The conductor resistance is a series connection. Parallel and serial connection
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 using the following example. Let's take three constant resistances, the values of which are known (for example, R1 == 10 Ohms, R 2 = 20 Ohms and R 3 = 50 Ohms). Let's connect them 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 the internal resistance of the 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 current I1 in the first branch, and I 2 in the second. Adding these two ammeter readings, we get a total current equal in value 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 relationship, which is of great practical importance, is called branched chain law.
Let us now consider what the relationship between the currents in the branches will be.
Let's turn on the voltmeter between points a and b and see what it shows us. Firstly, the voltmeter will show the voltage of the current source 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 a branch, the less current will flow through it, and, conversely, the less resistance of a branch, the greater the 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 larger than the diameter of pipes C, more water will pass through pipe B at the same time than through pipe B, which offers greater 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 The total resistance of the 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 will be approximately the same in both cases.
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.
In electrical circuits, elements can be connected according to various circuits, including serial and parallel connections.
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. The main feature of this connection is that all conductors belong to one wire, there are no branches. The same electric current will flow through each of the conductors. 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 in a 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 electrical 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 it is necessary to connect 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 the 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 an electrical circuit, sections of a circuit can have both series and parallel connections, intertwined with each other. But all the laws discussed above for certain types of connections 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 comes off after the point.
Next you need to depict the space between the points. Along the upper conductor there are 3 resistances with conventional values 2, 3, 4. From below there will be a current with index 5. The first 3 resistances 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 a series connection.
- Next, for the parallel circuit.
- And finally for the sequential circuit.
In a similar way, any complex circuit can be decomposed into separate circuits, including connections of not only conductors in the form of resistances, but also capacitors. To learn to master calculation techniques for different types of schemes, you need to practice in practice by completing several tasks.
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 a limited number of sources and a fairly 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. In this case, electrical appliances can be compared to people in a round dance, and their hands holding each other are the wires connecting the 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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Series, parallel and mixed connections of resistors. A significant number of receivers included in the electrical circuit (electric lamps, electric heating devices, etc.) can be considered as some elements that have a certain resistance. This circumstance gives us the opportunity, when drawing up and studying electrical circuits, to replace specific receivers with resistors with certain resistances. There are the following methods resistor connections(receivers of electrical energy): serial, parallel and mixed.
Series connection of resistors.
For serial connection several resistors, the end of the first resistor is connected to the beginning of the second, the end of the second to the beginning of the third, etc. With this connection, all elements of the series circuit pass
the same current I.
The serial connection of receivers is illustrated in Fig. 25, a.
.Replacing the lamps with resistors with resistances R1, R2 and R3, we get the circuit shown in Fig. 25, b.
If we assume that Ro = 0 in the source, then for three series-connected resistors, according to Kirchhoff’s second law, we can write:
E = IR 1 + IR 2 + IR 3 = I(R 1 + R 2 + R 3) = IR eq (19)
Where R eq =R 1 + R 2 + R 3.
Consequently, the equivalent resistance of a series circuit is equal to the sum of the resistances of all series-connected resistors. Since the voltages in individual sections of the circuit are according to Ohm’s law: U 1 =IR 1 ; U 2 = IR 2, U 3 = IR 3 and in this case E = U, then for the circuit under consideration
U = U 1 + U 2 + U 3 (20)
Consequently, the voltage U at the source terminals is equal to the sum of the voltages at each of the series-connected resistors.
From these formulas it also follows that the voltages are distributed between series-connected resistors in proportion to their resistances:
U 1: U 2: U 3 = R 1: R 2: R 3 (21)
that is, the greater the resistance of any receiver in a series circuit, the greater the voltage applied to it.
If several, for example n, resistors with the same resistance R1 are connected in series, the equivalent resistance of the circuit Rek will be n times greater than the resistance R1, i.e. Rek = nR1. The voltage U1 on each resistor in this case is n times less than the total voltage U:
When receivers are connected in series, a change in the resistance of one of them immediately entails a change in the voltage at the other receivers connected to it. When the electrical circuit is turned off or broken, the current in one of the receivers and in the other receivers stops. Therefore, series connection of receivers is rarely used - only in the case when the voltage of the electrical energy source is greater than the rated voltage for which the consumer is designed. For example, the voltage in the electrical network from which subway cars are powered is 825 V, while the nominal voltage of the electric lamps used in these cars is 55 V. Therefore, in subway cars, electric lamps are switched on in series, 15 lamps in each circuit.
Parallel connection of resistors. In parallel connection several receivers, they are connected between two points of the electrical circuit, forming parallel branches (Fig. 26, a). Replacing
lamps with resistors with resistances R1, R2, R3, we get the circuit shown in Fig. 26, b.
When connected in parallel, the same voltage U is applied to all resistors. Therefore, according to Ohm’s law:
I 1 =U/R 1; I 2 =U/R 2 ; I 3 =U/R 3.
Current in the unbranched part of the circuit according to Kirchhoff’s first law I = I 1 +I 2 +I 3, or
I = U / R 1 + U / R 2 + U / R 3 = U (1/R 1 + 1/R 2 + 1/R 3) = U / R eq (23)
Therefore, the equivalent resistance of the circuit under consideration when three resistors are connected in parallel is determined by the formula
1/R eq = 1/R 1 + 1/R 2 + 1/R 3 (24)
By introducing into formula (24) instead of the values 1/R eq, 1/R 1, 1/R 2 and 1/R 3 the corresponding conductivities G eq, G 1, G 2 and G 3, we obtain: the equivalent conductance of a parallel circuit is equal to the sum of the conductances of parallel connected resistors:
G eq = G 1 + G 2 + G 3 (25)
Thus, as the number of resistors connected in parallel increases, the resulting conductivity of the electrical circuit increases, and the resulting resistance decreases.
From the above formulas it follows that currents are distributed between parallel branches in inverse proportion to their electrical resistance or directly proportional to their conductivity. For example, with three branches
I 1: I 2: I 3 = 1/R 1: 1/R 2: 1/R 3 = G 1 + G 2 + G 3 (26)
In this regard, there is a complete analogy between the distribution of currents along individual branches and the distribution of water flows through pipes.
The given formulas make it possible to determine the equivalent circuit resistance for various specific cases. For example, with two resistors connected in parallel, the resulting circuit resistance is
R eq =R 1 R 2 /(R 1 +R 2)
with three resistors connected in parallel
R eq =R 1 R 2 R 3 /(R 1 R 2 +R 2 R 3 +R 1 R 3)
When several, for example n, resistors with the same resistance R1 are connected in parallel, the resulting circuit resistance Rec will be n times less than the resistance R1, i.e.
R eq = R1/n(27)
The current I1 passing through each branch, in this case, will be n times less than the total current:
I1 = I/n (28)
When the receivers are connected in parallel, they are all under the same voltage, and the operating mode of each of them does not depend on the others. This means that the current passing through any of the receivers will not have a significant effect on the other receivers. Whenever any receiver is turned off or fails, the remaining receivers remain on.
valuable. Therefore, a parallel connection has significant advantages over a serial connection, as a result of which it is most widely used. In particular, electric lamps and motors designed to operate at a certain (rated) voltage are always connected in parallel.
On DC electric locomotives and some diesel locomotives, traction motors must be switched on at different voltages during speed control, so they switch from a series connection to a parallel connection during acceleration.
Mixed connection of resistors. Mixed compound This is a connection in which some of the resistors are connected in series, and some in parallel. For example, in the diagram of Fig. 27, and there are two series-connected resistors with resistances R1 and R2, a resistor with resistance R3 is connected in parallel with them, and a resistor with resistance R4 is connected in series with a group of resistors with resistances R1, R2 and R3.
The equivalent resistance of a circuit in a mixed connection is usually determined by the conversion method, in which a complex circuit is converted into a simple one in successive steps. For example, for the diagram in Fig. 27, and first determine the equivalent resistance R12 of series-connected resistors with resistances R1 and R2: R12 = R1 + R2. In this case, the diagram in Fig. 27, but is replaced by the equivalent circuit in Fig. 27, b. Then the equivalent resistance R123 of parallel-connected resistances and R3 are determined using the formula
R 123 = R 12 R 3 / (R 12 + R 3) = (R 1 + R 2) R 3 / (R 1 + R 2 + R 3).
In this case, the diagram in Fig. 27, b is replaced by the equivalent circuit of Fig. 27, v. After this, the equivalent resistance of the entire circuit is found by summing the resistance R123 and the resistance R4 connected in series with it:
R eq = R 123 + R 4 = (R 1 + R 2) R 3 / (R 1 + R 2 + R 3) + R 4
Series, parallel and mixed connections are widely used to change the resistance of starting rheostats when starting an electric power plant. p.s. DC.
Almost everyone who worked as an electrician had to solve the issue of parallel and series connection of circuit elements. Some solve the problems of parallel and series connection of conductors using the “poke” method; for many, a “fireproof” garland is an inexplicable but familiar axiom. However, all these and many other similar questions are easily solved by the method proposed at the very beginning of the 19th century by the German physicist Georg Ohm. The laws discovered by him are still in effect today, and almost everyone can understand them.
Basic electrical quantities of the circuit
In order to find out how a particular connection of conductors will affect the characteristics of the circuit, it is necessary to determine the quantities that characterize any electrical circuit. Here are the main ones:
Mutual dependence of electrical quantities
Now you need to decide, how all of the above quantities depend on one another. The rules of dependence are simple and come down to two basic formulas:
- I=U/R.
- P=I*U.
Here I is the current in the circuit in amperes, U is the voltage supplied to the circuit in volts, R is the resistance of the circuit in ohms, P is the electrical power of the circuit in watts.
Suppose we have a simple electrical circuit, consisting of a power source with voltage U and a conductor with resistance R (load).
Since the circuit is closed, current I flows through it. What value will it be? Based on the above formula 1, to calculate it we need to know the voltage developed by the power source and the load resistance. If we take, for example, a soldering iron with a coil resistance of 100 Ohms and connect it to a lighting socket with a voltage of 220 V, then the current through the soldering iron will be:
220 / 100 = 2.2 A.
What is the power of this soldering iron? Let's use formula 2:
2.2 * 220 = 484 W.
It turned out to be a good soldering iron, powerful, most likely two-handed. In the same way, by operating with these two formulas and transforming them, you can find out the current through power and voltage, voltage through current and resistance, etc. How much, for example, does a 60 W light bulb in your table lamp consume:
60 / 220 = 0.27 A or 270 mA.
Lamp filament resistance in operating mode:
220 / 0.27 = 815 Ohms.
Circuits with multiple conductors
All the cases discussed above are simple - one source, one load. But in practice there can be several loads, and they are also connected in different ways. There are three types of load connection:
- Parallel.
- Consistent.
- Mixed.
Parallel connection of conductors
The chandelier has 3 lamps, each 60 W. How much does a chandelier consume? That's right, 180 W. Let’s quickly calculate the current through the chandelier:
180 / 220 = 0.818 A.
And then her resistance:
220 / 0.818 = 269 Ohms.
Before this, we calculated the resistance of one lamp (815 Ohms) and the current through it (270 mA). The resistance of the chandelier turned out to be three times lower, and the current was three times higher. Now it’s time to look at the diagram of a three-arm lamp.
All lamps in it are connected in parallel and connected to the network. It turns out that when three lamps are connected in parallel, the total load resistance decreases by three times? In our case, yes, but it is private - all lamps have the same resistance and power. If each of the loads has its own resistance, then simply dividing by the number of loads is not enough to calculate the total value. But there is a way out of the situation - just use this formula: 
1/Rtotal = 1/R1 + 1/R2 + … 1/Rn.
For ease of use, the formula can be easily converted:
Rtot. = (R1*R2*… Rn) / (R1+R2+… Rn).
Here Rtotal. – the total resistance of the circuit when the load is connected in parallel. R1…Rn – resistance of each load.
Why the current increased when you connected three lamps in parallel instead of one is not difficult to understand - after all, it depends on the voltage (it remained unchanged) divided by the resistance (it decreased). Obviously, the power in a parallel connection will increase in proportion to the increase in current.
Serial connection
Now it’s time to find out how the parameters of the circuit will change if the conductors (in our case, lamps) are connected in series.
Calculating resistance when connecting conductors in series is extremely simple:
Rtot. = R1 + R2.
The same three sixty-watt lamps connected in series will already amount to 2445 Ohms (see calculations above). What are the consequences of increasing circuit resistance? According to formulas 1 and 2, it becomes quite clear that the power and current strength when connecting conductors in series will drop. But why are all the lamps dim now? This is one of the most interesting properties of series connection of conductors, which is very widely used. Let's take a look at a garland of three lamps familiar to us, but connected in series.
The total voltage applied to the entire circuit remained 220 V. But it was divided between each of the lamps in proportion to their resistance! Since we have lamps of the same power and resistance, the voltage is divided equally: U1 = U2 = U3 = U/3. That is, each of the lamps is now supplied with three times less voltage, which is why they glow so dimly. If you take more lamps, their brightness will drop even more. How to calculate the voltage drop across each lamp if they all have different resistances? To do this, the four formulas given above are sufficient. The calculation algorithm will be as follows:
- Measure the resistance of each lamp.
- Calculate the total resistance of the circuit.
- Based on the total voltage and resistance, calculate the current in the circuit.
- Based on the total current and resistance of the lamps, calculate the voltage drop across each of them.
Do you want to consolidate your acquired knowledge?? Solve a simple problem without looking at the answer at the end:
You have at your disposal 15 miniature light bulbs of the same type, designed for a voltage of 13.5 V. Is it possible to use them to make a Christmas tree garland that connects to a regular outlet, and if so, how?
Mixed compound
You, of course, easily figured out the parallel and serial connection of conductors. But what if you have something like this in front of you?
Mixed connection of conductors
How to determine the total resistance of a circuit? To do this, you will need to break the circuit into several sections. The above design is quite simple and there will be two sections - R1 and R2, R3. First, you calculate the total resistance of parallel-connected elements R2, R3 and find Rtot.23. Then calculate the total resistance of the entire circuit, consisting of R1 and Rtot.23 connected in series:
- Rtot.23 = (R2*R3) / (R2+R3).
- Rchains = R1 + Rtot.23.
The problem is solved, everything is very simple. Now the question is somewhat more complicated.
Complex mixed connection of resistances
How to be here? In the same way, you just need to show some imagination. Resistors R2, R4, R5 are connected in series. We calculate their total resistance:
Rtot.245 = R2+R4+R5.
Now we connect R3 in parallel to Rtot.245:
Rtot.2345 = (R3* Rtot.245) / (R3+ Rtot.245).
Rchains = R1+ Rtot.2345+R6.
That's it!
Answer to the problem about the Christmas tree garland
The lamps have an operating voltage of only 13.5 V, and the socket is 220 V, so they must be connected in series.
Since the lamps are of the same type, the network voltage will be divided equally between them and each lamp will have 220 / 15 = 14.6 V. The lamps are designed for a voltage of 13.5 V, so although such a garland will work, it will burn out very quickly. To realize your idea, you will need at least 220 / 13.5 = 17, and preferably 18-19 light bulbs.