Techniques for studying static switching circuits

As switching circuits in power systems proliferate there is an increasing need to understand and accurately model the interactions between the switching circuit and the power system to which it is connected. Two such interactions are the generation and propagation of harmonics and the existence of unstable operating regions. Harmonics are hard to analyze since they are affected by the power system and stability issues are hard to analyze since they require more sophisticated tools than those for harmonic calculations.; For the calculation of harmonics a new formulation for frequency plane analysis is developed which takes into account the coupling between different harmonic frequencies. Since voltage and current harmonics are present in a switching circuit a matrix formulation, called a Fourier matrix, is used. Each voltage or current is represented by a column vector and each impedance or admittance is represented by a matrix. The formulation is for a Thyristor Controlled Reactor (TCR) although the method is valid for other types of switching circuits. An admittance matrix will be developed satisfying I{dollar}sb{lcub}rm R{rcub}{dollar} = (Y{dollar}sb{lcub}rm TCR{rcub}{dollar}) V{dollar}sb{lcub}rm T{rcub}{dollar}, where I{dollar}sb{lcub}rm R{rcub}{dollar} and V{dollar}sb{lcub}rm T{rcub}{dollar} are, respectively, vectors of harmonic current through and harmonic voltage across the TCR. The off diagonal elements of (Y{dollar}sb{lcub}rm TCR{rcub}{dollar}) model the interaction between the voltage at one frequency to the current at another frequency.; The state equations for a switching circuit will also be developed. Again, a TCR will be used but the formulation will hold for other switching circuits as well. The equations constitute a set of linear differential equations with time varying, periodic coefficients. The theory of linear differential equations with periodic coefficients will be investigated and conditions for stability will be found.; Both tools prove quite useful and together form a unified set of techniques for analyzing switching circuits. The analysis of the state equations will give a qualitative assessment of the circuit behavior. Statements about the transient behavior of the solution, existence of a periodic solution, and the stability of the solution can all be made with this analysis. The analysis with the Fourier matrices will give a quantitative solution. It finds the Fourier components of the periodic solution where that solution exists.