Serial, parallel and mixed connection 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 electrical energy): serial, parallel and mixed.
Series connection of resistors.
At 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, 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 circuit under consideration
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. than more resistance 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 turned off or disconnected electrical circuit in one of the receivers and in the other receivers, the current stops. 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 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 the subway cars, electric lamps are switched on in series with 15 lamps in each circuit.
Parallel connection 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 resistance of the circuit Rek 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 electric locomotives direct current and some diesel locomotives, traction motors in the process of regulating the speed of movement must be switched on for various voltages, therefore, during acceleration, they switch from a serial connection to a parallel one.
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.
A parallel connection of resistances is such a connection when the beginnings of the resistances are connected into one common point, and the ends - in another.
The parallel connection of resistances is characterized by the following properties:
The voltages at the terminals of all resistances are the same:
U 1 \u003d U 2 \u003d U 3 \u003d U;
The conductance of all resistors connected in parallel is equal to the sum of the conductivities of the individual resistances:
1 / R \u003d 1 / R 1 + 1 / R 2 + 1 / R 3 \u003d R 1 R 2 + R 1 R 3 + R 2 R 3 / R 1 R 2 R 3,
where R - equivalent (resultant) resistance of three resistances (in this case R 1 , R 2 and R 3 ) .
To obtain the resistance of such a circuit, it is necessary to turn the fraction that determines the value of its conductivity. Therefore, the resistance of the parallel branching of the three resistors is:
R \u003d R 1 R 2 R 3 / R 1 R 2 + R 2 R 3 + R 1 R 3.
Equivalent resistance is such a resistance that can replace several resistances (connected in parallel or in series) without changing the magnitude of the current in the circuit.
To find the equivalent resistance in parallel connection, it is necessary to add the conductivities of all individual sections, i.e. find the total conductivity. The reciprocal of the total conductivity is the total resistance.
With a parallel connection, the equivalent conductivity is equal to the sum of the conductivities of the individual branches, therefore, the equivalent resistance in this case is always less than the smallest of the parallel-connected resistances.
In practice, there may be cases where the chain consists of more than three parallel branches. All obtained relationships remain valid for circuits consisting of any number of resistors connected in parallel.
Find the equivalent resistance of two resistors connected in parallel R1 and R2 (see fig.). The conductivity of the first branch is 1/R1 , conductivity of the second branch - 1/R2 . Total conductivity:
1/R = 1/R 1 + 1/R 2 .
Let's get to the common denominator:
1 / R \u003d R 2 + R 1 / R 1 R 2,
hence the equivalent resistance
R \u003d R 1 R 2 / R 1 + R 2.
This formula is used to calculate the total resistance of a circuit consisting of two resistances connected in parallel.
Thus, the equivalent resistance of two resistors connected in parallel is equal to the product of these resistances divided by their sum.
When connected in parallel n equal resistance R1 their equivalent resistance will be in n times less, i.e.
R \u003d R 1 / n.
In the circuit shown in the last figure, five resistances are included. R1 30 ohms each. Therefore, the total resistance R will
R \u003d R 1 / 5 \u003d 30/5 \u003d 6 ohms.
We can say that the sum of the currents approaching the nodal point A (in the first figure) is equal to the sum of the currents departing from it:
I \u003d I 1 + I 2 + I 3.
Consider how the current branching occurs in circuits with resistances R1 and R2 (second picture). Since the voltage at the terminals of these resistances is the same, then
U = I 1 R 1 and U = I 2 R 2 .
The left parts of these equalities are the same, therefore, the right parts are also equal:
I 1 R 1 \u003d I 2 R 2,
or
I 1 /I 2 \u003d R 2 /R 1,
Those. current with a parallel connection of resistances branches in inverse proportion to the resistances of the branches (or directly proportional to their conductivities). The greater the resistance of the branch, the less current in it, and vice versa.
Thus, from several identical resistors, you can get a common resistor with more power dissipation.
When unequal resistors are connected in parallel, the highest power is released in the highest resistance resistor.
Example 1. There are two resistors connected in parallel. Resistance R 1 \u003d 25 Ohm, and R 2 \u003d 50 ohm. Determine the total resistance of the circuit Rtot.
Solution. R total \u003d R 1 R 2 / R 1 + R 2 \u003d 25. 50 / 25 + 50 ≈ 16.6 ohms.
Example 2. There are three lamps in a tube amplifier, the filaments of which are connected in parallel. Filament current of the first lamp I 1 \u003d 1 ampere, second I 2 \u003d 1.5 amp and third I 3 = 2.5 amps. Define total current amplifier lamp circuits I common .
Solution. I total \u003d I 1 + I 2 + I 3 \u003d 1 + 1, 5 + 2, 5 = 5 amps.
Parallel connection of resistors is often found in radio equipment. Two or more resistors are connected in parallel in cases where the current in the circuit is too high and can cause excessive heating of the resistor.
An example of a parallel connection of consumers of electrical energy is the inclusion of electric lamps of a conventional lighting network, which are connected in parallel. The advantage of parallel connection of consumers is that turning off one of them does not affect the operation of others.
1. When connected in series conductors
1. The current strength in all conductors is the same:
I 1 = I 2 = I
2. General voltage U on both conductors is equal to the sum of the voltages U 1 and U 2 on each conductor:
U = U 1 + U 2
3. According to Ohm's law, voltage U 1 and U 2 on conductors are equal U 1 = IR 1 , U 2 = IR 2 a total voltage U = IR where R – electrical resistance the whole chain, then IR= IR 1 + IR 2. It follows from this
R= R 1 + R 2
When connected in series, the total resistance of the circuit is equal to the sum of the resistances of the individual conductors.
This result is valid for any number of series-connected conductors.
2. When connected in parallel conductors
1. Voltage U 1 and U 2 are the same on both conductors.
U 1 = U 2 = U
2. sum of currents I 1 + I 2 , flowing through both conductors is equal to the current in an unbranched circuit:
I = I 1 + I 2
This result follows from the fact that at the branching points of the currents (nodes A and B) charges cannot accumulate in the DC circuit. For example, to node A in time Δ t leaking charge IΔ t, and the charge flows away from the node in the same time I 1 Δ t + I 2Δ t. Consequently, I = I 1 + I 2 .
3. Writing based on Ohm's law
where R is the electrical resistance of the entire circuit, we get
With a parallel connection of conductors, the reciprocal of the total resistance of the circuit is equal to the sum of the reciprocals of the resistances of the parallel-connected conductors.
This result is valid for any number of conductors connected in parallel.
Formulas for series and parallel connection of conductors allow in many cases to calculate the resistance of a complex circuit consisting of many resistors. The figure shows an example of such a complex circuit and shows the sequence of calculations. The resistances of all conductors are specified in ohms (Ohm).
In practice, one current source in the circuit is not enough, and then the current sources are also interconnected to power the circuit. The connection of sources in the battery can be serial and parallel.
With a series connection, two adjacent sources are connected by opposite poles.
That is, for serial connection of batteries, to the ″plus″ electrical circuit connect the positive terminal of the first battery. The positive terminal of the second battery is connected to its negative terminal, etc. The negative terminal of the last battery is connected to the "minus" of the electrical circuit.
The battery resulting from a serial connection has the same capacity as that of a single battery, and the voltage of such a battery is equal to the sum of the voltages of the batteries included in it. Those. if the batteries have the same voltage, then the battery voltage is equal to the voltage of one battery multiplied by the number of batteries in the battery.
1. The EMF of the battery is equal to the sum of the EMF of individual sourcesε= ε 1 + ε 2 + ε 3
2 . The total resistance of the battery of sources is equal to the sum of the internal resistances of the individual sources r batteries = r 1 + r 2 + r 3
If n identical sources are connected to the battery, then the battery EMF ε= nε 1, and the battery resistance r = nr 1
3.
When connected in parallel, all positive and all negative poles of two or moren sources.
That is, when connected in parallel, the batteries are connected so that the positive terminals of all batteries are connected to one point of the electrical circuit (″plus″), and the negative terminals of all batteries are connected to another point in the circuit (″minus″).
Connect in parallel only sources With the same EMF. The battery resulting from a parallel connection has the same voltage as that of a single battery, and the capacity of such a battery is equal to the sum of the capacities of the batteries included in it. Those. if the batteries have the same capacity, then the capacity of the battery is equal to the capacity of one battery multiplied by the number of batteries in the battery.
1. The EMF of a battery of identical sources is equal to the EMF of one source.ε= ε 1 = ε 2 = ε 3
2.
The battery resistance is less than the resistance of a single source r batteries = r 1 /n
3.
The current strength in such a circuit according to Ohm's law
The electrical energy stored in the battery is equal to the sum of the energies of the individual batteries (the product of the energies of the individual batteries, if the batteries are the same), regardless of whether the batteries are connected in parallel or in series.
The internal resistance of batteries manufactured using the same technology is approximately inversely proportional to the capacity of the battery. Therefore, since with a parallel connection, the capacity of the battery is equal to the sum of the capacities of the batteries included in it, that is, it increases, then internal resistance decreases.
