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State budget educational institution secondary vocational education in the Penza region

Nikolsky Technological College named after A.D. Obolensky

Discipline: "Electrical Engineering"

On the topic: “Methods for calculating linear DC electrical circuits”

Completed

1st year student

Alekhin Andrey Alexandrovich

Nikolsk 2014

Introduction

The general task of analyzing an electrical circuit is that, based on given parameters (EMF, TMF, resistances), it is necessary to calculate currents, power, and voltage in individual sections.

Let us consider in more detail the methods of calculating electrical circuits.

1. Method of Kirchhoff equations

This method is the most general method solving the problem of electrical circuit analysis. It is based on solving a system of equations compiled according to Kirchhoff’s first and second laws regarding real currents in the branches of the circuit in question. Consequently, the total number of equations is equal to the number of branches with unknown currents. Some of these equations are compiled according to Kirchhoff's first law, the rest - according to Kirchhoff's second law. In a circuit containing q nodes, according to Kirchhoff’s first law, q equations can be composed. However, one of them (any one) is the sum of all the others. Consequently, there will be independent equations compiled according to Kirchhoff’s first law.

According to Kirchhoff's second law, the missing m equations must be composed, the number of which is equal.

To write equations according to Kirchhoff’s second law, it is necessary to select m circuits so that they ultimately include all branches of the circuit.

Let's consider this method using the example of a specific circuit (Fig. 1).

First of all, we select and indicate on the diagram the positive directions of currents in the branches and determine their number p. For the circuit under consideration, p = 6. It should be noted that the directions of the currents in the branches are chosen arbitrarily. If the accepted direction of any current does not correspond to the actual one, then the numerical value of this current is negative.

Therefore, the number of equations according to Kirchhoff’s first law is q - 1 = 3.

The number of equations compiled according to Kirchhoff's second law

m = p - (q - 1) = 3.

We select the nodes and circuits for which we will compose equations and designate them on the electrical circuit diagram.

Equations according to Kirchhoff's first law:

Equations according to Kirchhoff's second law:

By solving the resulting system of equations, we determine the branch currents. Calculation of an electrical circuit does not necessarily involve calculating currents based on the given EMF of voltage sources. Another formulation of the problem is also possible - calculating the emf of sources based on given currents in the branches of the circuit. The problem can also be of a mixed nature - the currents in some branches and the emf of some sources are specified. It is necessary to find currents in other branches and emfs of other sources. In all cases, the number of equations compiled must be equal to the number of unknown quantities. The circuit may also include energy sources specified in the form of current sources. In this case, the current of the current source is taken into account as the branch current when drawing up equations according to Kirchhoff’s first law.

The circuits for composing equations according to Kirchhoff's second law must be selected so that not a single calculated circuit passes through the current source.

Let's consider the electrical circuit diagram shown in Fig. 2.

We select the positive directions of the currents and plot them on the diagram. Total number There are five branches of the circuit. If we consider the current of the current source J to be a known value, then the number of branches with unknown currents is p = 4.

The circuit contains three nodes (q = 3). Therefore, according to Kirchhoff’s first law, it is necessary to construct q - 1 = 2 equations. Let's label the nodes in the diagram. The number of equations compiled according to Kirchhoff’s second law m = p - (q - 1) =2.

We select the circuits in such a way that none of them passes through the current source, and mark them on the diagram.

The system of equations compiled according to Kirchhoff’s laws has the form:

Solving the resulting system of equations, we find the currents in the branches. The method of Kirchhoff equations is applicable for calculating complex linear and non-linear linear circuits, and this is its dignity. The disadvantage of the method is that when calculating complex circuits it is necessary to compose and solve a number of equations equal to the number of branches p.

The final stage calculation - verification of the solution, which can be performed by drawing up a power balance equation.

The balance of power in an electrical circuit means the equality of the power developed by all energy sources of a given circuit and the power consumed by all receivers of the same circuit (the law of conservation of energy).

If in a section of the circuit ab there is an energy source with an emf and a current flows through this section, then the power developed by this source is determined by the product.

Each of the factors of this product can have a positive or negative sign relative to the direction ab. The product will have a positive sign if the signs of the calculated quantities coincide (the power developed by this source is given to the receivers of the circuit). The product will have a negative sign if the signs and are opposite (the source consumes power developed by other sources). An example would be a battery in charging mode. In this case, the power of this source (term) is included in the algebraic sum of the powers developed by all sources of the circuit, with a negative sign. The magnitude and sign of the power developed by the current source are determined similarly. If on a section of the circuit mn there is an ideal current source with current, then the power developed by this source is determined by the product. As in the EMF source, the sign of the product is determined by the signs of the factors.

Now you can write general form power balance equations

For the circuit presented in Fig. 2.2, the power balance equation has the form

2. Loop current method

The loop current method comes down to composing equations only according to Kirchhoff’s second law. The number of these equations is equal to one less than the number of equations required to calculate electrical circuits using the method of Kirchhoff's laws.

In this case, we assume that in each selected circuit independent design currents, called circuit currents, flow. The current of each branch is determined as the algebraic sum of the loop currents closing through this branch, taking into account the accepted directions of the loop currents and the signs of their magnitudes.

The number of loop currents is equal to the number of “cells” (elementary circuits) of the electrical circuit diagram. If the circuit under consideration contains a current source, then independent circuits must be selected so that the branch with the current source is included in only one circuit. For this circuit, the design equation is not drawn up, since the circuit current is equal to the source current.

Canonical form writing the equations of loop currents for n independent loops has the form

Circuit current of the nth circuit;

The algebraic sum of the EMF acting in the nth circuit, called the contour EMF;

Own resistance of the nth circuit, equal to the sum of all resistances included in the circuit under consideration;

Resistance belonging simultaneously to two circuits (in in this case circuit n and i) and is called the total or mutual resistance of these circuits. The first is the index of the contour for which the equation is drawn up. From the definition of mutual resistance it follows that resistances differing in the order of the indices are equal, i.e. .

Mutual resistance is assigned a plus sign if the loop currents flowing through them have the same directions, and a minus sign if their directions are opposite.

Thus, compiling equations for loop currents can be reduced to writing a symmetrical resistance matrix

and vector of contour EMF

When introducing the vector of the desired loop currents || equations (5) can be written in matrix form

System solution linear equations algebraic equations(5) for the nth circuit current can be found using Cramer’s rule

where is the main determinant of the system of equations corresponding to the circuit resistance matrix

The determinant is obtained from the main determinant by replacing the nth column of resistance with a column (vector) of loop EMF.

Let's consider the loop current method using the example of a specific electrical circuit diagram (Fig. 3).

The circuit consists of 3 elementary circuits (cells). Therefore, there are three independent loop currents. We arbitrarily choose the direction of the loop currents and plot them on the diagram. Contours can be selected not by cells, but there must be three of them (for a given circuit) and all branches of the circuit must be included in the selected circuits.

For a 3-circuit circuit, the loop current equation in canonical form has the form:

We find our own and mutual resistance and loop EMF.

Own circuit resistances

Let us remember that self-resistances are always positive.

Let us determine the mutual resistance, i.e. resistances common to two circuits.

The negative sign of mutual resistance is due to the fact that the loop currents flowing through these resistances are in opposite directions.

Loop EMF

We substitute the values ​​of the coefficients (resistances) into the equations:

By solving the system of equations (7), we determine the loop currents.

To unambiguously determine the branch currents, we select their positive directions and indicate them on the diagram (Fig. 3).

Branch currents

3. Method of nodal voltages (potentials)

The essence of the method is that the node voltages (potentials) of independent circuit nodes relative to one node, selected as a reference or basic node, are taken as unknowns. The potential of the base node is taken equal to zero, and the calculation is reduced to determining (q-1) nodal voltages existing between the other nodes and the base one.

The equations of nodal stresses in canonical form with the number of independent nodes n=q-1 have the form

The coefficient is called the intrinsic conductivity of the nth node. The intrinsic conductivity is equal to the sum of the conductivities of all branches connected to node n.

Coefficient

called mutual or internodal conductance. It is equal to the sum of the conductivities of all branches directly connecting nodes i and n, taken with a minus sign.

The right side of equations (9) is called the node current. The node current is equal to the algebraic sum of all current sources connected to the node in question, plus the algebraic sum of the products of the emf of the sources and the conductivity of the branch with the emf

In this case, the terms are written with a plus sign if the current of the current source and the emf of the voltage source are directed to the node for which the equation is being drawn up.

The given pattern for determining the coefficients significantly simplifies the compilation of equations, which comes down to writing a symmetric matrix of nodal parameters

and vectors of source nodal currents

Nodal stress equations can be written in matrix form

If any branch of a given circuit contains only an ideal source of EMF (the resistance of this branch is zero, i.e. the conductivity of the branch is infinity), it is advisable to select one of the two nodes between which this branch is connected as the base one. Then the potential of the second node also becomes known and equal in EMF value(taking into account the sign). In this case, for a node with a known node voltage (potential), an equation should not be drawn up and the total number of equations of the system is reduced by one.

By solving the system of equations (9), we determine the nodal voltages, and then, according to Ohm’s law, we determine the currents in the branches. So for a branch connected between nodes m and n, the current is equal to

In this case, those quantities (voltage, emf) whose direction coincides with the selected coordinate direction are recorded with a positive sign. In our case (11) - from node m to node n. The voltage between nodes is determined through the node voltages

Let us consider the method of nodal voltages using the example of an electrical circuit, the diagram of which is shown in Fig. 4.

We determine the number of nodes (in this example, the number of nodes is q=4) and designate them on the diagram.

Since the circuit does not contain ideal voltage sources, any node, for example node 4, can be selected as the base one.

Wherein.

For the remaining independent nodes of the circuit (q-1=3), we compose the equations of nodal voltages in canonical form.

We determine the coefficients of the equations.

Own conductivities of nodes

Mutual (internodal) conductances

We determine the nodal currents.

For 1st node

For 2nd node

For 3rd node

Substituting the values ​​of the coefficients (conductivities) and nodal currents into equations (12), we determine the nodal voltages

Before moving on to determining the branch currents, we set them in a positive direction and plot them on the diagram (Fig. 5).

Currents are determined by Ohm's law. So, for example, the current is directed from node 3 to node 1. The EMF of this branch is also directed. Hence

The currents of the remaining branches are determined by the same principle

Since then

4. Principle and method of application

The principle of imposition (superposition) is an expression of one of the basic properties linear systems of any physical nature and in relation to linear electrical circuits is formulated as follows: the current in any branch of a complex electrical circuit is equal to the algebraic sum of the partial currents caused by each source acting in the circuit electrical energy separately.

The use of the superposition principle makes it possible in many circuits to simplify the problem of calculating a complex circuit, since it is replaced by several relatively simple circuits, each of which has one energy source.

From the superposition principle follows the superposition method used to calculate electrical circuits.

In this case, the superposition method can be applied not only to currents, but also to voltages in individual sections of the electrical circuit, linearly related to the currents.

The principle of superposition cannot be applied to capacities, because they are not linear, but quadratic functions current (voltage).

The superposition principle does not apply to nonlinear circuits either.

Let us consider the calculation procedure using the superposition method using the example of determining currents in the circuit in Fig. 5.

We choose the direction of the currents arbitrarily and plot them on the diagram (Fig. 5).

If the proposed problem were solved by any of the methods (MZK, MKT, EOR), then it would be necessary to compile a system of equations. The superposition method makes it possible to simplify the solution of the problem, reducing it in fact to a solution according to Ohm’s law.

We divide this circuit into two subcircuits (according to the number of branches with sources).

In the first subcircuit (Fig. 6), we assume that only the voltage source is active, and the current of the current source is J=0 (this corresponds to a break in the branch with the current source).

In the second subcircuit (Fig. 7), only the current source operates. The EMF of the voltage source is taken equal to zero E=0 (this corresponds to a short circuit of the voltage source).

We indicate the direction of currents on the subcircuits. In this case, you should pay attention to the following: all currents indicated on the original diagram must be indicated on the subcircuits. For example, in the subcircuit of Fig. 6, resistances and are connected in series and the same current flows through them. However, the diagram must indicate currents and.

The calculation for the circuit (Fig. 6) can be performed using Ohm's law.

Currents in parallel branches are determined by the scatter formula

We determine the currents in the subcircuit shown in Fig. 7. Replacing previously parallel-connected resistances and equivalent

we get a diagram (Fig. 8).

Using the scatter formula, we determine the currents and

Based on the partial currents of the subcircuits (Fig. 2.6 and 2.7), we determine the currents of the original circuit (Fig. 5) as the algebraic sum of the partial currents.

In this case, the current is recorded with a minus sign, because its direction in the subcircuit is opposite to the direction of the current in the original circuit

electric circuit current kirchhoff

The directions of currents in the subcircuits coincide with the direction of the current in the original circuit. We determine the remaining currents in the same way.

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For practical calculations of electrical circuits, methods have been developed that make it possible to reduce the number of equations to be solved in comparison with the system of equations according to Kirchhoff’s laws. Let's look at these methods.

a) Method of nodal potentials. Two node method.

The number of equations using the nodal potential method is determined by the number of equations using Kirchhoff’s first law for the electrical circuit under consideration. In accordance with this method, it is necessary to first determine the potentials of all nodes of the electrical circuit, and then, using Ohm's law, determine the currents in the branches. In this case, one of the nodes of the electrical circuit, which is called supporting, is grounded, its potential becomes zero. The grounding node is selected arbitrarily. It is convenient to ground the node whose number has the highest value in a given electrical circuit.

A system of equations using the nodal potential method in the form of a matrix will have as many rows and columns as the number of equations that need to be written according to Kirchhoff’s first law for the electrical circuit in question. If in an electrical circuit there is a branch containing only an ideal source of emf. Then it is convenient to number the nodes of the electrical circuit so that the node number with highest values in a given electrical circuit, found himself at the node from which the EMF source departs. This node is taken as a reference node and grounded. Then the potential of the node into which the EMF source enters will be known and equal to the value of the source EMF.

Let's consider the use of the nodal potential method using an example.

Example.

It is advisable to use the method of nodal potentials when the number of equations according to the first Kirchhoff law for an electrical circuit is less than according to the second. Figure 2.1 shows an electrical circuit that meets these requirements.

Figure 2.1 – Electrical circuit diagram for calculation using the nodal potential method

The presented circuit contains 8 branches, 2 of which contain current sources, therefore, the number of equations according to Kirchhoff’s second law is equal to: 8 – 2 – 3 = 3 equations.

There are four nodes in a given circuit, therefore, according to Kirchhoff’s first law, it is necessary to write 4 – 1 = 3 equations, and there is a branch containing only an ideal source of emf. In this case, we number the node from which the EMF source departs with the number 4 and take it as a reference, the potential of which is zero. Let us denote grounding at φ 4 on the design diagram. The potential of the node into which the EMF source enters will be known and equal to the value of the source EMF.

Thus, there remain two unknown potentials and , to find them we use a system of two equations:

where are the own conductivities of nodes 1, 2, ..., S, respectively, which are defined as the sum of the conductances of the branches connected to the corresponding node. For the circuit in question; . In the system of equations (2.1), the intrinsic conductivities of the nodes along the main diagonal of the matrix will always have plus signs. If an electrical circuit is given for which, according to Kirchhoff’s first law, it will be necessary to write down a different number of equations, then the system of equations (2.1) should consist of rows and columns, the number of which is determined by the number of equations according to Kirchhoff’s first law.

– the sum of the conductivities of the branches connecting the node S with knot N, is always taken with a minus sign in the system of equations (2.1). For the circuit under consideration, the sum of the conductances of the branches between nodes 1 and 2, and, therefore, 2 and 1, is . The sum of the conductances of the branches between nodes 1 and 3 is equal to zero, therefore . The sum of the conductances of the branches between nodes 2 and 3.

Where is the algebraic sum of the products of the emf of the branches adjacent to the node S, on their conductivity; in this case, those products are taken with a “plus” sign, in the branches of which the emfs act in the direction of the node S, and with a minus sign – in the direction from the node S;

– algebraic sum of current sources connected to the node S, the sign before is determined according to the rule stated above. In our case , .

Thus, in the new notation, system of equations 2.1 will take next view

Having solved the resulting system (2.2) with respect to and, assuming known, we will find the currents in the branches of the electric circuit.

Currents in the branches of an electric circuit are determined according to Ohm’s law through the potentials obtained by solving system (2.2):

Two node method ( special case method of nodal potentials).

There are electrical circuits that have only two nodes (Figure 2.2). To calculate currents in such a circuit, the most rational calculation method is the two-node method.

Figure 2.2 – Diagram of an electrical circuit containing two nodes

Let's look at using the two-node method with an example.

Example.

For an electrical circuit (Figure 2.2) using the nodal potential method, we write the following expression:

Let us write the resulting expression for voltage:

Expression (2.3) is usually called two node method.

Currents in the branches of an electrical circuit are determined by Ohm's law as follows:

b) Method of loop currents and equivalent generator.

Loop current method also allows you to reduce the number of equations to be solved in comparison with the system of equations according to Kirchhoff’s laws. The number of equations using the loop current method is determined by the number of equations using Kirchhoff’s second law for the electrical circuit under consideration. The method is based on the property that the current in any branch can be represented as an algebraic sum of independent loop currents flowing through this branch. In accordance with this method, it is necessary to select loop currents in such a way that each of them passes through one current source, and the remaining loop currents are selected to pass through branches that do not contain current sources.

The system of equations using the loop current method in the form of a matrix will have as many rows and columns as the number of equations that need to be written according to Kirchhoff’s second law for the electrical circuit in question. If there is a current source in the electrical circuit, then a column will be added to the system of equations, if there are two, then two columns, etc.

Let's consider using the loop current method with an example.

Example.

It is advisable to use the loop current method when the number of equations according to Kirchhoff’s second law for an electrical circuit is less than according to the first. Figure 2.3 shows an electrical circuit that meets the specified requirements.

Figure 2.3 – Electrical circuit diagram for calculation using the loop current method

Solution.

There are four nodes in this chain, therefore, according to Kirchhoff’s first law, it is necessary to write 4 – 1 = 3 equations. The circuit under consideration contains seven branches, two of which are with current sources, therefore, according to Kirchhoff’s second law, the number of equations is equal to: 7 – 2 – 3 = 2. For a given circuit, the directions of bypass of the loop currents , , , are taken clockwise, and , i.e. To. bypassing the circuit does not coincide with the direction of the current source; , because bypassing the circuit coincides with the direction of the current source. Thus, the loop currents are considered known. Consequently, there remain two unknown loop currents ( and ), to find them we use a system of two equations:

where is the circuit’s own resistance m(the sum of the resistances of all branches included in the circuit m). For the circuit in question; . In the system of equations (2.4), the intrinsic resistances of the circuits along the main diagonal of the matrix will always have plus signs. If an electrical circuit is given for which, according to Kirchhoff’s second law, it will be necessary to write a different number of equations, then the number of equations in system (2.4) will change. The number of rows in system (2.4) is determined by the number of equations according to Kirchhoff’s second law, and the number of columns is equal to the sum of the number of equations according to Kirchhoff’s second law and the number of current sources.

– total circuit resistance m And l, is taken with a “plus” sign if the directions of the loop currents in a given branch coincide; otherwise, a “minus” sign is taken. In the circuit under consideration, the total resistance between circuits 1 and 2, and therefore 2 and 1, is . The directions of the loop currents in this branch do not coincide, therefore, the resistance will enter the equation with a minus sign. The resistances between circuits 1 and 3, as well as 1 and 4, are zero, therefore, and . Resistance between circuits 2 and 3. The directions of the loop currents in this branch do not coincide, therefore, the resistance will enter the equation with a minus sign. Resistance between circuits 2 and 4. The directions of the loop currents in this branch coincide, therefore, the resistance will enter the equation with a plus sign.

– algebraic sum of the emfs included in the circuit m. For this scheme, .

Thus, in the new notation, the system of equations 2.4 will take the following form

Having solved the resulting system (2.5) for and , considering and known, we will find the currents in the branches of the electric circuit.

Currents in the branches of the electrical circuit, through loop currents, are determined as follows:

Equivalent generator method unlike those presented above, it allows you to determine the current in only one selected branch, by simplifying the remaining part of the electrical circuit into a single-circuit unbranched circuit. In relation to the selected branch, the rest of the circuit is replaced by an equivalent source of EMF - a generator. The emf of this generator is equal to the voltage idle move on the terminals of the selected part of the circuit to which the branch with the determined current will be connected. The internal resistance of the generator will be equal to the input resistance with respect to the terminals of the selected branch.

Sequence of calculation using the equivalent generator method.

1. Select in the calculation circuit the branch whose current needs to be determined. The rest of the circuit is represented as an EMF source and internal resistance. Select the positive direction of current in the branch.

2. Having disconnected the selected branch, determine using any of the previously studied methods the voltage at the terminals of the remaining part of the circuit to which the branch with the determined current will be connected.

3. Determine the equivalent input resistance with respect to the terminals of the selected branch. In this case, replace the energy sources with their internal resistances and consider the resistance of the EMF sources to be equal to zero, and the resistance of the current sources to be equal to infinity.

4. Determine the current in the resulting unbranched circuit using Ohm’s law:

where , are the parameters of the branch with the desired current.

Example.

Let's consider using the equivalent generator method for the electrical circuit shown in Figure 2.4.

Figure 2.4 – Electrical circuit diagram for calculation using the equivalent generator method

When calculating, it is necessary to determine the current in the third branch. All parameters of the elements of the electrical circuit and its topology are considered known.

Solution.

1. Let us select in the calculation circuit a branch with a current that we will determine. Let's imagine the rest of the circuit as an EMF source with voltage and internal resistance (Figure 2.5). Let us choose the positive direction of the current in the third branch.

Figure 2.5 – Equivalent equivalent circuit

3. Disconnect the third branch and use any of the previously studied methods to determine the voltage at the terminals of the remaining part of the circuit (Figure 2.6).

Figure 2.6 – Electrical circuit diagram for calculation using the equivalent generator method

After disconnecting the third branch, the three electrical circuit splits into two independent ones. One with an EMF source, the other with a current source. In this case, you can find the voltage using Ohm's law. Let us conditionally ground node 3, then .

Let's find the potential of point 1 through the potential of node 3

Point 2 potential through point 3 potential

Thus,

4. Let's determine the equivalent input resistance with respect to the terminals of the selected branch. In this case, we replace the energy sources with their internal resistances and consider the resistance of the EMF sources to be equal to zero, and the resistance of the current sources to be equal to infinity

5. Let us determine the current in the third branch using Ohm’s law:

c) Overlay method.

The superposition method is applicable only for the calculation of linear circuits, the parameters of the elements of which do not depend on the values ​​of the flowing current or applied voltage. To calculate circuits using the superposition method, as many partial circuits are made as there are independent energy sources in the original circuit. IN private scheme only one source is left, all others replace them with internal resistances. The resulting branch current is equal to the algebraic sum of the partial currents caused by the action of each source separately. When ideal voltage sources are excluded, a short-circuited jumper is placed instead of the source, which corresponds to an emf equal to zero with zero internal resistance. The branch with the current source, on the contrary, opens, which corresponds to zero current at zero conductivity.

When calculating particular circuits, the currents flowing in the branches are designated by two indices. The lower index shows the number of the branch in which the current is determined, and the upper index shows the number of the source whose action causes the current. For example, − the current of the first branch caused by the action of the second source. When calculating particular circuits, it is often necessary to calculate currents in parallel branches and (see. practical lesson 1).

Let's look at using the overlay method with an example.

Example.

It is advisable to use the superposition method when calculating an electrical circuit that contains one or two, in extreme cases, three sources of electrical energy. Figure 2.7 shows an electrical circuit with two energy sources that meets the specified requirements.

Figure 2.7 – Electrical circuit diagram for calculation using the superposition method

To find the currents in the circuit of Figure 2.7, the superposition method is used to determine the currents in the particular circuits shown in Figures 2.8 and 2.9.

Figure 2.8 Figure 2.9

For the diagram in Figure 2.8:

For the diagram in Figure 2.9:

It should be noted that when operating in electrical diagram one source of electrical energy, as shown in Figures 2.8 and 2.9, partial currents flow in the branches of the circuit. Their direction is determined by the direction of the source of electrical energy operating in the electrical circuit. So, in Figure 2.8, the direction of the current is determined by the direction of the EMF source. From Figure 2.8 it can be seen that the current flows to node 1, where it is divided into current and current, after which these currents flow to node 2, where they are again combined into current. In Figure 2.9, the direction of the currents , , , is determined by the direction of the current source.

Thus, the currents in the original electrical circuit are determined based on the partial currents as follows:

The first and third currents have a minus sign, since their partial currents under the action of two energy sources have different directions. The partial currents of the second and fourth currents have the same directions, so the resulting current equations have a plus sign.

Conclusions from the lecture

To calculate the currents in the branches of an electrical circuit, in addition to Kirchhoff's laws, you can use the method of nodal potentials, the method of loop currents, the superposition method, and the equivalent generator method. The number of equations for the method of nodal potentials is the same as according to Kirchhoff’s first law; if the circuit contains only two nodes, the two-node method can be used. The number of equations for the loop current method is the same as according to Kirchhoff’s second law. It is advisable to use the superposition method when the electrical circuit contains no more than three sources of electrical energy. If you need to calculate the current in one branch, you can use the equivalent generator method.

Self-test questions

1. Formulate the basic principles of the nodal potential method.

2. What are the features of using the nodal potential method for circuits containing only an ideal source of emf in any of the branches?

3. How to find currents in branches using the two-node method?

4. Describe the main stages of the loop current method.

5. What are the features of using the loop current method for circuits containing a current source?

6. What are the advantages and disadvantages of the overlay method?

7. Explain the essence of the equivalent generator method?

Given: E1=28 V, E2= 16 V,

R1= R5 =R6= 30 Ohm,

R2=16 Ohm, R3=R4= 10 Ohm,

r01=2 Ohm, r02=1 Ohm.

Determine: I1, I2, I3, I4, I5, I6.

Do the following:

• 1) based on Kirchhoff’s laws, draw up a system of equations to determine currents in all branches of the circuit;
• 2) determine the currents in all branches of the circuit based on the loop current method;
• 3) determine the currents in all branches of the circuit based on the superposition method;

We compose a system of equations. The system must have as many equations as there are branches (unknown currents) in the chain.

There are six branches in a given chain, which means that the system must have six equations (m = 6). First, we compose equations for the nodes according to Kirchhoff’s first law. For a chain with n nodes, you can create (n-1) independent equations. There are four nodes in our chain (A, B, C, D), which means the number of equations: n-1 = 4-1 = 3.

We compose three equations for any 3 nodes.

node A: I2+I3=I1

node B: I4+I6=I2

node C: I5+I6=I3

There should be six equations in total in the system. There are already three. We make up the three missing ones for linearly independent contours.

We set out to bypass each contour and compose an equation according to Kirchhoff’s second law.

Circuit A - we take a counterclockwise bypass:

E1=I1 (r01+R1+R2)+I2

Circuit B - clockwise bypass:

Circuit C - counterclockwise bypass:

Determine the currents in all branches of the circuit using the loop current method.

The loop current method is based on the use of only Kirchhoff's second law. This will reduce the number of equations in the system by (n-1).

Where n is the number of nodes in the circuit. This is achieved by dividing the circuit into independent circuits and introducing for each cell circuit its own loop current, which is a calculated value. And so, in a given circuit, you can consider three cell circuits (ACDA, ABDA, CBDC) and introduce circuit currents for them.

Branches belonging to two adjacent circuits are called adjacent branches. In them, the actual current is equal to the algebraic sum of the circuit currents of adjacent circuits, taking into account their direction.

When compiling equations according to Kirchhoff’s second law, on the left side of the equality the EMF of the sources included in the circuit-cell is algebraically summed up, on the right side of the equality the voltages on the resistances included in this circuit are algebraically summed up, and the voltage drop on the resistances of the adjacent branch, determined by the contour line, is also taken into account current of the neighboring circuit. Based on this, the procedure for solving a DC circuit using the loop current method will look like this:

Arrows indicate the selected directions of loop currents, in the loops - cells. The direction of bypassing the contours is assumed to be the same;

As when solving a problem according to Kirchhoff’s laws, equations are drawn up here (we take a contour and walk around it along a given traversal direction) and solve a system of equations by the substitution method, or using determinants. We compose a system of equations according to Kirchhoff’s second law:

E1 =Ik1(r01+R1+R4+R2)-Ik2 R2-Ik3 R4

E2-E3 =Ik2(r02+R3+R6+R2)-Ik2 R2-Ik3 R6

E3 =Ik3(R4+R5+R3)-Ik1 R4-Ik2 R6

Let's substitute numerical values resistances and emf of sources into the resulting system of equations

• 28= Ik1 (2+30+10+16)- Ik2 16- Ik3 10
• 16-24= Ik2 (1+10+30+16)- Ik1 16- Ik3 30
• 24= Ik3 (10+30+10)- Ik1 10- Ik2 30
• 28= Ik1 58- Ik2 16- Ik3 10
• -8= - Ik1 16+ Ik2 57- Ik3 30
• 24= - Ik1 10- Ik2 30+ Ik3 50

We solve the compiled system of equations using Cramer’s methods

Calculation of unbranched circuits

The basis for the calculation of single-circuit (unbranched) electrical circuits containing sources of both types and consumers are Ohm's and Kirchhoff's laws.

If the consumer parameters in the circuit ( R) and voltage sources ( E) are given, the task is usually to determine the loop current. The positive direction of the desired current is chosen arbitrarily and the equation is drawn up:

It must be remembered that the “+” sign refers to the EMF of sources that act in the direction of the selected current.

Calculation of branched circuits with one source

A branched circuit with a single source is usually simplified by converting it into an unbranched circuit.

A mixed connection of energy receivers is a combination of series and parallel connections. General formula There is no equivalent connection to calculate, since there are many different connection schemes. When calculating, you need to select sections in the circuit connected in series or parallel and determine their equivalent resistance. The circuit is gradually simplified, leading to its simplest form, and the currents of the sections are determined using Ohm's law.

Example 2

In the circuit in Figure 4, the following quantities are known:

R1 =3Ohm; R2 =2Ohm; R3 =24Ohm; R4 =12Ohm; R5 =10Ohm; R6 =2Ohm;

Determine the equivalent resistance and currents of all sections.

Figure 4

Resistors R3 and R4 are connected in parallel. Let's find the equivalent resistance and simplify the circuit (See Figure 5)

Figure 5

Resistors R2 and R3,4 are connected in series. Let's find the equivalent resistance and simplify the circuit

(See Figure 6)

Figure 6

Resistors R2,3,4 and R5 are connected in parallel. Let's find the equivalent resistance and simplify the circuit.

(See Figure 7)

Figure 7

Resistors R1,R6 and R2,3,4,5 are connected in series. Let's find the equivalent resistance and simplify the circuit.

(See Figure 8)

Let's find the current strength in an unbranched section of the circuit

using Ohm's law.

Figure 8

To determine the currents in all sections, it is convenient to consider the circuits in reverse order.

notice, that

Let's find the voltage across these series-connected resistors.

Let us determine the currents in these sections.

Let's find the voltage in section R3.4

The voltages on the third and fourth resistors are the same and equal to 9.6V (the sections are parallel)

Equivalent transformation of triangle and star resistances

In the field of electrical measurements, an electrical circuit with a single power source is widely used, the circuit shown in Figure 9. A feature of this circuit is the presence of connections in it, called delta and star.

Triangle resistance is the connection of three branches forming a closed circuit with three nodes. In the circuit (Figure 9a) there are two triangles with resistances R1, R2, R3 and R3, R4, R5.

Star resistance is the connection of three branches that have a common node. The resistance star is formed by branches with resistances R2, R3, R5 and R1, R3, R4.

(See Figure 9a)

Any resistance triangle can be replaced by an equivalent star

(See Figure 9b). To move from a resistance triangle to an equivalent star, use the formulas:

In some electrical circuits, the calculation is simplified by replacing the three-pointed star with an equivalent resistance triangle. In this case, the inverse transformation formulas are used:

Figure 9

Figure 9

Calculation of branched circuits with multiple sources

If the configuration of a complex electrical circuit is known and the properties of all its constituent elements are specified, then the calculation of such a circuit usually comes down to determining the currents in the branches and the potentials of the nodes. Unlike the cases discussed above, a branched circuit with multiple sources requires special calculation methods. It should be noted that branched chains with a single source can also be calculated using the methods discussed below.

Kirchhoff equation method

The search for unknown quantities involves the compilation and joint solution of a system of equations written according to Kirchhoff’s I and II laws.

Calculation algorithm.

1. Determine the number of nodes, branches and independent circuits of the electrical circuit.

2. Designate the branch currents and arbitrarily choose their positive direction.

3. For nodes, compose equations according to Kirchhoff’s first law. There should be such equations ( n – 1). n is the number of nodes.

4. For each selected contour, compose equations according to Kirchhoff’s II law. There should be such equations p. p - number of independent circuits, p=m-(n-1)

5. Solve a system of m equations (the number of equations in the system must coincide with the number of branches).

6. Check the correctness of the calculation using the power balance.

Example 3. Determine the currents in individual sections of the circuit shown

in Figure 10.

E 1 = 95 V, r1 = 1 Ohm, E 2 = 69 V, r2 = 2 Ohm, R1 = 20 Ohm, R2 = 10 Ohm, R3 = 29 Ohm, R4 = 5 Ohm, R5 = 1 Ohm.

Solution:

1. The complex chain contains two nodes B and E (n=2), three branches BE, VBAE, VGDE (m=3), three circuits (ABVEA, VGDEV, ABVGDEA)

Figure 10

2. Let us arbitrarily designate the direction of the branch currents and the direction of bypassing the circuits. (See Figure 11)

3. Let’s make one equation according to Kirchhoff’s I law for node B: I1 + I2 = I3

4. Let’s compose two equations according to Kirchhoff’s II law, since р=3-(2-1).

ABVEA circuit: E 1- E 2 = I1(r1 +R1 + R3)- I2(r2+R2)

VGDEV circuit: E 2 = I2(r2+R2)+ I3(R4 + R5)

5. Let’s solve the system of equations:

Figure 11

Loop current method

The method of Kirchhoff equations (nodal and contour equations) in some cases leads to complex calculations. For example, when calculating a chain containing five branches, it is necessary to create five equations. The number of system equations can be reduced by using the loop current method.

The loop current method is one of the main methods for calculating complex electrical circuits, which is widely used in practice.

Calculation algorithm

1 Designate all branch currents and their positive direction.

2 Divide the diagram of a complex circuit into separate circuits-cells.

3 Assign to each circuit an arbitrarily directed circuit current, identical for all sections of a given circuit. (It is better to choose one positive direction for all loop currents).

4 Compose equations according to Kirchhoff’s second law, the number of equations should be equal to the number of loop currents.

5 Solve the resulting system of equations for loop currents using mathematical methods(Cramer method, Gauss method, etc.)

6 Determine the branch currents through the loop currents according to Kirchhoff’s first law.

7 If necessary, use the generalized Ohm’s law to determine the node potentials.

8 Check the correctness of the calculations using the power balance.

Example 4. Let's consider solving the previous problem using the loop current method.

1. Chose the direction of the branch currents

2. In this scheme we can define two

circuit-cells ABVEA, VGDEV.

3. We attribute circuit current II to circuit ABVEA, the positive direction of which coincides with

contour VGDEV-III, positive direction - clockwise.

4. Let’s create equations according to Kirchhoff’s II law:

5. Let's solve the system of equations:

Currents in the extreme branches of the electrical circuit coincide with the loop currents

I 1=I I=1A, I 3=I II=3A. We determine the current in the internal branch according to Kirchhoff’s first law

I2 = I 3 -I 1=2A

The results of solving the problem coincided with the answer obtained by solving the method of Kirchhoff equations.

Nodal potential method

The current in any branch of the circuit can be found using the generalized Ohm's law. In order to apply Ohm's law, it is necessary to know the value of the potentials of the circuit nodes. The method of calculating electrical circuits, in which the potentials of circuit nodes are taken as unknowns, is called the method of nodal potentials. The number of unknowns in the nodal potential method is equal to the number of equations that need to be compiled for the circuit according to Kirchhoff’s first law. The method of nodal potentials, like the method of loop currents, is one of the main calculation methods. In the case when P- 1 < p (n – number of nodes, p– number of independent circuits), this method is more economical than the circuit current method.

Calculation algorithm

1. Designate all branch currents and their positive direction.

2. Randomly select a reference node (the potential of this node is conventionally considered equal to zero) and number all the others ( n- 1)-e nodes.

3. Determine the own and general conductivities of the nodes, as well as nodal currents, i.e. calculate the coefficients in the system of equations. Intrinsic conductivity of the node (G ii ) represents arithmetic sum conductivities of all branches connected in i-th node.

Total conductance of the i-th and j-th nodes (G ij = G ji ) is the sum of the conductivities of the branches connected simultaneously to i- oh and j- omu nodes.

The conductivities of the branches with current sources are assumed to be equal to zero and are not included in the own and general conductivities!

Nodal current (J ii ) consists of two algebraic sums: the first contains the currents of the current sources contained in the branches connected in i- om node; the second is the product of the EMF of the voltage sources and the conductivity of the corresponding branches connected in i- om node. With a “+” sign this amount includes E And J sources whose action is directed to the node, with the sign “–” the rest.

4.Write the system of equations in the form

In this system, each node corresponds to a separate equation.

5.Solve the resulting system of equations with respect to the unknowns ( n – 1) potentials.

7.Check the correctness of the calculations using the power balance.

The calculation procedure does not depend on the type of sources operating in the circuit. But the calculation is simplified when idealized EMF sources are included between one or more pairs of nodes. Then the voltages between these pairs of nodes become known quantities determined by the conditions of the problem. To successfully solve such problems, it is necessary to correctly designate the reference node, for which only one of the nodes to which the branch with an idealized EMF source is connected can be selected.

If such branches q, then the number of equations in the system will be reduced to

k = n – 1–q.

Example 5. Let's calculate the electrical circuit using the potential method. (See Figure 13)

Figure 13

Overlay method

The superposition method is based on the superposition principle and is as follows: the current or voltage of an arbitrary branch or section of a branched DC electrical circuit is determined as the algebraic sum of the currents or voltages caused by each source separately.

When using this method, the task of calculating a branched electrical circuit with n sources comes down to a joint decision n single source circuits.

Solution algorithm

1. The original chain containing n sources, convert to n subcircuits, each of which contains only one of the sources, other sources are excluded as follows: the voltage sources are short-circuited, and the branches with current sources are broken. Internal resistances real sources play the role of consumers and therefore they must remain in the subcircuits.

2. Determine the currents of each of the subcircuits, specifying their direction in accordance with the polarity of the source. The calculation is carried out according to Ohm's law using the method of equivalent transformations of passive circuits.

3. Total current in any branch of the original circuit is defined as the algebraic sum of the currents of the auxiliary subcircuits, and when summing with the “+” sign, the currents of the subcircuits are taken, the direction of which coincides with the direction of the current in the original circuit, with the “–” sign - the rest.

The advantages of the method include the fact that the calculation is carried out in parts, where the current and voltage components are determined quite simply. The method is recommended for use in circuits containing 2-3 sources.

The electrical circuit diagram of the original problem contains two sources of emf, therefore this task can also be solved by the method of superimposing currents.

1. We transform the circuit (see Figure 14) so ​​that only the first source remains in the circuit; the second source is not ideal, so we replace it with a resistor with resistance r2 (see Figure 15).

Let's find partial currents.

Figure 14

Figure 15

2. We transform the circuit (see Figure 14) so ​​that only the second source remains in the circuit; the first source is not ideal, so we replace it with a resistor with resistance r1 (see Figure 16).

Let's find partial currents.

Figure 16

Let's determine the true currents:

Nodal voltage method

Electric energy consumers are connected in parallel. Often the total power of the switched on receivers becomes greater than what the energy source can supply to the network. In such cases, at a constant voltage, energy sources are switched on in parallel. This results in a chain that contains two nodes. The voltage between nodes A and B is called nodal voltage. It is convenient to calculate such a circuit using the nodal voltage method.

Calculation algorithm

1. Indicate the direction of the currents on the diagram (select the direction of the currents towards one of the nodes).

3. Determine nodal voltage:

If the direction of the EMF is opposite to the direction of the current in the branch, it will enter the formula with a minus sign.

4. Find the current in the branches:

Example7

The problem discussed earlier can also be solved using the nodal voltage method.

1. Let's designate nodes A and B in the diagram. We indicate the direction of the currents. (See Figure 17)

2. Let's calculate the conductivity of each branch:

Figure 17

3. Let's determine the nodal voltage:

4. Let's find the currents of the branches:

A MAGNETIC FIELD

SUMMARY ON THE TOPIC:

METHODS FOR CALCULATING DC ELECTRICAL CIRCUITS

Introduction

The general task of analyzing an electrical circuit is that, based on given parameters (EMF, TMF, resistances), it is necessary to calculate currents, power, and voltage in individual sections.

Let us consider in more detail the methods of calculating electrical circuits.

1. Method of Kirchhoff equations

This method is the most general method for solving the problem of electrical circuit analysis. It is based on solving a system of equations compiled according to Kirchhoff’s first and second laws regarding real currents in the branches of the circuit in question. Therefore, the total number of equations p equal to the number of branches with unknown currents. Some of these equations are compiled according to Kirchhoff's first law, the rest - according to Kirchhoff's second law. In a diagram containing q nodes, according to Kirchhoff’s first law, we can compose q equations. However, one of them (any one) is the sum of all the others. Consequently, independent equations compiled according to Kirchhoff’s first law will be

.

According to Kirchhoff's second law, the missing m equations, the number of which is equal to

.

To write equations according to Kirchhoff's second law, you must select m contours so that they ultimately include all the branches of the circuit.

Let's consider this method using a specific circuit as an example (Fig. 1).

First of all, we select and indicate on the diagram the positive directions of currents in the branches and determine their number p. For the considered scheme p= 6. It should be noted that the directions of currents in the branches are chosen arbitrarily. If the accepted direction of any current does not correspond to the actual one, then the numerical value of this current is negative.

Therefore, the number of equations according to Kirchhoff’s first law is equal to q – 1 = 3.

The number of equations compiled according to Kirchhoff's second law

m = p - (q – 1) = 3.

We select the nodes and circuits for which we will compose equations and designate them on the electrical circuit diagram.

Equations according to Kirchhoff's first law:

Equations according to Kirchhoff's second law:

By solving the resulting system of equations, we determine the branch currents. Calculation of an electrical circuit does not necessarily involve calculating currents based on the given EMF of voltage sources. Another formulation of the problem is also possible - calculating the emf of sources based on given currents in the branches of the circuit. The problem can also be of a mixed nature - the currents in some branches and the emf of some sources are specified. It is necessary to find currents in other branches and emfs of other sources. In all cases, the number of equations compiled must be equal to the number of unknown quantities. The circuit may also include energy sources specified in the form of current sources. In this case, the current of the current source is taken into account as the branch current when drawing up equations according to Kirchhoff’s first law.

The circuits for composing equations according to Kirchhoff's second law must be selected so that not a single calculated circuit passes through the current source.

Let's consider the electrical circuit diagram shown in Fig. 2.

We select the positive directions of the currents and plot them on the diagram. The total number of circuit branches is five. If we consider the current of the current source J known quantity, then the number of branches with unknown currents p = 4.

The circuit contains three nodes ( q= 3). Therefore, according to Kirchhoff’s first law, it is necessary to compose q– 1 = 2 equations. Let's label the nodes in the diagram. The number of equations compiled according to Kirchhoff's second law m = p - (q – 1) =2.

We select the circuits in such a way that none of them passes through the current source, and mark them on the diagram.

The system of equations compiled according to Kirchhoff’s laws has the form:

Solving the resulting system of equations, we find the currents in the branches. The method of Kirchhoff equations is applicable to the calculation of complex both linear and nonlinear circuits, and this is its advantage. The disadvantage of the method is that when calculating complex circuits it is necessary to compose and solve a number of equations equal to the number of branches p .

The final stage of the calculation is checking the solution, which can be done by drawing up a power balance equation.

The balance of power in an electrical circuit means the equality of the power developed by all energy sources of a given circuit and the power consumed by all receivers of the same circuit (the law of conservation of energy).

If in the circuit section ab there is an energy source with EMF

and a current flows through this section, then the power developed by this source is determined by the product.

Each of the factors of this product can have a positive or negative sign relative to the direction ab. Work

will have a positive sign if the signs of the calculated values ​​coincide (the power developed by this source is given to the receivers of the circuit). The product will have a negative sign if the signs and are opposite (the source consumes power developed by other sources). An example would be a battery in charging mode. In this case, the power of this source (term) is included in the algebraic sum of the powers developed by all sources of the circuit, with a negative sign. The magnitude and sign of the power developed by the current source are determined similarly. If in a section of the circuit mn there is an ideal current source with a current, then the power developed by this source is determined by the product. As in the EMF source, the sign of the product is determined by the signs of the factors.

Now we can write down the general form of the power balance equation

.

For the circuit presented in Fig. 2.2, the power balance equation has the form

.

2. Loop current method

The loop current method comes down to composing equations only according to Kirchhoff’s second law. The number of these equations is equal to

, there are fewer equations than the number of equations required to calculate electrical circuits using the method of Kirchhoff’s laws.

In this case, we assume that in each selected circuit independent design currents, called circuit currents, flow. The current of each branch is determined as the algebraic sum of the loop currents closing through this branch, taking into account the accepted directions of the loop currents and the signs of their magnitudes.