Application of serial and parallel connection of conductors. Parallel and series connection of resistances

Separate conductors electrical circuit can be connected to each other in series, parallel and mixed. In this case, series and parallel connections of conductors are the main types of connections, and mixed compound it is their totality.

A series connection of conductors is such a connection when the end of the first conductor is connected to the beginning of the second, the end of the second conductor is connected to the beginning of the third, and so on (Figure 1).

Figure 1. Diagram of serial connection of conductors

The total resistance of a circuit consisting of several conductors connected in series is equal to the sum of the resistances of the individual conductors:

r = r 1 + r 2 + r 3 + … + r n.

Current in individual areas series circuit is the same everywhere:

I 1 = I 2 = I 3 = I.

Video 1. Serial connection conductors

Example 1. Figure 2 shows an electrical circuit consisting of three series-connected resistances r 1 = 2 Ohm, r 2 = 3 Ohm, r 3 = 5 ohms. It is necessary to determine the readings of voltmeters V 1 , V 2 , V 3 and V 4 if the current in the circuit is 4 A.

Whole circuit resistance

r = r 1 + r 2 + r 3 = 2 + 3 + 5 = 10 Ohm.

Figure 2. Scheme for measuring voltages in individual sections of the electrical circuit

In resistance r 1 when current flows there will be a voltage drop:

U 1 = I × r 1 = 4 × 2 = 8 V.

Voltmeter V 1 included between points A And b, will show 8 V.

In resistance r 2 there is also a voltage drop:

U 2 = I × r 2 = 4 × 3 = 12 V.

Voltmeter V 2 included between points V And G, will show 12 V.

Voltage drop in resistance r 3:

U 3 = I × r 3 = 4 × 5 = 20 V.

Voltmeter V 3 included between points d And e, will show 20 V.

If a voltmeter is connected at one end to a point A, the other end to the point G, then it will show the potential difference between these points, equal to the sum of the voltage drops in the resistances r 1 and r 2 (8 + 12 = 20 V).

So the voltmeter V, measuring the voltage at the terminals of the circuit and connected between the points A And e, will show the potential difference between these points or the sum of the voltage drops in the resistances r 1 , r 2 and r 3 .

This shows that the sum of the voltage drops in individual sections of the electrical circuit is equal to the voltage at the circuit terminals.

Since in a series connection the circuit current is the same in all sections, the voltage drop is proportional to the resistance of a given section.

Example 2. Three resistances of 10, 15 and 20 ohms are connected in series, as shown in Figure 3. The current in the circuit is 5 A. Determine the voltage drop across each resistance.

U 1 = I × r 1 = 5 ×10 = 50 V,
U 2 = I × r 2 = 5 ×15 = 75 V,
U 3 = I × r 3 = 5 ×20 = 100 V.

Figure 3. Example 2

The total voltage of the circuit is equal to the sum of the voltage drops in individual sections of the circuit:

U = U 1 + U 2 + U 3 = 50 + 75 + 100 = 225 V.

Parallel connection of conductors

A parallel connection of conductors is a connection when the beginnings of all conductors are connected to one point, and the ends of the conductors to another point (Figure 4). The beginning of the circuit is connected to one pole of the voltage source, and the end of the circuit is connected to the other pole.

The figure shows that when conductors are connected in parallel, there are several paths for current to pass. Current flowing to branch point A, spreads further over three resistances and is equal to the sum of the currents leaving this point:

I = I 1 + I 2 + I 3 .

If the currents arriving at the branching point are considered positive, and the currents leaving are negative, then for the branching point we can write:

that is, the algebraic sum of currents for any nodal point in the circuit is always equal to zero. This relationship connecting the currents at any branch point in the circuit is called Kirchhoff's first law. The definition of Kirchhoff's first law can be expressed in another formulation, namely: the sum of currents flowing into a node of an electrical circuit is equal to the sum of currents flowing out of this node.

Video 2. Kirchhoff's first law

Usually, when calculating electrical circuits, the direction of the currents in the branches connected to any branch point is unknown. Therefore, in order to be able to write down the equation of Kirchhoff’s first law, before starting to calculate the circuit, it is necessary to arbitrarily select the so-called positive directions of currents in all its branches and designate them with arrows on the diagram.

Using Ohm's law, you can derive a formula for calculating the total resistance when connecting consumers in parallel.

Total current arriving at a point A, is equal to:

The currents in each of the branches have the following values:

According to the formula of Kirchhoff's first law

I = I 1 + I 2 + I 3

Taking out U on the right side of the equality outside the brackets, we get:

Reducing both sides of the equality by U, we get the formula for calculating the total conductivity:

g = g 1 + g 2 + g 3 .

Thus, with a parallel connection, it is not the resistance that increases, but the conductivity.

Example 3. Define total resistance three parallel-connected resistances, if r 1 = 2 Ohm, r 2 = 3 Ohm, r 3 = 4 ohms.

Example 4. Five resistances of 20, 30, 15, 40 and 60 Ohms are connected in parallel to the network. Determine the total resistance:

It should be noted that when calculating the total resistance of a branch, it is always less than the smallest resistance included in the branch.

If the resistances connected in parallel are equal to each other, then the total resistance r circuit is equal to the resistance of one branch r 1 divided by the number of branches n:

Example 5. Determine the total resistance of four parallel-connected resistances of 20 ohms each:

To check, let's try to find the branching resistance using the formula:

As you can see, the answer is the same.

Example 6. Let it be necessary to determine the currents in each branch when they are connected in parallel, shown in Figure 5, A.

Let's find the total resistance of the circuit:

Now we can depict all the branches in a simplified manner as one resistance (Figure 5, b).

Voltage drop between points A And B will:

U = I × r= 22 × 1.09 = 24 V.

Returning again to Figure 5, we see that all three resistances will be energized at 24 V, since they are connected between the points A And B.

Considering the first branch of the branching with resistance r 1, we see that the voltage in this section is 24 V, the resistance of the section is 2 Ohms. According to Ohm's law for a section of a circuit, the current in this section will be:

Second branch current

Third branch current

Let's check using Kirchhoff's first law

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

Always about her we're talking about 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 same value, and in the second - the voltages are equal. This means they can be reduced. That is, the following expressions are obtained:

1. R total = R 1 + R 2 (for series connection of conductors).
2. 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 serial arrangement of conductors in a circuit. Therefore, it is imperative to know them. You just need to remember about work and power 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. That's why electric charge 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 in specific example(see picture).

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 equal to current in each series consumer), you 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.

If we need an electrical appliance to work, we must connect it to. In this case, the current must pass through the device and return again to the source, that is, the circuit must be closed.

But connecting each device to a separate source is feasible mainly in laboratory conditions. In life you have to deal with limited quantity sources and quite a large number of current consumers. Therefore, connection systems are created that allow one source to be loaded with a large number of consumers. Systems can be as complex and branched as desired, but they are based on only two types of connections: serial and parallel connection of conductors. Each type has its own characteristics, pros and cons. Let's look at both of them.

Series connection of conductors

Series connection of conductors is the inclusion of several devices in an electrical circuit in series, one after another. Electrical appliances in in this case can be compared to people in a round dance, and their hands holding each other are wires connecting devices. The current source in this case will be one of the participants in the round dance.

The voltage of the entire circuit when connected in series will be equal to the sum of the voltages on each element included in the circuit. The current strength in the circuit will be the same at any point. And the sum of the resistances of all elements will be the total resistance of the entire circuit. Therefore, series resistance can be expressed on paper as follows:

I=I_1=I_2=⋯=I_n ; U=U_1+U_2+⋯+U_n ; R=R_1+R_2+⋯+R_n ,

The advantage of a series connection is the ease of assembly, but the disadvantage is that if one element fails, the current will be lost in the entire circuit. In such a situation, the inoperative element will be like a key in the off position. An example from life of the inconvenience of such a connection will probably be remembered by all older people who decorated Christmas trees with garlands of light bulbs.

If at least one light bulb in such a garland failed, you had to go through them all until you found the one that had burned out. In modern garlands this problem has been solved. They use special diode light bulbs, in which, when they burn out, the contacts are fused together, and the current continues to flow unhindered.

Parallel connection of conductors

When connecting conductors in parallel, all elements of the circuit are connected to the same pair of points, we can call them A and B. A current source is connected to the same pair of points. That is, it turns out that all elements are connected to the same voltage between A and B. At the same time, the current is, as it were, divided among all loads depending on the resistance of each of them.

The parallel connection can be compared to the flow of a river, on the way of which a small hill has arisen. In this case, the water goes around the hill on both sides, and then again merges into one stream. It turns out to be an island in the middle of the river. So the parallel connection is two separate channels around the island. And points A and B are the places where the common river bed is separated and reconnected.

The current voltage in each individual branch will be equal to the total voltage in the circuit. The total current of the circuit will be the sum of the currents of all individual branches. But the total resistance of the circuit in a parallel connection will be less than the current resistance on each of the branches. This happens because the total cross-section of the conductor between points A and B seems to increase due to an increase in the number of parallel connected loads. Therefore, the overall resistance decreases. A parallel connection is described by the following relations:

U=U_1=U_2=⋯=U_n ; I=I_1+I_2+⋯+I_n ; 1/R=1/R_1 +1/R_2 +⋯+1/R_n ,

where I is the current, U is the voltage, R is the resistance, 1,2,...,n are the numbers of the elements included in the circuit.

A huge advantage of a parallel connection is that when one of the elements is turned off, the circuit continues to function. All other elements continue to work. The downside is that all devices must be rated for the same voltage. It is in a parallel manner that 220 V network sockets are installed in apartments. This connection allows you to connect various devices to the network completely independently of each other, and if one of them fails, this does not affect the operation of the others.

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Usually everyone finds it difficult to answer. But this riddle, when applied to electricity, is solved quite definitely.

Electricity begins with Ohm's law.

And if we consider the dilemma in the context of parallel or serial connections - considering one connection to be a chicken and the other to be an egg, then there is no doubt at all.

Because Ohm's law is the very original electrical circuit. And it can only be consistent.

Yes, they came up with a galvanic cell and didn’t know what to do with it, so they immediately came up with another light bulb. And this is what came out of it. Here, a voltage of 1.5 V immediately flowed as current, in strict compliance with Ohm's law, through the light bulb to the back of the same battery. And inside the battery itself, under the influence of the sorceress-chemistry, the charges again ended up at the original point of their journey. And therefore, where the voltage was 1.5 volts, it remains that way. That is, the voltage is always the same, and the charges are constantly moving and successively pass through the light bulb and the galvanic cell.

And it is usually drawn on the diagram like this:

According to Ohm's law I=U/R

Then the resistance of the light bulb (with the current and voltage that I wrote) will be

R= 1/U, WhereR = 1 Ohm

And the power will be released P = I * U , that is, P=2.25 Vm

In a series circuit, especially with such a simple and undeniable example, it is clear that the current that runs through it from beginning to end is the same all the time. And if we now take two light bulbs and make sure that the current runs first through one and then through the other, then the same thing will happen again - the current will be the same in both the light bulb and the other. Although different in size. The current now experiences the resistance of two light bulbs, but each of them has the same resistance as it was, and remains the same, because it is determined solely by the physical properties of the light bulb itself. We calculate the new current again using Ohm's law.

It will turn out to be equal to I=U/R+R, that is, 0.75A, exactly half of the current that was at first.

In this case, the current has to overcome two resistances, it becomes smaller. As can be seen from the glow of the light bulbs - they are now burning at full intensity. And the total resistance of a chain of two light bulbs will be equal to the sum of their resistances. Knowing arithmetic, in a particular case you can use the action of multiplication: if N identical light bulbs are connected in series, then their total resistance will be equal to N multiplied by R, where R is the resistance of one light bulb. The logic is impeccable.

And we will continue our experiments. Now let's do something similar to what we did with light bulbs, but only on the left side of the circuit: add another galvanic element, exactly the same as the first. As you can see, now our total voltage has doubled, and the current has returned to 1.5 A, which is signaled by the light bulbs, which light up again at full power.

We conclude:

• When an electrical circuit is connected in series, the resistances and voltages of its elements are summed up, and the current on all elements remains unchanged.

It is easy to verify that this statement is true for both active components (galvanic cells) and passive ones (light bulbs, resistors).

That is, this means that the voltage measured across one resistor (it is called the voltage drop) can be safely summed up with the voltage measured across another resistor, and the total will be the same 3 V. And at each of the resistances it will be equal to half - then there is 1.5 V. And this is fair. Two galvanic cells produce their voltages, and two light bulbs consume them. Because in a voltage source, the energy of chemical processes is converted into electricity, which takes the form of voltage, and in light bulbs the same energy is converted from electrical into heat and light.

Let's return to the first circuit, connect another light bulb in it, but differently.

Now the voltage at the points connecting the two branches is the same as on the galvanic element - 1.5 V. But since the resistance of both bulbs is also the same as it was, the current through each of them will flow 1.5 A - "full glow" current.

The galvanic cell now supplies them with current at the same time, therefore, both of these currents flow out of it at once. That is, the total current from the voltage source will be 1.5 A + 1.5 A = 3.0 A.

What is the difference between this circuit and the circuit when the same light bulbs were connected in series? Only in the glow of light bulbs, that is, only in current.

Then the current was 0.75 A, but now it is immediately 3 A.

It turns out that if we compare it with the original circuit, then when the light bulbs were connected in series (scheme 2), there was more resistance to the current (which is why it decreased, and the light bulbs lost their luminosity), and parallel connection provides LESS resistance, although the resistance of the light bulbs remains unchanged. What's the matter?

But the fact is that we forget one interesting truth, that every sword is a double-edged sword.

When we say that a resistor resists current, we seem to forget that it still conducts current. And now that the light bulbs have been connected in parallel, their overall ability to conduct current rather than resist it has increased. Well, and, accordingly, a certain amount G, by analogy with resistance R and should be called conductivity. And it must be summed up in a parallel connection of conductors.

Well here she is

Ohm's law will then look like

I = U* G&

And in the case of a parallel connection, the current I will be equal to U*(G+G) = 2*U*G, which is exactly what we observe.

Replacement of circuit elements with a common equivalent element

Engineers often need to recognize currents and voltages in all parts of circuits. But real electrical circuits can be quite complex and branched and can contain many elements that actively consume electricity and are connected to each other in completely different combinations. It's called calculation electrical diagrams. It is done when designing the energy supply of houses, apartments, and organizations. In this case, it is very important what currents and voltages will act in the electrical circuit, if only in order to select appropriate wire sections, loads on the entire network or its parts, and so on. And how complicated they can be electronic circuits, containing thousands, or even millions of elements, I think everyone understands.

The very first thing that suggests itself is to use the knowledge of how voltage currents behave in such simple network connections as serial and parallel. They do this: instead of a serial connection found on the network of two or more active consumer devices (like our light bulbs), draw one, but so that its resistance is the same as both. Then the picture of currents and voltages in the rest of the circuit will not change. Similarly with parallel connections: instead of them, draw an element whose CONDUCTIVITY would be the same as both.

Now, if we redraw the circuit, replacing the serial and parallel connections with one element, we will get a circuit called an “equivalent equivalent circuit.”

This procedure can be continued until we are left with the simplest one - with which we illustrated Ohm’s law at the very beginning. Only instead of the light bulb there will be one resistance, which is called the equivalent load resistance.

This is the first task. It allows us to use Ohm's law to calculate the total current in the entire network, or the total load current.

This is a complete calculation of the electrical network.

Examples

Let the chain contain 9 active resistances. It could be light bulbs or something else.

A voltage of 60 V is applied to its input terminals.

The resistance values ​​for all elements are as follows:

Find all unknown currents and voltages.

It is necessary to follow the path of searching for parallel and serial sections of the network, calculating their equivalent resistances and gradually simplifying the circuit. We see that R 3, R 9 and R 6 are connected in series. Then their equivalent resistance R e 3, 6, 9 will be equal to their sum R e 3, 6, 9 = 1 + 4 + 1 Ohm = 6 Ohm.

Now we replace the parallel piece of resistance R 8 and R e 3, 6, 9, getting R e 8, 3, 6, 9. Only when connecting conductors in parallel will the conductivity have to be added.

Conductivity is measured in units called siemens, the reciprocal of ohms.

If we turn the fraction over, we get resistance R e 8, 3, 6, 9 = 2 Ohm

Exactly the same as in the first case, we combine resistances R 2, R e 8, 3, 6, 9 and R 5 connected in series, obtaining R e 2, 8, 3, 6, 9, 5 = 1 + 2 + 1 = 4 Ohm.

There are two steps left: obtain a resistance equivalent to two resistors for parallel connection of conductors R 7 and R e 2, 8, 3, 6, 9, 5.

It is equal to R e 7, 2, 8, 3, 6, 9, 5 = 1/(1/4+1/4)=1/(2/4)=4/2 = 2 Ohm

On last step sum up all the series-connected resistances R 1, R e 7, 2, 8, 3, 6, 9, 5 and R 4 and get a resistance equivalent to the resistance of the entire circuit R e and equal to the sum of these three resistances

R e = R 1 + R e 7, 2, 8, 3, 6, 9, 5 + R4 = 1 + 2 + 1 = 4 Ohm

Well, let’s remember in whose honor the unit of resistance we wrote in the last of these formulas was named, and use his law to calculate the total current in the entire circuit I

Now, moving in the opposite direction, towards increasing complexity of the network, we can obtain currents and voltages in all chains of our fairly simple circuit according to Ohm’s law.

This is how apartment power supply schemes are usually calculated, which consist of parallel and serial sections. Which, as a rule, is not suitable in electronics, because a lot of things work there differently, and everything is much more intricate. And such a circuit, for example, when you don’t understand whether the connection of conductors is parallel or serial, is calculated according to Kirchhoff’s laws.

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, parallel connection of devices is used in home electrical networks.

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

Electrical networks typically use serial parallel and mixed connections of 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.