Energy and Power Engineering, 2013, 5, 1101-1104
doi:10.4236/epe.2013.54B210 Published Online July 2013 (http://www.scirp.org/journal/epe)
Calculation and Design of Dry-type Air-core Reactor
Yan Li1, Zhenh ai Zhang1, Longnv Li1,Guoli Li1, Manhua Jiang2
1Research Institute of Special Electrical Machines, Shenyang University of Technology, Shenyang, China
2Tebian Electric Apparatus Shenyang Transformer Group Co., Ltd., Shenyang, China
Received March, 2013
ABSTRACT
Based on the method of compoun d and additional cond itions under the conditio ns of the equal temperature rise and the
equal potential drop (P.D.) of resistance, the application of design software of dry-type air-core reactor is introduced in
this thesis. The analytical methods of th e inductance are also given. This approach is proved entirely feasible in theory
through the simplification with Bartky transformation, and is able to quickly and accurately calculate reactor inductance.
This pape r p r esents the analytica l methods of the loss of dry-type air-core reactor as well.
Keywords: Dry-type Air-core Reactor; Bartky Transformation; Compound and Additional Conditions; Software
Design
1. Introduction
With the advantages of light weight, good linearity, high
mechanical strength, low noise and easy maintenance,
dry-type air-core reactor is a equipment widely used for
the current limit, voltage regulator and reactive power
compensation in power system [1].
The factors, such as inductance, temperature rise,
height and others, should be considered comprehensively
in the design of dry-type air-core reactor. The mutual
affection and restraint of various factors make the design
of dry-type air-core reactor more complicated. In order to
simplify the design and improve the efficiency, it is nec-
essary to take appropriate simplification and process. The
design adopts the simplified processing mathematical
model which is based on the additional constraints. And
this paper developed the design software of dry-type
air-core reactor.
2. Calculation Methods
In the premise of known voltage, capacity, radius and
height cond itions, th en the desig n of the dry-typ e air-cor e
reactor could obtain parameters of turns, inductance,
temperature rise, current, etc. by using relevant theorems
and formulas. It is difficult to directly calculate the out-
come. Because of the shortage of given conditions. Thus
it’s necessary to simplify it without affecting its accuracy.
The encapsulating width of the reactor is so far smaller
than the coil height that the reactor is considered as the
ideal solenoid in the calculation of the inductor. The re-
sistance is much smaller than the inductance of the reac-
tor, so that wire could be considered as an ideal conduc-
tor, which means we can ignore the affection of resis-
tance. To make the design to meet certain characteristics,
such as making the reactor have a minimum temperature
rise, minimum supplies, minimum loss, simple structure
and convenient manufacturing, we can use the additional
constraints of equivalent temperature rise, height and
resistance drop method, which could not only make the
design more reasonable and achieve structure optimiza-
tion, but also reduce the computing time and improve
efficiency. This article uses additional constraints of en-
trapment between temperature rise, layer winding resis-
tance drop to achieve minimal supplies, losses and tem-
perature rise of reactor.
2.1. The Calculation of the Self and Mutual
Inductances
The inductance of the reactor is the most important pa-
rameter in the dry-type air-core reactor and its calcula-
tion accuracy directly affects the accuracy of the other
parameters of the reactor, and then affects the perform-
ance of the reactor. According to the reference [2], the
mutual inductance of any two coils is :
32
01212121
121 123 124
M 2 (R R)n n[(,,)
(, ,)(, ,)(, ,)
i
iii
CRR Z
CRRZ CRRZCRRZ



 (1)
12
12 22
012 12
222 2
12 12
1
(, ,)22cos
2cossin
i
RR
CRR ZRR RR
RRZ RRd






 (2)

Y. LI ET AL.
1102
12
1
12
2
12
3
12
4
22
22
22
22
H
H
ZS
H
H
ZS
H
H
ZS
H
H
ZS
 


 



 





(3)
where 1、2
nis the number of turns of the two windings
per unit length，1
R、2 R, respectively are the radius、
the distance、the height of the two windings and
nS

presents the horizontal angle.
From the above equation, 12 is only related
with the radius, height, and relative position of two
windings, but regardless of the number of turns of the
windings. After analysis, we can know that 12
is a elliptic integral and the computation of its direct in-
tegral is complex and not easy to achieve. But it could be
simplified and then be solved with the Applied Numeri-
cal Solution according to the Bartky transformation me-
thod proposed in reference [3]. After using simplifica-
tion and the Bartky Transformation, the expression is:
(, ,)
i
CRRZ
(, ,)
i
CRRZ
222
0
2
222 22 2
2
(,,,,)[sincos
sin ]
cos sincos sin
B
CABCqkA B
d
Cqk





 



(4)
where
2
12
12 12
2
2
22
22
12
22
12
, 0<1; ;
4
1
1;
3(1 )31
()
; 1;
()
[]
RR z
qq
RR RR
k
Ak B
kk
RR Z
kCk
RR Z






 




;
when 1, namely q = 0, the parameters need to be
adjusted to 2
RR
A
AC; ; . BBC 0C
By the Bartky transformation method, the complicated
elliptic integral could be simplified to be easily solved
with a more accurate calculation.
2.2. The Calculation of the Encapsulation
1112 2111
112 2
112 2
....................
..... ....
...................
mm N
ii iiimmii
mmmm mmN
Suppose the number of encapsulations in reactor is m,
and regard each encapsulation as a coil. Set as the
self-inductance of the ith encapsulation, ij
i
L
M
as the
mutual inductance of the ith and jth encapsulation, i
as the resistance of the ith encapsulation and i
R
I

as AC
current of the ith encapsulation and
N
U as the voltage
of both ends of reactor. With the basic principle of the
circuit, it is possible to calculate the equivalent height,
turns number and current of the encapsulation, and then
obtain other parameters.
Under general conditions, the resistance is much
smaller than the inductor. In the case of ignoring encap-
sulation resistor, the equation can be equivalent to the
following.
1(1,2,3,)
mN
ijijj
j
U
WW f Iim



 (6)
when solving the number of turns, equation 6 is a nonli-
near equation. It is difficult to directly calculate, but we
can take advantage of some simplifications under certain
constraints. Under the conditions of the equivalent tem-
perature rise, we can get the following equation [4].
1
11
()
m
ijij
j
i
mm
kjkj
kj
aaf
N
I
I
aaf




 (7)
11
2
1
11
1
()
()
1
()
mm
kjkj
kj
N
im
Njij
j
mm
N
kjkj
mkj
jij
j
aaf
U
WIaf
Laaf
af











(8)
2
11 max
0.36
()()(
mm pi
N
kjkj
kj ii
kJ
S
aaf kH




 )
(9)
where
N
N
N
U
L
I

 is the rated inductance,
N
S is the
rated capacity and max

is the maximum allowable
temperature rise.
N
j
LIj MIj MIRIU
 jMI jMI
j
LIj MIRIU
j
LIjM IjLIRIU
 

 
 
 
 
 







 




From the above equation, we can see that in some cas-
es of ignoring the resistor, nonlinear equations could be
converted into linear equations to solve through the addi-
tional conditions of the equivalent temperature rise,
which can greatly simplify the solution process. Under
the condition of knowing the height of dry hollow reactor,
encapsulation thickness, the calculated inductance and
the maximum temperature of encapsulation, the mini-
mum encapsulation number and the turns number of the
respective encapsulating and current, which satisfy the



(5)

Y. LI ET AL. 1103
requirement, could be calculated. The temperature rise
and loss of the encapsulation can be determined after
determining the encapsulation turns and current numbers,
and then some parameters, such as AWG and the height
and width of encapsulation, could be confirmed.
2.3. Calculation of the Layer Current and the
Yurns Number
The calculation of the layer current an d the turns number
is similar to the calculation of the encapsulation. If we
regard each layer as a coil, its layer equations are nonlinear
known from the circuit principle. The resistance is so
much smaller than the inductance that the pressure drop
of the resistance could be ignored. Thus it’s unnecessary
to solve nonlinear equations, but directly obtain the num-
ber of turns, the current, inductance, thickness and loss
parameters of each layer under the constraints of resis-
tance drop method.
Under the constraints of resistance drop method, each
layer current and turns number can be obtained as fol-
lowing:
1
11
nci cjij
jij
iN
nn ck cjkj
kj kj
ss f
rr
I
I
ss f
rr





(10)
11
1
nn ck cj
N
kj
kj kj
incj ij
jj
ss
Lf
rr
Wsf
r





(11)
Where ci
s
is the conductor sectional area of the ith layer
of the coil and i is the radius distance between the ith
layer and the axis.
r
From the above equation, we can see that by the way
of ignoring the resistor and through the additional condi-
tions of the temperature rise, nonlinear equations could
be converted into linear equations to solve, which can
greatly simplify the solution process. Under the condition
of knowing AWG, radius, calculate the inductance and
rated current, the turns number and current of each layer,
which satisfy the requirement, could be calculated. Then
other parameters can be obtained, such as the resistor, the
current density, inductance of each layer.
3. Software Design
The idea for design is as following: According to the
known basic parameters of known, Bartky transformation
is applied to solve the mutual inductance coefficient ij
f
.
Under the constraint condition of the encapsulation
equivalent temperature rise, we can get the encapsulation
number meeting the minimum encapsulating condition
according to the equation 8.Then the turn number and
current of each encapsulation could be obtained on the
basis of the equation 7 and 8. Under the comprehensive
constraint condition such as AWG, height, ampere den-
sity, encapsulation thickness and equal resistance drop
method, the turn number and current of every layer could
be obtained by the formula 10, 11. And then the wire
gauge, axial and radial parallel conductors, encapsulation
height and current density can be confirmed.
In the computation of the wire gauge, the current and
the turn number of each layer as well as current density,
the correct parameters must be chosen to iterate compu-
tation and the two results are also supposed to be com-
pared to confirm a reasonable error range for a accurate
calculation result. If using the wire gauge of the conduc-
tor as iteration parameters, the two results would mutate
because of the inconsecutive change of the wire gauge of
the conductor, thus there is no guarantee for the conver-
gence of the wire gauge. Therefore, the wire gauge
should not be chosen as iteration parameter. If choose the
height or thickness of each layer whose change is not
obvious in Iteration calculations, the change of the pa-
rameters could not be accurately determined, and so the
precision of calculation was poor. Consequently, the
height or thickness of each layer should not be also cho-
sen as iteration parameter. The current density of the
conductor not only changes continuously but also con-
verges well, and its variation in the process of iteration is
apparent, thus various parameters can be accurately de-
termined. According to the mentioned above, this paper
chooses the current density as the iterative parameter.
The program flowchart is shown in Figures 1 (a) and
(b).
4. Example
From the software interface, we can see that the basic
parameters such as rated voltage, rated capacity, rated
current, winding height and winding diameter need to be
input to the interface. The other options can be default
values or modified values, which makes the design more
flexible, convenient, and applicab le.
For example, A dry-type air-core reactor with the fol-
lowing parameters: rated voltage is 10 kV, rated current
is 400 A, rated capacity is 185 kvar, winding height is
0.9 m, winding diameter is 0.8 m, inductance is 3.7674
mH. The design results are shown in Figure 3.
The design results are presented in Figure 3. in detail,
which include encapsulation number, encapsulation loss,
temperature rise, the number of axial and radial parallel
conductors, wire gauge, the thickness, current density,
the number of turns, inductance, the turn number of
each layer, inductance, current and height. These pa-
rameters can fully meet the requirements of a dry-type
air-core reactor.

Y. LI ET AL.
1104
B
Assume the rad ial
p
arallel conductor numbe
r
curent density
is too large
Yes
Adjust the radial number of
parallel conductors
Curent density
is reasonable
No: Adjust the initial
parameters
A
Yes
Calculate and resul t out
p
ut
end
The AWG is
too small A
Yes: Adjust the
height
No
Figure 3. Output result.
5. Conclusions
In this paper, the analytical method is used to calculate
the inductance of the dry-type air-core reactor, the Bartky
transformation is used to simplify it, and the constraints
of the temperature rise method and the resistance drop
method are used to optimize it. Users only need to enter
the basic parameters the design scheme with high accu-
racy and fast calculation can be obtained. The developed
design software interface is simple and easy to use.
REFERENCES
[1] Z. G. Liu, J. H. Wang and W. P. Wang, “Development
and Application of Dry-type Air-core Reactor Design
Software,” Electric Machines and Control, Vol. 6, No. 7,
2003, pp. 103-106.
Figure 1. Program flow chart (b).
[2] X. L. Wei and S. Ma, “Analytic Calculation Method for
the Magnetic Field of Air-core Power Reactor with Mul-
ti-Layer in Parallel,” Transformer, Vol. 2, 1993, pp.
12-15.
[3] T. H. Fawzi and P. E. Burke, “The Accurate Computation
of Self and Mutual Inductances of Circular Coils”. IEEE
Trans Power Syst. Vol. 97, No. 2, 1978, pp. 464-468.
[4] Z. Min, “Relevant Issues and Computer Aided Design of
Hollow Electric Reactor,” Harbin: Harbin University of
Science and Technology, Mar 2003.
Figure 2. Software interface diagram.