analysis of electrical circuits with variable load regime parameters projective geometry method pdf

The circuit regime analysis is one of the main problems for electric circuit theory.
The finding of the actual (absolute) value of regime parameters (voltage, current,
power, and transformation ratio for different parts of a circuit) is the simplest
analysis task. If a circuit has variable elements (loads and voltage regulators),
additional analysis tasks appear.

The interest in such circuits is defined, in particular, by the state and tendencies of development of power electronics, modular power supply, or distributed power supply systems with renewable power sources. Similar devices, in general, represent the complex multiple input and multiple output systems and their loads can change from the short circuit to open circuit and further give energy. In turn, the loads can be subdivided into high priority and additional (ballast) loads. For definiteness, it is possible to accept that such systems, for circuit theory, present linear mesh circuits of a direct current or multi-port networks.

We will consider some of the arising additional tasks of analysis. For example, it
is important to confront operating regime parameters with characteristic values; that
is, to represent these parameters in the normalized or relative form. In this case, the
informational content of these parameters is increasing; it is possible to appreciate
qualitative characteristics of an operating regime or its effectiveness, to compare
regimes of different circuits, and to set a necessary regime.

The other task of analysis is the determination of the dependence of the regime
parameter changes on the respective change of element’s parameters (for example,
the problem of the recalculation of load currents). Thus, it is necessary to set the
form of these changes reasonably, that is, to determine whether these changes are
increments or any other expressions.

Another task of analysis is the definition of the view or character of such an active circuit with a changeable element (as a power source concerning load); that is, this circuit shows more property of a voltage source or current source. In the electric circuit theory, a range of circuit’s properties, theorems, and methods is well known, and their use simplifies the decision of these problems.
However, the known approaches do not completely disclose the properties of such circuits, which reduces the effectiveness of analysis.

The method of analysis for a circuit with variable element parameters is developed by the author. For interpretation of changes or “kinematics” of circuit regimes, projective geometry is used. For example, the known expression has the typical fractionally linear view for functional dependence of current (or voltage) via resistance. It gives the grounds for considering this expression as a projective transformation. The projective transformations preserve an invariant; there is a cross-ratio of four points (a ratio of two proportions) or four values of current and resistance. The value of this invariant is preserved for all the variables (as a current, voltage, and resistance) and for parts or sections of a circuit.