[James_H._Harlow]_Electric_Power_Transformer_Engin(BookSee.org)

nected. This steady-state voltage distribution is very nearly (but not identical) to the turns ratio. For the
simple winding shown in Figure 3.10.1, the distribution is known by inspection and shown in Figure
3.10.3, but for more complex windings it can be determined by finding the voltage distribution of the
inductive network and ignoring the effect of winding capacitances.
3.10.2.4 Transient-Voltage Distribution
Figure 3.10.2 shows the transient response for this simple coil. The transient response is the voltage the
coil experiences when the voltage within the coil is in transition between the initial voltage distribution
and the steady-state distribution. It is the very same idea as pulling a rubber-band away from its stretched
(but steady-state position) and then letting it go and monitoring its movement in space and time. What
is of considerable importance is the magnitude and duration of these transient voltages and the ability
of the transformer’s insulation structure to consistently survive these voltages.
The coil shown in Figure 3.10.1 is a very simple structure. The challenge facing transformer design
engineers since the early 1900s has been to determine what this voltage distribution would be for a
complex winding structure of a commercial transformer design.
3.10.3 Determining Transient Response
3.10.3.1 History
Considerable research has been devoted to determining the transformer’s internal transient-voltage
distribution. These attempts started at the beginning of this century and have continued at a steady pace
for almost 100 years [12–14]. The earliest attempts at using a lumped-parameter network to model the
transient response was in 1915. Until the early 1960s, these efforts were of limited success due to
computational limitations encountered when solving large numbers of coupled stiff differential equations.
During this period the problem was attacked in the time domain using either a standing-wave approach
or the traveling-wave method. The problem was also explored in the frequency domain and the resultant
individual results combined to form the needed response in the time domain. Abetti introduced the idea
of a scale model, or analog, for each new design. Then, in the 1960s, with the introduction of the highspeed
digital computer, major improvements in computational algorithms, detail, accuracy, and speed
were obtained.
In 1956 Waldvogel and Rouxel [16] used an analog computer to calculate the internal voltages in the
transformer by solving a system of linear differential equations with constant coefficients resulting from
a uniform, lossless, and linear lumped-parameter model of the winding, where mutual and self-inductances
were calculated assuming an air core. One year later, McWhirter et al. [17], using a digital and
analog computer, developed a method of determining impulse voltage stresses within the transformer
windings; their model was applicable, to some extent, for nonuniform windings. But it was not until
1958 that Dent et al. [18] recognized the limitations of the analog models and developed a digital
computer model in which any degree of nonuniformity in the windings could be introduced and any
form of applied input voltage applied. During the mid 1970s, efforts at General Electric [19–21] focused
on building a program to compute transients for core-form winding of completely general design. By
the end of the 1970s, an adequate linear lossless model of the transformer was available to the industry.
However, adequate representation of the effects of the nonlinear core and losses was not available.
Additionally, the transformer models used in insulation design studies had little relationship with
lumped-parameter transformer models used for system studies.
Wilcox improved the transformer model by including core losses in a linear-frequency-domain model
where mutual and self-impedances between winding sections were calculated considering a grain-oriented
conductive core with permeability r [22]. In his work, Wilcox modeled the skin effect, the losses
associated with the magnetizing impedance, and a loss mechanism associated with the effect of the flux
radial component on the transformer core during short-circuit conditions. Wilcox applied his modified
modal theory to model the internal voltage response on practical transformers. Vakilian [23] modified
White’s inductance model [19] to include the saturable characteristics of the core and established a system
of ordinary differential equations for the linearized lumped-parameter induction resistive and capacitive
(LRC) network. These researchers then used Gear’s method to solve for the internal voltage response in
time. During the same year, Gutierrez and Degeneff [24,25] presented a transformer-model reduction
technique as an effort to reduce the computational time required by linear and nonlinear detailed
transformer models and to make these models small enough to fit into the electromagnetic transients
program (EMTP). In 1994 de Leon and Semlyen [26] presented a nonlinear three-phase-transformer
model including core and winding losses. This was the first attempt to combine frequency dependency
and nonlinearity in a detailed transformer model. The authors used the principle of duality to extend
their model to three-phase transformers.
3.10.3.2 Lumped-Parameter Model
Transient response is a result of the flow of energy between the distributed electrostatic and electromagnetic
characteristics of the device. For all practical transformer winding structures, this interaction is
quite complex and can only be realistically investigated by constructing a detailed lumped-parameter
model of the winding structure and then carrying out a numerical solution for the transient-voltage
response. The most common approach is to subdivide the winding into a number of segments (or groups
of turns). The method of subdividing the winding can be complex and, if not addressed carefully, can
affect the accuracy of the resultant model. The resultant lumped-parameter model is composed of
inductances, capacitances, and losses. Starting with these inductances, capacitances, and resistive elements,
equations reflecting the transformer’s transient response can be written in numerous forms. Two
of the most common are the basic admittance formulation of the differential equation and the statevariable
formulation. The admittance formulation [27] is given by:
Is ()
1 G s C
n E()
s
(3.10.1)
s
The general state-variable formulation is given by Vakilian [23] describing the transformer’s lumpedparameter
network at time t:
L 0 0 die
/ dt
C
0 0 den
/ dt
0 0 U
dfe
/ dt
r
T
r
t
T 0
ie
G
Is
t en
T
0
(3.10.2)
where the variables in Equation 3.10.1 and Equation 3.10.2 are:
[i e ] = vector of currents in the winding
[e n ] = model’s nodal voltages vector
[f e ] = windings’ flux-linkages vector
[r] = diagonal matrix of windings series resistance
[T] = windings connection matrix
[T] t = transpose of [T]
[C] = nodal capacitance matrix
[U] = unity matrix
[I(s)] = Laplace transform of current sources
[E(s)] = Laplace transform of nodal voltages
[ n ] = inverse nodal inductance matrix = [T][L] –1 [T] t
[L] = matrix of self and mutual inductances
[G] = conductance matrix for resistors connected between nodes
[I s ] = vector of current sources
In a linear representation of an iron-core transformer, the permeability of the core is assumed constant
regardless of the magnitude of the core flux. This assumption allows the inductance model to remain
© 2004 by CRC Press LLC
© 2004 by CRC Press LLC