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