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On Developing the Theory of Resonant Processes in Power Transformer Windings. Part 2. Frequency Responses of a Circuit with Four PI Sections

In the first part of the article, the results from theoretical studies of frequency responses in a homogeneous chain circuit containing two PI sections were presented, and conclusions about the voltage resonance occurrence conditions were drawn. A circuit with two PI sections has a single independent node and one natural frequency, whereas transformer windings are more complex oscillatory circuits and have a much larger number of natural oscillation frequencies. The second part of the article presents the results from studies of frequency responses in a homogeneous chain circuit containing four PI sections, three independent nodes, and three natural oscillation frequencies. Analytical expressions for the admittances of the equivalent circuit individual parts and the voltages at the intermediate nodes, as well as expressions for the natural frequencies are obtained. Using an analysis of the frequency dependences of the admittances of the equivalent circuit parts, the conditions and frequency ranges under which voltage resonance may occur at the first, second, and third natural frequencies, are shown. It has been demonstrated that for the considered resonant circuits there is a critical frequency above which the conditions for voltage resonance are not satisfied. Formulas for impedances and voltages at intermediate nodes of a chain circuit with an arbitrary number of PI sections are given.