Wavelet Transform is a mathematical tool for time frequency analysis of non stationary signals like the current waveform in electric traction system. The locomotives used in traction system have different types of traction motors –dc and three phase ac motors with different methods of speed control. Hence technologies will be different depending on the type of locomotives. The nature of the load in electric traction system is very dynamic due to the acceleration, constant speed operation and deceleration of the locomotives. Due to the variation in the type and running pattern of locomotives harmonics induced into the system varies with time. In this paper, wavelet transform is applied to find out the total harmonic distortion of the measured current from a traction substation. Also a comparison is made by applying discrete Fourier transform to the instantaneous values of measured current.

Preliminary recalling the idea of generalized sinusoid, fundamental concepts of cisoidal algebra are then exposed. The approach is regarded as a generalization of the classic phasors analysis and it is proposed for the analysis of circuits and electromagnetic fields in damped steady state conditions. In this way is possible to reformulate the energetic balance by means of the concept of contra-rotating generalized vectors, obtaining a possible unified expression for the balance itself.

The concept of magneto-electric root mean square (RMS) is extended to the motional cases, where electromechanical energy conversion processes are concurrently present. Emerge, with undiminished physical meaning, the concepts of mechanical RMS of force f(t) and velocity u(t). In this way is possible, regarding the instantaneous conversion relation pm(t)=f(t)·u(t)=e(t)·i(t)=pe(t) in terms of RMS values, to correlate the two coupled systems (the electrical and the mechanical), in order to consider the two necessary and distinct sizing powers: the first associated to the mechanical movement of masses and the second to the movement of electric charges in the conductors. With these results and supposing that the electromechanical energy conversion process is periodic, it is possible to perform an energetic and power quality analysis in frequency domain. What emerges is a theory formally amenable to the Budeanu one. And, despite the critics already well highlighted against this approach, a "mechanical inactive power" that can be considered as an index of electromechanical power quality. Views from the two distinct electric and magnetic ports, the proposed approach is conceptually unitary: taking into account the Lagrangian approach under which the theory of electromechanical conversion is due, the dynamic factors F and E are both forces; similarly, the kinetic factors U and / are the two speeds which perform the development of power: the first associated with the geometric coordinates x, the second to electrical coordinate q.

After the representation of sinusoidal voltage and current, at the generic electrical port, in terms of counter-rotating vectors, the corresponding energy balance is formalized. The consequence is an instantaneous complex power which unifies the single-phase and the three-phase cases. The classic expressions of the active, reactive and apparent powers are obtained as mean values, on the temporal period, of the complex power expression. The extension of this approach to the periodic case, i.e. under non-sinusoidal or deformed conditions, leads to the classic Budeanu's treatment, but with an immediate algorithm clarity. The disagreement between the instantaneous complex power expression and the energy conservation condition, expressed by the Tellegen's relation, deprives this quantity of every topological and physical content. Incisive for the analytical relationships elaboration, this quantity has a purely formal character.

In this paper the analysis of unsymmetrical transversal faults that can occur on a four-phase networks is presented by means of Fortescue symmetrical component transformation (SCT). The various types of fault are analyzed and the connections between sequence networks and the related sequence equations are used to evaluate the phase fault currents. The obtained results are compared with those obtained for three-phase and six-phase systems. The related considerations are also extended to the generic n-phase systems.

In recent years, literature has presented an increasing number of contributions showing a rising interest in four-phase symmetric networks with reference to power electrical systems and electromechanic conversion and drivers. In this paper, a complete methodological analysis of four-phase systems, based on the phasorial and instantaneous sequence components, is presented for sinusoidal and dynamic conditions. This approach allows for the formalization of the theoretical and applicative methodologies necessary for the development of four-phase systems in a systematic and unified way.

In the present paper the analysis based on state equations method is applied to the transient study of interrupted single- and three-phase networks. The proposed approach is independent on the mathematical model adopted to represent the electric arc. In the three-phase case, the Clarke transformation allows us to perform the energy analysis of the three-phase electric arc during any balanced or unbalanced transient conditions. The Clarke energy balance related to the imaginary power concept permits to design the breaker under study without the use of empirical coefficients normally adopted in literature.

The analysis performed in this paper is related to a four-quadrant ac converter used to drive the motors of a railway locomotive in modern traction systems. The paper numerically analyses the starting transient and the instantaneous harmonic content of the input current. The current obtained by the analytical relationships is then studied by means of the wavelet analysis, putting into evidence the starting transient component and the superimposed noise. It is clearly stated in the work how this methodology can extend to the transient analysis the evaluation of the harmonic content evolution during the transient phase and how any change in the power supply frequency influence the system dynamic. These results cannot be found as well by applying the more traditional Fourier transform.

The analysis of the three-phase transverse-electromagnetic (TEM) line based on the Park approach represents, tinder the theoretical and the applicative point of view, an important analytical tool. For these reasons, this paper presents the analysis of lossy (quasi-TEM) three-phase transmission lines in terms of three-phase vectors based on the Park transformation. The three-phase transmission-line transient analysis presented in this paper emphasizes the conceptual contents specific to the Park approach. Furthermore, applied to some examples, it gives very important results for the practical analysis of a lossy three-phase transmission line.

An innovative three-phase distributed model is presented for asynchronous machines operating at high-frequency. The model is useful to study the propagation of surge along the stator windings of the machines that is excited by PWM-inverter source or fault waves that occur in the connected line. The new model is derived from single-phase models traditionally considered in literature. The use of time-space Clarke vectors allow the introduction of mutual coupling between phase winding and the integration of the model extending the methods that had been developed for the single-phase case by substituting real time-space variables with complex functions. A numerical method useful to simulate the distributed model is presented too. This is based on Laplace transformation of the Clarke waves. Finally, the first numerical results, carried out by applying a unitary steep-surge and a standard IEEE wave for insulation test to the obtained equivalent distributed networks, confirm the model validity and permit to underline the required low computational cost of the approach.

An innovative three-phase distributed model is presented for asynchronous machines operating at high-frequency. The model is useful to study the propagation of surge along the stator windings of the machines that is excited by PWMinverter source or fault waves that occur in the connected line. The new model is derived from single-phase models traditionally considered in literature. The use of time-space Clarke vectors allow the introduction of mutual coupling between phase winding and the integration of the model extending the methods that had been developed for the single-phase case by substituting real time-space variables with complex functions. A numerical method useful to simulate the distributed model is presented too. This is based on Laplace transformation of the Clarke waves. Finally, the first numerical results, carried out by applying a unitary steep-surge and a standard IEEE wave for insulation test to the obtained equivalent distributed networks, confirm the model validity and permit to underline the required low computational cost of the approach. Key-Words: Machines surge, Clarke transformation, High frequency model, Three-phase distributed model

PurposeTo provide a unified analytical tool, based on Park transformation, for the theoretical and practical analysis of a lossy three‐phase transmission line.Design/methodology/approachThe results obtained in the study of TEM waves propagation in two‐wire line can be extended to a symmetric m‐wire line by employing the modal analysis. This approach relates the dynamic of m‐wire guided field to the propagation of m modal voltages and currents acting on m single‐wire decoupled transverse electromagnetic (TEM) lines. In the symmetric three‐phase system case, the modal analysis includes, as a particular case for m=3, the symmetric component theory. In previous papers, the authors applied the Park transformation to study the wave propagation of (TEM) three‐phase symmetrical lines. The formulation proposed and tested considers the lossless TEM wave propagation of a three‐phase line without consideration to the dissipations phenomena present in the line itself. Taking into account the obtained results, the extension of the developed approach to the lossy three‐phase transmission line transient analysis is very useful on both theoretical and practical points of view.FindingsThe symmetrical three‐phase line Park model for the lossy transmission line transient analysis, regarded as vector formulation of the line modal analysis, has been presented. The proposed examples highlight how, thanks to the Park model, the dynamic analysis of the three‐phase line in distorted and unsymmetrical systems becomes an integral part of the more general and well‐established power electric system dynamic theory.Originality/valueThe three‐phase transmission line transient analysis presented in this paper emphasizes the conceptual contents, specific to the Park approach. Furthermore, it gives some results very important for the practical analysis of a lossy three‐phase transmission line.

High-speed traction studies need to carry out detailed equations for formalising a complete electromechanical model, representing the dynamic coupling of electrical and mechanical phenomena. Starting from Lagrange and Park theories, and after recalling the general criteria for the set-up of the two distinct mechanical and electrical sub-systems, the new formulisation of a complete dynamic model is presented, with specific reference to a train dedicated to high-speed services. Simulations are carried out relating to the steady state condition at speed of 100 km/hr. Representative of the dynamics of each single component, the investigation focuses in particular on the parts of transmission (rotor, cardan joint, gearcase) and the relevant torques associated to the short circuit dynamics.

The classic phasorial formalism employed to represent the single-phase systems in the time domain can be extended to the three-phase systems by means of the Park approach. It analytically extends the phasorial approach to the Gauss plane in terms of three-phase vectors, so that the power phenomena, which in the single-phase systems are represented by real quantities, are expressed, in the three-phase systems, by complex quantities. The power factor concept is extended as well, and the imaginary power can be introduced as an extension to the three-phase systems, of the reactive power concept. By analyzing the Park three-phase vectors in the frequency domain, a decomposition of the power factor into two terms can be attained: the first term takes into account the displacement effects, while the second term takes into account the distortion effects. Examples are given to show the meaning of the proposed decomposition in practical situations.

This paper deals with the line voltage drop calculation in presence of disturbances in the electric network, like un-balance, harmonic, and interharmonic components. The Park approach permits to define a calculation expression with a structure very close to the classical one typically employed in case of balanced systems and also permits to underline the role of the imaginary power that quantifies in one term only the effects of disturbances on the voltage drop. In order to text the effectiveness of the proposed methodology, two applications on real industrial plants are presented in the second part of the paper. The comparison with the classical theory and calculation made by using commercial software gives rise to interesting comments and discussion on the proposed approach.

The sensorless determination of the dynamic hysteresis loop of magnetic materials, and in particular those of magnetic plates, can be obtained under ac symmetric conditions in a relatively simple way. This paper proposes a new digital method and instrument for the sensorless evaluation of the dynamic hysteresis loop under ac asymmetric conditions, in the presence of a dc polarization of the magnetic flux. The method is based on the determination of the parameters of a simplified linear model of the magnetic circuit, which preserve the energy equivalence. The results of some experimental work are given in order to validate the proposed method.

Park approach. belongs to the usual procedures specific of the modal analysis. Under this point of view, it can be therefore used in the analysis of three-phase static components, in particular in the study of transmission lines. In this case, the approach replaces the matrix formulation, specific of the modal analysis, with the space-vectors formulation. In this way, we obtain complex time-functions, similar to those obtained for the single-phase systems, that allow one to synthesize the whole logic modal processes in a straightforward and physically sound way, much more significant than the matrix formulation. For the same reason, the energetic approach appears more effective. Expressed in a complex form valid under any condition, it refers, in particular, to the imaginary power concept. This makes the computation possible, under the power quality point of view, of the effects due to the propagation of the harmonic and sequence instantaneous components along the line.