Basic methods for calculating DC electrical circuits. DZ - Calculation of a complex DC circuit. Equivalent circuit method
In electrical engineering, it is generally accepted that a simple circuit is a circuit that reduces to a circuit with one source and one equivalent resistance. You can collapse a circuit using equivalent transformations of serial, parallel, and mixed connections. The exception is circuits containing more complex star and delta connections. Calculation of DC circuits produced using Ohm's and Kirchhoff's laws.
Example 1
Two resistors are connected to a 50 V DC voltage source, with internal resistance r = 0.5 Ohm. Resistor values R 1 = 20 and R2= 32 Ohm. Determine the current in the circuit and the voltage across the resistors.
Since the resistors are connected in series, the equivalent resistance will be equal to their sum. Knowing it, we will use Ohm's law for a complete circuit to find the current in the circuit.
Now knowing the current in the circuit, you can determine the voltage drop across each resistor.
There are several ways to check the correctness of the solution. For example, using Kirchhoff's law, which states that the sum of the emf in the circuit is equal to the sum of the voltages in it.
But using Kirchhoff's law it is convenient to check simple circuits that have one circuit. A more convenient way to check is power balance.
The circuit must maintain a power balance, that is, the energy given by the sources must be equal to the energy received by the receivers.
The source power is defined as the product of the emf and the current, and the power received by the receiver as the product of the voltage drop and the current.
The advantage of checking the power balance is that you do not need to create complex cumbersome equations based on Kirchhoff’s laws; it is enough to know the EMF, voltages and currents in the circuit.
Example 2
Total current of a circuit containing two resistors connected in parallel R 1 =70 Ohm and R 2 =90 Ohm, equals 500 mA. Determine the currents in each of the resistors.
Two resistors connected in series are nothing more than a current divider. We can determine the currents flowing through each resistor using the divider formula, while we do not need to know the voltage in the circuit; we only need the total current and the resistance of the resistors.
Currents in resistors 
In this case, it is convenient to check the problem using Kirchhoff’s first law, according to which the sum of currents converging at a node is equal to zero.
If you do not remember the current divider formula, then you can solve the problem in another way. To do this, you need to find the voltage in the circuit, which will be common to both resistors, since the connection is parallel. In order to find it, you must first calculate the circuit resistance
And then the tension
Knowing the voltages, we will find the currents flowing through the resistors
As you can see, the currents turned out to be the same.
Example 3
In the electrical circuit shown in the diagram R 1 =50 Ohm, R 2 =180 Ohm, R 3 =220 Ohm. Find the power released by the resistor R 1, current through resistor R 2, voltage across resistor R 3 if it is known that the voltage at the circuit terminals is 100 V.
To calculate the DC power dissipated by resistor R 1, it is necessary to determine the current I 1, which is common to the entire circuit. Knowing the voltage at the terminals and the equivalent resistance of the circuit, you can find it.
Equivalent resistance and current in the circuit
Hence the power allocated to R 1
Electric DC circuits and methods for their calculation
1.1. Electric circuit and its elements
Electrical engineering examines the structure and operating principle of basic electrical devices used in everyday life and industry. In order for an electrical device to work, an electrical circuit must be created, the task of which is to transfer electrical energy to this device and provide it with the required operating mode.
An electrical circuit is a set of devices and objects that form a path for electric current, the electromagnetic processes in which can be described using the concepts of electric current, EMF (electromotive force) and electric voltage.
For analysis and calculation, an electrical circuit is graphically represented in the form of an electrical diagram containing symbols of its elements and methods of connecting them. The electrical diagram of the simplest electrical circuit that ensures the operation of lighting equipment is shown in Fig. 1.1.
All devices and objects that are part of the electrical circuit can be divided into three groups:
1) Sources of electrical energy (power).
A common property of all power sources is the conversion of some type of energy into electrical energy. The sources in which the conversion of non-electrical energy into electrical energy occurs are called primary sources. Secondary sources are those sources that have electrical energy at both the input and output (for example, rectifiers).
2) Consumers of electrical energy.
A common property of all consumers is the conversion of electricity into other types of energy (for example, a heating device). Sometimes consumers call it a load.
3) Auxiliary elements of the circuit: connecting wires, switching equipment, protection equipment, measuring instruments, etc., without which the real circuit does not work.
All elements of the circuit are covered by one electromagnetic process.
In the electrical diagram in Fig. 1.1 electrical energy from the EMF source E, which has an internal resistance r 0, is transmitted through the control rheostat R to consumers (load): light bulbs EL 1 and EL 2 with the help of auxiliary circuit elements.
1.2. Basic concepts and definitions for an electrical circuit
For calculation and analysis, a real electrical circuit is represented graphically in the form of a calculated electrical circuit (equivalent circuit). In this diagram, real circuit elements are depicted by symbols, and auxiliary circuit elements are usually not depicted, and if the resistance of the connecting wires is much less than the resistance of other circuit elements, it is not taken into account. The power source is shown as a source of EMF E with internal resistance r 0, real consumers of direct current electrical energy are replaced by their electrical parameters: active resistances R 1, R 2, ..., R n. Using resistance R, the ability of a real circuit element to irreversibly convert electricity into other types, for example, thermal or radiant, is taken into account.
Under these conditions, the diagram in Fig. 1.1 can be presented in the form of a calculated electrical circuit (Fig. 1.2), in which there is a power source with EMF E and internal resistance r 0, and consumers of electrical energy: control rheostat R, light bulbs EL 1 and EL 2 are replaced by active resistances R, R 1 and R 2.
The EMF source in the electrical circuit (Fig. 1.2) can be replaced by a voltage source U, and the conditional positive direction of the voltage U of the source is set opposite to the direction of the EMF.
When calculating, several main elements are distinguished in the electrical circuit diagram.
A branch of an electrical circuit (circuit) is a section of a circuit with the same current. A branch can consist of one or more elements connected in series. Scheme in Fig. 1.2 has three branches: branch bma, which includes elements r 0 , E, R and in which current I arises; branch ab with element R 1 and current I 1; branch anb with element R 2 and current I 2 .
An electrical circuit (circuit) node is a junction of three or more branches. In the diagram in Fig. 1.2 – two nodes a and b. Branches attached to the same pair of nodes are called parallel. Resistances R 1 and R 2 (Fig. 1.2) are in parallel branches.
A circuit is any closed path passing along several branches. In the diagram in Fig. 1.2, three circuits can be distinguished: I – bmab; II – anba; III – manbm, in the diagram the arrow shows the direction of bypassing the circuit.
Conditional positive directions of EMF of power sources, currents in all branches, voltages between nodes and at the terminals of circuit elements must be set to correctly write equations describing processes in an electrical circuit or its elements. In the diagram (Fig. 1.2) we indicate with arrows the positive directions of the EMF, voltages and currents:
a) for EMF sources - arbitrarily, but it should be taken into account that the pole (source terminal) to which the arrow is directed has a higher potential relative to the other pole;
b) for currents in branches containing EMF sources - coinciding with the direction of the EMF; in all other branches arbitrarily;
c) for voltages - coinciding with the direction of the current in a branch or element of the circuit.
All electrical circuits are divided into linear and nonlinear.
An element of an electrical circuit whose parameters (resistance, etc.) do not depend on the current in it is called linear, for example an electric furnace.
A nonlinear element, such as an incandescent lamp, has a resistance, the value of which increases with increasing voltage, and therefore the current supplied to the lamp.
Consequently, in a linear electrical circuit all the elements are linear, and an electrical circuit containing at least one nonlinear element is called nonlinear.
1.3. Basic laws of DC circuits
Calculation and analysis of electrical circuits is carried out using Ohm's law, Kirchhoff's first and second laws. Based on these laws, a relationship is established between the values of currents, voltages, EMF of the entire electrical circuit and its individual sections and the parameters of the elements that make up this circuit.
Ohm's law for a circuit section
The relationship between current I, voltage UR and resistance R of section ab of the electrical circuit (Fig. 1.3) is expressed by Ohm’s law
Rice. 1.3 In this case, Ohm’s law for a section of the circuit will be written as:
Ohm's law for the entire circuit
This law determines the relationship between the emf E of a power source with internal resistance r 0 (Fig. 1.3), the current I of the electrical circuit and the total equivalent resistance R E = r 0 + R of the entire circuit:
.A complex electrical circuit, as a rule, contains several branches, which can include their own power sources, and its operating mode cannot be described only by Ohm's law. But this can be done on the basis of Kirchhoff’s first and second laws, which are a consequence of the law of conservation of energy.
Kirchhoff's first law
At any node of an electrical circuit, the algebraic sum of currents is zero
,where m is the number of branches connected to the node.
When writing equations according to Kirchhoff’s first law, currents directed to a node are taken with a “plus” sign, and currents directed from the node are taken with a “minus” sign. For example, for node a (see Fig. 1.2) I - I 1 - I 2 = 0.
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
,where n is the number of EMF sources in the circuit;
m – number of elements with resistance Rk in the circuit;
U k = R k I k – voltage or voltage drop on the kth element of the circuit.
For the circuit (Fig. 1.2), we write the equation according to Kirchhoff’s second law:
If voltage sources are included in the electrical circuit, then Kirchhoff’s second law is formulated as follows: the algebraic sum of the voltages on all control elements, including EMF sources, is equal to zero
.When writing equations according to Kirchhoff’s second law, you must:
1) set conditional positive directions of EMF, currents and voltages;
2) select the direction of traversal of the contour for which the equation is written;
3) write down the equation using one of the formulations of Kirchhoff’s second law, and the terms included in the equation are taken with a “plus” sign if their conditional positive directions coincide with the circuit bypass, and with a “minus” sign if they are opposite.
Direct current is particles with a charge moving in a certain direction. In another way, current can be called quantities such as voltage that are constant both in direction and in value.
Let's consider its characteristics, application, as well as DC electrical circuits. We will answer the questions of how an electrical circuit is studied, how it is calculated, and some others.
From plus to minus or vice versa?
In the source, electrons move from negative to positive. Despite the fact that everyone knows about this, it is customary to consider the direction from plus to minus. I wonder why? They explain to us that this is how it happened historically. But is this really so? After all, this “history” took shape in a completely insignificant period of time.
In direct current, the main laws of electrical engineering apply: Ohm's law and Kirchhoff's laws. The current was previously called galvanic, since it was obtained as a result of a galvanic reaction. When they began to conduct electricity into houses, there were fierce debates about what kind of current to introduce: direct or alternating. The second one won the “War”, as it turned out to be less expensive. It is much easier to transmit over long distances thanks to its easy transformation.
How is direct current obtained?
But direct current has not disappeared from use either. direct current are found, for example, in batteries.
The current is generated through electromagnetic induction, after which it is rectified by a collector. This reaction is produced by a generator, which also produces direct current. Electrical circuits of direct current can be transformed from alternating current due to converters and rectifiers.
Application area
The use of this type is quite wide. In most household appliances at home, for example, a computer modem, a mobile phone charger, an electric kettle or a food processor, it is direct current that operates. DC electrical circuits are generated and converted by a car generator and any portable device. All industrial engines operate on it, and in some countries even high-voltage electrical transmission lines. It is even used in some medical devices.
Direct current is safer, since death can occur with an electric shock of 300 mA, and with alternating current - already at 50-100 mA.
Electrical circuit
Communication is provided by all devices through which the transmission, distribution and transformation of thermal, electromagnetic, light and other types of energy information is carried out. The processes are described as current and voltage.
Basic elements of DC electrical circuits
The main elements are receivers and sources of energy information connected by conductors. In sources, various types of energy are converted into electrical energy. And in receivers, on the contrary, electricity is converted into other forms.
Circuits where the conversion, transmission and receipt of electrical energy occurs at a constant value of voltage and current throughout the entire time are called direct current circuits. Where the process occurs with a variable value - alternating current circuits.
To calculate and study a direct current electrical circuit (laboratory work usually serves for these purposes), an equivalent circuit is used, that is, an idealized circuit to calculate the real one. To get it, you need to replace all elements of the circuit. Physical processes must be expressed in every mathematical description.
Resistive elements
The resistor is one of the receivers of the electrical circuit. It is characterized by active resistance, which is measured in Ohms. Resistive resistances, or, as they are also called, active ones, are introduced into equivalent circuits to take into account the conversion of electromagnetic energy into other types.
Calculation of complex DC electrical circuits is carried out by setting the positive direction of all currents and voltages. They choose the direction of their node, which has a high potential, towards a node with less potential.
When the resistance does not depend on the current, the resistor is called linear, and the electrical circuit is called linear resistive. The current-voltage characteristic is expressed through a linear function passing through the origin.
When analyzing such circuits, the simplification principle is often used, which consists of replacing complex sections of the electrical circuit with simple ones. But the current and voltage should not change. Then the chain will collapse to its simplest form. The connected resistive elements must be converted in parallel and in series.
Serial and parallel connection
In a series connection, the current in all elements has the same value. Here the voltage is determined by the sum of all included resistances multiplied by I, that is:
U=(R1+R2+RN)I=RI.
In a parallel connection, a constant voltage is applied, but the current is the sum of the currents on each of the elements. Therefore, it can be represented as the product of voltage and the equivalent conductivity of the active elements. And it, in turn, is equal to the sum of the conductivities of the elements. This is what direct current consists of.
DC electrical circuits also contain voltage and current sources.
Sources
The independent voltage (EMF, current) from the resistance of the external circuit is called its source. The EMF (voltage) source is measured at no-load, that is, where the current in the source is zero. In equivalent circuits, the resistor takes into account the thermal energy losses that are released from the source. If it is zero and the current source is infinity, this is an ideal source. The real always has a finite meaning.
The external characteristics are as follows: for sources of EMF and voltage, the dependence arises on the flowing current, and for a current source - on the voltage at the terminals.
Real sources have linear and nonlinear sections. Let's consider methods for calculating linear DC electrical circuits. They are described in Ohm's law for a complete circuit, where I=E/(Rh+Rbh). Then U= E- RbhI. From these formulas, internal resistance and internal conductivity are derived:
- Rbh=∆U/∆I;
- Gbh=∆I/∆U.
Nonlinear DC electrical circuits are calculated based on Kirchhoff's law. Calculation methods for linear and nonlinear circuits are different. Therefore, the latter are not considered within the scope of this article.
Instruments for measuring linear section
The DC circuit contains sources. And the instruments that measure it are: a voltmeter for measuring the voltage on a section of the circuit and an ammeter for sequential connection in the circuit. At zero value and conductivity the devices are ideal.
Switching methods become clearer when considering them using resistance measurements. According to Ohm's law R=U/I.
We know that real devices do not have zero value. Therefore, there are only two options for their inclusion:
- the internal resistance of the voltmeter is several times greater than the ammeter being measured - such that a decrease in the voltage on it does not reduce the decrease in the measured resistance, and the voltage measured by the voltmeter must correspond to the operating range;
- The internal resistance of a voltmeter is commensurate with what is being measured, while that of an ammeter is significantly less than what is being measured.
Experiment and tasks for the test
Appropriate generators are used to measure voltage and current. Their internal resistance is measured using switches.
A voltmeter and an ammeter are included in block AB1.
Special circuits are used to measure resistance. In the source of electromotive force, the internal resistance must be turned off.
In the recommended task that the test should have, DC electrical circuits are studied by determining the parameters of the source of electromotive force, the current source, measuring resistance, studying the inclusion of parallel and series resistances, and current-voltage characteristics.
It is the determination of some parameters based on initial data, from the conditions of the problem. In practice, several methods for calculating simple circuits are used. One of them is based on the use of equivalent transformations to simplify the circuit.
Equivalent transformations in an electrical circuit mean the replacement of some elements with others in such a way that the electromagnetic processes in it do not change, and the circuit is simplified. One type of such transformation is the replacement of several consumers connected in series or parallel with one equivalent one.
Several series-connected consumers can be replaced by one, and its equivalent resistance is equal to the sum of the consumer resistances, . For n consumers we can write:
rе = r1 +r2+…+rn,
where r1, r2, ..., rn are the resistances of each of the n consumers.
With a parallel connection of n consumers, the equivalent conductivity gе is equal to the sum of the conductivities of individual elements connected in parallel:
gе= g1 + g2 +…+ gn .
Considering that conductivity is the reciprocal of resistance, the equivalent resistance can be determined from the expression:
1/re = 1/r1 + 1/r2 +…+ 1/rn,
where r1, r2, ..., rn are the resistances of each of the n consumers connected in parallel.
In the particular case when two consumers r1 and r2 are connected in parallel, the equivalent circuit resistance is:
rе = (r1 x r2)/(r1 + r2)
Transformations in complex circuits where elements are missing explicitly (Figure 1) begin with replacing the elements included in the original circuit with a triangle with equivalent elements connected by a star.
Figure 1. Conversion of circuit elements: a - connected by a triangle, b - into an equivalent star
In Figure 1, a, the triangle of elements is formed by consumers r1, r2, r3. In Figure 1, b, this triangle is replaced by equivalent elements ra, rb, rc, connected by a star. To prevent potential changes at points a, b, c of the circuit, the resistances of equivalent consumers are determined from the expressions:
Simplification of the original circuit can also be accomplished by replacing the elements connected by a star with a circuit in which consumers .
In the diagram shown in Figure 2, a, we can distinguish a star formed by consumers r1, r3, r4. These elements are included between points c, b, d. In Figure 2, b, between these points there are equivalent consumers rbc, rcd, rbd, connected by a triangle. The resistances of equivalent consumers are determined from the expressions:
Figure 2. Transformation of circuit elements: a - connected by a star, b - into an equivalent triangle
Further simplification of the circuits shown in Figures 1, b and 2, b can be carried out by replacing sections with serial and parallel connections of elements with their equivalent consumers.
In the practical implementation of the method of calculating a simple circuit, using transformations, sections in the circuit with parallel and serial connections of consumers are identified, and then the equivalent resistance of these sections is calculated.
If the original circuit does not explicitly contain such sections, then using the previously described transitions from a triangle of elements to a star or from a star to a triangle, they are revealed.
These operations allow you to simplify the circuit. Having applied them several times, they come to a form with one source and one equivalent energy consumer. Next, using , the currents and voltages in sections of the circuit are calculated.
Calculation of complex DC circuits
When calculating a complex circuit, it is necessary to determine some electrical parameters (primarily currents and voltages on the elements) based on the initial values specified in the problem statement. In practice, several methods for calculating such circuits are used.
To determine branch currents, you can use: a method based on direct application, the method of nodal voltages.
To check the correctness of the calculation of currents, it is necessary to compile. It follows that the algebraic sum of the powers of all power sources in the circuit is equal to the arithmetic sum of the powers of all consumers.
The power of a power source is equal to the product of its emf and the amount of current flowing through this source. If the direction of the emf and the current in the source coincide, then the power is positive. Otherwise it is negative.
The power of the consumer is always positive and is equal to the product of the square of the current in the consumer by the value of its resistance.
Mathematically, the power balance can be written as follows:
where n is the number of power supplies in the circuit; m – number of consumers.
If the power balance is maintained, then the current calculation is done correctly.
In the process of drawing up a power balance, you can find out in what mode the power source operates. If its power is positive, then it transfers energy to an external circuit (for example, like a battery in discharge mode). When the source power value is negative, the latter consumes energy from the circuit (battery in charging mode).
A complex electrical circuit is a circuit with several closed circuits, with any placement of power sources and consumers in it, which cannot be reduced to a combination of serial and parallel connections.
The basic laws for calculating circuits, along with Ohm's law, are Kirchhoff's two laws, using which one can find the distribution of currents and voltages in all sections of any complex circuit.
In § 2-15 we were introduced to one method for calculating complex circuits, the superposition method.
The essence of this method is that the current in any branch is the algebraic sum of the currents created in it by all alternately acting electrical currents. d.s. chains.
Let's consider the calculation of a complex chain using the method of nodal and contour equations or equations according to Kirchhoff's laws.
To find the currents in all branches of the circuits, it is necessary to know the resistances of the branches, as well as the magnitudes and directions of all e. d.s.
Before drawing up equations according to Kirchhoff’s laws, you should arbitrarily set the directions of the currents in the branches, showing them on the diagram with arrows. If the selected direction of the current in any branch is opposite to the actual one, then after solving the equations this current is obtained with a minus sign.
The number of necessary equations is equal to the number of unknown currents; the number of equations compiled according to Kirchhoff’s first law must be one less than the number of nodes in the chain, the remaining equations are compiled according to Kirchhoff’s second law. When composing equations according to Kirchhoff’s second law, you should choose the simplest contours, and each of them should contain at least one branch that was not included in the previously compiled equations.
Let's look at the calculation of a complex chain using two Kirchhoff equations using an example.
Example 2-12. Calculate the currents in all branches of the circuit in Fig. 2-11, if e. d.s. sources, and the resistance of branches.
Neglect the internal resistances of the sources.
Rice. 2-11. Complex electrical circuit with two power sources.
Arbitrarily selected directions of currents in the branches are shown in Fig. 2-11.
Since there are three unknown currents, it is necessary to create three equations.
With two nodes in the chain, one node equation is required. Let's write it for point B:
4 We will write the second equation, traversing the ABVZZZA contour in a clockwise direction,
We will write the third equation by going around the AGVZZZA contour in a clockwise direction,
Replacing the letter designations with numerical values in equations (2-49) and (2-50), we obtain:
Replacing the current in the last equation with its expression in equation (2-48), we obtain;
Multiplying equation (2-52a) by 0.3 and adding it with equation (2-51), we get.