the coil shape
approaches a circle. As the inner diameter of the coil increases,
the inductance also rises as follows,L nt2 d ave   2.46 2
ln   0.20  
2    (12),

d out  din
2

(13),

d out  din
d out  din(14),

Spirald ave 
24c



12c

where dave is the average diameter and  is the fill ratio [13].

6c(b) InductanceFig. 9. Frequency characteristics of the coils.

3) Quality factor
Fig. 10 presents the FEM analysis results of the quality
factor for varying coil shapes. The quality factor of the
proposed 12c coil is the highest. In comparison to the spiralcoil, the 12c shape demonstrates a quality factor that is at least
10.9% higher. From this result, the optimum coil shape, whichhas 12 corners and three-splitting windings, is introduced, as
shown in Fig. 3.
IV. EXPERIMENTAL RESULTS
Fig. 11 shows prototypes of the spiral and proposed 12c
coil with the common parameters listed in TableⅡ for ACresistance evaluation. The equivalent series resistance andinductance of these coils are measured by the LCR meter(IM3533).

Fig. 10. Quality factor of the coils at a frequency of 100 kHz.

distribution compared to the spiral coil. Furthermore, as thenumber of corners in the polygon increases, the variation in thecurrent density distribution is reduced.
2) Frequency characteristics
Fig. 9 shows the frequency characteristics of resistance andinductance for the proposed coil and the spiral coil. As theouter diameter of the spiral coil is equal to the circumscribedcircle of the proposed coil, the length of the spiral coil patternis longer than that of the proposed coil. Consequently, theresistance and inductance values of the spiral coil are larger
because of the difference in pattern length. Moreover, as thenumber of corners in the polygon increases, approaching acircle, the difference in length decreases.
From Fig. 9 (a), the proposed coil effectively suppresses
the increase of AC-resistance. Furthermore, comparing theresistance values of the proposed coil with 6c, 12c, and 24c
shapes, it is evident that the 12c and 24c shapes more

A. AC-resistanceFig. 12 shows the frequency characteristic of the ACresistance of the coils with reference to the resistance at 1 kHz.
From Fig. 12, the proposed 12c coil effectively suppresses theincrease of AC-resistance at high frequencies. At a frequencyof 100 kHz, the 12c coil achieves a reduction in resistance of at

Spiral coil

Proposed coil12cFig. 11. Prototypes of the spiral and proposed 12c coil.

Spiral12c

V. CONCLUSION
This paper proposes a new coil pattern formed on printed
circuit boards to suppress the AC-resistance. The proposed
structure involves splitting the patterns constituting the coil inthe width direction and shaping the coil into a polygon.
Simulation results demonstrate that by employing a 12c coilshape and dividing the patterns into three parts, the proposed
approach achieves a resistance reduction of over 16.7%
compared to a spiral coil, and improves the quality factor by
10.9%, as confirmed by FEM analysis. Practical experiments
have revealed that the optimized 12c coil improves resistanceby 24.6% and the Q factor by 18.7%.

Fig. 12. Frequency characteristic of the AC-resistance of the coils.

REFERENCES[1]Spiral12cFig. 13. Frequency characteristic of the inductance of the coils.

12c

SpiralFig. 14. Frequency characteristic of the quality factor of the coils.

least 24.6% compared to the spiral coil, where the resistance ofthe spiral and 12c coil at 1 kHz is 495.4 m and 401.1 m.
B. InductanceFig. 13 shows the frequency characteristic of the inductanceof the coils. Fig. 13 has the same characteristics as shown in
Fig. 9 (b). The inductance values of both of the coils remain
nearly constant regardless of frequency.
C. Quality Factor
Fig. 14 shows the frequency characteristic of the quality
factor of the coils. In comparison to the spiral coil, the 12c coildemonstrates a quality factor that is at least 18.7% higher.

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