Father of the lecture. Fundamentals of the theory of electrical circuits

Foreword
Conventions
Introduction
Chapter one. Basic definitions, laws, elements and parameters electrical circuits
1-1. Electrical circuit
1-2. Positive directions of current and voltage
1-3. Instant Power and Energy
1-4. Resistance
1-5. Inductance
1-6. Capacity
1-7. Replacing Physical Devices with Idealized Circuit Elements
1-8. Source e. d.s. and current source
1-9. Linear electrical circuits
1-10. Basic definitions related to the electrical circuit
1-11. Volt-ampere characteristic of the circuit section with the source
1-12. Potential distribution along a circuit with resistances and voltage sources
1-13. Kirchhoff's laws
1-14. Tasks and questions for self-examination
Chapter two. Harmonic current circuits
2-1. Harmonic vibrations
2-2. Generation of sinusoidal e. d.s.
2-3. Mean and effective values ​​of the function
2-4. Performance harmonic vibrations as projections of rotating vectors
2-5. Harmonic current in resistance
2-6. Harmonic current in inductor
2-7. Harmonic current in capacitance
2-8. Serial connection r, L, C
2-9. Parallel connection r, L, C
2-10. Power in the harmonic current circuit
2-11. Tasks and questions for self-examination
Chapter three. Application complex numbers to the calculation of electrical circuits (method of complex amplitudes)
3-1. Representation of harmonic functions using complex quantities
3-2. Ohm's and Kirchhoff's laws in complex form
3-3. Relationship between resistances and conductivities of a circuit section
3-4. complex form power records
3-5. The condition for transmitting the maximum average power from the source to the receiver is
3-6. The condition for the source to transmit maximum power at a given receiver power factor
3-7. Power balance
3-8. Potential (topographic) diagram
3-9. Tasks and questions for self-examination
Chapter Four. Converting electrical circuit diagrams. Method geometric places
4-1. Serial and Parallel Connections
4-2. mixed connection
4-3. Equivalent circuit sections with series and parallel connections
4-4. Convert triangle to equivalent star
4-5. Convert star to equivalent triangle
4-6. Equivalent voltage and current sources
4-7. Converting Diagrams with Two Nodes
4-8. Transferring sources in a schema
4-9. Converting symmetrical circuits
4-10. Graphic image dependences of complex quantities on the parameter
4-11. View transformation
4-12. Diagrams of resistance and conductance of the simplest electrical circuits
4-13. View transformation
4-14. Tasks and questions for self-examination
Chapter five. Methods for calculating complex electrical circuits
5-1. Applying Kirchhoff's Laws to Calculate Complex Circuits
5-2. Loop current method
5-3. Nodal stress method
5-4. overlay method
5-5. Input and transfer conductances and resistances
5-6. Reversibility (or reciprocity) theorem
5-7. Compensation theorem
5-8. Theorem on the change in currents in an electric circuit when the resistance changes in one branch
5-9. Equivalent source theorem
5-10. Application of matrices to the calculation of electrical circuits
5-11. Some features of the calculation of electrical circuits with capacitances
5-12. Dual chains
5-13. Electromechanical analogies
5-14. Tasks and questions for self-examination
Chapter six. Inductively coupled electrical circuits
6-1. Basic provisions and definitions
6-2. Polarities of inductively coupled coils; e. d.s. mutual induction
6-3. Complex form of calculation of a circuit with mutual induction
6-4. Inductive coupling coefficient. Leakage inductance
6-5. Equations and equivalent circuits for a transformer without a ferromagnetic core
6-6. Energy of inductively coupled windings
6-7. Transformer input impedance
6-8. autotransformer
6-9. Tasks and questions for self-examination
Chapter seven. Single oscillation circuit
7-1. Oscillatory (resonant) circuits
7-2. Series oscillatory circuit. Stress resonance
7-3. Frequency characteristics of series resonant circuit
7-4. Parallel oscillatory circuit. Current resonance
7-5. Varieties of a parallel oscillatory circuit
7-6. Elements of the oscillatory circuit
7-7. Tasks and questions for self-examination
Chapter eight. Related oscillatory circuits
8-1. Communication types
8-2. Coupling resistance and insertion resistance
8-3. Vector charts
8-4. Coupling coefficient
8-5. Set up linked contours. Energy ratios
8-6. Resonance curves of coupled circuits. Bandwidth
8-7. Tasks and questions for self-examination
Chapter nine. Three-phase current circuits
9-1. Three-phase electrical circuits
9-2. Star and delta connection
9-3. Symmetrical operation of a three-phase circuit
9-4. Unbalanced operation of a three-phase circuit
9-5. Power of an unbalanced three-phase circuit
9-6. Rotating magnetic field
9-7. The principle of operation of asynchronous and synchronous motors
9-8. Tasks and questions for self-examination
Chapter ten. Periodic non-sinusoidal processes
10-1. Trigonometric form of the Fourier series
10-2. Cases of symmetry
10-3. Transfer of origin
10-4. Complex form of the Fourier series
10-5. Application of the Fourier series to the calculation of a periodic non-sinusoidal process
10-6. Effective and mean values ​​of a periodic non-sinusoidal function
10-7. Power in the circuit of periodic non-sinusoidal current
10-8. Coefficients characterizing periodic non-sinusoidal functions
10-9. Tasks and questions for self-examination
Chapter Eleven. Circuits with ferromagnetic cores at constant magnetic flux
11-1. Purpose and types of magnetic circuits
11-2. Basic laws of the magnetic circuit and properties of ferromagnetic materials
11-3. Unbranched magnetic circuit
11-4. Branched magnetic circuit
11-5. Tasks and questions for self-examination
Chapter twelve. chains alternating current with ferromagnetic elements
12-1. Some features of AC circuits with ferromagnetic elements
12-2. Basic properties ferromagnetic materials in alternating fields
12-3. Coil with ferromagnetic core
12-4. Ferromagnetic Core Transformer
12-5. Tasks and questions for self-examination
Chapter thirteen. Transient processes in linear circuits with lumped parameters (classical method)
13-1. The occurrence of transients
13-2. Commutation laws and initial conditions
13-3. Forced and free modes
13-4. Transient process in the circuit r, L
13-5. Transient process in the circuit r, C
13-6. Transient process in the circuit r, L, C
13-7. Calculation of the transient process in a branched circuit
13-8. Tasks and questions for self-examination
Chapter fourteen. Applying the Laplace Transform to Transient Calculation
14-1. General information
14-2. direct conversion Laplace. Original and Image
14-3. Pictures of some simple functions
14-4. Basic properties of the Laplace transform
14-5. Finding the original from an image using inverse transformation Laplace
14-6. Decomposition theorem
14-7. Tables of originals and images
14-8. Application of the Laplace transform to the solution of differential equations of electrical circuits
14-9. Accounting for non-zero initial conditions equivalent source method
14-10. Inclusion formulas
14-11. Transient Calculation Using Overlay Formulas
14-12. Finding in a closed form the steady response of a circuit to a periodic non-sinusoidal acting function
14-13. Tasks and questions for self-examination
Chapter fifteen. Spectral method
15-1. Temporal and spectral representation of signals
15-2. non-periodic signals. The Fourier integral as a limiting case of the Fourier series
15-3. Relationship between discrete and continuous spectra
15-4. Cases of symmetry of non-periodic functions
15-5. Energy distribution in the spectrum
15-6. Relationship between Fourier transform and Laplace transform
15-7. Properties of the Fourier Transform
15-8. Spectra of some typical non-periodic signals
15-9. Generalized form of the Fourier integral
15-10. Special cases
15-11. Finding a signal from given frequency characteristics of the real and imaginary components of the spectrum
15-12. Application spectral method for calculating transients
15-13. The condition of undistorted signal transmission through linear system
15-14. Passing a signal through a linear system with limited bandwidth
15-15. Tasks and questions for self-examination
Chapter sixteen. Circuits with distributed parameters
16-1. Primary parameters of a uniform line
16-2. Differential Equations homogeneous line
16-3. Periodic mode in a homogeneous line
16-4. Secondary parameters of a homogeneous line
16-5. line without distortion
16-6. Lossless line
16-7. Lossless line operation modes. standing waves
16-8. Line input impedance
16-9. Lossless line power
16-10. Line as a matching transformer
16-11. Resistance matching by connecting line segments in parallel
16-12. Pie charts for lossless line
16-13. Line as an element of a resonant circuit
16-14. Transient processes in circuits with distributed parameters
16-15. Investigation of transient processes in circuits with distributed parameters using the Laplace transform
16-16. Tasks and questions for self-examination
Chapter seventeen. Bipolar networks
17-1. Definition and classification of bipolar networks
17-2. Single-element reactive two-terminal networks
17-3. Two-element reactive two-terminals
17-4. Multi-element reactive two-terminal networks
17-5. General expression for the resistance of a passive multi-element reactive two-terminal network
17-6. Canonical schemes of reactive two-terminal networks
17-7. The sign of the frequency derivative of the resistance or conductivity of a reactive two-terminal network
17-8. Chain circuits of reactive two-terminal networks
17-9. Potentially equivalent two-port networks and conditions for their equivalence
17-10. Potentially - reverse two-terminal networks and conditions for their mutual inverse
17-11. Multi-element lossy two-terminal networks containing elements of two types
17-12. Evenness of the active and oddness of the reactive components of resistance relative to frequency. Sign of active resistance and active conductivity
17-13. Relationship between the frequency characteristics of the active and reactive components of the resistance or conductivity of a two-terminal network
17-14. Tasks and questions for self-examination
Chapter eighteen. Quadripoles
18-1. Basic definitions and classification of quadripoles
18-2. Systems of equations of a quadripole
18-3 Quadripole equations in the form
18-4. Open circuit and short circuit parameters
18-5. Quadripole equivalent circuits
18-6. Input impedance of a quadripole with arbitrary load
18-7. Characteristic parameters of a quadripole
18-8. Insertion loss of a quadripole
18-9. Transmission function
18-10. Cascade connection of four-terminal networks based on the matching of characteristic impedances
18-11. Equations of complex quadripoles in matrix form
18-12. Single-element quadripoles
18-13. L-shaped quadripole
18-14. T-shaped and U-shaped quadripoles
18-15. Symmetrical bridge quadripole
18-16. Ideal transformer as a four-terminal
18-17. Feedback
18-18. Tasks and questions for self-examination
Chapter nineteen. Electrical filters
19-1. Basic definitions and classification of electrical filters
19-2. Reactive filter pass condition
19-3. Type k filters
19-4. T type filters
19-5. Inductively coupled circuits as a filter system
19-6. Bridge filters, piezoelectric resonators
19-7. Non-inductive filters
19-8. Tasks and questions for self-examination
Chapter Twenty. Synthesis of linear electrical circuits
20-1. Characteristics of synthesis problems
20-2. Investigation of a two-terminal network at a complex frequency
20-3. Resistance and conductance as positive actual function
20-4. Conditions for the physical realizability of a function
20-5. Methods for constructing a two-terminal network according to a given frequency response
20-6. Investigation of a quadripole at a complex frequency
20-7. Tasks and questions for self-examination
Applications
I. Signal Graph Method
II. Relationships between coefficients of a quadripole
III. Determinants Expressed in Terms of Quadripole Coefficients
IV. Originals and images according to Laplace
Literature
Alphabetical index

Name In: Fundamentals of the theory of circuits. 1975.

The book outlines general methods of analysis and synthesis and a description of the properties of linear electrical circuits with lumped and distributed parameters at constant, alternating, periodic and transient currents and voltages. The properties and methods for calculating steady-state and transient processes in nonlinear electrical and magnetic circuits of direct and alternating current are considered. All provisions of the theory are illustrated by practical examples.

TABLE OF CONTENTS

Preface to the fourth edition.
Introduction.
Section 1 LINEAR ELECTRIC CIRCUITS WITH LOCALIZED. PARAMETERS
Chapter 1.
Basic laws and methods for calculating electrical circuits at constant currents and voltages.
1-1. Elements of electrical circuits and electrical circuits.
1-2. Equivalent circuits for energy sources.
1-3. Ohm's law for a chain section with e. d.s.
1-4. Potential distribution along an unbranched electrical circuit.
1-5. Power balance for the simplest unbranched circuit.
1-6. Application of Kirchhoff's laws for the calculation of branched circuits.
1-7. Method of nodal potentials.
1-8. Loop current method.
1-9. Circuit State Equations in Matrix Form.
1-10. Converting linear electrical circuits.
Chapter 2
The main properties of electrical circuits at direct currents and voltages
2-1. The principle of imposition.
2-2. reciprocity property.
2-3. Input and mutual conductivities and resistances of branches; voltage and current transfer coefficients.
2-4. Application of topological methods for the calculation of circuits.
2-5. Topological formulas and rules for determining the transmission of an electrical circuit.
2-6. The compensation theorem.
2-7. Linear relationships between voltages and currents.
2-8. Theorem on mutual increments of currents and voltages.
2-9. General remarks about bipolar networks.
2-10. The active two-pole theorem and its application to the calculation of branched circuits.
2-11. Transfer of energy from an active two-terminal network to a passive one.
Chapter 3
Basic concepts of sinusoidal current circuits
3-1. alternating currents.
3-2. The concept of alternating current generators.
3-3. sinusoidal current.
3-4. Operating current, e. d.s. and tension.
3-5. Depiction of sinusoidal functions of time by vectors and complex numbers.
3-6. Addition of sinusoidal functions of time.
3-7. Electrical circuit and its scheme.
3-8. Current and voltage in series connection of resistance, inductance and capacitance.
3-9. resistance.
3-10. Phase difference of voltage and current.
3-11. Voltage and currents with parallel connection of resistance, inductance and capacitance.
3-12. Conductivity.
3-13. Passive bipolar.
3-14. Power.
3-15. Power in resistance, inductance and capacitance.
3-16. Power balance.
3-17. Signs of power and the direction of energy transfer.
3-38. Determining the parameters of a passive two-terminal network using an ammeter, voltmeter and wattmeter.
3-19. Conditions for transferring maximum power from an energy source to a receiver.
3-20. The concept of the surface effect and the effect of proximity.
3-21. Parameters and equivalent circuits of capacitors.
3-22. Parameters and equivalent circuits of inductive coils and resistors.
Chapter 4
Calculation of circuits at sinusoidal currents.
4-1. On the applicability of circuit calculation methods direct current to the calculations of sinusoidal current circuits.
4-2. Serial connection of receivers.
4-3. Parallel connection of receivers.
4-4. Mixed connection of receivers.
4-5. Complex branched chains.
4-6. Topographic charts.
4-7. The duality of electrical circuits.
4-8. Signal graphs and their application for the calculation of chains.
Chapter 5
Resonance in electrical circuits
5-1. Resonance in an unbranched circuit.
5-2. Frequency characteristics of an unbranched circuit.
5-3. Resonance in a circuit with two parallel branches.
5-4. Frequency characteristics of the parallel circuit.
5-5. The concept of resonance in complex circuits.
Chapter 6
Circuits with mutual inductance.
6-1. Inductively coupled circuit elements.
6-2. Electromotive force of mutual induction.
6-3. Series connection of inductively coupled circuit elements.
6-4. Parallel connection of inductively coupled circuit elements.
6-5. Calculations of branched circuits in the presence of mutual inductance.
6-6. Equivalent replacement of inductive connections.
6-7. Transfer of energy between inductively coupled circuit elements.
6-8. Transformer without steel core (air transformer).
Chapter 7
Pie charts.
7-1. Complex Equations straight line and circle.
7-2. Pie-diagrams for an unbranched circuit and for an active two-terminal network.
7-3. Pie charts for any branched chain.
Chapter 8
Multipole and quadripole networks with sinusoidal currents and voltages.
8-1. Quadripoles and their basic equations.
8-2. Determination of coefficients of quadripoles.
8-3. Quadripole mode under load.
8-4. Equivalent circuits of quadripoles.
8-5. Basic equations and equivalent circuits for an active quadripole.
8-6. An ideal transformer is like a four-pole.
8-7. Equivalent circuits with ideal transformers for a four-terminal network.
8-8. Equivalent circuits of a transformer with a steel magnetic core.
8-9. Calculations of electrical circuits with transformers.
8-10. Graphs of passive quadripoles and their simplest connections.
Chapter 9
Circuits with electronic and semiconductor devices in linear mode.
9-1. Tube triode and its parameters.
9-2. Equivalent circuits of a tube triode.
9 3. Transistors (semiconductor triodes).
9 4. Equivalent circuits of transistors.
9 5. The simplest electrical circuits with non-reciprocal elements and their directed graphs.
Chapter 10
Three-phase circuits
10-1. The concept of multi-phase power supplies and multi-phase circuits.
10-2. Star and polygon connections.
10-3. Symmetric mode of a three-phase circuit.
10-4. Some properties of three-phase circuits with different connection schemes.
10-5. Calculation of symmetrical modes of three-phase circuits.
10-6. Calculation of asymmetric modes of three-phase circuits with static load.
10-7. Voltages on the phases of the receiver in some special cases.
10-8. Equivalent circuits of three-phase lines.
10-9. Power measurement in three-phase circuits.
10-10. Rotating magnetic field.
10-11. Principles of operation of asynchronous and synchronous motors.
Chapter 11
Method of symmetrical components.
11-1. Symmetrical components of the three-phase system of quantities.
11-2. Some properties of three-phase circuits in relation to the symmetrical components of currents and voltages.
11-3. Resistances of a symmetrical three-phase circuit for currents of various sequences.
11-4. Determination of currents in a symmetrical circuit.
11-5. Symmetrical components of voltages and currents in an asymmetrical three-phase circuit.
11-6. Calculation of a circuit with an asymmetric load.
11-7. Calculation of a circuit with an asymmetrical section in the line.
Chapter 12
non-sinusoidal currents.
12-1. Non-sinusoidal e. d.s., voltages and currents.
12-2 Decomposition of a periodic non-sinusoidal curve into a trigonometric series.
12-3. Maximum, effective and average values ​​of non-sinusoidal periodic e. d.s., voltages and currents.
32-4. Coefficients characterizing the shape of non-sinusoidal periodic curves.
12-5. Non-sinusoidal curves with a periodic envelope.
12-6. Valid values ​​e. d.s., voltages and currents with periodic envelopes.
12-7. Calculation of circuits with non-sinusoidal periodic e. d.s. and currents.
12-8. Resonance at non-sinusoidal e. d.s. and currents.
12-9. Power of periodic non-sinusoidal currents.
12-10. Higher harmonics in three-phase circuits.
Chapter 13
Classical method for calculating transients
13-1. The emergence of transient processes and the laws of switching.
13-2. Transitional, forced and free processes.
13-3. Short circuit R, L.
13-4. Turning on the circuit to, L to a constant voltage.
13 5. Turning on the circuit r, L on a sinusoidal voltage.
13-6. Short circuit g, C.
13-7. Turning on the circuit r, C to a constant voltage.
13-8. Turning on the circuit g, C to a sinusoidal voltage.
13-9. Transients in an unbranched circuit r, L, C.
13-10. Aperiodic discharge of a capacitor.
13-11. Limiting case of aperiodic capacitor discharge.
13-12. Periodic (oscillatory) discharge of a capacitor.
13-13. Turning on the circuit r, L, C for a constant voltage.
13-14. The general case of calculating transient processes by the classical method.
13-15. Turning on a passive two-terminal network for a continuously changing voltage (Duhamel's formula or integral).
13-16. Turning on a passive two-terminal network for voltage of any form.
13-17. Temporal and impulse transient characteristics.
13-18. Writing the convolution theorem using the impulse response.
13-19. Transient processes during current surges in inductors and voltages across capacitors.
13-20. Determination of the transient process and steady state under the influence of periodic voltage or current pulses.
Chapter 14
Operator method for calculating transient processes.
14-1. Application of the Laplace transform to the calculation of transients.
14-2. Ohm's and Kirchhoff's laws in operator form.
14-3. Equivalent operator schemes.
14-4. Transient processes in circuits with mutual inductance.
34-5. Reduction of calculations of "transitional processes to zero initial conditions.
14-6. Determination of free currents by their images.
14-7. Inclusion formulas.
14-8. Calculation of transient processes by the method of state variables.
14-9. Determination of the forced mode of the circuit when exposed to a periodic non-sinusoidal voltage.
Chapter 15
Frequency method for calculating transient processes.
15-1. Fourier transform and its main properties.
15-2. Ohm's and Kirchhoff's laws and equivalent circuits for frequency spectra.
15-3. Approximate method for determining the original by real frequency response (trapezoidal method).
15-4. On the transition from Fourier transforms to Laplace transforms.
15-5. Comparison various methods calculation of transient processes in linear electrical circuits.
Chapter 16
Chain circuits and frequency electrical filters.
Characteristic impedances and constant transmission of an asymmetric quadripole.
Characteristic impedance and transmission constant of a symmetrical quadripole.
Inserted and working permanent transmissions.
Chain schemes.
Frequency electrical filters.
Low frequency filters.
high frequency filters.
Band filters.
Barrier filters.
Constant M filters.
L-shaped filter as an example of a single-ended filter. Non-inductive (iln r, C) filters.
Chapter 17
Synthesis of electrical circuits.
17-1. general characteristics synthesis tasks.
17-2. Transfer function of a quadripole. Chains of the minimum phase.
17-3. Input functions of circuits. Positive real functions.
17-4. Reactive bipolar.
17-5. Frequency characteristics of reactive two-terminal networks.
17-6. Synthesis of reactive two-terminal networks. Foster method.
17-7. Synthesis of reactive two-terminal networks. Cauer method.
17-8. Synthesis of two-terminal networks with losses. Foster method.
17-9. Synthesis of two-terminal networks with losses. Cauer method.
17-10. The concept of the synthesis of quadripoles.
Section 2. LINEAR CIRCUITS WITH DISTRIBUTED PARAMETERS.
Chapter 18
Harmonic processes in chains with distributed parameters.
18-1. Currents and voltages in long lines.
18-2. Equations of a homogeneous line.
18-3. Steady state in a homogeneous line.
18-4. Homogeneous line equations with hyperbolic functions.
18-5. Characteristics of a homogeneous line.
18-6. Line input impedance.
18-7. Wave reflection coefficient.
18-8. Matched line load.
18-9. Line without distortion.
18-10. Idling, short circuit and load mode of the line with losses.
18-11. Lossless lines.
18-12. standing waves.
18-13. The line is like a quadripole.
Chapter 19
Transient processes in circuits with distributed parameters.
19-1. Occurrence of transient processes in circuits with distributed parameters.
19-2. Common decision homogeneous line equations.
19-3. The emergence of waves with a rectangular front.
19-4. Common cases finding waves arising during switching.
19-5. Reflection of a wave with a rectangular front from the end of the line.
19-6. General Method determination of reflected waves.
19-7. Qualitative consideration of transient processes in lines containing lumped capacitances and inductances.
19-8. Multiple will reflections with a rectangular front from active resistance.
19-9. Wandering waves.
Section 3 Nonlinear circuits.
Chapter 20
Nonlinear electrical circuits at direct currents and voltages.
20-1. Elements and equivalent circuits of the simplest non-linear circuits.
20-2. Graphic method calculation of unbranched circuits with non-linear elements.
20-3. Graphical method for calculating circuits with parallel connection of non-linear elements.
20-4. Graphical method for calculating circuits with a mixed connection of non-linear and linear elements.
20-5. Application of equivalent circuits with sources e. d.s. to study the regime of nonlinear circuits.
20-6. Volt-ampere characteristics of non-linear active two-pole networks.
20-7. Examples of calculation of branched electrical circuits with non-linear elements.
20-8. Application of the theory of active two-pole, four-pole and six-pole for the calculation of circuits with linear and non-linear elements.
20-9. Calculation of branched non-linear circuits iterative method(method of successive approximations).
Chapter 21
Magnetic circuits at direct currents.
21-1. Basic concepts and laws of magnetic circuits.
21-2. Calculation of unbranched magnetic circuits.
21-3. Calculation of branched magnetic circuits.
21-4. Calculation of the magnetic circuit of the ring permanent magnet with air gap.
21-5. Calculation of an unbranched inhomogeneous magnetic circuit with a permanent magnet.
Chapter 22
General characteristics of nonlinear AC circuits and methods for their calculation
22-1. Nonlinear two-terminal networks and quadripoles at alternating currents.
22-2. Determination of operating points on the characteristics of non-linear two-terminal and quadripole networks.
22-3. Phenomena in non-linear alternating current circuits.
22-4. Methods for calculating non-linear AC circuits.
Chapter 23
Nonlinear circuits with sources of e. d.s. and current of the same frequency.
23-1. General characteristics of circuits with sources of e. d.s. the same frequency.
23-2. The shape of the current curve in a circuit with valves.
23-3. The simplest rectifiers.
23-4. Waveforms of current and voltage in circuits with non-linear reactances.
23-5. Frequency triplers.
23-6. Forms of current and voltage curves in circuits with thermistors.
23-7. Replacing real non-linear elements with conditionally non-linear ones.
23-8. Accounting for the real properties of steel magnetic cores.
23-9. Calculation of the current in a coil with a steel magnetic circuit.
23-10. The concept of the calculation of conditionally nonlinear magnetic circuits.
23-11. The phenomenon of ferroresonance.
23-12. Surge Protectors.
Chapter 24
Nonlinear circuits with sources of e. d.s., and currents of various frequencies.
24-1. General characteristics of nonlinear circuits with sources of e. d.s. different frequencies.
24-2. Valves in circuits with constant and variable e. d.s.
24-3. Controlled valves in the simplest rectifiers and DC-to-AC converters.
24-4. Coils with steel magnetic circuits in circuits with constant and variable e. d.s.
24-5. frequency doubler.
24-6. Harmonic balance method.
24-7. The influence of the constant e. d.s. on the variable component of the current in circuits with non-linear inertial resistances.
24-8. The principle of obtaining modulated oscillations.
24-9. The influence of the constant component on the variable in circuits with non-linear inductances.
24-10. Magnetic power amplifiers.
Chapter 25
Transient processes in nonlinear circuits.
25-1. General characteristics of transient processes in nonlinear circuits.
25-2. Switching on a coil with a steel magnetic circuit for direct voltage.
25-3. Turning on a coil with a steel magnetic circuit for a sinusoidal voltage.
25-4. Impulse action in circuits with ambiguous nonlinearities.
25-5. The concept of simple storage devices.
25-6. Image of transients on the phase plane.
25-7. Oscillatory capacitance discharge through a non-linear inductance
Chapter 26
Self-oscillations
26-1. Nonlinear resistors with a falling section of the characteristic.
26-2. The concept of mode stability in a circuit with non-linear resistors.
26-3. Relaxation oscillations in a circuit with negative resistance
26-4. Close to sinusoidal oscillations in a circuit with negative resistance.
26-5. Phase trajectories of processes in a circuit with negative resistance.
26-6. Phase trajectories of processes in the generator of sinusoidal oscillations.
26-7. Determination of the amplitude of self-oscillations by the method of harmonic balance.
Applications.
Bibliography.
Subject index.

electrical circuit a set of devices designed for the transmission, distribution and mutual conversion of electrical (electromagnetic) and other types of energy and information is called, if the processes occurring in the devices can be described using the concepts of electromotive force (emf s), current and voltage
The main elements of the electrical circuit are sources and receivers. electrical energy(and information) that are interconnected by wires.

In sources of electrical energy ( galvanic cells, batteries, electric machine generators, etc.) chemical, mechanical, thermal energy or other types of energy is converted into electrical energy, receivers of electrical energy (electrothermal devices, electric lamps, resistors, electric motors etc.), on the contrary, electrical energy is converted into heat, light, mechanical, etc.
Electrical circuits in which the receipt of electrical energy in sources, its transmission and conversion in receivers occur at currents and voltages that are constant in time, are usually called DC circuits.

This article is for those who are just starting to study the theory of electrical circuits. As always, we will not go into the jungle of formulas, but we will try to explain the basic concepts and the essence of things that are important for understanding. So, welcome to the world of electrical circuits!

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Electrical circuits

is a set of devices through which electric current flows.

Consider the simplest electrical circuit. What does it consist of? It has a generator - a current source, a receiver (for example, a light bulb or an electric motor), as well as a transmission system (wires). In order for a circuit to become a circuit, and not a set of wires and batteries, its elements must be interconnected by conductors. Current can only flow in a closed circuit. Let's give another definition:

- These are interconnected current source, transmission lines and receiver.

Of course, source, sink and wires is the simplest option for an elementary electrical circuit. In reality, different chains include many more elements and auxiliary equipment: resistors, capacitors, knife switches, ammeters, voltmeters, switches, contact connections, transformers and more.

Classification of electrical circuits

By appointment, electrical circuits are:

• Power electrical circuits;
• Electrical control circuits;
• Electrical measurement circuits;

Power circuits designed for the transmission and distribution of electrical energy. It is the power circuits that conduct current to the consumer.

Also, the circuits are divided according to the strength of the current in them. For example, if the current in the circuit exceeds 5 amperes, then the circuit is power. When you click on the kettle plugged into the outlet, you close the power circuit.

Electrical control circuits are not power and are designed to actuate or change the operation parameters of electrical devices and equipment. An example of a control circuit is monitoring, control and signaling equipment.

Electrical measurement circuits designed to record changes in the parameters of electrical equipment.

Calculation of electrical circuits

To calculate a circuit means to find all the currents in it. There are different methods for calculating electrical circuits: Kirchhoff's laws, the method of loop currents, the method of nodal potentials, and others. Consider the application of the method of loop currents on the example of a specific circuit.

First, we select the circuits and denote the current in them. The direction of the current can be chosen arbitrarily. In our case, clockwise. Then for each contour we will compose equations according to the 2nd Kirchhoff law. The equations are compiled as follows: The loop current is multiplied by the loop resistance, the products of the current of other loops and the total resistances of these loops are added to the resulting expression. For our schema:

The resulting system is solved by substituting the initial data of the problem. The currents in the branches of the original circuit are found as the algebraic sum of the loop currents

Definition 1

The theory of electrical circuits is considered to be a complex of the most general patterns, which is used to describe processes in electrical circuits.

The theory of electrical circuits is based on two postulates:

• the initial assumption of the theory of electrical circuits (implies that in any electrical devices all processes can be described by such concepts as "voltage" and "current");
• the initial assumption of the theory of electrical circuits (assumes that the current strength at any point of the conductor cross section will be the same, while the voltage between the two points taken in space will change according to a linear law).

Basic concepts in the theory of electrical circuits

The electrical circuit consists of:

• current sources (generators);
• consumers of electromagnetic energy (receivers).

Remark 1

A source is a device that creates currents and voltages. As such, devices such as batteries, generators, oriented to the conversion different types energy (chemical, thermal, etc.) into electrical energy.

The theory of electrical circuits is based on the principle of modeling. At the same time, real electrical circuits are replaced by some idealized model, which consists of interconnected elements.

Definition 2

Elements are understood as idealized models different devices, which are assigned certain electrical properties with a display with a given accuracy of phenomena occurring in real devices.

Passive elements in the theory of electric circuit

Passive elements in the theory of electrical circuits include resistance, which represents its idealized element, which will characterize the conversion of electromagnetic energy into any other form of energy, which implies its possession exclusively of the property of irreversible energy dissipation. The model that mathematically describes the properties of resistance is determined by Ohm's law:

Here $R$ and $G$− are the parameters of the section of the circuit, which are called resistance and conductivity, respectively.

Instantaneous power that goes into the resistance:

Definition 3

A real element, in its properties approaching resistance, is called a resistor.

Inductance is an idealized electrical circuit element that characterizes the energy magnetic field stored on the network. The capacitance is an idealized electrical circuit element that characterizes the energy electric field.

Active elements in the theory of electric circuit

The active elements in the theory of electrical circuits include the source of EMF. An idealized current source, or current generator, is an energy source whose current will not be dependent on the voltage at its terminals.

In the case of an unlimited increase in the resistance of a circuit connected to an ideal source of electric current, the power developed by it and, accordingly, the voltage at its terminals will also increase indefinitely. A finite power current source is depicted in the format of an ideal source with an internal resistance connected in parallel.

It is important that the input terminals of voltage controlled sources are open, while those of current controlled sources are short-circuited.

There are 4 types of dependent sources:

• a voltage source that is controlled by voltage (INUN);
• current controlled voltage source (INUT);
• voltage controlled current source (ITUN);
• a current source that is controlled by current (ITUT).

In INUN, the input resistance will be infinitely large, and the output voltage is associated with the input equality $U_2=HUU_1$, where $HU$ is the voltage transfer coefficient. INUN is considered an ideal voltage amplifier.

In INUT, the input current is controlled by the output voltage $U_2$, while the input conductivity is infinitely large:

Where $HZ$ is the transfer resistance.

In ITUN, the output current $I_2$ is controlled respectively by the input voltage $U_1$, where $I_1=0$ and the current $I_2$ is related to $U_1$ by the equation $I_2=HYU_1$, where $HY$ is the transfer conductivity.

In ITUT, the control current is $I_1$, and the controlled current is $I_2$. $U_1=0$, $I_2=HiI_1$, where $Hi$ is the current transfer coefficient. ITUT presents an ideal current amplifier.

Description of work and calculation (simulation) of electrical devices can be carried out on the basis of the electromagnetic field theory. This approach leads to complex mathematical models (systems of partial differential equations) and is mainly used in the analysis of microwave devices and antennas.

It is much easier and more convenient to model electrical devices based on the equations of electrical equilibrium of currents and voltages. On this basis built electrical circuit theory.

Charge, current, voltage, power, energy

electric charge called the source of the electric field through which the charges interact with each other. Electric charges can be positive (ions) or negative (electrons and ions). Opposite charges attract, and like charges repel. The amount of charge is measured in coulombs (K).

The magnitude (strength) of the current is equal to the ratio of an infinitesimal charge (amount of electricity)
transferred to this moment time through the cross section of the conductor in an infinitesimal time interval
to the size of this interval,

. (1.1)

The current is measured in amperes (A), values ​​in milliamps (1 mA = 10 -3 A), microamps (1 μA = 10 -6 A) and nanoamps (1 nA = 10 -9 A) are widely used in technology, are given in Appendix 1.

Electrical potentialsome point is a value equal to the ratio of potential energy , which has a charge at this point, to the magnitude of the charge,

. (1.2)

Potential energy is equal to the energy spent on the transfer of charge from a given point with a potential to a point with zero potential.

If a is the potential of point 2, and - points 1, then tension

the distance between points 2 and 1 is

. (1.3)

Voltage is measured in volts (V), using values ​​in kilovolts (kV), millivolts (mV), and microvolts (µV).

Current and voltage are characterized by the direction indicated by an arrow, as shown in fig. 1.1. They are set arbitrarily. before the start of settlements . It is desirable that the current and voltage for one circuit element would have the same polo-

Rice. 1.1 residential directions. Designations can

have indexes, e.g. voltage
between points 1 and 2 in fig. 1.1.

The numerical values ​​of current and voltage are characterized by a sign. If the sign is positive, then this means that the true positive direction is the same as the given one, otherwise they are opposite.

The movement of charges in an electric circuit is characterized by energy and power. To move an infinitesimal charge
between points 1 and 2 with voltage
in the circuit in Fig. 1.1 it is necessary to expend infinitesimal energy
equal to

, (1.4)

then the energy of the circuit in the time interval from before taking into account (1.1) is determined by the expression

. (1.5)

At direct current
and voltage
the energy is equal and increases indefinitely with time. This also applies to the general expression (1.5), which makes the energy of the circuit a rather inconvenient technical characteristic.

Instant Power
time dependent and may positive(the circuit consumes energy from the outside) and negative(the circuit gives off previously accumulated energy).

Average power always non-negative if there are no sources of electrical energy inside the circuit.

Energy is measured in joules (J), while instantaneous and average power is measured in watts (W).

1.3. Electrical circuit elements

An element is an indivisible part of an electrical circuit. In the physical circuit (radio receiver) there are physical elements (resistors, capacitors, inductors, diodes, transistors, etc.). They have complex properties and the mathematical apparatus for their exact description based on the theory of the electromagnetic field.

When calculating an electrical circuit, it is necessary to develop sufficiently accurate, simple and convenient from an engineering point of view. models physical elements, which we will refer to as elements.

Engineering models in electrical engineering are built on the basis of physical concepts of the relationship between current and voltage in them. The properties of resistive two-pole (with two terminals) elements are described current-voltage characteristics (VAC)- the dependence of the current through the element from the voltage applied to it . This dependence can be linear (for a resistor in Fig. 1.2a) or non-linear (for a semiconductor diode in Fig. 1.2b).

Elements with a rectilinear CVC are called linear, otherwise - non-linear. Similarly, capacitive elements are considered, for which the pendant-voltage characteristic is used (dependence of the accumulated charge on the applied voltage), and inductive elements using the weber-ampere characteristic (dependence magnetic flux from the current flowing through the element).

1.4. Models of the main linear elements of the circuit

The main linear elements of an electrical circuit are a resistor, a capacitor, and an inductor. Their conventional graphic designations are shown in fig. 1.3 (names of physical elements are indicated above, and their models are indicated below).

Resistance (resistor model) in accordance with fig. 1.4 is built on the basis of Ohm's law in the classical formulation,

, (1.10)

G de is a model parameter called resistance, a -conductivity,

. (1.11)

Rice. 1.4

As can be seen from (1.10), resistance is a linear element (with a rectilinear CVC). Its parameter is resistance - measured in Ohms (Ohm) or off-system units - kiloohms (kOhm), megaohms (Mohm) or gigaohms (GOhm). Conductivity is determined by expression (1.11), is inverse to the resistance and is measured in 1/Ohm. Element resistance and conductivity do not depend on current and voltage values.

In resistance, current and voltage are proportional to each other, have the same shape.

The instantaneous power of the electric current in the resistance is

As you can see, the instantaneous power in the resistance cannot be negative, that is, the resistance is always consumes power (energy), converting it into heat or other forms, for example, into electromagnetic radiation. Resistance is a model of a dissipative element that dissipates electrical energy.

Capacitance (capacitor model) in accordance with Fig. 1.5 is formed based on the fact that the charge accumulated in it is proportional to the applied voltage,

. (1.13)

Model parameter - capacity- does not depend

Rice. 1.5 of current and voltage and is measured in farads

(F). The capacitance value of 1 F is very large, in practice values ​​in microfarads (1 μF = 10 -6 F), nanofarads (1 nF = 10 -9 F) and picofarads (1 pF = 10 -12 F) are widely used.

Substituting (1.13) into (1.1), we obtain model for instantaneous values ​​of current and voltage

.

From (1.14) we can write the inverse expression for the model,

The instantaneous electric power in the tank is equal to

. (1.16)

If the voltage is positive and increases with time (its derivative Above zero), then the instantaneous power positive and capacity accumulates the energy of the electric field. A similar process takes place if the voltage is negative and continues to decrease.

If the capacitance voltage is positive and falls (negative and grows), then the instantaneous power negative, and the capacity gives to the external circuit previously stored energy.

Thus, a container is an element that accumulates electrical energy (like a jar in which water accumulates and from which it can pour out), there are no energy losses in the tank.

The energy accumulated in the tank is determined by the expression

Inductance (inductor model) is formed based on the fact that the flux linkage
equal to the product of the magnetic flux (in webers) per number of coil turns is directly proportional to the current flowing through it. (Fig. 1.6),

, (1.18)

where is a model parameter called inductance and is measured in henries (H).

Rice. 1.6 The value of 1 H is a very large in-

inductance, therefore off-system units are used: millihenry (1 mH = 10 -3 H), microhenry (1 μH = 10 -6 H) and nanohenry (1 nH = 10 -9 H).

A change in the flux linkage in an inductor causes electromotive force(emf) self-induction
equal to

(1.19)

and directed opposite to current and voltage, then
and the model of the inductor for instantaneous current and voltage takes the form

You can write the inverse expression of the model,

The instantaneous electrical power in the inductance is

. (1.22)

If the current is positive and rising, or negative and falling, then the instantaneous power positive and inductance accumulates the energy of the magnetic field. If the inductance current is positive and falls (negative and rises), then the instantaneous power negative, and inductance gives to the external circuit previously stored energy.

Thus, inductance (like capacitance) is an element that only accumulates energy, there is no energy loss in the inductance.

The energy stored in the inductor is

Ohm's laws for circuit elements

The considered models of electrical circuit elements, which determine the relationship between the instantaneous values ​​of currents and voltages, will be further called Ohm's laws for circuit elements, although Ohm's law itself applies only to resistance.

These ratios are summarized in Table. 1.1. They are linear mathematical operations and only apply to linear elements.

In non-linear elements, the connection between current and voltage is much more complicated and, in general, can be described by non-linear integro-differential equations, for which there are no general solution methods.

Table 1.1

Ohm's laws in circuit elements for instantaneous values ​​of current and voltage

Calculation of current and voltage in circuit elements

As an example, we will calculate the voltage on the circuit elements for a given dependence of the current on time, shown in fig. 1.7.

Mathematically, this relationship can be written

Rice. 1.7 as

(1.24)

It must be remembered that in (1.24) the time measured in milliseconds and the current - milliamps.

Then in the one shown in Fig. 1.4. resistance at
kΩ voltage is
(Fig. 1.8a) and power
(Fig. 1.8b). The shapes of the time diagrams of current and voltage in the resistance coincide, and the product of two straight-line dependencies
and
gives parabolic power curves
.

In a container (Fig. 1.5)
µF instantaneous values ​​of current and voltage are interconnected by expressions (1.14) or (1.15). For current (Fig. 1.7) of the form (1.24) from

(1.25)

we get the formula for the voltage across the capacitance in volts

(1.26)

Calculation at
1 ms is obvious. At

integral (1.25) is written in the form

(1.27)

On the time interval
ms integral (1.25) has the form

and is a constant. timing diagram
shown in fig. 1.9. As can be seen, in the time interval
ms, while the current pulse is active, the capacitor is charged, and then the voltage of the charged capacitance does not change. On fig. 1.10a shows the time dependence of the instantaneous power

Rice. 1.9 (1.16), and in fig. 1.10b - accumulation

lenoy in energy capacity
(1.17). As you can see, the capacitance only accumulates energy, since the discharge does not occur (the current of the form in Fig. 1.7 takes only positive values).

To get the power formula
it is necessary to multiply expressions (1.24) and (1.26) by the corresponding

time intervals (we get a polynomial of the third degree ).

Energy
is determined from (1.17) by substituting (1.26), which leads to polynomials of the fourth degree .

For inductance fig. 1.6
H at the current shown in fig. 1.7 voltage
is determined by expression (1.20)

, (1.29)

then after substituting (1.24) for
in volts we get

(1.30)

This dependence is shown in fig. 1.11. When graphical differentiation of rectilinear dependences in fig. 1.7 we obtain constants on the corresponding time intervals, which corresponds to Fig. 1.11.

Power is determined by expression (1.22), then for
in milliwatts we get

(1.31)

Addiction
shown in fig. 1.12a. The energy accumulated in the inductance is calculated by the formula (1.23), then the graph
has the form shown in Fig. 1.12b.

As can be seen, the instantaneous power increases in direct proportion with increasing current in the time interval from 0 to 1 ms, and the energy accumulated in the inductance grows according to a quadratic law. When the current starts to drop at
, then the voltage
and power
become negative (Fig. 1.11 and Fig. 1.12a), which means that the inductance gives off the previously accumulated energy, which begins to decrease according to a quadratic law (Fig. 1.12b).

The calculation of signals and energy characteristics in the circuit elements R, L and C can be carried out using the MathCAD program.

Ideal Signal Sources

Electrical signals (currents and voltages) occur in the circuit when exposed to sources. Physical sources are batteries and accumulators that generate direct current and voltage, alternating voltage generators various shapes and other electronic devices. A voltage (potential difference) appears on their clamps (poles) and current flows through them due to electrochemical processes or other complex physical phenomena. In physics, their generalized action is characterized electromotive force (EMF).

To calculate electrical circuits, you need models signal sources. The simplest of them are ideal springs.

A graphical representation (designation) of an ideal voltage source is shown in fig. 1.13 in the form of a circle with an arrow indicating the positive direction of the EMF
. A voltage is applied to the poles of the source
, which for the indicated positive directions is equal to the EMF,

(1.32)

If we change the positive

direction of emf or voltage (make them counter), will appear in the formula minus sign.

A load is connected to the source and then current flows through it
. Source properties permanent voltage or current are described by it current-voltage characteristic (VAC)– dependence of current on voltage
. An ideal voltage source with an emf equal to has a current-voltage characteristic shown in Fig. 1.14. If an AC signal source is considered, then from the current all its para-

Rice. 1.14 meters.

As can be seen, with increasing current at constant voltage the power delivered by an ideal voltage source to the load tends to infinity. This is a consequence of the chosen ideal model(VAC shape) and its disadvantage, since any physical source cannot deliver infinite power.

Graphical representation of an ideal current source
shown in fig. 1.15a in the form of a circle, inside which the positive direction of the current is indicated. When a load is connected, a voltage appears on the poles of the source
with the indicated positive direction.

On fig. 1.15b shows the CVC of an ideal DC source . And for this model, with increasing voltage, the power given by the source to the load tends to infinity.

1.8. Fundamentals of the topological description of the circuit

electrical circuit called a set of interconnected sources, consumers and converters of electrical energy, the processes in which are described in terms of current and voltage.

A physical electrical circuit (electronic device) consists of physical elements - resistors, capacitors, inductors, diodes, transistors and a large number of others. electronic elements. Each of them has a conventional graphic designation in accordance with the standard - unified system design documentation (ESKD). The connection of these elements to each other is graphically represented circuit diagram circuits (filter, amplifier, TV). Example circuit diagram transistor amplifier is shown in fig. 1.16.

Now we will not discuss the operation of the amplifier and

the meaning of its elements, but we only note the conditional graphic designations of the elements used, which are shown separately in Fig. 1.17. The bold dot marks the electrical connections of the elements.

Rice. 1.17 As you can see, graphic

the designations of the resistor and capacitor coincide with the designations of their models - resistance and capacitance, while the designations of others are different.

They are used to calculate circuits. equivalent circuits or equivalent circuits, which show the connections of models of elements that form an electrical circuit. Each physical element of the circuit diagram is replaced by a corresponding model, which may consist of one or more of the simplest ideal models (resistance, capacitance, inductance, or signal sources). Examples of models of physical elements are shown in fig. 1.18.

The resistor and capacitor are most often presented as their ideal models with the same conventional graphic symbols. An inductor can be represented by an ideal inductance, but in some cases it is necessary to take into account its loss resistance . In this case, the inductor model is represented by a series connection of an ideal inductance and resistance, as shown in Fig. 1.18.

On fig. 1.19 as an example, a schematic diagram of a parallel connection of an inductor and a capacitor is shown (such a circuit is called parallel oscillatory circuit) and the equivalent circuit of this circuit (the inductor is replaced by

nena follower-

connection 1.19

ideal inductive

ness and resistance).

The equivalent circuit of a circuit is its topological description. From a geometric point of view, the following main elements can be distinguished in it:

AT etv- serial connection of several, including one, bipolar elements, including signal sources;

- node- connection point of three or more branches;

- circuit- a closed connection of two or more branches.

On fig. 1.20 shows an example of an equivalent circuit circuit with the designation of branches, nodes (thick dots) and contours (closed lines). As you can see, a node can represent

is not a single connection point, but several (a distributed node enclosed by a dotted line).

In the theory of circuits, the number of nodes of the equivalent circuit is essential and the number of branches . For the circuit in fig. 1.20 available
nodes and
branches, one of which contains only an ideal current source.

1.9. Chain element connections

Two-pole elements of an electrical circuit can be interconnected in various ways. There are two simple connections: serial and parallel.

Consistent They call such a connection of two-terminal networks, in which the same current flows through them. His example is shown in Fig. 1.21. The circuit in Fig. 1.21 includes passive (R&C) and active (ideal voltage sources
and
) ele-

Rice. 1.21

delivers the same current
.

AT complex chain(for example, in Fig. 1.20) you can select simple fragments (branches) with serial connection elements (branch with source
, passive branches
and
).

Doesn't make sense connect in series two ideal current sources or an ideal voltage source with an ideal current source.

Parallel call the connection of two or more branches with the same pair of nodes, while the voltages on the parallel branches are the same. An example is shown in fig. 1.22. If the branches contain one element each, then they speak of a parallel connection of elements. For example, in fig. 1.22 ideal current source
and resistance Fig. 1.22

connected in parallel.

Doesn't make sense connect in parallel an ideal voltage source or an ideal voltage source with an ideal current source.

mixed call the connection of elements (branches) of the circuit, which cannot be considered as serial or parallel. For example, the diagram in Fig. 1.21 is a series connection of elements, and in fig. 1.22 - parallel connection of branches, although in the branches
and
elements are connected in series.

The scheme in fig. 1.20 is a typical representative of a mixed compound, and only separate fragments with simple compounds can be distinguished in it.

1.10. Kirchhoff's laws for instantaneous signal values

The two Kirchhoff laws establish electrical equilibrium equations between currents in the nodes and voltages in the contours of the circuit.

Algebraic summation is understood as the addition or subtraction of the corresponding quantities.

Another formulation of Kirchhoff's first law can also be used: the sum of the instantaneous values ​​of the currents flowing into the node is equal to the sum of the instantaneous values ​​of the outgoing currents.

An example circuit diagram is shown in fig. 1.23, it repeats the scheme in fig. 1 20 indicating positive directions and designations of currents and voltages in all elements, as well as node numbers (in circles).

There are four nodes in the circuit, and for each of them it is possible to write the equation of the first Kirchhoff law for the instantaneous values ​​of the branch currents,

Node 1:
;

Node 2:
;

Node 3:
.

It is easy to see that if we sum up the equations for the knots
and multiply the result by -1, then we get the equation for node 0. Therefore, one of the equations (any) is linearly dependent on the others, and must be excluded. Thus, the system of equations according to the first Kirchhoff law for the circuit in Fig. 1.23 can be written as

Obviously, other versions of this system of equations can be written, but they will all be equivalent.

The physical justification for Kirchhoff's first law is the principle of non-accumulation of charge in a chain node. At any moment in time, the charge entering the node from the incoming currents must be equal to the charge leaving the node due to the outflowing currents.

To select signs in algebraic sums, you must specify positive direction of contour traversal(Mostly chosen clockwise). Then, if the direction of voltage or EMF coincides with the direction of bypass, then in algebraic sum a plus sign is written, otherwise a minus sign.

Independent called contours that differ from each other by at least one branch.

In the diagram in fig. 1.23
,
(one branch contains an ideal current source) and
. Then it has
independent contours. As can be seen, the total number of contours is much larger .

We choose the following independent contours:

C 1 ,R 2 ,C 2 ,C 3 ,

C 3 R 3 ,L,R 4 ,

with a positive clockwise bypass direction and for them we write the equations of the second Kirchhoff law in the form

(1.34)

You can also choose other independent circuits, for example,

C 1 ,R 2 ,C 2 ,C 3 ,

E,R 1 ,R 2 ,C 2 ,C 3 ,

and for them write down the equations of the second Kirchhoff law, which will be equivalent to the system (1.34).

Kirchhoff's second law is based on a fundamental law of nature - the law of conservation of energy. The sum of voltages on the elements of a closed circuit is equal to the work of transferring a unit charge in the passive elements of the circuit, and the sum of the EMF is equal to the work of external forces in ideal voltage sources to transfer the same unit charge into them. Since as a result the charge returned to the starting point, these works should be the same.

1.11. Real signal sources

The ideal sources of voltage and current considered above are not always suitable for the formation of adequate models of electronic devices. The main reason for this is the ability to transfer infinite power to the load. In this case, complicated models of signal sources are used, which are called real.

The equivalent circuit (model) of a real voltage source is shown in fig. 1.24. It contains an ideal voltage source
and internal resistance real-

n source . A load resistance is connected to the source
. According to Kirchhoff's second law, we can write

, (1.35)

and according to Ohm's law for resistance

Rice. 1.24 leniya

. (1.36)

Substituting (1.36) into (1.35) we get

,

whence follows the equation for the current-voltage characteristic of a real voltage source

, (1.37)

the graph of which for constant values ​​of current and voltage is shown in fig. 1.25. The dotted line shows the current-voltage characteristic of an ideal voltage source. As you can see, in a real source, the maximum current limited, a

Rice. 1.25 means the power given off by it is not

may be endless.

At a constant voltage, the power given by a real source (Fig. 1.24) to the load is equal to

. (1.38)

Addiction
at
In and
Ohm is shown in fig. 1.26. As you can see, the maximum power of a real source is limited.

chena and equal
at
. Rice. 1.26

Current-voltage characteristic of a real voltage source at
tends to characterize the ideal source fig. 1.14. Thus, one can define an ideal voltage source as real source fromzero internal resistance(internal resistance of an ideal voltage source zero).

The equivalent circuit of a real current source is shown in fig. 1.27. It contains an ideal current source and internal resistance , the load is connected to the source
. The equation of the first Kirchhoff law for one of the nodes of the chain fig. 1.27 has the form

. (1.39) Fig. 1.27

Ohm's law
, then from (1.39) we obtain an expression for the current-voltage characteristic of a real current source

. (1.40)

For direct current, this dependence is shown in fig. 1.28. As you can see, the maximum voltage supplied by the source to the load is limited by the value
with infinite load resistance. Power constant

Rice. 1.28 current given to the load is equal to

. (1.41)

It looks like Fig. 1.26, the corresponding schedule for
mA and
Oh, build your own. The maximum power is reached at
and is equal to
.

With the internal resistance tending to infinity the current-voltage characteristic of a real current source tends to the characteristic of an ideal source (Fig. 1.15b). Then the ideal source can be considered as real withendless internal resistance.

Comparing the current-voltage characteristics of real voltage and current sources in fig. 1.25 and fig. 1.28, it is easy to verify that they can be the same under the conditions

(1.42)

This means that these sources under condition (1.42)

are equivalent, that is, in equivalent circuits of electrical circuits a real voltage source can be lured by a real current source and vice versa. For ideal sources, such a replacement is impossible.

1.12. System of electric circuit equations

for instantaneous values ​​of currents and voltages

On the basis of Ohm's and Kirchhoff's laws, it is possible to form a system of equations relating the instantaneous values ​​of currents and voltages. To do this, you must perform the following steps (let's consider them using the example of the circuit in Fig. 1.29).

Relationship equations between current and voltage in the elements or branches of the circuit are called subsystem of component equations. The number of equations is equal to the number of passive elements or circuit branches. As you can see, the subsystem includes differential or integral relationships between currents and voltages.

In the example under consideration, for nodes 1, 2 and 3, these equations have the form, for example, (1.32)

(1.44)

Total formed
equations.

In the diagram in fig. 1.29 selected three independent circuits are marked circular lines with an arrow indicating the positive direction of traversal. For them, the equations of the second Kirchhoff law have the form (1.34)

(1.45)

The total number of equations is
.

Equations formed according to the first and second Kirchhoff laws are called subsystem of topological equations, since they are determined by the scheme (topology) of the circuit. The total number of equations in it is equal to the number of branches that do not contain ideal current sources.

The set of subsystems of component and topological equations form complete system electrical circuit equations for instantaneous values ​​of currents and voltages, which is a complete circuit model.

From the component equations, it is not difficult to express all the voltages through the currents of the branches, then for the circuit in Fig. 1.29 from (1.43) we obtain

(1.46)

(1.46’)

Substituting (1.46) into the equations of the second Kirchhoff law of the form (1.45), we obtain a system of equations for the branch currents

(1.47)

The considered approach to the formation of the equations of electrical equilibrium of the circuit is called branch current method. The number of equations obtained is equal to the number chain branches, not containing ideal current sources.

As you can see, the model of a linear circuit for instantaneous values ​​of currents and voltages of the form (1.43), (1.44), (1.45) or (1.47) is linear system of integro-differential equations.

1.13. Tasks for independent solution

Task 1.1. Voltage
on the container C changes as shown in fig. 1.30. Get an expression for the capacitance current
, instant power
and stored energy
, on-

build graphs semi-Fig. 1.30

functions.

Task 1.2. Voltage
on resistance R changes, as shown in Fig. 1.31. Get expression for capacitance voltage
, build a graph
(through
necessary op-

redistribute current
,

and then - strain- Fig. 1.31

ing
).

Task 1.3. Voltage
on a parallel connection of resistance R and inductance L changes, as shown in fig. 1.32. Write an expression for the total current
, plot its graph (required

find the branch currents, and 1.32

the more their sum is the current
).

Task 1.4. In the circuit diagrams shown in fig. 1.33, determine the number of nodes and branches, the number of equations according to the first and second Kirchhoff laws.

Task 1.5. For circuits whose equivalent circuits are shown in fig. 1.33, write down the complete systems of equations according to Ohm's law, the first and second Kirchhoff's laws for the instantaneous values ​​of the currents and voltages of the elements.

Task 1.6. For the circuit shown in fig. 1.34, write down the complete system of equations according to Ohm's and Kirchhoff's laws for the instantaneous values ​​of the currents and voltages of the elements.