Serial and parallel connection. Types of conductor connections
In electrical circuits, elements can be connected according to various schemes, including they have consistent and parallel connection.
Serial connection
With this connection, the conductors are connected to each other in series, that is, the beginning of one conductor will be connected to the end of the other. Main Feature of this connection is that all conductors belong to one wire, there are no branches. The same current will flow through each of the conductors. electricity. But the total voltage on the conductors will be equal to the combined voltages on each of them.
Consider a number of resistors connected in series. Since there are no branches, the amount of charge passing through one conductor will be equal to the amount of charge passing through the other conductor. The current strength on all conductors will be the same. This is the main feature of this connection.
This connection can be viewed differently. All resistors can be replaced with one equivalent resistor.
The current across the equivalent resistor will be the same as the total current flowing through all resistors. The equivalent total voltage will be the sum of the voltages across each resistor. This is the potential difference across the resistor.
If you use these rules and Ohm's law, which applies to each resistor, you can prove that the resistance of the equivalent common resistor will be equal to the sum of the resistances. The consequence of the first two rules will be the third rule.
Application
A serial connection is used when you need to purposefully turn on or off a device; the switch is connected to it in a series circuit. For example, an electric bell will only ring when it is connected in series with a source and a button. According to the first rule, if there is no electric current on at least one of the conductors, then there will be no electric current on the other conductors. And vice versa, if there is current on at least one conductor, then it will be on all other conductors. A pocket flashlight also works, which has a button, a battery and a light bulb. All these elements must be connected in series, since the flashlight needs to shine when the button is pressed.
Sometimes a serial connection does not achieve the desired goals. For example, in an apartment where there are many chandeliers, light bulbs and other devices, you should not connect all the lamps and devices in series, since you never need to turn on the lights in each of the rooms of the apartment at the same time. For this purpose, serial and parallel connections are considered separately, and a parallel type of circuit is used to connect lighting fixtures in the apartment.
Parallel connection
In this type of circuit, all conductors are connected in parallel to each other. All the beginnings of the conductors are connected to one point, and all the ends are also connected together. Let's consider a number of homogeneous conductors (resistors) connected along parallel circuit.
This type of connection is branched. Each branch contains one resistor. The electric current, having reached the branching point, is divided into each resistor and will be equal to the sum of the currents at all resistances. The voltage across all elements connected in parallel is the same.
All resistors can be replaced with one equivalent resistor. If you use Ohm's law, you can get an expression for resistance. If, with a series connection, the resistances were added, then with a parallel connection, the inverse values of them will be added, as written in the formula above.
Application
If we consider connections in domestic conditions, then in an apartment lighting lamps and chandeliers should be connected in parallel. If we connect them in series, then when one light bulb turns on, we turn on all the others. With a parallel connection, we can, by adding the corresponding switch to each of the branches, turn on the corresponding light bulb as desired. In this case, turning on one lamp in this way does not affect the other lamps.
All electric household devices in the apartment are connected in parallel to a network with a voltage of 220 V, and connected to the distribution panel. In other words, parallel connection is used when connection is required electrical devices independently of each other. Serial and parallel connections have their own characteristics. There are also mixed compounds.
Current work
The series and parallel connections discussed earlier were valid for voltage, resistance and current values being the fundamental ones. The work of the current is determined by the formula:
A = I x U x t, Where A– current work, t– flow time along the conductor.
To determine operation with a series connection circuit, it is necessary to replace the voltage in the original expression. We get:
A=I x (U1 + U2) x t
We open the brackets and find that in the entire diagram, the work is determined by the amount at each load.
We also consider a parallel connection circuit. We just change not the voltage, but the current. The result is:
A = A1+A2
Current power
When considering the formula for the power of a circuit section, it is again necessary to use the formula:
P=U x I
After similar reasoning, the result is that series and parallel connections can be determined by the following power formula:
P=P1 + P2
In other words, for any circuit, the total power is equal to the sum of all powers in the circuit. This can explain that it is not recommended to turn on several powerful electrical devices in an apartment at once, since the wiring may not withstand such power.
The influence of the connection diagram on the New Year's garland
After one lamp in a garland burns out, you can determine the type of connection diagram. If the circuit is sequential, then not a single light bulb will light up, since a burnt out light bulb breaks the common circuit. To find out which light bulb has burned out, you need to check everything. Next, replace faulty lamp, the garland will function.
When using a parallel connection circuit, the garland will continue to work even if one or more lamps have burned out, since the circuit is not completely broken, but only one small parallel section. To restore such a garland, it is enough to see which lamps are not lit and replace them.
Series and parallel connection for capacitors
With a series circuit, the following picture arises: charges from the positive pole of the power source go only to the outer plates of the outer capacitors. , located between them, transfer charge along the circuit. This explains the appearance of equal charges with different signs on all plates. Based on this, the charge of any capacitor connected in a series circuit can be expressed by the following formula:
q total = q1 = q2 = q3
To determine the voltage on any capacitor, you need the formula:
Where C is capacity. The total voltage is expressed by the same law that is suitable for resistances. Therefore, we obtain the capacity formula:
С= q/(U1 + U2 + U3)
To make this formula simpler, you can reverse the fractions and replace the ratio of the potential difference to the charge on the capacitor. As a result we get:
1/C= 1/C1 + 1/C2 + 1/C3
The parallel connection of capacitors is calculated a little differently.
The total charge is calculated as the sum of all charges accumulated on the plates of all capacitors. And the voltage value is also calculated according to general laws. In this regard, the formula for the total capacitance in a parallel connection circuit looks like this:
С= (q1 + q2 + q3)/U
This value is calculated as the sum of each device in the circuit:
C=C1 + C2 + C3
Mixed connection of conductors
IN electrical diagram sections of the circuit can have both serial and parallel connections, intertwined with each other. But all the laws discussed above for certain types of compounds are still valid and are used in stages.
First you need to mentally decompose the diagram into separate parts. For a better representation, it is drawn on paper. Let's look at our example using the diagram shown above.
It is most convenient to depict it starting from the points B And IN. They are placed at some distance from each other and from the edge of the sheet of paper. From the left side to the point B one wire is connected, and two wires go off to the right. Dot IN on the contrary, it has two branches on the left, and one wire goes off after the point.
Next you need to depict the space between the points. Along the upper conductor there are 3 resistances with conditional meanings 2, 3, 4. A current with index 5 will flow from below. The first 3 resistors are connected in series in the circuit, and the fifth resistor is connected in parallel.
The remaining two resistances (the first and sixth) are connected in series with the section we are considering B-C. Therefore, we supplement the diagram with 2 rectangles on the sides of the selected points.
Now we use the formula for calculating resistance:
- The first formula for the series connection.
- Next, for the parallel circuit.
- And finally for the sequential circuit.
In a similar way, you can decompose any complex circuit, including connections not only of conductors in the form of resistances, but also of capacitors. To learn how to calculate using different types schemes, you need to practice in practice by completing several tasks.
When solving problems, it is customary to transform the circuit so that it is as simple as possible. To do this, equivalent transformations are used. Equivalent are those transformations of a part of an electrical circuit circuit in which the currents and voltages in the non-transformed part remain unchanged.
There are four main types of conductor connections: series, parallel, mixed and bridge.
Serial connection
Serial connection- this is a connection in which the current strength throughout the entire circuit is the same. A striking example serial connection is old Christmas tree garland. There the light bulbs are connected in series, one after another. Now imagine one light bulb burns out, the circuit is broken and the rest of the light bulbs go out. The failure of one element leads to the shutdown of all the others, this is significant drawback serial connection.
When connected in series, the resistances of the elements are summed up.
Parallel connection
Parallel connection- this is a connection in which the voltage at the ends of the circuit section is the same. Parallel connection is the most common, mainly because all the elements are under the same voltage, the current is distributed differently and when one of the elements exits, all the others continue to work.
In a parallel connection, the equivalent resistance is found as:
In the case of two parallel connected resistors 
In the case of three resistors connected in parallel:
Mixed compound
Mixed compound– a connection, which is a collection of serial and parallel connections. To find the equivalent resistance, you need to “collapse” the circuit by alternately transforming parallel and serial sections of the circuit.
First, let's find the equivalent resistance for the parallel section of the circuit, and then add to it the remaining resistance R 3 . It should be understood that after the conversion, the equivalent resistance R 1 R 2 and resistor R 3 are connected in series.
So, that leaves the most interesting and most complex connection of conductors.
Bridge circuit
The bridge connection diagram is shown in the figure below.
In order to collapse the bridge circuit, one of the bridge triangles is replaced with an equivalent star.
And find the resistances R 1, R 2 and R 3.
When solving problems, it is customary to transform the circuit so that it is as simple as possible. To do this, equivalent transformations are used. Equivalent are those transformations of a part of an electrical circuit circuit in which the currents and voltages in the non-transformed part remain unchanged.
There are four main types of conductor connections: series, parallel, mixed and bridge.
Serial connection
Serial connection- this is a connection in which the current strength throughout the entire circuit is the same. A striking example of a series connection is an old Christmas tree garland. There the light bulbs are connected in series, one after another. Now imagine one light bulb burns out, the circuit is broken and the rest of the light bulbs go out. The failure of one element leads to the shutdown of all the others; this is a significant disadvantage of a serial connection.
When connected in series, the resistances of the elements are summed up.
Parallel connection
Parallel connection- this is a connection in which the voltage at the ends of the circuit section is the same. Parallel connection is the most common, mainly because all the elements are under the same voltage, the current is distributed differently and when one of the elements exits, all the others continue to work.
In a parallel connection, the equivalent resistance is found as:
In the case of two parallel connected resistors
In the case of three resistors connected in parallel:
Mixed compound
Mixed compound– a connection, which is a collection of serial and parallel connections. To find the equivalent resistance, you need to “collapse” the circuit by alternately transforming parallel and serial sections of the circuit.
First, let's find the equivalent resistance for the parallel section of the circuit, and then add to it the remaining resistance R 3 . It should be understood that after the conversion, the equivalent resistance R 1 R 2 and resistor R 3 are connected in series.
So, that leaves the most interesting and most complex connection of conductors.
Bridge circuit
The bridge connection diagram is shown in the figure below.
In order to collapse the bridge circuit, one of the bridge triangles is replaced with an equivalent star.
And find the resistances R 1, R 2 and R 3.
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 electrical circuit. When connecting conductors in parallel, two or more elements are connected by their contacts on both sides, respectively. Connecting them to general scheme carried out precisely by these nodal points.
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General form
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 familiarizing yourself with general principles for calculating the elements of complex chains, you can go to specific examples 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 task 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.
So comparatively in a 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.
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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 any number 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.
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Mixed scheme
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.
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 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 series. Obviously, the current in such a circuit will be the same at all its points.
Rice 1
How to determine the total resistance of a circuit if we already know all the resistances included in it in series? 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 finally have R = R1 + R2 + R3
Thus, we came to the conclusion that when resistances are connected in series, the total resistance of the entire circuit is equal to the sum of the resistances of the individual sections.
Let's check this conclusion for 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 in series (Fig. 2) and connect them to a current source whose EMF is 60 V (neglected).
Rice. 2. Example of series connection of three 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: 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 thus obtained 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 internal resistance current source.
Having now closed the key switch K, we can verify from the instruments 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 are included in another common point b. By then connecting points a and b with a current source, we obtain a closed electrical 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, since 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. Each of these currents will go along 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 connecting resistances in parallel, a branched circuit is obtained. Let's see what the relationship between the currents in the circuit we have compiled 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.
This means that before its branching (to point a) it is equal to the current strength after the branching of the circuit (after point b).
We will now turn on the ammeter in turn 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 second - I 2. By adding these two ammeter readings, we get the total current, in magnitude equal to current I until the branching (to point a).
Hence, 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 ratio has a large practical significance, 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. First, the voltmeter will show the voltage of the current source as it is connected, as can be seen in Fig. 3, directly to the terminals of the current source. Secondly, the voltmeter will show the voltage drops U1 and U2 across resistances R1 and R2, since it is connected to the beginning and end of each resistance.
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 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 we can write: U1 = I1R1 and U2 = I2R2, but since 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 second branch, how many times the resistance of the first branch is less (or greater) than the resistance of the second branches.
So we have come to the important 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 using 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 out in inverse 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 a smaller resistance (R1 = 10 Ohms), and a smaller part (R2 = 150 mA) went through a larger resistance (R 2 = 20 Ohm).
This branching of current in parallel branches is similar to the flow of liquid through pipes. Imagine pipe A, which at some point branches into two pipes B and C of different diameters (Fig. 4). Since the diameter of pipe B is greater than the diameter of pipes C, then through pipe B into the same time will pass more water than through pipe B, which offers more resistance to the flow of water.
Rice. 4
Let us now consider what the total resistance of an external circuit consisting of two parallel-connected resistances will be equal to.
Underneath this total resistance external circuit must be understood as a resistance that could replace 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 two 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 branch point), U is the voltage of the external circuit, R is the resistance of the external circuit, i.e. equivalent resistance.
Similarly, 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
Since in 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)
Now dividing both sides of the equality by U, we will finally have 1 / R = 1 / R1 + 1 / R2
Remembering that conductivity is the reciprocal of resistance, we can say that in the resulting formula 1/R is the conductivity of the external circuit; 1 / R1 conductivity of the first branch; 1/R2 is the conductivity of the second 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.
Hence, to determine the equivalent resistance of resistances connected in parallel, it is necessary to determine the conductivity of the circuit and take its 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 smallest of the resistances connected in parallel.
Considering the case of parallel connection of resistances, we took the simplest circuit, consisting of two branches. However, in practice there may be cases when the chain consists of three or more parallel branches. What to do in these cases?
It turns out that all the relationships we obtained remain valid for a circuit consisting of any number of parallel-connected resistances.
To see this, consider 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).
Rice. 5. Circuit with three resistances connected in parallel
Applying the formula 1 / R = 1 / R1 + 1 / R2 for this circuit, we can write 1 / R = 1 / R1 + 1 / R2 + 1 / R3 and, substituting known values, we get 1 / R = 1 / 10 + 1 /20 + 1/60
Let's add these fractions: 1/R = 10/60 = 1/6, i.e. the conductivity of the circuit is 1/R = 1/6 Therefore, equivalent resistance R = 6 Ohm.
Thus, equivalent resistance is less than the smallest of the resistances connected in parallel in the circuit, i.e. less than resistance R1.
Let's now see whether this resistance is really equivalent, that is, one that could replace 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 therefore 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 three parallel-connected resistances known to us, one equivalent resistance of 6 Ohms is connected.
I = U / R = 12 / 6 = 2 A
As we can see, the resistance R = 6 Ohm we found is indeed equivalent for this circuit.
You can also verify this using measuring instruments 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 the same.
In practice, there may also be parallel connections for which it is possible to calculate the equivalent resistance more simply, i.e., without first determining the conductivities, 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 transformed as follows: 1/R = (R2 + R1) / R1 R2 and, solving the equality with respect to R, obtain 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.