Electric current is an ordered movement of an uncompensated electric charge. If this movement occurs in the conductor, then the electric current is called conduction current. Electric current can cause Coulomb forces. The field of these forces is called Coulomb and is characterized by the intensity E cool.
The movement of charges can also arise under the influence of non-electric forces, called external (magnetic, chemical). E st is the field strength of these forces.
An ordered movement of electric charges can also arise without the action of external forces (diffusion, chemical reactions in a current source). For the generality of the reasoning, in this case we will also introduce an effective external field E st.
Full work on moving the charge in the circuit section:
Let us divide both parts of the last equation by the value of the charge moving through this section.
.
Potential difference in the circuit section.
Voltage in a section of a circuit is a value equal to the ratio of the total work done when moving a charge in this section to the magnitude of the charge. Those. VOLTAGE IN THE SECTION OF THE CIRCUIT IS THE TOTAL WORK OF MOVING A SINGLE POSITIVE CHARGE IN THE SECTION.
EMF in this area is called a value equal to the ratio of the work done by non-electric energy sources when moving a charge to the value of this charge. EMF IS THE WORK OF OUTSIDE FORCES TO MOVEMENT A SINGLE POSITIVE CHARGE IN A PART OF THE CHAIN.
Third-party forces in an electric circuit work, as a rule, in current sources. If there is a current source in a section of the circuit, then such a section is called non-uniform.
The voltage at an inhomogeneous section of the circuit is equal to the sum of the potential difference at the ends of this section and the EMF of the sources in it. In this case, the EMF is considered positive if the direction of the current coincides with the direction of action of external forces, i.e. from source minus to plus.
If there are no current sources in the area of interest to us, then in this and only in this case the voltage is equal to the potential difference.
In a closed circuit, for each of the sections that form a closed circuit, you can write:
Because the potentials of the initial and final points are equal, then .
Therefore, (2),
those. the sum of the voltage drops in a closed circuit of any electrical circuit is equal to the sum of the emf.
Let us divide both parts of equation (1) by the length of the section.
Where is the strength of the total field, is the strength of the external field, is the strength of the Coulomb field.
For a homogeneous section of the chain.
Current density means - Ohm's law in differential form. THE CURRENT DENSITY IN A HOMOGENEOUS SECTION OF THE CIRCUIT IS DIRECTLY PROPORTIONAL TO THE STRENGTH OF THE ELECTROSTATIC FIELD IN THE CONDUCTOR.
If a Coulomb and external field acts on a given section of the circuit (an inhomogeneous section of the circuit), then the current density will be proportional to the total field strength:
. Means, .
Ohm's law for an inhomogeneous chain section: THE CURRENT STRENGTH IN A NON-UNIFORM SECTION OF THE CIRCUIT IS DIRECTLY PROPORTIONATE TO THE VOLTAGE IN THIS SECTION AND IS INVERSELY PROPORTIONATE TO ITS RESISTANCE.
If the direction E c t and E cool coincide, then the EMF and the potential difference have the same sign.
In a closed circuit, V=O, because the Coulomb field is conservative.
From here:
where R is the resistance of the external part of the circuit, r is the resistance of the internal part of the circuit (ie current sources).
Ohm's law for a closed circuit: THE CURRENT IN A CLOSED CIRCUIT IS DIRECTLY PROPORTIONATE TO THE EMF OF SOURCES AND IS INVERSE PROPORTIONATELY TO THE FULL RESISTANCE OF THE CIRCUIT.
KIRCHHOFF'S RULES.
For the calculation of branched electrical circuits, Kirchhoff's rules are used.
The point in a circuit where three or more wires intersect is called a node. According to the law of conservation of charge, the sum of currents entering and leaving the node is equal to zero. . (Kirchhoff's first rule). THE ALGEBRAIC SUM OF THE CURRENTS PASSING THROUGH THE NODE IS ZERO.
The current entering the node is considered positive, leaving the node - negative. The directions of the currents in the sections of the circuit can be chosen arbitrarily.
Equation (2) implies that WHEN BYPASSING ANY CLOSED LOOP, THE ALGEBRAIC SUM OF VOLTAGE DROPS IS EQUAL TO THE ALGEBRAIC SUM OF EMF IN THIS CIRCUIT , - (Kirchhoff's second rule).
The direction of the contour bypass is chosen arbitrarily. The voltage in a section of the circuit is considered positive if the direction of the current in this section coincides with the direction of bypassing the circuit. EMF is considered to be positive if, during the bypass along the circuit, the source passes from the negative pole to the positive one.
If the chain contains m nodes, then m-1 equation can be made according to the first rule. Each new equation must include at least one new element. The total number of equations compiled according to the Kirchhoff rules must match the number of segments between the nodes, i.e. with the number of currents.
The relationship between the basic electrical quantities most used in electrical engineering is Ohm's law, established by the German physicist Georg Ohm, empirically, in 1826. With its help, a relationship is established between the voltage (electromotive force), the resistance of the elements of this circuit, the strength of the passing current.
Electrical parameters that are described by Ohm's law:
- The current strength is determined by the amount of charge passing through the conductor for some time, denoted by the letter I, the unit of measurement is ampere (A). Included in the basic units of the international system C;
- Electrical voltage, the unit of measurement is a volt, the concept was introduced by the same Georg Ohm. The volt can be expressed in terms of the work of moving the charge, the power released at a current of 1 ampere, has reference sources in the form of highly stable galvanic cells. Often indicated as a potential difference, in some cases the concept of electromotive force (EMF) is used. The letters U, V can be used for designation;
- R - resistance (electrical), indicates the properties of the conductor that hinder the passage of current. Significantly depends on the conductor material and temperature. The unit of measurement is 1 ohm, the designation is Ohm or Ω.
The classic formulation of Ohm's law: The current in a circuit is directly proportional to the voltage and inversely proportional to the resistance.
This expression is valid for an electrical circuit that does not contain an additional electromotive force that provides an electric current, a circuit defined as homogeneous. In most cases, this formula is used. In practice, it is often required to calculate the value of the current flowing through some element with a known resistance, for this it is enough to measure the voltage drop (potential difference) at the terminals of this device, for example, a resistor. Given any two values, the unknown can be calculated; in the same way, in addition to the quantities included in the expression, the electric power is determined.
Important! When calculating, only one dimension is used - integer values of volts, amperes, ohms or their corresponding multiples and submultiples.
Heterogeneous chain
Ohm's law for a separate section of the circuit does not take into account the presence of a power source, its properties are not included in the calculations. For a circuit called inhomogeneous, containing EMF of any kind and its source, the internal resistance of the power supply itself should be added to the well-known formula:
Here E is the EMF of the voltage source, r is its internal resistance. Name options - Ohm's law for an inhomogeneous section of a circuit, for a complete or closed circuit. The expression differs little from the above - instead of voltage, there is an EMF and the resistance of the power source.
It should be noted that the concept of internal resistance makes sense only for chemical current sources, in the case of using other devices, such as any kind of power supply without batteries, they talk about the output resistance and load capacity of this unit.
In practical applications, Ohm's law for an inhomogeneous section of the circuit in this form is rarely used, mainly for measuring the internal resistance of the battery itself, other batteries.
The law also applies to alternating voltage if the resistance is an active load. With its help, the effective (rms) parameters of the circuit are determined. In the case of an inductive, capacitive or complex load and for different frequencies, the resistance is reactive, significantly different from that measured by the usual method - an ohmmeter.
Ohm's law is derived by practical means, so it cannot be fundamental, but accurately describes the relationship between the most commonly used electrical quantities.
Video
The current strength in a homogeneous section of the circuit is directly proportional to the voltage at a constant section resistance and inversely proportional to the section resistance at a constant voltage.
whereU - tension in the area, R- section resistance.
Ohm's law for an arbitrary section of the circuit containing a direct current source.
whereφ 1 - φ 2 + ε \u003d U voltage in a given section of the circuit,R - electrical resistance of a given section of the circuit.
Ohm's law for a complete circuit.
The current strength in a complete circuit is equal to the ratio of the electromotive force of the source to the sum of the resistances of the external and internal sections of the circuit.
whereR - electrical resistance of the external section of the circuit,r - electrical resistance of the internal section of the circuit.
Short circuit.
It follows from Ohm's law for a complete circuit that the current strength in a circuit with a given current source depends only on the resistance of the external circuit R.
If a conductor with resistance R is connected to the poles of the current source<< r, то тогда только ЭДС источника тока и его сопротивление будут определять значение силы тока в цепи. Такое значение силы тока будет являться предельным для данного источника тока и называется током короткого замыкания.
Electrical resistance (R) is a physical quantity numerically equal to the ratio
voltage at the ends of the conductor to the strength of the current passing through the conductor.
The resistance value for a circuit section can be determined from the Ohm's law formula for a circuit section.
However, the resistance of the conductor does not depend on the strength of the current in the circuit and voltage, but is determined only by the shape, size and material of the conductor.
where l - conductor length (m), S - cross-sectional area (sq.m),
r (ro) - resistivity (Ohm m).
Resistivity
Shows what the resistance of a conductor made of a given substance is,
1 m long and with a cross section of 1 sq. m.
Resistivity unit in SI system: 1 ohm m
However, in practice, the thickness of the wires is much less than 1 m2,
therefore, an off-system unit of measurement of resistivity is more often used:
The unit of resistance in the system in SI:
The resistance of the conductor is 1 ohm, if, with a potential difference at its ends of 1 V,
a current of 1 A flows through it.
The reason for the presence of resistance in the conductor is the interaction of moving electrons with the ions of the crystal lattice of the conductor. Due to the difference in the structure of the crystal lattice for conductors made of different substances, their resistances differ from each other.
N39
Serial and Parallel Connections in electrical engineering, there are two main ways to connect elements of an electrical circuit. When connected in series, all elements are connected to each other so that the section of the circuit that includes them does not have a single node. With a parallel connection, all elements included in the chain are united by two nodes and have no connections with other nodes, unless this contradicts the condition.
When the conductors are connected in series, the current strength in all conductors is the same.
With a parallel connection, the voltage drop between two nodes that combine the elements of the circuit is the same for all elements. In this case, the reciprocal of the total resistance of the circuit is equal to the sum of the reciprocals of the resistances of the conductors connected in parallel.
With a series connection of conductors, the current strength in any part of the circuit is the same:
The total voltage in the circuit when connected in series, or the voltage at the poles of the current source, is equal to the sum of the voltages in the individual sections of the circuit:
N40
Electromotive force(EMF) - a scalar physical quantity that characterizes the work of external (non-potential) forces in sources of direct or alternating current. In a closed conducting circuit, the EMF is equal to the work of these forces in moving a single positive charge along the circuit.
EMF can be expressed in terms of the electric field strength of external forces (). In a closed loop () then the EMF will be equal to:
, where is the contour length element.
EMF, like voltage, is measured in volts. We can talk about the electromotive force in any part of the circuit. This is the specific work of external forces not in the entire circuit, but only in this section. The EMF of a galvanic cell is the work of external forces when moving a single positive charge inside the cell from one pole to another. The work of external forces cannot be expressed in terms of the potential difference, since external forces are non-potential and their work depends on the shape of the trajectory. So, for example, the work of external forces when moving a charge between the current terminals outside the source itself is equal to zero.
[edit] EMF of induction
The cause of the electromotive force can be a change in the magnetic field in the surrounding space. This phenomenon is called electromagnetic induction. The value of the EMF induction in the circuit is determined by the expression
where is the flux of the magnetic field through a closed surface bounded by a contour. The "−" sign in front of the expression shows that the induction current created by the induction EMF prevents a change in the magnetic flux in the circuit
n41
The work of an electric current shows how much work was done by an electric field when moving charges through a conductor.
Knowing two formulas:
I = q/t ..... and ..... U = A/q
you can derive a formula for calculating the work of an electric current:
The work of an electric current is equal to the product of the current and the voltage
and for the duration of current flow in the circuit.
The unit of measure for the work of electric current in the SI system:
[A] = 1 J = 1A. b. c
The power of the electric current shows the work done by the current per unit of time.
and is equal to the ratio of the work done to the time during which this work was done.
(power in mechanics is usually denoted by the letter N, in electrical engineering - by letter R)
because A = IUt, then the power of the electric current is equal to:
The unit of electric current power in the SI system:
[P] = 1 W (watt) = 1 A. B
N42
Semiconductor- a material that, in terms of its specific conductivity, occupies an intermediate place between conductors and dielectrics and differs from conductors in a strong dependence of specific conductivity on impurity concentration, temperature and exposure to various types of radiation. The main property of a semiconductor is an increase in electrical conductivity with increasing temperature.
Semiconductors are substances whose band gap is on the order of a few electron volts (eV). For example, a diamond can be classified as wide gap semiconductors, and indium arsenide - to narrow-gap. Semiconductors include many chemical elements (germanium, silicon, selenium, tellurium, arsenic, and others), a huge number of alloys and chemical compounds (gallium arsenide, etc.). Almost all inorganic substances of the world around us are semiconductors. The most common semiconductor in nature is silicon, which makes up almost 30% of the earth's crust.
Depending on whether the impurity atom donates or captures an electron, impurity atoms are called donor or acceptor atoms. The nature of an impurity can change depending on which atom of the crystal lattice it replaces, in which crystallographic plane it is embedded.
The conductivity of semiconductors is highly dependent on temperature. Near the temperature of absolute zero, semiconductors have the properties of dielectrics.
N43
Magnetic phenomena were known in the ancient world. The compass was invented over 4500 years ago. It appeared in Europe around the 12th century AD. However, it was only in the 19th century that the connection between electricity and magnetism was discovered and the idea of magnetic field .
The first experiments (carried out in 1820), which showed that there is a deep connection between electrical and magnetic phenomena, were the experiments of the Danish physicist H. Oersted. These experiments showed that a magnetic needle located near a current-carrying conductor is subject to forces that tend to turn it. In the same year, the French physicist A. Ampère observed the force interaction of two conductors with currents and established the law of interaction of currents.
According to modern concepts, conductors with current exert a force on each other not directly, but through the magnetic fields surrounding them.
The sources of the magnetic field are moving electric charges (currents). A magnetic field arises in the space surrounding current-carrying conductors, just as an electric field arises in a space surrounding motionless electric charges. The magnetic field of permanent magnets is also created by electric microcurrents circulating inside the molecules of a substance (Ampère's hypothesis).
Scientists of the 19th century tried to create a theory of the magnetic field by analogy with electrostatics, introducing into consideration the so-called magnetic charges two characters (for example, northern N and southern S poles of a magnetic needle). However, experience shows that isolated magnetic charges do not exist.
The magnetic field of currents is fundamentally different from the electric field. A magnetic field, unlike an electric field, exerts a force only on moving charges (currents).
To describe the magnetic field, it is necessary to introduce the force characteristic of the field, which is similar to the vector of the electric field strength. Such a characteristic is magnetic induction vector which defines the forces acting on currents or moving charges in a magnetic field.
For positive vector direction the direction is taken from the south pole S to the north pole N of the magnetic needle, freely oriented in the magnetic field. Thus, by examining the magnetic field created by a current or a permanent magnet, using a small magnetic needle, it is possible to determine the direction of the vector at each point in space. Such a study allows us to visualize the spatial structure of the magnetic field. Similarly to lines of force in electrostatics, one can construct lines of magnetic induction , at each point of which the vector is directed tangentially.
N44
From Ampère's law it follows that parallel conductors with electric currents flowing in one direction attract, and in opposite directions they repel. Ampère's law is also called the law that determines the force with which the magnetic field acts on a small segment of a current-carrying conductor. The expression for the force with which the magnetic field acts on the volume element of a conductor with current density , located in a magnetic field with induction , in the International System of Units (SI) has the form:
.
If the current flows through a thin conductor, then , where is the “length element” of the conductor - a vector, equal in absolute value and coinciding in direction with the current. Then the previous equality can be rewritten as follows:
The direction of the force is determined by the rule for calculating the cross product, which is convenient to remember using the left hand rule.
The Ampere force modulus can be found by the formula:
where is the angle between the magnetic induction and current vectors.
The force is maximum when the conductor element with current is located perpendicular to the lines of magnetic induction ():
N45
Consider a circuit with a current formed by fixed wires and a movable jumper sliding along them with a length l(Fig. 2.17). This circuit is in an external uniform magnetic field perpendicular to the plane of the circuit. With the direction of current shown in the figure I, the vector is codirectional with .
per current element I(moving wire) length l Ampere's force is acting to the right:
Let the conductor l will move parallel to itself at a distance d x. This will do the work:
| , | (2.9.1) |
Work , performed by a current-carrying conductor when moving, numerically is equal to the product of current and magnetic flux, crossed by this conductor.
The formula remains valid if a conductor of any shape moves at any angle to the lines of the magnetic induction vector.
Lorentz force
The force exerted by a magnetic field on a moving electrically charged particle.
where q is the particle charge;
V - charge speed;
B - magnetic field induction;
a is the angle between the charge velocity vector and the magnetic induction vector.
The direction of the Lorentz force is determined onleft hand rule:
If you put your left hand so that the perpendicular to the velocity component of the induction vector enters the palm, and four fingers are located in the direction of the velocity of the positive charge (or against the direction of the velocity of the negative charge), then the bent thumb will indicate the direction of the Lorentz force
. 
Since the Lorentz force is always perpendicular to the charge velocity, it does no work (i.e. does not change the magnitude of the charge velocity and its kinetic energy).
If a charged particle moves parallel to the magnetic field lines, then Fl \u003d 0, and the charge in the magnetic field moves uniformly and rectilinearly.
If a charged particle moves perpendicular to the magnetic field lines, then the Lorentz force is centripetal
and creates a centripetal acceleration equal to
In this case, the particle moves in a circle.
.
According to Newton's second law: the Lorentz force is equal to the product of the mass of the particle and the centripetal acceleration
then the radius of the circle
and the period of charge circulation in a magnetic field
Since the electric current is an ordered movement of charges, the action of a magnetic field on a current-carrying conductor is the result of its action on individual moving charges.
MAGNETIC PROPERTIES OF SUBSTANCE
The magnetic properties of a substance are explained according to the Ampère hypothesis by closed currents circulating inside any substance:
Inside the atoms, due to the movement of electrons in orbits, there are elementary electric currents that create elementary magnetic fields.
That's why:
1. if the substance does not have magnetic properties - elementary magnetic fields are unoriented (due to thermal motion);
2. if the substance has magnetic properties - the elementary magnetic fields are equally directed (oriented) and the substance's own internal magnetic field is formed.
Electromagnetic induction- the phenomenon of the occurrence of an electric current in a closed circuit when the magnetic flux passing through it changes.
Electromagnetic induction was discovered by Michael Faraday on August 29 [ source unspecified 253 days] 1831. He found that the electromotive force that occurs in a closed conducting circuit is proportional to the rate of change of the magnetic flux through the surface bounded by this circuit. The magnitude of the electromotive force (EMF) does not depend on what causes the change in the flux - a change in the magnetic field itself or the movement of a circuit (or part of it) in a magnetic field. The electric current caused by this EMF is called the induction current.
According to Faraday's law of electromagnetic induction, when the magnetic flux changes, penetrating the electrical circuit, a current is excited in it, called induction. The magnitude of the electromotive force responsible for this current is given by the equation:
where the minus sign means that the induced emf acts so that the induced current prevents the flux from changing. This fact is reflected in Lenz's rule.
N48
So far, we have considered changing magnetic fields, without paying attention to what is their source. In practice, most often magnetic fields are created using various kinds of solenoids, i.e. multi-turn circuits with current.
There are two possible cases here: when the current in the circuit changes, the magnetic flux penetrating: a ) the same circuit ; b ) adjacent circuit.
The induction emf that occurs in the same circuit is called EMF self-induction, and the phenomenon itself self-induction.
If the induction emf occurs in a neighboring circuit, then they talk about the phenomenon mutual induction.
It is clear that the nature of the phenomenon is the same, and different names are used to emphasize the place of origin of the induction EMF.
The phenomenon of self-induction discovered by the American scientist J. Henry.
According to the law of electromagnetic induction 
But ΔФ=LΔI, Consequently: 
N49
An electric motor is simply a device for efficiently converting electrical energy into mechanical energy.
This transformation is based on magnetism. Electric motors use permanent magnets and electromagnets, and use the magnetic properties of various materials to create these amazing devices.
There are several types of electric motors. We note two main classes: AC and DC.
AC (Alternating Current) class electric motors require an alternating current or voltage source to operate (you can find such a source in any electrical outlet in the house).
Electric motors of class DC (Direct Current) require a source of direct current or voltage for operation (you can find such a source in any battery).
Universal motors can be operated from any type of source.
Not only the design of the motors is different, the way to control the speed and torque is different, although the principle of energy conversion is the same for all types.
.Conductors that obey Ohm's law are called linear.
Graphical dependence of current strength on voltage (such graphs are called volt-ampere characteristics, abbreviated VAC) is depicted by a straight line passing through the origin. It should be noted that there are many materials and devices that do not obey Ohm's law, such as a semiconductor diode or a gas discharge lamp. Even for metal conductors at sufficiently high currents, a deviation from Ohm's linear law is observed, since the electrical resistance of metal conductors increases with increasing temperature.
1.5. Series and parallel connection of conductors
Conductors in DC electrical circuits can be connected in series and in parallel.
When the conductors are connected in series, the end of the first conductor is connected to the beginning of the second, etc. In this case, the current strength is the same in all conductors , but the voltage at the ends of the entire circuit is equal to the sum of the voltages across all the wires connected in series. For example, for three conductors connected in series 1, 2, 3 (Fig. 4) with electrical resistances , and we get:
 Rice. 4. |
.
According to Ohm's law for a chain section:
U 1 = IR 1, U 2 = IR 2, U 3 = IR 3 and U=IR(1)
where is the total resistance of a section of a circuit of series-connected conductors. From the expression and (1) we will have 
. In this way,
R \u003d R 1 + R 2 + R 3 . (2)
When the conductors are connected in series, their total electrical resistance is equal to the sum of the electrical resistances of all conductors.
From relations (1) it follows that the voltages on the series-connected conductors are directly proportional to their resistances:
 Rice. five. |
When conductors 1, 2, 3 are connected in parallel (Fig. 5), their beginnings and ends have common points of connection to the current source.
In this case, the voltage on all conductors is the same, and the current strength in an unbranched circuit is equal to the sum of the current strengths in all parallel-connected conductors 
.
For three conductors connected in parallel with resistances , and based on Ohm's law for a section of the circuit, we write
Denoting the total resistance of a section of an electrical circuit of three parallel-connected conductors through , for the current strength in an unbranched circuit, we obtain
, (5)
then from expressions (3), (4) and (5) it follows that:
. (6)
When conductors are connected in parallel, the reciprocal of the total resistance of the circuit is equal to the sum of the reciprocals of the resistances of all parallel-connected conductors.
The parallel switching method is widely used to connect electric lighting lamps and household appliances to the electrical network.
1.6. Resistance measurement
What are the features of measuring resistance?
When measuring low resistances, the measurement result is affected by the resistance of the connecting wires, contacts and contact thermo-emf. When measuring high resistances, it is necessary to take into account volume and surface resistances and take into account or eliminate the influence of temperature, humidity and other causes. Measurement of the resistance of liquid conductors or conductors with high humidity (ground resistance) is carried out on alternating current, since the use of direct current is associated with errors caused by the phenomenon of electrolysis.
Measurement of the resistance of solid conductors is carried out at direct current. Since, on the one hand, errors associated with the influence of the capacitance and inductance of the measurement object and the measuring circuit are excluded, on the other hand, it becomes possible to use magnetoelectric system devices with high sensitivity and accuracy. Therefore, megohmmeters are produced at direct current.
1.7. Kirchhoff rules
Kirchhoff rules– relationships that are performed between currents and voltages in sections of any electrical circuit.
Kirchhoff's rules do not express any new properties of a stationary electric field in conductors with current compared to Ohm's law. The first of them is a consequence of the law of conservation of electric charges, the second is a consequence of Ohm's law for an inhomogeneous section of the circuit. However, their use greatly simplifies the calculation of currents in branched circuits.
Kirchhoff's first rule
In branched chains, nodal points can be distinguished ( nodes ), in which at least three conductors converge (Fig. 6). The currents flowing into the node are considered to be positive; arising from the node - negative.
In the nodes of the DC circuit, no accumulation of charges can occur. This implies Kirchhoff's first rule:the algebraic sum of the strengths of the currents converging in the node is equal to zero:
Or in general terms:
In other words, how much current flows into the node, so much flows out of it. This rule follows from the fundamental law of conservation of charge.
Kirchhoff's second rule
In a branched chain, you can always select a certain number of closed paths, consisting of homogeneous and heterogeneous sections. Such closed paths are called loops. . Different currents can flow in different parts of the selected circuit. On fig. 7 shows a simple example of a branched chain. The circuit contains two nodes a and d, in which the same currents converge; so only one of the nodes is independent (a or d).
The circuit contains one independent node (a or d) and two independent circuits (for example, abcd and adef)
Three contours can be distinguished in the circuit abcd, adef and abcdef. Of these, only two are independent (for example, abcd and adef), since the third does not contain any new sections.
Kirchhoff's second rule is a consequence of the generalized Ohm's law.
Let us write the generalized Ohm's law for the segments that make up one of the contours of the circuit shown in fig. 8, for example, abcd. To do this, for each section, you need to set positive current direction And positive direction of contour traversal. When writing the generalized Ohm's law for each of the sections, it is necessary to observe certain "rules of signs", which are explained in Fig. 8.
For sections of the contour abcd, the generalized Ohm's law is written as:
for plotbc: 
for plot da: 
Adding the left and right sides of these equalities and taking into account that 
, we get:
Similarly, for the contour adef one can write:
According to Kirchhoff's second rule:
in any simple closed circuit, arbitrarily chosen in a branched electrical circuit, the algebraic sum of the products of the current strengths and the resistances of the corresponding sections is equal to the algebraic sum of the EMF present in the circuit:
,
where is the number of sources in the circuit, is the number of resistances in it.
When drawing up the stress equation for the loop, you need to choose the positive direction of bypassing the loop.
If the directions of the currents coincide with the selected direction of bypassing the circuit, then the current strengths are considered positive. EMF are considered positive if they create currents co-directional with the direction of bypassing the circuit.
A special case of the second rule for a circuit consisting of one circuit is Ohm's law for this circuit.
The procedure for calculating branched DC circuits
The calculation of a branched DC electrical circuit is performed in the following order:
arbitrarily choose the direction of currents in all sections of the circuit;
write down independent equations, according to the first Kirchhoff rule, where is the number of nodes in the chain;
arbitrarily closed contours are chosen so that each new contour contains at least one section of the circuit that is not included in the previously selected contours. They write down the second rule of Kirchhoff for them.
In a branched chain containing nodes and sections of the chain between neighboring nodes, the number of independent equations corresponding to the contour rule is .
Based on the Kirchhoff rules, a system of equations is compiled, the solution of which allows you to find the current strengths in the branches of the circuit.
Example 1:
Kirchhoff's first and second rules written for all independent nodes and circuits of a branched circuit, together give the necessary and sufficient number of algebraic equations for calculating the values of voltages and currents in an electrical circuit. For the circuit shown in Fig. 7, the system of equations for determining three unknown currents , and has the form:
,
,
.
Thus, the Kirchhoff rules reduce the calculation of a branched electrical circuit to the solution of a system of linear algebraic equations. This solution does not cause fundamental difficulties, however, it can be very cumbersome even in the case of fairly simple circuits. If, as a result of the solution, the current strength in some section turns out to be negative, then this means that the current in this section goes in the direction opposite to the chosen positive direction.
All applied electrical engineering is based on one dogma - this is Ohm's law for a circuit section. Without understanding the principle of this law, it is impossible to start practicing, because this leads to many mistakes. It makes sense to refresh this knowledge, in the article we will recall the interpretation of the law compiled by Ohm for a homogeneous and inhomogeneous section and a complete chain.
Classic wording
This is a simple interpretation, known to us from school.
The formula in integral form will look like this:
That is, by raising the voltage, we thereby increase the current. While an increase in a parameter such as "R" leads to a decrease in "I". Naturally, in the figure, the resistance of the circuit is shown by one element, although it can be a series, parallel (up to arbitrary) connection of several conductors.
We will not give the law in differential form, since in this form it is used, as a rule, only in physics.
Accepted units of measure
Please note that all calculations must be carried out in the following units of measurement:
- voltage - in volts;
- current in amperes
- resistance in ohms.
If you meet other values, then they will need to be converted to generally accepted ones.
Formulation for the complete chain
The interpretation for the complete circuit will be somewhat different than for the section, since the law drawn up by Ohm still takes into account the parameter "r", this is the resistance of the EMF source. The figure below illustrates such a scheme.
Given the "r" EMF, the formula will appear in the following form:
Note that if "R" is made equal to 0, then it becomes possible to calculate the "I" that occurs during a short circuit.
The voltage will be less than the EMF, it can be determined by the formula:
Actually, the voltage drop is characterized by the parameter "I * r". This property is characteristic of many galvanic power supplies.
Non-uniform section of the DC circuit
By this type is meant a section where, in addition to an electric charge, other forces are affected. An image of such a site is shown in the figure below.
The formula for such a section (generalized law) will have the following form:
Alternating current
If the circuit connected to alternating current is equipped with a capacitance and / or inductance (coil), the calculation is made taking into account the values of their reactances. A simplified form of the law would look like this:
Where "Z" represents the impedance, it is a complex value consisting of active (R) and passive (X) resistances.
Practical use
Video: Ohm's Law for a chain section - the practice of calculating chains.
Actually, this law can be applied to any section of the chain. An example is shown in the figure.
Using such a plan, you can calculate all the necessary characteristics for an unbranched section. Let's consider more detailed examples.
Finding the current
Let us now consider a more specific example, for example, it became necessary to find out the current flowing through an incandescent lamp. Terms:
- Voltage - 220 V;
- R filament - 500 ohms.
The solution to the problem will look like this: 220V / 500 Ohm \u003d 0.44 A.
Consider another problem with the following conditions:
- R=0.2 MΩ;
- U=400 V.
In this case, first of all, it will be necessary to perform the conversion: 0.2 MΩ = 200000 Ohm, after which you can proceed to the solution: 400 V / 200000 Ohm = 0.002 A (2 mA).
Voltage calculation
For the solution, we will also use the law drawn up by Ohm. So the task is:
- R=20 kOhm;
- I=10 mA.
Let's transform the original data:
- 20 kOhm = 20000 Ohm;
- 10mA=0.01A.
Solution: 20000 Ohm x 0.01 A = 200 V.
Don't forget to convert the values, because quite often the current can be specified in milliamps.
Resistance.
Despite the fact that the general view of the method for calculating the parameter "R" resembles finding the value of "I", there are fundamental differences between these options. If the current can vary depending on the other two parameters, then R (in practice) has a constant value. That is, in its essence, it is represented as an unchanging constant.
If the same current (I) passes through two different sections, while the applied voltage (U) is different, then, based on the law we are considering, we can say with confidence that where the low voltage "R" will be the smallest.
Consider the case when different currents and the same voltage in unrelated sections. According to Ohm's law, a large current strength will be characteristic of a small parameter "R".
Let's look at a few examples.
Suppose there is a circuit to which the voltage U=50 V is applied, and the current consumed I=100 mA. To find the missing parameter, 50 V / 0.1 A (100 mA) should be used, in the end the solution will be - 500 ohms.
The current-voltage characteristic allows you to clearly demonstrate the proportional (linear) dependence of the law. The figure below is a graph for a section with a resistance of one ohm (almost like a mathematical representation of Ohm's law).
Image of current-voltage characteristic, where R=1 ohm
Image of the current-voltage characteristic The vertical axis of the graph displays the current I (A), the horizontal axis shows the voltage U (V). The graph itself is presented as a straight line, which clearly displays the dependence on resistance, which remains unchanged. For example, at 12 V and 12 A, "R" will be equal to one ohm (12 V / 12 A).
Please note that only positive values are displayed on the given current-voltage characteristic. This indicates that the circuit is designed to allow current to flow in one direction. Where a reverse direction is allowed, the graph will continue to negative values.
Note that the equipment, the current-voltage characteristic of which is displayed as a straight line, is called linear. The same term is used to refer to other parameters.
In addition to linear equipment, there are various devices whose “R” parameter can vary depending on the current strength or applied voltage. In this case, Ohm's law cannot be used to calculate the dependence. This type of equipment is said to be non-linear, so its volt-ampere characteristics will not be displayed as straight lines.
Output
As mentioned at the beginning of the article, all applied electrical engineering is based on Ohm's law. Ignorance of this basic dogma can lead to incorrect calculation, which, in turn, will cause an accident.
The training of electricians as specialists begins with the study of the theoretical foundations of electrical engineering. And the first thing they should remember is Ohm's law, since almost all calculations of the parameters of electrical circuits for various purposes are made on its basis.
Understanding the basic law of electrical engineering will help you better understand the operation of electrical equipment and its main components. This will have a positive effect on maintenance during operation.
Independent verification, development, as well as experimental study of equipment components - all this is greatly simplified if Ohm's law is used for a circuit section. In this case, it is not required to carry out all measurements, it is enough to take some parameters and, after simple calculations, obtain the necessary values.
