Method of equivalent transformations in an alternating current circuit. Methods for calculating linear electrical circuits

An electrical circuit with a series connection of resistances (Figure 1.3, a) is replaced by a circuit with one equivalent resistance Rek (Figure 1.3, b), equal to the sum of all circuit resistances:

Rek = R1 + R2 +…+ Rn = , (1.5)

where R1, R2 ... Rn are the resistances of individual sections of the circuit.

Figure 1.3 Electrical circuit with series connection of resistances

In this case, the current I in the electric circuit remains unchanged, all resistances are flowed around by the same current. Voltages (voltage drops) on the resistances when they are connected in series are distributed in proportion to the resistances of individual sections:

U1/R1 = U2/R2 = … = Un/Rn.

With a parallel connection of resistances, all resistances are under the same voltage U (Figure 1.4). It is advisable to replace an electrical circuit consisting of parallel-connected resistances with a circuit with an equivalent resistance Rek, which is determined from the expression

where is the sum of the values ​​reciprocal to the resistances of sections of parallel branches of the electrical circuit;

Rj - resistance of the parallel section of the circuit;

n is the number of parallel branches of the chain.

Figure 1.4 Electric circuit with parallel connection of resistances

The equivalent resistance of a circuit section consisting of identical resistances connected in parallel is Rek = Rj / n. When two resistances R1 and R2 are connected in parallel, the equivalent resistance is defined as

and the currents are distributed inversely with these resistances, while

U = R1I1 = R2I2 = ... = RnIn.

With a mixed connection of resistances, i.e. in the presence of sections of the electrical circuit with series and parallel connection of resistances, the equivalent resistance of the circuit is determined in accordance with the expression

In many cases, it also makes sense to convert the resistances connected by a triangle (Figure 1.5) to an equivalent star (Figure 1.5).

Figure 1.5 Electrical circuit with delta and star connection

In this case, the resistance of the rays of an equivalent star is determined by the formulas:

R1 = ; R2 = ; R3 = ,

where R1, R2, R3 are the resistances of the rays of the equivalent resistance star;

R12, R23, R31 are the resistances of the sides of the equivalent resistance triangle. When replacing a resistance star with an equivalent resistance triangle, its resistance is calculated by the formulas:

R31 = R3 + R1 + R3R1/R2; R12 = R1 + R2 + R1R2/R3; R23 = R2 + R3 + R2R3/R1.

2.2. Parallel connection of elements
electrical circuits

On fig. 2.2 shows an electrical circuit with a parallel connection of resistances.

Rice. 2.2

Currents in parallel branches are determined by the formulas:

where - conductivity of the 1st, 2nd and nth branches.

According to Kirchhoff's first law, the current in the unbranched part of the circuit is equal to the sum of the currents in the parallel branches.

The equivalent conductivity of an electric circuit consisting of n elements connected in parallel is equal to the sum of the conductivities of the elements connected in parallel.
The equivalent resistance of the circuit is the reciprocal of the equivalent conductivity

Let the electrical circuit contain three resistors connected in parallel.
Equivalent conductivity

The equivalent resistance of a circuit consisting of n identical elements is n times less than the resistance R of one element

Let's take a circuit consisting of two resistors connected in parallel (Fig. 2.3). The resistance values ​​and the current in the unbranched part of the circuit are known. It is necessary to determine the currents in the parallel branches.

Rice. 2.3 Equivalent circuit conductance

,

and the equivalent resistance

Circuit input voltage

Currents in parallel branches

Similarly

The current in the parallel branch is equal to the current in the unbranched part of the circuit, multiplied by the resistance of the opposite, alien parallel branch and divided by the sum of the resistances of the alien and its parallel branches.

2.3.Transformation of the resistance triangle
into an equivalent star

There are circuits in which there are no resistances connected in series or in parallel, for example, the bridge circuit shown in Fig. 2.4. It is impossible to determine the equivalent resistance of this circuit relative to the branch with the EMF source using the methods described above. If the triangle of resistances R1-R2-R3 included between nodes 1-2-3 is replaced by a three-beam resistance star, the rays of which diverge from point 0 to the same nodes 1-2-3, the equivalent resistance of the resulting circuit is easily determined.

Rice. 2.4 The resistance of the beam of an equivalent resistance star is equal to the product of the resistances of the adjacent sides of the triangle, divided by the sum of the resistances of all sides of the triangle.
In accordance with said rule, the resistance of the rays of the star is determined by the formulas:

The equivalent connection of the resulting scheme is determined by the formula

The resistances R0 and Rλ1 are connected in series, and the branches with resistances Rλ1 + R4 and Rλ3 + R5 are connected in parallel.

2.4.Transformation of the resistance star
into an equivalent triangle

Sometimes, to simplify the circuit, it is useful to convert the resistance star into an equivalent triangle.
Consider the diagram in Fig. 2.5. Let's replace the star of resistances R1-R2-R3 with an equivalent triangle of resistances RΔ1-RΔ2-RΔ3 connected between nodes 1-2-3.

2.5. Resistance Star Conversion
into an equivalent triangle

The resistance of a side of an equivalent resistance triangle is equal to the sum of the resistances of the two adjacent rays of the star plus the product of the same resistances, divided by the resistance of the remaining (opposite) beam. The resistances of the sides of a triangle are determined by the formulas:

The equivalent resistance of the converted circuit is

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The method of equivalent transformations consists in replacing the electrical circuit or part of it with a simpler electrical circuit. In this case, the currents and voltages in the unconverted part of the circuit must remain unchanged. Any series connection can include an arbitrary number of resistances (resistors) and EMF sources, as well as no more than one current source.

H the presence of more than one current source in the connection is excluded due to a logical contradiction, since in a series connection, the same current flows through all the elements and this current is equal to the current of the source. If there are several current sources, then they must form several different currents, which is impossible due to the nature of their connection. The presence of a source in a connection only means that the current in this connection is given, therefore, without prejudice to the generality of the conclusions, the current source can be moved outside the connection and not considered. Then, in the general case, the connection will include m resistances and n EMF sources (Fig. a). Without changing the operating mode of the connection, they can be moved so that two groups of elements are formed: resistances and EMF sources (Fig. b). For this circuit, the Kirchhoff equation can be written as:

U=IR1+IR2+…+IRm+E1+…-En-1+En=I(R1+R2+…Rm)+E1…-En-1+En=IR+E

Thus, any series connection of elements can be represented by a series connection of one resistance R and one source of EMF E Moreover, the total resistance of the connection is equal to the sum of all resistances

and the total EMF is the algebraic sum

6. Method of nodal potentials

The current in any branch of the circuit can be found using Ohm's law for a circuit section containing an EMF. In order to be able to apply Ohm's law, it is necessary to know the potentials of the nodes of the circuit. The method of calculating electrical circuits, in which the potentials of the nodes of the circuit are taken as unknowns, is called the method of nodal potentials. Let's say there are n nodes in the circuit. Since any (one) point of the circuit can be grounded without changing the current distribution in it, one of the nodes of the circuit can be mentally grounded, that is, its potential can be taken equal to zero. In this case, the number of unknowns decreases from n to n-1. The number of unknowns in the method of nodal potentials is equal to the number of equations that must be compiled for the circuit according to the first Kirchhoff law. In the case when the number of nodes without unity is less than the number of independent circuits in the circuit, this method is more economical than the method loop currents. Kirchhoff's first law: The algebraic sum of the currents for each node in a branched circuit is zero I1+I2+I3+…+In=0

7.Two node method

Often there are schemes containing only two nodes. The most rational method for calculating the currents in them is the two-node method. The method of two nodes is understood as a method for calculating electrical circuits, in which the voltage between the two nodes of the circuit is taken as the desired (with its help, then the currents of the branches are determined). The circuit has two nodes. We take the potential of point 2 equal to zero φ2 = 0. Let's make a nodal equation for node 1.

φ1(g1+g2+g3)- φ2(g1+g2+g3)=E1g1-E3g3

U12= φ1- φ2= φ1= (E1g1-E3g3)/g1+g2+g3, where

g1=1/R1, g2=1/R2, g3=1/R3 – branch conductivities

AT general view

The denominator of the formula is the sum of the conductivities of the branches connected in parallel. In the numerator - the algebraic sum of the products of the EMF of sources and the conductivity of the branches in which these EMFs are included. The EMF in the formula is written with a plus sign if it is directed to node 1, and with a minus sign if it is directed from node 1. After calculating the potential value φ1, we find the currents in the branches using Ohm's law for the active and passive branches.

8 .Loop current method

When calculating by the method of loop currents, it is assumed that each independent loop of the circuit has its own loop current. The equations are made with respect to the loop currents, after which the branch currents are determined through them. Thus, the method of loop currents can be defined as a calculation method in which loop currents are taken as the desired ones. The number of unknowns in this method is equal to the number of equations that had to be compiled for the circuit according to the second Kirchhoff law: the algebraic sum of the products of the resistance of each of the sections of any closed circuit of a branched circuit direct current on the current strength in this section is equal to the algebraic sum of the EMF along this circuit. I1R1 + I2R2 \u003d E1 + E2

The currents in the resistances R1 and R2 are equal to the corresponding circuit currents. The current in the resistance R3, which is common to both circuits, is equal to the difference between the circuit currents I11 and I22, since these currents are directed oppositely to the branches with R3. Independent circuits are selected, and arbitrary directions of the circuit currents are set. In our case, these currents are directed clockwise. The direction of loop bypass coincides with the direction of loop currents. The equations for these circuits are as follows: I11(R1+Ri1)+I11R3-I22R3=E1,

I22(Ri2-R2)+I22R3-I11R3=-E2 of this circuit is called the self-resistance of the circuit. Self-resistance of the circuit circuits R11=R1+Ri1+R3, R22=Ri2+R2+R3 The resistance R3, belonging to two circuits at the same time, is called the total resistance of these circuits. R12=R21=R3 where R12 is the total resistance between the first and second circuits; R21 is the total resistance between the second and first circuits. E11 = E1 and E22 = E2 are the loop emfs. I11R11+I22R12=E11, I11R21+I22R22=E22 Self-resistances always have a plus sign.

The total resistance has a minus sign if the loop currents in this resistance are directed opposite to each other, and a plus sign if the loop currents in the total resistance coincide in direction. Solving the equations jointly, we find the loop currents I11 and I22, then we pass from the loop currents to the currents in the branches. I1=I11, I2=I22,I3=I11-I22.

9. Overlay method. This method is valid only for linear electrical circuits and is especially effective when it is required to calculate currents for various EMF values ​​and source currents while the circuit resistances remain unchanged. This method is based on the principle of superposition (superposition), which is formulated as follows: the current in the k -th branch of a linear electrical circuit is equal to the algebraic sum of the currents caused by each of the sources separately. Analytically, the superposition principle for a circuit containing n EMF sources and m current sources , is expressed

ratio: Here - the complex of input conductivity of the k - th branch, numerically equal to the ratio of the current to the EMF in this branch at zero EMF in the remaining branches; - the complex of mutual conductivity of the k - th and i - th branches, numerically equal to the ratio of the current in the k - th branch and EMF in the i-th branch with zero EMF in the remaining branches. The input and mutual conductivities can be determined experimentally or analytically, using their indicated semantic interpretation, in this case, which directly follows from the reciprocity property. Similarly, the current transfer coefficients are determined, which, unlike conductivities, are dimensionless quantities.

The proof of the principle of superposition can be carried out on the basis of the method of loop currents.

If we solve the system of equations compiled by the method of loop currents for any loop current, for example, we get (2), where

-the determinant of the system of equations compiled by the method of loop currents; - the algebraic complement of the determinant. Each of the EMF in (2) is the algebraic sum of the EMF in the branches of the i-th circuit. If now all the loop EMFs in (2) are replaced by the algebraic sums of the EMFs in the corresponding branches, then after grouping the terms, an expression for the loop current will be obtained in the form of an algebraic sum of the components of the currents caused by each of the EMFs of the branches separately. Since the system of independent circuits can always be chosen so that the considered hth branch will enter only one circuit, i.e. loop current will be equal to the actual current of the h-th branch, then the superposition principle is valid for currents of any branches and, therefore, the validity of the superposition principle has been proven. internal resistances, and calculate the components of the desired currents in these circuits. After that, the results obtained for the corresponding branches are summarized - these will be the desired currents in the branches of the original circuit.

The analysis of any electrical circuit begins with the construction of its model, which is described by the equivalent circuit.

AT electrical diagrams the following simplest connections of passive elements are distinguished: series, parallel, connection in the form of a triangle and in the form of a three-beam star. Before starting circuit analysis, it is desirable to carry out preliminary equivalent circuit transformations. The essence of such transformations is to replace some part of the circuit with another one, electrically equivalent to it, but with a structure more convenient for calculation. More often than others, two types of such transformations are used: replacement of series and parallel connected elements with one equivalent; transformation of a three-beam star into a triangle and vice versa.

The equivalent resistance of series-connected elements is equal to the arithmetic sum of their resistances:

. (1.26)

The equivalent conductivity of resistive elements connected in parallel is equal to the arithmetic sum of their conductivities:

. (1.27)

When transforming a triangle (Fig. 1.14) into a star (Fig. 1.15), with given resistances of the sides of the triangle RAB, RBV, RBA, the equivalent resistances of the rays of the star RA, RB, RB are determined.

Rice. 1.14. Circuit Diagram - Triangle

Rice. 1.15. Circuit diagram - star

The equivalent resistances of the rays of a star are:

When transforming a star into an equivalent triangle for given RA, RB, RB, the equivalent resistances are determined as follows.

Kirchhoff's first law

At any node of the electrical circuit, the algebraic sum of the currents is zero

Kirchhoff's second law

In any closed circuit of an electrical circuit, the algebraic sum of the EMF is equal to the algebraic sum of the voltage drops in all its sections

Calculation of an electrical circuit using Kirchhoff's laws. Power balance

Based on Ohm's and Kirchhoff's laws, absolutely any electrical circuit can be calculated. Other circuit calculation methods are designed solely to reduce the amount of computation required.

Sequencing:

The directions of currents in the branches are arbitrarily assigned.

Arbitrarily assign directions to bypass the contours.

Write U - 1 equation according to Kirchhoff's I law. (Y is the number of nodes in the chain).

Write B - Y + 1 equation according to Kirchhoff's II law. (B is the number of branches in the chain).

Solve a system of equations for currents and specify the magnitude of the voltage drops on the elements.

Notes:

When compiling equations, the terms are taken with a "+" sign if the direction of bypassing the circuit coincides with the direction of the voltage drop, current or EMF. AT otherwise with a "-" sign.

If negative currents are obtained when solving the system of equations, then the chosen direction does not coincide with the real one.

You should choose those contours in which there are the fewest elements.

The correctness of the calculations can be checked by composing power balance. In an electrical circuit, the sum of the powers of the power sources is equal to the sum of the powers of the consumers:

It should be remembered that one or another source of the circuit may not generate energy, but consume it (the process of charging batteries). In this case, the direction of the current flowing through the section with this source is opposite to the direction of the EMF. Sources in this mode should enter the power balance with the "-" sign.

Loop current method

One of the electrical circuit analysis methods is loop current method. It is based on Kirchhoff's second law.

Real current in a certain branch is determined by the algebraic sum of the loop currents, in which this branch is included. Finding real currents is the primary task of the loop current method.

1. We arbitrarily choose the directions of real currents I1-I6.

2. We select three circuits, and then indicate the direction of the circuit currents I11, I22, I33. We will choose a clockwise direction.

3. We determine the own resistance of the circuits. To do this, we add the resistances in each circuit.

R11=R1+R4+R5=10+25+30= 65 ohm

R22=R2+R4+R6=15+25+35 = 75 Ohm

R33=R3+R5+R6=20+30+35= 85 ohm

Then we determine the common resistances, the common resistances are easy to detect, they belong to several circuits at once, for example, the resistance R4 belongs to circuit 1 and circuit 2. Therefore, for convenience, we denote such resistances by the numbers of the circuits to which they belong.

R12=R21=R4=25 ohm

R23=R32=R6=35 ohm

R31=R13=R5=30 ohm

4. We proceed to the main stage - the compilation of a system of equations for loop currents. The left side of the equations includes voltage drops in the circuit, and the right side EMF of the sources of this circuit.

Since we have three contours, therefore, the system will consist of three equations. For the first circuit, the equation will look like this:

The current of the first circuit I11, we multiply by its own resistance R11 of the same circuit, and then subtract the current I22 multiplied by the total resistance of the first and second circuits R21 and the current I33 multiplied by the total resistance of the first and third circuit R31. This expression will be equal to the EMF E1 of this circuit. We take the EMF value with a plus sign, since the bypass direction (clockwise) coincides with the EMF direction, otherwise it would be necessary to take it with a minus sign.

We do the same actions with two other circuits and as a result we get a system:

In the resulting system, we substitute the already known resistance values ​​​​and solve it in any known way.

5. The last step is to find the real currents, for this you need to write expressions for them.

The loop current is equal to the actual current, which belongs only to this loop.. That is, in other words, if the current flows in only one circuit, then it is equal to the circuit.

But, you need to take into account the direction of bypass, for example, in our case, the current I2 does not coincide with the direction, so we take it with a minus sign.

The currents flowing through the common resistances are defined as the algebraic sum of the circuit ones, taking into account the direction of the bypass.

For example, current I4 flows through resistor R4, its direction coincides with the direction of bypassing the first circuit and opposite to the direction of the second circuit. So, for him, the expression will look like

And for the rest

Method of equivalent transformations

Some complex electrical circuits contain multiple sinks but only one source. Such chains can be calculated by the method of equivalent transformations. This method is based on the possibility of converting two series-connected or parallel-connected resistors R1 and R2 to one equivalent Req. The condition for equivalent conversion should be the preservation of the current and voltage of the section under consideration: I = Ieq, U = Ueq. For the initial section of the circuit, according to Kirchhoff's II law, taking into account Ohm's law for each of the two series-connected elements: U = U1 + U2 = R1I + R2I = (R1 + R2)I. For an equivalent element according to Ohm's law: Ueq = Reqv * Ieq. Taking into account the conditions of the equivalent transformation U = Ueq = (R1 + R2)I = (R1 + R2)Ieq = Reqv* Ieq. Hence Req = (R1 + R2). This ratio determines the resistance of an element equivalent to two series-connected elements. For two parallel-connected elements according to Kirchhoff's I law, taking into account Ohm's law for each of the two parallel-connected elements: I = I1 + I2 = U/R1 + U/R2 = U(1/R1 + 1/R2). For an equivalent element according to Ohm's law: Ieq = Ueq / Req. Taking into account the conditions of the equivalent transformation I = Ieq = U(1/R1 + 1/R2) = Ueq(1/R1 + 1/R2) = Ueq/Req, hence 1/Req = 1/R1 + 1/R2 (1.59) or Req = (R1 R2)/(R1 + R2). This ratio defines the resistance of an element equivalent to two parallel connected elements. The relations make it possible to carry out stage-by-stage equivalent transformations of a complex electrical circuit with several receivers and to calculate such a circuit. At given parameters of all elements of the circuit (E, R1, R2, R3), the calculation can be carried out by the method of equivalent transformations as follows. At the first stage of conversion, two resistors R1 and R2 connected in parallel are replaced by one equivalent with the resistance Req12 equal to Req12 = (R1 * R2)/(R1 + R2). (1.61) In this case, an equivalent circuit is formed, which contains two resistors Req12 and R3 connected in series. The voltage Uab in the equivalent circuit corresponds to the voltage Uab in the original circuit, and the current in the equivalent circuit corresponds to the current in the unbranched part of the original circuit. At the second stage of conversion, two series-connected resistors Req12 and R2 are replaced by one equivalent with resistance Req123 equal to Req123 = Req12 + R3. In this case, a simple equivalent circuit is formed, which contains one resistor Req123. The current in this circuit corresponds to the current in the unbranched part of the original circuit and is determined by Ohm's law: I = Uac/ Req123 = E/ Req123. Further calculation is carried out according to Ohm's law, following the stages of equivalent transformations in reverse order. For equivalent circuit: Uab = I* Req12 ; Ubc = I* R3 . For the original circuit: I1 = Uab/R1 ; I2 = Uab/R2. Thus, the described method of equivalent transformations makes it possible to calculate a complex electrical circuit without reducing the problem to solving a system of equations, but by successive calculations. However, this method is applicable to circuits containing only one source of EMF