[James_H._Harlow]_Electric_Power_Transformer_Engin(BookSee.org)

where
L = inductance, H
= flux linkage caused by current I, weber turns
I = current producing flux linkages, A
d = first derivative of
dI = first derivative of I
L is referred to as self-inductance if the current producing the flux is the same current being linked.
L is referred to as mutual inductance if the current producing the flux is other than the current being
linked [3]. Grover [34] published an extensive work providing expressions to compute the mutual and
self-inductance in air for a large number of practical conductor and winding shapes.
One of the most difficult phenomena to model is the magnetic flux interaction involving the different
winding sections and the iron core. Historically, this phenomenon has been modeled by dividing the
flux into two components: the common and leakage flux. The common flux dominates when the
transformer behavior is studied under open-circuit conditions, and the leakage flux dominates the
transient response when the winding is shorted or loaded heavily. Developing a transformer model
capable of representing the magnetic-flux behavior for all conditions the transformer will see in factory
test and in service requires the accurate calculation of the mutual and self-inductances.
3.10.5.2 Transformer Inductance Model
Until the introduction of the computer, there was a lack a practical analytical formulas to compute the
mutual and self-inductances of coils with an iron core. Rabins [35] developed an expression to calculate
mutual and self-inductances for a coil on an iron core based on the assumption of a round core leg and
infinite core yokes, both of infinite permeability. Fergestad and Heriksen [36] improved Rabins’s inductance
model in 1974 by assuming an infinite permeable core except for the core leg. In their approach,
a set of state-variable equations was derived from the classic lumped-parameter model of the winding.
White [19,37] derived an expression to calculate the mutual and self-inductances in the presence of
an iron core with finite permeability under the assumption of an infinitely long iron core. White’s
inductance model had the advantage that the open-circuit inductance matrix could be inverted [37].
White derived an expression for the mutual and self-inductance between sections of a transformer
winding by solving a two-dimensional problem in cylindrical coordinates for the magnetic vector potential
assuming a nonconductive and infinitely long open core. He assumed that the leakage inductance of
an open-core configuration is the same as the closed core [19,37]. r r r
Starting from the definition of the r magnetic r vector potential BA
and A 0
and using
Ampere’s law in differential form, Jf H
, White solved the following equation:
2 Arz
r ,
r
oJrz
,
(3.10.12)
The solution broke into two parts: the air-core solution and the change in the solution due to the
insertion of the iron core, as shown in the following equations:
r r
r Aor, z A
r
1r, z , 0 r R
r
c
Arz , {
Aor, z A2
r, z , r Rc
r
A (, r z ) r
0
is the solution when the core is present, and A (, r z ) r
1
and A (, r z ) 2
are the r solutions when
the iron core is added. r Applying Fourier series to Equation 3.10.12, the solution for A (, r z ) 0
was found
first and then A (, r z ) 2
.
Knowing the magnetic vector potential allows the flux linking a filamentary turn at (r,z) to be determined
by recalling (r,z) = Arz (, ) dl. The flux for the filamentary turn is given
r r
by:
(3.10.13)
The flux in air, 0 (r,z), can be obtained from known formulas for filaments in air [49], therefore it is
only necessary to obtain the change in the flux linking the filamentary turn due to the iron core. If the
mutual inductance L ij between two coil sections is going to be calculated, then the average flux linking
section i need to be calculated. This average flux is given by
(3.10.14)
Knowing the average flux, the mutual inductance can be calculated using the following expression
NN
i jave
Lij
I
j
White’s final expression for the mutual inductance between two coil segments is:
where
(3.10.15)
(3.10.16)
(3.10.17)
and where L ijo is the air-core inductance; r = 1/ r is the relative reluctivity; and I 0 (R c ), I 1 (R c ), K 0 (R c ),
and K 1 (R c ) are modified Bessel functions of first and second kind.
3.10.5.3 Inductance Model Validity
The ability of the inductance model to accurately represent the magnetic characteristic of the transformer
can be assessed by the accuracy with which it reproduces the transformer electrical characteristics, e.g.,
the short-circuit and open-circuit inductance and the pseudo-final (turns ratio) voltage distribution. The
short-circuit and open-circuit inductance of a transformer can be determined by several methods, but
the simplest is to obtain the inverse of the sum of all the elements in the inverse nodal inductance matrix,
n . This has been verified in other works [38,39]. The pseudo-final voltage distribution is defined in a
work by Abetti [15]. It is very nearly the turns-ratio distribution and must match whatever voltage
distribution the winding arrangement and number of turns dictates. An example of this is available in
the literature [39].
3.10.6 Capacitance Model
3.10.6.1 Definition of Capacitance
Capacitance is defined as:
r v r
rz , 2rAorz , A2rz , orz , 2rA2rz
,
ave
L L 2N N 1v R
ij ijo i j r o c
0
v v
Ri
Zi
r,
z dzdr
R
i
Zi
v
H R R
i i i
I0 Rc
I1
Rc
F
v 1v R I R K R d
r r c 1 c 0 c
R
1 1
i
2 Hi
F
[ v xK x dx
R R
1( ) ] sin( )
R
H
2
i i i
i
R
1
i
2 H
j
[ v xK x dx
dij
R R
1( ) ] sin( )cos( )
R
H
2
v
v
i i i
j
C
Q V
(3.10.18)
© 2004 by CRC Press LLC
© 2004 by CRC Press LLC