Mathematical problems and computer simulation in power engineering: methodical instructions for laboratory works, 9786010446915

The methodological instructions for the discipline «Mathematical prob-lems and computer simulation in power engineering»

464 120 3MB

English Pages [60] Year 2020

Report DMCA / Copyright

DOWNLOAD FILE

Polecaj historie

Mathematical problems and computer simulation in power engineering: methodical instructions for laboratory works,
 9786010446915

Citation preview

AL-FARABI KAZAKH NATIONAL UNIVERSITY

MATHEMATICAL PROBLEMS AND COMPUTER SIMULATION IN POWER ENGINEERING Methodical instructions for laboratory works

Almaty «Qazaq University» 2020

UDC 519.7(075) LBC 22.18я73 M 39 Recommended for publication by the decision of the Academic Council of the Faculty of Physics and Technology, Editorial and Publishing Council of Al-Farabi Kazakh National University (Protocol No.4 dated 19.06.2020) Reviewers: Candidate of science degree on physics and mathematics, Associate professor K.A. Iskakova Candidate of science degree on physics and mathematics, Associate professor S.A. Zhaugasheva

M 39

Mathematical problems and computer simulation in power engineering: methodical instructions for laboratory works / K.Е. Nurgaliyeva, M.K. Isanova, S.K. Kodanova [et al.]. ‒ Almaty: Qazaq University, 2020. ‒ 59 p. ISBN 978-601-04-4691-5 The methodological instructions for the discipline «Mathematical problems and computer simulation in power engineering» are intended for use as a methodological textbook on laboratory works. Laboratory works using MatCAD and Multisim programs will help students to learn the basic principles of computer modeling, solve mathematical problems in electric power engineering and use topological and matrix-vector methods. Each work is provided with a brief theoretical introduction, description of the order of work and control questions. The manual is intended for students studying in the specialty of Electric Power Engineering.

UDC 519.7(075) LBC 22.18я73 ISBN 978-601-04-4691-5

© Nurgaliyeva K.Е., Isanova M.K., Kodanova S.K., Masheeva R., Yerimbetova L.T., 2020 © Al-Farabi KazNU, 2020

2

TABLE OF CONTENTS Foreword......................................................................... ............................ 4 Chapter I. Fundamentals of simulation in Multisim environment and Mathcad software.............................. .................................................. 5 Laboratory work 1.1. Introduction to Multisim. Building an equivalent circuit of the electrical network............................. 5 Laboratory work 1.2. The topological method of analysis of electric circuits............................................ ........................................... 9 Chapter II. Matrix notation of basic laws of electric circuits. An analysis of the stable state in linear electric circuits ............................. 17 Laboratory work 2.1. Calculation of electric circuits with the help of state equations....................................... ................................... 17 Laboratory work 2.2. Mesh analysis of electric circuits ............................ 20 Laboratory work 2.3. Nodal analysis of electric circuits ............................ 24 Chapter III. Approximation of characteristics of nonlinear elements of electric circuits............................. ........................................... 27 Laboratory work 3.1. The least-square method for nonlinear dependences ......................................................................... 27 Laboratory work 3.2. Approximation of the nonlinear dependences by the spline method .................................................................................. 31 Chapter IV. Analysis of transitional processes in electric systems by differential equations …………….. ........................................ 33 Laboratory work 4.1. Analysis of transitional processes in electric systems by simple numerical Euler method for the solution of differential equations.......................... ................................................... 33 Laboratory work 4.2. Analysis of transitional processes in electric systems by modified numerical Euler method for the solution of differential equations.......................... ................................................... 40 Laboratory work 4.3. Analysis of transitional processes in electric systems by numerical Euler-Cauchy method for the solution of differential equations.......................... ................................................... 45 Appendix 1 ................................................................................................ 51 Appendix 2 ................................................................................................ 54 Appendix 3 ................................................................................................ 57

3

FOREWORD Today, rapid development of computers and their integration in all fields, allows us to modernize solution and mathematical simulation of stationary and transient processes in electrotechnical and energy systems. In addition, the process is based on analytical analysis and information on parameter systems and topological links with the following elements, which should be considered as a model. Projecting a model, we can analyze the system status in different modes of operation. The methods of electric and magnetic circuit analysis are based on the methods of matrix calculations, graph theory, numerical solution of nonlinear algebraic and differential equations. The manual includes the theoretical material of the lectures, which are read according to the course work plan, the theoretical basis, the tasks of the laboratory works, the algorithm and methods of their execution, the methodological instructions for performing laboratory works. The appendix contains variants of the calculation schemes and tables of values. The focus is mostly on practical training for computer modeling, especially for working in Mathcad and Multisim environment. The authors hope that the proposed methodological manual will help students to study the discipline «Mathematical Problems and Computer Modeling in Electrical Power Engineering».

4

CHAPTER I FUNDAMENTALS OF SIMULATION IN MULTISIM ENVIRONMENT AND MATHCAD SOFTWARE

Laboratory work 1.1 INTRODUCTION TO MULTISIM. CONSTRUCTION OF AN EQUIVALENT CIRCUIT OF THE ELECTRICAL NETWORK The aim is to learn how to make an equivalent circuit for electric power networks in the Multisim environment. Tasks: ‒ Introducing the Multisim Environment ‒ Constructing a given electric circuit in the Multisim environment ‒ Learning how to work with a multimeter, to measure current and voltage. Brief theory: Multisim circuit simulation systems are designed to simulate electrical and electronic circuits. Multisim is also an electronic lab that makes circuit research available in the lab. It also protects from real laboratory mistakes. We use a linear device library to study linear circuits. We can select them in dialog boxes. After clicking the «View» toolbar, Tools and Components panels will open. There are several types of components in the Multisim environment: Family Components – a stock of production settings; Virtual Components – a virtual component store, which the client sets by himself; Rated Virtual Components – virtual components with restrictions, the simulation scheme will be alerted if the input parameters 5

are exceeded (the component window is in the Value window), it will fail. 3D Virtual Parts – (3D) three-dimensional virtual components. There are 17 different components in the component resource (Fig.1.1), in the window that opens, select the Group menu and select the required component from the drop-down list. For example, in the Basic window (to the left of the column) you can select a Resistor. In the second column, you will find the value of the impedance (for example, 1k for 1kOhm), specify the percentage of deviation from the nominal and press the OK button with the mouse. The graphic image of the component is displayed at the top of the third column of Figure 1.1 and is displayed on the screen when designing the wiring diagram with the mouse button selection. You can set other options in the dialog box by double-clicking the component graphic image. For example, the temperature of the research, the set parameters. The main resistor parameter values (1k) cannot be changed, it can be replaced (in the Replace menu) by another resistor and the resistor parameters are selected according to the normal values.

Figure 1.1. Component resource

6

The list of elements in the component resource is given in Table 1.1. Important and remarkable: to activate the schema click on the button at the top right as follows: Table 1.1

Component resource Source – Energy source Basic – Basic components Diode – Diode Transistors – Transistors Analog – Analog microcircuits TTL – Logical microcircuits CMOS – CMOS microcircuits Misc Digital – Digital microcircuits Mixed – Analog-to-digital converters

Indicator – Lighting devices Power – Power supplies Miscellantous – Mixed type components Advanced Peripherals – Peripheral devices RF – Radio elements Electromechanical – Electromechanical devices MCU Modul – The key elements of digital electronics Hierarchical Block – Block circuits

Process of the work: 1. The basic information for the electrical circuit is given in the sample, draw your sample in your lab notebook and write down the parameters (Appendix 1). 2. Build a physical model as an electrical circuit in the Multisim environment and measure the values of currents and voltages. The examples are shown in Figures 1.2 and 1.3. Fill in the obtained values in Table 1.2. Table 1.2

Physical model values measured with a multimeter Number of items Current value I, A Voltage value, U, В

1

2

...

...

...

...

...

...

3. Save the result in the Word using the «print screen». Report on your lab work, and sum up your work. 7

Figure 1.2. Measuring current with a multimeter

Figure 1.3. Measuring voltage with a multimeter

8

Control questions: 1. What is the Multisim Circuit Simulation System designed for? 2. How do we find the circuit elements in Multisim? 3. How do you simulate Multisim circuits? 4. How does a multimeter work in Multisim? References: 1. Marchenko A.L. Fundamentals of Electronics. Textbook for High Schools, 2009. – 185 p. (in Russian)

Laboratory work 1.2 THE TOPOLOGICAL METHOD OF ANALYSIS OF ELECTRIC CIRCUITS The aim is to get familiar with topological methods of analyzing equivalent schemes with the help of the graph, to learn how to constract incidence matrixes. Tasks: ‒ Draw the graph of the given electric circuit. Show the tree and the co-tree. ‒ With the help of the directed graph construct the incidence matrices in Mathcad. Proof the correctness of your matrices. Brief theory: In technology, the term node is commonly used to refer to a junction of two or more branches. A loop is a closed path of branches that begins from one node and ends again at the same node. A mesh is a loop that does not contain any other loop inside. The structural scheme of the electric circuit can be shown by a directed graph, which means that the interconnection of nodes and branches in the graph is sketched by nodes and edges of the graphs. The elements of electric circuits like emf or current source, resistances and other elements are not shown in the graph. By the direction of the edges the currents flow and voltage decreasing directions are defined. Let us take as an example a scheme shown in Figure 1.4 and draw its graph. 9

Figure 1.4. An electric circuit and its graph

It should be noticed that the branch with the current source is not shown in the graph. Usually the directions of edges are co-directed with the current direction. The edges of the graph are divided into a tree and a co-tree. For one electric scheme there are a lot of versions of how to draw a tree and a co-tree. A tree is an oriented connected subgraph of an oriented connected graph containing all the nodes of the graph, but, containing no loops. The branches of the tree are called twigs. The remaining branches of the graph are called links or chords. The links form a subgraph, not necessarily connected, are called a co-tree. Co-tree is the complement of the tree. There is a co-tree for every tree. See the example in Figure 1.5. Here (1,3,6) are chords and (2,4,5) are trees.

Figure 1.5. Tree and chords

10

There are several incidence matrices that are important in developing various networks matrices such as bus impedance matrix, branch admittance matrix, etc., using singular or nonsingular transformation. These various incidence matrices are basically derived from the connectivity or incidence of an element to a node, path, cutset or loop. The following incidence matrices are of interest in power network analysis. Element Node Incidence Matrix П0 shows the incidence of elements to nodes in the connected graph. The incidence or connectivity is indicated as in Figure 1.6. The element-node incidence matrix will have the dimension pij where 'i' is the number of elements and ‘j’ is the number of nodes in the graph. So, the element of the matrix is equal to 1 when the edge directed to the node and equal to -1 in the opposite case. If there is a no connection between the node and the edge the element is equal to zero.

j

edges

n

ПП 0 0= o

= d e s

i

Pij

1, if edge is directed to node

pij 

–1, if edge is counter directed to node

 

j

j

0, if there is no connection between edge and node

Figure 1.6. Element Node Incidence Matrix

The network usually contains a reference as a reference node. In fact, any node of the connected graph can be selected as the reference node. The matrix obtained by deleting the column corresponding to the reference node in the element node is called bus incidence matrix П. Thus, the dimension of this matrix is [m,n-1], where n is the number of nodes, m is the number of branches. Basic Loop incidence matrix Г shows the incidence of the elements of the connected graph to the basic loops. The incidence of the element is indicated as it shown in Figure 1.7. 11

Г=

e d g e s

j

loops

i

g ij

1, if branch is co-directed with loop

g ij  –1, if branch is counter-directed with loop

0, if there no connection between node and loop

Figure 1.7. Basic Loop incidence matrix

There is a ratio between bus incidence matrix and basic loop incidence matrix: П‧Г = 0.

(1.1)

Let us take n – the number of nodes, m – the number of branches and L – the number of independent loops. The current source vector should be constructed by nodes:  J1    J J   2.     Jn 

(1.2)

Here it should be noticed that if the current source is directed to the node, it should be taken with «+» sign and in the opposite case with «-» sign. Branch Admittance Y and Branch Impedance Z Matrices are square matrices with column and row numbers equal and let us take m as the number of branches.  Z1 0 Z 2 Z    0 0

0

12

0  0   Y 1 .   Zm 

(1.3)

The current in branches:  I1    I I  2 .      Im 

(1.4)

 Е1    Е Е   2 .      Еm 

(1.5)

Emf in branches:

Progress of work: 1. For the given variant from Appendix 1 take your own scheme and draw the directed graph. Show tree and co-tree. 2. Construct the bus incidence matrix П in your workbook. 3. Construct the Г basic loop incidence matrix in your workbook. 4. In Mathcad construct the bus incidence matrix П and the Г basic loop incidence matrix. For this purpose use Calculator Toolbar (Figure 1.8). Type «П» in Mathcad and choose : = from Calculator.

Figure 1.8. «Assignment» command

13

In order to construct matrices we need to use Vector and Matrix Toolbar (Figure 1.9).

Figure 1.9. Vector and Matrix Toolbar

In Matrix insrument pannel choose matrix constructor (shown by the arrow in Figure 1.10).

Figure 1.10. Matrix insrument pannel

Show the number of rows and columns in the appearing window and press «ok» (Figure 1.11). Consruct your matrix.

Figure 1.11. The number of rows and columns

14

5. With the help of multiplication in the Calculator find П‧Г on Mathcad (Figure 1.12). You will find the result as it is shown in Figure 1.13. With the help of «prt sc» button save your work in Word.

Figure 1.12. Multiplication in Calculator

Figure 1.13. Results

6. Consruct the current source, emf vectors and Branch Impedance Matrix. 7. Write a report on your laboratory work and make a conclusion. Control questions: 1. What is graph, tree,and co-tree? How to draw it? 2. What is the bus incidence matrix, how to construct it? 3. What is the basic loop incidence matrix, how to construct it? 4. How to construct the current source, emf vectors and Branch Impedance Matrix?

15

5. What do you know about the ratio between bus incidence matrix and basic loop incidence matrix? References: 1. Sivokobylenko V.F. Matematicheskoe modelirovanie v jelektrotehnike i jenergetike. – Doneck: RVA DonNTU, 2005. – 350 p 2. Matematicheskie zadachi jelektrojenergetiki: uchebnik dlja stud. Vuzov/pod red. V.A. Venikov – 2-e izd. Pererab. I dop. – M: Vyssh. Shkola, 1981. – 288 р. 3. Filjaev K.Ju. Matematicheskie zadachi jenergetiki: uchebno-metod. Kompleks, 2005. – Cheljabinsk.

16

CHAPTER II MATRIX NOTATION OF BASIC LAWS OF ELECTRIC CIRCUIT. ANALYSIS OF THE STABLE STATE IN LINEAR ELECTRIC CIRCUITS

Laboratory work 2.1 CALCULATION OF ELECTRIC CIRCUITS WITH THE HELP OF STATE EQUATIONS The aim is to learn how to use familiar topological methods of analyzing equivalent schemes with the help of Ohm’s and Kirchhoff’s laws Tasks: ‒ Write the Ohm’s law in matrix form using topology. ‒ Write the Kirchhoff’s current law in matrix form using topology. ‒ Write the Kirchhoff’s voltage law in matrix form using topology. Brief theory: Ohm’s law for branches is written as follows: (2.1)

Z  I U  E

here Z is the resistance matrix, I, U, E are current, voltage and EMF vectors, correspondingly. This equation can be rewritten as follows: (2.2)

Z  I  U  E

or with the help of conductivity matrix Y   Z  : 1

I  Y   U  E  . 17

(2.3)

The number of branches is p, then equation (2.2) would be written as:  Z11 I1  Z12 I 2  ....  Z1 p I p  U1  E1  Z I  Z I  ....  Z I  U  E  21 1 22 2 2p p 2 2 .  ......    Z p1I1  Z p 2 I 2  ....  Z pp I p  U p  E p

(2.4)

Kirchhoff’s currents law (KCL):

ПI  J 0.

(2.5)

Here П is the node incidence matrix, I is the edge current vector, J is the vector of current sources. If n is the number of nodes, m is the number of branches, the dimension of vector would be [1,n-1]. The vector representation of Kirchhoff’s voltage law (KVL): Г ТU  0 ,

(2.6)

there Г Т is a transposed basic loop incidence matrix, U is a voltage vector. The Ohm’s law, KVL and KCL can be written as joint blockmatrix as follows:  1 Z  E   U    . 0 П        J   ГЕ 0   I   0     

(2.7)

There are 2m unknowns and 2m equations, where m equations are Ohm’s law for m branches, (n-1) equations give KCL for (n-1) nodes and m-(n-1) equations represent KVL for m-(n-1) independent meshes. In total, we have m+(n-1)+m-(n-1) = 2m equations. Equation (2.7) can be solved by MathCAD totally, but its accuracy would be less and calculating time more in comparison with the next two. 18

1st method It is based on calculation of currents. For this aim we have to remove voltage. Then we can solve not 2m equations but m equations. To do this, we need to multiply both parts of Ohm’s law (2.1) by Г Т – transposed basic loop incidence matrix: Г Т  Z  I  Г Т U  Г Т  E .

(2.8)

Taking into account KVL (2.6) and KCL (2.5):  П   J   Т I    Т  .  Г Z  Г E

(2.9)

So, we can obtain р currents and then using Ohm’s law find voltages across all branches. 2nd method Using Ohm’s law in the form (2.3) multiply both parts by П node incidence matrix:

П  I   П  Y U  П  Y  E .

(2.10)

Taking into account KCL (2.5):

 П Y   J  П Y  E   Т   U    . 0  Г   

(2.11)

Using these р equations we can find voltages and then with the help of Ohm’s law we can calculate currents through the branches. Progress of work: 1. Take the initial values from the appendix 1. 2. Using the Ohm’s and Kirchhoff’s laws construct the mathematical model and by using Given Find  in Mathcad find curst rents and voltages according to the 1 or 2nd method. 19

3. Using the Ohm’s and Kirchhoff’s laws construct the mathematical model and by using the matrix method in Mathcad find currents and voltages according to the 1st or 2nd method. 4. Fill in Table 2.1 Table 2.1

Calculated values for the electric circuit Number of branches Current I, A (Given Voltage U, В (Given

1

2

...

...

...

...

...

...

Find ) Find )

Current I, A (matrix method) Voltage U, В (matrix method)

5. Compare the obtained values with the values in laboratory work 1.1. Control questions: 1. What kind of equations is contained in the joint block-matrix? 2. How many equations are there in Ohm’s law? 3. How many equations are there in Kirchhoff’s current law? 4. How many equations are there in Kirchhoff’s voltage law? 5. What kind of methods can be used to increase the accuracy of computation? References: 1 Sivokobylenko V.F. Matematicheskoe modelirovanie v jelektrotehnike i jenergetike. – Doneck: RVA DonNTU, 2005. – 350 p. 2 Matematicheskie zadachi jelektrojenergetiki: uchebnik dlja stud. Vuzov/pod red. V.A. Venikov – 2-e izd. Pererab. I dop. – M.: Vyssh. Shkola, 1981. – 288 р. 3 Filjaev K.Ju. Matematicheskie zadachi jenergetiki: uchebno-metod. Kompleks, 2005. – Cheljabinsk.

Laboratory work 2.2 MESH ANALYSIS OF ELECTRIC CIRCUITS The aim of this lab work is to learn how to use familiar topological methods of analyzing equivalent schemes with the help of independent currents and mesh analysis. 20

Tasks: ‒ Write the independent current equations for analysis of the given equivalent schemes using familiar topological methods ‒ Write the mesh currents equations for analysis of the given equivalent schemes using familiar topological methods Brief theory: If we decrease the equation number to n = p-(q-1), it will make our computation easier and more accurate. In order to find independent current equations we will construct the edge current vector in the sequence of the chords and the tree:

I   I ch , I t    I ind , I dep  . Т

Т

(2.12)

Here the currents of chordes are independent, the current of the tree is dependent. In construction of other vectors and matrixes we will preserve the aforementioned order. The Node Incidence Matrix will be written as follows:

It can be represented as constructed from two submatrixes П = [П1, П2], here:

П  I  J Kirchhoff’s current law will be rewritten as: П1  I ind  П2  I dep   J . 21

(2.13)

From this equation we can define the dependent currents:

I dep    П2   П1  I ind   П2   J . 1

1

(2.14)

Taking into account equation (2.14),

 I ind  I ind   I   dep     1 1 ind  I     П2   П1  I   П2   J  1 0   ind   . I  I   1 1  J   П   П  П   2 1 2   

Here

1   . Бi   1    П   П  2 1 

(2.15)

This matrix is called the current transformation matrix. The independent currents equation is written as:

 0  Г Т  Z  Бi  I ind  Г Т  E  Г Т  Z    Jk . 1   ( П2 ) 

(2.16)

Using this equation one can obtain the chord currents. Then using the transformation matrix the tree current can be obtained. Using the independent current equations we can find the mesh currents equation. For this aim we need to make the transformation operation symmetrical, that’s Бi  Г . Progress of work: 1. Find the initial data on electric circuit in Appendix 1. 2. For the given electric circuit construct the independent current equations and determine the value of currents and voltages in Mathcad. 22

3. For the given electric circuit construct the mesh current equations and determine the value of currents and voltages in Mathcad. 4. Write found values in Table 2.2. Table 2.2

Values determined by solving equations Edges number Current value I, A (independent current) Voltage value, U, V (independent current) Current value I, A (mesh analysis) Voltage value, U, V (mesh analysis)

1

2

...

...

...

...

...

...

5. Compare the results of the current work with the results of Laboratory work 1.1 and Laboratory work 2.1. Control questions: 1. What is the importance of independent current equations? 2. What is the dimension of matrixes and vectors in independent current analysis? 3. How is the current transformation matrix defined? 4. How is the mesh resistance matrix defined? 5. What is the diference between independent current equations and mesh current equations? References: 1 Sivokobylenko V.F. Matematicheskoe modelirovanie v jelektrotehnike i jenergetike. – Doneck: RVA DonNTU, 2005. – 350 p. 2 Matematicheskie zadachi jelektrojenergetiki: uchebnik dlja stud. Vuzov/pod red. V.A. Venikov – 2-e izd. Pererab. I dop. – M: Vyssh. Shkola, 1981. – 288 р. 3 Filjaev K.Ju. Matematicheskie zadachi jenergetiki: uchebno-metod. Kompleks, 2005. – Cheljabinsk.

23

Laboratory work 2.3 NODAL ANALYSIS OF ELECTRIC CIRCUIT The aim of this work is to learn how to use familiar topological methods of analyzing equivalent schemes with the help of independent potentials and nodal analysis Tasks: ‒ Write the independent potential equations for analysis of the given equivalent schemes using familiar topological methods ‒ Write the nodal potential equations for analysis of the given equivalent schemes using familiar topological methods Brief theory: The independent potential equations are derived on the base of Kirchhoff’s voltage law Г Т U  0 , for (n-1) unknowns we will solve (n-1) equations. The basis node potential is equal to zero, the voltage vector is constructed in the sequence of the chords and tree: U  U ch ,U tr   U ind ,U dep  Т

Т

(2.17)

KVL is written as: Т

0

(2.18)

U dep  ( Г 2Т ) 1  Г1Т  U ind

(2.19)

Г1Т , Г 2Т U ind ,U dep

Hence:

Voltage vector:

  U ind  U  1     Г 2Т   Г1Т  U ind    24

We can find the voltage transformation matrix: 1   . БU    1    Г 2Т   Г1Т   

(2.20)

Using KCL and Ohm’s law we can find independent voltages:

Y т U ind  J K  J KE ,

(2.21)

here Y m  П  Y  БU are independent admittances, J KE  П  Y  E are equivalent current sources. Using this equation we can find the chord voltages. Then, using the transformation matrix we can find the tree voltages. Using independent voltage equation we can obtain the nodal voltage equation. With this aim we need to make transformation operation symmetrical, i.e. БU  П Т . Progress of work: 1. Find the initial data on electric circuit in Appendix 1. 2. For the given electric circuit construct the independent voltage equations and determine the values of currents and voltages in Mathcad. 3. For the given electric circuit construct the nodal voltages equations and determine the values of currents and voltages in Mathcad. 4. Write the found values in Table 2.3 Table 2.3

Values determined by solving equations Edges number Voltage value, U, V (Independent voltages method) Current value I, A (Independent voltages method)

1

2

...

25

...

...

...

...

...

Voltage value, U, V (nodal voltages method) Current value I, A (nodal voltages method)

5. Compare the results of the current work with the results of Laboratory work 1.1, Laboratory work 2.1.and. Laboratory work 2.2. Control questions: 1. What is the importance of independent voltage equations? 2. What is the dimention of matrixes and vectors in independent voltage analysis? 3. Can you define the voltage transformation matrix? 4. How is the nodal admittances matrix defined? 5. What is the diference between independent voltage equations and nodal voltage equations? References: 1. Sivokobylenko V.F. Matematicheskoe modelirovanie v jelektrotehnike i jenergetike. – Doneck: RVA DonNTU, 2005. – 350 p. 2. Matematicheskie zadachi jelektrojenergetiki: uchebnik dlja stud. Vuzov/pod red. V.A. Venikov – 2-e izd. Pererab. I dop. – M.: Vyssh. Shkola, 1981. – 288 р. 3. Filjaev K.Ju. Matematicheskie zadachi jenergetiki: uchebno-metod. Kompleks, 2005. – Cheljabinsk.

26

CHAPTER III APPROXIMATION OF CHARACTERISTICS OF NONLINEAR ELEMENTS OF ELECTRIC CIRCUITS

Laboratory work 3.1 THE LEAST-SQUARE METHOD FOR NONLINEAR DEPENDENCES The purpose of the work:the least square approximation of the characteristics of nonlinear elements of electric circuit Tasks: ‒ Tablesize the given description. ‒ Approximation of VAC’s by the least square method. ‒ Display the result on the graph. Brief theory: To describe in analytical form the nonlinear dependence given by a graphical form the methods of approximation or interpolation are used. The least square method has often been used to determine the interpolation coefficient. Approximation by the least square method ‒ This method is used if the number of points of the original curve exceeds the degree of the polynomial. This method is based on the determination of the square deviation of the iterative functions from the curve.

S   U kисх  U kитер  n

2

(3.1)

k 1

n d d d S 0 S  2 U kисх   U kинтер (ik , a, b, c, d )   U kинтер  , da da da k 1 n d d d S  0 (3.2) S  2 U kисх   U kинтер (ik , a, b, c, d )   U kинтер  , db db da k 1 n d d d S 0 S  2  U kисх  U kинтер (ik , a, b, c, d )  U kинтер , dc dc da k 1 ……………………..



 



27





here U kисх are the initial values of voltage corresponding to the given currents. U kитер are interpolated values determined by interpolation function. a, b, c, d are unknown coefficients, which can be obtained by the minimum condition of S. It is calculated in MathCad by the following description Minerr (var1, var2, ...), on the basis of the system equation 3.2, her evar1, var2, ... are unknown coefficients, which should be determined. Working process: 1. The characteristics of the nonlinear elements are given in the graphical form, (Application No. 2). The numerical values of the quantity should be typed as a table. Values of VAC

Table 3.1

U, В I,А Values of Weber-ampere characteristics

Table 3.2

, В I,А

2. On the basis of the values given in the table in MathCAD the Ur, Ir, L , IL vectors can be built. For example:

28

3. It is necessary to approximate by the least square method. For this purpose we must give the initial values of the approximation coefficients. It is defined by the least square approximation.

4. To use the least square approximation method we need to write a program. It should be noted that: the given row is written in MathCAD in the following form: 1..20, which means – 1,2,3,4,…,20. To define it we need to type in MathCAD the mark[;] For example: 1..20

– 1, 2, 3 ... 20

20..1

– 20, 19, 18 ... 1

1,1.1..5 – 1, 1.1, 1.2 ...5 5,3..-7

– 5, 3, 1, -1 ...-7

5. Obtain the interpolation coefficients by Minerr (var1, var2, ...):

29

6. Draw the graph on the basis of Figure 3.1.

Figure 3.1. Approximated graph

7. Analyze the obtained results Control questions: 1. What is the difference of approximation and interpolation? 2. What is the least square method based on? References: 1. Sivokobylenko V.F. Mathematical modeling in electrical engineering and energy. – Donetsk: PBA DonNTU, 2005. – 350 p.

30

Laboratory work 3.2 APPROXIMATION OF NONLINEAR DEPENDENCES BY THE SPLINE METHOD The purpose of the work: the spline approximation of the characteristics of nonlinear elements of electric circuit Tasks: ‒ Tablesize the given description. ‒ Approximation of VAC’s by the spline method. ‒ Display the result on the graph. Brief theory: The method of the spline approximation is used in many software enviroments and in graphical editors. Spline approximation Spline approximation in MathCAD can be defined as cspline(vx, vy), it gives the vy vector corresponding to the dimension of the vх. The function interp(vs, vx, vy, x) defines the interpolation function vs. Working process: 1. The characteristics of the nonlinear elements are given in the graphical form, (Appendix No.2). The numerical values of the characteristics should be typed as a table. Values of VAC

Table 3.3

U, В I, А Values of Weber-ampere characteristics

Table 3.4

 , Wb I, А

1. On the basis of the values given in the table, we build Ur, Ir, L , IL vectors in MathCAD. The example of it is given in example 3.2: 31

2. Approximate by the spline cspline(vx, vy), and by the interp(vs, vx, vy, x) we can find the interpolation coefficients, draw the graphs (Figure 3.3).

Figure 3.2. The vectors of given values

Figure 3.3. Spline approximation

3. Analyze the obtained results Control questions: 1. Why is Approximation required? 2. What is the spline approximation? References: 1. Sivokobylenko V.F. Mathematical modeling in electrical engineering and energy. – Donetsk: PBA DonNTU, 2005. – 350 p.

32

CHAPTER IV ANALYSIS OF TRANSITIONAL PROCESSES IN ELECTRIC SYSTEMS BY DIFFERENTIAL EQUATIONS

In general, the differential equation is written as follows:

dy  f (x, y) , dx where

(4.1)

y  (y1 , y 2 ,... y n ) tr

f  ( f1 , f 2 ,... f n )tr x is an independent variable (time, current, etc.). The simplest quantitative method of solving the Cauchy problem for equation (4.1) is the Euler's method. If we replace the first order derivative in this differential equation with the following y  yk 1  yk and x  xk 1  xk , then

yk 1  yk  f (x k , y k ). h

(4.2)

will be obtained. Here the derivative f ( x, y ) is determined in the first step. Laboratory work 4.1 ANALYSIS OF TRANSITIONAL PROCESSES IN ELECTRIC SYSTEMS BY SIMPLE NUMERICAL EULER METHOD FOR THE SOLUTION OF DIFFERENTIAL EQUATIONS Purpose of work: Solving of transitional processes in electric systems by using simple Euler method for differential equations. 33

Tasks: ‒ Compose the system of differential equations using the Ohm and Kirchhoff laws for the given circuit; ‒ Solve the mathematical model of AC electric circuits using the simple Euler method in the Mathcad system; ‒ Create a given circuit using Multisim, compare it with the results obtained in the Mathcad system. Brief theory: We obtain an expression for the Euler method from the formula (4.2), i.e., if the step h and yk , xk are known, we obtain yk 1 from the following equation

yk 1  yk  h  f (x k , y k ).

(4.3)

This calculation (4.3) is given in the following example. Example 4.1. Given: R = 10 Оhm active resistance, L(i ) nonlinear inductance and P power circuit with a key. Find the current in electrical circuit when the key is locked u  3000V . Initial conditions: i1(0)  0 A, t  0 s, L(i  0)  1 H , h  0.005 s.

Figure 4.1. Electrical circuit scheme

Figure 4.2. L = f(i) dependency graph

Let's make a differential equation according to the second law of Kirchhoff: di ; dt di u R i   . dt L(i ) L(i )

u  R  i  L(i ) 

34

(4.4)

From equation (4.4), we can determine current at each moment of time. 3000 10  0  )  15 A, 1 1 3000 10 15 i (2)  15  0.005  (  )  29.394 A, 0.99 0.99 3000 10  29.394 i (3)  29.394  0.005  (  )  43.271 A. 0.975 0.975

i (1)  0  0.005  (

(4.5)

We will continue the calculation until the value i ( k 1) is equal i ( k ) or until the precision exceeds the accuracy. We are programming in the Mathcad to automate this operation. You can see it in the following example. Example 4.2. Given: L1 = 1 H, R1 = 1 Ohm, L2 = 2 H, R2 = 2 Ohm, R = 5 Ohm, С = 0.2 F, e1 = 10cos(w*t) V, e2 = 5cos(w*t+0.1) V, 1   3.14 . s

Initial conditions: i10   0 A, E10   10 V, i20   0 A, 0  E 20   4.975 V, u c  1 V, h  0.1s .

Figure 4.3. Electrical circuit scheme

We construct differential equation for the given circuit parts: 35

 di1 1  dt  L  e1  R1  i1  u c  Ri1  i2 ; 1   di2 1   e2  R2  i2  u c  Ri2  i1 ;   dt L2  du c 1   i1  i2 ;  C  dt

(4.6)

Below is an example of the calculation model in the Mathcad programming environment for this circuit. Let's set the initial values of example 4.2 as follows:

Let's use replication operators Add Line and while from the Programming window for calculation of transition processes (Figure 4.4).

Figure 4.4. Window of operators «Add Line» and «while»

36

Let's write the main part of example 4.2 as follows:

Let's build up a picture of obtained results, that is, branch currents (Figures 4.5-4.7). Let's save the obtained result for drawing the images as text in the Matchad as follows WRITEPRN(«Eiler.pm») : = Method_Eiler, and then use any image building program (Origin, Excell, etc.). 37

Figure 4.5. The time change of the first branch currents

Figure 4.6. The time change of the second branch currents

Figure 4.7. The time change of the current in the condenser

38

Work progress: 1. Using the initial data for electrical circuits given according to the variant, draw the scheme of your variant in the laboratory notebook and write the parameters (Appendix 3). 2. Build the circuit in the Multisim system in accordance with the scheme, measure the current values I1 , I 2 , I 3 . 3. Using the Ohm and Kirchhoff laws construct differential equations for the circuit, take the examples 4.1 and 4.2 as a template, and calculate by the simple Euler method in the Mathcad system. 4. Compare the results obtained through the Multisim and the Mathcad system. The values of the branch currents in the different times, obtained by the simple Euler method

t  t I1 I2 I3

t

2t

3t

Table 4.1

4t

..., I n

Control questions: 1. Write the differential type of the active barrier, inductivity and capacity in the equation for transient processes. 2. Describe the transient processes in the electric circuit with the inductance elements. 3. Describe the transient processes in electric circuits with a capacity elements. 4. Explain the simple Euler method. References: 1 V.Ph. Sivokobylenko. Mathematical modeling in electrical engineering and power engineering. – Donetsk: RVA DonNTU, 2005. – 350 p. 2 V.Y. Lubchenko, S.V. Khokhlova. Mathematical modeling in the tasks of electric power. Methodical Manual – Novosibirsk, 2006 – 20 p.

39

Laboratory work 4.2 ANALYSIS OF TRANSITIONAL PROCESSES IN ELECTRIC SYSTEMS BY MODIFIED NUMERICAL EULER METHOD FOR THE SOLUTION OF DIFFERENTIAL EQUATIONS Purpose of the work: Solving of transitional processes in electric systems by using modified Euler method for differential equations. Tasks: ‒ Compose the system of differential equations using the Ohm law and Kirchhoff laws for the given circuit. ‒ Solve the mathematical model of AC electric circuits using the modified Euler method in the Mathcad system ‒ Create a given circuit using Multisim, compare it with the results obtained in the Mathcad system. Brief theory: Let's write the following equation for the modified Euler method:

yk 1  yk  h  f (x k 1 , y k 1 ),

(4.7)

where, on the right side of the equation yk 1 is the unknown variable, which is obtained by the simple Euler method. Let's compose a system of differential equations for circuits characterizing transitional processes to the examples below and solve by modified Euler method. Example 4.3. Let's solve example 4.1 given in Laboratory Work 4.1 by the modified Euler method. Suppose that Kirchhoff's second law yields the following differential equations: di ; dt di u R i   . dt L(i ) L(i )

u  R  i  L(i ) 

40

(4.8)

Initial conditions: i1(0)  0 A, t  0 s, L(i  0)  1 H , h  0.005 s.

using the equation (4.7) we write the equation (4.8) using the finite difference method: (k )

i ( k )  i ( k 1) u R i   ; h L(i ) L(i ) u R i(k )  h   h  i ( k )  i ( k 1) ; L(i ) L(i ) u  h  i ( k 1) L(i ) i(k )  . R 1 h L(i )

(4.9)

Let's find i (1) from the first step, then

i*(1)

3000  0.005  0  1  14.29 A; 10 1   0.005 1

(4.10)

Let's define the value of the current for L(i  14.29)  0.99 H :

i

(1)

3000  0.005  0  0.99  14.42 A. 10 1  0.005 0.99

(4.11)

We find i (2) from the second step for L* (i  14.42)  0.98H i*(2)

3000  0.005  14.42  0.98  28.28 A; 10 1  0.005 0.98 41

(4.12)

Let's define the value of the current for L(i  28.28)  0.97 H :

i*(2)

3000  0.005  14.42  0.97  28.42 A; 10  0.005 1 0.97

(4.13)

We continue the calculation until the value i ( k 1) is equal to i ( k ) or exceeds the accuracy. We are programming in the Mathcad system to automate this operation. You can see it in the following example. Example 4.4. Let's program in the Mathcad system using the modified Euler method Example 4.2. given in Laboratory Work 4.1.  di1 1  dt  L  e1  R1  i1  u c  Ri1  i2 ; 1   di2 1   e2  R2  i2  u c  Ri2  i1 ;   dt L2  du c 1   i1  i2 ;  C  dt

(4.14)

For these electrical circuits, the calculated model is given in the Mathcad system.

42

43

Figure 4.8. Change of branch currents in electric circuits with time

Work progress: 1. Using the initial data for electrical circuits given according to the variant, draw the scheme of your variant in a laboratory notebook and write the parameters (Appendix 3). 2. Build the circuit in the Multisim system in accordance with the scheme, measure the currents values I1 , I 2 , I 3 . 3. Using the Ohm and Kirchhoff laws construct differential equations for the circuit, take the examples 4.3 and 4.4 as a template, and calculate by the modified Euler method in the Mathcad system. 4. Compare the results obtained through the Multisim and the Mathcad system. The values of the branch currents in the different times, obtained by the modified Euler method

t  t

t

3t

2t

I1

I2

I3 ..., I n

44

4t

Table 4.2

Control questions: 1. How is the free current described in transitional processes? 2. Explain the modified Euler method. 3. What determines the time of the transition process? References: 1. V. Ph. Sivokobylenko. Mathematical modeling in electrical engineering and power engineering. – Donetsk: RVA DonNTU, 2005. – 350 p. 2. V.Y. Lubchenko, S.V. Khokhlova. Mathematical modeling in the tasks of electric power. Methodical Manual – Novosibirsk, 2006 – 20 p.

Laboratory work 4.3 ANALYSIS OF TRANSITIONAL PROCESSES IN ELECTRIC SYSTEMS BY NUMERICAL EULER-CAUCHY METHOD FOR THE SOLUTION OF DIFFERENTIAL EQUATIONS Purpose of the work: Solution of transitional processes in electric systems by using Euler-Cauchy method for differential equations. Tasks: ‒ Compose the system of differential equations using the Ohm law and Kirchhoff laws for the given circuit. ‒ Solve the mathematical model of AC electric circuits using the Euler-Cauchy method in the Mathcad system ‒ Create a given circuit using Multisim, compare it with the results obtained in the Mathcad system. Brief theory: The Euler-Cauchy method proposed a modification to improve the accuracy of the simple Euler method. The basis of the modification (4.3) is the use of a more accurate value of the product in the computational expression. The basis of the modification is the use of a more accurate derivative f ( x, y ) in the expression (4.3). It defines it as the average value of the derivative between the initial and final calculation step. 1  f ( x, y )   [ f ( xk , yk )  f ( xk 1 , yk 1 )]  2  45

(4.15)

However, y* k 1 in this expression is unknown to us. We define its approximate value with the expression of the simple Euler method (4.15). That is, the Euler-Cauchy method at each step of the calculation (4.16) yk 1  yk  h  f ( xk , yk ) 1) First, we define y* k 1 with the expression (4.16) by the Euler method. 2) After that, calculate the desired value using the more accurate value of the derivative defined by the expression (4.15), i.e., 1  f ( x, y )    h  [ f ( xk , yk )  f ( xk 1 , yk 1 )]  2  

(4.17)

This is a working Euler-Cauchy expression. Let's compose a system of differential equations for circuits characterizing transitional processes in the examples below and solve by the Euler-Cauchy method. Example 4.5. Let's solve the example 4.1 given in Laboratory Work 4.1 by the Euler-Cauchy method. According to the second Kirchhoff law write the following differential equations: di ; dt di u R i   . dt L(i ) L(i )

u  R  i  L(i ) 

(4.18)

Initial conditions: i1(0)  0 A, t  0 c, L(i  0)  1 Гн, h  0.005 c.

Let's make the following calculations using the Euler-Cauchy method's algorithm:

1 i ( n 1)  i ( n )   h  f [t ( n ) , i ( n ) ]  f [t ( n 1) , i ( n 1) ] 2 ( n 1) (n) ix  i  h  f [t ( n ) , i ( n ) ] 46

ix (1)  i (0)  h 

U  R  i (0) 3000  10  0  0  0.03   90 A L 1

Lx (1)  0.95H 3000  10  0 A  3000 L(i (0) ) C 3000  10  0 A f (t (1) , ix (1) )   2210.5 0.95 C 1 i (1)  0   0.03  (3000  2210.5)  78.15 A 2 (1) L  0.97 H 3000  10  78.15 ix (2)  78.15  0.03   146.76 А 0.97 L(2)  0.72 H 3000  10  78.15 A f (t (1) , i (1) )   2287.1 0.97 C 3000  10 146.76 A f (t (2) , ix (2) )   2128.3 0.72 C 1 (1) i  78.15   0.03  (2287  2128.3)  144.38 A 2 f (t (0) , i (0) ) 

We continue the calculation until the value i ( k 1) is equal to i ( k ) or exceeds the accuracy. We are programming in the Mathcad system to automate this operation. You can see it in the following example. Example 4.6. Let's program in the Mathcad system by the EulerCauchy method Example 4.1. given in Laboratory Work 4.1.  di1 1  dt  L  e1  R1  i1  u c  Ri1  i2 ; 1   di2 1   e2  R2  i2  u c  Ri2  i1 ;   dt L2  du c 1   i1  i2 ;  C  dt

(4.19)

For these electrical circuits, the calculated model is given in the Mathcad system. 47

48

Figure 4.9. Change of branch currents in electric circuits with time

Work progress: 1. Using the initial data for electrical circuits given according to the variant, draw the scheme of your variant in a laboratory notebook and write the parameters (Appendix 3). 2. Build the circuit in the Multisim system in accordance with the scheme, measure the currents values I1 , I 2 , I 3 . 3. Using the Ohm and Kirchhoff laws construct differential equations for the circuit, take the examples 4.5 and 4.6 as a template, and calculate by the Euler-Cauchy method in the Mathcad system. 4. Fill the branch currents defined by the Euler-Cauchy method in Mathcad system at different times in Table 4.3. 5. Fill in Table 4.4 branch currents determined by the simple Euler, modified Euler, and Euler-Cauchy methods at different times and compare the graphs. 49

Table 4.3 The values of the branch currents in the different moments of time, obtained by the Euler-Cauchy method

t  t I1

3t

2t

t

4t

I2

I3

..., I n Table 4.4 Branch current values at different times determined by different methods

t  t I1 , Euler

t

2t

3t

4t

I1 , Modified Euler I1 , Euler –Cauchy

Control questions: 1. What is the nature of transitional processes, why do they occur? 2. Explain the Euler-Cauchy method. 3. What is the difference between the accuracy of Euler and Euler-Cauchy methods, and how is this explained? References: 1 V. Ph. Sivokobylenko. Mathematical modeling in electrical engineering and power engineering. – Donetsk: RVA DonNTU, 2005. – 350 p. 2 V.Y. Lubchenko, S.V. Khokhlova. Mathematical modeling in the tasks of electric power. Methodical Manual – Novosibirsk, 2006 – 20 p.

50

Appendix 1

LINEAR ELECTRIC CIRCUITS Variants

E = 35 kV, J = 100A, Z1 = 10Оhm, Z2 = 15 Оhm, Z3 = 10 Оhm, Z4 = 5 Оhm, Z5 = 20 Оhm, Z6 = 5 Оhm

E = 110 kV, J = 110A, Z1 = 10 Оhm, Z2 = 5 Оhm, Z3 = 15 Оhm, Z4 = 10 Оhm, Z5 = 5 Оhm, Z6 = 20 Оhm, Z7 = 5 Оhm

Figure 1

Figure 2

E = 35 kV, J = 100A, Z1 = 10 Оhm, Z2 = 15 Оhm, Z3 = 10 Оhm, Z4 = 5 Оhm, Z5 = 20 Оhm, Z6 = 5 Оhm

Figure 3

51

E = 110 kV, J = 110A, Z1 = 10 Оhm, Z2 = 5 Оhm, Z3 = 15 Оhm, Z4 = 10 Оhm, Z5 = 5 Оhm, Z6 = 20 Оhm, Z7 = 5 Оhm Figure 4

E = 35 kV, J = 100A, Z1 = 10 Оhm, Z2 = 15 Оhm, Z3 = 10 Оhm, Z4 = 5 Оhm, Z5 = 20 Оhm, Z6 = 5 Оhm

E = 110 kV, J = 110A, Z1 = 10 Оhm, Z2 = 5 Оhm, Z3 = 15 Оhm, Z4 = 10 Оhm, Z5 = 5 Оhm, Z6 = 20 Оhm, Z7 = 5 Оhm

Figure 5

Figure 6

E = 35 kV, J = 100A, Z1 = 10 Оhm, Z2 = 15 Оhm, Z3 = 10 Оhm, Z4 = 5 Оhm, Z5 = 20 Оhm, Z6 = 5 Оhm, Z7 = 15 Оhm, Z8 = 10 Оhm

E = 110 kV, J = 110A, Z1 = 10 Оhm, Z2 = 5 Оhm, Z3 = 15 Оhm, Z4 = 10 Оhm, Z5 = 5 Оhm, Z6 = 20 Оhm, Z7 = 5 Оhm

Figure 7

Figure 8

52

E = 35 kV, J = 100A, Z1 = 10 Оhm, Z2 = 15 Оhm, Z3 = 10 Оhm, Z4 = 5 Оhm, Z5 = 20 Оhm, Z6 = 5 Оhm Figure 9

E = 110 kV, J = 110A, Z1 = 10 Оhm, Z2 = 5 Оhm, Z3 = 15 Оhm, Z4 = 10 Оhm, Z5 = 5 Оhm, Z6 = 20 Оhm Figure 10

53

Appendix 2

APPROXIMATION Variants

Figure 1

Figure 2

Figure 3

Figure 4

54

Figure 5

Figure 6

Figure 7

Figure 8

55

Figure 9

Figure 10

Figure 11

Figure 12

56

Appendix 3

ANALYSIS OF TRANSITIONAL PROCESSES Variants

  314 1 s Variant

Scheme

1

R1, Оhm

R2, Оhm

L, H

С, μF

1

14

15

340

15

2

14

360

17

4

18

370

5

11

20

390

e(t )  2  380  sin(  t  1000 )

2 e(t )  2  400  sin(  t  200 )

3 e(t )  2  400  sin(  t  100 )

4 e(t )  2  250  cos(  t  1000 )

57

5

12

13

22

380

20

8

28

420

9

18

28

410

19

10

30

430

3

16

18

350

e(t )  2  70  sin(  t  1300 )

6 e(t )  2  3400  sin(  t  100 )

7 e(t )  2  410  sin(  t  1100 )

8 e(t )  2 120  sin(  t )

9

i (t )  2  40  sin(  t  200 )

58

Еducational issue

Nurgaliyeva Kuralay Yerkenovna Isanova Moldir Kenesovna Kodanova Sandugash Kulmagambetovna Masheeva Ranna Uytbayevna Yerimbetova Lazzat Tastanbekovna MATHEMATICAL PROBLEMS AND COMPUTER SIMULATION IN POWER ENGINEERING Methodical instructions for laboratory works Editor L.E. Strautman Typesetting G. Kaliyeva Cover design B. Malayeva Cover photo «Sfedor» is from www.pixabay.com.

IB No.13765

Signed for publishing 21.08.2020. Format 60x84 1/16. Offset paper. Digital printing. Volume 3,75 printer’s sheet. 70 copies. Order No.11213. Publishing house Qazaq University Al-Farabi Kazakh National University KazNU, 71 Al-Farabi, 050040, Almaty Printed in the printing office of the Qazaq University Publishing House.

59

«ҚАЗАҚ УНИВЕРСИТЕТІ» баспа үйінің жаңа кітаптары

Искаков Б.А. Сборник тестов и задач по физике: сборник / Б.А. Иска­ков, Д.А. Иска­ко­ва. – Алматы: Қазақ университе­ті, 2019. – 100 с. ISBN 978-601-04-4375-4 Сборник содержит тесты и задачи по физике на примере заданий ЕНТ и Республиканской олимпиады «аль-Фараби». Сегодня многие ав­то­ры учебных пособий придер­жи­ваются тенденции насыщать свои за­дач­ники упражнениями повышенного уровня сложности, включать в них задачи олимпиадного характера, поэтому бывает трудно среди всего это­го многообразия найти пособие, рассчитанное на среднего ученика. Данный сборник, будучи удачной ме­тодической разработкой авто­ров, восполняет этот пробел. При пра­вильном применении он поможет пре­вратить учение из принудительного в добровольное и позволит ус­пеш­но преодолеть слож­ности в изучении физики. Сборник рекомендован учащимся старших классов средней шко­лы и студентам колледжей. Искaков Б.A. Введение в физику космических лучей: учебное пособие / Б.А. Искаков, Е.М. Тaутaев. – Алматы: Қазақ универ­си­те­ті, 2020. – 90 с. ISBN 978-601-04-4429-4 В учеб­ном по­со­бии ос­вещaет­ся сов­ре­мен­ное сос­тоя­ние нaуки о Все­лен­ной и ее вaжней­шей состaвной чaсти – фи­зи­ки кос­ми­чес­ких лу­чей. Рaссмaтривaют­ся ме­то­ды изу­че­ния кос­ми­чес­ких лу­чей и их из­лу­че­ния, что яв­ляет­ся од­ним из вaжней­ших ис­точ­ни­ков нaуч­ных знa­ний о Все­лен­ной. Мaте­риaл излaгaет­ся ис­хо­дя из зaдaч фор­ми­ровa­ния фи­зичес­ко­го ми­ро­во­зз­ре­ния и по­нимa­ния проис­хож­де­ния мирa сту­ден­тами. Учебное по­со­бие преднaзнaче­но сту­ден­там вузов, по­лучaющим обрaзовa­ние по ес­те­ст­вен­но-мaтемaти­чес­ким и тех­ни­ческим нaпрaвле­ниям. Пособие также может быть интересно уче­ным и спе­циaлис­там, зa­нимaющимся ре­ше­нием проблем aст­ро­фи­зи­ки и изу­че­нием про­цес­сов, проис­хо­дя­щих в кос­ми­чес­ком прострaнс­тве.

Кітаптарды сатып алу үшін «Қазақ университеті» баспа үйінің маркетинг және сату бөліміне хабарласу керек. Байланыс тел: 8 (727) 377-34-11, колл-центр: 8 (727) 377-33-99. E-mail: [email protected], cайт: www.magkaznu.kz, интернет-магазин: www.magkaznu.com