Parallel connection of resistances in a DC circuit. Ohm's law

When solving problems, it is customary to transform the circuit so that it is as simple as possible. For this, equivalent transformations are used. Such transformations of a part of an electrical circuit are called equivalent, in which currents and voltages in its unconverted part remain unchanged.

There are four main types of wire connection: serial, parallel, mixed and bridged.

Serial connection

Serial connection- this is a connection in which the current strength is the same throughout the entire circuit. A striking example of a serial connection is the old Christmas tree garland. There the bulbs are connected in series, one after the other. Now imagine, one light bulb burns out, the circuit is broken, and the rest of the light bulbs go out. Failure of one element leads to the shutdown of all the others, this is a significant drawback of a serial connection.

When connected in series, the resistances of the elements are summed up.

Parallel connection

Parallel connection- this is a connection in which the voltage at the ends of the circuit section is the same. Parallel connection is the most common, mainly because all the elements are under the same voltage, the current is distributed in different ways and when one of the elements leaves, all the others continue their work.

When connected in parallel, the equivalent resistance is found as:

In the case of two resistors connected in parallel

In the case of three resistors connected in parallel:

Mixed connection

Mixed connection- a connection that is a collection of serial and parallel connections. To find the equivalent resistance, you need to “roll up” the circuit by alternating parallel and serial sections of the circuit.

First, we find the equivalent resistance for the parallel section of the circuit, and then add the remaining resistance R 3 to it. It should be understood that after conversion, the equivalent resistance R 1 R 2 and the resistor R 3 are connected in series.

So, the most interesting and most difficult connection of conductors remains.

Bridge circuit

The bridge connection diagram is shown in the figure below.

In order to fold the bridge circuit, one of the triangles of the bridge is replaced with an equivalent star.

And find resistances R 1, R 2 and R 3.

Almost everyone who was engaged in electrical engineering had to solve the issue of parallel and serial connection of circuit elements. Some solve the problems of parallel and series connection of conductors by the "poke" method, for many "fireproof" garland is an inexplicable, but familiar axiom. Nevertheless, all these and many other similar questions are easily solved by the method proposed at the very beginning of the 19th century by the German physicist Georg Ohm. The laws discovered by him are still in effect, and almost everyone can understand them.

Basic electrical quantities of the circuit

In order to find out how this or that connection of conductors will affect the characteristics of the circuit, it is necessary to determine the values ​​that characterize any electrical circuit. Here are the main ones:

Mutual dependence of electrical quantities

Now you need to decide, as all of the above values ​​depend on one another. The dependency rules are simple and boil down to two basic formulas:

• I = U / R.
• P = I * U.

Here I is the current in the circuit in amperes, U is the voltage supplied to the circuit in volts, R is the resistance of the circuit in ohms, P is the electrical power of the circuit in watts.

Suppose we are dealing with a simple electrical circuit consisting of a power supply with voltage U and a conductor with resistance R (load).

Since the circuit is closed, current I flows through it. What value will it be? Based on the above formula 1, to calculate it, we need to know the voltage developed by the power supply and the load resistance. If we take, for example, a soldering iron with a 100 Ohm coil resistance and connect it to a 220 V lighting socket, then the current through the soldering iron will be:

220/100 = 2.2 A.

What is the power of this soldering iron? Let's use formula 2:

2.2 * 220 = 484 W.

A good soldering iron turned out, powerful, most likely two-handed. In the same way, operating with these two formulas and transforming them, you can find out the current through power and voltage, voltage through current and resistance, etc. How much, for example, does a 60W light bulb consume in your desk lamp:

60/220 = 0.27 A or 270 mA.

Resistance of the lamp spiral in operation:

220 / 0.27 = 815 ohms.

Multiple conductor circuits

All the cases discussed above are simple - one source, one load. But in practice, there can be several loads, and they are also connected in different ways. There are three types of load connection:

1. Parallel.
2. Consistent.
3. Mixed.

Parallel connection of conductors

The chandelier has 3 lamps, each with 60 watts. How much does a chandelier consume? That's right, 180 watts. First, quickly calculate the current through the chandelier:

180/220 = 0.818 A.

And then her resistance:

220 / 0.818 = 269 ohms.

Before that, we calculated the resistance of one lamp (815 Ohm) and the current through it (270 mA). The resistance of the chandelier turned out to be three times lower, and the current - three times higher. And now it's time to take a look at the three-arm lamp diagram.

All lamps in it are connected in parallel and connected to the network. It turns out that when three lamps are connected in parallel, the total load resistance has decreased threefold? In our case, yes, but it is private - all lamps have the same resistance and power. If each of the loads has its own resistance, then a simple division by the number of loads is not enough to calculate the total value. But even here there is a way out - it is enough to use this formula:

1 / Rtot. = 1 / R1 + 1 / R2 +… 1 / Rn.

For ease of use, the formula can be easily transformed:

Rtot. = (R1 * R2 *… Rn) / (R1 + R2 +… Rn).

Here Rtot... - the total resistance of the circuit when the load is connected in parallel. R1… Rn - resistances of each load.

Why the current increased when you connected three lamps in parallel instead of one is easy to understand - after all, it depends on the voltage (it remained unchanged) divided by the resistance (it decreased). It is obvious that the power in parallel connection will increase in proportion to the increase in current.

Serial connection

Now it's time to figure out how the parameters of the circuit will change if the conductors (in our case, the lamps) are connected in series.

The calculation of resistance in series connection of conductors is extremely simple:

Rtot. = R1 + R2.

The same three sixty-watt lamps, connected in series, will already amount to 2445 ohms (see calculations above). What will be the consequences of increasing the resistance of the circuit? According to formulas 1 and 2, it becomes quite clear that the power and current strength will drop when the conductors are connected in series. But why are all the lamps burning dim now? This is one of the most interesting properties of daisy chaining and is widely used. Let's take a look at a garland of three familiar to us, but series-connected lamps.

The total voltage applied to the entire circuit remained 220 V. But it was divided between each of the lamps in proportion to their resistance! Since we have lamps of the same power and resistance, the voltage is divided equally: U1 = U2 = U3 = U / 3. That is, three times less voltage is now applied to each of the lamps, which is why they glow so dimly. Take more lamps - their brightness will drop even more. How to calculate the voltage drop across each of the lamps if they all have different resistances? For this, the four formulas given above are sufficient. The calculation algorithm will be as follows:

1. Measure the resistance of each of the lamps.
2. Calculate the total resistance of the circuit.
3. From the total voltage and resistance, you calculate the current in the circuit.
4. Based on the total current and resistance of the lamps, you calculate the voltage drop across each of them.

Want to consolidate your knowledge? Solve a simple problem without looking at the answer at the end:

You have at your disposal 15 miniature bulbs of the same type, designed for a voltage of 13.5 V. Is it possible to make a Christmas tree garland out of them, connected to a regular outlet, and if possible, how?

Mixed connection

You, of course, easily figured out the parallel and serial connection of conductors. But what if you have something like this in front of you?

Mixed connection of conductors

How to determine the total resistance of a circuit? To do this, you need to split the circuit into several sections. The above construction is quite simple and there will be two sections - R1 and R2, R3. First, you calculate the total resistance of the parallel connected elements R2, R3 and find Rtot. 23. Then calculate the total resistance of the entire circuit consisting of R1 and Rtot.23 connected in series:

• Rtot. 23 = (R2 * R3) / (R2 + R3).
• Rchain = R1 + Rtotal 23.

The problem is solved, everything is very simple. And now the question is a little more complicated.

Complex mixed connection of resistances

How to be here? Likewise, you just need to show some imagination. Resistors R2, R4, R5 are connected in series. We calculate their total resistance:

Rtotal. 245 = R2 + R4 + R5.

Now we connect R3 in parallel to Rtot. 245:

Rtot.2345 = (R3 * Rtot.245) / (R3 + Rtot.245).

Rchain = R1 + Rtotal 2345 + R6.

That's all!

The answer to the Christmas tree garland problem

The lamps have an operating voltage of only 13.5 V, and in the 220 V outlet, so they must be connected in series.

Since the lamps are of the same type, the mains voltage will be divided equally between them and on each bulb there will be 220/15 = 14.6 V. The lamps are designed for a voltage of 13.5 V, so although such a garland will work, it will burn out very quickly. To implement the idea, you need a minimum of 220 / 13.5 = 17, and preferably 18-19 bulbs.

A serial connection is called a connection of circuit elements in which the same current I appears in all elements included in the circuit (Fig. 1.4).

Based on the second Kirchhoff's law (1.5), the total voltage U of the entire circuit is equal to the sum of the voltages in individual sections:

U = U 1 + U 2 + U 3 or IR equiv = IR 1 + IR 2 + IR 3,

whence follows

R eq = R 1 + R 2 + R 3.

Thus, when the elements of the circuit are connected in series, the total equivalent resistance of the circuit is equal to the arithmetic sum of the resistances of the individual sections. Consequently, a circuit with any number of series-connected resistances can be replaced with a simple circuit with one equivalent resistance R eq (Fig. 1.5). After that, the calculation of the circuit is reduced to determining the current I of the entire circuit according to Ohm's law

and the above formulas calculate the voltage drop U 1, U 2, U 3 in the corresponding sections of the electrical circuit (Fig. 1.4).

The disadvantage of sequential switching of elements is that if at least one element fails, the operation of all other elements of the circuit stops.

Electric circuit with parallel connection of elements

Parallel connection is called such a connection in which all consumers of electrical energy included in the circuit are under the same voltage (Fig. 1.6).

In this case, they are attached to two nodes of the circuit a and b, and on the basis of the first Kirchhoff's law, it can be written that the total current I of the entire circuit is equal to the algebraic sum of the currents of the individual branches:

I = I 1 + I 2 + I 3, i.e.

whence it follows that

.

In the case when two resistances R 1 and R 2 are connected in parallel, they are replaced by one equivalent resistance

.

From relation (1.6), it follows that the equivalent conductance of the circuit is equal to the arithmetic sum of the conductances of individual branches:

g eq = g 1 + g 2 + g 3.

As the number of parallel connected consumers increases, the conductivity of the circuit g eq increases, and vice versa, the total resistance R eq decreases.

Voltages in an electrical circuit with parallel-connected resistances (Fig. 1.6)

U = IR equiv = I 1 R 1 = I 2 R 2 = I 3 R 3.

Hence it follows that

those. the current in the circuit is distributed between the parallel branches in inverse proportion to their resistances.

According to the parallel connected circuit, consumers of any power, designed for the same voltage, operate in nominal mode. Moreover, the inclusion or shutdown of one or several consumers does not affect the work of the rest. Therefore, this scheme is the main scheme for connecting consumers to a source of electrical energy.

Electrical circuit with mixed connection of elements

Mixed is a connection in which there are groups of parallel and series-connected resistances in the circuit.

For the circuit shown in Fig. 1.7, the calculation of the equivalent resistance begins at the end of the circuit. To simplify calculations, let us assume that all resistances in this circuit are the same: R 1 = R 2 = R 3 = R 4 = R 5 = R. Resistances R 4 and R 5 are connected in parallel, then the resistance of the circuit section cd is equal to:

.

In this case, the original circuit (Fig. 1.7) can be represented in the following form (Fig. 1.8):

In the diagram (Fig.1.8), the resistance R 3 and R cd are connected in series, and then the resistance of the circuit section ad is equal to:

.

Then the diagram (Fig. 1.8) can be represented in an abbreviated version (Fig. 1.9):

In the diagram (Fig. 1.9), the resistance R 2 and R ad are connected in parallel, then the resistance of the circuit section ab is

.

The circuit (Fig. 1.9) can be presented in a simplified version (Fig. 1.10), where the resistances R 1 and R ab are connected in series.

Then the equivalent resistance of the original circuit (Fig. 1.7) will be equal to:

As a result of the transformations, the original circuit (Fig. 1.7) is presented in the form of a circuit (Fig. 1.11) with one resistance R eq. The calculation of currents and voltages for all elements of the circuit can be done according to Ohm's and Kirchhoff's laws.

LINEAR CIRCUITS OF SINGLE-PHASE SINUSOIDAL CURRENT.

Obtaining a sinusoidal EMF. ... Main characteristics of sinusoidal current

The main advantage of sinusoidal currents is that they allow the most economical production, transmission, distribution and use of electrical energy. The expediency of their use is due to the fact that the efficiency of generators, electric motors, transformers and power lines in this case is the highest.

To obtain sinusoidally varying currents in linear circuits, it is necessary that e. etc. with. also changed according to the sinusoidal law. Consider the process of the appearance of a sinusoidal EMF. The simplest generator of sinusoidal EMF can be a rectangular coil (frame), which rotates uniformly in a uniform magnetic field with an angular velocity ω (fig. 2.1, b).

Magnetic flux penetrating the coil during coil rotation abcd induces (induces) in it on the basis of the law of electromagnetic induction EMF e ... The load is connected to the generator using brushes 1 pressing against two slip rings 2 which, in turn, are connected to the coil. The value of the hover in the coil abcd e. etc. with. at each moment in time proportional to the magnetic induction V, the size of the active part of the coil l = ab + dc and the normal component of the speed of its movement relative to the field vn:

e = Blvn (2.1)

where V and l are constants, a vn is a variable depending on the angle α. Expressing the speed v n through the linear speed of the coil v, we get

e = Blv sinα (2.2)

In expression (2.2), the product Blv= const. Therefore, e. d.c. induced in a coil rotating in a magnetic field is a sinusoidal function of the angle α .

If the angle α = π / 2, then the product Blv in formula (2.2) is the maximum (amplitude) value of the induced emis- sion. etc. with. E m = Blv... Therefore, expression (2.2) can be written in the form

e = Emsinα (2.3)

Because α there is a rotation angle per time t, then, expressing it in terms of the angular velocity ω , you can write α = ωt, a formula (2.3) can be rewritten as

e = Emsinωt (2.4)

where e- instantaneous value of e. etc. with. in a coil; α = ωt- the phase characterizing the value of e. etc. with. at this point in time.

It should be noted that instant e. etc. with. during an infinitely small period of time can be considered a constant value, therefore, for instantaneous values ​​of e. etc. with. e, voltages and and currents i the laws of direct current are valid.

Sinusoidal quantities can be plotted as sinusoids and rotating vectors. When depicting them with sinusoids on the ordinate on a certain scale, the instantaneous values ​​of the quantities are plotted, on the abscissa - time. If a sinusoidal value is depicted as rotating vectors, then the length of the vector on a scale reflects the amplitude of the sinusoid, the angle formed with the positive direction of the abscissa axis at the initial time is equal to the initial phase, and the rotation speed of the vector is equal to the angular frequency. The instantaneous values ​​of sinusoidal values ​​are the projection of the rotating vector onto the ordinate axis. It should be noted that the counterclockwise direction of rotation is considered to be the positive direction of rotation of the radius vector. In fig. 2.2 graphs of instantaneous values ​​of e. etc. with. e and e ".

If the number of pole pairs of magnets p ≠ 1, then for one revolution of the coil (see.Fig.2.1) occurs p full cycles of change of e. etc. with. If the angular frequency of the coil (rotor) n revolutions per minute, the period will decrease by pn once. Then the frequency of e. d. s., i.e. the number of periods per second,

f = Pn / 60

From fig. 2.2 it is seen that ωТ = 2π, where

ω = 2π / T = 2πf (2.5)

The value ω proportional to the frequency f and equal to the angular velocity of rotation of the radius vector is called the angular frequency. The angular frequency is expressed in radians per second (rad / s) or 1 / s.

Graphically shown in Fig. 2.2 e. etc. with. e and e " can be described by expressions

e = Emsinωt; e "= E"msin (ωt + ψe ") .

Here ωt and ωt + ψe "- phases characterizing the values ​​of e. etc. with. e and e " at a given moment in time; ψ e "- the initial phase, which determines the value of e. etc. with. e " at t = 0. For e. etc. with. e the initial phase is zero ( ψ e = 0 ). Injection ψ always counted from the zero value of the sinusoidal value during its transition from negative to positive values ​​to the origin (t = 0). In this case, the positive initial phase ψ (Figure 2.2) is laid to the left of the origin (towards negative values ωt), and the negative phase - to the right.

If two or more sinusoidal quantities varying with the same frequency, the beginning of the sinusoids do not coincide in time, then they are shifted relative to each other in phase, that is, they do not coincide in phase.

Angle difference φ equal to the difference between the initial phases is called the phase angle. Phase shift between sinusoidal quantities of the same name, for example, between two e. etc. with. or two currents, denote α ... The phase angle between the sinusoids of current and voltage or their maximum vectors is denoted by the letter φ (fig. 2.3).

When for sinusoidal quantities, the phase difference is ±π , then they are opposite in phase, if the phase difference is equal ± π / 2 then they are said to be in square. If for sinusoidal quantities of the same frequency, the initial phases are the same, then this means that they are in phase.

Sinusoidal voltage and current, graphs of which are shown in Fig. 2.3 are described as follows:

u = Umsin (ω t +ψ u) ; i = Imsin (ω t +ψ i) , (2.6)

and the phase angle between current and voltage (see Fig. 2.3) in this case φ = ψ u - ψ i.

Equations (2.6) can be written differently:

u = Umsin (ωt + ψi + φ) ; i = Imsin (ωt + ψu - φ) ,

insofar as ψ u = ψ i + φ and ψ i = ψ u - φ .

From these expressions it follows that the voltage is ahead of the current in phase by an angle φ (or the current lags behind the voltage by an angle φ ).

Forms of presentation of sinusoidal electrical quantities.

Any sinusoidally changing electrical quantity (current, voltage, EMF) can be presented in analytical, graphic and complex forms.

one). Analytical presentation form

I = I m Sin ( ω t + ψ i), u = U m Sin ( ω t + ψ u), e = E m Sin ( ω t + ψ e),

where I, u, e- the instantaneous value of the sinusoidal current, voltage, EMF, i.e. the values ​​at the considered moment of time;

I m , U m , E m- amplitudes of sinusoidal current, voltage, EMF;

(ω t + ψ ) - phase angle, phase; ω = 2 π / T- angular frequency, characterizing the rate of phase change;

ψ i, ψ u, ψ e - the initial phases of current, voltage, EMF are counted from the point of transition of the sinusoidal function through zero to a positive value before the start of time ( t= 0). The initial phase can be either positive or negative.

Graphs of instantaneous values ​​of current and voltage are shown in Fig. 2.3

The initial phase of the voltage is shifted to the left from the origin and is positive ψ u> 0, the initial phase of the current is shifted to the right from the origin and is negative ψ i< 0. Алгебраическая величина, равная разности начальных фаз двух синусоид, называется сдвигом фаз φ ... Phase displacement between voltage and current

φ = ψ u - ψ i = ψ u - (- ψ i) = ψ u + ψ i.

Using the analytical form for calculating circuits is cumbersome and inconvenient.

In practice, one has to deal not with instantaneous values ​​of sinusoidal quantities, but with effective ones. All calculations are carried out for rms values, rms values ​​(current, voltage) are indicated in the passport data of various electrical devices, most electrical measuring instruments show rms values. RMS current is the equivalent of a direct current, which in the same time generates the same amount of heat in the resistor as an alternating current. The effective value is related to the amplitude simple ratio

2). Vector the form of representation of a sinusoidal electrical quantity is a vector rotating in a Cartesian coordinate system with the origin at point 0, the length of which is equal to the amplitude of the sinusoidal quantity, the angle relative to the x-axis is its initial phase, and the rotation frequency is ω = 2πf... The projection of this vector onto the y-axis at any moment in time determines the instantaneous value of the value under consideration.

Rice. 2.4

The set of vectors depicting sinusoidal functions is called a vector diagram, Fig. 2.4

3). Complex The presentation of sinusoidal electrical quantities combines the clarity of vector diagrams with accurate analytical calculations of circuits.

Rice. 2.5

We represent the current and voltage in the form of vectors on the complex plane, Fig. 2.5 The abscissa axis is called the axis of real numbers and denotes +1 , the ordinate axis is called the axis of imaginary numbers and is denoted + j... (In some textbooks, the real axis is denoted by Re, and the imaginary axis is Im). Consider vectors U and I at the moment t= 0. Each of these vectors corresponds to a complex number, which can be represented in three forms:

a). Algebraic

U = U’+ jU"

I = I’ – jI",

where U", U", I", I"- projections of vectors on the axes of real and imaginary numbers.

b). Indicative

where U, I- modules (lengths) of vectors; e- the base of the natural logarithm; rotational factors, since multiplication by them corresponds to the rotation of vectors relative to the positive direction of the real axis by an angle equal to the initial phase.

v). Trigonometric

U = U(Cos ψ u + j sin ψ u)

I = I(Cos ψ i - j sin ψ i).

When solving problems, the algebraic form (for operations of addition and subtraction) and exponential form (for operations of multiplication and division) are mainly used. The connection between them is established by the Euler formula

e jΨ = cos ψ + j sin ψ .

Unbranched electrical circuits

Parallel and serial connection of conductors - methods of switching an electrical circuit. Electrical circuits of any complexity can be represented by means of these abstractions.

Definitions

There are two ways to connect conductors, it becomes possible to simplify the calculation of a circuit of arbitrary complexity:

• The end of the previous conductor is connected directly to the beginning of the next - the connection is called serial. A chain is formed. To turn on the next link, you need to break the electrical circuit by inserting a new conductor there.
• The beginnings of the conductors are connected by one point, the ends - by another, the connection is called parallel. A bundle is usually called a branching. Each individual conductor forms a branch. Common points are called nodes of the electrical network.

In practice, a mixed connection of conductors is more common, some are connected in series, some in parallel. You need to break the chain in simple segments, solve the problem for each separately. Any complex electrical circuit can be described by parallel, serial connection of conductors. This is done in practice.

Using parallel and serial connection of conductors

Terms applied to electrical circuits

The theory acts as the basis for the formation of solid knowledge, few know how the voltage (potential difference) differs from the voltage drop. In terms of physics, an internal circuit is called a current source, which is outside is called an external one. The demarcation helps to correctly describe the distribution of the field. The current does work. In the simplest case, heat generation according to the Joule-Lenz law. Charged particles, moving towards a lower potential, collide with the crystal lattice, give up energy. Resistance heating occurs.

To ensure movement, it is necessary to maintain a potential difference at the ends of the conductor. This is called the voltage of the section of the circuit. If you just place a conductor in the field along the lines of force, the current will flow, it will be very short-lived. The process will end with the onset of equilibrium. The external field will be balanced by its own field of charges in the opposite direction. The current will stop. An external force is needed for the process to become continuous.

The current source acts as such a drive for the movement of the electric circuit. To maintain the potential, work is done within. Chemical reaction, as in a galvanic cell, mechanical forces - a hydroelectric generator. The charges inside the source move in the direction opposite to the field. The work of outside forces is being done on this. You can rephrase the above formulations, say:

• The outer part of the circuit, where the charges move, carried away by the field.
• The inner part of the circuit, where the charges move against the tension.

The generator (current source) is equipped with two poles. The one with the lower potential is called negative, the other is called positive. In the case of alternating current, the poles are continually reversed. The direction of movement of charges is inconsistent. The current flows from the positive pole to the negative pole. The movement of positive charges goes in the direction of decreasing potential. According to this fact, the concept of a potential drop is introduced:

The drop in the potential of a section of the chain is called a decrease in the potential within the segment. Formally, this is tension. It is the same for branches of a parallel circuit.

A voltage drop also means something else. The value characterizing the heat loss is numerically equal to the product of the current and the active resistance of the section. Ohm's and Kirchhoff's laws, considered below, are formulated for this case. In electric motors, transformers, the potential difference can differ significantly from the voltage drop. The latter characterizes the resistance loss, while the former takes into account the full operation of the current source.

When solving physical problems, for simplicity, the motor can include an EMF, the direction of which is opposite to the effect of the power source. The fact of energy loss through the reactive part of the impedance is taken into account. School and university physics courses differ from reality. That is why students open their mouths and listen to the phenomena that take place in electrical engineering. In the period preceding the era of the industrial revolution, the main laws were discovered, the scientist must combine the role of a theoretician and a talented experimenter. This is openly stated in the prefaces to the works of Kirchhoff (the works of Georg Ohm have not been translated into Russian). The teachers literally lured people with additional lectures, flavored with visual, amazing experiments.

Ohm's and Kirchhoff's laws in relation to serial and parallel connection of conductors

Ohm's and Kirchhoff's laws are used to solve real-life problems. The first deduced equality in a purely empirical way - experimentally - the second began by mathematical analysis of the problem, then checked the guesses with practice. Here are some information that helps to solve the problem:

Calculate the resistance of elements in series and parallel connection

The algorithm for calculating real circuits is simple. Here are some theses on the topic under consideration:

1. When connected in series, resistances are summed, with parallel - conductivity:
   1. For resistors, the law is rewritten in an unchanged form. With a parallel connection, the total resistance is equal to the product of the original, divided by the total. With sequential - the denominations are summed up.
   2. Inductance acts as reactance (j * ω * L), behaves like a normal resistor. In terms of writing the formula, it is no different. Nuance, for any purely imaginary impedance, that you need to multiply the result by the operator j, the angular frequency ω (2 * Pi * f). When the inductance coils are connected in series, the ratings are summed up, when in parallel, the reciprocal values ​​are added.
   3. The apparent resistance of the capacitance is written in the form: -j / ω * С. It is easy to notice: adding the values ​​of the series connection, we get the formula, exactly as for resistors and inductors it was with parallel. The opposite is true for capacitors. With parallel connection, the denominations are added, with sequential - reciprocal values ​​are added.

Theses are easily extended to arbitrary cases. The voltage drop across two open silicon diodes is equal to the sum. In practice, it is 1 volt, the exact value depends on the type of semiconductor element, characteristics. Power supplies are considered in a similar way: when connected in series, the ratings are added. Parallel is often found in substations, where transformers are placed in a row. The voltage will be one (controlled by the equipment), divided between the branches. The transformation ratio is strictly equal, blocking the occurrence of negative effects.

For some, the case is difficult: two batteries of different ratings are connected in parallel. The case is described by the second Kirchhoff's law, physics cannot imagine any complexity. If the values ​​of the two sources are not equal, the arithmetic mean is taken, if we neglect the internal resistance of both. Otherwise, the Kirchhoff equations are solved for all contours. The unknowns will be currents (three in total), the total number of which is equal to the number of equations. For a complete understanding, they brought a drawing.

An example of solving the Kirchhoff equations

Let's see the image: according to the condition of the problem, the source E1 is stronger than E2. We take the direction of the currents in the circuit from sound considerations. But if they put it down incorrectly, after solving the problem, one would turn out with a negative sign. Then the direction had to be changed. Obviously, current flows in the external circuit as shown in the figure. We compose the Kirchhoff equations for three circuits, here is what follows:

1. The work of the first (strong) source is spent on creating current in the external circuit, overcoming the neighbor's weakness (current I2).
2. The second source does not perform useful work in the load, it fights with the first one. You cannot say otherwise.

Including batteries of different ratings in parallel is certainly harmful. What is observed at the substation when using transformers with different transfer ratios. Equalizing currents do no useful work. Different batteries connected in parallel will begin to function effectively when the strong one dies down to the level of the weak one.

Series, parallel and mixed connections of resistors. A significant number of receivers included in the electrical circuit (electric lamps, electric heating devices, etc.) can be considered as some elements that have a certain resistance. This circumstance gives us the opportunity, when drawing up and studying electrical circuits, to replace specific receivers with resistors with specific resistances. There are the following methods resistor connections(receivers of electrical energy): serial, parallel and mixed.

Series connection of resistors. With serial connection of several resistors, the end of the first resistor is connected to the beginning of the second, the end of the second - to the beginning of the third, etc.
the same current I.
Series connection of receivers is illustrated in Fig. 25, a.
Replacing the lamps with resistors with resistances R1, R2 and R3, we get the circuit shown in Fig. 25, b.
If we assume that Ro = 0 in the source, then for three series-connected resistors, according to the second Kirchhoff's law, we can write:

E = IR 1 + IR 2 + IR 3 = I (R 1 + R 2 + R 3) = IR eq (19)

where R eq =R 1 + R 2 + R 3.
Consequently, the equivalent resistance of the series circuit is equal to the sum of the resistances of all series-connected resistors. Since the voltages in individual sections of the circuit according to Ohm's law: U 1 = IR 1; U 2 = IR 2, U 3 = IR s and in this case E = U, then for the circuit under consideration

U = U 1 + U 2 + U 3 (20)

Consequently, the voltage U at the source terminals is equal to the sum of the voltages at each of the series-connected resistors.
It also follows from these formulas that voltages are distributed between series-connected resistors in proportion to their resistances:

U 1: U 2: U 3 = R 1: R 2: R 3 (21)

that is, the greater the resistance of any receiver in the series circuit, the greater the voltage applied to it.

If several, for example n, resistors with the same resistance R1 are connected in series, the equivalent resistance of the circuit Rek will be n times greater than the resistance R1, i.e. Rek = nR1. The voltage U1 across each resistor in this case is n times less than the total voltage U:

When the receivers are connected in series, a change in the resistance of one of them immediately entails a change in the voltage on the other receivers connected to it. When you turn off or break the electrical circuit in one of the receivers and in the other receivers, the current stops. Therefore, serial connection of receivers is rarely used - only when the voltage of the source of electrical energy is greater than the rated voltage for which the consumer is designed. For example, the voltage in the electric network from which the subway cars are powered is 825 V, while the nominal voltage of the electric lamps used in these cars is 55 V. Therefore, in the metro cars, the electric lamps turn on 15 lamps in series in each circuit.
Parallel connection of resistors. Parallel connection of several receivers, they are switched on between two points of the electrical circuit, forming parallel branches (Fig. 26, a). Replacing

lamps with resistors with resistances R1, R2, R3, we get the circuit shown in Fig. 26, b.
When connected in parallel, the same voltage U is applied to all resistors.Therefore, according to Ohm's law:

I 1 = U / R 1; I 2 = U / R 2; I 3 = U / R 3.

The current in the unbranched part of the circuit according to the first Kirchhoff's law I = I 1 + I 2 + I 3, or

I = U / R 1 + U / R 2 + U / R 3 = U (1 / R 1 + 1 / R 2 + 1 / R 3) = U / R eq (23)

Consequently, the equivalent resistance of the circuit under consideration when three resistors are connected in parallel is determined by the formula

1/R eq = 1 / R 1 + 1 / R 2 + 1 / R 3 (24)

Introducing into the formula (24) instead of the values ​​1 / R eq, 1 / R 1, 1 / R 2 and 1 / R 3 corresponding to the conductivity G eq, G 1, G 2 and G 3, we get: the equivalent conductance of the parallel circuit is equal to the sum of the conductances of the resistors connected in parallel:

G eq = G 1 + G 2 + G 3 (25)

Thus, with an increase in the number of resistors connected in parallel, the resulting conductivity of the electrical circuit increases, and the resulting resistance decreases.
From the above formulas it follows that the currents are distributed between the parallel branches inversely proportional to their electrical resistances or directly proportional to their conductivities. For example, with three branches

I 1: I 2: I 3 = 1 / R 1: 1 / R 2: 1 / R 3 = G 1 + G 2 + G 3 (26)

In this respect, there is a complete analogy between the distribution of currents in individual branches and the distribution of water flows through pipes.
The above formulas make it possible to determine the equivalent circuit resistance for various specific cases. For example, with two resistors connected in parallel, the resulting circuit resistance is

R eq = R 1 R 2 / (R 1 + R 2)

with three resistors connected in parallel

R eq = R 1 R 2 R 3 / (R 1 R 2 + R 2 R 3 + R 1 R 3)

When several, for example n, resistors with the same resistance R1 are connected in parallel, the resulting resistance of the circuit Rek will be n times less than the resistance R1, i.e.

R eq = R1 / n(27)

The current I1 passing through each branch, in this case, will be n times less than the total current:

I1 = I / n (28)

When the receivers are connected in parallel, they are all under the same voltage, and the operating mode of each of them does not depend on the others. This means that the current passing through any of the receivers will not significantly affect the other receivers. At any shutdown or failure of any receiver, the remaining receivers remain on.

chenny. Therefore, a parallel connection has significant advantages over a serial connection, as a result of which it has become the most widespread. In particular, electric lamps and motors designed to operate at a certain (nominal) voltage are always connected in parallel.
On DC electric locomotives and some diesel locomotives, traction motors must be switched on under different voltages during speed control, so they switch from serial to parallel during acceleration.

Mixed connection of resistors. Mixed connection is called a connection in which some of the resistors are connected in series, and some in parallel. For example, in the diagram in Fig. 27, and there are two series-connected resistors with resistances R1 and R2, a resistor with resistance Rc is connected in parallel to them, and a resistor with resistance R4 is connected in series with a group of resistors with resistances R1, R2 and R3.
The equivalent circuit resistance in a mixed connection is usually determined by the transformation method, in which a complex circuit is transformed in successive stages into the simplest one. For example, for the circuit in Fig. 27, and first determine the equivalent resistance R12 of series-connected resistors with resistances R1 and R2: R12 = R1 + R2. In this case, the diagram in Fig. 27, and is replaced by the equivalent circuit in Fig. 27, b. Then the equivalent resistance R123 of parallel connected resistances and R3 is determined by the formula

R 123 = R 12 R 3 / (R 12 + R 3) = (R 1 + R 2) R 3 / (R 1 + R 2 + R 3).

In this case, the diagram in Fig. 27, b is replaced by the equivalent circuit in Fig. 27, c. After that, the equivalent resistance of the entire circuit is found by summing the resistance R123 and the resistance R4 connected in series with it:

R eq = R 123 + R 4 = (R 1 + R 2) R 3 / (R 1 + R 2 + R 3) + R 4

Serial, parallel and mixed connections are widely used to change the resistance of starting rheostats when starting e. p. from. direct current.