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Файл:The elements of the electrical circuit theory. Tutorial
.pdf
Ministry of Science and Higher Education of the Russian Federation
Federal State Budgetary Educational Institution of Higher Education
NOVOSIBIRSK STATE TECHNICAL UNIVERSITY
A. V. BLANC
THE ELEMENTS
OF THE ELECTRICAL
CIRCUIT THEORY
TUTORIAL
NOVOSIBIRSK
2025
UDC 621.3.011.7(075.8)
B64
Reviewers:
Associate professor Yu. V. Morozov, PhD (Eng.)
PhD (Eng.) V. V. Grechkin
The tutorial was prepared at the Department of Theoretical Foundations
of Electrical Engineering for students studying the electrical circuit theory
Blanc A. V.
B64 The elements of the electrical circuit theory : tutorial /
A.V. Blanc. – Novosibirsk : NSTU Publisher, 2025. – 120 p.
ISBN 978-5-7782-5501-2
This tutorial is the collection of the lectures on the initial knowledge about the
linear electric circuits, and it covers only the linear direct current circuits and the linear single-phase sinusoidal circuits. At the same time, the methods, which are usually
used for analyzing the linear direct current circuits, are considered in the most detail,
since although the linear direct current circuits are the easiest electric circuits, but in
studying them, the important knowledge is able to be obtained that can be extraordinarily useful in analyzing the more complex electric circuits. When considering the
linear single-phase sinusoidal circuits, the attention is paid mainly to the specific features of these circuits, which differentiate them from the linear direct current circuits.
UDC 621.3.011.7(075.8)
Alexey V. Blanc
THE ELEMENTS OF THE ELECTRICAL CIRCUIT THEORY
In the author’s edition
Managing Editor I.P. Brovanova
Art Director A.V. Ladyzskaya
Tax credit-the Russian Classification of product
Edition corresponds to 3000 95 OK 93-005 (OKP)
Signed Print 10.11.2025. Format 60 84 1/16. Newsprint. Uch.-ed. l. 6,9. Pecs. l. 7,5
ISBN 978-5-7782-5501-2 Blanc A. V., 2025
Novosibirsk State
Technical University, 2025
Quantity 50 copies. Ed. No. 157. Order No. 232
Printed in the Printing House
of the Novosibirsk State Technical University
630073, Novosibirsk, Russia, 20, Karl Marx prospect
DTP N.V. Gavrilova
PREFACE
The book, which is now offered to the reader, has the significantly
smaller volume compared to the most other books on the circuit theory that
have been published for many years. Generally speaking, it is the mere collection of twenty lectures on the initial knowledge about the linear electric
circuits, and it covers only the linear direct current circuits and the linear
single-phase sinusoidal circuits.
At the same time, the methods, which are usually used for analyzing the
linear direct current circuits, are considered in the most detail. Since, although the linear direct current circuits are the easiest electric circuits, but in
studying them, the important knowledge is able to be obtained that can be
extraordinarily useful in analyzing the more complex electric circuits that
work in the wide variety of the modes. In other words, the analysis of the
linear direct current circuits lays the foundation for the further successful
study of the more complex methods of the analysis of the electric circuits.
The linear single-phase sinusoidal circuits are considered in less detail.
When considering them, the attention is paid mainly to the specific features
of these circuits, which differentiate them from the linear direct current circuits. For example, the great attention is paid to the behavior of the current
and the voltage in the reactive elements of the sinusoidal circuit, the features
of the power balance in the sinusoidal circuit. In addition, the properties of
the complex numbers (on the use of which the symbolic method of the analysis of the sinusoidal circuit is based) are considered in detail.
In some chapters, the narrative is more brief but more detail in other
chapters. In any case, the offered book is only the introduction to the circuit
theory, which must be followed by the study of the more complex, specialized tutorial literature.
1. THE ELECTRIC CIRCUIT AND ITS ELEMENTS
Let us consider the electric circuit as the aggregate of the elements for
the conversion of the electric energy (Fig. 1.1). In the electric circuit, there
are energy sources and receivers. There are also branches and nodes.
Fig. 1.1
The energy sources (that are subdivided into the EMF sources and the
current sources shown in Fig. 1.2) are the circuit elements in which nonelectric energy is converted to electric energy.
Fig. 1.2
4
The energy receivers subdivided into the resistance elements and the reactance elements. In the resistance elements (Fig. 1.3), electric energy is
converted into thermal energy. The reactance elements subdivided into the
inductance coils and the capacitors (Fig. 1.4). In the inductance coils, energy accumulates in the magnetic field. In the capacitors, energy accumulates
in the electrostatic field.
Fig. 1.3
Fig. 1.4
Fig. 1.5
5
There are certain energy parameters (the electric potential, the voltage
and the current) that are used to describe the energy distribution in the electric circuit.
The electric potential (hereinafter denoted as φ, its dimensionality is
Volt) is the function that determines the energy distribution between the
elements of the electric circuit (Fig. 1.5). The potential is the multiplevalued function that can only be determined up to a certain constant. Therefore, before analyzing the circuit, it is necessary to set the potential of some
point in the circuit (usually the potential of some point is zero).
If there are no the sources and the receivers between two points of the
branch, the potentials of these two points are equal to each other.
The voltage (hereinafter denoted as U, its dimensionality is Volt) is the
difference of the potentials between two points of the electric circuit
(Fig. 1.6, a). In the electric circuit, the voltage is shown as the arrow directing from the higher potential to the lower potential. As for the letter U, the
first subscript corresponds to the higher potential, the second subscript corresponds to the lower potential.
The voltage is a vector quantity. The voltage sign changes when the arrow direction changes or when the order of subscript changes (Fig. 1.6 b).
The current (hereinafter denoted as I, its dimensionality is Ampere) is
shown as the arrow in the branch (Fig. 1.7).
a) b)
Fig. 1.6
6
Nota bene! The current (as well as the voltage) is a vector quantity directed from the higher potential to the lower potential. Unlike the current
and the voltage, the EMF source is directed from the lower potential to the
higher potential.
Fig. 1.7
The current is related to the branches and nodes of the circuit as follows.
The branch is the element of the circuit trough that the same current flows.
The node is the connection of at least three branches.
Nota bene! It is necessary to note that the potential distribution and the
directions of currents and voltages are unknown before analyzing the circuit. Therefore, the directions of currents and voltages are set arbitrarily before analyzing. If the current (or the voltage) obtained during the analyzing
is negative, this means that its true direction is opposite to the direction setting before the analyzing.
The current in the branch is associated with the voltage by the unique
dependence that is called the volt-ampere characteristic (Fig. 1.8). The voltampere characteristic can have any form (Fig. 1.8 a), and in particular, can
be linear (Fig. 1.8 b).
The element of the electric circuit is called the linear element if its voltampere characteristic is linear. The electric circuit, which consists only of
linear elements, is called the linear electric circuit.
7
a) b)
Fig. 1.8
The expression (known as Ohm’s law for the passive branch) is true for
linear elements:
UIR
(1.1)
where R is the proportionality coefficient between the current and the voltage that is called the resistance of the element (its dimensionality is Ohm).
In addition, the resistance can be defined as the tangent of the angle between the volt-ampere characteristic and the current axis (see Fig. 1.8 b):
tanR . (1.2)
The quantity that is the inverse to the resistance is called the conductance (hereinafter denoted as g).
In linear circuits, the branch resistances are constant, they are determined only by the physical properties of the conductors and do not depend
on the sources, the currents and the voltages.
If the sources of the electric circuit generate voltages and currents that
do not change over time, this circuit is called the direct current circuit. In the
direct current circuit, the resistance of inductance coils is zero and the resistance of capacitors is infinity.
Next, the linear direct current circuits are describes.
8
2. THE EMF SOURCES
AND THE CURRENT SOURCES
The electromotive force (EMF) is the maximal voltage that can be generated at the source output by extraneous forces if there are no currents in
the circuit. Extraneous forces can be, for example, the chemical reactions of
the galvanic battery or the mechanical moment of the electric machine operating in generator mode.
For ease of analysis, the energy sources are usually represented either by
the ideal EMF source or by the ideal current source. The ideal EMF source
and the ideal current source are also called the sources of infinite power.
The volt-ampere characteristic of the ideal EMF source is shown in
Fig. 2.1 a. The main distinguishing feature of the ideal EMF source is that
the voltage at its output is equal to EMF regardless of the load current. The
volt-ampere characteristic of the ideal current source is shown in Fig. 2.1 b.
The ideal current source keeps its current invariable regardless of the voltage at its output.
a b
Fig. 2.1
If we apply formulas (1.1) and (1.2) to the volt-ampere characteristics of
these ideal sources, we conclude that the resistance of the ideal EMF source
is zero and the resistance of the ideal current source is infinity. Let us further call the resistance of the sources the internal resistance.
9
The real source of electrical energy has some finite internal resistance.
R
s
s
The volt-ampere characteristic of the real source is shown in Fig. 2.2 and
can be determined by the following expression:
where
is the internal resistance of the real source;
in
UU IR (2.1)
nl in
U is the no-load
nl
voltage of the real source.
Fig. 2.2
When the real source is not connected to the load, there is, at its output,
the no-load voltage
U that is equal to its EMF. If we connect the output
nl
terminals of the real source together, the voltage at the output will be zero,
and the current at the output will be equal to the short-circuit current
I .
c
If we compare the volt-ampere characteristic of the real source (see
Fig. 2.2) with the volt-ampere characteristics of the ideal sources (see
Fig. 2.1), we conclude that the real source (Fig. 2.3 a) can be modeled either
using the ideal EMF source and the series-connected internal resistance
(Fig. 2.3 b), or using the ideal current source and the parallel-connected internal resistance (Fig. 2.3 c).
The expression is true for the internal resistance of the real source:
U
RI
nl
in
. (2.2)
c
10
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