D.C. Polarized Current Transformers For The Measurement Of Harmonic Noise: Numerical and Experimental Analysis

Abstract

The paper describes the method employed for calculating the parameters of the model of a current transformer (C.T.) designed to detect the harmonic content of unidirectional currents generated by static convertors on their D . C . side. The paper also analyses the experimental results obtained during tests carried out on a prototype C.T.. comparing them with the values calculated from the theoretical model. I M R O D U f l I ON The detection of the harmonic content of unidirectional currents generated by static convertors presents considerable difficulties in certain cases, particularly whet the harmonic current amplitudes are of the order of 10to lo-+ times the D.C. component of the current. During an earlier study 121 an algorithm has been developed for the design of a C.T. employing an air gap in its core. This device makes possible the measurement of harmonic currents only. and presents the double advantage of filtering out the D.C. and providing D.C. isolation between the measuring circuit and the power system. At the same time, it assures an acceptable ratio error over the frequency range 10+104 Hz. In I21 it is also described the development of a mathematical model for the evaluation of the ratio error over the entire frequency range mentioned. This model takes into account the influence of magnetic phenomena (eddy currents in laminations and secondary leakage inductance) as well as the effect of parasitic capacitances. The present study considers the problem of calculating the design parameters of the C.T. model. Particular attention is given to the calculation of the equivalent parasitic capacitance, arriving at the determination of a simplified expression for calculating the main resonance frequency in high-frequency operation. The paper shows many test results, obtained on a suitably constructed C.T. prototype. These consist of frequency response curves (at different A . C . and D.C. current amplitudes), examples of transformation of some waveforms and curves showing the transient response to a step input. These results, compared with those obtained by numerical simulation, confirm the quality of the model adopted and of the method used for calculating its parameters. The results also yield further informations i n guiding and improving the design. CALCULATION PARAKtXERS The lumped-parameter electrical model adopted for the study of the C.T. operation is the well-known I-type equivalent circuit. Given the width of the band covered (10+104 Hz). the parameters cannot be considered as constant. Ihis applies particularly to the derived branch parameters (GO, L o ) . which are sensitive to the influence of eddy currents. As for the distributed capacitances of the windings. which affect the performance at high frequencies, an approximate representation 1s possible, enabling the calculation of the first resonance frequency. 731s consists o f representing the overall capacitance effect by means of a single equivalent parasitic capacitance C, placed immediately after the ideal transformer 12, 31. Estimatinp the equivalent parasitic capacitance A possible evaluation criterion i a based on the formulation of an energy-type equivalence between the total dielectric energy distributed among the network of single parasitic capacitances and the energy pertaining to the capacitance C in the equivalent circuit. For this purpose, the expression for the dielectric energy distributed in the secondary winding is formulated with reference to a working situation in which it can be claimed that the inter-turn capacitances do not substantially affect the voltage distribution between the winding turns, due to the induced e.m.f.s and to the voltage drops. This assumption, which is perfectly acceptable in low and medimfrequency operation, leads to the evaluation of the total equivalent capacitance C. This value of capacitance is also considered valid for the determination of the first resonance frequency. It is clear that this representation, employing a single capacitance, does not permit the calculation of further resonance phenomena, which can be observed experimentally at even higher frequencies. On the other hand, it is the identification of the first resonance frequency which is the factor of major interest, insofar as it determines the upper limit of the acceptable range of operation for the C.T.. The basic element. on which the calculation is founded, is the capacitance per unit length between adjacent turns. This parameter can be estimated by using the expression for calculating the capacitance between the two parallel conductors of a bifilar line. If “dc” denotes the diameter of the metallic wire and “h” the distance between the axes of the wires. the capacitance per unit length C’ is given by: . 7 T c r . r o [F/ml . ‘’ tn(2-h/dc 1) Eq.(l) is exact when the distance h is large compared with the diameter dc. As a first approximation, however, it is assumed to be valid also in the case being examined. In addition. the dielectric constant Lr which appears in eq.(1) is taken as a suitable value between those for air and f o r the insulant, depending upon the distance h between the wires. Thus, the capacitance between two adjacent turns, calculated on the basis of the average length C m 2 of a turn in the secondary winding. is: