Theoretical fundamentals of electrical engineering. Problem book Part I. DC and AC linear electrical circuits

MINISTRY OF TRANSPORT OF THE RUSSIAN FEDERATION
FEDERAL STATE AUTONOMOUS EDUCATIONAL INSTITUTION OF HIGHER EDUCATION
"RUSSIAN UNIVERSITY OF TRANSPORT"



Department of power supply on electric railways



THEORETICAL FUNDAMENTALS OF ELECTRICAL ENGINEERING



PART I DC AND AC LINEAR ELECTRICAL CIRCUITS




PROBLEM BOOK










Moscow - 2022

MINISTRY OF TRANSPORT OF THE RUSSIAN FEDERATION
   FEDERAL STATE AUTONOMOUS EDUCATIONAL INSTITUTION OF HIGHER EDUCATION
"RUSSIAN UNIVERSITY OF TRANSPORT"



Department of power supply on electric railways



THEORETICAL FUNDAMENTALS OF ELECTRICAL ENGINEERING



PART I DC AND AC LINEAR ELECTRICAL CIRCUITS



Problem Book
Advanced problems For students of the specialty 23.05.05 "Train support systems" programs "Railway Power supply" "Telecommunications and railway transport networks" "Automation and telemechanics in railway transport"

Moscow - 2022

    UDK 621.3


     T 44

     Theoretical fundamentals of electrical engineering. Problem book Part I. DC and AC linear electrical circuits: Advanced problems for students of electrical specialties and specializations of the university. Revised edition/Vlasov S.P., Volyntsev V.V., Kosarev B.I., Kruchinin E.V., Simakov A.V. -M.: RUT (MIIT), 2022. - 180 pages.

     The Problem book is intended for in-depth study of the TFEE course and can be useful in preparing students for participation in electrical Olympiads. The Problem book contains problems of increased complexity with solutions and without solutions for linear circuits of direct and sinusoidal current.


© RUT (MIIT), 2022

    PREFACE
    This Collection of advanced problems for TFEE is a significantly supplemented and revised edition of the Problem book 2019 [1]. Both editions contain problems of increased complexity for linear DC circuits (1st section) and linear AC circuits (2nd section).
    Most of the problems of the Problem book [1] were compiled by Associate Professors of the Department of TFEE MIIT V.F. Klimov, A.P. Milyutin, A.I. Shchurov, V.S. Vorotnikov, etc., and were proposed for solving to participants of electrical Olympiads at MIIT in different years.
    In addition to these problems, the present Problem book includes the problems proposed for solving to the participants of the Moscow City Olympiads on TFEE in MPEI [2], includes some problems from well-known TFEE problem books edited by K.M. Polivanov [3], L.A. Bessonov [4], P.A. Ionkin [5], O.E. Goldin [6], M.R. Shebes. Also we used some materials of the textbook “The fundamentals of circuit theory”/G.V. Zeveke, P.A. Ionkin, A.V. Netushil, S.V. Strakhov [8].
    Except mentioned above, here are problems of increased complexity from educational publications on electrical engineering of the Bauman Moscow State Technical University (National Research University) [9], Ivanovo state Power University named Lenin [10], Almaty University of Power Engineering and Telecommunication named Gumarbek Daukeev [11], Igor Sikorsky National Technical University of Ukraine (NTUU) and Donetsk National Technical University (DonNTU) [12].
    The increased complexity of the problems in the Collection is mainly associated with the unusual formulation and with the non-standard approach to solving, despite simplicity of the problem circuits.

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    Every problem has the answer. Most problems are provided with the recommendations for its solution and, often, a solution is proposed in a concise, laconic form. As a rule, problems without solution are similar to problems with solution and are intended to develop self-solving skills.
    The authors hope that working with the problem book will help students increase their level of understanding of theoretical positions, such as the input and mutual conductivities of branches, the linear relations in electrical circuits, the compensation theorem, the equivalent generator theorem, the load matching with the electrical circuit, the voltage and current resonances, the complex resonances in the branched circuits and in the circuits with the presence of mutually inductive connections, etc.
    The authors believe that this Problem book will be useful to teachers in organizing the independent work of students of electrical specialties during the semester, as well as in preparing the most successful students for the TFEE Olympiads.
    All comments and wishes on the content of the Problem book should be sent to the authors at the Department of power supply on electric railways RUT (MIIT).

Authors.

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    In problems, unless otherwise specified, we consider electricity-measuring instruments as ideal instruments of electrodynamic system.

    1. DC LINEAR ELECTRICAL CIRCUITS



             Problem 1.1.
     For the circuit shown in fig.1.1, it is known:
     at series connection of resistors r₁-r₂ (fig. 1.1a) the consumption power in the second resistor was 5 times more P
than in the first resistor. What is the resistors power ratio 2^- at

parallel connection r1//r2 (fig.1.1b)?


Fig. 1.1

    Answer: —.


    Solution


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    1. At series connection of resistors
P2 = 5 . So,
Pi
P2 = r2 • I" = % = 5
P    r • I² r     .
  ₁  r₁ •    r₁
    2. At parallel connection of resistors
P₂ _ U² /r₂ _ r _ 1
P = rr = ъ = 5 .



           Problem 1.2.
    For the circuit whose graph is shown in fig. 1.2, considering the currents I₁, I₂, I₄, I₅ been known, find the current I3.



Fig. 1.2

    Answer: I₃ = I +1₂.

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     Solution
     Kirchhoff's 1st law is valid not only for any node of the circuit, but for any closed volume that cuts the branches of the electric circuit with its surface. For a «flat» circuit, the cutting surface is a closed line. Let us use the 1st Kirchhoff’s law for the section in the dashed loop cutting the 1st, 2nd and 3rd branches (fig. 1.2).
I +1₂ -1₃ = 0 or I₃ = I +1₂.



            Problem 1.3.
    For the circuit shown in fig. 1.3, it is known:
    Triple-poles A1 and A2 are active.
    r = 100Q; rᵥ = 1000Q; | = 0,1A; I₂ = 0,2A.
    Calculate voltmeter reading.


Fig. 1.3

    Answer: 27.3 V.


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    Solution
    Let us use the Kirchhoff's 1st law for the section in dashed loop in fig. 1.3.
I₃ = I + i₂ = 0,3A.
    Then

UV =

r • rv r + rv

• I3 ⁼

100 4000 ---------0,3 100 +1000

« 27,3V.

            Problem 1.4.
    Determine the resistance value r at which the change in resistance r does not cause a change in currents in the other circuit elements (fig. 1.4).

Fig. 1.4

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Answer: r

(r5 ⁺ r6 И r3 • r7 - r2 • r8 ) r6 •( r6 ⁺ r8 )⁻r3 •( r5 ⁺ r7 )


    Solution
    The solution is based on the fact that the change in the resistor r does not affect the currents in the circuit only if there is no current in resistor r , that is, the circuit will operate in the balanced bridge (measuring bridge) mode.
    Convert the triangle cbd to an equivalent star (fig. 1.4.1).

Fig. 1.4.1

     In this circuit

_    .   r5 • r0   .
r50      ,     ,   ;
       r₅ +r₆ +r₀

       r • r
    —⁶—0—.
r60         ;
    r₅ +r₆ +r₀

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