INSTRUCTIONS AND PROPHECIES OF THE Blessed MOTHER ALIPIA GOLOSEEVSKY, Kyiv...
With the simultaneous inclusion of several receivers of electricity in the same network, these receivers can be easily considered simply as elements of a single circuit, each of which has its own resistance.
In some cases, this approach turns out to be quite acceptable: incandescent lamps, electric heaters, etc. - can be perceived as resistors. That is, devices can be replaced by their resistances, and it is easy to calculate the parameters of the circuit.
The method of connecting power receivers can be one of the following: serial, parallel or mixed type of connection.
serial connection
When several receivers (resistors) are connected in a serial circuit, that is, the second terminal of the first is connected to the first terminal of the second, the second terminal of the second is connected to the first terminal of the third, the second terminal of the third to the first terminal of the fourth, etc., then when such a circuit is connected to power source, a current I of the same value will flow through all elements of the circuit. This idea is illustrated in the figure below.
Replacing the devices with their resistances, we convert the pattern into a circuit, then the resistances from R1 to R4, connected in series, will each take on certain voltages, which in total will give the EMF value at the power supply terminals. For simplicity, hereinafter we will depict the source as a galvanic cell.
Having expressed the voltage drops through the current and through the resistances, we obtain an expression for the equivalent resistance of the series circuit of receivers: the total resistance of the series connection of resistors is always equal to algebraic sum all the resistances that make up this circuit. And since the voltages on each of the sections of the circuit can be found from Ohm's law (U = I * R, U1 = I * R1, U2 = I * R2, etc.) and E = U, then for our circuit we get:
The voltage at the power supply terminals is equal to the sum of the voltage drops across each of the receivers connected in series that make up the circuit.
Since the current flows through the entire circuit of the same value, it will be fair to say that the voltages on series-connected receivers (resistors) are proportional to each other in proportion to the resistances. And the higher the resistance, the higher the voltage applied to the receiver will be.
For a series connection of resistors in the amount of n pieces with the same resistance Rk, the equivalent total resistance of the entire circuit will be n times greater than each of these resistances: R = n * Rk. Accordingly, the voltages applied to each of the resistors in the circuit will be equal to each other, and will be n times less than the voltage applied to the entire circuit: Uk \u003d U / n.
The following properties are characteristic of the series connection of electric power receivers: if the resistance of one of the circuit receivers is changed, then the voltages on the remaining receivers of the circuit will change; if one of the receivers breaks, the current will stop in the entire circuit, in all other receivers.
Due to these features serial connection is rare, and is only used where the mains voltage is higher than the rated voltage of the receivers, in the absence of alternatives.
For example, a voltage of 220 volts can power two series-connected lamps of equal power, each of which is rated for a voltage of 110 volts. If these lamps at the same rated supply voltage will have different rated power, then one of them will be overloaded and most likely will burn out instantly.
Parallel connection
Parallel connection of receivers involves the inclusion of each of them between a pair of points in the electrical circuit so that they form parallel branches, each of which is powered by a source voltage. For clarity, we will again replace the receivers with their electrical resistances in order to obtain a scheme by which it is convenient to calculate the parameters.
As already mentioned, in the case parallel connection each of the resistors is subjected to the same voltage. And in accordance with Ohm's law, we have: I1=U/R1, I2=U/R2, I3=U/R3.
Here I is the source current. The first Kirchhoff law for this circuit allows you to write an expression for the current in its unbranched part: I = I1 + I2 + I3.
Hence, the total resistance for the parallel connection between the circuit elements can be found from the formula:
The reciprocal of resistance is called conductivity G, and the formula for the conductivity of a circuit consisting of several parallel-connected elements can also be written: G \u003d G1 + G2 + G3. The conductivity of the circuit in the case of a parallel connection of the resistors forming it is equal to the algebraic sum of the conductivities of these resistors. Therefore, when parallel receivers (resistors) are added to the circuit, the total resistance of the circuit will decrease, and the total conductivity will increase accordingly.
Currents in a circuit consisting of parallel-connected receivers are distributed between them in direct proportion to their conductivities, that is, inversely proportional to their resistances. Here we can draw an analogy from hydraulics, where the flow of water is distributed through the pipes in accordance with their sections, then a larger section is similar to a lower resistance, that is, a greater conductivity.
If the circuit consists of several (n) identical resistors connected in parallel, then the total resistance of the circuit will be n times lower than the resistance of one of the resistors, and the current through each of the resistors will be n times less than the total current: R = R1 / n; I1 = I/n.
A circuit consisting of receivers connected in parallel, connected to a power source, is characterized in that each of the receivers is energized by the power source.
For an ideal source of electricity, the statement is true: when resistors are connected or disconnected in parallel to the source, the currents in the remaining connected resistors will not change, that is, if one or more receivers in the parallel circuit fail, the rest will continue to work in the same mode.
Due to these features, a parallel connection has a significant advantage over a serial connection, and for this reason, it is the parallel connection that is most common in electrical networks. For example, all electrical appliances in our homes are designed to parallel connection to a household network, and if you turn off one, then it will not harm the rest.
Comparison of series and parallel circuits
A mixed connection of receivers is understood as such a connection when some or several of them are connected to each other in series, and the other part or several are connected in parallel. In this case, the entire chain can be formed from various connections of such parts to each other. For example, consider the diagram:
Three resistors connected in series are connected to a power source, two more are connected in parallel to one of them, and the third is connected in parallel to the entire circuit. To find the total resistance, the circuits go through successive transformations: a complex circuit is successively reduced to a simple form, sequentially calculating the resistance of each link, and in this way the total equivalent resistance is found.
For our example. First, the total resistance of two resistors R4 and R5 connected in series is found, then the resistance of their parallel connection with R2, then they are added to the obtained value of R1 and R3, and then the resistance value of the entire circuit is calculated, including the parallel branch R6.
Various methods of connecting power receivers are used in practice for various purposes in order to solve specific tasks. For example, a mixed connection can be found in smooth charge circuits in powerful power supplies, where the load (capacitors after the diode bridge) is first powered in series through a resistor, then the resistor is shunted by relay contacts, and the load is connected to the diode bridge in parallel.
Andrey Povny
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 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 total resistance R:
Substituting I, I 1 and I 2 into formula (2), we get.
Series connection of resistances
We 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 serial. Of course, the current in such a circuit will be the same at all its points.
Rice one . Series connection of resistances
How to find the total resistance of the circuit, if we already know all the resistances included in it in turn? 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) .
Dividing now both parts of the equality by I , we will have R = R1 + R2 + R3
Thus, we concluded 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 \u003d= 10 Ohms, R 2 \u003d 20 Ohms and R 3 \u003d 50 Ohms). Let's connect them one by one (Fig. 2) and connect to a current source, the EMF of which is 60 V (we neglect the internal resistance of the current source).
Rice. 2. Example of serial connection of 3 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 \u003d 10 + 20 + 50 \u003d 80 Ohms.
Let's find the current in the circuit according to Ohm's law: 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 got 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 now, we can verify by the devices 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. Then connecting points a and b with a current source, we get a closed electronic 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, because 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. Any 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, with a parallel connection of resistances, a branched circuit comes out. Let's see what will be the ratio between the currents in the circuit we have compiled.
Let's 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.
Means current in the circuit before branching(up to point a) is equal to current after circuit branching(after point b).
We will now turn on the ammeter alternately in each branch of the circuit, remembering the readings of the device. Let the ammeter show the current I1 in the first branch, and I 2 in the 2nd branch. Adding these two ammeter readings, we get a total current equal in magnitude to the current I to a branch (to point a).
Properly, 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 has a huge practical value, bears the title branched chain law.
Consider now 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. Firstly, the voltmeter will show the voltage of the current source, because it is connected, as can be seen from fig. 3, specifically to the current source terminals. Secondly, the voltmeter will show the voltage drops U1 and U2 across the resistances R1 and R2 because it is connected to the start and end of each resistance.
As follows, 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 because U1 = U2, then I1R1 = I2R2 .
Applying the rule of proportion to this expression, we get I1 / I2 \u003d U2 / U1 i.e. the current in the first branch will be so many times more (or less) than the current in the 2nd branch, how many times the resistance of the first branch is less (or more) resistance of the 2nd branch.
Thus, we have come to the fundamental conclusion that when the resistances are connected in parallel, the total current of the circuit branches into currents that are inversely proportional to the resistance values of the parallel branches. In other words, the greater the resistance of the branch, the less current will flow through it, and, conversely, the lower the resistance of the branch, the greater the 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 2 parallel-connected resistances R1 and R 2 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 \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 back in proportion to the resistances.
Indeed, R1 \u003d= 10 Ohm is half R 2 \u003d 20 Ohm, while I1 \u003d 300 mA is twice as large as I2 \u003d 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 the smallest resistance (R1 \u003d 10 Ohm), and the smallest part (R2 \u003d 150 mA) went through the greater resistance (R 2 = 20 ohms).
Such branching of the current in parallel branches is similar to the flow of water through pipes. Imagine pipe A, which in some place branches into two pipes B and C of different diameters (Fig. 4). Because the diameter of pipe B is greater than the diameter of pipes C, then through pipe B to the same time will pass more water than through pipe B, which offers more resistance to the water clot.
Rice. 4
Let us now consider what the total resistance of the external circuit, consisting of 2 parallel-connected resistances, will be equal to.
Underneath it the total resistance of the external circuit must be understood as such a resistance that could change both resistances connected in parallel at a given circuit voltage, without changing the current before branching. Such resistance is called equivalent resistance.
Let's go back to the circuit shown in Fig. 3, and let's see what the equivalent resistance of 2 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
Because with a parallel connection U \u003d U1 \u003d U2, then 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 sides of the equality by U, we will have 1 / R = 1 / R1 + 1 / R2
Remembering that conductivity is the reciprocal of resistance, we can say that in the acquired formula 1 / R is the conductivity of the outer circuit; 1 / R1 conductivity of the first branch; 1 / R2 - conductivity of the 2nd 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.
Properly, to find the equivalent resistance of the resistances connected in parallel, you need to find the conductivity of the circuit and take the value that is 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 smaller of the resistances connected in parallel.
Considering the case of a parallel connection of resistances, we took a more ordinary circuit, consisting of 2 branches. But in practice, there may be cases when the chain consists of 3 or more parallel branches. How to act in these cases?
It turns out that all the relationships we have acquired remain valid for a circuit consisting of any number of parallel-connected resistances.
To verify 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). R = 1 / 6 As follows, equivalent resistance R = 6 ohm.
In such a way, 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 see now whether this resistance is really equivalent, that is, one that could change 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 as it follows, the voltage at 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 3 resistances recognizable to us in parallel, one equivalent resistance of 6 Ohms is included.
I \u003d U / R \u003d 12 / 6 \u003d 2 A
As we see, the resistance R = 6 Ohm we found is indeed equivalent for this circuit.
This can also be verified on measuring devices, if we assemble a circuit with the resistances we have taken, measure the current in the external circuit (before branching), then change the resistances connected in parallel 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 easier to calculate the equivalent resistance, i.e., without previously 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. with a parallel connection of 2 resistances, the equivalent resistance of the circuit is equal to the product of the resistances connected in parallel, divided by their sum.
Series and parallel connection of conductors are the main types of connection of conductors encountered in practice. Since electrical circuits, as a rule, do not consist of homogeneous conductors of the same cross section. How to find the resistance of the circuit, if the resistance of its individual parts is known.
Let's consider two typical cases. The first of these is when two or more resistance conductors are connected in series. In series means that the end of the first conductor is connected to the beginning of the second, and so on. With this inclusion of conductors, the current strength in each of them will be the same. But the voltage on each of them will be different.
Figure 1 - serial connection of conductors
The voltage drop across the resistances can be determined based on Ohm's law.
Formula 1 - Voltage drop across resistance
The sum of these voltages will be equal to the total voltage applied to the circuit. The voltage on the conductors will be distributed in proportion to their resistance. That is, you can write
Formula 2 - Relationship Between Resistance and Voltage
The total resistance of the circuit will be equal to the sum of all the resistances connected in series.
Formula 3 - calculation of the total resistance in parallel connection
The second case is when the resistances in the circuit are connected in parallel to each other. That is, there are two nodes in the circuit and all conductors with resistance are connected to these nodes. In such a circuit, the currents in all branches are generally not equal to each other. But the sum of all currents in the circuit after the branching will be equal to the current before the branching.
Figure 2 - Parallel connection of conductors
Formula 4 - ratio between currents in parallel branches
The current strength in each of the branched circuits also obeys Ohm's law. The voltage on all conductors will be the same. But the current strength will be separated. In a circuit consisting of conductors connected in parallel, the currents are distributed in proportion to the resistances.
Formula 5 - Distribution of Currents in Parallel Branches
To find the total resistance of the circuit in this case, it is necessary to add the reciprocal of the resistances, that is, the conductance.
Formula 6 - Resistance of conductors connected in parallel
There is also a simplified formula for the particular case when two identical resistances are connected in parallel.
All electrical circuits use resistors, which are elements with a precisely set resistance value. Due to the specific qualities of these devices, it becomes possible to adjust the voltage and current in any part of the circuit. These properties underlie the operation of almost all electronic appliances and equipment. So, the voltage for parallel and series connection of resistors will be different. Therefore, each type of connection can only be used under certain conditions, so that one or another circuit diagram able to fully perform its functions.
Series connection voltage
When connected in series, two or more resistors are connected in a common circuit in such a way that each of them has contact with the other device at only one point. In other words, the end of the first resistor is connected to the beginning of the second, and the end of the second - to the beginning of the third, and so on.
A feature of this circuit is the passage through all connected resistors of the same value electric current. With an increase in the number of elements in the considered section of the circuit, the flow of electric current becomes more and more difficult. This is due to the increase in the total resistance of the resistors when they are connected in series. This property is reflected by the formula: R total \u003d R 1 + R 2.
The voltage distribution, in accordance with Ohm's law, is carried out for each resistor according to the formula: V Rn \u003d I Rn x R n. Thus, as the resistance of the resistor increases, the voltage across it also increases.
Parallel voltage
When connected in parallel, the inclusion of resistors in electrical circuit is performed in such a way that all elements of resistance are connected to each other by both contacts at once. One point, which is an electrical node, can connect several resistors at the same time.
Such a connection assumes the flow of a separate current in each resistor. The strength of this current is inversely proportional to . As a result, there is an increase in the total conductivity of a given section of the circuit, with a general decrease in resistance. In the case of a parallel connection of resistors with different resistances, the value of the total resistance in this section will always be lower than the smallest resistance of a single resistor.
In the presented diagram, the voltage between points A and B is not only the total voltage for the entire section, but also the voltage supplied to each individual resistor. Thus, in the case of a parallel connection, the voltage applied to all resistors will be the same.
As a result, the voltage in parallel and series connection will be different in each case. Due to this property, there is a real opportunity to adjust this value in any part of the chain.
