The formula for the total resistance in a series connection. Electrical circuit with series connection of elements
Individual conductors of an electrical circuit can be connected to each other in series, in parallel and mixed. In this case, the series and parallel connection of conductors are the main types of connections, and the mixed connection is their combination.
A series connection of conductors is such a connection when the end of the first conductor is connected to the beginning of the second, the end of the second conductor is connected to the beginning of the third, and so on (Figure 1).
Figure 1. Scheme of serial connection of conductors
The total resistance of a circuit consisting of several series-connected conductors is equal to the sum of the resistances of the individual conductors:
r = r 1 + r 2 + r 3 + … + rn.
Current on separate sections serial circuit is the same everywhere:
I 1 = I 2 = I 3 = I.
Video 1. Serial connection of conductors
Example 1. Figure 2 shows an electrical circuit consisting of three resistors connected in series r 1 = 2 ohm, r 2 = 3 ohm, r 3 = 5 ohm. It is required to determine the readings of voltmeters V 1 , V 2 , V 3 and V 4 if the current in the circuit is 4 A.
Whole circuit resistance
r = r 1 + r 2 + r 3 \u003d 2 + 3 + 5 \u003d 10 ohms.
Figure 2. Scheme for measuring voltages in individual sections of the electrical circuit
In resistance r 1 when current flows, there will be a voltage drop:
U 1 = I × r 1=4×2=8V.
Voltmeter V 1 included between points A And b, will show 8 V.
In resistance r 2 there is also a voltage drop:
U 2 = I × r 2 = 4 × 3 = 12V.
Voltmeter V 2 included between points V And G, will show 12 V.
Voltage drop in resistance r 3:
U 3 = I × r 3 = 4 × 5 = 20 V.
Voltmeter V 3 included between dots d And e, will show 20 V.
If the voltmeter is connected at one end to the point A, the other end to the point G, then it will show the potential difference between these points, equal to the sum of the voltage drops in the resistances r 1 and r 2 (8 + 12 = 20 V).
So the voltmeter V, measuring the voltage at the circuit terminals and connected between the points A And e, will show the potential difference between these points or the sum of the voltage drops in the resistances r 1 , r 2 and r 3 .
This shows that the sum of the voltage drops in individual sections of the electrical circuit is equal to the voltage at the circuit terminals.
Since with a series connection, the circuit current is the same in all sections, the voltage drop is proportional to the resistance of this section.
Example 2 Three resistances of 10, 15 and 20 ohms are connected in series as shown in Figure 3. The current in the circuit is 5 A. Determine the voltage drop across each resistance.
U 1 = I × r 1 = 5 × 10 = 50 V,
U 2 = I × r 2 = 5 × 15 = 75 V,
U 3 = I × r 3 = 5 × 20 = 100 V.
Figure 3. Example 2
The total voltage of the circuit is equal to the sum of the voltage drops in the individual sections of the circuit:
U = U 1 + U 2 + U 3 = 50 + 75 + 100 = 225 V.
Parallel connection of conductors
A parallel connection of conductors is such a connection when the beginnings of all conductors are connected to one point, and the ends of the conductors to another point (Figure 4). The beginning of the circuit is connected to one pole of the voltage source, and the end of the circuit is connected to the other pole.
It can be seen from the figure that at parallel connection conductors for the passage of current there are several ways. Current flowing to the branch point A, spreads further over three resistances and is equal to the sum of the currents leaving this point:
I = I 1 + I 2 + I 3 .
If the currents coming to the branching point are considered positive, and the outgoing currents are negative, then for the branching point we can write:
that is, the algebraic sum of the currents for any nodal point chain is always zero. This relation, which connects the currents at any branching point in the circuit, is called Kirchhoff's first law. The definition of the first Kirchhoff's law can sound in another formulation, namely: the sum of the currents flowing into the node of the electrical circuit is equal to the sum of the currents flowing from this node.
Video 2. Kirchhoff's first law
Usually, when calculating electrical circuits, the direction of currents in the branches connected to any branching point is unknown. Therefore, in order to be able to record the equation of the first Kirchhoff law, it is necessary to arbitrarily choose the so-called positive directions of currents in all its branches before starting the calculation of the circuit and designate them with arrows in the diagram.
Using Ohm's law, you can derive a formula for counting total resistance with parallel connection of consumers.
Total current coming to the point A, is equal to:
The currents in each of the branches have the following values:
According to the formula of Kirchhoff's first law
I = I 1 + I 2 + I 3
Bringing out U on the right side of the equation outside the brackets, we get:
Reducing both sides of the equality by U, we get the formula for calculating the total conductivity:
g \u003d g 1 + g 2 + g 3.
Thus, with a parallel connection, it is not the resistance that increases, but the conductivity.
Example 3 Determine the total resistance of three resistors connected in parallel if r 1 = 2 ohm, r 2 = 3 ohm, r 3 = 4 ohms.
Example 4 Five resistances 20, 30, 15, 40 and 60 ohms are connected in parallel in the network. Determine the total resistance:
It should be noted that when calculating the total branching resistance, it always turns out to be less than the smallest resistance included in the branching.
If the resistances connected in parallel are equal to each other, then the total resistance r circuit is equal to the resistance of one branch r 1 divided by the number of branches n:
Example 5 Determine the total resistance of four parallel-connected resistances of 20 ohms each:
To check, let's try to find the branching resistance using the formula:
As you can see, the answer is the same.
Example 6 Let it be required to determine the currents in each branch with their parallel connection, shown in Figure 5, A.
Find the total resistance of the circuit:
Now we can depict all branches in a simplified way as one resistance (Figure 5, b).
Voltage drop in the section between points A And B will:
U = I × r= 22 × 1.09 = 24 V.
Returning again to Figure 5, we see that all three resistances will be energized at 24 V, since they are connected between the points A And B.
Considering the first branch branch with resistance r 1, we see that the voltage in this section is 24 V, the resistance of the section is 2 ohms. According to Ohm's law for a section of the circuit, the current in this section will be:
Current of the second branch
Third branch current
Let's check according to the first law of Kirchhoff
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 compiling and studying electrical circuits replace specific receivers with resistors with specific resistances. Distinguish the following ways resistor connections(receivers 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.
Hence, equivalent resistance 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 \u003d IR 1; U 2 \u003d IR 2, U 3 \u003d IR s and in this case E = 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. That's why serial connection receivers are 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 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: equivalent conductivity parallel circuit is equal to the sum of the conductances of 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 DC electric locomotives and some diesel locomotives, traction motors in the process of regulating the speed of movement must be turned on under various voltages, so they switch from serial to parallel during overclocking.
mixed connection 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, and is replaced equivalent circuit rice. 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.
Parallel connections of resistors, the calculation formula of which is derived from Ohm's law and Kirchhoff's rules, are the most common type of inclusion of elements in an electrical circuit. When conductors are connected in parallel, two or more elements are connected by their contacts on both sides, respectively. Connecting them to general scheme carried out by these nodal points.
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General form
Features of inclusion
The conductors included in this way are often part of complex chains, which, in addition, contain a serial connection of individual sections.
The following features are typical for such an inclusion:
- The total voltage in each of the branches will have the same value;
- Flowing in any of the resistances electricity always inversely proportional to their value.
In a particular case, when all resistors connected in parallel have the same nominal values, the “individual” currents flowing through them will also be equal to each other.
Calculation
The resistances of a number of conductive elements connected in parallel are determined by the well-known form of calculation, which involves the addition of their conductivities (the reciprocal of the resistance values).
The current flowing in each of the individual conductors, in accordance with Ohm's law, can be found by the formula:
I= U/R (one of the resistors).
After getting acquainted with general principles calculation of elements of complex chains, you can go to concrete examples solving problems of this class.
Typical Connections
Example #1
Often, in order to solve the problem facing the designer, it is required to obtain a specific resistance as a result by combining several elements. When considering the simplest version of such a solution, let's assume that the total resistance of a chain of several elements should be 8 ohms. This example needs to be considered separately for the simple reason that there is no 8 ohm value in the standard series of resistances (there are only 7.5 and 8.2 ohms).
The solution to this simplest problem can be obtained by connecting two identical elements with resistances of 16 ohms each (such ratings exist in the resistive series). According to the above formula, the total resistance of the chain in this case is calculated very simply.
It follows from it:
16x16 / 32 \u003d 8 (Ohm), that is, just as much as it was required to receive.
So comparatively in a simple way it is possible to solve the problem of forming a total resistance equal to 8 ohms.
Example #2
As another typical example formation of the required resistance, we can consider the construction of a circuit consisting of 3 resistors.
The total R value of such an inclusion can be calculated using the formula for series and parallel connection in conductors.
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In accordance with the values \u200b\u200bof the ratings indicated in the picture, the total resistance of the chain will be equal to:
1/R = 1/200+1/220+1/470 = 0.0117;
R \u003d 1 / 0.0117 \u003d 85.67 Ohm.
As a result, we find the total resistance of the entire chain obtained by connecting three elements in parallel with nominal values of 200, 240 and 470 ohms.
Important! This method is also applicable when calculating an arbitrary number of conductors or consumers connected in parallel.
It should also be noted that with this method of switching on elements of various sizes, the total resistance will be less than that of the smallest denomination.
Calculation of combined schemes
The considered method can also be used in calculating the resistance of more complex or combined circuits consisting of a whole set of components. They are sometimes called mixed, since both methods are used at once when forming chains. A mixed connection of resistors is shown in the figure below.
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mixed scheme
To simplify the calculation, we first divide all the resistors by type of inclusion into two independent groups. One of them is a serial connection, and the second is a parallel type connection.
From the above diagram, it can be seen that the elements R2 and R3 are connected in series (they are combined in group 2), which, in turn, is connected in parallel with the resistor R1 belonging to group 1.
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 circuit. Their accumulation would lead to a change in the electric field strength and, consequently, to a change in the current strength. That's why
\(~I = I_1 = I_2 .\)
Ammeter A measures the current in the circuit and has a small internal resistance (R A → 0).
Included voltmeters V 1 and V 2 measure voltage U 1 and U 2 on resistance R 1 and R 2. Voltmeter V measures the input to the terminals Μ And N voltage U. Voltmeters show that when connected in series, the voltage U equal to the sum of the voltages in the individual sections of the circuit:
\(~U = U_1 + U_2 . \qquad (1)\)
Applying Ohm's law for each section of the circuit, we get:
\(~U = IR ; \ U_1 = IR_1 ; \ U_2 = IR_2 ,\)
Where R is the total resistance of a series-connected circuit. Substituting U, U 1 , U 2 into formula (1), we have
\(~IR = IR_1 + IR_2 \Rightarrow R = R_1 + R_2 .\)
n resistors connected in series is equal to the sum of the resistances of these resistors:
\(~R = R_1 + R_2 + \ldots R_n\) , or \(~R = \sum_(i=1)^n R_i .\)
If the resistances of the individual resistors are equal to each other, i.e. R 1 = R 2 = ... = R n, then the total resistance of these resistors when connected in series in n times the resistance of one resistor: R = nR 1 .
When resistors are connected in series, the relation \(~\frac(U_1)(U_2) = \frac(R_1)(R_2)\), i.e. 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 to points Μ And N connected voltmeter. 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:
\(~U = U_1 = U_2 .\)
The ammeter shows that the current I in the unbranched part of the circuit is equal to the sum of the current strengths I 1 and I 2 in parallel connected conductors R 1 and R 2:
\(~I = I_1 + I_2 . \qquad (2)\)
This also follows from the conservation law electric charge. We apply Ohm's law for individual sections of the circuit and the entire circuit with a common resistance R:
\(~I = \frac(U)(R) ; \ I_1 = \frac(U)(R_1) ; \ I_2 = \frac(U)(R_2) .\)
Substituting I, I 1 and I 2 into formula (2), we get:
\(~\frac(U)(R) = \frac(U)(R_1) + \frac(U)(R_2) \Rightarrow \frac(1)(R) = \frac(1)(R_1) + \ frac(1)(R_2) .\)
The reciprocal of the resistance of a circuit consisting of n resistors connected in parallel is equal to the sum of the reciprocals of the resistances of these resistors:
\(~\frac 1R = \sum_(i=1)^n \frac(1)(R_i) .\)
If the resistance of all n resistors connected in parallel are the same and equal R 1 then \(~\frac 1R = \frac(n)(R_1)\) . Whence \(~R = \frac(R_1)(n)\) .
The resistance of a circuit consisting of n resistors connected in parallel n times less than the resistance of each of them.
When resistors are connected in parallel, the relation \(~\frac(I_1)(I_2) = \frac(R_2)(R_1)\), i.e. the current strengths in the branches of a parallel-connected circuit are inversely proportional to the resistances of the branches.
Literature
Aksenovich L. A. Physics in high school: Theory. Tasks. Tests: Proc. allowance for institutions providing general. environments, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsia i vykhavanne, 2004. - C. 257-259.
Let us check the validity of the formulas shown here on a simple experiment.
Take two resistors MLT-2 on 3 And 47 ohm and connect them in series. Then we measure the total resistance of the resulting circuit digital multimeter. As you can see, it is equal to the sum of the resistances of the resistors included in this chain.
Measurement of total resistance in series connection
Now let's connect our resistors in parallel and measure their total resistance.
Resistance measurement in parallel connection
As you can see, the resulting resistance (2.9 ohms) is less than the smallest (3 ohms) included in the chain. From this comes another well-known rule which can be put into practice:
When resistors are connected in parallel, the total resistance of the circuit will be less than the smallest resistance included in this circuit.
What else needs to be considered when connecting resistors?
Firstly, Necessarily their rated power. For example, we need to find a replacement resistor for 100 ohm and power 1 W. Take two resistors of 50 ohms each and connect them in series. What power dissipation should these two resistors be rated for?
Since the same current flows through the resistors connected in series. D.C.(let's say 0.1 A), and the resistance of each of them is 50 ohm, then the power dissipation of each of them must be at least 0.5W. As a result, each of them will have 0.5W power. In sum, this will be the same 1 W.
This example is rather rough. Therefore, if in doubt, it is worth taking resistors with a power margin.
Read more about the power dissipation of the resistor.
Secondly, when connecting, it is worth using the same type of resistors, for example, the MLT series. Of course, there is nothing wrong with taking different ones. This is just a recommendation.
