H. Heydari*
S.M. Pedramrazi
Regular paper
Impact of Mechanical Forces in a
25kA Current Injection Transformer
Since Current Injection transformer (CIT) systems are within the major group of the standard type
test equipments in electrical industry, their performances are very important. When designing a
high current devices, there are many factors to be considered from which, unharmful effects of
mechanical forces due to high current (25kA) passing through secondary winding of a CIT must be
ensured. Since the electromagnetic forces is proportional to the square of current, up to the present
time, as far as the authors are aware no report on impact of mechanical forces considerations for
current injection devices has been made. This paper attempts to fill this void in our knowledge by
considering various issues related to the mechanical forces in a 25kA CIT, using 2D
electromagnetic fields (EMFs) simulation tools (Ansys 8.1) to calculate stray reactance fields and
the mechanical forces in a single turn secondary winding of the CIT.
Keywords Current Injection Transformer (CIT), Mechanical Forces, Leakage Flux, Finite
Element Method (FEM).
1. INTRODUCTION
Electrical equipment such as circuit breakers, protective relays, and meters are routinely
tested to verify proper operation of current sensing elements. This testing is performed
using high-current, low-voltage test equipment that provides a means of adjusting the value
of current and also of measuring the operating time of the device under test. When
designing a high current devices, there are many factors to be considered from which,
unharmful effects of mechanical forces due to high current (25kA) passing through
secondary winding of a CIT must be ensured. As the current densities rise the
electromagnetic forces acting on the turns of the coil become very significant. CIT systems
are within the major group of the standard type test equipments in electrical industry and
their overall performances are very crucial. Secondary voltage of CIT is usually low, (less
than 10 volts) whereas, the output current is very high (1kA-100kA). The CITs use for
testing equipments such as circuit breakers, that must carry short-circuit current. Testing
can assure that your product meets or exceeds performance standards. During CITs
operation, the mechanical vibration of windings and thermal expansion of wires take place
simultaneously.
However, as the current densities rise the transformer windings are exposed to excessive
electromagnetic forces [1, 2]. Under this condition, the winding will undergo substantial
geometry changes in the coil shape leading to coil/insulation damage and consequently
impedance change.
The relative displacements between wire insulation and other component parts of the
winding may injure the insulation, leading to turn faults [3-5]. In this condition the behavior
of transformer windings must be grasped precisely and reflected in their strength design for
better reliability [6]. The vibration is influenced by force, inertia, material property, friction,
etc [7]. When the winding of transformer carry current, a leakage flux field is produced
*Center of Excellence for Power System of Automation & Operation, Electrical Engineering Department,
Iran University of Science and Technology
[email protected], [email protected]
Copyright © JES 2007 on-line : http://journal.esrgroups.org/jes/
between the concentric windings (i.e. high voltage and low voltage windings). The
principal component of the leakage flux is axial. The current carried by the winding
conductors (circumferential direction) interact with the leakage flux and produces forces as
shown in Fig.1.
Fig. 1. Direction of Forces generated due to interaction of Flux and Current
The nature of the leakage flux is such that it is mostly in the axial direction (except in the
ends of the windings) and hence the maximum radial force occurs around the mid-height of
the coils. Radial forces act outward on the outer winding and inward on the inner winding
(see Fig.2).
Fig. 2 Leakage flux pattern and direction of forces acting on high current and low current
sides.
Near the ends of the windings, a large part of the leakage flux bends towards the core
leg and thereby causes a radial component of the flux. The interaction of this radial
component of flux with the winding current produces axial forces that tend to compress the
winding conductors along the vertical axis. However, there are both radial forces and axial
forces generated in a transformer winding. During the short circuit condition, large amount
of currents flow through the windings. The forces generated in a transformer subject to
short circuit conditions are magnified many times more compared to those under normal
operating conditions. For example, a transformer with 6 percent of leakage impedance will
carry 16 times the rated current during short circuit and this will result in forces more than
250 times those under normal operating conditions. Also, the peak forces are much higher
than the RMS forces. The axial vibration of a transformer winding was studied by massspring system and the effect of magnetic center misalignment was taken into account [7]. A
nonlinear dynamic model for the axial vibration of the transformer winding was
established. In this model, the electromagnetic forces not only vary with time, but also with
the displacement of windings [6]. The simulation of copper and core losses of the CIT from
which temperature rise of secondary windings together with thermal limitation were
calculated [8]. Numerical calculation of short circuit electromagnetic forces on the
transformer winding in a 17MVA transformer is investigated in [9]. The estimation of the
force magnitudes and directions was carried out and compared with those from short circuit
condition [10]. The skin and proximity effects were also considered and their effects on the
force diffusivity were studied [11,12,13]. Up to the present time, as far as the authors are
aware no report on impact of mechanical forces considerations for current injection devices
has been made. This paper attempts to fill this void in our knowledge by considering
various issues related to the mechanical forces in a 25kA CIT, using 2D electromagnetic
fields (EMFs) simulation tools (Ansys 8.1) to calculate stray reactance fields and the
mechanical forces in a single turn secondary winding of the CIT.
2. FEM MATHEMATICAL ANALYSIS
The simulations are based on the edge element method and mainly magnetic vector
potential method [14-15]. The former method constitutes the theoretical foundation of low
frequency EMF element
The vector potential method is applicable for both 2-D and 3-D EMFs. Considering static
and dynamic fields and neglecting displacement currents (quasi-stationary limit), the
following subset of Maxwell’s equations apply:
∇×{H } = {J },
(1)
∂B
},
∂t
∇.{B } = 0.
∇×{E } = −{
(2)
(3)
The basic equation to be solved is in the form of:
•
[C ]{u}+[K ]{u} = {J i }.
(4)
The terms of this equation are defined below; the edge field formulation matrices are
obtained from these terms follow with boundary conditions.
-Degree of freedom:
⎧⎪{Ae }⎫⎪
{u} = ⎨
⎬,
⎪⎩{ve } ⎭⎪
(5)
where {Ae } and {ve } are magnetic vector potentials and time integrated electric scalar
potential respectively.
Coefficient matrices:
⎡[K AA ] [0]⎤
⎥,
[K ]= ⎢ vA
⎢⎣[K ] [0] ⎥⎦
(6)
[K AA ]=[K L ]+[K N ]+[K G ],
[K ]=
L
∫ (∇×[N
A
(7)
] ) [v ](∇×[N A ] −[N A ][σ ] ({v}×∇×[N A ] ))d (vol ),
T T
T
T
(8)
Vol
[K G ]=
∫ (∇⋅[N
A
]T )T [v ](∇⋅[N A ]T )d (vol ),
(9)
Vol
dvh
T
T T
T
T
2 ({B } (∇×[N A ] )) ({B } (∇×[N A ] )) d (vol ),
d (B )
vol
(10)
[K VA ]= − ∫ (∇[N ]T )T [σ ]{v}×∇×[N A ]T d (vol ),
(11)
[K N ]= 2 ∫
⎡[C AA ]
[C Av ] ⎤
⎥,
[C ]= ⎢ Av T
[C vv ] ⎥
⎢⎣[C ]
⎦
AA
[C ]= ∫ [N A ][σ ] [N A ]T d (vol ),
(12)
(13)
vol
[C Av ]= ∫ [N A ][σ ] ∇{N }T d (vol ),
(14)
vol
[C vv ]= ∫ (∇{N }T )T [σ ]∇{N }T d (vol ),
(15)
vol
-Applied loads:
⎧{J A }⎫
⎪
⎪
{J i } = ⎨ t ⎬ ,
{I } ⎪
⎭
⎩⎪
{J A } = {J S } +{J pm },
{J } = ∫ {J s }[N A ] d (vol ),
S
T
(16)
(17)
(18)
vol
{J pm } = ∫ (∇×[N A ]T )T {Hc} d (vol ),
(19)
vol
{I t } = ∫ {J t }[N A ]T d (vol ),
(20)
vol
dvh
, [σ ] , and {v} are
d (B )2
matrix of element shape function for {A} , vector of element shape function for {V } ,
where [N A ] , {N } , {J s } , {J t } , vol , {H c } , vo , [v ] ,
source current density vector, total current density vector, volume of the element, coercive
force vector, reluctivity of free space, partially orthotropic reluctivity matrix, derivative of
reluctivity with respect to the magnitude of magnetic field squared, orthotropic
conductivity, and velocity vector respectively.
The magnetic field density is the first derived result. It is defined as the curl of the
magnetic vector potential. This evaluation is performed at the integration points using the
element shape function:
{B} =∇×[N A ]T {Ae },
(21)
where, {B} , and {Ae } are magnetic field density, and nodal magnetic vector potential
respectively.
3 CIT DESIGN PRINCIPLE
According to Faraday's law,
dB
(22)
dt
where, N is the number of turns and Ai is the effective cross sectional area of the magnetic
core. In the case of laminated and tap-wound cores, this is less than the physical area AC
due to interlamination space and insulation. The form factor K is defined as the ratio of the
rms value of the applied voltage waveform to V
V
(23)
K = rms .
V
Combining (22) and (23) yields
Vrms = KfNBm Ai
(24)
eind = − NAi
where, f is the frequency of eind and T is the period of eind . Equation (24) is the classic
equation for voltage in a transformer where, K for sinusoidal waveform is K=4.44.
Parameters of case study CIT transformer as below:
Secondary
voltage
of
the
CIT,
V2 =5 (V )
and
output
current
is,
I 2 =25 (kA) → Q2 =125 (kVA) .
The volts per turn ratio, V/T for a conventional transformer is determined by using (25),
where R is 0.7 for single phase and 0.55 for three phase. In (40), P is the rated power in
kVA, %IZ means the per unit impedance in percentage, f is the frequency and Q is unity for
single phase.
1
⎧
⎫2
0.376
P
⎪ ⎛ f ⎞
V = R × ⎨⎪
×⎜ ⎟
× Q (v )
⎬
1
T
⎪ (% IZ / 5) 2 ⎪ ⎝ 60 ⎠
⎩
⎭
(25)
If voltage per turn to be as:
Et =5 (V / T )
Maximum flux density obtains as:
Et
5
φm =
=
= 22.5( mwb)
4.44 f 4.44 × 50
For flux density Bmax = 1.3 (T ) we have [17]:
22.5 × 103
= 0.0173(m 2 ), K f = 0.9 →
1.3
0.0173
= 192.2 (cm )
A =
c
0.9
A = 13.84 × 13.84 (cm )
c
Since the magnetic flux flow through the outer limbs is half of that of the central limb, so
that their cross sectional area is also half of that of the central leg, that is:
b = 2a → a = 6.92(cm)
Ai =
2
2
If the current densities of the primary and secondary windings are δ1 = 2.25 × 106 ( A
δ 2 = 100 × 106 ( A
m2
m2
)
) respectively, then the window area, Aw obtained as (26):
Q = 4.44 × f × Bm × K w ×
125 × 103 = 4.44 × 50 × 1.3 × 0.328 ×
δ1δ 2
× Ai × Aw
δ1 + δ 2
(26)
100 × 2.25
2
× 106 × 0.0173 × Aw Aw = 345 (cm )
100 + 2.25
The cross section area of primary winding is:
I
2
AP = P = 138.8mm ,
δ1
and the cross section area of secondary winding is:
I
2
AS = S = 250 mm .
δ2
The main design parameters of the CIT are shown in Table 1.
Ratings
Core
Windings
Table 1. Design parameters of the CIT
125kVA
Capacity
400/5V
Voltage
312.5/25000A
Current
50Hz
Frequency
Single-phase
Phase
Materials
30M5
Nominal Bmax
1.6T
Effective Cross Sectional Area
14100mm2
Copper
Material
3.2mm×22mm
Cross Section
2.25A/mm2
Jc
5.0V
V/N
4 THE LEAKAGE FLUX AND MECHANICAL FORCES ON THE CIT WINDINGS
Generally, the flow of flux density in a medium is three dimensions, r, z and ϕ . The flux
density resultant BR (t) is given by:
2 (t)
BR (t) = Br2 (t) + B 2z (t) + Bϕ
(27)
Where: r, z and ϕ denote the directions of the flux. The A vector potential is needed to
calculate magnetic flux and Because of symmetric of transformer [9],
So,
∂A
=0
∂ϕ
(28)
Bϕ = 0
(29)
In this study we only use 2-D simulation and the ϕ dimension is assumed to be constant. A
computer aided method that is based on magnetic vector potential and finite element
equations [6] was used to obtain the leakage magnetic field distribution around the coil
winding of 25kA CIT.
∂A
∂z
A ∂A
Bz = +
r
∂r
Br = −
(30)
(31)
The section in which the magnetic field is to be calculated is divided up into small
triangular elements. Within one element, the vector potential A is considered to vary
linearly and α is constant coefficient.
A = α1 + α2r + α3 z
(32)
This is equivalent to a constant radial component of flux density within the element.
Br = −
∂A
= −α3
∂z
(33)
However, the axial component varies with the location.
Bz =
A ∂A A
+
= + α2
r ∂r
r
(34)
But in the numerical solution, the triangles are made small enough, so that the axial
component Bz at the centroid can be taken as representative for the triangle as a whole.
Bz =
Ac
+ α2
rc
(35)
Because of the linear variation within the element, the vector potential Ac at the centroid is
the average of the values at the three vertices.
Ac =
(
1
A + Am + An
3 l
)
(36)
Total stray flux density:
BT = B 2r + B 2z
(37)
From (30), (32) and (34):
⎛A ⎞
2α
BT = α 22 + α 32 + 2 A c + ⎜ c ⎟
rc
⎝ rc ⎠
2
(38)
Equation (32) written for each of the three vertices produces three equations, which are
solved with respect to the two coefficients α 2 and α 3 ( α1 is not needed).
α2 =
α3 =
1
[ A l (z m − z n ) + A m (z n − z l ) + A n (z l − z m )]
D
1
[ A (rn − rm ) + A m (rl − rn ) + A n (rm − rl )]
D l
(39)
(40)
where
D = (rm z n − rn z m ) + (rn z l − rl z n ) + (rl z m − rm z l )
(41)
If l, m and n are assigned in a counter clockwise direction, D is positive, and equal to twice
the area of the triangle.
3.
FINITE ELEMENT METHOD
The FEM of the CIT is performed in order to verify the effectiveness of the theoretical
equations used in the design process and validate the design parameters. The FEM in the
CIT involves two stages, magnetic flux distribution within the CIT core, and mechanical
forces being imposed by the stray flux upon the windings of the CIT.
4.1 Simulation Results of the Flux Distribution within the Core and Windings
In this section, the simulation results obtained using Ansys software are discussed. The
color plotting of the magnetic flux density within the core and windings obtained as a result
of simulation process are explained. The color plot of the magnetic flux density obtained as
a result of 2D magneto static simulation is shown in Fig.3.
The flux density at any point of the CIT is found from this color plot. It is seen from the
plot pattern that the main flux paths are not confined within magnetic core, but they detour
through the windings some of which cross the windows and the rest return back to the core.
These flux leakages will impose mechanical forces upon the windings which could have
harm effects on them.
Fig.3. Axial and Radial stray field in primary winding
Fig.4 shows the stray flux density plots passing through the windings. It shows that the
most of the stray fields in top and bottom of the primary and secondary windings are
diverted opposite to each other, into the left and right yoke of the core. This is due to the
fact that, the current flowing into primary and secondary windings is 180o out of phase
with each other. However, it can also be seen that the rest of these fields are crossing the
windows and join the fluxes in the outer limbs of the core.
Fig. 4. Stray field vector in top of primary and secondary Windings
Fig.5 and Fig.6 show the variations along vertical and horizontal axes, of radial stray field
along the primary winding approaching the gap between the two windings. The radial
components of the magnetic field alternate between the maximum positive (top of the
windings) and maximum negative (bottom of the windings), while in the midway of top and
bottom, the stay fields are reduced to nearly to zero. The maximum values of the stray
fields alternated at the top and bottom of the winding are about ±9mT.
Fig.6 shows the axial stray field variations along the gap between two windings. Unlike
the radial fields, the axial stray fields are minimal (5.2mT) at both vertical ends of
windings, while they are increased (9.7mT) around their centre. Figures. 5 and 6 show that
at the position where the radial field component attains maximum, but the axial component
is minimal, and vice versa.
4.
MECHANICAL FORCES
In case of high current injection transformers, the main coupling between the mechanical
and magnetic field results from the interaction between the magnetic stray field of currentcarrying coil conductors and the electric current in conductive parts of the vibration
winding structure
Generally Electromagnetic forces are calculated as flux density times current times length.
F = BLI
(42)
As there are stray radial and axial field components within the windings, so the
corresponding forces are divided into two radial and axial components. The radial force due
to axial flux tends to separate the two windings, whereas the axial force due to radial flux
tends to compress the windings. The total radial forces give rise to tensile stresses in the
outer winding, but compressive and bending stresses in the inner winding.
Fig.5. Radial stray field variation along the primary winding
Fig.6. Axial stray field variation along the primary winding
However, bending stresses can be reduced by increasing the number of axial spacers
between the core leg and the winding. The maximum axial force per unit volume occurs at
the winding end. It tends to bend individual disks axially between radial spacers. Axial
forces build up down through the winding.
The mechanical force due to 25kA in the CIT is presented in this section. As the result
shows the axial force imposed upon the coils is negligible because the coils are
symmetrical, so this proves the validity of the model. Since the axial stray field almost
covers the whole area of the windings the radial force component also is also applied
throughout the windings.
Fig.7 shows the electromagnetic force vectors on windings area. It can be seen that the
whole area of the windings are under the radial forces, while axial forces pressurize only
the top and bottom areas of the windings. These phenomena are shown in Figures 8 and 9,
where in Fig.8, the axial forces are only applied to two ends of windings and it is negligible
in the other area. While Fig.9 shows that the radial forces are distributed throughout the
winding area, whereas they are at two ends of the windings become weaker. At the position
where the radial field component attains maximum, the axial component is minimal, and
vice versa.
Fig.7. Electromagnetic force vectors on windings area
The Table.2 shows the axial and radial components values of primary and secondary
windings.
Table.2: Electromagnetic forces values on windings
Winding
Radial component
Axial component
Primary
33N
0.0003N
Secondary
1624N
-5.8N
The calculated values of the pressure from the forces and winding areas are shown in
Table.3.
Table.3: Pressure value on copper conductor
Winding
Stress
Primary
0.075MPa
Secondary
0.0232MPa
It can be seen from the Table 3 that the stress value is smaller than allowable stress so the
windings are impervious versus the radial forces. Skin effect and proximity effect may be
significant as far as the force distribution is concerned but the have no significant effect on
the total force [11].
Figure.8. Axial force distribution in windings area
Fig. 9. Radial force distribution in windings area
5 CONCLUSIONS
This paper presented a 25kA CIT design model based on finite elements accounting for
dominant effects of "magnetic field distributions" on the mechanical stresses. The turns of
windings are effective parameter on electromagnetic forces. It was found that in the CIT
despite 25kA current in secondary winding the mechanical force is not harmful to the
winding, since the value of the ampere/turn is not noticeable. For a good design, the
number of high current winding turn and its limitation is a key factor. Because, applied
stresses on the winding cross section are proportional to forces, so in this study the forces
were low and were not harmful to the windings of the 25kA CIT.
The present approach has been described with reference to exemplary embodiments.
However, it will be readily apparent to those skilled in the art that it is possible to embody
the phenomena in specific electromagnetic systems other than that described above without
departing from the spirit of the concept.
ACKNOWLEDGMENTS
The authors would like to thank Prof. Mohsseni from University of Tehran for his critical
and extensive comments led to a transparent and a useful representation of this work. The
financial support of the Tehran Regional Electric Company; Energy Ministry of Iran is
gratefully acknowledged. The authors would also like to express their sincere gratitude to
the editorial board and the referees for their constructive comments which enhanced the
quality of the paper.
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