Resistance in the serial chain. Consecutive and parallel compound of resistors

This is consistently called such a connection of the chain elements, in which the same current I (Fig. 1.4) occurs in all the elements in the chain. 1.4).

Based on the second law of Kirchhoff (1.5), the total voltage U of the entire circuit is equal to the amount of stresses in the individual sections:

U \u003d u 1 + U 2 + U 3 or IR eq \u003d Ir 1 + Ir 2 + Ir 3,

where follows

R eq \u003d R 1 + R 2 + R 3.

Thus, with a sequential connection of the chain elements, the total equivalent chain resistance is equal to the arithmetic amount of the resistance of individual sections. Consequently, a chain with any number of consecutable resistances can be replaced with a simple chain with one equivalent resistance R EQ (Fig. 1.5). After that, the calculation of the chain is reduced to the determination of the current of the entire chain by the law of Oma

and according to the above formulas, the drop in the voltages U 1, U 2, U 3 at the respective sections of the electrical circuit (Fig. 1.4) are calculated.

The disadvantage of the sequential inclusion of the elements is that at the failure of at least one element, the operation of all other elements of the chain stops.

Electrical circuit with parallel connection elements

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

In this case, they are attached to the two nodes of the chain A and B, and on the basis of the first law of Kirchhof, it can be written that the total current I of the entire chain is equal to the algebraic amount of current branches:

I \u003d i 1 + i 2 + i 3, i.e.

from where it follows that

.

In the case when two resistances R 1 and R 2 are included in parallel, they are replaced with one equivalent resistance.

.

From relation (1.6), it follows that the equivalent conductivity of the chain is equal to the arithmetic amount of the conductivity of individual branches:

g EQ \u003d G 1 + G 2 + G 3.

As the number of parallel to the included consumers, the conductivity of the chain g is increasing, and vice versa, the general resistance R EQ decreases.

Voltages in the electrical circuit with parallel resistances (Fig. 1.6)

U \u003d IR eq \u003d i 1 R 1 \u003d i 2 R 2 \u003d i 3 R 3.

Hence it follows that

those. The current in the chain is distributed between the parallel branches inversely proportional to their resistance.

By parallel, the scheme turned on in the nominal mode of consumers of any power, designed for the same voltage. Moreover, the inclusion or shutdown of one or more consumers is not reflected at the work of the rest. Therefore, this scheme is the main circuit of connecting consumers to the source of electrical energy.

Electric circuit with mixed connection elements

A mixed is called such a compound in which there are groups in parallel and consistently included resistances.

For the chain presented in Fig. 1.7, the calculation of equivalent resistance begins with the end of the scheme. To simplify the calculations, we will take that all resistance in this scheme are the same: R 1 \u003d R 2 \u003d R 3 \u003d R 4 \u003d R 5 \u003d R. Resistance R 4 and R 5 are included in parallel, then the resistance of the CD circuit section is equal to:

.

In this case, the initial scheme (Fig. 1.7) can be represented as follows (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 AD chain section is:

.

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

In the diagram (Fig. 1.9) resistance R 2 and R AD are connected in parallel, then the resistance of the AB chain section is equal

.

The scheme (Fig. 1.9) can be represented in a simplified version (Fig. 1.10), where R 1 and R AB resist are included in series.

Then the equivalent resistance of the source scheme (Fig. 1.7) will be:

As a result of the transformations, the source scheme (Fig. 1.7) is represented as a scheme (Fig. 1.11) with one resistance of R eq. Calculation of currents and stresses for all elements of the scheme can be made according to the laws of Ohm and Kirchhoff.

Linear chains of single-phase sinusoidal current.

Getting sinusoidal EMF. . The main characteristics of the sinusoidal current

The main advantage of sinusoidal currents is that they allow the most economically producing production, transmission, distribution and use of electrical energy. The feasibility of using them is due to the fact that the efficiency of the generators, electric motors, transformers and power lines in this case turns out to be the highest.

To obtain in linear circuits of sinusoidally changing currents, it is necessary that e. d. s. Also changed according to the sinusoidal law. Consider the process of sinusoidal EDC. The simplest sinusoidal emf generator can serve as a rectangular coil (frame), evenly rotating in a homogeneous magnetic field with an angular velocity. ω (Fig. 2.1, b.).

Piercing coil Magnetic flow during coil rotation abcd. leads (induces) in it on the basis of the law of electromagnetic induction of EDC e. . The load is connected to the generator using brushes. 1 pressed to two contact rings 2 which, in turn, are connected to the coil. The value induced in the coil abcd. e. d. s. At every moment of time in proportion to magnetic induction IN, the size of the active part of the coil l. = aB + dC and the normal component of the speed of moving it relative to the field v. N.:

e. = Blv N. (2.1)

where IN and l. - Permanent values, a v. N. - variable depending on the angle α. Expressing the speed V. N. through the linear speed of the coil v., get

e. = BLV · SINα. (2.2)

In terms (2.2), the work Blv \u003d const. Consequently, e. s., induced in the coil rotating in the magnetic field, is a sinusoidal angle function α .

If corner α \u003d π / 2, then the work Blv In formula (2.2) there is a maximum (amplitude) value of the induced er d. s. E M \u003d. Blv. Therefore, the expression (2.2) can be written as

e \u003d E. M.sINα. (2.3)

As α there is an angle of turn during t., then expressing it through the angular speed ω can be recorded α \u003d Ωt., A formula (2.3) rewrite in the form

e \u003d E. M.sinωt. (2.4)

where e. - Instant value of er d. s. in the coil; α \u003d Ωt. - Phase characterizing the value of e. d. s. At present.

It should be noted that the instantaneous er d. s. For an infinitely small period of time, it can be considered a permanent value, so for instant values \u200b\u200bof er. d. s. e., voltages and and currents i. Fair DC laws.

Sinusoidal values \u200b\u200bcan be graphically portrayed with sinusoids and rotating vectors. When you depicting their sinusoids on the ordinate at a certain scale, instantaneous values \u200b\u200bof values \u200b\u200bare deposited, on the abscissa - time. If the sinusoidal value is depicted by rotating vectors, the length of the vector on the scale reflects the amplitude of the sinusoids, an angle formed with the positive direction of the abscissa axis, at the initial moment of time is the initial phase, and the velocity speed of the vector is equal to the angular frequency. Instant values \u200b\u200bof sinusoidal values \u200b\u200bThere are projections of a rotating vector on the ordinate axis. It should be noted that the positive direction of rotation of the radius-vector is considered to be the direction of rotation counterclockwise. In fig. 2.2 The graphs of instantaneous values \u200b\u200bof er are built. d. s. e. and e ".

If the number of pairs of magnet poles p ≠ 1., then for one turnover of the coil (see Fig. 2.1) occurs p. Full cycles change e. d. s. If the angular frequency of the coil (rotor) n. revolutions per minute, then the period will decrease in pN. time. Then the frequency e. d. p., i.e. the number of periods per second,

f. = PN. / 60

From fig. 2.2 shows that ωt \u003d 2π.From!

ω \u003d 2π / t \u003d 2πf (2.5)

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

Graphically depicted in fig. 2.2 er. d. s. e. and e " You can describe expressions

e \u003d E. M.sinωt; E "\u003d E" M.sin (ωt + ψ E ") .

Here ωt. and ωt + ψ E " - Phases characterizing the values \u200b\u200bof e. d. s. e. and e " at a given point in time; ψ E " - The initial phase that determines the value of e. d. s. e " at T \u003d 0. For er d. s. e. The initial phase is zero ( ψ E. = 0 ). Angle ψ It is always counted from the zero value of the sinusoidal value when it moves from negative values \u200b\u200bto positive before the origin (T \u003d 0). At the same time, the positive initial phase ψ (Fig. 2.2) postpone from the beginning of the coordinates (towards negative values ωt.), and the negative phase is right.

If two or more sinusoidal values, varying with the same frequency, the beginning of the sinusoid does not coincide in time, then they are shifted relative to each other by phase, i.e. they do not coincide in phase.

The difference of corners φ equal to the difference of initial phases, called the phase shift angle. The phase shift between the same name sinusoidal values, for example, between two e. d. s. or two currents denoted α . The phase shift angle between current and voltage sinusoids or their maximum vectors are denoted by the letter φ (Fig. 2.3).

When the sinusoidal values \u200b\u200bof the phase difference is equal to ±π then they are opposite to phase, if the phase difference is equal ± π / 2, they say that they are in the quadrature. If the sinusoidal values \u200b\u200bof one frequency of the initial phases are the same, then this means that they coincide in phase.

Sinusoidal voltage and current, whose graphs are presented in Fig. 2.3, are described as follows:

u \u003d U. M.sin (ω t +.ψ U.) ; i \u003d I. M.sin (ω t +.ψ I.) , (2.6)

wherein the phase shift angle between the current and voltage (see Fig. 2.3) in this case φ = ψ U. - ψ I..

Equations (2.6) can be written otherwise:

u \u003d U. M.sin (ωt + ψ I. + φ) ; i \u003d I. M.sin (ωt + ψ U. - 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 phase current to the angle φ (or current lags behind the phase from the voltage at the angle φ ).

Forms of presentation of sinusoidal electrical values.

Any, sinusoidally changing, electrical value (current, voltage, EMF) can be represented in analytical, graphic and complex species.

one). Analytical Form of representation

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. - instantaneous value of sinusoidal current, voltage, EMF, i.e. values \u200b\u200bin the present time in time;

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

(Ω · T. + ψ ) - phase angle, phase; ω \u003d 2 · π / T. - angular frequency characterizing the rate of phase change;

ψ I, ψ u, ψ E is the initial phases of current, voltage, EMFs are counted from the transition point of the sinusoidal function through zero to a positive value before the start of time starts ( t. \u003d 0). The initial phase may have both positive and negative meaning.

Graphs of instantaneous current and voltage values \u200b\u200bare shown in Fig. 2.3.

The initial phase of the voltage is shifted to the left of the beginning of the reference and is positive ψ u\u003e 0, the initial phase of the current is shifted to the right from the beginning of the reference and is negative ψ I.< 0. Алгебраическая величина, равная разности начальных фаз двух синусоид, называется сдвигом фаз φ . Shift phases between voltage and current

φ = ψ U - ψ I \u003d. ψ U - (- ψ i) \u003d. ψ U +. ψ i.

The use of an analytical form for calculating the chains is cumbersome and uncomfortable.

In practice, it is necessary to deal with the instantaneous values \u200b\u200bof sinusoidal values, but with valid. All calculations are carried out for existing values, in passport data of various electrical devices, active values \u200b\u200b(current, voltage) are indicated, most electrical instruments show acting values. The current current is an equivalent of direct current, which in the same time allocates the same amount of heat in the resistor as an alternating current. Actual value is associated with an amplitude simple ratio.

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

Fig. 2.4.

The combination of vectors depicting sinusoidal functions are called a vector diagram, rice. 2.4.

3). Comprehensive The presentation of sinusoidal electrical values \u200b\u200bcombines the clarity of vector diagrams with accurate analytical calculations of chains.

Fig. 2.5

Current and voltage will be shown in the form of vectors on the complex plane, Fig.2.5 Axis of the abscissa is called the axis of valid numbers and denote +1 , the axis of the ordinate is called the axis of imaginary numbers and denote + J.. (In some textbooks, the axis of real numbers denote Re., and the axis are imaginary - IM.). Consider vectors U. and I. At the time of time t. \u003d 0. Each of these vectors corresponds to a complex number that can be represented in three forms:

but). Algebraic

U. = U.’+ ju."

I. = I.’ – ji.",

where U.", U.", I.", I."- Projections of vectors on the axis of valid and imaginary numbers.

b). Indicative

where U., I. - modules (lengths) of vectors; e. - the basis of the natural logarithm; swivel multipliers, since. Multiplication on them corresponds to the rotation of vectors relative to the positive direction of the valid axis by an angle equal to the initial phase.

in). Trigonometric

U. = U.· (COS. ψ U +. j.sin. ψ U)

I. = I.· (COS. ψ I - j.sin. ψ i).

When solving tasks, the algebraic form (for addition and subtraction operations) and an indicative form (for multiplication and division operations) are used. Communication between them is established by Euler Formula

e. j. · Ψ \u003d cos ψ + j.sin. ψ .

Unbreakable electric chains

In electrical circuits, elements can be connected via various schemes, including they have a serial and parallel connection.

Serial connection

With this connection, the conductors are connected to each other sequentially, that is, the beginning of one conductor will connect with the end of another. The main feature of this compound is that all conductors belong to one wire, there are no branches. Through each of the conductors will flow the same electric current. But the total voltage on the conductors will be equal to the tensions on each of them.

Consider a certain number of resistors connected in series. Since there are no branchings, the amount of passing charge through one conductor will be equal to the number of charge passed through another conductor. Current forces on all conductors will be the same. This is the main feature of this compound.

This connection can be considered otherwise. All resistors can be replaced by one equivalent resistor.

The current on an equivalent resistor will coincide with the total current flowing through all resistors. Equivalent general voltage will be folded from stresses on each resistor. This is the difference in potentials on the resistor.

If you take advantage of these rules and the law of OMA, which is suitable for each resistor, one can prove that the resistance of the equivalent general resistor will be equal to the amount of resistance. The consequence of the first two rules will be the third rule.

Application

The serial connection is used when it is necessary to purposefully turn on or off any device, the switch connects to it by a sequential scheme. For example, an electrical bell will ring only when it is sequentially connected to the source and button. According to the first rule, if the electric current is missing at least on one of the conductors, it will not be on other conductors. And vice versa, if the current is at least one conductor, it will be on all other conductors. A pocket flashlight is also operating, in which there is a button, battery and light bulb. All these items must be connected sequentially, as it is necessary that the flashlight shines when the button is pressed.

Sometimes the serial connection does not lead to the necessary goals. For example, in an apartment where there is a lot of chandeliers, light bulbs and other devices, you should not connect all lamps and devices sequentially, since it is never necessary to simultaneously include the light in each of the apartments. To do this, the sequential and parallel compound is considered separately, and the parallel type of scheme is used to connect light fixtures.

Parallel connection

In this form of the scheme, all conductors are connected in parallel with each other. All starts of the conductors are combined at one point, and all the ends are also connected together. Consider a certain amount of homogeneous conductors (resistors) connected via a parallel scheme.

This type of compound is branched. Each branch contains one resistor. The electric current, reaching the point of branching, is divided into each resistor, and will be equal to the sum of currents at all resistances. The voltage on all elements connected in parallel is the same.

All resistors can be replaced by one equivalent resistor. If you use the Ohm's law, you can get an expression of resistance. If with a sequential connection of the resistance, the resistance was folded, then the values \u200b\u200bof the reverse them will fold with parallel, as recorded in the formula above.

Application

If we consider compounds in household conditions, then in the apartment lighting lamps, chandeliers must be connected in parallel. If they are connected sequentially, when you turn on the same bulb, we will turn on all the others. With the parallel compound, we can, add the corresponding switch to each of the branches, to include the corresponding light bulb as desired. In this case, such inclusion of one lamp does not affect the rest of the lamps.

All electrical household devices in the apartment are connected in parallel to the network with a voltage of 220 V, and connected to the distribution panel. In other words, a parallel connection is used if you need to connect electrical devices independently of each other. Sequential and parallel compound have its own characteristics. There are also mixed connections.

TKA operation

The sequential and parallel compound, discussed earlier, was fair for the values \u200b\u200bof the voltage, resistance and strength of the current, which are the main. The current is determined by the formula:

A \u003d i x u x twhere BUT - current operation, T. - flow time on the conductor.

To determine the operation with a sequential compound scheme, it is necessary to replace the voltage in the initial expression. We get:

A \u003d i x (u1 + u2) x t

We reveal the brackets and get that the whole work is determined by the amount on each load.

Similarly, we consider the parallel compound scheme. Only changing no longer, but current strength. The result is:

A \u003d A1 + A2

Current power

When considering the power formula, the chain section must be used by the formula:

P \u003d u x i

After similar reasoning, it turns out the result that the sequential and parallel compound can be determined by the following power formula:

P \u003d p1 + p2

In other words, with any schemes, the total power is equal to the sum of all capacities in the scheme. This can be explained that it is not recommended to include several powerful electrical devices in the apartment, since the wiring may not withstand such power.

Influence of the connection scheme on the New Year Girland

After burning one lamp in the garland, you can determine the type of connection scheme. If the scheme is consistent, it will not be lit a single light bulb, since the burned light bulb breaks the total chain. To find out what kind of light bulb is burned, you need to check everything. Next, replace the faulty lamp, the garland will function.

When using a parallel compound circuit, the garland will continue to work, even if one or more lamps burned, as the circuit is not completely broken, but only one small parallel area. To restore such a garland, it is enough to see which lamps are not lit, and replace them.

Sequential and parallel connection for capacitors

With a consistent scheme, such a picture occurs: charges from the positive pole of the power supply are only on the outer plates of extreme capacitors. located between them transmit chain charge. This explains the appearance on all plates of equal charges with different signs. Based on this, the charge of any capacitor connected by a sequential scheme can be expressed by such a formula:

q total \u003d Q1 \u003d Q2 \u003d Q3

To determine the voltage on any condenser, the formula is necessary:

Where C is a container. The total voltage is expressed as the same law that is suitable for resistance. Therefore, we obtain the formula of the container:

C \u003d Q / (U1 + U2 + U3)

To make this formula easier, you can turn the fraction and replace the ratio of the potential difference to the container charge. As a result, we get:

1 / C \u003d 1 / C1 + 1 / C2 + 1 / C3

A little differently calculated parallel connection of capacitors.

The total charge is calculated as the sum of all charges that have accumulated on the plates of all capacitors. And the voltage value is also calculated by general laws. In this regard, the formula of the total capacity with a parallel compound scheme looks like this:

C \u003d (Q1 + Q2 + Q3) / U

This value is calculated as the sum of each device in the diagram:

C \u003d C1 + C2 + C3

Mixed connection of conductors

In the electrical circuit, the sections of the chain can have a consecutive and parallel compound intertwined. But all the laws discussed above for certain types of compounds are still valid, and are used in stages.

First you need to mentally decompose the scheme into separate parts. For a better presentation, it is drawn on paper. Consider our example on the scheme shown above.

It is convenient to portray it, starting with points B. and IN. They are set up at some distance among themselves and from the edge of the paper sheet. On the left side to the point B. One wire is connected, and two wires depart on the right. Point IN On the contrary, it has two branches on the left, and after the point leaves one wire.

Next you need to portray space between points. On the upper conductor, there are 3 resistance with conventional values \u200b\u200b2, 3, 4. The current will go with the index 5. The first 3 resists are included in the scheme sequentially, and the fifth resistor is connected in parallel.

The remaining two resistance (first and sixth) are connected consistently with the site under consideration. B-B.. Therefore, the scheme is complementary by 2 rectangles on the sides of the selected points.

Now we use the formula for calculating the resistance:

• The first formula for a sequential type of connection.
• Next, for a parallel scheme.
• And finally for serial scheme.

Similarly, any complicated scheme can be decomposed into separate schemes, including connections not only for conductors in the form of resistance, but also capacitors. To learn to own the methods for calculating different types Schemes must be accessed in practice by performing several tasks.

Resistance to conductors. Parallel and serial connection of the conductors.

Electrical resistance - The physical quantity characterizing the properties of the conductor to prevent the passage of electric current and equal relationships at the ends of the conductor to the strength of the current flowing through it. Resistance for alternating current circuits and for variables of electromagnetic fields is described by the concepts of impedance and wave resistance. Resistance (resistor) is also called radio component intended for administration to the electrical circuits of active resistance.

Resistance (often indicated by the letter R. or r.) It is considered, at certain limits, a constant value for this conductor; It can be calculated as

R. - resistance;

U. - the difference of electrical potentials (voltage) at the ends of the conductor;

I. - The strength of the current flowing between the edges of the conductor under the action of the potential difference.

With a sequential connection Conductors (Fig. 1.9.1) The power of the current in all conductors is the same:

According to the law of Ohm, voltage U. 1 I. U. 2 on the conductors are equal

With a sequential connection, the total resistance of the circuit is equal to the sum of the resistance of individual conductors.

This result is valid for any number of sequentially connected conductors.

With parallel compound (Fig. 1.9.2) voltage U. 1 I. U. 2 On both guides the same:

This result follows from the fact that at the branching points (nodes A. and B.) In the DC circuit, charges cannot accumulate. For example, to the node A. During Δ. t. Fees charge I.Δ t., and breaks away from the knot for the same time charge I. 1 Δ. t. + I. 2 Δ. t.. Hence, I. = I. 1 + I. 2 .

Recording on the basis of the law of Ohm

With parallel connection of the conductors, the value of the total chain resistance is equal to the sum of the reverse resistances of parallel conductor turned on.

This result is fair for any number of parallel conductor included.

Formulas for sequential and parallel connection of conductors allow in many cases to calculate the resistance of a complex chain consisting of many resistors. In fig. 1.9.3 An example of such a complex circuit is given and a sequence of calculations is indicated.

It should be noted that not all complex chains consisting of conductors with different resistances can be calculated using formulas for serial and parallel connections. In fig. 1.9.4 An example of an electrical circuit is shown, which cannot be calculated by the above method.

) Today we are talking about possible ways to connect resistors, in particular about a sequential connection and parallel.

Let's start with the consideration of the chains, the elements of which are connected sequence. And at least we will consider only resistors as the elements of the chain in this article, but the rules relating to stresses and currents at different compounds will be fair for other elements. So, the first chain, which we will disassemble as follows:

Here we have a classic case serial connection - Two successively included resistors. But we will not ride ahead and calculate the overall resistance of the chain, and for beginnings, consider all the voltages and currents. So, the first rule is that currents flowing across all conductors with a sequential connection are equal to each other:

And to determine the general voltage in a sequential connection, the voltage on the individual elements must be summed up:

At the same time, the software for stresses, resistance and currents in this chain are just the following ratios:

Then it will be possible to use the following expression to calculate the general voltage:

But for general stress, the Ohm law is also fair:

Here is the overall chain resistance, which, based on two formulas for general voltage, is:

Thus, with a sequential connection of the resistors, the total chain resistance will be equal to the sum of the resistance of all conductors.

For example, for the next chain:

General resistance will be equal:

The number of elements of the value does not have, the rule by which we determine the overall resistance will operate in any case 🙂 and if with a sequential connection all resistance are equal (), then the total chain resistance will be:

In this formula, equals the number of chain elements.

With a consistent compound of resistors, we figured out, let's move to parallel.

With parallel voltage connection on the conductors are equal:

And for the currents rightly the following expression:

That is, the total current is branched into two components, and its value is equal to the sum of all components. According to the law of Ohm:

Substitute these expressions in the general current formula:

And according to the law of Oma Current:

We equate these expressions and get a formula for the overall resistance of the chain:

This formula can be recorded somewhat differently:

In this way,with parallel connection of the conductors, the value of the total chain resistance is equal to the sum of the reverse resistances of parallel conductor turned on.

A similar situation will be observed with larger number of conductors connected in parallel:

In addition to parallel and consistent compounds of resistors, there is still mixed connection. It is already clear from the name that with such a connection in the chain there are resistors, connected both in parallel and sequentially. Here is an example of such a chain:

Let's calculate the overall chain resistance. Let's start with resistors and - they are connected in parallel. We can calculate the overall resistance for these resistors and replace them in the scheme with a single sole resistor:

« Physics - Grade 10 »

What does the dependence of the current flow in the conductor from the voltage on it?
What is the dependence of the current strength in the conductor from its resistance?

From the current source, energy can be transferred over the wires to devices that consume energy: an electric lamp, radio and others. For this make up electrical chains Various complexity.

The most simple and frequently encountered conductor connections include sequential and parallel compounds.

Sequential connection of conductors.

With a sequential connection, the electrical circuit does not have branching. All conductors include in the chain alternately after each other. Figure (15.5, a) shows the sequential connection of two conductors 1 and 2, having resist R 1 and R 2, these can be two lamps, two windings of the electric motor, etc.

The power of the current in both conductors is the same, i.e.

I 1 \u003d i 2 \u003d I. (15.5)

In the conductors, the electrical charge in the case of direct current does not accumulate, and through any cross-section of the conductor over a certain time passes the same charge.

The voltage at the ends of the chain section under consideration is made of voltages on the first and second conductors:

Applying Ohm's law for the entire site as a whole and for sections with resistances of conductors R1 and R2, it can be proved that the total resistance of the entire section of the chain with a sequential connection is:

R \u003d R 1 + R 2. (15.6)

This rule can be applied for any number of sequentially connected conductors.

Voltages on the conductors and their resistance with a sequential connection are associated with the relation

Parallel connection of conductors.

Figure (15.5 b) shows a parallel connection of two conductors 1 and 2 resistances R 1 and R 2. In this case, the electrical current I branches into two parts. The current strength in the first and second conductor is denoted by I 1 and I 2.

Since at a point of a - branching of conductors (such a point is called a node) - the electrical charge does not accumulate, then the charge coming into a unit of time to the node is equal to the charge, leaving the node during the same time. Hence,

I \u003d i 1 + i 2. (15.8)

The voltage U at the ends of the conductors, connected in parallel, is equally, as they are attached to the same chain points.

In the lighting network, a voltage 220 V is usually maintained on this stress, instruments consuming electrical energy are calculated. Therefore, a parallel connection is the most common way to connect different consumers. In this case, the failure of one device is not reflected at the work of the rest, whereas with a sequential connection, the failure of one device opens the chain. Applying the Ohm law for the entire site as a whole and for the conductors of the conductor resistances R 1 and R 2, it can be proved that the value inverse the impedance of the AB portion is equal to the amount of quantities, inverse resistances of individual conductors:

From here it follows that for two conductors

The voltages on parallel to the connected conductors are equal: i 1 R 1 \u003d i 2 R 2. Hence,

We draw attention to the fact that if in some of the plots of the chain on which there is a constant current, in parallel to one of the resistors to connect the condenser, the current through the capacitor will not go, the chain on the site with the capacitor will be open. However, there will be a voltage equal to the resistor between the capacitor, and the charge Q \u003d Cu is accumulated on the plates.

Consider the R - 2R resistance chain, called the matrix (Fig. 15.6).

On the last (right) link of the matrix, the voltage is divided in half due to equality of resistance, on the previous link, the voltage is also divided into half, since it is distributed between the resistance resistor R and two parallel resistors resistances 2r, etc., this idea - voltage division - lies in The basis of the transformation of the binary code into constant voltage, which is necessary for the operation of computers.