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Engineering, Technology & Applied Science Research Vol. 12, No. 1, 2022, 8223-8227 8223 
 

www.etasr.com Minh et al.: A Novel Approach for the Modeling of Electromagnetic Forces in Air-Gap Shunt Reactors  

 

A Novel Approach for the Modeling of 

Electromagnetic Forces in Air-Gap Shunt Reactors  
 

Hung Bui Duc  

School of Electrical Engineering 
Hanoi University of Science and Technology 

Vietnam 

hung.buiduc@hust.edu.vn 

Tu Pham Minh 

School of Electrical Engineering 
Hanoi University of Science and Technology 

Vietnam 

tu.phaminh@hust.edu.vn 

Tuan Phung Anh 

School of Electrical Engineering 

Hanoi University of Science and Technology 
Vietnam 

tuanphunganh1@hust.edu.vn 

Vuong Dang Quoc 

School of Electrical Engineering 

Hanoi University of Science and Technology 
Hanoi, Vietnam 

vuong.dangquoc@hust.edu.vn 
 

 

Abstract-Shunt reactors are usually used in electrical systems to 

imbibe reactive powers created by capacitive powers on the lines 

when the system is operating on low or no loads. Moreover, they 

are also used to balance reactive powers and maintain the 

stability of a specified voltage. In general, the air gaps of a 

magnetic circuit shunt reactor are arranged along the iron core 

to reduce the influence of fringing and leakage fluxes. Therefore, 
non-magnetic materials made of ceramics or marbles are often 

used in air gaps to separate the iron core packets. The direction 

of the fringing flux is perpendicular to the laminations, so the 

core packets of the shunt reactor are generally made from 

radially laminated silicon steels. Due to the alternating 

electromagnetic field through the core, a periodically altered 

electromagnetic force is produced between the core packets, 
tending to compress the ceramic spacers. This electromagnetic 

force causes vibration and noise in the core. In this research, a 

finite element approach based on the Maxwell stress tensor was 

developed to compute the magnetic flux density and the 
electromagnetic forces appearing in a shunt reactor. 

Keywords-shunt reactor; air gaps; magnetic flux density; 
electromagnetic force; Maxwell stress tensor 

I. INTRODUCTION  

A Shunt Reactor (SR) is an important component widely 
used in power transmission systems to improve stability and 
efficiency. Parasitic capacitance occurs in a line when the 
system is working under low or no load. Especially for a long 
line, its value is very large and leads to a voltage increase, 
causing an overvoltage at its end. Hence, to maintain voltage 
stability according to the regulations and balance the reactive 
power in electrical systems, the SR is proposed to absorb the 
excess reactive power created by the line capacitance [1-4]. In 
addition, in order to reduce the magnetic flux and to avoid the 
saturation of the magnetic circuit, air gaps are added in the iron 
core of the SR to increase the reluctance values. On the other 
hand, fringing fluxes appear around the air gaps [6-9], 

increasing the winding inductance and causing nonuniform 
magnetic fields in the iron core packets of the SR. 
Nonmagnetic materials such as ceramic or marble are often 
used in air gaps to separate the iron core packets. The fringing 
flux departs and reenters the lamination perpendicularly, so the 
core packets of the SR are generally made from radially 
laminated silicon steels. Because of the alternating 
electromagnetic field through the core, a periodically altering 
electromagnetic force is produced between the core packets, 
tending to compress the ceramic spacers. The force on the core 
packets is a Maxwell force that causes vibration and noise. 
Maxwell forces are acting at twice the power frequency due to 
the sinusoidal flux passing through the windings at the power 
frequency. 

Many researchers have investigated transient situations and 
magnetic fields in SR or calculated the air-gap reluctance and -
inductance of winding for magnetic circuits [6-12]. In [13], a 
method was introduced to calculate the leakage fields in air-
core reactors using a 3D reluctance network. Testing problems 
of gapped-core reactors were presented in [14, 15]. Some 
researchers investigated the temperature field distribution [16, 
17]. In this research, a Finite Element Method (FEM) based on 
Maxwell stress tensor theory [18] was developed for an iron 
core gapped SR to compute the magnetic flux density and the 
electromagnetic force on the core packets. These fields are the 
inherent reason for the vibration and noise and tend to 
compress the ceramic spacers. 

II. METHODOLOGY  

A. The Structure of a Gapped-Core Shunt Reactor 

The structure of an SR is illustrated in Figure 1. The mid 
limb with the non-magnetic gaps is enclosed by the winding. 
The winding is also enclosed by a frame of core steel providing 
the return path for the magnetic field. Three single-phase SRs 
are connected to provide a three-phase transmission line. A 

Corresponding author: Hung Bui Duc 



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www.etasr.com Minh et al.: A Novel Approach for the Modeling of Electromagnetic Forces in Air-Gap Shunt Reactors  

 

three-phase SR can also be used, having 3 or 5 limbed cores. 
Figure 1(b) shows a three-limbed core SR having a strong 
magnetic coupling between the 3 phases. A 5-limbed core is 
shown in Figure 1(c), where the phases are magnetically 
independent due to the enclosing magnetic frame formed by the 
two yokes and the two unwound side-limbs. The gapped core 
limbs are built of core packets. The core packet type is further 
classified into parallel and radially laminated core types by the 
core packet laminating methods. The parallel laminated core 
type has steel plates in parallel in the same direction as the 
power transformer core, and the radially laminated core type is 
constructed as a cylindrical core packet by laminating steel 
plates in the radial direction arranged in a wedge-shaped 
pattern [14]. The core packet is filled with epoxy resin and 
molded into one solid unit. 

 

 
Fig. 1.  Modeling of the shunt reactor: (a) single-phase, (b) 3-phase 3-
limbed core, (c) 3-phase 5-limbed core [6]. 

The vibration in the SRs is usually higher than in 
transformers as a result of the magnetic forces between the 
consecutive core packets. Therefore, the core is designed to 
eliminate excessive vibrations. The non-magnetic gaps are 
established by using cylindrical spacers to take the large 
magnetic forces appearing between the discs in a gapped core 
SR. The height of the distance spacer is equal to the height of 
the air gap. The core packets are stacked and cemented together 
to form a core limb column. Dimensional stability and core 
tightness can be fixed by an epoxy impregnated polyester 
material and a fiberglass cloth of a few millimeters between the 
last limb packet and the top/bottom yoke. The yokes and side 
limbs, normally with a rectangular cross-section, make up the 
magnetic circuit. 

B. Parameters of the Shunt Reactor 

A single-phase SR of 35MVAr (500/√3kV and 50Hz) was 
considered to calculate the electromagnetic force in the core 
packets. A simple model of this single-phase SR is shown in 
Figure 2. The volume of the air gaps was determined by the 
equations describing the magnetic circuit model. It should be 
noted that this volume depends on the main parameters of the 
shunt reactor, i.e. reactive power, magnetic inductance, 
winding inductance, frequency, and energy storage in the 
winding space air gap. Iron materials have usually a high 
magnetic permeability (µ=µr.µ0) compared to the air gap 
permeability (µ0). Therefore, the reluctance of the magnetic 
core is very small, compared to the reluctance of the air gaps, 
and can be neglected in the equivalent magnetic circuit. In 
addition, the winding resistance is normally very small 
compared to the inductance and can be neglected when 
defining the parameters of the electric circuits. From the 
relationship between the Magnetomotive Force (MMF), 
magnetic flux, and reluctances of the magnetic circuit, the 
magnetic flux density on the core can be determined via the 
reactive power and the volume of the air gap as: 

�� � � ��	
.�.
�    (1) 
where Bm, Vg, and µ0 are the maximum flux density, the air gap 
volume, and air permeability respectively. The obtained results 
of the SR of 35MVAr are shown in Table I [10] 

 

  
Fig. 2.  Model of the shunt reactor. 

TABLE I.  SR RESULTS BY THE ANALYTIC METHOD 

Parameters Notation Results 

Reactive power Q (MVAr) 35 

Rated voltage U (kV) 500/√3 
Rated current I (A) 121,24 

Totalinductance L (H) 7.5788 

Core dimension Dc (mm) 673 

Height of core Hc (mm) 1758 

Total air gap length lg (mm) 363 

Turn number N (turn) 2166 

Width of winding Ww (mm) 248 

Height of winding Hw (mm) 1488 

 

C. Calculation of the Electromagnetic Force  

In the gapped core of the SR, vibrations can be very high 
due to the magnetic forces between the consecutive core 
packets. Thus, the ceramic spacers and any other gap materials 
should tolerate these forces without any significant shrinkage. 
The electromagnetic force acting on the core packets can be 
computed by either the Maxwell stress tensor approach or the 
virtual work method [20]. This force is given by: 

� � ∆�∆� � �� ���
 ��,    (2) 
where Ac is the cross-sectional area, and ∆x is the translation of 
a body. The surface force density is therefore given by: 

F� � ��� � �� ���
   (N/m2)    (3) 
The higher the operating flux density, the higher the 

attractive force is. Therefore, the operation flux density is 
typically lower than the value used sometimes in power 
transformers. According to (2), it is necessary to increase the 
total volumic air gap to reduce the magnetic flux density. The 
force between the core packets can be calculated by (3) if the 
flux distribution is uniform. However, as the flux distribution in 
the SR is non-uniform, the finite element approach is generally 
applied with the virtual work principles. The magnetic flux 
density can be determined through the MMF and the air gap 
parameters as:  

Hc

Dc

Hw

Ww

Hy

Dy

Wy



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www.etasr.com Minh et al.: A Novel Approach for the Modeling of Electromagnetic Forces in Air-Gap Shunt Reactors  

 

� � ��� � � !."#�     (4) 
The surface force density can also be calculated by: 

F� =
�

�

�
!
�"�

#�
    (N/m

2
)   (5) 

The Maxwell stress tensor approach is widely used for 
computing electromagnetic forces. When ignoring the 
magnetostriction, the distribution of the magnetic forces in the 
core packets can be defined as: 

�$ = −
�

�
H.H	∇�    (6) 

On the material surface, Fv can be written in terms of 
normal and tangential components as: 

�$ � − ��)H*� + H,�-	∇�  � − �� .H*� + �/���0∇�  					� − �� )H*�	∇� − B,�	∇2-	  
(7) 

since  
3��� � −42, where reluctivity is 2 � ��. 

The force density at the surface can be defined as: 

�5 � − �� )H*�	)� − �- − B,�	)2 − 2--6    (8) 
where n is the unit normal vector to the surface. In the gapped-
core SR, the force density between two packets can be 
determined by (8), which when the fringing effect is negligible 
(Ht=0) becomes: �5 � �� )B,�	)2 − 2--6    (9) 
The reluctivity 2 of the core material in (9) is negligible. 

Thus, the magnitude of the force per unit area perpendicular to 
the surface can be expressed as: 

�7 � �� ���
    (N/m2)    (10) 
This force is a function of the square of the flux density. 

The higher the operating flux density, the higher the forces of 
attraction are. 

D. Modeling of the Shunt Reactor via a Finite Element 

Approach 

Maxwell's equations and the constitutive laws of the 
problem are written as [10-13]:  

∇ × 9 � − :�:*     (11) ∇	•	B � 0    (12) 
∇ × < � = + :>:*     (13) B � ∇ × A    (14) 

The value of the magnetic flux density B corresponding to 
the magnetic field intensity H is: � � �<    (15) 

The magnetization forces were calculated by applying the 
Maxwell stress tensor locally. The Maxwell stress tensor for 
magnetic fields is a 3×3 matrix and has a unit of N/m

2
: 

@ � A�B<B − �� �<�C<B�D<B 			
�B<C�C<C − ���<�D<C 			

�B<D�C<D�D<D − ���<EF
6B6C6DG    (16) 

This is also a suitable approach for computing the force 
distributions. The divergence of the stress tensor H$ is defined 
by the volume force density in (16a), while the volume force is 
defined in (16b): f$ � ∇ ∙ @    (16a) �$ � K ∇ ∙ @LM$     (16b) 
The force on the surface of the core packets can be obtained 

by: 

�7 � K @LN7     (17) 
The tangential Ft and the normal Fn components of the 

force F per unit area on an elemental surface can be expressed 
in terms of the tangential and normal field components as: �* � �,<*    (18a) �, � �� . ��
 �,� − � <*�0    (18b) 
Both the normal component of B and the tangential 

component of H should be determined. In this part, the FEM 
was used to determine the modeled SR parameters, utilizing the 
Ansys Maxwell software [20]. The accuracy and the validity of 
FEM have been proven in recent studies [10, 12, 19]. Due to 
the anisotropy of the magnetic properties, a set of 3D laminated 
coordinate systems was defined considering the anisotropy of 
magnetic properties for laminated silicon steel in the Rolling 
Direction (RD), the Transverse Direction (TD), and the 
Lamination Direction (LD). The models of all parts (yokes, 
core packets, windings) in the 3D coordinate system were 
created, and core packets are shown in Figure 3(a). Then, the 
lamination direction of the yokes and all core packets was 
chosen in the 3D laminated coordinate systems. The core 
packet of the SR was generally made from radially laminated 
silicon steels, as shown in Figure 3(c). The mapping between 
the laminated and the rectangular spatial coordinate systems is 
presented in Table II. 

III. RESULTS AND DISCUSSION 

The mapping distributions of the magnetic flux density in 
the SR core are presented in Figure 4. The magnetic flux 
density in the SR model is shown in Figure 4(a)-(b). The 
distribution of the magnetic flux density on a core packet, 
generally made from radially laminated silicon steels as Figure 
3(c), is presented in Figure 4(c). The magnetic flux density is 
unevenly distributed, especially in the core packets. The 
magnetic flux density around the core’s corner is larger than in 
its interior due to the characteristics of the SR, which has air 
gaps between the core packets to increase the reluctance of the 
magnetic circuit. However, the fringing flux around the air gaps 



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increases the magnetic flux around the iron core packets. 
Together with the fringing flux, it is directed to the laminated 
silicon steel in the transverse direction, which has worse 
magnetic properties than the rolling direction. The magnetic 
flux density along the lines Y1-Y2 and Y3-Y4 is shown in 
Figure 5. 

 

 
Fig. 3.  3D model of the single-phase SR and the radially laminated silicon 
steels. 

TABLE II.  MAPPING BETWEEN THE TWO COORDINATE SYSTEMS 

Part RD TD LD 

Core packets z-axis Perpendicular z-axis Radial as a fan-shaped 

Top and bottom yokes x-axis z-axis y-axis 

Side yokes z-axis x-axis y-axis 

 

 
Fig. 4.  The magnetic flux density on the magnetic circuit and iron core 
packets. 

 
Fig. 5.  Magnetic flux density along the lines Y1-Y2 and Y3-Y4 on the 
surface of a core packet. 

In the same way, the magnetic flux density around the 
outermost edge is highlighted in Figure 5. It is significantly 
larger than the rest, while this magnetic flux density is directly 
related to the electromagnetic force acting on the core packets. 
This electromagnetic force component will act on the ceramic 
plates that separate the core packets. The distribution of the 
force density on the surface of the core packets is shown in 
Figure 6. In the gapped-core SR, the magnetic force acts 
between the core packets like close electromagnets and can be 
interpreted as attractive forces that try to minimize the gap 
length. Due to the very high magnetic attraction forces between 
the core packets, the ceramic spacers and any other air gap 
material should tolerate these forces without any significant 

shrinking. As shown in (14), these forces are a function of the 
square of the flux density, and thus the higher the operating 
flux density, the higher the forces of attraction are. The 
operating flux density is typically lower than the value usually 
used in power transformers. The cut values of the 
electromagnetic force density on the lines Y1-Y2 and Y3-Y4 
on the core’s surface are shown in Figure 7. It can be observed 
that these forces have the same value but in the opposite 
direction. 

 

 
Fig. 6.  Distribution of  the force density on the surface of the core packets. 

 
Fig. 7.  Distribution of force density along the lines Y1-Y2 and Y3-Y4 on 
the core surface. 

Figures 5 and 7 show that the electromagnetic force results 
are similar to the magnetic field results. The electromagnetic 
force around the outermost edge is also significantly larger than 
in the rest. Figure 8 shows the post-processing normal forces 
acting on the top and bottom surfaces of the core packets. The 
positive normal forces are linked to the top surfaces of the core 
packets, whereas the negative normal forces are linked to the 
bottom surfaces. As shown in Figures 6 and 8, the forces acting 
on the top and bottom surfaces of each core packet are in the 
opposite directions and have approximately the same 
magnitude, resulting in a net force acting close to zero on each 
core packet. The ceramic spacers are in contact with the bottom 
surfaces of the upper core and the top surfaces of the lower 
core packets. Due to the two surface forces acting on the 
ceramic spacer in the opposite directions, these spacers suffer 
huge compression forces and should withstand approximately 
257.5kN/m

2
 or 26.3Ton/m

2
. If the flux density on the iron cores 

is doubled, these forces will increase 4 times to over 



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100Ton/m
2
. To withstand the very high forces acting on the 

ceramic spacers, their height has to be precisely uniform to 
ensure a uniform force distribution on all the ceramic spacers. 
Usually, the spacers are first glued to the core packet and then 
grinded to be equal. This result explains why during the design 
of shunt reactors, the flux density of their core should be 
smaller than the core of the power transformer, even using the 
same magnetic material. 

 

 
Fig. 8.  Force density distribution along the lines Y1-Y2 and Y3-Y4 on the 
surface of the core packets. 

IV. CONCLUSION 

This study investigated the electromagnetic forces on the 

core packets of a shunt reactor by both the virtual work 
method and the Maxwell stress tensor approach. In both 

methods, the normal forces are a function of the square of the 

flux density. Finite element approach, using Ansys Maxwell 
software, was applied to compute the magnetic flux density 

distribution and the normal surface forces density on the core 

packets. The forces acting on the top and bottom surfaces of 
each core packet are in opposite directions and have 
approximately the same magnitude, resulting in a near-to-zero 

net force acting on each core packet. However, as the surface 

forces act on the ceramic spacers in opposite directions, they 

become huge compression forces. Therefore, these results will 
be useful to researchers, design engineers, and manufacturers 

when choosing the appropriate magnetic flux density on the 

core in order to control the electromagnetic forces acting on 
the spacers between the core packets. 

ACKNOWLEDGMENT 

This study was funded by the Hanoi University of Science 

and Technology (HUST) under grant number T2021-PC-006. 

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