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Name: Fundamentals of circuit theory. 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 for 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 direct and alternating current. All provisions of the theory are illustrated with practical examples.
TABLE OF CONTENTS
Preface to the fourth edition.
Introduction.
Section 1 LINEAR ELECTRICAL CIRCUITS WITH CONCENTRATED. PARAMETERS
Chapter 1.
Basic laws and methods for calculating electrical circuits at constant currents and voltages.
1-1. Elements of electrical circuits and electrical diagrams.
1-2. Equivalent circuits for energy sources.
1-3. Ohm's law for a section of a circuit 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. Equations of state of a circuit in matrix form.
1-10. Conversion of linear electrical circuits.
Chapter 2.
Basic properties of electrical circuits at constant currents and voltages
2-1. The principle of superposition.
2-2. Property of reciprocity.
2-3. Input and mutual conductances and resistances of branches; voltage and current transfer coefficients.
2-4. Application of topological methods for circuit calculations.
2-5. Topological formulas and rules for determining the transmission of an electrical circuit.
2-6. 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 two-terminal networks.
2-10. The theorem about the active two-terminal network 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 about sinusoidal current circuits
3-1. Alternating currents.
3-2. The concept of alternating current generators.
3-3. Sinusoidal current.
3-4. Effective current, e. d.s. and tension.
3-5. Representation of sinusoidal functions of time by vectors and complex numbers.
3-6. Addition of sinusoidal functions of time.
3-7. Electric circuit and its diagram.
3-8. Current and voltage when connecting resistance, inductance and capacitance in series.
3-9. Resistance.
3-10. Phase difference between voltage and current.
3-11. Voltage and currents when connecting resistance, inductance and capacitance in parallel.
3-12. Conductivity.
3-13. Passive two-terminal network.
3-14. Power.
3-15. Power in resistance, inductance and capacitance.
3-16. Power balance.
3-17. Power signs and 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 the energy source to the receiver.
3-20. The concept of the surface effect and the proximity effect.
3-21. Parameters and equivalent circuits of capacitors.
3-22. Parameters and equivalent circuits of inductive coils and resistors.
Chapter 4.
Calculation of circuits with sinusoidal currents.
4-1. On the applicability of methods for calculating DC circuits to calculating 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 diagrams.
4-7. Duality of electrical circuits.
4-8. Signal graphs and their application to calculate Circuits.
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 a 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 couplings.
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. Circular diagrams for an unbranched circuit and for an active two-terminal network.
7-3. Pie charts for any branched circuit.
Chapter 8.
Multi-terminal and four-terminal 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 four-terminal networks.
8-5. Basic equations and equivalent circuits for an active quadripole.
8-6. An ideal transformer is like a four-terminal network.
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 multiphase power supplies and multiphase circuits.
10-2. Star and polygon connections.
10-3. Symmetrical mode of three-phase circuit.
10-4. Some properties of three-phase circuits with different connection diagrams.
10-5. Calculation of symmetrical modes of three-phase circuits.
10-6. Calculation of asymmetrical 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. Operating principles of asynchronous and synchronous motors.
Chapter 11.
Method of symmetrical components.
11-1. Symmetrical components of a three-phase system of quantities.
11-2. Some properties of three-phase circuits in relation to symmetrical components of currents and voltages.
11-3. Resistances of a symmetrical three-phase circuit for currents different 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. Effective values of 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 with 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.
Classic method transient calculations
13-1. Occurrence of transient processes and commutation laws.
13-2. Transitional, forced and free processes.
13-3. Short circuit in circuit R, L.
13-4. Turning on the circuit k, L to constant voltage.
13 5. Switching on the circuit r, L for sinusoidal voltage.
13-6. Short circuit of circuit g, C.
13-7. Turning on circuit r, C to constant voltage.
13-8. Switching on circuit g, C to sinusoidal voltage.
13-9. Transient processes in an unbranched chain r, L, C.
13-10. Aperiodic capacitor discharge.
13-11. Limiting case of aperiodic capacitor discharge.
13-12. Periodic (oscillatory) discharge of a capacitor.
13-13. Turning on the r, L, C circuit to constant voltage.
13-14. General case of calculating transient processes using the classical method.
13-15. Switching on a passive two-terminal network for a continuously varying voltage (Duhamel’s formula or integral).
13-16. Switching on a passive two-terminal network for voltage of any form.
13-17. Time and impulse transient characteristics.
13-18. Writing the convolution theorem using impulse response.
13-19. Transient processes during current surges in inductors and voltages on capacitors.
13-20. Determination of the transient process and steady state when exposed to periodic voltage or current pulses.
Chapter 14.
Operator method for calculating transient processes.
14-1. Application of the Laplace transform to the calculation of transient processes.
14-2. Ohm's and Kirchhoff's laws in operator form.
14-3. Equivalent operator circuits.
14-4. Transient processes in circuits with mutual inductance.
34-5. Reducing calculations of "transient processes to zero" initial conditions.
14-6. Determination of free currents from 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 a circuit when exposed to periodic non-sinusoidal voltage.
Chapter 15.
Frequency method for calculating transient processes.
15-1. Fourier transform and its basic properties.
15-2. Ohm's and Kirchhoff's laws and equivalent circuits for frequency spectra.
15-3. An approximate method for determining the original using a 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 resistances and transmission constant of an asymmetrical four-port network.
Characteristic impedance and transmission constant of a symmetrical four-port network.
Introduced and working constant transfers.
Chain diagrams.
Frequency electric filters.
Low pass filters.
High-pass filters.
Bandpass filters.
Blocking filters.
Constant M filters.
An L-shaped filter is an example of a single-ended filter. Non-induction (iln r, C) filters.
Chapter 17.
Synthesis of electrical circuits.
17-1. general characteristics synthesis problems.
17-2. Transfer function of a quadripole network. Minimum phase circuits.
17-3. Input functions of circuits. Positive real functions.
17-4. Reactive two-terminal networks.
17-5. Frequency characteristics of reactive two-terminal networks.
17-6. Synthesis of reactive two-terminal networks. Foster's method.
17-7. Synthesis of reactive two-terminal networks. Cauer's method.
17-8. Synthesis of two-terminal networks with losses. Foster's method.
17-9. Synthesis of two-terminal networks with losses. Cauer's method.
17-10. The concept of synthesis of quadripoles.
Section 2. LINEAR CIRCUITS WITH DISTRIBUTED PARAMETERS.
Chapter 18.
Harmonic processes in circuits 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. Equations of a homogeneous line 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. No-load, 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 equations of a homogeneous line.
19-3. The appearance of waves with a rectangular front.
19-4. General cases of 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 for determining reflected waves.
19-7. Qualitative consideration of transient processes in lines containing lumped capacitances and inductances.
19-8. Multiple reflections of will with a rectangular front from active resistance.
19-9. Wandering waves.
Section 3 Nonlinear circuits.
Chapter 20
Nonlinear electrical circuits at constant currents and voltages.
20-1. Elements and equivalent circuits of the simplest nonlinear circuits.
20-2. Graphical method for calculating unbranched circuits with nonlinear elements.
20-3. Graphical method for calculating circuits with parallel connection of nonlinear elements.
20-4. Graphical method for calculating circuits with a mixed connection of nonlinear and linear elements.
20-5. Application of equivalent circuits with energy sources. d.s. to study the mode of nonlinear circuits.
20-6. Current-voltage characteristics of nonlinear active two-terminal networks.
20-7. Examples of calculations of branched electrical circuits with nonlinear elements.
20-8. Application of the theory of active two-terminal, four-terminal and six-terminal networks for the calculation of circuits with linear and nonlinear elements.
20-9. Calculation of branched nonlinear chains using the iterative method (method of successive approximations).
Chapter 21.
Magnetic circuits at constant 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 an air gap.
21-5. Calculation of an unbranched inhomogeneous magnetic circuit with a permanent magnet.
Chapter 22.
General characteristics of nonlinear alternating current circuits and methods for their calculation
22-1. Nonlinear two-terminal and four-terminal networks with alternating currents.
22-2. Determination of operating points on the characteristics of nonlinear two-terminal and four-terminal networks.
22-3. Phenomena in nonlinear alternating current circuits.
22-4. Methods for calculating nonlinear alternating current circuits.
Chapter 23.
Nonlinear circuits with energy sources. d.s. and current of the same frequency.
23-1. General characteristics of circuits with energy sources. d.s. the same frequency.
23-2. Current waveform in a circuit with valves.
23-3. The simplest rectifiers.
23-4. Shapes of current and voltage curves in circuits with nonlinear reactances.
23-5. Frequency triplers.
23-6. Shapes of current and voltage curves in circuits with thermistors.
23-7. Replacement of real nonlinear elements with conditionally nonlinear ones.
23-8. Accounting real properties steel magnetic cores.
23-9. Calculation of current in a coil with a steel magnetic core.
23-10. The concept of calculating conditionally nonlinear magnetic circuits.
23-11. The phenomenon of ferroresonance.
23-12. Surge Protectors.
Chapter 24.
Nonlinear circuits with energy sources. d. s, and current of various frequencies.
24-1. General characteristics of nonlinear circuits with energy sources. 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-AC converters.
24-4. Coils with steel magnetic cores in circuits with constant and variable e. d.s.
24-5. Frequency doubler.
24-6. Harmonic balance method.
24-7. The influence of constant e. d.s. on the alternating component of current in circuits with nonlinear inertia-free resistances.
24-8. The principle of obtaining modulated oscillations.
24-9. The influence of a constant component on a variable in circuits with nonlinear 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 coils with a steel magnetic core to constant voltage.
25-3. Switching on a coil with a steel magnetic core for sinusoidal voltage.
25-4. Pulse action in circuits with ambiguous nonlinearities.
25-5. The concept of the simplest storage devices.
25-6. Image of transient processes on the phase plane.
25-7. Oscillatory discharge of capacitance through nonlinear inductance
Chapter 26.
Self-oscillations
26-1. Nonlinear resistors with a falling part of the characteristic.
26-2. The concept of mode stability in a circuit with nonlinear 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 processes in a circuit with negative resistance.
26-6. Phase trajectories of processes in a sinusoidal oscillation generator.
26-7. Determination of the amplitude of self-oscillations by the harmonic balance method.
Applications.
Bibliography.
Subject index.
Electrical circuit is a set of devices intended for the transmission, distribution and mutual conversion of electrical (electromagnetic) and other types of energy and information, if the processes occurring in the devices can be described using the concepts of electromotive force (emds), current and voltage
The main elements of an electrical circuit are sources and receivers electrical energy(and information) that are connected to each other by wires.
In sources of electrical energy (galvanic cells, batteries, electric machine generators, etc.) chemical, mechanical, thermal energy or energy of other types is converted into electrical energy, by receivers of electrical energy (electrothermal devices, electric lamps, resistors, electric motors, etc.), on the contrary, electrical energy is converted into thermal, light, mechanical, etc.
Electrical circuits in which the receipt of electrical energy in sources, its transmission and transformation in receivers occur at constant currents and voltages are usually called DC circuits.
Legend basic quantities
Preface
Part one. Linear electrical circuits
Chapter 1. Basic properties and transformations of electrical circuits
§ 1.1. Topology (geometry) of the electrical circuit
§ 1.2. Equivalent circuits of electrical energy sources
§ 1.3. Equivalent conversions of electrical energy sources
§ 1.4. Converting Two-Node Diagrams Containing Sources
§ 1.5. Basic properties and theorems of linear electrical circuits
§ 1.6. Dual elements and diagrams
§ 1.7. Algorithm graphic construction dual planar scheme
§ 1.8. Electrostatic circuits
§ 1.9. Methods for calculating electrostatic circuits
§ 1.10. Basic quantities characterizing harmonic current
§ 1.11. Complex method
§ 1.12. Calculation algorithm complex method
§ 1.13. Complex numbers
§ 1.14. Basic complex quantities and laws characterizing harmonic voltage (current)
§ 1.15. Passive elements in a harmonic current circuit
§ 1.16. Connections and transformations of passive elements
§ 1.17. Examples equivalent transformations
§ 1.18. Series connection of elements
§ 1.19. Parallel connection of elements
§ 1.20. Resonances in linear electrical circuits
§ 1.21. Two-terminal networks
§ 1.22. Harmonic current circuit power
§ 1.23. Vector diagrams simplest circuits
§ 1.24. Pie chart for quadripole currents
§ 1.25. Topographic diagram
§ 1.26. Circuits with mutual inductance
§ 1.27. Consistent series connection of inductively coupled coils
§ 1.28. Back-to-back series connection of inductively coupled coils
§ 1.29. Parallel connection of inductively coupled coils. 46
§ 1.30. Experienced determination mutual inductance
§ 1.31. Transformer without ferromagnetic core (air transformer)
§ 1.32. Calculation of branched circuits with mutual induction
Chapter 2. Non-harmonic currents
§ 2.1. Fourier series for some periodic nonharmonic functions
§ 2.2. Non-harmonic curves with periodic envelope
§ 2.3. Basic quantities and coefficients of non-harmonic current
§ 2.4. Calculation of circuits with periodic non-harmonic currents
§ 2.5. Measurement of non-harmonic currents and voltages
Chapter 3. Three-phase current circuits
§ 3.1. Three phase generator
§ 3.2. Symmetrical mode in three-phase circuits
§ 3.3. Neutral bias voltage when connecting an uneven load to a star
§ 3.4. Determination of currents in a three-phase circuit
§ 3.5. Conversion of three-phase circuit with mixed load
§ 3.6. Method of symmetrical components
§ 3.7. Phase multiplier
§ 3.8. Resistance of a symmetrical three-phase circuit to currents of various sequences
§ 3.9. Longitudinal and transverse asymmetry of a three-phase circuit
§ 3.10. Longitudinal asymmetry of a three-phase circuit
§ 3.11. Types of longitudinal asymmetry
§ 3.12. Transverse unbalance of three-phase circuit
§ 3.13. Types of transverse asymmetry
§ 3.14. Algorithm for calculating an asymmetrical three-phase circuit
Chapter 4. Methods for calculating electrical circuits
§ 4.1. Calculation of circuits according to Ohm's law
§ 4.2. Calculation of circuits using Kirchhoff equations
§ 4.3. Matrix form of writing Kirchhoff's equations
§ 4.4. Loop current method
§ 4.5. Matrix form of writing equations using the loop current method
§ 4.6. Nodal potential method
§ 4.7. Matrix form of writing equations using the nodal potential method
§ 4.8. Two node method
§ 4.9. Overlay method
§ 4.10. Equivalent source method
§ 4.11. Compensation method
Chapter 5. Topological methods for calculating electrical circuits
§ 5.1. Basic concepts and definitions
§ 5.2. Topological graph matrices
§ 5.3. Writing equations electrical diagram in matrix form
§ 5.4. Finding the determinant of a circuit using topological formulas
§ 5.5. Signal graphs
§ 5.6. Algorithm for constructing a signal graph based on the system linear equations
§ 5.7. Drawing up a system of equations using a signal graph
§ 5.8. Transformation of signal graphs
§ 5.9. Topological rule for determining the transfer of a graph (Mason's formula)
§ 5.10. Signal graphs of quadripole equations
§ 5.11. Signal graphs of connections of quadripoles
Chapter 6. Quadripoles
§ 6.1. Basic definitions
§ 6.2. Passive quadripole equations
§ 6.3. Quadripole equations in A-form (basic equations)
§ 6.4. Equivalent circuits and parameters of passive quadripoles
§ 6.5. Quadripole connections
§ 6.6. Characteristic parameters of quadripoles
§ 6.7. Transfer function (transmission coefficient or amplitude-phase characteristic) of a four-port network
§ 6.8. Attenuation Constant Units
Chapter 7. Electrical filters
§ 7.1. Classification
§ 7.2. Electrical reactive circuit filters
§ 7.3. Type k reactive filters
§ 7.4. T type reactive filters
§ 7.5. Non-induction filters (RC filters)
Chapter 8. Transient processes in linear electrical circuits
§ 8.1. Calculation methods
§ 8.2. Commutation laws
§ 8.3. Classic method
§ 8.4. The nature of the free process depending on the roots characteristic equation
§ 8.5. Drawing up a characteristic equation
§ 8.6. Determining the degree of a characteristic equation
§ 8.7. Initial conditions ( initial values currents and voltages at t=0
§ 8.8. Determination of dependent initial conditions
§ 8.9. Determination of initial conditions for free components of currents and voltages
§ 8.10. Algorithm for calculating transient processes using the classical method
§ 8.11. Transient processes in the simplest circuits
§ 8.12. Operator method
§ 8.13. Equivalent operator schemes for circuit elements with non-zero initial conditions
§ 8.14. Ohm's law and Kirchhoff's laws in operator form. Equivalent operator circuits
§ 8.15. Finding the original from the image
§ 8.16. Table of originals and images according to Laplace
§ 8.17. Basic operator transformations according to Laplace
§ 8.18. Algorithm for calculating transient processes using the operator method
§ 8.19. Calculation of free components by the operator method
§ 8.20. Calculation of transient processes by the Duhamel integral method
§ 8.21. Unit and transition functions
§ 8.22. The action of single step and single pulse sources on inductive and capacitive elements
§ 8.23. Algorithm for calculating transient processes using the Duhamel integral method
§ 8.24. Bringing the circuit to zero initial conditions
§ 8.25. Frequency method
§ 8.26. Basic properties of one-way Fourier transform
§ 8.27. Spectral characteristics of some functions
§ 8.28. Fourier series and integral
§ 8.29. Algorithm for calculating transient processes using the frequency method
§ 8.30. State variable method
§ 8.31. Matrix form of writing equations using the state variable method
§ 8.32. Drawing up differential equations of state using Kirchhoff's equations
§ 8.33. Drawing up differential equations of state using the superposition method
Chapter 9. Steady-state processes in long lines (circuits with distributed constants)
§ 9.1. General information
§ 9.2. Long line parameters 157
§ 9.3. Dependence on the geometric dimensions of the simplest lines
§ 9.4. Equations of a homogeneous long line with losses
§ 9.5. Input impedance of a long line with losses
§ 9.6. Long line without loss
§ 9.7. Lossless Long Line Input Impedance
§ 9.8. Standing waves
§ 9.9. Properties of distribution of effective values of voltage and current along a line without losses at
§ 9.10. Line without distortion
§ 9.11. Line matched to load
§ 9.12. Matching the lossless line with the load
§ 9.13. Measuring line
§ 9.14. Artificial line
§ 9.15. Long line with parameters variable along the length
Chapter 10. Transients in Long Lossless Lines
§ 10.1. Incident and reflected waves
§ 10.2. Wave reflection from the end of the line
§ 10.3. Multiple reflections of waves when connecting a source DC voltage to the line
§ 10.4. Equivalent circuit for determining currents and voltages at line nodes
§ 10.5. Distribution of voltage and current along lines connected through L or C
§ 10.6. Waves when branches are turned on and off
Chapter 11. Synthesis of linear electrical circuits
§ 11.1. General information
§ 11.2. Definition, properties and signs of a positive real function
§ 11.3. Signs of positivity and materiality of a rational function
§ 11.4. Positive real functions Z(p) and Y(p) of the simplest two-terminal networks
§ 11.5. Implementation of reactive two-terminal networks by decomposing the input function into simple fractions(implementation of two-terminal networks according to Foster)
§ 11.6. Foster expansion of the imaginary input function Z(p)
§ 11.7. Foster expansion of the imaginary input function Y (p)
§ 11.8. Implementation of real positive input functions having poles and zeros on the imaginary axis and real positive semi-axis
§ 11.9. Expansion of the input function into a continued fraction (implementation of two-terminal networks according to Cauer)
§ 11.10. Synthesis of quadripoles
§ 11.11. Transfer functions of a quadripole
§ 11.12. Implementation of LC and RC four-terminal networks using a bridge circuit
§ 11.13. Required properties parameters of a passive four-port network during its synthesis
§ 11.14. Features of the voltage transfer function of quadripoles Ni
§ 11.15. Implementation of LC and RC four-terminal networks using a chain circuit
Part two. Nonlinear electrical circuits
Chapter 12. Nonlinear elements
§ 12.1. General information
§ 12.2. Resistive elements
§ 12.3. Bipolar resistive elements
§ 12.4. Controlled bipolar resistive elements
§ 12.5. Controlled three-pole resistive elements
§ 12.6. Calculation of nonlinear DC circuits
§ 12.7. Two node method
§ 12.8. Static and differential resistance
§ 12.9. Equivalent replacement of a nonlinear resistive element with a linear resistive element and an electrical source. d.s.
§ 12.10. Calculation of a branched circuit with nonlinear elements
Chapter 13. Nonlinear inductive and capacitive elements
§ 13.1. Nonlinear inductive elements
§ 13.2. Magnetization curves B(H) of ferromagnetic materials
§ 13.3. Losses in a real inductive element
§ 13.4. Basic quantities and dependencies characterizing the magnetic field
§ 13.5. Formal analogy between electric and magnetic DC circuits
§ 13.6. Calculation of the magnetic circuit at DC. Direct task
§ 13.7. Calculation of a magnetic circuit at direct current. Inverse problem
§ 13.8. Unbranched permanent magnet magnetic circuit
§ 13.9. Coil with ferromagnetic core
§ 13.10. Nonlinear circuits with a controlled inductive element
§ 13.11. Magnetic power amplifier
§ 13.12. Transformer with ferromagnetic core
§ 13.13. Peak transformer
§ 13.14. Nonlinear capacitive elements
§ 13.15. Resonance phenomena in nonlinear circuits
Chapter 14. Approximation of nonlinear characteristics
§ 14.1. Approximation functions
§ 14.2. Approximation of characteristics of nonlinear elements
§ 14.3. Piecewise linear approximation of current-voltage characteristics
§ 14.4. Equivalent circuits for ideal elements with piecewise linear characteristics
§ 14.5. AC Rectification
§ 14.6. Determining the coefficients of the approximating function
Chapter 15. Analytical methods analysis of periodic processes in nonlinear circuits
§ 15.1. General information
§ 15.2. Harmonic linearization method (frequency method)
§ 15.3. Harmonic balance method
§ 15.4. Slowly varying amplitude method
§ 15.5. Piecewise method linear approximation
§ 15.6. Analytical approximation method
Chapter 16. Graphical methods analysis of periodic processes in nonlinear circuits
§ 16.1. Calculation by characteristic for instantaneous values
§ 16.2. Calculation according to the characteristic for the first harmonic
§ 16.3. Calculation according to characteristic for effective values
Chapter 17. Methods for calculating transient processes in nonlinear circuits
§ 17.1. Methods for calculating transient processes in circuits with one nonlinear reactive element
§ 17.2. Linear approximation method
§ 17.3. Piecewise linear approximation method
§ 17.4. Analytical approximation method
§ 17.5. Sequential Interval Method
§ 17.6. Graphical integration method
§ 17.7. Phase plane method
Chapter 18. Self-oscillations
§ 18.1. General information
§ 18.2. Relaxation oscillations
§ 18.3. Almost harmonic vibrations
§ 18.4. Stability of the equilibrium state
§ 18.5. Sustainability in small things
§ 18.6. Algorithm for obtaining linearized equations for the quantity under study
§ 18.7. A. M. Lyapunov’s theorem on establishing stability in small autonomous nonlinear systems
§ 18.8. Hurwitz stability criterion
Chapter 19. Electric circuits with variable parameters
§ 19.1. General information
§ 19.2. Elements with variable parameters
§ 19.3. Circuit with resistive element
§ 19.4. Circuit with inductive element
§ 19.5. Circuit with capacitive element
§ 19.6. Variable Circuit Analysis
§ 19.7. Parametric oscillations
List of recommended literature
Subject index
a) basic literature:
1. Popov V.P. Basics of circuit theory. – M.: graduate School, 1985. –496 p.,
2. Popov V.P. Basics of circuit theory. – M.: Higher School, 2013. –696 p.
3. Beletsky A.F. Theory of linear electrical circuits. – St. Petersburg: Lan, 2009. – 544 p.
4. Bakalov V.P., Dmitrikov V.F., Kruk B.I. Basics of Circuit Theory:
Textbook for universities; Ed. V.P. Bakalova. - 2nd ed., revised. and additional
- M.: Radio and Communications, 2000. - 592 p.
5. Dmitrikov V.F., Bakalov V.F., Kruk B.I. Basics of Circuit Theory:
Hotline - Telecom, 2009. – 596 p.
6. Shebes M.R., Kablukova M.V. Problem book on the theory of linear
electrical circuits. –M: Higher School, 1986. –596 p.
1b) additional literature
Baskakov S.I. Radio circuits and signals: Textbook. for universities for special purposes
"Radio engineering". - M.: Higher School, 1988. - 448 p.
Frisk V.V. Basics of circuit theory./ Tutorial. – M.: IP RadioSoft,
2002. – 288 p.
Electrical circuit
An electrical circuit is a collection of elements and
devices forming a path or paths for electric current,
electromagnetic processes in which can be described by
help concepts “ electricity” and “electrical voltage”.
Electrical circuit elements
Sources
Receivers
2
Classification of electrical circuits
ViewPassive and active
Two-terminal and
multipole networks
With focused and
distributed
parameters
Continuous and discrete
With constant and
variable parameters
Linear and nonlinear
Sign
Energy properties
Number of external terminals
Spatial
localization of parameters
the nature of the processes
element properties
Operator type
3
CURRENT, VOLTAGE and ENERGY IN AN ELECTRICAL CIRCUIT
. i(t) = dq(t)/dt.
[A]
u12 = φ1 - φ2
[B]
[W]
4IDEALIZED PASSIVE ELEMENTS OF AN ELECTRIC CIRCUIT
Conversion of electrical energy in electrical circuit elements
irreversible transformation of electrical energy into other types of energy;
accumulation of energy in an electric field;
accumulation of energy in a magnetic field;
conversion of non-electrical energy into electrical energy
Resistance
uR(t) = R iR(t)
Ohm's law
iR(t)
= G uR(t)
5wk =
.
Pk = dwk / dt = uR IR
I
R st A = uA / iA
R diff A = du / di
A
6wC =
Capacity
.
qC(t) = C uC(t)
iC = C duC / dt,
iC = C duC / dt,
pC = iC uC = C uC d uC/dt
1 t1
iC(t)dt
uC(t = t1) =
WITH
[F]
. wC = C uC2(t1)/2 > 0
Cst = qC /uc
Sdif. = dqC/duc
7
Inductance
..
Inductance
Ψ(t) = L iL(t)
uL(t) = Ψ(t)/ dt
uL = L diL/dt
1 t1
u L(t)dt
iL(t1) =
L
pL = iL uL = L iL diL/dt
wL =
t1
[Gn]
pL (t)dt = L iL2(t1)/2 > 0
Lst = Ψ/iL
,
Ldiff = d Ψ/d iL.
8
IDEALIZED ACTIVE ELEMENTS OF AN ELECTRIC CIRCUIT
Independent voltage sourceIndependent current source
9
10. Dependent (controlled) sources of electrical energy
2.1Dependent (controlled) sources of electrical energy
Name
Designations
Voltage source controlled
voltage
(INUN), u2 = k1u1
Voltage source
current controlled (INUT), u2 = k2 i1
Current source controlled
voltage
(ITUN), i2 = k3 u1
Current controlled current source
(ITUT), i2 = k4 i1
10
11. Electrical circuit diagrams
principled;replacement (calculated);
functional (block diagram)
Equivalent circuits real elements electrical circuit
i/ ikz - u / uхх = 1
u = uхх - (uхх / ikз) I = uхх - Ri i
I = ikz - (ikz / uхх) u = ikz - Gi u
11
12.
j = ikz, Gi = 1/ RiE = ikz Ri
Ri
=
uhx / ikz
Connections of electrical circuit elements
sequential
parallel
mixed
12
13.
startriangle
Elements of electrical circuit topology
0 0 0
1 1 1
0 1 0 1 0 1
À0
0 0 1 1 1
0
1
0
0
0
1
1
13
14. BASIC LAWS AND THEOREMS OF THE THEORY OF ELECTRIC CIRCUITS
Kirchhoff's first law (law of currents)At any time algebraic sum instantaneous current values in
all branches of the electrical circuit that have a common node is equal to zero
z
i
k 1 k
Node No.
The equation
= 0
0
(1)
(2)
-i1 + i2 + i3 + i4
-i3 - i4 + i5 - j =
0
(3)
-i5 + i6 + j = 0
(0)
i1 – i2 – i6 = 0
14
15.
Consequences1)
Zk
=
Zе = jе =
jk
n
k 1
2) Zk
3)
Zk
Ck
Ñý k 1 Ñk .
n
Lk
4) Zk
1/Le =
n
jk
Rk
Gý k 1 Gk .
n
1 Lk
k 1
combined into one element.
Kirchhoff's second law (stress law)
Algebraic sum of instantaneous voltage values of all branches included in
the composition of an arbitrary circuit of an electrical circuit at any time is equal to
zero.
15
16.
sïu
k 1 k
sï
u l 1 el
sà
k 1 k
In any circuit of an electrical circuit, the algebraic sum of instantaneous
voltage drop values on passive elements is equal to
algebraic sum of the instantaneous values of the emf acting in this circuit.
uz1+uz3+uz2 = e ;
-uz2 – uz3+uz4 + uz5 = 0;
uz1 + uz4 + uz5 = e.
Consequences
1)
2)
Zk
Zk
Ek
Rk
16
17.
.,3)
Zk
Lk
4)
Zk
Ck
.
Parallel connected elements of the same name can be
combined into one element.
THE PRINCIPLE OF SUPERPOSITION AND THE ANALYSIS METHOD BASED ON IT
ELECTRICAL CIRCUITS (SUPERPOSITION METHOD)
The response of a linear electrical circuit y(t) to the influence x(t) in the form of a linear
combinations of simpler influences xk(t), is a linear
combination of reactions of this chain to each of the influences separately - yk(t), i.e.
at
x(t) =
n
k 1
k xk t
n
y(t) =
k 1
k y k t
Where
k
- constant coefficients,
xk(t) - k-th component of the impact.
17
18.
Overlay methodTheorems on an active two-terminal network. Equivalent generator method
18
19.
Equivalent Voltage Source TheoremLinear electrical circuit considered relative to its two
clamps, can be replaced by an EE voltage source turned on
in series with resistance Re. Setting voltage uе source
voltage is equal to the open circuit voltage uхх at the considered
terminals (branch Rн is open), and the resistance Re is equal to the resistance
between these terminals, calculated under the assumption that branch Rн
is open and all voltage sources contained in the circuit are replaced
Equivalent Current Source Theorem
A linear electrical circuit, considered relative to its two terminals,
can be replaced by a current source jе connected in parallel with the conductivity
Ge. Source setting current jе equal to current short circuit the one under consideration
pairs of terminals, conductivity Ge is equal to the input (from the side of terminals 1.1′)
circuit conductivity N, calculated under the assumption that branch Rн is open
and all voltage sources contained in the circuit are replaced
short-circuiting jumpers, and the circuits of all current sources are open.
19
20.
Equivalent voltage source method, calculation procedureare specified by the direction of the current in the branch Rн;
open the Rн branch and find the open circuit voltage (in the general case
taking into account the emf e in the branch Rн) uхх = uе = φ1 - φ1′ + e;
determine the input resistance Rin = Re circuit N from the terminal side
1.1′, branch Rн is open;
using the formula i uõõ Râõ Rí, the current in the branch Rн is determined and the voltage on it is determined using the formula un = Rнi.
Equivalent current source method, calculation procedure
are specified by the direction of the current in the branch Rн;
short-circuit the branch Rн and find the short circuit current between
clamps
1.1′ ikz = jе;
determine the input conductivity Gin = Ge circuit N from the terminal side
1.1′, branch Rн is open;
i irp Gí Gâõ Gí
according to the formula
determine the current in the branch Rн and according to the formula un =
Rni is the voltage across it.
20
21.
.Energy relationships in a linear electrical circuit
Tellegen's theorem
With a consistent choice of current directions and
voltage in the branches of the circuit graph sum
products of voltage uk and current ik all
branches of a directed chain graph in any
n
moment of time is equal to zero, i.e. , k 1 uk ik 0
or in matrix form: uТ i= 0, where uТ = (u1…
uk …um), iТ = (i1…ik …im) – voltage vectors
and branch currents, respectively.
Power Balance Equations
n
k 1
pk 0
nÏ
R i k 1 ek ik k 1 uk jk
2
nèí
P=
I 2 Rн
no
и 1 k k
=
Ri Rн 2
E 2 Rн
dP dRí E 2 Ri Rí 2Rí Ri Rí Ri Rí
2
4
Pmax E 2 4 Ri
21
22.
η =.
Rn R Rn I 2 Ri I 2 Rn I 2 Rn Ri Rn
4. General methods electrical circuit analysis
Kirchhoff equation method
: -i1 + i2 + i3 = 0,
u1 + u2 = e,
u4 - u5 = 0.
u1 = R1 i1 - e,
u3 = R3 i3,
-i3 + i4 - i5 = 0
-u2 + u3 + u4 = 0,
u2 = R2 i2,
u4 = R4 i4.
22
23.
Loop current methodCalculation procedure
1.
2.
3.
4.
5.
6.
7.
Define a system of independent circuits
Set the directions of the loop currents
Determine the matrix of circuit resistances and the vector of circuit EMF
Write down a system of contour equations and solve it
Determine branch currents
Determine branch voltages
Check the correctness of the solution
Circuit resistance matrix
Rк = (Rji), j , i 1, q
q - order of the system of contour equations, q = n – (m – 1),
for circuits with current sources q = n – (m – 1)- nit, n, m – number
branches and nodes in the circuit, nit – number of branches containing current sources
23
24.
the intrinsic resistance Rjj of the j-th circuit is the sum of resistancesall branches included in this circuit;
the mutual resistance of the j-th and i-th circuits is called the resistance Rji,
equal to the sum resistances of the branches common to these circuits. Mutual
resistance has a plus sign if the loop currents of the j-th and i-th flow
through branches common to these circuits in the same direction, if in
opposite directions, then Rji has a minus sign. If j-th and i-th
the circuits do not have common branches, then their mutual resistance is zero.
Rк =
the contour emf of the jth loop ejj is the algebraic sum of the emf
all voltage sources included in this circuit. If the direction
EMF of any source included in j-th circuit, coincides with
direction of the loop current of this circuit, then the corresponding EMF
enters ejj with a plus sign, otherwise with a minus sign.
e ê e11....eii ...eqq
Ò
24
25.
ExampleR11 R12 R13 R1 R2 R4
R ê R21 R22 R23
R2
R
R4
31 R32 R33
eê
Ò
R2
R2 R3 R5
R5
R4
R5
R4 R5 R6
E1 ,0, E 2
25
26.
Contour equations,
R to i to ek
i to i11...i jj ...iqq
T
- vector of loop currents
R11i11 R12i22 ... R1i iii ... R1q iqq e11.
………………………..
R j1i11 R j 2 i22 ... R ji iii ... R jq iqq e jj.
…………………………
Rq1i11 Rq 2i22 ... Rqiiii ... Rqqiqq eqq.
R2
R4
R1 R2 R4
R
R
R
R
R
2
2
3
5
5
R
R5
R4 R5 R6
4
i11 e1
i 22 0
i e
33 2
26
27.
Nodal stress methodui0= φi- φ0
ui j = φi - φj = φi- φ0 - (φi- φ0) = ui0 - uj0
Calculation procedure
if necessary, implement equivalent
converting voltage sources into sources
current;
set the directions of branch currents;
write down the matrix of nodal conductivities and the vector
nodal currents;
write down a system of nodal equations and solve it;
determine the voltages and currents of the circuit branches;
check the correctness of the solution.
27
28.
Nodal conductivity matrixGу = (Gji),
j , i 1, р
P – order of the system of nodal equations, р = m – 1, m – number of nodes in the chain, for chains with
“voltage sources” p = m – 1 – nin, nin - the number of branches that include
only voltage sources are included.
own conductivity Gii of the i-th node of the electrical circuit is called
the sum of the conductances of all branches connected to this node;
mutual conductance of the i-th and j-th nodes Gij is the sum of the conductivities of all
branches connected between these nodes, taken with a minus sign;
if there are no branches in the chain connected between the i-th and j-th nodes, then them
mutual conductivity is zero.
Gу =
28
29.
the nodal current of the i-th node jii is the algebraic sum of the driving currentsall current sources connected to this node. If the current of any source
directed to the i-th node, then it is included in this sum with a plus sign, if from the node, then
it enters jii with a minus sign.
jуТ =
j
11
...jii...jpp
Example
G11 G12 G13 G2 G4 G5 5
G y G21 G22 G23
-G5
G G G
-G 2
31 32 33
-G5
G 3 G 5 G6
-G3
-G3
G1 G2 G3
-G 2
29
30.
,jуТ =
0...
j...G1e
Nodal equations
G y u y jу
u u 01...u 0i ...u 0 p - vector of nodal voltages
T
G11u 01 G12u 02 ... G1i u 0i ... G1 p u 0 p j11.
……………………………………………
Gi1u 01 Gi 2 u 02 ... Gii u 0i ... Gip u 0 p jii.
……………………………………………
G p1u 01 G p 2 u 02 ... G pi i0i ... G pp u 0 p j pp.
G 2 G 4 G5 5
-G 5
-G 2
-G 5
G 3 G 5 G6
-G 3
u 01 0
-G 3
u 02 j
G1 G2 G3 u 03 G1e
-G 2
30
31.
3. Electric circuits under harmonic influencex(t) = Xm cos (ω t +) =
Xm sin (ω t +
+
Harmonic voltages and currents in electrical
chains
u(t) = Um cosω t = Umsin (ω t +
u(t) = Umсos (ω t -
) = Umsin ω t
u(t) = Umcos (ω t +
) = - Umsin ω t
Harmonic parameters
Xm - amplitude, ω - frequency,
fluctuations.
,ω = 2
- initial phase of harmonic
f, f = 1/ T - cyclic frequency, T - oscillation period,
X = Xm /√2 - effective (rms) value
harmonic vibration
31
32.
1)2)
Complex amplitude and complex impedance. Ohm's laws and
Kirchhoff in complex form
- complex amplitude
32
33.
Kirchhoff's first lawcurrents converging in an arbitrary node of an electrical circuit is zero.
Kirchhoff's second law
In the steady harmonic mode, the sum of the complex amplitudes of all
voltages acting in an arbitrary circuit of an electrical circuit is equal to
zero.
When summing the complex values of currents and voltages,
the same sign rules as when summing their instantaneous values
33
34.
COMPLEX RESISTANCEComplex resistance of the passive section of the electrical circuit –
this is the ratio of complex amplitudes (complex acting
values) of voltage and current acting at the terminals of this section
chains, i.e.
,
This expression is called Ohm's law in complex form. In him:
z(ω) and φ(ω) – module and argument z(jω). Dependence of z(ω) on frequency
called amplitude-frequency response (AFC)
two-terminal network, dependence φ(ω) – its phase-frequency
characteristic (phase response)
The reciprocal of the complex resistance is called
complex conductivity of a two-terminal network, i.e.
34
35.
Complex resistances of passive two-pole elements,
Resistance
u R t U m R cos t
Capacity
35
36.
InductanceComplex replacement circuits for elements
36
37.
Symbolic method of electrical circuit analysisExample
x(t)
u(t) = Umсos (ω t +)
i(t) = ?
e
37
38.
UmEnergy ratios
39.
Power Balance EquationAnalysis of simple circuits
Serial RL circuit
39
40.
Serial RC circuitSerial RLC circuit
40
41.
Parallel RLC circuit=
f = fp
f< fp
f > fp
42.
4243.
FREQUENCY CHARACTERISTICS OF ELECTRICAL CIRCUITSInput and transfer frequency characteristics
System circuit function
Input system functions
Transfer system functions
- voltage transfer function - current transfer function –
- transfer resistance - transfer conductivity -
43
44.
With harmonic influence, the system functions of the circuit are calledinput and transmission frequency characteristics
- complex response amplitude
- complex amplitude of influence
- frequency response,
- FCHH
The hodograph of the complex frequency response is
locus of complex numbers
when changing frequency
from 0 to ∞.
44
45.
Frequency characteristics of passive two-pole elementsResistance
=
Inductance
45
46.
CapacityFrequency characteristics of RL and RC circuits
46
47.
Input frequency responseGear frequency characteristics
Resonance in electrical circuits
Phenomenon sharp increase amplitude of the circuit response when approaching
frequency of influence to some well-defined values
called resonance.
By resonance we mean such a mode of operation of an electrical circuit,
containing capacitance and inductance, in which reactive
the input resistance and conductance components are zero.
47
48.
Series oscillating circuitC Z e
Z11` j Z 11` R j L 1
0
1
L.C.
,
f0
1
2LC
j
0 L 1 Ñ L C
0
Ratio of the effective voltage value on the reactive element
circuit to the effective voltage value on the circuit at the resonant
frequency is called the quality factor of the circuit.
49.
p2QDetuning
absolute
0 ,
relative
generalized
f f f 0 ;
f f
0
0
Q 0 2Q 2Q
0
0
,
f and fp are the values of the current and resonant frequencies, respectively. At resonance
.
all detunings are equal to zero, at f< fp они принимают отрицательные значения,
for f > fp – positive.
Input frequency response
C Z e
Z11` j Z 11` R j L 1
j
49
50.
frequency responseC
Z j Z R L 1
2
2
R 1 2
L 1 C
arctg
arctg
FCHH
R
U
U
I
e j U I e j I
Z j Z
I
I0
1
2
I U arctan
50
51.
,,
Gear frequency characteristics
Complex stresses on circuit elements
U C
U C e
jC
U L U L e j
L
U R U R e
R
I0
j
1
j I 90
I
e
U 1Q
e j I 90
C
1 2
C 1 2
LI 0
1
j I 90
j I 90
j L I
e
U 1Q
e
2
2
p 1
1
j R
R I
R I0
1 2
e j I
51
52.
SelectivityThe ability of an electrical circuit to isolate oscillations of individual frequencies
from the sum of oscillations of different frequencies is called selectivity.
The frequency range in which the transmission coefficient decreases no more than
than √2 times compared to his maximum value, called
bandwidth
52
53.
Parallel oscillatory circuit53
54.
=At
0
1
,
L.C.
f0
1
2LC
=
Input frequency response
=
ρ
54
55.
=frequency response
FCHH
Z
ρ
=
Gear frequency characteristics
by voltage
55
56.
by currentFor a low loss circuit
56
57.
Influence of generator internal resistance57
58.
Frequency characteristics of coupled circuitsTwo circuits are said to be connected if the excitation electrical vibrations V
one of them leads to oscillations in the other.
Based on the type of element through which the connection is made, circuits are distinguished:
with transformer connection;
with inductive coupling;
capacitively coupled;
with combined (inductive-capacitive) coupling.
According to the method of connecting the connection element, circuits are distinguished:
With external communications;
with intercom.
58
59.
Complex substitution schemes1
2
Coefficient of communication
transformer connection -
internal inductive coupling internal capacitive coupling -
60.
Equivalent circuit 1Designations
60
61.
Types of resonanceFirst private
Second private
Difficult
Difficult
61
62.
AtZsv = jXsv
A – coupling factor
Normalized relative to
1. K< d, (A < 1)
Frequency response of current I2
weak connection
-
2. K > d, (A > 1)
-
strong connection
3. K = d, (A = 1)
-
critical link
62
63.
6363
64.
Electric circuits with mutual inductanceФ21 - magnetic flux penetrating the second coil and created by current
the first coil (the mutual induction flux of the first coil);
Ф12 - magnetic flux penetrating the first coil and created by current
the second coil (the mutual induction flux of the second coil);
Фр1 - leakage flux of the first coil;
Фр1 - leakage flux of the second coil.
Ф11 - self-induction flux of the first coil, Ф11 = Ф21 + Фр1
Ф22 - self-induction flux of the first coil, Ф22 = Ф12 + Фр2
f1, f2
- full streams, penetrating each of the coils
Ф1 = Ф11 ± Ф12
Ф2 = Ф22 ± Ф21
64
65.
Ψ = wФ = L iL1 = Ψ
11
⁄ i1
L2 = Ψ
22
⁄ i2
Ψ ij
M12 = Ψ
12
⁄ i2
M21 = Ψ
21
⁄ i1
= wi Фij
Law of Electromagnetic Induction
e
= - dΨ ⁄ dt
= -
(dΨ ⁄ di)(di ⁄ dt)
EMF induced in coupled coils
Coil terminal voltages
65
66.
Clamps of the same nameClamps of the same name are those clamps of magnetically coupled elements when
with the same direction of currents relative to these terminals (both currents “enter”,
or both currents “come out” from these terminals) magnetic fluxes both
elements are directed according to
Magnetic coupling coefficient
66
67.
Analysis of electrical circuits with mutual inductanceComponent equations for coupled inductances in complex form
(1)
System of equations electrical balance
(0)
67
68.
Equivalent transformations of circuits with coupled inductancesSerial connection
Parallel connection
Decoupling of magnetic circuits
68
69.
Fundamentals of the theory of quadripolesDefinitions and classification
Quadrupole - an electrical circuit of any complexity having four
external clamp.
Classification of quadripoles
- passive and active
-linear and nonlinear
- balanced and unbalanced
- symmetrical and asymmetrical
- by the nature of the elements included in
The composition of a quadripole network is distinguished:
69
70.
reactive quadripolesRC quadripoles
ARC quadripoles, etc.
-depending on the structure,
quadripoles are distinguished:
pavements
staircases
L-shaped
T-shaped
U-shaped, etc.
Transmission equations for quadripoles
Relations that connect complex voltages and currents acting
at the terminals of the four-terminal network are called transmission equations.
Dependents
Variables
U1, U2
I1, I2
U2, I2
U1, I1 U1, I2 I1, U2
Dependents
Variables
I1, I2
U1, U2
U1, I1
U2, I2 I1, U2
System
parameters
Y
Z
A
B
F
U1, I2
H
70
71.
7172.
Communication equationsTwo or more four-port networks with equal matrices at all frequencies
primary parameters are called equivalent.
The primary parameters of a quadripole network can be determined using
experiments of idle and short circuit on its terminals
Primary parameters of composite quadripoles
A quadripole is called composite if it can be represented
as a connection of several simpler (elementary) quadripoles.
72
73.
If, when connecting elementary quadripoles, nochanges in the relationships between voltages and currents, then the primary
the parameters of a composite four-port network can be expressed in terms of
primary parameters of the original quadripoles.
Connections of quadripoles that satisfy this condition
are called regular.
The following five main types are known
quadripole connections:
cascade;
parallel;
sequential;
Parallel-serial;
series-parallel.
Cascade connection
73
74.
Parallel connectionSerial connection
75.
Parallel-serial connectionSerial-parallel connection
75
76.
5. Mode of non-harmonic influences1. Classic method of analysis
X(t) - impact
Y(t) -reaction
Calculation procedure
1 write down the differential equation of the circuit
*
n - order of the electrical circuit
76
76
77.
Examplei(t) = iR = iL
uR + uL = e(t)
uL
=
+RI=
2. Solution differential equation chains
-
free and forced components of the chain reaction
77
78.
=a) simple (different) real roots
b) equal real roots
c) pairwise complex conjugate roots
Example
=
79.
(**)-partial solution of equation (*).
3. At the final stage of the analysis, the integration constants Ak are determined
To do this, substitute the values into equalities (**)
, as well as initial
conditions and solve the resulting equation.
79
80.
Integral representations of signals.Spectral representations of non-harmonic signals. (Generalized Fourier series)
Definitions:
1. Signal energy -
2. Scalar product two signals
=
=
3. Two signals are called orthogonal if their scalar product
equals zero.
Generalized Fourier series for the signal S(t) in an orthogonal basis
(V(t)) has the form:
81.
Fourier series for a periodic signalPeriodic signal
=
On the interval
let's ask
orthogonal basis (V(t))
the following type
Spectral decomposition
.
.
.
.
.
.
.
.
.
.
.
81
82.
Fourier integral=
Inverse Fourier Transform
82
83.
Decomposition theoremIf F(p) can be represented as a ratio of two polynomials in p,
having no common roots
1)
Moreover, the degree of the polynomial N(p) is higher than the degree of the polynomial M(p), and
the equation N(p) = 0 has no multiple roots, then
And
at real values roots of the equation N(p) = 0,
, represents the sum of n exponents
Complex conjugate roots correspond to exponentially decreasing
law of harmonic oscillation.
2)
If the equation N(p) = 0 has one root, equal to zero, i.e.
That
83
84.
Laplace transformDirect
Reverse
= 0
=0
Calculation methods
=0
1. Integration using the theorem
deductions
2. Original tables - image
3. Expansion of L(p) simple fractions followed by
using tables original - image
84
85.
Time Domain Representations of SignalsAt
85
86.
Circuit representationsPerformance
signal
Circuit Description
S(t)
Equivalent circuit (design diagram)
F
Integrated equivalent circuit
L(p)
Operator substitution scheme
A complex equivalent circuit follows from the design diagram of the circuit by replacing the harmonics
energy, their complex amplitudes, and circuit elements - their complex resistances.
The operator equivalent circuit follows from the design circuit diagram by replacing the harmonic
oscillations describing the setting voltages and currents of independent sources of electrical
energy, their L-images, and the circuit elements – their operator resistances.
Operator equivalent circuit of capacitance
86
87.
Operator equivalent circuit of inductanceOperator resistive equivalent circuit
System functions of electrical circuits
ω
Input system functions
Input operator impedance
Input operator conductivity
87
88.
Transfer system functionsOperator voltage transfer function
Operator transfer function for current
Operator transfer resistance
Operator transfer conductivity
Determination methods
1. Based on the differential equation of the circuit
This equation in operator form has the form:
88
89.
ExampleDefine
A) Input operator conductivity
B) Operator room transfer function By
voltage
A)
B)
89
90.
2. Based on the analysis of operator circuit equivalent circuitsBy replacing bipolar elements in a given electrical circuit with their operator equivalent circuits, and
specifying currents and voltages of independent sources of electrical energy by their L-images, we obtain
operator equivalent circuit of a given circuit. When writing the electrical equilibrium equations for
L-images of independent variables can use all methods that are used for this purpose
in the symbolic method of electrical circuit analysis. It is clear that in this case the complex amplitudes of reactions and
influences should be replaced by their L-images, and complex resistances (conductivity) -
operator resistances (conductivities). As a result of the analysis of the operator diagram
chain substitution, the L-image of the required reaction of the chain is determined and after dividing it by the L-image
input influence - the desired system characteristic of the circuit.
Example
90
91.
Replacing the operator p with jω in the expression for H(p), we obtain the complex input orcircuit transfer function
Pulse and transient characteristics of an electrical circuit
Response of an electrical circuit to an impact in the form of a δ-function
is called the impulse response of this circuit -
Reaction of an electrical circuit to an impact as a function of a single
the jump is called the transition characteristic of this circuit -
Impulse and complex transfer functions of an electrical circuit
are connected by a pair of Fourier transforms, i.e.
91
92.
Impulse and operator transfer functions of an electrical circuitare connected by a pair of Laplace transforms, i.e.
Frequency (spectral) method of analysis of electrical circuits
Necessary:
define a comprehensive spectral density impact -
determine the complex transfer function of the circuit determine the complex spectral density of the circuit reaction -
Determine the reaction of a circuit in the time domain -
92
93.
Example93
94.
Conditions for distortion-free transmission of signals through an electrical circuit,
If the spectrum of input influence S(t)-
then the spectrum
-
And
As follows from the last expression, a distortion-free electrical circuit has a constant frequency response
for any values of w, the phase response of this circuit is linear.
Complex transfer function of a multi-link electrical circuit.
94
95.
Operator method for electrical circuit analysisNecessary:
determine L-image impact
determine the operator transfer function of the circuit – Н(р)
determine the L-image of the chain reaction -
determine the response of a circuit in the time domain -
Example
95
96.
Timing Method for Electrical Circuit AnalysisDuhamel integral
1.
2.
96
97.
3.
4.
If the impact is described by two different functions acting on different
sections of the time axis i.e.
97
98.
Procedure for calculating the reaction of a chain, Necessary:
determine either impulse or transient response chains
using one of the forms of writing the Duhamel integral, determine the desired reaction of the chain
Example
H(p) =
Differentiating electrical circuits
98
99.
ψ(ω)=
H(p) =
=
=
=
- circuit time constant
H(p)
=
At R<< 1/pC
Therefore, at R<< 1/
WITH
100.
voltage taken from the resistor of a series RC circuithas a form close to the derivative of the action.
The transient response of the RC circuit has the form
sequential RC circuit is called practically differentiating if
upper operating band frequency
input frequencies. For the signal shown above,
Active Differentiation Circuit
at μ =
H
τ=
100
101.
=Integrating electrical circuits
ψ(ω) =
H(p) =
101
102.
==
=
τ
H(p)
At R >> 1/pC
therefore, for R >>
the voltage removed from the capacitor has a shape close to the integral of
impact.
The transition characteristic has the form
a series RC circuit is called practically integrating if
τ
0.1R
lower frequency of the operating frequency band
impact
102
103.
HActive integrating circuit
at μ = ∞
H
The book consists of two parts and is a textbook on the basics of the theory of electrical circuits, intended to provide methodological assistance to university students studying in the direction of "Radio Engineering" with their independent work on mastering the circuit theory course. Unlike previous editions, this edition of the textbook includes as electronic application a collection of problems on the basics of circuit theory, which was previously published as a separate book. The book outlines the fundamentals of the theory of linear electrical circuits with lumped and distributed parameters in steady-state and transient modes, as well as the fundamentals of the analysis of nonlinear resistive circuits at direct current and under harmonic influence. Circuits with controlled sources, non-reciprocal four-terminal networks, ideal operational amplifiers, resistance converters and active filters are considered.
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