[James_H._Harlow]_Electric_Power_Transformer_Engin(BookSee.org)

L ijo = N i N j ijo (3.10.31)
1.4
where ijo is the average flux cutting each conductor in section i due to the current I j , and where N is the
number of turns in each section, then the inductance (L ij , H) as a function of frequency is:
1.3
L
ij
Lijo
b
tanh( 1
j)
b
( 1
j)
a
a
(3.10.32)
The impedance (Z prox,ij , ) is obtained by multiplying the inductance by the complex variable s. Using
the same notation as in Equation 3.10.29, the impedance of the conductor due to the proximity effect
is given as
H
PER UNIT CAPACITANCE
1.2
1.1
1
BEFORE AGING
AFTER AGING
Z s L ijo
prox
tanh
(3.10.33)
ij
3.10.7.2 Core Losses
The effect of eddy currents in the core have been represented in various works [26,48,49] by the welltested
formula:
where
(3.10.34)
x= d j
(3.10.35)
2
and where
l = length of the core limb (axial direction), m
d = thickness of the lamination, m
= permeability of the material, H/m
N = number of turns in the coil
A = total cross-sectional area of all laminations
= frequency, rad/sec
This formula represents the equivalent impedance of a coil wound around a laminated iron core limb.
The expression was derived [49] by solving Maxwell’s equations assuming the electromagnetic field
distribution is identical in all laminations and an axial component of the magnetic flux.
The total hysteresis loss in core volume, V, in which the flux density is everywhere uniform and varying
cyclically at a frequency of , can be expressed as:
where
N A
Z 4 2
x x
ld
2 tanh
P h = total hysteresis loss in core
= constant, a function of material
V = core volume
= flux density
n = exponent, dependent upon material, 1.6 to 2.0
= frequency, rad/sec
P h = 2 V n max (3.10.36)
PER UNIT CONDUCTANCE, LOG(G)
0.9
0.8
10 2 10 3 10 4 10 5 10 6 10 7
FREQUENCY, LOG(F)
10 4
10 3
10 2
10 1
10 0
AFTER AGING
10 2 10 3 10 4 10 5 10 6 10 7
FREQUENCY, LOG(F)
FIGURE 3.10.7 Top) Oil-soaked paper capacitance as a function of frequency; bottom) Oil-soaked paper conductance
as a function of frequency.
3.10.7.3 Dielectric Losses
The capacitive structure of a transformer has associated with it parallel losses. At low frequency, the effect
of capacitance on the internal voltage distribution can be ignored. As such, the effect of the losses in the
dielectric structure can be ignored. However, at higher frequencies the losses in the dielectric system can
have a significant effect on the transient response. Batruni et al. [50] explore the effect of dielectric losses
on the impedance-vs.-frequency characteristic of the materials in power transformers. These losses are
frequency dependent and are shown in Figure 3.10.7.
3.10.8 Winding Construction Strategies
BEFORE AGING
3.10.8.1 Design
The successful design of a commercial transformer requires the selection of a simple structure so that
the core and coils are easy to manufacture. At the same time, the structure should be as compact as
possible to reduce required materials, shipping concerns, and footprint. The form of construction should
© 2004 by CRC Press LLC
© 2004 by CRC Press LLC