main-content

## Über dieses Buch

This book introduces readers to electric circuits with variable loads and voltage regulators. It defines invariant relationships for numerous parameters, and proves the concepts characterizing these circuits. Moreover, the book presents the fundamentals of electric circuits and develops circuit theorems, while also familiarizing readers with generalized equivalent circuits and using projective geometry to interpret changes in operating regime parameters. It provides useful expressions for normalized regime parameters and changes in them, as well as convenient formulas for calculating currents.

This updated and extended third edition features new chapters on the use of invariant properties in two-port circuits, invariant energy characteristics for limited single-valued two-port circuits, and on testing projective coordinates. Given its novel geometrical approach to real electrical circuits, the book offers a valuable guide for engineers, researchers, and graduate students who are interested in basic electric circuit theory and the regulation and monitoring of power supply systems.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
The typical structure of power supply systems is considered. The equivalent circuits for its base units as active two-poles, passive, and active two-ports allow introducing the regime parameters by traditional methods in the relative view at change of loads and parameters of these circuits. The features and disadvantages of these methods are shown.
A. Penin

### Chapter 2. Operating Regimes of an Active Two-Pole. Display of Projective Geometry

Abstract
For the interpretation of changes or “kinematics” of circuit regimes, some information about affine and projective geometry is provided by examples of simple circuits. It allows revealing the invariant properties of circuits, that is, such expressions, which turn out identical to all the regime parameters. These invariant expressions can be assumed as the basis of well-founded definition of regimes in the relative form.
A. Penin

### Chapter 3. Generalized Equivalent Circuit of an Active Two-Pole with a Variable Element

Abstract
The generalized equivalent circuits, which develop the known theorems, are formulated. It appears that the load straight line at various values of a changeable element (resistor) of an active two-pole passes into a bunch of these lines. The bunch center coordinates do not depend on this changeable element. It is proposed to use as the parameters of the generalized equivalent generator such a load current and voltage, which proved the current across this element equal to zero. The application of projective coordinates instead of resistance values allows obtaining suitable formulas of the recalculation of the load current, to define the scales for the load and variable element. The influence of the changeable element on properties of the generalized equivalent generator can be used in practice, for example, for a parametric stabilization of load current.
A. Penin

### Chapter 4. Two-Port Circuits

Abstract
The correlations between various regime parameters for different parts or sections of a circuit are considered. The transfer from one section to another is interpreted as an affine (for voltage and current) transformation and as a projective (for conductance) transformation. As it was shown in Chap. 2, such transformations possess the invariants in the form of an affine ratio of three points and a cross-ratio of four points accordingly. Therefore, these invariants express the above correlations.
Effectiveness of two-ports is considered too.
A. Penin

### Chapter 5. Invariant Energy Characteristics for Limited Single-Valued Working Area of Two-Ports

Abstract
Besides of fractionally linear expressions of two-port circuits, there are quadratic fractional ones for some parameters as cubic curves. A load power and efficiency via the load resistance are examples of these curves. Cross-ratios are carried out in a limited unambiguous or single-valued working area of these characteristics too. Such well-founded introduction of regime parameter shows that the form of expressions for the subsequent values depends on a type of the regime. Therefore, there will also be not correctly the introduction of formal increments, divisions. Changes of regime parameters are proved, and direct formulas of recalculation are proposed. The obtained results have a practical interest to the recalculation of a capacitor store voltage and power using the input current samples.
A. Penin

### Chapter 6. Operating Regimes of an Active Multi-port

Abstract
Some features of electric circuits, which contain few loads, are shown. There is a mutual influence of these loads; additional characteristic regimes besides familiar regimes, as the short circuit and open circuit, take place. Therefore, these features bring up the problem of choice of the scales for load parameters. The problem of recalculation of the load currents is considered. Changes of regime parameters are introduced by a cross-ratio of four points. Easy-to-use formulas of the recalculation of the currents, which possess the group properties at change of load conductivities, are obtained. It allows expressing the final values of the currents through the intermediate changes of the currents and conductivities.
A. Penin

### Chapter 7. Active Multi-ports with Variable Elements

Abstract
The recalculation of load currents is considered. The regime changes are defined by variable elements of a circuit. Generalized equivalent circuits are introduced for active multi-ports by the results of Chap. 3. The results of Chap. 6 are used for interpretation of changes or “kinematics” of the circuit regimes. Changes of regime parameters are also introduced by a cross-ratio of four points. Easy-to-use formulas of the recalculation of the currents, which possess the group properties at change of conductivity of the loads, are obtained.
A. Penin

### Chapter 8. Passive Multi-port Circuits

Abstract
The invariant properties of input–output of circuits, established in Chap.4, are generalized for multi-ports. A mutual influence of loads of passive multi-ports is investigated by affine and projective transformations too. The direct and reverse formulas of recalculation of currents and non-uniform coordinates are obtained in the form of fractionally linear expressions of identical type. The results allow separating or restoring two signals at output via input currents of the four-port circuit or the three-wire line inputs without determination of their transmission parameters.
A. Penin

### Chapter 9. Hierarchy of Projective Coordinate Systems

Abstract
The projective coordinates for calculation of load currents, presented in Chaps. 68, are analyzed. These coordinates, as a reference triangle, are introduced with the aid of characteristic regime points. Usually, the working values of load currents are much less, than the values of the characteristic regime currents. Therefore, the recalculation of load currents can lead to inadmissible errors. On the other hand, the high characteristic regime currents present technical difficulties when testing such circuits. The reasonable choice of more convenient projective coordinate systems, when the currents of the corresponding characteristic regimes are close to the working values of load currents, is given. The definitions of regime and its invariants, group properties of the regime changes, remain in these coordinate systems.
A. Penin

### Chapter 10. Regulation of Load Voltages

Abstract
The restriction of load power, two-valued regulation characteristic, and interference of several loads are observed in power supply systems with limited power of voltage source. In this chapter, the definition of regime in an invariant form through different parameters of changes of transformation ratio and voltage load is grounded for these circuits with two loads. The approach for interpretation of changes or “kinematics” of load regime is presented by using the conformal and hyperbolic plane. To simplify the task and to reveal the basic moments of influence of limited source power, the static regulation characteristics and idealized models of voltage converters are considered. Geometrical interpretation of a simplified model of multi-channel power supply system allows to base definition of operating regime parameters.
A. Penin

### Chapter 11. Stabilization of Load Voltages

Abstract
In the present chapter, the results of Chap. 10 for interpretation of changes or “kinematics” of load regimes are used. Obtained geometrical transformations describe the movement of an operating point along regulation and stabilization trajectories.
A. Penin

### Chapter 12. Pulse-Width Modulation Regulators

Abstract
In the present chapter, the results of Chap. 10 are developed for power supply systems with $$PWM$$ boost and buck–boost converters. Regime parameters of converter are determined relatively to the maximum permissible load voltage and the control pulse width. Regime change and correspondence between various regime parameters are examined as projective transformations. That makes possible to validate the regime definition, to restrict the range of their variation on the forward branch of regulation curve, to realize the linearization of this regulation characteristic, and to get the feedforward control.
A. Penin

### Chapter 13. Power-Source and Power-Load Elements

Abstract
The concepts of power-source and power-load elements are considered. The consideration of losses shows the two-valued characteristics of such elements. The single-valued areas of their characteristics are determined reasonably. The invariants of regime parameters are considered by using the cross-ratio.
A. Penin

### Chapter 14. Quasi-resonant Voltage Converter with Self-limitation of Load Current. Similarity of Load Characteristics of Some Electronic Devices

Abstract
The load characteristics of regulable quasi-resonant voltage converter as an active two-port with self-limitation of current are considered. The equivalent generator is introduced of this two-port. The load approximation expressions and deviation from the maximum load power point are considered. The attention to similarity of characteristics of this converter and some electronic devices is paid. In particular, a convenient linearly hyperbolic approximation of solar cell characteristics is presented. The direct analytical calculation of load regime leads to a quadratic equation that importantly simplifies the calculations in the real time. Also, this model makes it possible to soundly determine deviation from the maximum load power and to compare the effectiveness of solar cells with different parameters.
A. Penin

### Backmatter

Weitere Informationen