The current in the electrical circuit passes through the conductors from the voltage source to the load, that is, to lamps, appliances. In most cases, copper wires are used as conductors. A circuit can have several elements with different resistances. In the instrument circuit, conductors can be connected in parallel or in series, and there can also be mixed types.
A circuit element with resistance is called a resistor, the voltage of this element is the potential difference between the ends of the resistor. Parallel and series electrical connection of conductors is characterized by a single principle of operation, according to which the current flows from plus to minus, respectively, the potential decreases. On wiring diagrams, the wiring resistance is taken as 0, since it is negligible.
Parallel connection assumes that the elements of the circuit are connected to the source in parallel and are switched on at the same time. Serial connection means that the resistance conductors are connected in strict sequence one after the other.
When calculating, the idealization method is used, which greatly simplifies understanding. In fact, in electrical circuits, the potential gradually decreases in the process of moving through the wiring and elements that are included in a parallel or series connection.
Serial connection of conductors
The serial connection scheme implies that they are switched on in a certain sequence, one after the other. Moreover, the current strength in all of them is equal. These elements create a total voltage on the site. Charges do not accumulate in the nodes of the electrical circuit, since otherwise a change in voltage and current would be observed. With a constant voltage, the current is determined by the value of the resistance of the circuit, therefore, in a series circuit, the resistance changes if one load changes.
The disadvantage of such a scheme is the fact that in the event of failure of one element, the rest also lose the ability to function, since the circuit is broken. An example is a garland that does not work if one light bulb burns out. This is a key difference from a parallel connection, where the elements can function separately.
The series circuit assumes that, due to the single-level connection of conductors, their resistance is equal at any point in the network. The total resistance is equal to the sum of the voltage reduction of the individual elements of the network.
With this type of connection, the beginning of one conductor is connected to the end of another. The key feature of the connection is that all conductors are on the same wire without branches, and one electric current flows through each of them. However, the total voltage is equal to the sum of the voltages on each. You can also consider the connection from a different point of view - all conductors are replaced by one equivalent resistor, and the current on it is the same as the total current that passes through all resistors. The equivalent total voltage is the sum of the voltage values across each resistor. This is the potential difference across the resistor.
Using a serial connection is useful when you want to specifically turn on and off a specific device. For example, an electric bell can only ring when there is a connection to a voltage source and a button. The first rule says that if there is no current on at least one of the elements of the circuit, then it will not be on the rest. Accordingly, if there is current in one conductor, it is in the others. Another example would be a battery-powered flashlight, which only shines when there is a battery, a working bulb, and a pressed button.
In some cases, a serial scheme is not practical. In an apartment where the lighting system consists of many lamps, sconces, chandeliers, you should not organize a scheme of this type, since there is no need to turn the lights on and off in all rooms at the same time. For this purpose, it is better to use a parallel connection in order to be able to turn on the light in individual rooms.
Parallel connection of conductors
In a parallel circuit, conductors are a set of resistors, one end of which is assembled into one node, and the other ends into a second node. It is assumed that the voltage in the parallel type of connection is the same in all parts of the circuit. Parallel sections of the electrical circuit are called branches and pass between two connecting nodes, they have the same voltage. This voltage is equal to the value on each conductor. The sum of the reciprocal resistances of the branches is also the inverse of the resistance of a separate section of the parallel circuit circuit.
With parallel and series connections, the system for calculating the resistances of individual conductors is different. In the case of a parallel circuit, the current flows through the branches, which increases the conductivity of the circuit and reduces the total resistance. When several resistors with similar values \u200b\u200bare connected in parallel, the total resistance of such an electrical circuit will be less than one resistor a number of times equal to the number.
Each branch has one resistor, and the electric current, when it reaches the branching point, is divided and diverges to each resistor, its final value is equal to the sum of the currents on all resistances. All resistors are replaced with one equivalent resistor. Applying Ohm's law, the value of the resistance becomes clear - in a parallel circuit, the values reciprocal of the resistances on the resistors are summed up.
With this circuit, the current value is inversely proportional to the resistance value. The currents in the resistors are not interconnected, so if one of them is turned off, this will in no way affect the others. For this reason, such a scheme is used in many devices.
Considering the possibilities of using a parallel circuit in everyday life, it is advisable to note the lighting system of the apartment. All lamps and chandeliers must be connected in parallel, in which case turning one of them on and off does not affect the operation of the other lamps. Thus, by adding a switch for each light bulb to a branch of the circuit, the corresponding lamp can be turned on and off as needed. All other lamps work independently.
All electrical appliances are connected in parallel to a 220 V power grid, then they are connected to. That is, all devices are connected regardless of the connection of other devices.
Laws of series and parallel connection of conductors
For a detailed understanding in practice of both types of compounds, we present formulas that explain the laws of these types of compounds. The power calculation for parallel and series connection type is different.
In a series circuit, there is the same current strength in all conductors:
According to Ohm's law, these types of conductor connections are explained differently in different cases. So, in the case of a series circuit, the voltages are equal to each other:
U1 = IR1, U2 = IR2.
In addition, the total voltage is equal to the sum of the voltages of individual conductors:
U = U1 + U2 = I(R1 + R2) = IR.
The total resistance of the electrical circuit is calculated as the sum of the active resistances of all conductors, regardless of their number.
In the case of a parallel circuit, the total voltage of the circuit is similar to the voltage of the individual elements:
And the total strength of the electric current is calculated as the sum of the currents that are available in all conductors located in parallel:
To ensure the maximum efficiency of electrical networks, it is necessary to understand the essence of both types of connections and apply them appropriately, using the laws and calculating the rationality of practical implementation.
Mixed connection of conductors
Series and parallel resistance connections can be combined in one electrical circuit if necessary. For example, it is allowed to connect parallel resistors in series or in a group, this type is considered combined or mixed.
In such a case, the total resistance is calculated by taking the sum of the values for the parallel connection in the system and for the series connection. First you need to calculate the equivalent resistances of the resistors in the series circuit, and then the elements of the parallel circuit. A serial connection is considered a priority, and circuits of this combined type are often used in household appliances and appliances.
So, considering the types of connections of conductors in electrical circuits and based on the laws of their functioning, one can fully understand the essence of the organization of circuits of most household electrical appliances. With parallel and series connections, the calculation of resistance and current strength indicators is different. Knowing the principles of calculation and formulas, you can competently use each type of circuit organization to connect elements in the best way and with maximum efficiency.
Series, parallel and mixed connection of resistors. A significant number of receivers included in the electrical circuit (electric lamps, electric heaters, 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 ways resistor connections(receivers of electrical energy): serial, parallel and mixed.
Series connection of resistors.
When connected in series 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, a
the same current I.
Serial connection of receivers explains fig. 25 a.
.Replacing the lamps with resistors with resistances R1, R2 and R3, we obtain the circuit shown in fig. 25, b.
If we assume that Ro = 0 in the source, then for three series-connected resistors, according to the second Kirchhoff law, we can write:
E \u003d IR 1 + IR 2 + IR 3 \u003d I (R 1 + R 2 + R 3) \u003d IR eq (19)
Where R eq =R1 + R2 + R3.
Therefore, 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 according to Ohm's law: U 1 =IR 1; U 2 \u003d IR 2, U 3 \u003d IR h and in this case E \u003d U, then for the considered circuit
U = U 1 + U 2 + U 3 (20)
Therefore, the voltage U at the source terminals is equal to the sum of the voltages across each of the resistors connected in series.
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)
i.e., 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 Rec will be n times greater than the resistance R1, i.e. Rec = nR1. The voltage U1 across 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 voltage on the other receivers connected to it. When the electrical circuit is turned off or broken, the current stops in one of the receivers and in the other receivers. Therefore, serial connection of receivers is rarely used - only 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 the 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 with 15 lamps in each circuit.
Parallel connection of resistors. When connected in parallel several receivers, they are switched on between two points of the electrical circuit, forming parallel branches (Fig. 26, a). Replacing
lamp 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 \u003d U / R 3.
The current in the unbranched part of the circuit according to the first Kirchhoff law I \u003d I 1 +I 2 +I 3, or
I \u003d U / R 1 + U / R 2 + U / R 3 \u003d U (1 / R 1 + 1 / R 2 + 1 / R 3) \u003d 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/R1 + 1/R2 + 1/R3 (24)
Introducing into formula (24) instead of the values 1/R eq, 1/R 1 , 1/R 2 and 1/R 3 the corresponding conductivity G eq, G 1 , G 2 and G 3 , we get: the equivalent conductance of a parallel circuit is equal to the sum of the conductances of the resistors connected in parallel:
G eq = G 1 + G 2 + G 3 (25)
Thus, with an increase in the number of resistors connected in parallel, the resulting conductivity of the electrical circuit increases, and the resulting resistance decreases.
It follows from the above formulas that the currents are distributed between the parallel branches in inverse proportion to their electrical resistances or in direct proportion to their conductivities. For example, with three branches
I 1: I 2: I 3 = 1/R 1: 1/R 2: 1/R 3 = G 1 + G 2 + G 3 (26)
In this regard, there is a complete analogy between the distribution of currents in individual branches and the distribution of water flows through pipes.
The above 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
R eq \u003d R 1 R 2 / (R 1 + R 2)
with three resistors connected in parallel
R eq \u003d 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 Reck 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 receivers are connected in parallel, they are all under the same voltage, and the mode of operation of each of them does not depend on the others. This means that the current flowing through any of the receivers will not significantly affect the other receivers. With any shutdown or failure of any receiver, the remaining receivers remain on.
chennymi. Therefore, a parallel connection has significant advantages over a serial connection, as a result of which it has become the most widespread. 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 in the process of speed control must be switched on for different voltages, so they switch from serial to parallel connection during acceleration.
Mixed connection of resistors. mixed connection a connection is called in which part of the resistors is connected in series, and part in parallel. For example, in the diagram of Fig. 27, but there are two resistors connected in series 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 stages. For example, for the circuit 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 scheme of Fig. 27, but is replaced by the equivalent circuit of fig. 27, b. Then, the equivalent resistance R123 of the resistors connected in parallel and R3 is determined by the formula
R 123 \u003d R 12 R 3 / (R 12 + R 3) \u003d (R 1 + R 2) R 3 / (R 1 + R 2 + R 3).
In this case, the scheme of Fig. 27, b is replaced by the equivalent circuit of fig. 27, c. After that, 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 during start-up e. p.s. direct current.
Let's take three constant resistances R1, R2 and R3 and include them in the circuit so that the end of the first resistance R1 is connected to the beginning of the second resistance R 2, the end of the second - to the beginning of the third R 3, and to the beginning of the first resistance and to the end of the third we bring conductors from a current source (Fig. 1).
Such a 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 the 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 get IR = I(R1 + R2 + R3) .
Now dividing both sides of the equality by I , we finally have R = R1 + R2 + R3
Thus, we came to the conclusion that when the 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 on 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 to a current source, the EMF of which is 60 V (neglect).
Rice. 2. An example of a series connection of three resistances
Let's calculate what readings the devices should give, 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 ohms.
Let's find the current in the circuit: 60/80 \u003d 0.75 A
Knowing the current in the circuit and the resistance of its sections, we determine the voltage drop in each section of the circuit U 1 = 0.75x10 = 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 obtained in such a way that U \u003d 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.
By closing the key switch K, we can verify by 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 - in another common point b. Connecting then points a and b with a current source, we get a closed electrical circuit. Such a connection of resistances is called a parallel connection.
Fig 3. Parallel connection of resistances
Let's trace the current flow in this circuit. From the positive pole of the current source through the connecting conductor, the current will reach point a. At point a, it branches, 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 as I1 and I 2, respectively. Each of these currents will go along its 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 the resistances are connected in parallel, a branched circuit is obtained. Let's see what will be the ratio between the currents in the circuit we have compiled.
We turn on the ammeter between the positive pole of the current source (+) and point a and note its readings. Having then turned on the ammeter (shown in the dotted line in the figure) in the wire connecting point b with 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 (up 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. Adding these two ammeter readings, we get a total current equal in magnitude to the current I to a branch (to point a).
Hence, the strength of the current flowing to the branch point is equal to the sum of the strengths of the currents flowing from this point. I = I1 + I2 Expressing this in a formula, we get
This ratio, which is of great practical importance, is called branched chain law.
Let us now consider what will be the ratio between the currents in the branches.
Let's turn on a voltmeter between points a and b and see what it will show us. First, the voltmeter will show the voltage of the current source, as it is connected, as can be seen from fig. 3 directly to the current source terminals. Secondly, the voltmeter will show the voltage drops U1 and U2 across the resistors R1 and R2 as it is connected to the start and end of each resistance.
Therefore, when the 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 the resistance R1, U2 - voltage drop across the resistance R2. Recall that the voltage drop in 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 \u003d IR.
Therefore, for each branch, you 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 \u003d U2 / U1 i.e. the current in the first branch will be so 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.
Thus, we have come to the important conclusion that when the resistances are connected in parallel, the total current of the circuit branches into currents 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 more current will flow through this branch.
We will verify the correctness of this dependence in the following example. Let's assemble a circuit consisting of two parallel-connected resistances R1 and R 2 connected to a current source. Let R1 = 10 ohms, R2 = 20 ohms and U = 3 V.
Let us first calculate what the ammeter included in each branch will show us:
I1 = U / R1 = 3 / 10 = 0.3 A = 300 mA
I 2 \u003d U / R 2 \u003d 3 / 20 \u003d 0.15 A \u003d 150 mA
Total circuit current I = I1 + I2 = 300 + 150 = 450 mA
Our calculation confirms that when the resistances are connected in parallel, the current in the circuit branches inversely proportional to the resistances.
Indeed, R1 == 10 ohms is half R 2 = 20 ohms, while I1 = 300 mA is twice as much as I2 = 150 mA. The total current in the circuit I \u003d 450 mA branched into two parts so that most of it (I1 \u003d 300 mA) went through a smaller resistance (R1 \u003d 10 Ohm), and a smaller part (R2 \u003d 150 mA) went through a larger resistance (R 2 = 20 ohms).
Such branching of the current in parallel branches is similar to the flow of liquid through pipes. Imagine pipe A, which in some place 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 pipe C, which has more resistance to the flow of water.
Rice. 4
Let us now consider what the total resistance of an external circuit consisting of two resistors connected in parallel will be equal to.
Underneath it The total resistance of the external circuit should be understood as such a resistance that could replace both resistances connected in parallel at a given circuit voltage, without changing the current until the branching. This kind of resistance is called equivalent resistance.
Let's go back to the circuit shown in Fig. 3, and see what the equivalent resistance of two resistors connected in parallel will be. Applying Ohm's law to this circuit, we can write: I \u003d U / R, where I is the current in the external circuit (up to the branching point), U is the voltage of the external circuit, R is the resistance of the external circuit, i.e. equivalent resistance.
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 the branches; R1 and R2 - branch resistances.
Branched chain law: I = I1 + I2
Substituting the values of the currents, we get U / R = U1 / R1 + U2 / R2
Since with a parallel connection U \u003d U1 \u003d U2, we can write U / R \u003d U / R1 + U / R2
Taking U on the right side of the equality out of brackets, we get U / R = U (1 / R1 + 1 / R2 )
Dividing now both parts of the equality by U , we 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 - 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 the resistances connected in parallel, it is necessary to determine the conductivity of the circuit and take the value of its reciprocal.
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 a 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 have obtained remain valid for a circuit consisting of any number of resistors connected in parallel.
To see this, consider the following example.
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 resistors 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 the known values, we get 1 / R = 1 / 10 + 1 / 20 + 1 / 60
Let's add this fraction: 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 the resistance R1.
Let's now see if this resistance is really equivalent, that is, one that could replace the resistances of 10, 20 and 60 ohms connected in parallel, without changing the current strength before the circuit branching.
Let's 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 \u003d 12 / 20 \u003d 1.6 A I 3 \u003d U / R1 \u003d 12 / 60 \u003d 0.2 A
We obtain the total current in the circuit using the formula I \u003d I1 + I2 + I3 \u003d 1.2 + 0.6 + 0.2 \u003d 2 A.
Let's check according to the formula of Ohm's law, whether a current of 2 A will turn out in the circuit if instead of three parallel resistances known to us, one equivalent resistance of 6 Ohms is included.
I \u003d U / R \u003d 12 / 6 \u003d 2 A
As you can see, the resistance R = 6 Ohm we found is indeed equivalent for this circuit.
This can also be verified on measuring instruments, if we assemble a circuit with the resistances we have taken, 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 readings of the ammeter in both cases will be approximately the same.
In practice, there may also be parallel connections, for which it is easier to calculate the equivalent resistance, i.e., without first determining the conductivities, immediately find the resistance.
For example, if two resistances R1 and R2 are connected in parallel, then the formula 1 / R \u003d 1 / R1 + 1 / R2 can be converted as follows: 1 / R \u003d (R2 + R1) / R1 R2 and, solving equality with respect to R, get R \u003d 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.
Consistent such a connection of resistors is called when the end of one conductor is connected to the beginning of another, etc. (Fig. 1). With a series connection, the current strength in any part of the electrical circuit is the same. This is because charges cannot accumulate at the nodes of the chain. Their accumulation would lead to a change in the electric field strength and, consequently, to a change in the current strength. That's why
Ammeter A measures the current in the circuit and has a low internal resistance (R A 0).
The included voltmeters V 1 and V 2 measure the voltage U 1 and U 2 across the resistances R 1 and R 2 . The voltmeter V measures the voltage U supplied to the M and N terminals. Voltmeters show that when connected in series, the voltage U is equal to the sum of the voltages in the individual sections of the circuit:
Applying Ohm's law for each section of the circuit, we get:
where R is the total resistance of the series connected circuit. Substituting U, U 1 , U 2 into formula (1), we have
The resistance of a circuit consisting of n resistors connected in series is equal to the sum of the resistances of these resistors:
If the resistances of the individual resistors are equal to each other, i.e. R 1 \u003d R 2 \u003d ... \u003d R n, then the total resistance of these resistors when connected in series is n times the resistance of one resistor: R \u003d nR 1.
When resistors are connected in series, the relation is true
those. The voltages across resistors are directly proportional to the resistances.
Parallel such a connection of resistors is called when one end of all resistors is connected to one node, the other ends to another node (Fig. 2). A node is a point in a branched circuit at which more than two conductors converge. When resistors are connected in parallel, a voltmeter is connected to points M and N. It shows that the voltages in individual sections of the circuit with resistances R 1 and R 2 are equal. This is explained by the fact that the work of the forces of a stationary electric field does not depend on the shape of the trajectory:
The ammeter shows that the current I in the unbranched part of the circuit is equal to the sum of the currents I 1 and I 2 in parallel-connected conductors R 1 and R 2:
This follows from the law of conservation of electric charge. We apply Ohm's law for individual sections of the circuit and the entire circuit with a total resistance R:
Substituting I, I 1 and I 2 into formula (2), we get.
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 the connection of each device to a separate source is feasible, mainly in laboratory conditions. In life, one has to deal with a limited number of sources and a rather large number of current consumers. Therefore, they create connection systems that allow loading one source with a large number of consumers. At the same time, systems can be arbitrarily complex and branched, but they are based on only two types of connection: serial and parallel connection of conductors. Each type has its own characteristics, pros and cons. Let's consider them both.
Serial connection of conductors
Serial connection of conductors is the inclusion of several devices in an electrical circuit in series, one after another. Electrical appliances in this case can be compared with 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 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 serial connection is the ease of assembly, and the disadvantage is that if one element fails, then the current will disappear in the entire circuit. In such a situation, a non-working element will be like a key in the off position. An example from life of the inconvenience of such a connection will surely be remembered by all older people who decorated Christmas trees with garlands of light bulbs.
If at least one light bulb failed in such a garland, you had to sort through them all until you find the one that burned out. In modern garlands, this problem is solved. They use special diode bulbs, in which, when burned out, the contacts are fused together, and the current continues to flow unhindered.
Parallel connection of conductors
With a parallel connection of conductors, all elements of the circuit are connected to the same pair of points, you 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 into all loads, depending on the resistance of each of them.
A parallel connection can be compared to the flow of a river, on the way of which a small hill has arisen. In this case, water goes around the hill from two sides, and then again merges into one stream. It turns out an island in the middle of the river. So a parallel connection is two separate channels around the island. And points A and B are the places where the common riverbed is disconnected and reconnected.
The voltage in each individual branch will be equal to the total voltage in the circuit. The total circuit current will be the sum of the currents of all individual branches. But the total resistance of the circuit when connected in parallel will be less than the current resistance on each of the branches. This is because the total cross section of the conductor between points A and B, as it were, increases due to an increase in the number of loads connected in parallel. Therefore, the total resistance decreases. 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 - current strength, U - voltage, R - resistance, 1,2, ..., n - numbers of elements included in the circuit.
A huge plus of a parallel connection is that when one of the elements is turned off, the circuit continues to function further. All other elements continue to work. The downside is that all devices must be designed for the same voltage. It is in a parallel way that 220 V network sockets are installed in apartments. Such a connection allows you to include various devices in 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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