A device’s transient response is a result of the flow of energy between the distributed electrostatic and electromagnetic structures of the device. For all practical transformer-winding structures, this interaction is quite complex and can only be realistically investigated by constructing a detailed 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) and build a lumped-parameter model. The method of subdividing the winding can be complex and, if not addressed carefully, affects the accuracy of the resultant model. The resultant lumped-parameter model is composed of inductances, capacitance, 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 state-variable formulation. The admittance formulation is given by Degeneff (1977):
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 constant for the entire computation. Above equations are based on this assumption and that the various elements in the model are also not frequency-dependent. Work in the last decade has addressed both the nonlinear characteristics of the core and the frequency-dependent properties of the materials. Much progress has been made, but their inclusion adds considerably to the model’s complexity and the computational difficulty. If the core is nonlinear, the permeability changes as a function of the material properties, past history, and instantaneous flux magnitude. Therefore, the associated inductance model is time-dependent. The basic strategy for solving the transient response of the nonlinear model in mentioned Eqs. is to linearize the transformer’s nonlinear magnetic characteristics at an instant of time based on the flux in the core at that instant.
Two other model formulations should be mentioned. de Leon addressed the transient problem using a model based on the idea of duality (de Leon and Semlyen, 1994). The FEM has found wide acceptance in solving for electrostatic and electromagnetic field distributions. In some instances, it is very useful in solving for the transient distribution in coils and windings of complex shape.
Every model of a physical system is an approximation. Even the simplest transformer has a complex winding and core structure and as such possesses an infinite number of resonant frequencies. A lumped parameter model, or for that matter, any model is at best an approximation of the actual device of interest. A lumped-parameter model containing a structure of inductances, capacitances, and resistances will produce a resonance frequency characteristic that contains the same number of resonant frequencies as nodes in the model. The transient behavior of a linear circuit (the lumped-parameter model) is determined by the location of the poles and zeros of its terminal impedance characteristic. It follows then that a detailed transformer model must possess two independent characteristics to faithfully reproduce the transient behavior of the actual equipment. First, it must include accurate values of R, L, and C, reflecting the transformer geometry. This fact is well appreciated and documented. Second, the transformer must be modeled with sufficient detail to address the bandwidth of the applied wave shape.
In a valid model, the highest frequency of interest would have a period at least ten times larger than the travel time in the largest winding segment in the model. If this second characteristic is overlooked, a model can produce results that appear valid, but may have little physical basis. A final issue is the manner in which the transformer structure is subdivided. If care is not taken, the manner in which the model is constructed will itself introduce significant errors and the computation will be mathematically robust but an inaccurate approximation of the physical reality.
McNutt et al. (1974) suggested a method of obtaining a reduced order transformer model by star ting w ith the detailed model and appropriately combining series and shunt capacitances. This suggestion was extended by de Leon and Semlyen (1992). This method is limited to linear models and cannot be used to eliminate large propor tions of the detailed models wi thout affecting the resulting model’s accuracy.
Degeneff (1978) proposed a terminal model developed from information from the transformer’s nameplate and capacitance measured among the terminals. This model is useful below the first resonant frequency but lacks the necessary accuracy at higher frequencies for system. Dommel et al. (1982) proposed a reduced model for EMTP described by branch impedance or admittance matrix calculated from open- and short-circuit tests. TRELEG and BCTRAN matrix models for EMTP can be applied only for ver y low-frequency studies. Morched et al. (1992) proposed a terminal transformer model, composed of a synthesized LRC network, where the nodal admittance matrix approximates the nodal admittance matrix of the actual transformer over the frequency range of interest. This method is appropriate only for linear models.