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Parallel connection of 4 resistors. Serial and parallel connection

Series and parallel connection of resistors

Series connection of resistors

Serial connection is the connection of two or more resistors in the form of a circuit in which each individual resistor is connected to another individual resistor at only one point.

Total resistance Rtot

With this connection, the same electric current passes through all resistors. The more elements there are in a given section of an electrical circuit, the more difficult it is for current to flow through it. Therefore, when resistors are connected in series, their total resistance increases, and it is equal to the sum of all resistances.

Series voltage

The voltage in a series connection is distributed to each resistor according to Ohm's law:

That is, the greater the resistance of the resistor, the greater the voltage drops across it.

Parallel connection of resistors

Parallel connection- This is a connection in which resistors are connected to each other by both contacts. As a result, several resistors can be connected to one point (electrical node).

Total resistance Rtot

With this connection, a separate current will flow through each resistor. The strength of this current will be inversely proportional to the resistance of the resistor. As a result, the total conductivity of such a section of the electrical circuit increases, and the total resistance, in turn, decreases.

Thus, when connecting resistors with different resistances in parallel, the total resistance will always be less than the value of the smallest individual resistor.

Formula for total conductivity when connecting resistors in parallel:

Formula for equivalent total resistance when connecting resistors in parallel:

For two identical resistors, the total resistance will be equal to half of one individual resistor:

Accordingly, for n identical resistors, the total resistance will be equal to the value of one resistor divided by n.

Parallel voltage

The voltage between points A and B is both the total voltage for the entire circuit section and the voltage across each resistor individually. Therefore, with a parallel connection, the same voltage will drop across all resistors.

Electric current in parallel connection

A current flows through each resistor, the strength of which is inversely proportional to the resistance of the resistor. In order to find out how much current flows through a certain resistor, you can use Ohm's law:

Mixed connection of resistors

A mixed connection is a section of a circuit where some resistors are connected in series and some in parallel. In turn, a mixed connection is of serial and parallel types.

Total resistance Rtot

o The circuit is divided into sections with only parallel or only series connections. o Calculate the total resistance for each individual section.

o Calculate the total resistance for the entire mixed circuit.

This is what it will look like for Scheme 1:

There is also a faster way to calculate the total resistance for a mixed connection. You can, in accordance with the diagram, immediately write the formula as follows:

o If resistors are connected in series, add them.

o If resistors are connected in parallel, use the symbol "||". o Substitute the formula for a parallel connection where the symbol "||" appears.

This is what it will look like for Scheme 1.

The current in the circuit flows through the conductors to the load from the source. Copper is most often used as such elements. A circuit may have several electrical receivers. Their resistances vary. In an electrical circuit, conductors can be connected in parallel or in series. There are also mixed types. The difference between each of them should be known before choosing the electrical circuit structure.

Conductors and circuit elements

Current flows through conductors. It follows from the source to the load. In this case, the conductor must easily release electrons.

A conductor that has resistance is called a resistor. The voltage of this element is the potential difference between the ends of the resistor, which is consistent with the direction of power flow.

The serial and parallel connection of conductors is characterized by one general principle. Current flows in the circuit from plus (it is called the source) to minus, where the potential becomes less and less. In electrical diagrams, the resistance of the wires is considered to be zero, since it is negligibly small.

Therefore, when calculating a serial or parallel connection, they resort to idealization. This makes them easier to learn. In real circuits, the potential gradually decreases as it moves along the wire and elements that have a parallel or series connection.

Series connection of conductors

If there is a series combination of conductors, the resistances are turned on one after another. In this position, the current strength in all elements of the circuit is the same. Series-connected conductors create a voltage in the area that is equal to their sum on all elements.

Charges do not have the opportunity to accumulate at the nodes of the circuit. This would lead to a change in the electric field voltage and current.

In the presence of constant voltage, the current will depend on the resistance of the circuit. Therefore, with a series connection, the resistance will change due to a change in one load.

Series connection of conductors has a disadvantage. If one of the circuit elements breaks down, the operation of all its other components will be interrupted. For example, as in a garland. If one light bulb burns out, the entire product will not work.

If the conductors were connected in series in a circuit, their resistance at each point will be the same. The resistance in the sum of all circuit elements will be equal to the sum of the voltage reduction in the circuit sections.

Experience can confirm this. The series connection of resistances is calculated using instruments and mathematical verification. For example, three constant resistances of known magnitude are taken. They are connected in series and connected to a 60 V power supply.

After this, the expected indicators of the devices are calculated if the circuit is closed. According to Ohm's law, there is a current in the circuit, which will allow us to determine the voltage drop in all its sections. After this, the results obtained are summed up and the total value of the reduction in resistance in the external circuit is obtained. The series connection of resistances can be confirmed approximately. If we do not take into account the internal resistance created by the energy source, the voltage drop will be less than the sum of the resistances. Using instruments, you can verify that equality is approximately maintained.

Parallel connection of conductors

When connecting conductors in series and parallel in a circuit, resistors are used. A parallel connection of conductors is a system in which some ends of all resistors converge into one common node, and the other ends into another node. More than two conductors converge at these points in the circuit.

With this connection, the same voltage is applied to the elements. Parallel sections of a chain are called branches. They pass between two nodes. Parallel and serial connections have their own properties.

If there are branches in the electrical circuit, then the voltage on each of them will be the same. It is equal to the voltage on the unbranched section. At this point, the current strength will be calculated as the sum of it in each branch.

A value equal to the sum of the inverses of the resistances of the branches will also be the inverse of the resistance of the parallel connection section.

Parallel connection of resistances

Parallel and series connections differ in the calculation of the resistance of its elements. When connected in parallel, the current branches out. This increases the conductivity of the circuit (reduces the total resistance), which will be equal to the sum of the conductances of the branches.

If several resistors of the same value are connected in parallel, then the total resistance of the circuit will be less than one resistor as many times as they are included in the circuit.

Serial and parallel connection of conductors have a number of features. In a parallel connection, the current is inversely proportional to the resistance. The currents in resistors do not depend on each other. Therefore, turning off one of them will not affect the operation of the others. Therefore, many electrical appliances have this type of connection of circuit elements.

Mixed

Parallel and series connections of conductors can be combined in the same circuit. For example, elements connected in parallel can be connected in series with another resistor or group of resistors. This is a mixed compound. The total resistance of the circuits is calculated by separately summing the values for the parallel connected unit and for the series connection.

Moreover, the equivalent resistances of series-connected elements are first calculated, and then the total resistance of parallel sections of the circuit is calculated. Serial connection in calculations takes priority. These types of electrical circuits are quite common in various devices and equipment.

Having familiarized yourself with the types of connections of circuit elements, you can understand the principle of organization of circuits of various electrical devices. Parallel and serial connections have a number of features in the calculation and operation of the entire system. Knowing them, you can correctly use each of the presented types to connect elements of electrical circuits.

Serial connection – it is the connection of two or more resistors in the form of a circuit in which each individual resistor is connected to another individual resistor at only one point.

Parallel connection – This is a connection in which resistors are connected to each other by both contacts. As a result, several resistors can be connected to one point (electrical node).

2) Total resistance Rtot

Total resistance Rtot

Thus, when connecting resistors with different resistances in parallel, the total resistance will always be less than the value of the smallest individual resistor.

Formula for equivalent total resistance when connecting resistors in parallel:

For two identical resistors, the total resistance will be equal to half of one individual resistor:

Accordingly, for n identical resistors, the total resistance will be equal to the value of one resistor divided by n.

3) Electrical conductivity, electrical conductivity, conductivity, the ability of a body to pass electric current under the influence of an electric field, as well as a physical quantity that quantitatively characterizes this ability. Bodies that conduct electric current are called conductors, in contrast to insulators...
The basic unit of resistance is Ohm. Specific conductivity is the reciprocal of resistance; it is measured in Siemens, formerly called mho. In relation to bulk substances, it is more convenient to talk about special conductivity, usually called specific conductivity.
Specific conductivity is the conductivity measured between opposite sides of a 1 cm cube of a substance. The unit for this type of measurement is Siemens/cm. When measuring water conductivity, more accurate μS/cm (microsiemens) and mS/cm (millisiemens) are often used.
The corresponding units of resistance (or resistivity) are Ohm/cm, MegaOhm/cm and kiloOhm/cm. When measuring ultrapure water, MegaOhm/cm is more often used, as this gives more accurate results. The resistance of less pure water, such as tap water, is measured in kiloOhm/cm.

4) The total resistance in a series connection is equal to the sum of the resistances Rsum=R1+R2+R3...
The same current flows through all resistances (I). Therefore, you calculate the current as the ratio of the source voltage U to Rtotal.

Power

P=U*I or P=I*I*R (since U=I*R).

P1=I*I*R1
P2=I*I*R2
P3=I*I*R3

5) the power of the electric current in a circuit consisting of parallel connected sections,
equal to the sum of capacities in individual areas:

With a parallel connection, each lamp is connected to its rated voltage of 220 V. At the same time, each lamp receives its own rated current, providing a given glow in accordance with the rated power. power depends on the resistance of the filament. The higher the resistance of the thread, the lower the current and, accordingly, the lower the rated power.
with a series connection, the same current flows in each lamp. and the voltage is distributed depending on the proportion of the resistance of each lamp in relation to the resistance of the entire circuit.
For a circuit of two lamps, the total voltage is divided.
the voltage on a 40 W lamp will be 220X60:(40+60)=132; IN.
the voltage on a 60 W lamp will be 220X40:(40+60)=80; IN.

Parallel connections of resistors, the calculation formula for which is derived from Ohm's law and Kirchhoff's rules, are the most common type of inclusion of elements in an electrical circuit. When connecting conductors in parallel, two or more elements are connected by their contacts on both sides, respectively. Their connection to the general circuit is carried out precisely by these nodal points.

Features of inclusion

Conductors connected in this way are often part of complex chains that, in addition, contain a series connection of individual sections.

The following features are typical for such inclusion:

The total voltage in each of the branches will have the same value;
The electric current flowing in any of the resistances is always inversely proportional to the value of their nominal value.

In the particular case when all resistors connected in parallel have the same nominal values, the “individual” currents flowing through them will also be equal to each other.

Calculation

The resistances of a number of conductive elements connected in parallel are determined using a well-known form of calculation, which involves the addition of their conductivities (the reciprocal of the resistance values).

The current flowing in each of the individual conductors in accordance with Ohm's law can be found by the formula:

I= U/R (one of the resistors).

After becoming familiar with the general principles of calculating the elements of complex chains, you can move on to specific examples of solving problems of this class.

Typical Connections

Example No. 1

Often, in order to solve the problem facing the designer, it is necessary to ultimately obtain a specific resistance by combining several elements. When considering the simplest version of such a solution, let’s assume that the total resistance of a chain of several elements should be 8 Ohms. This example requires separate consideration for the simple reason that in the standard series of resistances there is no nominal value of 8 Ohms (there are only 7.5 and 8.2 Ohms).

The solution to this simplest problem can be obtained by connecting two identical elements with resistances of 16 Ohms each (such ratings exist in the resistive series). According to the formula given above, the total resistance of the chain in this case is calculated very simply.

It follows from it:

16x16/32=8 (Ohm), that is, exactly as much as was required.

In this relatively simple way, it is possible to solve the problem of forming a total resistance equal to 8 Ohms.

Example No. 2

As another typical example of the formation of the required resistance, we can consider the construction of a circuit consisting of 3 resistors.

The total R value of such a connection can be calculated using the formula for series and parallel connections in conductors.

In accordance with the nominal values indicated in the picture, the total resistance of the chain will be equal to:

1/R = 1/200+1/220+1/470 = 0.0117;

R=1/0.0117 = 85.67 Ohm.

As a result, we find the total resistance of the entire chain obtained by connecting three elements in parallel with nominal values of 200, 240 and 470 Ohms.

Important! This method is also applicable when calculating an arbitrary number of conductors or consumers connected in parallel.

It should also be noted that with this method of connecting elements of different sizes, the total resistance will be less than that of the smallest value.

Calculation of combined circuits

The considered method can also be used when calculating the resistance of more complex or combined circuits consisting of a whole set of components. They are sometimes called mixed, since both methods are used at once when forming chains. A mixed connection of resistors is shown in the figure below.

To simplify the calculation, we first divide all resistors according to the type of connection into two independent groups. One of them is a serial connection, and the second is a parallel type connection.

From the above diagram it can be seen that elements R2 and R3 are connected in series (they are combined into group 2), which, in turn, is connected in parallel with resistor R1, which belongs to group 1.

For elements from group 2, the value of the total resistance is found as the sum of R2 and R3:

R (2+3) = R2 + R3.

To obtain the final result, we reduce the circuit to the form obtained by connecting two resistances in parallel. After this, the total value for the entire circuit as a whole is calculated according to the formula already discussed earlier.

In conclusion, we note that to carry out calculation operations that fall into the category of complex connections, you can use the same techniques. They are based on the same Ohm’s law and Kirchhoff’s rules, known from school. The main thing is to correctly use all the formulas described above.

Video

Moreover, these can be not only conductors, but also capacitors. It is important here not to get confused about what each of them looks like on the diagram. And only then apply specific formulas. By the way, you need to remember them by heart.

How can you differentiate between these two compounds?

Look carefully at the diagram. If you imagine the wires as a road, then the cars on it will play the role of resistors. On a straight road without any branches, cars drive one after another, in a chain. The series connection of conductors looks the same. In this case, the road can have an unlimited number of turns, but not a single intersection. No matter how the road (wires) twist, the machines (resistors) will always be located one after another, in one chain.

It's a completely different matter if a parallel connection is considered. Then the resistors can be compared to athletes at the start line. They each stand on their own path, but their direction of movement is the same, and the finish line is in the same place. The same goes for resistors - each of them has its own wire, but they are all connected at some point.

Formulas for current strength

It is always discussed in the topic “Electricity”. Parallel and series connections have different effects on the value in resistors. Formulas have been derived for them that can be remembered. But it’s enough just to remember the meaning that is put into them.

So, the current when connecting conductors in series is always the same. That is, in each of them the current value is not different. An analogy can be drawn by comparing a wire with a pipe. The water always flows in it the same way. And all obstacles in her path will be swept away with the same force. Same with current strength. Therefore, the formula for the total current in a circuit with resistors connected in series looks like this:

I total = I 1 = I 2

Here the letter I denotes the current strength. This is a common designation, so you need to remember it.

The current in a parallel connection will no longer be a constant value. Using the same analogy with a pipe, it turns out that water will split into two streams if the main pipe has a branch. The same phenomenon is observed with current when a branching wire appears in its path. Formula for total current at:

I total = I 1 + I 2

If the branching is made up of more than two wires, then in the above formula there will be more terms by the same number.

Formulas for voltage

When we consider a circuit in which the conductors are connected in series, the voltage across the entire section is determined by the sum of these values on each specific resistor. You can compare this situation with plates. One person can easily hold one of them; he can also take the second one nearby, but with difficulty. One person will no longer be able to hold three plates in their hands next to each other; the help of a second person will be required. And so on. People's efforts add up.

The formula for the total voltage of a circuit section with a series connection of conductors looks like this:

U total = U 1 + U 2, where U is the designation adopted for

A different situation arises when considering When the plates are stacked on top of each other, they can still be held by one person. Therefore, there is no need to fold anything. The same analogy is observed when connecting conductors in parallel. The voltage on each of them is the same and equal to that on all of them at once. The formula for total voltage is:

U total = U 1 = U 2

Formulas for electrical resistance

You no longer need to memorize them, but know the formula of Ohm’s law and derive the necessary one from it. From this law it follows that voltage is equal to the product of current and resistance. That is, U = I * R, where R is resistance.

Then the formula you need to work with depends on how the conductors are connected:

sequentially, which means we need equality for the voltage - I total * R total = I 1 * R 1 + I 2 * R 2;
in parallel, it is necessary to use the formula for current strength - Utot / Rtot = U 1 / R 1 + U 2 / R 2 .

What follows are simple transformations, which are based on the fact that in the first equality all currents have the same value, and in the second, the voltages are equal. This means they can be reduced. That is, the following expressions are obtained:

R total = R 1 + R 2 (for series connection of conductors).
1 / R total = 1 / R 1 + 1 / R 2 (for parallel connection).

As the number of resistors that are connected to the network increases, the number of terms in these expressions changes.

It is worth noting that parallel and series connections of conductors have different effects on the total resistance. The first of them reduces the resistance of the circuit section. Moreover, it turns out to be smaller than the smallest of the resistors used. With a serial connection, everything is logical: the values are added, so the total number will always be the largest.

Current work

The previous three quantities make up the laws of parallel connection and series arrangement of conductors in a circuit. Therefore, it is imperative to know them. About work and power, you just need to remember the basic formula. It is written like this: A = I * U * t, where A is the work done by the current, t is the time it passes through the conductor.

In order to determine the overall work for a series connection, it is necessary to replace the voltage in the original expression. The result is the equality: A = I * (U 1 + U 2) * t, opening the brackets in which it turns out that the work on the entire section is equal to their sum on each specific current consumer.

The reasoning is similar if a parallel connection scheme is considered. Only the current strength must be replaced. But the result will be the same: A = A 1 + A 2.

Current power

When deriving the formula for the power (designation “P”) of a section of the circuit, you again need to use one formula: P = U * I. After similar reasoning, it turns out that parallel and serial connections are described by the following formula for power: P = P 1 + P 2.

That is, no matter how the circuits are drawn up, the total power will be the sum of those involved in the work. This explains the fact that you cannot connect many powerful devices to your apartment’s network at the same time. She simply cannot withstand such a load.

How does the connection of conductors affect the repair of a New Year's garland?

Immediately after one of the bulbs burns out, it will become clear how they were connected. When connected in series, none of them will light up. This is explained by the fact that a lamp that has become unusable creates a break in the circuit. Therefore, you need to check everything to determine which one is burned out, replace it - and the garland will start working.

If it uses a parallel connection, it does not stop working if one of the bulbs fails. After all, the chain will not be completely broken, but only one parallel part. To repair such a garland, you do not need to check all the elements of the circuit, but only those that do not light up.

What happens to a circuit if it includes capacitors rather than resistors?

When they are connected in series, the following situation is observed: charges from the pluses of the power source are supplied only to the outer plates of the outer capacitors. Those that are between them simply transfer this charge along the chain. This explains the fact that identical charges appear on all plates, but with different signs. Therefore, the electric charge of each capacitor connected in series can be written as follows:

q total = q 1 = q 2.

In order to determine the voltage on each capacitor, you will need to know the formula: U = q / C. In it, C is the capacitance of the capacitor.

The total voltage obeys the same law that is valid for resistors. Therefore, replacing the voltage with the sum in the capacitance formula, we get that the total capacitance of the devices must be calculated using the formula:

C = q / (U 1 + U 2).

You can simplify this formula by reversing the fractions and replacing the voltage-to-charge ratio with capacitance. We get the following equality: 1 / C = 1 / C 1 + 1 / C 2 .

The situation looks somewhat different when the capacitors are connected in parallel. Then the total charge is determined by the sum of all charges that accumulate on the plates of all devices. And the voltage value is still determined according to general laws. Therefore, the formula for the total capacitance of parallel-connected capacitors looks like this:

C = (q 1 + q 2) / U.

That is, this value is calculated as the sum of each of the devices used in the connection:

C = C 1 + C 2.

How to determine the total resistance of an arbitrary connection of conductors?

That is, one in which successive sections replace parallel ones, and vice versa. All the laws described are still valid for them. You just need to apply them step by step.

First, you need to mentally unfold the diagram. If it’s difficult to imagine, then you need to draw what you get. The explanation will become clearer if we consider it with a specific example (see figure).

It is convenient to start drawing it from points B and C. They need to be placed at some distance from each other and from the edges of the sheet. One wire approaches point B from the left, and two are already directed to the right. Point B, on the contrary, on the left has two branches, and after it there is one wire.

Now you need to fill the space between these points. Along the top wire you need to place three resistors with coefficients 2, 3 and 4, and the one with the index equal to 5 will go below. The first three are connected in series. They are parallel with the fifth resistor.

The remaining two resistors (the first and sixth) are connected in series with the considered section of the BV. Therefore, the drawing can simply be supplemented with two rectangles on either side of the selected points. It remains to apply the formulas to calculate the resistance:

first the one given for the serial connection;
then for parallel;
and again for consistency.

In this way, you can deploy any, even very complex, scheme.

Problem on serial connection of conductors

Condition. Two lamps and a resistor are connected in a circuit one behind the other. The total voltage is 110 V and the current is 12 A. What is the value of the resistor if each lamp is rated at 40 V?

Solution. Since a series connection is considered, the formulas of its laws are known. You just need to apply them correctly. Start by finding out the voltage across the resistor. To do this, you need to subtract the voltage of one lamp twice from the total. It turns out 30 V.

Now that two quantities are known, U and I (the second of them is given in the condition, since the total current is equal to the current in each series consumer), we can calculate the resistance of the resistor using Ohm’s law. It turns out to be equal to 2.5 ohms.

Answer. The resistor's resistance is 2.5 ohms.

Parallel and serial problem

Condition. There are three capacitors with capacities of 20, 25 and 30 μF. Determine their total capacitance when connected in series and in parallel.

Solution. It's easier to start with In this situation, all three values just need to be added. Thus, the total capacitance is equal to 75 µF.

The calculations will be somewhat more complicated when these capacitors are connected in series. After all, you first need to find the ratio of one to each of these containers, and then add them to each other. It turns out that one divided by the total capacity is equal to 37/300. Then the desired value is approximately 8 µF.

Answer. The total capacitance for a series connection is 8 µF, for a parallel connection - 75 µF.