376
-vi-
v2
vN
Transactions
on
Power Delivery,
Vol.
7
No.1,
January
1992
EFFICIENT CALCULATION OF ELEMENTARY PARAMETERS
OF
TRANSFORMERS
-Lll LIZ
...
Lw- -i1-
Lz1 LZZ
LW
i2
:
dt
:
-
(1)
.
-.
.
.. ..
LN1 LNZ
'**
LNN iN
Francisco de Leon Adam Semlyen
Department
of
Electrical Engineering
University
of
Toronto
Toronto, Ontario, Canada,
M5S
1A4
Abstract.
-
Very efficient procedures for computing elementary parame-
ters (turn leakage inductances and capacitances) in a transformer are
presented. The
turns
are
used
as
a calculation base to permit modeling at
very high frequencies. Turn-to-turn
(or
loop) leakage inductances are
obtained by an image method. The charge simulation method is used for
finding the capacitances between
tums and from
turns
to
ground. The
new methods are very efficient compared with the use of the technique of
finite elements and
atp.
also
remarkably accurate. Thus, the
short
circuit
(or
test) leakage inductance can be obtained from turn-to-turn informa-
tion. Examples of calculated parameters
are
given for illustration. For
validation, the results are compared with the Parameters obtained using
finite elements and tests. The elementary parameters can be used
to
create reduced order computational models for the calculation of tran-
sient phenomena.
Keywords:
Electromagnetic transients, Transformer modeling, Leakage
inductance of transformers, Capacitance of transformers.
INTRODUCTION
For the study of electromagnetic transients in power systems, com-
ponent models which
are
valid for a wide fresuency range
are
needed.
Synchronous machines and transmission lines have adequate models
that
are
almost universally accepted. However,
no
power transformer
model, appropriate for a wide range of frequencies. is yet available. One
unsolved related problem is the accurate and efficient calculation of the
model parameters (iductances and capacitances, resistances
and
conduc-
tances). The present paper intends
to
contribute in
this
direction.
The purpose of
this
paper is the calculation of the parameters that
are
necessary for the construction of
a
transient model for transformers.
It presents methodologies for computing the leakage inductances and
capacitances of transformers. The approach is based on the representa-
tion of the winding by its individual
turns,
in contrast to existing
methods, used by transformer designers, which take a global geometrical
approach for windings
or
sections. The new method is simple and gen-
eral, and particularly appropriate for the resolution needed for the calcu-
lation of transients. The complete model, which includes the windings
as
well
as
the iron core magnetization, will be presented in a subsequent
paper.
We compute the parameters on a turn-to-turn basis. The resulting
model is thus adequate for high frequency transients, but it can be
reduced
to
lower order for studies of slower transients. We assume
axisymmetricsll geometry
and
infinite iron core permeability.
In
this
way, we separate the physical phenomena occurring inside the core from
the phenomena in the window of the transformer (air, insulation and con-
ductors).
A
widely used procedure for estimating parameters is the technique
of finite elements. For a high frequency model,
this
would require
lengthy computations, mainly due to postprocessing, and tedious manual
work
to
enter the geometric data. We propose alternative methods for
the computation of the parameters.
91
K;
002-6
P1dP.D
A
paper reconnended and approved
by the
IEEE
Transformers Committee
of
the
IEEE
Power Engineering Society
for
presentation at the
IXE/PES
1991 dinter ;leetin,?,
!Jew
York,
:;e# York,
February
3-7,
1991
.
Xanuscript submitted
January
25,
1990:
made available
for
printing
i.ovcrll:ier
6,
1990.
It has been recognized for a long time that
the
use of self and
mutual inductances for the calculation of low frequency transients may
lead to computational difficulties. The main reason is:
For computing transients using
an
approach based on self and mutual
inductances we will have to solve a very ill-conditioned set of equations.
The equations
are
ill-conditioned because the coils (or
turns)
are very
tightly coupled due to the presence of the (unsaturated
or
lightly
saturated) iron-core and,
as
a
consequence, the elements in the induc-
tance matrix
L
are
almost identical.
To
surmount
this
difficulty, we deal with the leakage inductances
between
tums.
One could, theoretically, compute the leakage inductances between
turns
by subtracting mutual inductances from self inductances, but taking
the difference of two nearly equal numbers leads to results of low accu-
racy. Recently [1],[2], there have been significant advances in the
analytical calculation of self and mutual impedances in transformers.
Although the analytical expressions obtained require numerical evalua-
tion, the self and mutual impedances are calculated with accuracy. How-
ever, quantities related to leakage impedances (differences between
self
and mutual impedances)
are
not calculated with the same degree of accu-
racy, in accordance with the remark made above.
To
get accurate leak-
age related quantities, the authors of
[
11 and [2] rely on tests to compute
some adjustment parameters. There is, thus, a need
to
calculate leakage
inductances accurately and efficiently using a direct approach. To
this
end, and with the aim of avoiding the use of finite elements for comput-
ing leakage inductances, a very efficient and sufficiently accurate image
method is proposed in
this
paper.
For the first time in the case of transformers. to our knowledge, the
charge simulation approach for computing the capacitances is used. The
purpose of using
this
method is to give a
less
time-consuming alternative
to the use of finite elements.
This
technique has been employed before
for air-cored reactors
[31
and other electrode configurations [41,[51. The
method is very efficient compared with the method of finite elements
when we compute elementary capacitances, and its accuracy is impres-
sive.
PARAMETER CALCULATION
317
In an actual transformer, the iron permeability is not infinity nor is the
magnetizing current
zero,
but the
turns
are
so
tightly coupled that the
matrix
L
is ill-conditioned (since its elements are almost equal). There-
fore,
one
is compelled
to
work with leakage inductances in order
to
have
a well behaved model.
To
derive a model based on leakage inductances,
a procedure similar to using a slack node in power system studies is fol-
lowed. In
this
case, we take a turn as a reference
(turn
N)
and use it
to
assure that condition
(2)
is met. Then
N-1
-iN-1
(24
iN
=
-
Cik=-il-iz-
k=l
Using equation (2a) to incorporate the last term into the matrix, we
obtain
The elements of the above loop inductance matrix
L',
as
functions of the
elements of the matrix
L,
are:
(4)
We note that the elements of
L'
can be obtained from leakage inductance
tests.
This
is what permits
us
to use an alternative method of images
to
calculate the elements of the loop inductance matrix
L'.
Indeed, if
i
=
j,
then equation
(4)
becomes (since
Li,
=
4)
which is the definition of the leakage inductance between
turns
i
and
N.
L'ij
=
Lq
-
LW
-
LN~
+
LNN
L'ii
=
Lii
+
LNN
-
2
LW
Thus
L'ii
=
L,&
j&l
For
i
#
j,
adding
f
'hLii
and
f
".,j
to
(4),
we get
Thus, in terms of leakage inductances, we have
L"
1.1
'
=
'h
[(
L"
+
Lm
-2
LiN
)
+
(Ljj
+Lm-2
LjN
)
L'ij
=
%(
LIcak
W
+
Llcak
fl
-
LIcak
ij
)
The voltage difference
vi
-
VN
between the two
-
(L,+Ljj-2Lij)l
i#j
(6)
turn voltages will be
called loop voltage.
L' is called the loop inductance matrix because it is
related to the loops formed by two
turns
connected in opposition. In this
connection, the magnetic flux due
to
the
current flowing through one of
the
turns
and the flux produced by the current in the other
turn
are
in
opposite direction. For example, loop
i
is the path formed by turns
i
and
N
when they
are
connected in opposition (see Figure 1). The elements of
L'
can be obtained in two ways:
a) Following standard tests
(this
is an indirect approach): we perform
N(N-1)/2
short
circuit tests for
all
possible pairs of
turns
(i.e.
loops) and in
this
way obtain the leakage fluxes (inductances).
Then we use equations
(5)
and
(6)
to get the elements of the matrix.
This
is the approach that we will follow, since we can obtain the
loop leakage inductances efficiently and accurately (see next sub-
section).
Using non-standard tests (this a direct method): based on equation
(3)
we perform N-1 short circuit tests, using
turn
N
as a reference
and measuring one leakage flux (inductance) for the self terms and
N-2 linkage fluxes (inductances) for the mutual terms.
b)
I
"N
I
-
Figure
1.
Loop
i
formed by
turns
i
and
N
in opposition.
Image Conductors
Inspired by the successful
use
of complex depths (images) in the
calculation of parameters for transmission lies
[6]
(which permits
to
avoid the relatively cumbersome procedure based on Carson's formulae)
we have substituted the finite elements method in the calculation of leak-
age inductances for transformers, by a method based on images (see
Figure
2).
As
the magnetic permeability of the iron is very
high
(infinity
in
our
case) compared with the permeability of the air, the magnetic field
is everywhere perpendicular to the iron. Thus the surface of the core is
an equipotential for the magnetic scalar potential. In
this
way,
the
iron-
core surface could be considered
as
a cylindrical mirror for the magnetic
field.
To
compute the N(N-1)/2 leakage inductances that we need
to
evaluate the elements
of
the loop inductance matrix
L'
(equations
(5)
and
(6)).
we will need
to
know the magnetic vector potential that
has
only a
tangential component,
Al.
The calculation
of
A$
requires the evaluation
of elliptic integrals. These functions
are
readily available in many
mathematical libraries (e.g.
IMSL)
or they can
be
programmed with a
fraction of the effort needed for preparing a data file for finite elements.
The magnetic vector potential for a circular filament with unit
current is
[7]
where
k=,/-
2
i
(a+r)'
(8)
E
(k)
=
elliptic integral of second kind and argument
k
K
(k)
=
elliptic integral of first kind and argument
k
r
=
radius of observation filament
a
=
radius of excited filament
z
=
vertical separation between the two filaments
Figure
2.
Image method for the loops formed
by turns
j
and
k
around a core of radius
R.
378
We compute the linkage flux (self and mutual) for the system
formed by the
turns
(in air) and their images (without core) by calculat-
ing the vector potential and then integrating around the circular contour
(Neumann's formula);
see
Figure 2. Then, the self-inductance is
where
LpZe
=Lint
+
21t
(~i-b;
)
[A&~i-bi.O)
+APEC(ai-bi,O)
]
(9)
Lh
=
internal inductance
=
k/8x
ai
=
radius of turn
i
bi
=
conductor radius of
turn
i
A
9(ai-bi,0)
=
vector potential due
to
the current in the actual turn
APge(ai-bi,O)
=
vector potential due to the current in the image
conductor
The mutual inductance is
L?
=(2mi)
[A~(u~,z)+AP~~(u,,z)I
(loa)
(
1 Ob)
=(2~j)
[
A+(u~,z)+AP~~(u~,z)
]
where
z=z.-zz.
11
To obtain the leakage inductances we use
L,,,~
ij
=
~pge
+
~kwe
-
2
~Lw8'
(1
1)
Note that, in
this
case, it
is
possible
to
obtain the leakage inductances by
subtracting the mutual inductances from the self-inductances, since the
former
are
at least one order of magnitude smaller than the latter. This
is
not the case when we have the mutual and self-inductances with the iron
core and actual geometry. The self- and mutual inductances from the
image method given in equations
(9)
and (10) are not related
U)
the ones
in the transformer (equation (1)). However, their differences (in the leak-
age inductance sense:
Lii
+
Ljj
-
2
Li,)
are
very close,
as
demonstrated
later.
We have two parameters for adjusting the leakage inductance
value:
a) the radius of the image conductor, and
b) the current flowing through it.
By analogy between the iron leg surface
and
a curved mirror, the
radius of the image conductor can be obtained using the location of an
optical image
(see
Figure 2)
1
21
Rr
ri
=
-
---
To adjust the current flowing in the image conductor, we have
computed the leakage inductance of various conductor configurations.
As
a conclusion, when we are interested in the terminal behavior of the
transformer, the best value for
this
current is estimated to be
2.5
times
the current flowing in the actual conductor. The current in the image
conductors should
flow
in the same direction
as
the current in the
turns.
A
current of 1.0 gives
good
results when the conductors
are
not close to
the leg or to the yokes, or when the conductors
are
close to each other.
For extreme cases, the best current
in
the image conductors is
4.0.
How-
ever, using a current of 2.5 for
all
the image conductors, we get max-
imum errors off 17.7% for a standard design. Moreover, when we com-
pute the test (or total) leakage inductance, the errors compensate and the
resulting leakage inductance is very accurate. Two geometrical mnge-
ments were used for comparison
(see
Figure
3):
A)
leg only (no yoke: open geometry), and
B)
window (closed geometry).
All
comparisons were made against a 2-D finite element program
(axisymmetric). We did not have a
3-D
field calculation package for
comparison with the real geometry. However, the two extreme cases
(leg only and window) gave close limits, thus confirming the accuracy of
the method.
An
example of the calculation can be found in the results
section.
Turn-to-Turn Capacitances
As
a more efficient alternative to the method of finite elements, the
capacitances were calculated by a charge simulation approach. The basic
principle behind it is to assume that a potential difference
(v
=
1,
for con-
venience) is applied between one
turn and
all
the others connected
to
each other and to the
core.
Since
all
metallic surfaces are equipotentials.
we can evaluate a number of simulated charges that
will
produce
the
boundary conditions to be met. The charges are on circular filaments and
are
assumed to be placed inside the metal (see Figure
4).
We have to use
a number of rings equal to
the
total number
of
charges,
n,
to serve
as
reference points with specified potential. We can apply a potential
difference
v
(=1) between the core and a conductor, but the potential
YO
of the core (potential with respect to infinity) is not yet known;
see
Fig-
ure
4.
Thus, for the boundary points on the
core
and on the conductors
with no excitation, we have
vi
=
VO;
for the boundary points on the
excited conductor, we have
vi
=
YO
+
1.
In
this
way, for each boundary
point
i
we can write an equation as follows:
n
0
for non-excited conductors
(13)
J'1
Since the total charge has to be zero, we have
n
C
qj=O
]=1
Arranging the equation in matrix form for all boundaries, we have a sys-
tem of n+l equations
-:I
[e,]
=
I]
I
LJ.
I
I
32.4
I
21
Not
to
scale
-
Dimensions in
cm
I
I
Figure
3.
Example: 12-turn transformer.
core
surface
\"ia
Nmj
v
+1
Boundarypoints
rn
Simulated charges
x
Testingpoints
Figure
4.
Charge simulation method.
319
where
and,
PI,,
=
potential coefficients (see [7])
k,,,
=
argument defined in equation
(8)
r,
=
radius of observation ring i
a,
=
radius of charged ring
j
z,,,
=
vertical separation between the two rings i,
j
n
=
total number of charges
N
=
number
of
turns
K(k,,)
=
elliptic integral of first kind and argument
k
The elements of the excitation vector
v
in equation (15) are zero,
according to equation (13). with the exception of the corresponding
entries for the excited conductor that
are
1. Note that we can use a dif-
ferent number of charges in each conductor. From the definition of self-
and mutual capacitances we can see that:
a) The total charge computed inside each non-excited conductor is the
capacitance of that conductor with respect to the excited one
(mutual capacitance).
The charge inside the excited conductor is the self-capacitance. It is
the sum of all mutual capacitances and the capacitance from
this
conductor to ground (core).
In
this
way, a column of the turn capacitance matrix C'w" is computed
from each solution of the equation.
A new set of points (filament rings), different from those used
to
compute the charges, are taken
to
test the uniformity of the potential at
the surface of the conductors (see Figure 4). There is no systematic tech-
nique to find a suitable charge-boundary arrangement. However, even
with little prior experience, one can get highly accurate results by trial
and error in a reasonable amount of time.
Care
should be taken
to
avoid
placing a charge in the location of a boundary or test point, since doing
so
would lead
to
singularity of the elliptic integral
K(k)
as
k
+
1.
Also,
two
charges should not occupy the same point in order to prevent the
potential coefficients matrix from becoming singular.
When we have several insulating materials (e.g. paper and oil or
air), the dielectric constant
E
in equation (16) must be the equivalent
dielectric constant for the arrangement. Its proper choice has to be based
on tests and experience.
b)
TEST LEAKAGE INDUCTANCE
Previously, we have calculated the leakage inductances for simple
loops formed by two turns. In
this
section we will show how the total or
test leakage inductance for a two winding transformer is calculated from
the simple loops information.
This test leakage inductance corresponds
to
the leakage inductance that we would measure in a bucking test (when
the ampere-turns of the two windings are equal and of opposite direc-
tion).
This
test can
be
easily performed only when the two windings
being tested have an equal number of turns. It should be mentioned that
a short circuit test gives almost identical results since the magnetizing
current in
this
test is very small.
Note that the leakage inductance is usually defined for a pair of
windings (two windings at a time). For a transformer with more than
two
windings, a matrix similar to our loop inductance matrix
L'
has
to
be
constructed
as
shown inreferences
[8],
[9], [lo], [ll], [12].
The voltage-current equation for the simple loops of equation (3)
is, in matrix form
d.
Vloop
=
L'
-$
'loop
Recall that the above equation has dimension N-1.
Np and Ns. Thus, equation (2) becomes
Each winding consists of a number of turns connected in series, say
(18)
(
18a)
Np ip +Ns is
=O
Np ip
=
-
Ns is
Equation (18) expresses the bucking test condition
Based on
this
relation, we can introduce a distribution vector a, of
dimension N, showing the relative current magnitudes in the turns during
the bucking test. The
first
Np elements of a
are
normalized to 1 while
the remaining Ns
are
equal to:
a
=
-
Np
I
Ns. Thus, we have
(19)
aT
=
[I, 1, . . .
,I
,a,a,
*
.
.
,a
I
We note that the N-1 loop currents coincide with the first N-1 turn
currents. Therefore, the corresponding loop-test distribution vector
a' is
of order N-1, and its elements equal
to
the first N-1 elements of
a.
We obtain the test leakage inductance from equation (17) by apply-
ing the power-invariant transformation
i/mp
=
a'
imt
(20)
v,,,
=
alT
This
yields
where
L,,,
=a'
L'
a'
Developing
this
product we obtain
(22)
We note that equation (24) is similar
to
the equation used for computing
leakage inductances
(Lii
+Lj,
-
2
Lii).
The test voltage, obtained from
equation (21), is
Two concrete examples of
this
calculation are presented in Appendix 1.
In the next section, the results are compared with those obtained with the
technique of finite elements and by tests.
RESULTS
Several turn configurations were used to test the alternative
methods described here for the computation of parameters.
As
an exam-
ple consider the transformer data given in Figure 3, where we show the
conductors in the extreme positions (1,2,
..., 12).
All
other conductors
have inductances and capacitances in between those of the conductors
shown in
this
figure.
Leakage Inductances
In
the next table 11 leakage inductances (in henry) out of the total
of
66
are presented and compared against the results from those obtained
by the method of finite elements. The following table was obtained
using a current of 2.5 in all image conductors.
FNTE
ELEMENTS
LEAKAGE
11
IMAGE
I
I
INDUCTANCE
Y
1.2)
Y
1.3)
Y
1.5)
U
1.6)
Y
1.7)
Y
1.8)
U
1.4)
Y
1.9)
U
1.10)
MFIHOD
0.83390606
033448c-05
030403~05
03449145
03161oGo5
0.18691e-05
01107345
03232145
032-5
0.74017~46 -12.7
0308pe-05
03818Ec-05
037053~45 14.7
0.1587&-05 -17.7
0.18WcXIS -17.0
0328E9r.M
0331W-05 0.6
m
ZEna
0.74052e-06 -126
0.4113k-05 18.7
0.40111c-05 241
0.&787c-05 52.7
O.16030e-05 -16.6
0.1823Oe-05 -15.6
0.428Mcas 23.1
0.6ns9e-05 49.1
0.42389c-o~
zia
1.12) 034545e-05 039059~45 11.6 0.676Sk-05
48.9
We
are
presenting one of the worst cases. The other
55
leakage
inductances have errors in the same range or smaller. We adjusted the
leakage inductances calculated with the image method to be closer to the
geometry of case A (leg only) than
to
the geometry of case B (closed
window) because the actual geometry of a transformer in the region at
the top of the windings shows close to
90%
air and 10% iron for the
leakage flux path (see Figure
5),
as
the yokes cover only a fraction
of
the
top of the windings. Some of the values obtained with the image method
lie in the region between the two geometries but, in general, the values
are
slightly smaller than those of geometry A.
a)
We can adjust the image currents
to
treat independently:
the turns near the leg or yoke, which requires large image currents
=
4.0)
380
LEAKAGE
[NDUCTANCE
L(
132)
L(
133)
L( 1.4)
L( 1.5)
L(
1.6)
L(
197)
L(
1.8)
L(
1.9)
L(
1,lO)
L(
1,11)
L(
1,12)
LEG
YOKE
WGE
CURRENT %Error
1.0
2.5
2.5 -4.8
2.5 1.4
3.0 -0.3
3.5 -1.7
0.0
0.4
0.0
1.4
2.5
1.7
2.5 0.6
3.5 -0.1
3.5 -0.7
I=
1.0
WI”GS
Figure
5.
Region at the top of the windings.
b) the
turns
close to each other, which requires
small
image cumnts
c) the
turns
far from each other and from the iron, which requires no
image current
(i-,,
=
0.0)
The results obtained by adjusting the image cumnts
as
described above
are shown
in
the next table.
(ikge
=
1.0)
In
this
way we have obtained very accurate turn-to-turn leakage
inductances. This adjustment requires some extra computational effort,
mainly for evaluating the distances between conductors and distances
from conductors to the leg and yokes.
This
process has to be followed if
we want accurate local responses (e.g. internal voltage distribution dur-
ing a transient). If we are interested only in the terminal behavior we
believe, however, that
this
sophistication is not necessary. As we will
show next, the errors innodud by using a common rule, of a multiplier
equal to
2.5,
for the current in the image conductors, compensate each
other when we calculate the test leakage inductance.
We computed the test leakage inductance using the arrangement
shown in Figure 6. The step by step process is described in Appendix 1.
We used for comparison the
two
configurations with finite elements (A
and
B)
and with a familiar approximate equation which assumes that the
field is axial [13]:
The results are shown in the following table (with geometry A as refer-
ence).
-
1.57
Image Method 1.436oOE-5
We performed the bucking test (windings connected in series opposition)
and the short circuit test on a 2 kVA transformer, 110/110 V, with two
windings of 11
8
turns in two layers.
In
Appendix 2 the geometrical data
are presented. The leakage inductance obtained from tests is 4.5
x
IO4
H.
The same result was obtained in both tests (bucking and short circuit).
Using the method described above (and in Apndix
1)
to simulate the
bucking test (equation (24)) we got 4.3
x
10
H.
The
error is less that
5%.
@
i-;
turn
10
cucuit
turn
11
turn
12
e,
Figure 6. Test leakage inductance
for the 12-turn transformer.
As
we can see the errors obtained in the calculation of leakage
inductances for
turns
compensate each other when we lump the
turns
to
form a winding. Note also that there is not much difference between the
geometries (A) and
(B)
using finite elements to calculate the parameters.
This
means that the effect of the yokes in the leakage inductance is very
small. Therefore, we do not introduce large errors when we neglect the
yokes
in
the image method. The assumption that the magnetic field is
axial (equation (26)) leads to a larger error.
Capacitances
For the 12-turn transformer (Figure 3) no more than 30 simulated
charges in the core and 4 charges
per
conductor are needed
to
obtain very
good potential profiles along the surfaces. The maximum emr found
was around 1% of the specified potential in the testing points. The first
column of the capacitance matrix for this example is presented in the
next table. The assumed geometry is leg only, no yoke (case A). The
values with closed window (case B) are 20% to 25% greater than those
shown in the table. But, as mentioned before, the real geometry is closer
to case A than to case
B
(see
Figure
5).
TURN TO
TURN
CAPACITANCES
[F]
CHARGE
TURNS
I
SIMULATION
0.1251667e-09
-0.196562 le- 10
-0.9477953e-11
-0.1966649e-I0
-0.196562le-I0
-0.9477953e-11
-0.5964382e-11
-0.6836441e-11
-0.5550385e-11
-0.5964382e-11
-0.92645501~-11
-0.681533Oe-11
These values were calculated with
E
=
1. For a transformer with dif-
ferent insulating materials we should multiply the values of the capaci-
tance by the equivalent dielectric constant of the insulation.
Another test for
the
capacitances was to use a transformer with a
very tall leg and the conductors were placed (radially) close to the sur-
face and axially far apart from each other.
In
this way, the geometry
resembles a multi-conductor transmission line over a flat (perfect con-
dsctor) plane. Then we compared the capacitances obtained for
this
transformer with the capacitances calculated with equations for a
transmission line. As the height over the ground in the transmission line
formulae, we used the radius of the turn minus the radius of the core of
the transformer
(hi
=
ai
-
R),
and
as
the length of the conductor, we used
the length of the turn
(li
=
2
x
ai).
The errors were found to be less than
1%.