Find all currents and voltages in the chain. Calculation of simple electrical chains. Basic laws of DC chains

In practice, a number of methods have been developed for determining and calculating constant current schemes, which provides the ability to reduce the laborious process of calculating difficult electrical chains. The basic laws with which the characteristics of almost every scheme are determined are Kirchhoff's postulates.

Ways to calculate electrical circuits

The calculation of the electrical circuits is branched into a plurality of methods used in practice, namely: the equivalent transformation method, a reception based on OMA and Kirchhoff postulates, a method of overlays, a method of contour currents, a method of nodal potentials, a method of identical generator.

The process of calculating the electrical circuit consists of several mandatory steps that allow you to quickly and accurately produce all calculations.

Before you know or calculate the necessary parameters, the calculated electrical circuit is transferred schematically on paper, where the symbolic notation of the elements included in its composition and their compounds are contained.

All elements and devices are divided into three categories:

Power sources. The main sign this element It is the conversion of non-electrical energy into electric. These energy sources are referred to as primary sources of energy. Secondary energy sources are such devices on the inputs and the outputs of which there is electrical energy. These include rectifier devices or voltage transformers;
Devices that consume electrical energy. Such elements convert electrical energy into any other, be it light, sound, heat, and the like species;
The auxiliary elements of the chain to which the wires are connected, switching equipment, protection and other similar elements.

Also to basic concepts electrical circuit relate:

The branch of the electrical circuit is a plot of a chain with the same current. The composition of such a branch can include one or more consistently connected elements;
The electrical circuit assembly is a point of connecting three and more scheme branches;
The circuit of the electrical circuit, which is any closed path passing through several branches.

Method of calculation according to the laws of Ohm and Kirchhoff

These laws allow you to find out the current strength and find the relationship between current values, stresses, EMF of the entire chain and single sections.

Ohma law for a plot of chain

According to the law of Oka, the current ratio, voltage and resistance of the chain looks like:

Based on this formula, it is possible to find the current strength by expression:

Ur - voltage or voltage drop on the resistor;
I - Current in the resistor.

Ohm law for full chain

In the Ohm law, the value of the internal resistance of the power supply is additionally used for the full chain. Find current strength taking into account internal resistance is possible by expression:

I \u003d E / R \u003d E / R0 + R, where:

E - EMF power supply;
rO - internal resistance of the power supply.

Since the complex electrical circuit consisting of several branches and has a number of power devices in its structure, cannot be described by the Ohm law, then the 1st and 2nd Kirchoff law apply.

First Law of Kirchhoff

The Law of Kirchhoff states that the sum of currents flowing into the node is equal to the amount of currents arising from it, it looks like:

Σmik \u003d 0, where M is the number of branches supplied to the node.

According to the Circhoff law, the currents flowing into the node are used with the "+" sign, and the currents arising from the node - with the sign "-".

The second law of Kirchhoff

From the second law of Kirchhof, it follows that the amount of stress drops on all elements of the chain is equal to the amount of EDC chain, it looks like:

Σnek \u003d σmrkik \u003d σmuk, where:

n - the number of EMF sources in the circuit;
m is the number of elements with RK resistance in the circuit;
UK \u003d RKIK - voltage or voltage drop on the k-volume element of the circuit.

Before applying the second law of Kirchhoff, check the following requirements:

Specify relatively positive directions of EDC, currents and stresses;
Specify the direction of circuit bypass described by the equation;
Using one of the interpretations of the 2nd Kirchhoff law, the characteristics of the equation included in the equation are used with the "+" sign, if there are relatively positive directions similar to the contour bypass, and with "-", if they are multidirectional.

From the 2nd Circle Law, an expression of the balance of capacity, according to which the power of power sources at any time is equal to the sum of the capacities spent at all parts of the chain. The capacity balance equation is:

Electrical circuit transformation method

Elements in electrical circuits can be connected in parallel, consistently, mixed method and according to the "Star" schemes, triangle. The calculation of such schemes is simplified by replacing several resistance to equivalent resistance, and further calculations are already carried out according to the law of Oma or Kirchhoff.

Under the mixed connection of the elements, the simultaneous presence in the diagram and sequential, and parallel connection of the elements is implied. At the same time, the resistance of the mixed compound is calculated after converting the circuit to an equivalent chain using the formulas shown in Fig. above.

There is also a connection of the "star" and "triangle" elements. To find the equivalent resistance, it is necessary to initially convert the "Triangle" scheme in the "Star" scheme. The picture below, the resistance is equal:

R1 \u003d R12R31 / R12 + R31 + R23,
R2 \u003d R12R23 / R12 + R31 + R23,
R3 \u003d R31R23 / R12 + R31 + R23.

Additional chain calculation methods

Everything additional methods Calculation of chains in one way or another are or based on the first and second laws of Kirchhoff. These methods include:

The contour current method is based on the introduction of additional quantities of contour currents satisfying the 1st Circhoff law;
The method of nodal potentials - it is used to use the potentials of all nodes of the circuit and then according to known potentials of currents in all branches. The method is based on the first law of Kirchhoff;
The equivalent generator method - this method provides a solution to the task, how to find a current only in one or several branches. The essence of the method is that any electrical circuit with respect to the under study can be represented as an equivalent generator;
The overlay method is based on the fact that the current in the chain or the scheme branch is equal to the algebraic amount of currents inspected by each source separately.

The main part of the calculation methods is aimed at simplifying the procedure for determining currents in the branches of the scheme. These activities are carried out either by simplifying the systems of equations by which calculations are carried out or simplifying the scheme itself. Based on, first of all, at Kirchhoff's postulates, any of the methods answers the question: how to determine the current strength and voltage of the electrical circuit.

Video

In the electrical engineering it is believed that the simple chain is a chain that comes down to a chain with one source and one equivalent resistance. You can roll the chain using equivalent transformations of sequential, parallel and mixed connections. Exceptions are chains containing more complex connections with a star and a triangle. DC Calculation Calculation Performed with the help of the Law of Ohm and Kirchhoff.

Example 1.

Two resistors are connected to a source of constant voltage of 50 V, with internal resistance r. \u003d 0.5 Ohm. Resistance resistors R 1 \u003d. 20 I. R 2 \u003d. 32 ohm. Determine the current in the chain and voltage on the resistors.

Since the resistors are connected in series, the equivalent resistance will be equal to their sum. Knowing it, we use the Ohm's law for the full chain to find the current in the chain.

Now knowing the current in the chain, you can determine the voltage drops on each of the resistors.

You can check the correctness of the solution in several ways. For example, using the Kirchhoff law, which states that the amount of EDC in the circuit is equal to the amount of stresses in it.

But with the help of the Kirchhoff law it is convenient to check simple chains having one contour. More in a convenient way Checks is the balance of power.

In the chain, the balance of capacity must be observed, that is, the energy given to the sources should be equal to the energy received by the receivers.

The source power is defined as a product of the EMF on the current, and the power obtained by the receiver as a product of the voltage drop on the current.

The advantage of testing the balance of capacity is that it does not need to be complex bulky equations based on the laws of Kirchhoff, it is enough to know EDC, voltage and currents in the chain.

Example 2.

Common circuit current containing two connected parallel resistor R. 1 \u003d 70 ohms and R. 2 \u003d 90 ohms, equal to 500 mA. Determine currents in each of the resistors.

Two successively connected resistors are nothing but a current divider. It is possible to determine the currents flowing through each resistor using the divider formula, while we do not need to know the voltage in the chain, only the total current and resistance of the resistors will be required.

Toki in resistors

In this case, it is convenient to test the task using the first Circhoff law, according to which the sum of currents of the convergent, in the node is zero.

If you do not remember the current divider formula, then you can solve the problem with another way. To do this, it is necessary to find a voltage in the chain that will be common to both resistors, since the connection is parallel. In order to find it, you must first calculate the chain resistance

And then tension

Knowing voltages, we find currents flowing through resistors

As you can see, currents turned out the same.

Example 3.

In the electrical circuit shown in the diagram R. 1 \u003d 50 ohms, R. 2 \u003d 180 ohms, R. 3 \u003d 220 ohms. Find the power allocated on the resistor R. 1, current through the resistor R. 2, voltage on the resistor R. 3, if it is known that the voltage on the clips of the chain of 100 V.

To calculate the DC power allocated on the R 1 resistor, it is necessary to determine the current I 1, which is common to the entire chain. Knowing the voltage on the clips and the equivalent chain resistance, you can find it.

Equivalent resistance and current in the chain

Hence the power allocated on R 1

Example 1.

Since the resistors are connected in series, the equivalent resistance will be equal to their sum. Knowing it, we use the Ohm's law for the full chain to find the current in the chain.

Now knowing the current in the chain, you can determine the voltage drops on each of the resistors.

You can check the correctness of the solution in several ways. For example, using the Kirchhoff law, which states that the amount of EDC in the circuit is equal to the amount of stresses in it.

But with the help of the Kirchhoff law it is convenient to check simple chains having one contour. A more convenient way to check is the balance of capacity.

In the chain, the balance of capacity must be observed, that is, the energy given to the sources should be equal to the energy received by the receivers.

The source power is defined as a product of the EMF on the current, and the power obtained by the receiver as a product of the voltage drop on the current.

The advantage of testing the balance of capacity is that it does not need to be complex bulky equations based on the laws of Kirchhoff, it is enough to know EDC, voltage and currents in the chain.

Example 2.

Common circuit current containing two connected parallel resistor R. 1 \u003d 70 ohms and R. 2 \u003d 90 ohms, equal to 500 mA. Determine currents in each of the resistors.

Toki in resistors

In this case, it is convenient to test the task using the first Circhoff law, according to which the sum of currents of the convergent, in the node is zero.

And then tension

Knowing voltages, we find currents flowing through resistors

As you can see, currents turned out the same.

Example 3.

Equivalent resistance and current in the chain

Hence the power allocated on R 1

Basics\u003e Tasks and Answers\u003e Permanent Electric Current

Methods for calculating DC chains

The chain consists of branches, has nodes and Current sources. The following formulas are suitable for calculating chains containing and sources of voltage and sources of current. They are also valid for those special cases: when there are only voltage sources or only sources of current in the circuit.

Application of Kirchhoff laws. Usually in the chain are known all sources of EDS and sources of currents and all resistance. In this case, the number of unknown currents is set equal. For each branch set by a positive current direction.
The number of interconnected equations compiled according to the first Circhoff law is equal to the number of nodes without a unit. The number of interconnected equations compiled according to the second law of Kirchoff,

In the preparation of equations on the second law of Kirchhoff, independent contours that do not contain current sources should be selected. The total number of equations composed of the first and second laws of Kirchhoff is equal to the number Unknown currents.
Examples are given in the objectives of the section.

Method of contour currents (Maxwell). This method allows to reduce the number of system equations to the number to the formula defined by the formula (0.1.10). It is based on the fact that the current in any branch of the chain can be represented as an algebraic amount of contour currents occurring along this branch. When using this method, choose and denote contour currents (at least one selected contour current) should be passed on any branch). The theory is known that the total number of contour currents. It is recommended to choose Contour currents so that each of them passes through one current source (these contour currents can be considered coinciding with the corresponding current sources current and they are usually given to the terms of the task), and the remaining Contour currents to choose passing on branches that do not contain current sources. To determine the latest contour currents according to the second Circhoff law for these contours, they constitute the equations in this form:

where - own contour resistancen. (The sum of the resistance of all branches included in the contourn); - general contour resistancen and L, and if the directions of contour currents in the total branch for contoursn and L coincide, then positive , otherwise negatively; - Algebraic amount of EDC included in the branches forming the contourn; - general resistance of the branch of the contourn. With a circuit containing current source.
Examples are given in the objectives of the section.

Method of nodal stresses. This method allows you to reduce the number of system equations to the number of in equal to the number of nodes without one

The essence of the method is that at first the solution of the system of equations (0.1.13) determine the potentials of all nodes of the scheme, and the currents of the branches connecting the nodes are found using the Ohm law.
In the preparation of equations according to the components of the nodal stresses, it is first supposed to zero the potential of any node (it is called basic). To determine the potentials of the remaining nodes are the following system of equations:

Here - the sum of the conductivity of the branches attached to the node S; - the sum of the branches of the branches directly connecting the node S with the Q node; - Algebraic amount of the works of EMF branches adjacent to the nodes. , on their conductivity; At the same time, with the "+" sign, the EMFs are taken, which act in the direction of the node S, and with the sign "-" - in the direction from the node S; - Algebraic sum of current sources attached to the node S; At the same time, with the "+" sign, those currents are taken to the nodes. , and with the sign "-" - in the direction from the node s.
The nodal voltage method is recommended to use in cases where the number of equations is less than the number of equations drawn up by the contour current method.
If in the diagram, some nodes are connected by the ideal sources of EDC, the number in the equations component according to the node-voltage method decreases:

where - The number of branches containing only ideal sources of EDC.
Examples are given in the objectives of the section.
Private case-duplex scheme. For schemes having two nodes (for definiteness nodes A andb. ), nodal voltage

where - the algebraic amount of the works of EMF branches (EMF is considered positive if they are directed to the node A, and negative, if from the node A to the nodeb. ) on the conductivity of these branches; - current sources (positive if they are directed to the node A, and are negative, if directed from the node A to the nodeb); - Amount conductivity of all branches connecting nodes A andb.

The principle of imposition. If in the electrical circuit in the specified values \u200b\u200bof the sources and current sources current, the calculation of currents based on the principle of overlay is as follows. The current in any branch can be calculated as an algebraic amount of currents caused in it by each EMF source separately and the current passing through the same branch from each current source. It should be borne in mind that when currents are calculated caused by any one source of EDS or current, the remaining EMF sources in the circuit are replaced by short-circuited areas, and branches with current sources of the remaining sources are disconnected (branches with current sources are opened).

Equivalent circuit transformation. In all cases, the transformation of the replacement of some schemes by others, it is equivalent, should not lead to a change in currents or stresses in the regions of the circuit not subjected to the transformation.
Replacing sequentially connected resistances in one equivalent. Resistance are connected in series if they are streamlined with the same current (for example, resistance Connected sequentially (see Fig. 0.1.3), also consistent resistance).
n. sequentially connected resistance equal to the sum of these resistances

With a sequential connection n The resistances of the voltage on them are distributed directly in proportion to these resistance

In the particular case of two successively connected resistances

where u. - general voltage acting on a section of a chain containing two resistance (See Fig. 0.1.3).
Replacing parallel connected resistances in one equivalent. Resistance are connected in parallel if the weight they are attached to one part of the nodes, for example, resistance (See Fig. 0.1.3).
Equivalent chain resistance consisting ofn. parallel to the connected resistance (Fig. 0.1.4),

In the particular case of parallel compound of two resistances Equivalent resistance

With parallel connection n Resistance (Fig. 0.1.4, a) Currents in them are distributed inversely proportional to their resistance or directly proportional to their conductors

Current Each of them is calculated through the currentI. in the unbranched part of the chain

In the particular case of two parallel branches (Fig. 0.1.4, b)

Replacing the mixed resistance connection with one equivalent. Mixed connection is a combination of serial and parallel resistance connections. For example, resistance (Fig. 0.1.4, b) combined mixed. Their equivalent resistance

Formulas for conversion of the resistance triangle (Fig. 0.1.5, a) in an equivalent resistance star (Fig. 0.1.5, b), and vice versa, have this kind:

Equivalent source method (active two-pole metol, or idling and short-circuit method). The use of the method is advisable to determine the current in any one branch of the complex electrical circuit. Consider two options: a) the equivalent source of EMF and b) method of an equivalent current source.
With the method of equivalent source of EDC To find the currentI. in the arbitrary branch AB, the resistance of which R (Fig. 0.1.6, and, Letter A means an active two-pole), it is necessary to break off this branch (Fig. 0.1.6,b), and part of the chain connected to this branch, replace the equivalent source with EMF and internal resistance (Fig. 0.1.6, B).
EMF. This source is equal to the voltage on the clips of the open branch (idle voltage):

Calculation of schemes in idle mode (see Fig. 0.1.6, b) to determine It is carried out by any well-known method.
Interior resistance Equivalent EDC source is equal to the input resistance of the passive chain relative to the clamps A and B of the source scheme, from which all sources [EMF sources are replaced by short-circuited areas, and branches with current sources are disabled (Fig. 0.1.6, g); The letter P indicates the passive character of the chain], with an open AB branch. Resistance can be calculated directly according to the fig. 0.1.6, G.
Current in the desired branch of the scheme (Fig. 0.1.6, d) having resistance R is determined by the law of Ohm:

Calculation of electrical circuits of DC

Basic laws defining calculation of electrical chainare the laws of Kirchhoff.

Based on the laws of Kirchhoff, a number of practical methods have been developed calculation of DC electrical circuitsto reduce calculations when calculating complex schemes.

Significantly simplify calculations, and in some cases and reduce the complexity of the calculation, possibly with the help equivalent transformations schemes.

Convert parallel and consistent elements connections, a star connection to an equivalent "triangle" and vice versa. Replace the source of the current by the equivalent EDC source. Method of equivalent transformations theoretically you can calculate any chain, and at the same time use simple computing. Or determine the current in any one branch, without the calculation of currents of other sections of the chain.

In this article on theoretical foundations of electrical engineering Examples of calculating linear electrical circuits of DC using method of equivalent transformations typical schemes Compounds of sources and consumers of energy are given the calculated formulas.

Solving tasks

Task 1. For chain (Fig. 1), determine equivalent resistance relative to the entrance clamps a-G., if you know: R. 1 = R. 2 \u003d 0.5 Ohm, R. 3 \u003d 8 Ohm, R. 4 = R. 5 \u003d 1 ohms, R. 6 \u003d 12 ohms, R. 7 \u003d 15 ohms, R. 8 \u003d 2 ohms, R. 9 \u003d 10 ohms, R. 10 \u003d 20 ohms.

Let's start equivalent transformations Schemes from the branch of the most remote from the source, i.e. From the clamps a-G.:

Task 2. For chain (Fig. 2, but), determine the input resistance If you know: R. 1 = R. 2 = R. 3 = R. 4 \u003d 40 ohms.

Fig. 2.

The initial scheme can be reversed relative to the input clamps (Fig. 2, b.) What is seen that all resistance is included in parallel. Since the resistance values \u200b\u200bare equal, then to determine the value equivalent resistanceyou can use the formula:

where R. - the magnitude of the resistance, Ohm;

n. - The number of connected resistance parallel.

Task 3. Determine equivalent resistance Regarding the clamps a-B., if a R. 1 = R. 2 = R. 3 = R. 4 = R. 5 = R. 6 \u003d 10 ohms (Fig. 3, but).

We convert the connection "Triangle" f-D-C In the equivalent "star". Determine the magnitudes of the converted resistance (Fig. 3, b.):

By the condition of the problem of the magnitude of all resistances is equal, and therefore:

On the transformed scheme received parallel connection Branches between the nodes e-B., then equivalent resistance equally:

And then equivalent resistance The source scheme represents a serial connection of the resistance:

Task 4. In a given chain (Fig. 4, but) Input resistances branches A-b., c-d. and f-B.If it is known: R. 1 \u003d 4 ohm, R. 2 \u003d 8 Ohm, R. 3 \u003d 4 ohm, R. 4 \u003d 8 ohms, R. 5 \u003d 2 ohms, R. 6 \u003d 8 ohms, R. 7 \u003d 6 ohms, R. 8 \u003d 8 ohms.

To determine the input resistance of branches, all sources of EMF are excluded from the circuit. At the same time c. and d., as well as b. and f. Compared by spice, because Internal resistances of ideal voltage sources are zero.

Branch a-b. burst, and because resistance R a -b. \u003d 0, then the input resistance of the branch is equivalent to the equivalent resistance of the circuit relative to the points a. and b. (Fig. 4, b.):

Similarly method of equivalent transformations The input resistances of the branches are determined R CD. and R bf.. Moreover, when calculating the resistance, it is taken into account that the connection of the spice points a. and b. eliminates ("grows") from the resistance scheme R. 1 , R. 2 , R. 3 , R. 4 in the first case, and R. 5 , R. 6 , R. 7 , R. 8 in the second case.

Task 5. In the chain (Fig. 5) determine the equivalent transformation method toki. I. 1 , I. 2 , I. 3 I. make up balance capacity , if you know: R. 1 \u003d 12 ohms, R. 2 \u003d 20 ohms, R. 3 \u003d 30 ohms, U. \u003d 120 V.

Equivalent resistancefor parallel resistances:

Equivalent resistance all chains:

Current in the unbranched part of the scheme:

Voltage on parallel resistances:

Current in parallel branches:

Balance of capacity :

Task 6. In the chain (Fig. 6, but), determine method of equivalent transformations Ammeter readings , if you know: R. 1 \u003d 2 ohms, R. 2 \u003d 20 ohms, R. 3 \u003d 30 ohms, R. 4 \u003d 40 ohms, R. 5 \u003d 10 ohms, R. 6 \u003d 20 ohms, E. \u003d 48 V. Ammeter resistance can be considered equal to zero.

If resistance R. 2 , R. 3 , R. 4 , R. 5 Replace one equivalent resistance r e, the initial scheme can be presented in a simplified form (Fig. 6, b.).

The value of equivalent resistance:

Convert parallel connection Resistance R E. and R. 6 schemes (Fig. 6, b.), we get a closed circuit for which the second law of Kirchhoff You can record the equation:

from where current I. 1:

Voltage on the clips of parallel branches U. AB Express the equation law Ohm. For passive branch obtained by conversion R E. and R. 6:

Then the ammeter will show the current:

Task 7. Determine the currents of the branches of the scheme by the method of equivalent transformations (Fig. 7, but), if a R. 1 = R. 2 = R. 3 = R. 4 \u003d 3 ohms, J. \u003d 5 A, R. 5 \u003d 5 ohms.

Find all currents and voltages in the chain. Calculation of simple electrical chains. Basic laws of DC chains

Ways to calculate electrical circuits

Method of calculation according to the laws of Ohm and Kirchhoff

Ohma law for a plot of chain

Ohm law for full chain

First Law of Kirchhoff

The second law of Kirchhoff

Electrical circuit transformation method

Additional chain calculation methods

Video

Example 1.

Example 2.

Example 3.

Example 1.

Example 2.

Example 3.

Last

It is necessary to know