Journal of Transportation Technologies, 2011, 1, 94-101
doi:10.4236/jtts.2011.14012 Published Online October 2011 (http://www.SciRP.org/journal/jtts)
Transformer Characteristics of Linear
Motor-Transformer Apparatus
Nobuo Fujii1, Shuhei Kanamitsu1, Takeshi Mizuma2
1Faculty of Information Science and Electrical Engineering, Kyushu University, Fukuoka, Japan
2National Traffic S afety & Environment Laboratory, Tokyo, Japan
Received May 17, 201 1; revised July 12, 2011; accepted August 20, 2011
Abstract
The characteristics of linear transformer are studied analytically. The transformer is composed in one of
modes of linear motor-transformer apparatus proposed for future wireless light rail vehicle (LRV). The sec-
ondary (onboard) power factor can be adjusted at any value by an onboard converter. The equivalent circuit
is used to study the transferred power control. The parameters are determined by three-dimensional finite
element method (FEM) analysis for one pole-pair model. Under the rated primary (input) and secondary vol-
tage and current, which are specified for linear motor operation, the characteristics of the secondary power
factor are cleared. It is also shown that the input capacitor can improve the primary power factor and de-
crease the input power capacity, but does not change the efficiency. This linear transformer has the effi-
ciency of 91% and the input power factor of 0.87 when the apparatus without input capacitor is controlled at
the secondary power factor of 0.4.
Keywords: Linear Motor, LIM, Linear Transformer, Non-Contact Power Collection, Wireless LRV, FEM,
Finite Element Method, Power Factor, Equivalent Circuit
1. Introduction
New type of public transportation which is in harmony
with environment has been hoped for a future urban
transit system [1-3]. We have proposed the linear mo-
tor-transformer apparatus for overhead-wireless and
non-contact power collection of light rail vehicle (LRV),
which has functions the secondary current controlled
linear induction motor and linear transformer [2,3]. The
transformer and linear motor mode respectively can be
switched by only the signal of onboard converter, as
shown in Figure 1. The transformer mode with out thrust
is used for non-contact charge to onboard battery in
standstill at station. The charging power and the secon-
dary power factor respectively can be controlled by on-
board converter.
In the paper, the power supply characteristics of
transformer are studied analytically. The equivalent cir-
cuit of transformer is used to compute the characteristics.
The parameters in equivalent circuit are obtained by the
analysis using the three-dimensional finite element me-
thod (FEM). On this transformer, the desirable primary
power factor can be obtained by control the amplitude
and phase of secondary voltage and current.As functions
of secondary power factor, therefore, the characteristics
of primary power factor and efficiency are cleared under
the limit of primary voltage, primary current and secon-
dary current. The effect of serially or parallel connected
input capacitor to obtain a unity power factor at primary
side is also studied.
2. Analytical Model
Figure 1 shows configuration for each operating mode.
For economical configuration, the primary winding on
ground is the concentrated single-phase winding supplied
by commercial pow er source. The secondary winding on
board is a double-layer distributed winding with hex-
agonal shape in the end winding. The onboard converter
operates as a rectifier for the transformer with sin-
gle-phase and as a two-phase inverter for the linear mo-
tor respectively. Figure 1(a) shows the formation of lin-
ear transformer with single phase primary-single phase
secondary configuration, in which the thrust does not
generate at the position. The model with numerical v alue
for one pole-pair length is shown in Figure 2. The rated

95 N. FUJII ET AL.
Secondary two
s i ngle-phase
windings
+
-
Battery
C onve rter (T w o single-p hase rectifiers)
Ground
Primary
Vehicle
Commercial power
(a)
+
-
Battery
Secondary
two-p hase
winding
Primary
Converter (Two-phase inverter)
Vehi cle
Ground
Commercial power
(b)
Figure 1. Linear motor-transformer apparatus.(connection
for one pole-pair), (a) Linear-Transformer mode (position
for no thrust) (b) Linear-Motor mode (position for maxi-
mum thrust).
collecting power per vehicle is 200kW and the rated
power per one pole-pair length is 4.2kW in this design.
The main specifications are shown in Table 1. The rated
values of voltage and current are determined to obtain
the rated thrust in linear motor operation mode because
the apparatus is used as both transformer and motor.
Figure 3 shows the model of three-dimensional analy-
sis, in which the FEM tool named JMAG made in Japan
is used. The periodic method in the longitudinal direction
and the mirror image method in the lateral direction are
applied respectively.
10 25 20
60
26
100
20
70 34
2
4
12 426
Figure 2. Analytical model and dimension for one pole-pair
length.
Table 1. Specifications.
Rated collecting power / one pole-pair P2,rated=4.2 kW
Ground winding
Frequency 50 HZ
Turns of coil 32
Rated voltage / one pole-pair V1,rated=220 V
Rated current I1,rated=113 A
Pole pitch 0.140 m
Mechanical clearance 0.012 m
Width of core 0.300 m
Onboard winding
Turns of coil 8
Rated voltage / one set of winding V2,rated=82 V
Rated current I2,rated=110 A
Slots / pole 4
Slot pitch 0.035 m
Figure 3. Three-dimensional model of FEM.

N. FUJII ET AL.
96
3. Equivalent Circuit
The equivalent circu it of this transformer is expressed as
shown in Figure 4. The values of element parameters
can be estimated by using FEM analysis [4]. The basic
electrical circuit can be modified to the machinery
equivalent circuit with the leakage inductances, the ex-
citing inductance and the special corrected inductance
respectively [4]. Although the expression is intelligible,
it is difficult to estimate the equivalent turn ratio of pri-
mary to secondary exactly, as shown later in Figure 5.
Therefore, in the following study, the equivalent circuit
of Figure 4 is used, and the values of elements are dealt
with constant values.
When the primary and secondary members are in the
position of Figure 2 and the each secondary current is
controlled in the same manner, M1,a = M1,b and L2,a = L2,b.
The values of constants obtained by three-dimen- sional
FEM are shown in Table 2.
Figure 5 shows the ratio of primary to secondary on
voltage and current. These ratios are not constant and
considerably different from the turn ratio of windings.
b
M,1
1
r
1
L
1
I
a
L,2
b
I,2
a
V,2
b
L,2
2
r
2
M
a
M,1a
I,2
b
V,2
Primary Secondary
1
V
2
r
Figure 4. Equivalent circuit for electrical circuit.
Table 2. Values of constants.
Resistance of primary win din g (11 ) 5 C1 r 0.02357

Resistance of secondary winding
(11 ) 5 C2
r 0.01556

Reactance of primary winding 1 L  1.9678

Reactance of secondary winding 2 L  0.4404

Mutual reactance between primary and
secondary 1
M  0.4666

Mutual reactance between secondary and
secondary 2
M 
0.0718

0.2 0.4 0.6 0.8 1
2
4
6
8
0
Ratio of primary to
secondary voltage
V
1
/V
2
Secon dary power factor cos 
2
I
2
=110A
P
2
=4. 2kW
50A
90A
(a)
0.2 0.4 0.6 0.8 1
2
4
6
8
0Secondary power factor cos

2
I
2
=50A
Ratio of primary current
to secondary current
I
1
/I
2
P
2
=4.2kW
110A 90A
(b)
Figure 5. Ratio of voltage and current between primary and
secondary, (a) Voltage (b) Current.
4. Characteristics
In the following, the condition of rated collection power
of 4.2 kW is dealt with M1,a = M1,b = M1, L2,a = L2,b = L2,
I2a = I2b = I2 and V2a = V2b = V2 .
On this transformer, the converter is used to control
the power for charge and to obtain the desirable primary
power factor by control the secondary power factor. Al-
though the primary voltage is fixed in practical use, it is
hard to consider the fixed voltage in this stage because
the relation between primary and secondary voltage
changes as shown in Figure 5. The secondary power
factor controlled by the converter is, therefore, used as a
parameter to clear the characteristics of this type of
transformer.
Figure 6 denotes the secondary voltage curves as
functions of secondary power factor at the rated secon-
dary power. It is examined on the rated current of 110A,
90A and 50A as the secondary current. The secondary
voltage incr eases as secondary pow er factor or seco ndar y
current decreases. As the secondary voltage is limited
under the rated voltage of 82 V which is determined in
linear motor operation, the secondary power factor must
be over 0.23 at the rated secondary current. When the
secondary current is smaller, the secondary voltage in-
creases under the constant output of 4.2 kW, and the us-
able region of secondary power factor becomes narrow.
Figure 7 shows the primary voltage-secondary power
factor characteristics. The primary voltage is not propor-

97 N. FUJII ET AL.
tion to the secondary voltage. This is much different
from conventional transformer. On the rated secondary
current, the primary voltage decreases as secondary
power factor decreases in the power factor range higher
than 0.5. As the primary voltage is also limited to the
rated value of 220 V, the usable region of secondary
power factor is over 0.23.
Figure 8 represents the primary current characteristics.
The primary current increases as the secondary current
increases at secondary power factor of 1.0. On the other
hand, th e primar y current decr eases as second ary current
increases in the region of secondary power factor of 0.3.
This figure shows that the usable region of secondary
power factor is from 0.18 to 0.92 on the condition of
rated primary current.
From Figures 6-8, the usable region of secondary
power factor for control is determined to be from 0.23 to
0.92 from total conditions of rated primary voltage, cur-
rent and second ary voltage.
0.2 0.4 0.6 0.81
50
100
150
200
0
Secondary power factor cos 
2
I
2
=110A
Secondary voltage
V
2
[V]
P
2
=4.2kW
50A
90A V
2.rated
Figure 6. Secondary current—secondary power factor
curves.
0.2 0.4 0.6 0.8 1
100
200
300
400
0
Primary voltage V
1
[V]
Secondary power factor cos 
2
I
2
=110A
P
2
=4.2kW
50A
90A
V
1, r ated
Figure 7. Primary voltage—secondary power factor curves.
Figure 9 denotes the primary power factor character-
istics as functions of secondary power factor. When the
secondary power factor is controlled to be 1.0, the pri-
mary power factor is very low for any secondary current
because the magnetic coupling between primary and
secondary member is weak for large air gap. At the rated
secondary current, the primary power factor increases as
secondary power factor decreases in the region of sec-
ondary power factor bet w een 0.4 and 1. 0 .
The extreme value of primary power factor increases
as the secondary current increases. The maximum pri-
mary power factor can be 0.87 at the secondary power
factor of 0.4 for the rated secondary current.
Figure 10 shows the curves of input capacity as func-
tions of secondary power factor. The minimum capacity
is small as the secondary current is large. The rated sec-
ondary current of 110 A and the about secondary power
factor of 0.4 gives the minimum capacity. When the
secondary power factor is controlled to be 1.0, the input
capacity is about four times larger than the minimum
value.
0.2 0.4 0.6 0.8 1
50
100
150
200
0Secondary power factor cos

2
I
2
=110A
Primary curr en t
I
1
[A]
P
2
=4.2kW
50A
90A
I
1, r ated
Figure 8. Primary current—secondary power factor curves.
0.2 0.4 0.6 0.81
0.2
0.4
0.6
0.8
1
0
Primary p o w er factor co s

1
Secondary power factor cos 
2
I
2
=110A P
2
=4.2kW
50A
90A
Figure 9. Primary power factor—secondary power factor
curves.

N. FUJII ET AL.
98
Figure 11 shows the efficiency characteristics, which
is computed taking into consideration of only copper loss
in the primary and secondary windings. For the rated
secondary current of 110 A, in which the current density
is 2.21A/mm2, the efficiency is 91% when the secondary
power factor is 0.4.
Figure 12 represents the ratio between primary and
secondary copper loss. These losses have influence di-
rectly on the efficiency. The current densities of primary
and secondary winding are 2.00 A/mm2 and 2.21 A/mm2
respectively at rated current. When the secondary current
is the rated value of 110A, the ratio of secondary loss to
primary loss increase sharply as the secondary power
factor becomes smaller than 1, and the ratio is 5.7 at
secondary power factor of 0.4. As the influence of sec-
ondary resistance is largely on the efficiency, the smaller
secondary current density will bring higher efficiency.
5. Effect of Input Capacity
In the following, the effect of input capacitor is studied
to improve the input capacity. The cap acitor is connected
in series or parallel at the input side of linear transfo rmer,
as shown in Figure 13. In these cases, the input capacity
S1 is defined as V1I1 and the primary power factor is de-
fined as the ratio of input effective power to input ap-
parent power of V1I1.
5.1. Serial Capacitor
Figure 14 shows the primary power factor curves as
functions of capacitance in series connected input ca-
pacitor when the output power is the rated value and the
secondary power factor is kept at unity. The capacitance
with primary power factor of unity depends on value of
secondary current. In the following section, the cap acitor
with capacitance of 2.61mF for the rated secondary cur-
rent of 110 A, 2.43 mF for I2=90 A and 1.83mF for I2=50
A are used respectively for both primary and secondary
power factor of unity at the rated output power.
0.2 0.4 0.6 0.8 1
10
20
30
40
0
Primary apparent powe r S
1
[kVA]
Secondary po we r factor cos 
2
I
2
=110A
P
2
=4.2kW
50A90A
Figure 10. Primary apparent power—secondary power
factor curves.
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
0
Efficiency

Sec on dary po wer fac tor cos 
2
I
2
=110A
P
2
=4.2kW
50A
90A
Figure 11. Efficiency—secondary power factor curves.
0.2 0.4 0.6 0.81
1
2
3
4
5
6
7
8
0
Loss ratio of secondary to primary
P
2L
/P
1L
Secondary power factor cos 
2
I
2
=110A
P
2
=4.2kW
50A
90A
P
2L
=P
1L
Figure 12. Ratio of secondary loss to primary loss—second-
dary power factor curves.
1
r
1
L
2
L2
V
2
r
2
M
1
M
2
I
Primary Secondar y
L1
V
2
r
C
1
I
1
V
2
I
2
V
2
L
1
M
(a)
1
r
1
L
L
I1
2
r
2
M
Primary Secondary
1
V
2
r
C
1
I
1
M
1
M
2
L
2
L
2
I
2
I
2
V
2
V
(b)
Figure 13. Model with input capacitor, (a) Serial capacitor,
(b) Parallel capacitor.

99 N. FUJII ET AL.
In the connection of serial capacitor, the primary pow-
er factor is markedly improved as shown in Figure 15.
On the rated secondary current of 110A, the primary
power factor is a value above 0.9 in the wide region over
the secondary power factor of 0.3. On the input apparent
power shown in Figure 16, it is kept at nearly minimum
value in the region over secondary power factor of 0.3 at
the rated output power and the secondary current. How-
ever, the minimum value of primary apparent power in
serial capacitor connection is almost equal to that in the
case without input capacitor which is indicated in Figure
10.
Figure 17 represents the efficiency curves in serially
connected input capacitor. These values are quite equal
to those in case without input capacitor. The copper loss
in primary winding is determined by the value of curren t
in primary winding .
The input capacitor does not work to change the cur-
rent in winding which is determined by the design pa-
rameters of apparatus. The capacitor works decrease of
primary terminal voltage of apparatus.
123
0.90
0.95
1.00
I
2
=110A
Primary power factor cos

1
Capacitance of serial capacitor C [m F]
P
2
=4.2kW
co s 
2
=1.0
50A 90A
Figure 14. Primary power factor—capacitance of input
serial capacitor.
0.2 0.4 0.6 0.81
0.2
0.4
0.6
0.8
1
0
Prim ar y p ower f actor cos

1
Seconda ry power fac tor cos 
2
I
2
=110A
C=2.61mF
Serial capacitor
P
2
=4.2kW
50A
C=1.83mF
90A
C=2.43mF
Figure 15. Primary power factor—secondary power factor
in case with input serial capacitor.
0.2 0.4 0.6 0.8 1
10
20
30
40
0
Primary apparent power S
1
[kVA]
Sec ondary power fac tor cos 
2
I
2
=110A
Seria l ca pacitor
P
2
=4.2kW
50A
90A
Figure 16. Primary apparent power—secondary power
factor in case with input serial capacitor.
0.2 0.4 0.60.8 1
0.2
0.4
0.6
0.8
1
0
Efficiency

Secondary power factor cos 
2
I
2
=110A
Serial capacitor
P
2
=4.2kW
90A 50A
Figure 17. Efficiency-secondary power factor in case with
input serial capacitor.
5.2. Parallel Capacitor
Figure 18 indicates the capacitance for primary power
factor of unity at the rated output power and secondary
power factor of unity, which is 2.46mF for rated secon-
dary current, 2.22mF for I2=90A and 1.74mF at I2=50A
respectively. The capacitances are slightly smaller than
those in series capacitor.
Figure 19 shows the primary power factor as functions
of secondary power factor in case with input parallel ca-
pacitor. The primary power factor at rated secondary cur-
rent of 110A is about 1.0 in the region between 0.65 and
1.0 in secondary power factor. The primary apparent
power characteristics shown in Figure 20 are improved
significantly compared to the case without capacitor, as
the parallel capacitor works to reduce the input current
although it does not work to change the input voltage
which is equal t o the vol t age of primary wi ndi ng.

N. FUJII ET AL.
100
123
0.90
0.95
1.00
I
2
=110A
Primary p o wer factor co s

1
Capacitance of parallel capacitor C [mF]
P
2
=4.2kW
cos 
2
=1.0
50A 90A
Figure 18. Primary power factor—capacitance of input
parallel capacitor.
0.2 0.4 0.6 0.81
0.2
0.4
0.6
0.8
1
0
Primary power factor cos

1
Secondary power factor cos 
2
I
2
=110A
C=2.46mF
Parallel capaci tor
P
2
=4.2kW
50A
C=1.74mF
90A
C=2.22mF
Figure 19. Primary power factor—secondary power factor
in case with input parallel capacitor.
0.2 0.4 0.6 0.81
10
20
30
40
0
Primary apparent power S
1
[kVA]
Secondary pow er factor cos 
2
I
2
=110A
Parallel capacitor
P
2
=4.2kW
50A
90A
Figure 20. Primary apparent power—secondary power
factor in case with input parallel capacitor.
The efficiency characteristics do not change on com-
paring those without capacitor because neither the cur-
rent of I1L in Figure 13(b) nor the primary loss is
changed by the input capacitor.
5.3. Comparison
Figure 21 shows the comparison of input apparent power
among in cases wit h seri al capacitor, paral lel capacit or and
without capacitor. On the region of second- dary power
factor with minimum valu e of primary ap- parent power,
the region of serial capacitor is wide com- pared with
that of parallel capacitor. The minimum value in case
without capacitor is almost equal to that in case of serial
or parallel capacitor although the value is ob- tained at
the limited secondary power factor of about 0.4. In this
apparatus, the u sable region of secondary po wer factor is
over 0.23, which is obtained from the condition of rated
primary voltage, current and secondary voltag e for linear
motor operation.
Figure 22 indicates the capacity for serial capacitor
compared to that for parallel capacitor in the condition
rated secondary current with optimum capacitance for
input power factor. Considering the capacity and the us-
able region of secondary power factor, the serial capaci-
tor will be better than parallel capacitor. However, the
improvement of the apparent input power is not signify-
cant in the region between secondary power factor of 0.3
and 0.4, as shown in Figure 21.
0.2 0.4 0.6 0.81
10
20
30
40
0
Primary apparent power S
1
[kVA]
Secondary power factor cos 
2
P
2
=4.2kW
I
2
=110A
Parallel
capacitor
Serial
capacitor
Without
capacitor
Figure 21. Input apparent power in cases with serial capaci-
tor, parallel capacitor and without cap acitor respectively.
0.2 0.4 0.6 0.8 1
10
20
30
40
0
Ca p a city of capa c ito r S
C
[kVA]
Sec ondary powe r fact or cos 
2
P
2
=4.2kW
I
2
=110A
Para llel
cap acito r
Serial
capacitor
Figure 22. Capacity curves of input capacitor—secondary
power factor .

N. FUJII ET AL.
101
6. Conclusions
1) It is cleared that the secondary power factor can be
controlled in the value from 0.23 to 0.92 under the con-
ditions of rated primary voltage, current and secondary
voltage for linear motor operation, as this apparatus is
used for both linear transfo rmer and linear motor.
2) When the input capacitor is used in series or in par-
allel, the input power factor of unity can be obtained at
the secondary (output) power factor of unity. The effect
of input capacitor is recognized in the input apparent
power which can be kept at nearly minimum value in the
region over secondary power factor of 0.3 in serial ca-
pacitor or 0.4 in parallel capacitor at the rated output
power and the secondary current.
3) The serial input capacitor will be better than parallel
capacitor, considering the capacity and the usable region
of secondary power factor.
4) However, the improvement of the apparent input
power is not significant compared to the minimum value
in case without capacitor. The efficiency characteristics
do not change if the input capacitor is removed.
5) In the operation without input capacitor, the effi-
ciency is 91% and the input power factor is 0.87 when
the secondary power factor is controlled at 0.4.
7. References
[1] B. Yang, M. Henke and H. Grotstollen, “Pitch Analysis
and Control Design for the Linear Motor of a Railway
Carriage,” IEEE IAS Annual Meeting (IAS2001), Chicago,
October 2001, pp. 2360-2365.
[2] N. Fujii and T. Mizuma, “Device with Functions of Lin-
ear Motor and Non-Contact Power Collector for Wireless
Drive,” Transmission IEE of Japan, Vol. 126-D, No. 8,
August 2006, pp. 1113-1118 (in Japanese).
[3] N. Fujii and T. Mizuma, “Analytical Study of Special
Linear Motor-Transformer for Wireless Tram,” Confer-
ence Record of The 2008 IEEE Industry Applications
Conference, IAS62p1, October 2008, p. 7.
[4] N. Fujii, H. Ashitomi and T. Mizuma, “Equivalent Circuit
of Linear Transformer with Function of Linear Motor,”
ISEF 2009—XIV International Symposium on Electro-
magnetic Fields in Mechatronics, Electrical and Elec-
tronic Engineering, Arras, France, September 2009.