What is the difference between a serial connection and a parallel connection? Serial and parallel connection

1. Find the equivalent resistance of sections of the circuit with parallel connection of resistors. Figure 2. Series connection of resistors. To calculate the resistance of such connections, the entire circuit is divided into simple sections, consisting of resistors connected in parallel or in series.

This result follows from the fact that at the current branching points (nodes A and B) in the circuit direct current charges cannot accumulate. This result is valid for any number of conductors connected in parallel.

In Fig. 1.9.3 shows an example of such a complex circuit and indicates the sequence of calculations. It should be noted that not all complex circuits consisting of conductors with different resistances can be calculated using formulas for series and parallel connections.

When conductors are connected in series, the current in all conductors is the same. In a parallel connection, the voltage drop between the two nodes connecting the elements of the circuit is the same for all elements.

That is, the greater the resistance of the resistor, the greater the voltage drops across it. As a result, to one point ( electrical unit) several resistors can be connected. With this connection, a separate current will flow through each resistor. Force given current will be inversely proportional to the resistance of the resistor.

Thus, when parallel connection resistors with different resistances, the total resistance will always be less than the value of the smallest individual resistor. The voltage between points A and B is both the total voltage for the entire circuit section and the voltage across each resistor individually. A mixed connection is a section of a circuit where some resistors are connected in series and some in parallel.

The circuit is divided into sections with only parallel or only serial connections. The total resistance is calculated for each individual section. Calculate the total resistance for the entire mixed connection circuit. There are also more quick way calculating the total resistance for a mixed connection. If the resistors are connected in series, add them together.

That is, with a series connection, the resistors will be connected one after another. Figure 4 shows simplest example mixed connection of resistors. After calculating the equivalent resistances of the resistors, the circuit is redrawn. Usually a circuit of equivalent resistances connected in series is obtained.4. Figure 5. Calculation of the resistance of a circuit section with a mixed connection of resistors.

As a result, you will learn from scratch not only how to develop your own devices, but also how to interface various peripherals with them! A node is a branching point in a circuit at which at least three conductors are connected. Series connection of resistors is used to increase resistance.

Parallel voltage

As you can see, calculate the resistance of two parallel resistors much more convenient. Parallel connection of resistors is often used in cases where higher power resistance is needed. To do this, as a rule, resistors with the same power and the same resistance are used.

Total resistance Rtotal

This connection of resistances is called series. We thus obtained that U = 60 V, i.e. the non-existent equality of the emf of the current source and its voltage. We will now turn on the ammeter in turn in each branch of the circuit, remembering the readings of the device. Therefore, when resistances are connected in parallel, the voltage at the terminals of the current source is equal to the voltage drop across each resistance.

This branching of current in parallel branches is similar to the flow of liquid through pipes. Let us now consider what the total resistance of an external circuit consisting of two parallel-connected resistances will be equal to.

Let's return to the circuit shown in Fig. 3, and let’s see what the equivalent resistance of two parallel-connected resistances will be. Similarly, for each branch I1 = U1 / R1, I2 = U2 / R2, where I1 and I2 are the currents in the branches; U1 and U2 - voltage on branches; R1 and R2 - branch resistances.

This means that the total resistance of the circuit will always be lower than any resistor connected in parallel. 2. If these sections include resistors connected in series, then first calculate their resistance. By applying Ohm's law to a section of a circuit, it can be proven that the total resistance in a series connection is equal to the sum of the resistances of the individual conductors.

Series, parallel and mixed connections of resistors. A significant number of receivers included in 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 compiling and studying electrical diagrams 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. For serial connection 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. With this connection, all elements series circuit passes
the same current I.
The serial 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 Kirchhoff’s second 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 a series circuit is equal to the sum of the resistances of all series-connected resistors. Since the voltages in individual sections of the circuit are according to Ohm’s law: U 1 =IR 1 ; U 2 = IR 2, U 3 = IR h and v 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.
From these formulas it also follows that the 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 a 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 on each resistor in this case is n times less than the total voltage U:

When receivers are connected in series, a change in the resistance of one of them immediately entails a change in the voltage at the other receivers connected to it. When the electrical circuit is turned off or broken, the current in one of the receivers and in the remaining receivers stops. Therefore, series connection of receivers is rarely used - only in the case when the voltage of the electrical energy source is greater than the rated voltage for which the consumer is designed. For example, the voltage in the electrical network from which subway cars are powered is 825 V, while the nominal voltage of the electric lamps used in these cars is 55 V. Therefore, in subway cars, electric lamps are switched on in series, 15 lamps in each circuit.
Parallel connection of resistors. In parallel connection several receivers, they are connected 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.

Current in the unbranched part of the circuit according to Kirchhoff’s first 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)

Therefore, 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)

By introducing into formula (24) instead of the values ​​1/R eq, 1/R 1, 1/R 2 and 1/R 3 the corresponding conductivities G eq, G 1, G 2 and G 3, we obtain: equivalent conductivity parallel circuit equal to the sum of the conductances of parallel connected resistors:

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

Thus, as the number of resistors connected in parallel increases, the resulting conductivity of the electrical circuit increases, and the resulting resistance decreases.
From the above formulas it follows that currents are distributed between parallel branches in inverse proportion to their electrical resistance 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 regard, there is complete analogy between the distribution of currents along individual branches and the distribution of water flows through pipes.
The given 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 circuit resistance Rec 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 have a significant effect on the other receivers. Whenever any receiver is turned off or fails, the remaining receivers remain on.

valuable. That's why parallel connection has significant advantages over the sequential one, as a result of which it is most widely used. In particular, electric lamps and motors designed to operate at a certain (rated) voltage are always connected in parallel.
On DC electric locomotives and some diesel locomotives, traction motors must be switched on at different voltages during speed control, so they switch from a series connection to a parallel connection during acceleration.

Mixed connection of resistors. Mixed compound This is a connection in which some of the resistors are connected in series, and some in parallel. For example, in the diagram of Fig. 27, and there are two series-connected resistors with resistances R1 and R2, a resistor with resistance R3 is connected in parallel with 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 for a mixed connection is usually determined by the conversion method, in which complex chain are transformed into the simplest in successive stages. For example, for the diagram 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, but is replaced by the equivalent circuit in Fig. 27, b. Then the equivalent resistance R123 of parallel-connected resistances and R3 are determined using 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 of Fig. 27, v. After this, 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

Series, parallel and mixed connections are widely used to change the resistance of starting rheostats when starting an electric power plant. p.s. direct current.

In the previous summary, it was established that the current strength in a conductor depends on the voltage at its ends. If you change the conductors in an experiment, leaving the voltage on them unchanged, then you can show that when constant voltage at the ends of the conductor, the current strength is inversely proportional to its resistance. Combining the dependence of current on voltage and its dependence on conductor resistance, we can write: I = U/R . This law, established experimentally, is called Ohm's law(for a section of chain).

Ohm's law for a circuit section: The current strength in a conductor is directly proportional to the voltage applied to its ends and inversely proportional to the resistance of the conductor. First of all, the law is always true for solid and liquid metal conductors. And also for some other substances (usually solid or liquid).

Consumers of electrical energy (light bulbs, resistors, etc.) can be connected to each other in different ways in an electrical circuit. Dva main types of conductor connections : serial and parallel. And there are also two more connections that are rare: mixed and bridge.

Series connection of conductors

When connecting conductors in series, the end of one conductor will connect to the beginning of another conductor, and its end to the beginning of a third, etc. For example, connecting light bulbs in Christmas tree garland. When the conductors are connected in series, current passes through all the bulbs. In this case, the same charge passes through the cross section of each conductor per unit time. That is, the charge does not accumulate in any part of the conductor.

Therefore, when connecting conductors in series The current strength in any part of the circuit is the same:I 1 = I 2 = I .

The total resistance of series-connected conductors is equal to the sum of their resistances: R1 + R2 = R . Because when conductors are connected in series, their total length increases. It is greater than the length of each individual conductor, and the resistance of the conductors increases accordingly.

According to Ohm's law, the voltage on each conductor is equal to: U 1 = I* R 1 ,U 2 = I*R 2 . In this case, the total voltage is equal to U = I( R1+ R 2) . Since the current strength in all conductors is the same, and the total resistance is equal to the sum of the resistances of the conductors, then the total voltage on series-connected conductors is equal to the sum of the voltages on each conductor: U = U 1 + U 2 .

From the above equalities it follows that a series connection of conductors is used if the voltage for which the electrical energy consumers are designed is less than the total voltage in the circuit.

For series connection of conductors, the following laws apply: :

1) the current strength in all conductors is the same; 2) the voltage across the entire connection is equal to the sum of the voltages on the individual conductors; 3) the resistance of the entire connection is equal to the sum of the resistances of the individual conductors.

Parallel connection of conductors

Example parallel connection conductors serve to connect electrical energy consumers in the apartment. So, light bulbs, kettle, iron, etc. are switched on in parallel.

When connecting conductors in parallel, all conductors at one end are connected to one point in the circuit. And the second end to another point in the chain. A voltmeter connected to these points will show the voltage on both conductor 1 and conductor 2. In this case, the voltage at the ends of all parallel-connected conductors is the same: U 1 = U 2 = U .

When conductors are connected in parallel, the electrical circuit branches out. Therefore, part of the total charge passes through one conductor, and part through the other. Therefore, when connecting conductors in parallel, the current strength in the unbranched part of the circuit is equal to the sum of the current strength in the individual conductors: I = I 1 + I 2 .

According to Ohm's law I = U/R, I 1 = U 1 /R 1, I 2 = U 2 /R 2 . This implies: U/R = U 1 /R 1 + U 2 /R 2, U = U 1 = U 2, 1/R = 1/R 1 + 1/R 2 The reciprocal of the total resistance of parallel-connected conductors is equal to the sum of the reciprocals of the resistance of each conductor.

When conductors are connected in parallel, their total resistance is less than the resistance of each conductor. Indeed, if two conductors having the same resistance are connected in parallel G, then their total resistance is equal to: R = g/2. This is explained by the fact that when connecting conductors in parallel, their cross-sectional area increases. As a result, resistance decreases.

From the above formulas it is clear why electrical energy consumers are connected in parallel. They are all designed for a certain identical voltage, which in apartments is 220 V. Knowing the resistance of each consumer, you can calculate the current strength in each of them. And also the correspondence of the total current strength to the maximum permissible current strength.

For parallel connection of conductors, the following laws apply:

1) the voltage on all conductors is the same; 2) the current strength at the junction of the conductors is equal to the sum of the currents in the individual conductors; 3) the reciprocal value of the resistance of the entire connection is equal to the sum of the reciprocal values ​​of the resistance of individual conductors.

Content:

The flow of current in an electrical circuit is carried out through conductors, in the direction from the source to the consumers. Most of these circuits use copper wires and electrical receivers in a given quantity, having different resistances. Depending on the tasks performed, electrical circuits use serial and parallel connections of conductors. In some cases, both types of connections can be used, then this option will be called mixed. Each circuit has its own characteristics and differences, so they must be taken into account in advance when designing circuits, repairing and servicing electrical equipment.

Series connection of conductors

In electrical engineering great importance has a serial and parallel connection of conductors in an electrical circuit. Among them, a series connection scheme of conductors is often used, which assumes the same connection of consumers. In this case, inclusion in the circuit is performed one after another in order of priority. That is, the beginning of one consumer is connected to the end of another using wires, without any branches.

The properties of such an electrical circuit can be considered using the example of sections of a circuit with two loads. The current, voltage and resistance on each of them should be designated respectively as I1, U1, R1 and I2, U2, R2. As a result, relations were obtained that express the relationship between quantities as follows: I = I1 = I2, U = U1 + U2, R = R1 + R2. The data obtained are confirmed in practice by taking measurements with an ammeter and a voltmeter of the corresponding sections.

Thus, the series connection of conductors has the following individual features:

• The current strength in all parts of the circuit will be the same.
• The total voltage of the circuit is the sum of the voltages in each section.
• The total resistance includes the resistance of each individual conductor.

These ratios are suitable for any number of conductors connected in series. The total resistance value is always higher than the resistance of any individual conductor. This is due to an increase in their total length when connected in series, which also leads to an increase in resistance.

If you connect identical elements in series n, you get R = n x R1, where R is the total resistance, R1 is the resistance of one element, and n is the number of elements. Voltage U, on the contrary, is divided into equal parts, each of which is n times less general meaning. For example, if 10 lamps of the same power are connected in series to a network with a voltage of 220 volts, then the voltage in any of them will be: U1 = U/10 = 22 volts.

Conductors connected in series have a characteristic distinctive feature. If at least one of them fails during operation, the current flow stops in the entire circuit. The most striking example is when one burnt-out light bulb in a series circuit leads to failure of the entire system. To identify a burnt out light bulb, you will need to check the entire garland.

Parallel connection of conductors

In electrical networks, conductors can be connected different ways: series, parallel and combined. Among them, a parallel connection is an option when the conductors at the starting and ending points are connected to each other. Thus, the beginnings and ends of the loads are connected together, and the loads themselves are located parallel to each other. An electrical circuit may contain two, three or more conductors connected in parallel.

If we consider a series and parallel connection, the current strength in the latter can be studied using the following circuit. Take two incandescent lamps that have the same resistance and are connected in parallel. For control, each light bulb is connected to its own. In addition, another ammeter is used to monitor the total current in the circuit. The test circuit is supplemented by a power source and a key.

After closing the key, you need to monitor the readings of the measuring instruments. The ammeter on lamp No. 1 will show the current I1, and on lamp No. 2 the current I2. The general ammeter shows the current value equal to the sum of the currents of individual, parallel-connected circuits: I = I1 + I2. Unlike a series connection, if one of the bulbs burns out, the other will function normally. Therefore, in home electrical networks it is used parallel connection devices.

Using the same circuit, you can set the value of the equivalent resistance. For this purpose, a voltmeter is added to the electrical circuit. This allows you to measure the voltage in a parallel connection, while the current remains the same. There are also crossing points for the conductors connecting both lamps.

As a result of measurements, the total voltage for a parallel connection will be: U = U1 = U2. After this, you can calculate the equivalent resistance, which conditionally replaces all the elements in a given circuit. With a parallel connection, in accordance with Ohm's law I = U/R, the following formula is obtained: U/R = U1/R1 + U2/R2, in which R is the equivalent resistance, R1 and R2 are the resistances of both bulbs, U = U1 = U2 is the voltage value shown by the voltmeter.

One should also take into account the fact that the currents in each circuit add up to the total current strength of the entire circuit. In its final form, the formula reflecting the equivalent resistance will look like this: 1/R = 1/R1 + 1/R2. As the number of elements in such chains increases, the number of terms in the formula also increases. The difference in basic parameters distinguishes current sources from each other, allowing them to be used in various electrical circuits.

A parallel connection of conductors is characterized by a fairly low equivalent resistance value, so the current strength will be relatively high. This factor should be taken into account when plugging in a large number of electrical appliances. In this case, the current increases significantly, leading to overheating of cable lines and subsequent fires.

Laws of series and parallel connection of conductors

These laws concerning both types of conductor connections have been partially discussed earlier.

For a clearer understanding and perception in a practical sense, series and parallel connection of conductors, formulas should be considered in a certain sequence:

• A series connection assumes the same current in each conductor: I = I1 = I2.
• Parallel and series connection of conductors is explained in each case differently. For example, with a series connection, the voltages on all conductors will be equal to each other: U1 = IR1, U2 = IR2. In addition, with a series connection, the voltage is the sum of the voltages of each conductor: U = U1 + U2 = I(R1 + R2) = IR.
• Impedance a circuit when connected in series consists of the sum of the resistances of all individual conductors, regardless of their number.
• With a parallel connection, the voltage of the entire circuit is equal to the voltage on each of the conductors: U1 = U2 = U.
• The total current measured in the entire circuit is equal to the sum of the currents flowing through all conductors connected in parallel: I = I1 + I2.

In order to more effectively design electrical networks, you need to have a good knowledge of the series and parallel connection of conductors and its laws, finding the most rational practical application for them.

Mixed connection of conductors

In electrical networks, serial parallel and mixed compound conductors designed for specific operating conditions. However, most often preference is given to the third option, which is a set of combinations consisting of various types connections.

In such mixed circuits, serial and parallel connection of conductors is actively used, the pros and cons of which must be taken into account when designing electrical networks. These connections consist not only of individual resistors, but also rather complex sections that include many elements.

The mixed connection is calculated according to the known properties of series and parallel connections. The calculation method consists of breaking the circuit down into simpler components, which are calculated separately and then summed up with each other.

Series connection of resistances

Let's take three constant resistances R1, R2 and R3 and connect them to the circuit so that the end of the first resistance R1 is connected to the beginning of the second resistance R2, the end of the second is connected to the beginning of the third R3, and we connect conductors to the beginning of the first resistance and to the end of the third from the current source (Fig. 1).

This connection of resistances is called alternating. Of course, the current in such a circuit will be the same at all its points.

Rice 1 . Series connection of resistances

How to find the total resistance of a circuit if we already know all the resistances included in it one by one? Using the position that the voltage U at the terminals of the current source is equal to the sum of the voltage drops in the sections of the circuit, we can write:

U = U1 + U2 + U3

Where

U1 = IR1 U2 = IR2 and U3 = IR3

or

IR = IR1 + IR2 + IR3

Taking the equality I out of brackets on the right side, we obtain IR = I(R1 + R2 + R3) .

Now dividing both sides of the equality by I, we will have R = R1 + R2 + R3

Thus, we concluded that when resistances are alternately connected, the total resistance of the entire circuit is equal to the sum of the resistances of the individual sections.

Let's check this conclusion using the following example. Let's take three constant resistances, the values ​​of which are known (for example, R1 == 10 Ohms, R 2 = 20 Ohms and R 3 = 50 Ohms). Let's connect them one by one (Fig. 2) and connect them to a current source whose EMF is 60 V ( internal resistance current source is neglected).

Rice. 2. Example of alternate connection of 3 resistances

Let's calculate what readings should be given by the devices turned on, as shown in the diagram, if the circuit is closed. Let's determine the external resistance of the circuit: R = 10 + 20 + 50 = 80 Ohm.

Let's find the current in the circuit using Ohm's law: 60 / 80 = 0.75 A

Knowing the current in the circuit and the resistance of its sections, we determine the voltage drop for each section of the circuit U 1 = 0.75 x 10 = 7.5 V, U 2 = 0.75 x 20 = 15 V, U3 = 0.75 x 50 = 37 .5 V.

Knowing the voltage drop in the sections, we determine the total voltage drop in the external circuit, i.e. the voltage at the terminals of the current source U = 7.5 + 15 + 37.5 = 60 V.

We got it in such a way that U = 60 V, i.e. the non-existent equality of the emf of the current source and its voltage. This is explained by the fact that we neglected the internal resistance of the current source.

Having now closed the key switch K, we can verify from the devices that our calculations are approximately correct.

Let's take two constant resistances R1 and R2 and connect them so that the beginnings of these resistances are included in one common point a, and the ends - in another common point b. By then connecting points a and b with a current source, we obtain a closed electronic circuit. This connection of resistances is called a parallel connection.

Figure 3. Parallel connection of resistances

Let's trace the current flow in this circuit. From the positive pole of the current source, the current will reach point a along the connecting conductor. At point a it will branch, because here the circuit itself branches into two separate branches: the first branch with resistance R1 and the second with resistance R2. Let us denote the currents in these branches by I1 and I 2, respectively. Any of these currents will follow its own branch to point b. At this point, the currents will merge into one common current, which will come to the negative pole of the current source.

Thus, when resistances are connected in parallel, a branched circuit results. Let's see what the relationship between the currents in the circuit we have created will be.

Let's turn on the ammeter between the positive pole of the current source (+) and point a and note its readings. Having then connected the ammeter (shown in the dotted line in the figure) to the wire connecting point b to the negative pole of the current source (-), we note that the device will show the same amount of current.

Means current in the circuit before it branches(up to point a) is equal to current strength after circuit branching(after point b).

We will now turn on the ammeter alternately in each branch of the circuit, remembering the readings of the device. Let the ammeter show the current strength in the first branch I1, and in the 2nd branch - I 2. By adding these two ammeter readings, we get a total current equal in value to current I until the branching (to point a).

Properly, the strength of the current flowing to the branching point is equal to the sum of the currents flowing from this point. I = I1 + I2 Expressing this by the formula, we get

This relationship, which is of great practical importance, is called branched chain law.

Let us now consider what the relationship between the currents in the branches will be.

Let's turn on the voltmeter between points a and b and see what it shows us. Firstly, the voltmeter will show the voltage of the current source because it is connected, as can be seen from Fig. 3, specifically to the terminals of the current source. Secondly, the voltmeter will show the voltage drops U1 and U2 across resistances R1 and R2, because it is connected to the beginning and end of each resistance.

As follows, when connecting resistances in parallel, the voltage at the terminals of the current source is equal to the voltage drop across each resistance.

This gives us the right to write that U = U1 = U2.

where U is the voltage at the terminals of the current source; U1 - voltage drop across resistance R1, U2 - voltage drop across resistance R2. Let us remember that the voltage drop across a section of the circuit is numerically equal to the product of the current flowing through this section and the resistance of the section U = IR.

Therefore, for each branch you can write: U1 = I1R1 and U2 = I2R2, but because U1 = U2, then I1R1 = I2R2.

Applying the rule of proportion to this expression, we obtain I1 / I2 = U2 / U1 i.e. the current in the first branch will be as many times greater (or less) than the current in the 2nd branch, how many times the resistance of the first branch is less (or greater) resistance of the 2nd branch.

So, we have come to the fundamental conclusion that When resistances are connected in parallel, the total current of the circuit branches into currents that are inversely proportional to the resistance values ​​of the parallel branches. In other words, the greater the resistance of the branch, the less current will flow through it, and, conversely, the lower the resistance of the branch, the higher current will flow through this branch.

Let us verify the correctness of this dependence in the following example. Let's assemble a circuit consisting of two parallel-connected resistances R1 and R2 connected to a current source. Let R1 = 10 ohms, R2 = 20 ohms and U = 3 V.

Let's first calculate what the ammeter included in each branch will show us:

I1 = U / R1 = 3 / 10 = 0.3 A = 300 mA

I 2 = U / R 2 = 3 / 20 = 0.15 A = 150 mA

Total current in the circuit I = I1 + I2 = 300 + 150 = 450 mA

Our calculation confirms that when resistances are connected in parallel, the current in the circuit branches back in proportion to the resistances.

Indeed, R1 == 10 Ohm is half as much as R 2 = 20 Ohm, while I1 = 300 mA is twice as much as I2 = 150 mA. The total current in the circuit I = 450 mA branched into two parts so that most of it (I1 = 300 mA) went through the smallest resistance (R1 = 10 Ohms), and the smallest part (R2 = 150 mA) went through the larger resistance (R 2 = 20 Ohm).

This branching of current in parallel branches is similar to the flow of water through pipes. Imagine pipe A, which in some place branches into two pipes B and C of different diameters (Fig. 4). Because the diameter of pipe B is greater than the diameter of pipes B, then through pipe B to the same time will pass more water than through pipe B, which provides more resistance to the water clot.

Rice. 4

Let us now consider what the total resistance of the external circuit, consisting of 2 parallel-connected resistances, will be equal to.

Underneath this total resistance In the external circuit, you need to realize such a resistance that could be used to change both parallel-connected resistances at a given circuit voltage, without changing the current before branching. This resistance is called equivalent resistance.

Let's return to the circuit shown in Fig. 3, and let’s see what the equivalent resistance of 2 parallel connected resistances will be. Applying Ohm's law to this circuit, we can write: I = U/R, where I is the current in the external circuit (up to the branching point), U is the voltage of the external circuit, R is the resistance of the external circuit, i.e. equivalent resistance.

In the same way, for each branch I1 = U1 / R1, I2 = U2 / R2, where I1 and I 2 are the currents in the branches; U1 and U2 - voltage on branches; R1 and R2 - branch resistances.

According to the branched chain law: I = I1 + I2

Substituting the current values, we get U / R = U1 / R1 + U2 / R2

Because with a parallel connection U = U1 = U2, we can write U / R = U / R1 + U / R2

Taking U on the right side of the equality out of brackets, we get U / R = U (1 / R1 + 1 / R2)

Dividing now both sides of the equality by U, we will have 1 / R = 1 / R1 + 1 / R2

Remembering that conductivity is the reciprocal of resistance, we can say that in the acquired formula 1/R is the conductivity of the external circuit; 1 / R1 conductivity of the first branch; 1 / R2 - conductivity of the 2nd branch.

Based on this formula we conclude: with a parallel connection, the conductivity of the external circuit is equal to the sum of the conductivities of the individual branches.

Properly, to find the equivalent resistance of resistances connected in parallel, you need to find the conductivity of the circuit and take the reciprocal value.

It also follows from the formula that the conductivity of the circuit is greater than the conductivity of each branch, which means that the equivalent resistance of the external circuit is less than the smaller of the resistances connected in parallel.

Considering the case of parallel connection of resistances, we took a more ordinary circuit consisting of two branches. But in practice, there may be cases when the chain consists of 3 or more parallel branches. What to do in these cases?

It turns out that all the relationships we have acquired remain valid for a circuit consisting of any number of parallel-connected resistances.

To see this, let's look at the following example.

Let's take three resistances R1 = 10 Ohms, R2 = 20 Ohms and R3 = 60 Ohms and connect them in parallel. Let's determine the equivalent resistance of the circuit (Fig. 5). R = 1/6 As follows, equivalent resistance R = 6 Ohm.

In this way, equivalent resistance is less than the smaller of the resistances connected in parallel in the circuit, i.e. less than resistance R1.

Let's see now whether this resistance is really equivalent, that is, one that could change resistances of 10, 20 and 60 Ohms connected in parallel, without changing the current strength before branching the circuit.

Let us assume that the voltage of the external circuit, and as follows, the voltage across the resistances R1, R2, R3 is 12 V. Then the current strength in the branches will be: I1 = U/R1 = 12/10 = 1.2 A I 2 = U/ R 2 = 12 / 20 = 1.6 A I 3 = U/R1 = 12 / 60 = 0.2 A

We obtain the total current in the circuit using the formula I = I1 + I2 + I3 = 1.2 + 0.6 + 0.2 = 2 A.

Let's check, using the formula of Ohm's law, whether a current of 2 A will be obtained in the circuit if, instead of 3 parallel-connected resistances we recognize, one equivalent resistance of 6 Ohms is connected.

I = U / R = 12 / 6 = 2 A

As we see, the resistance we found R = 6 Ohms is indeed equivalent for this circuit.

You can also verify this using measuring devices if you assemble a circuit with the resistances we took, measure the current in the external circuit (before branching), then replace the parallel-connected resistances with one 6 Ohm resistance and measure the current again. The ammeter readings in both cases will be approximately similar.

In practice, you may also encounter parallel connections, for which it is easier to calculate the equivalent resistance, i.e., without first determining the conductivity, you can immediately find the resistance.

For example, if two resistances R1 and R2 are connected in parallel, then the formula 1 / R = 1 / R1 + 1 / R2 can be converted as follows: 1/R = (R2 + R1) / R1 R2 and, solving the equality with respect to R, get R = R1 x R2 / (R1 + R2), i.e. When two resistances are connected in parallel, the equivalent resistance of the circuit is equal to the product of the resistances connected in parallel divided by their sum.