In a series circuit, current flows through all elements. Series and parallel connection of resistances
Almost everyone who worked as an electrician had to solve the issue of parallel and series connection of circuit elements. Some solve the problems of parallel and series connection of conductors using the “poke” method; for many, a “fireproof” garland is an inexplicable but familiar axiom. However, all these and many other similar questions are easily solved by the method proposed at the very beginning of the 19th century by the German physicist Georg Ohm. The laws discovered by him are still in effect today, and almost everyone can understand them.
Basic electrical quantities of the circuit
In order to find out how a particular connection of conductors will affect the characteristics of the circuit, it is necessary to determine the quantities that characterize any electrical circuit. Here are the main ones:
Mutual dependence of electrical quantities
Now you need to decide, how all of the above quantities depend on one another. The rules of dependence are simple and come down to two basic formulas:
- I=U/R.
- P=I*U.
Here I is the current in the circuit in amperes, U is the voltage supplied to the circuit in volts, R is the circuit resistance in ohms, P is the electrical power of the circuit in watts.
Suppose we have a simple electrical circuit, consisting of a power source with voltage U and a conductor with resistance R (load).
Since the circuit is closed, current I flows through it. What value will it be? Based on the above formula 1, to calculate it we need to know the voltage developed by the power source and the load resistance. If we take, for example, a soldering iron with a coil resistance of 100 Ohms and connect it to a lighting socket with a voltage of 220 V, then the current through the soldering iron will be:
220 / 100 = 2.2 A.
What is the power of this soldering iron? Let's use formula 2:
2.2 * 220 = 484 W.
It turned out to be a good soldering iron, powerful, most likely two-handed. In the same way, operating with these two formulas and transforming them, you can find out the current through power and voltage, voltage through current and resistance, etc. How much, for example, does a 60 W light bulb in your table lamp consume:
60 / 220 = 0.27 A or 270 mA.
Lamp filament resistance in operating mode:
220 / 0.27 = 815 Ohms.
Circuits with multiple conductors
All the cases discussed above are simple - one source, one load. But in practice there can be several loads, and they are also connected in different ways. There are three types of load connection:
- Parallel.
- Consistent.
- Mixed.
Parallel connection of conductors
The chandelier has 3 lamps, each 60 W. How much does a chandelier consume? That's right, 180 W. Let’s quickly calculate the current through the chandelier:
180 / 220 = 0.818 A.
And then her resistance:
220 / 0.818 = 269 Ohms.
Before this, we calculated the resistance of one lamp (815 Ohms) and the current through it (270 mA). The resistance of the chandelier turned out to be three times lower, and the current was three times higher. Now it’s time to look at the diagram of a three-arm lamp.
All lamps in it are connected in parallel and connected to the network. It turns out that when three lamps are connected in parallel, the total load resistance decreases threefold? In our case, yes, but it is private - all lamps have the same resistance and power. If each of the loads has its own resistance, then simply dividing by the number of loads is not enough to calculate the total value. But there is a way out of the situation - just use this formula: 
1/Rtotal = 1/R1 + 1/R2 + … 1/Rn.
For ease of use, the formula can be easily converted:
Rtot. = (R1*R2*… Rn) / (R1+R2+… Rn).
Here Rtotal. – the total resistance of the circuit when the load is connected in parallel. R1…Rn – resistance of each load.
Why the current increased when you connected three lamps in parallel instead of one is not difficult to understand - after all, it depends on the voltage (it remained unchanged) divided by the resistance (it decreased). Obviously, the power in a parallel connection will increase in proportion to the increase in current.
Serial connection
Now it’s time to find out how the parameters of the circuit will change if the conductors (in our case, lamps) are connected in series.
Calculating resistance when connecting conductors in series is extremely simple:
Rtot. = R1 + R2.
The same three sixty-watt lamps connected in series will already amount to 2445 Ohms (see calculations above). What are the consequences of increasing circuit resistance? According to formulas 1 and 2, it becomes quite clear that the power and current strength when connecting conductors in series will drop. But why are all the lamps dim now? This is one of the most interesting properties of series connection of conductors, which is very widely used. Let's take a look at a garland of three lamps familiar to us, but connected in series.
The total voltage applied to the entire circuit remained 220 V. But it was divided between each of the lamps in proportion to their resistance! Since we have lamps of the same power and resistance, the voltage is divided equally: U1 = U2 = U3 = U/3. That is, each of the lamps is now supplied with three times less voltage, which is why they glow so dimly. If you take more lamps, their brightness will drop even more. How to calculate the voltage drop across each lamp if they all have different resistances? To do this, the four formulas given above are sufficient. The calculation algorithm will be as follows:
- Measure the resistance of each lamp.
- Calculate the total resistance of the circuit.
- Based on the total voltage and resistance, calculate the current in the circuit.
- Based on the total current and resistance of the lamps, calculate the voltage drop across each of them.
Do you want to consolidate your acquired knowledge?? Solve a simple problem without looking at the answer at the end:
You have at your disposal 15 miniature light bulbs of the same type, designed for a voltage of 13.5 V. Is it possible to use them to make a Christmas tree garland that connects to a regular outlet, and if so, how?
Mixed compound
You, of course, easily figured out the parallel and serial connection of conductors. But what if you have something like this in front of you?
Mixed connection of conductors
How to determine the total resistance of a circuit? To do this, you will need to break the circuit into several sections. The above design is quite simple and there will be two sections - R1 and R2, R3. First, you calculate the total resistance of parallel-connected elements R2, R3 and find Rtot.23. Then calculate the total resistance of the entire circuit, consisting of R1 and Rtot.23 connected in series:
- Rtot.23 = (R2*R3) / (R2+R3).
- Rchains = R1 + Rtot.23.
The problem is solved, everything is very simple. Now the question is somewhat more complicated.
Complex mixed connection of resistances
How to be here? In the same way, you just need to show some imagination. Resistors R2, R4, R5 are connected in series. We calculate their total resistance:
Rtot.245 = R2+R4+R5.
Now we connect R3 in parallel to Rtot.245:
Rtot.2345 = (R3* Rtot.245) / (R3+ Rtot.245).
Rchains = R1+ Rtot.2345+R6.
That's it!
Answer to the Christmas garland problem
The lamps have an operating voltage of only 13.5 V, and the socket is 220 V, so they must be connected in series.
Since the lamps are of the same type, the network voltage will be divided equally between them and each lamp will have 220 / 15 = 14.6 V. The lamps are designed for a voltage of 13.5 V, so although such a garland will work, it will burn out very quickly. To realize your idea, you will need at least 220 / 13.5 = 17, and preferably 18-19 light bulbs.
Content:Electrical circuits use different types of connections. The main ones are serial, parallel and mixed connection schemes. In the first case, several resistances are used, connected in a single chain one after another. That is, the beginning of one resistor is connected to the end of the second, and the beginning of the second to the end of the third, and so on, up to any number of resistances. The current strength in a series connection will be the same at all points and in all sections. To determine and compare other parameters of the electrical circuit, other types of connections that have their own properties and characteristics should be considered.
Series and parallel connection of resistances
Any load has resistance that prevents the free flow of electric current. Its path runs from the current source, through the conductors to the load. For normal current flow, the conductor must have good conductivity and easily give up electrons. This provision will be useful later when considering the question of what a serial connection is.
Most electrical circuits use copper conductors. Each circuit contains energy receivers - loads with different resistances. The connection parameters are best considered using the example of an external current source circuit consisting of three resistors R1, R2, R3. A serial connection involves the alternate inclusion of these elements in a closed circuit. That is, the beginning of R1 is connected to the end of R2, and the beginning of R2 is connected to the end of R3, and so on. There can be any number of resistors in such a chain. These symbols are used in calculations.
In all sections it will be the same: I = I1 = I2 = I3, and the total resistance of the circuit will be the sum of the resistances of all loads: R = R1 + R2 + R3. It remains only to determine what it will be like with a serial connection. According to Ohm's law, voltage represents current and resistance: U = IR. It follows that the voltage at the current source will be equal to the sum of the voltages at each load, since the current is the same everywhere: U = U1 + U2 + U3.
At a constant voltage value, the current in a series connection will depend on the resistance of the circuit. Therefore, if the resistance changes at least on one of the loads, the resistance in the entire circuit will change. In addition, the current and voltage across each load will change. The main disadvantage of a series connection is the cessation of operation of all elements of the circuit, if even one of them fails.
Completely different current, voltage and resistance characteristics are obtained when using a parallel connection. In this case, the beginnings and ends of the loads are connected at two common points. A kind of current branching occurs, which leads to a decrease in the total resistance and an increase in the total conductivity of the electrical circuit.
In order to display these properties, Ohm's law will again be needed. In this case, the current strength in a parallel connection and its formula will look like this: I = U/R. Thus, when connecting the nth number of identical resistors in parallel, the total resistance of the circuit will be n times less than any of them: Rtot = R/n. This indicates an inversely proportional distribution of currents in loads with respect to the resistances of these loads. That is, with an increase in parallel-connected resistances, the current strength in them will decrease proportionally. In the form of formulas, all characteristics are displayed as follows: current - I = I1 + I2 + I3, voltage - U = U1 = U2 = U3, resistance - 1/R = 1/R1 + 1/R2 + 1/R3.
At a constant voltage between the elements, the currents in these resistors are independent of each other. If one or more resistors are turned off from the circuit, this will not affect the operation of other devices that remain turned on. This factor is the main advantage of parallel connection of electrical appliances.
Circuits usually do not use only series and parallel resistances, but use them in a combined form known as . To calculate the characteristics of such circuits, the formulas of both options are used. All calculations are divided into several stages, when the parameters of individual sections are first determined, after which they are added up and the overall result is obtained.
Laws of series and parallel connection of conductors
The basic law used in the calculations of various types of connections is Ohm's law. Its main position is the presence in a section of the circuit of a current strength that is directly proportional to the voltage and inversely proportional to the resistance in this section. In the form of a formula, this law looks like this: I = U/R. It serves as the basis for carrying out calculations of electrical circuits connected in series or parallel. The order of calculations and the dependence of all parameters on Ohm's law are clearly shown in the figure. From here the formula for a series connection is derived.
More complex calculations involving other quantities require the use of . Its main position is that several series-connected current sources will have an electromotive force (EMF), which is the algebraic sum of the EMF of each of them. The total resistance of these batteries will be the sum of the resistances of each battery. If a parallel connection is made to the nth number of sources with equal EMF and internal resistances, then the total amount of EMF will be equal to the EMF at any of the sources. The value of internal resistance will be rв = r/n. These provisions are relevant not only for current sources, but also for conductors, including the formula for parallel connection of conductors.
In the case when the EMF of the sources will have different values, additional Kirchhoff rules are applied to calculate the current strength in different sections of the circuit.
Let's take three constant resistances R1, R2 and R3 and connect them to the circuit so that the end of the first resistance R1 is connected to the beginning of the second resistance R2, the end of the second to the beginning of the third R3, and we connect conductors to the beginning of the first resistance and to the end of the third from the current source (Fig. 1).
This connection of resistances is called series. Obviously, the current in such a circuit will be the same at all its points.
Rice 1
How to determine the total resistance of a circuit if we already know all the resistances included in it in series? Using the position that the voltage U at the terminals of the current source is equal to the sum of the voltage drops in the sections of the circuit, we can write:
U = U1 + U2 + U3
Where
U1 = IR1 U2 = IR2 and U3 = IR3
or
IR = IR1 + IR2 + IR3
Taking the equality I out of brackets on the right side, we obtain IR = I(R1 + R2 + R3) .
Now dividing both sides of the equality by I, we will finally have R = R1 + R2 + R3
Thus, we came to the conclusion that when resistances are connected in series, the total resistance of the entire circuit is equal to the sum of the resistances of the individual sections.
Let's check this conclusion using the following example. Let's take three constant resistances, the values of which are known (for example, R1 == 10 Ohms, R 2 = 20 Ohms and R 3 = 50 Ohms). Let's connect them in series (Fig. 2) and connect them to a current source whose EMF is 60 V (neglected).
Rice. 2. Example of series connection of three resistances
Let's calculate what readings should be given by the devices turned on, as shown in the diagram, if the circuit is closed. Let's determine the external resistance of the circuit: R = 10 + 20 + 50 = 80 Ohm.
Let's find the current in the circuit: 60 / 80 = 0.75 A
Knowing the current in the circuit and the resistance of its sections, we determine the voltage drop for each section of the circuit U 1 = 0.75 x 10 = 7.5 V, U 2 = 0.75 x 20 = 15 V, U3 = 0.75 x 50 = 37 .5 V.
Knowing the voltage drop in the sections, we determine the total voltage drop in the external circuit, i.e. the voltage at the terminals of the current source U = 7.5 + 15 + 37.5 = 60 V.
We thus obtained that U = 60 V, i.e., the non-existent equality of the emf of the current source and its voltage. This is explained by the fact that we neglected the internal resistance of the current source.
Having now closed the key switch K, we can verify from the instruments that our calculations are approximately correct.
Let's take two constant resistances R1 and R2 and connect them so that the beginnings of these resistances are included in one common point a, and the ends are included in another common point b. By then connecting points a and b with a current source, we obtain a closed electrical circuit. This connection of resistances is called a parallel connection.
Figure 3. Parallel connection of resistances
Let's trace the current flow in this circuit. From the positive pole of the current source, the current will reach point a along the connecting conductor. At point a it will branch, since here the circuit itself branches into two separate branches: the first branch with resistance R1 and the second with resistance R2. Let us denote the currents in these branches by I1 and I 2, respectively. Each of these currents will go along its own branch to point b. At this point, the currents will merge into one common current, which will come to the negative pole of the current source.
Thus, when connecting resistances in parallel, a branched circuit is obtained. Let's see what the relationship between the currents in the circuit we have compiled will be.
Let's turn on the ammeter between the positive pole of the current source (+) and point a and note its readings. Having then connected the ammeter (shown in the dotted line in the figure) to the wire connecting point b to the negative pole of the current source (-), we note that the device will show the same amount of current.
This means that before its branching (to point a) it is equal to the current strength after the branching of the circuit (after point b).
We will now turn on the ammeter in turn in each branch of the circuit, remembering the readings of the device. Let the ammeter show current I1 in the first branch, and I 2 in the second. Adding these two ammeter readings, we get a total current equal in value to current I until the branching (to point a).
Hence, the strength of the current flowing to the branching point is equal to the sum of the currents flowing from this point. I = I1 + I2 Expressing this by the formula, we get
This relationship, which is of great practical importance, is called branched chain law.
Let us now consider what the relationship between the currents in the branches will be.
Let's turn on the voltmeter between points a and b and see what it shows us. Firstly, the voltmeter will show the voltage of the current source as it is connected, as can be seen in Fig. 3, directly to the terminals of the current source. Secondly, the voltmeter will show the voltage drops U1 and U2 across resistances R1 and R2, since it is connected to the beginning and end of each resistance.
Therefore, when resistances are connected in parallel, the voltage at the terminals of the current source is equal to the voltage drop across each resistance.
This gives us the right to write that U = U1 = U2.
where U is the voltage at the terminals of the current source; U1 - voltage drop across resistance R1, U2 - voltage drop across resistance R2. Let us remember that the voltage drop across a section of the circuit is numerically equal to the product of the current flowing through this section and the resistance of the section U = IR.
Therefore, for each branch we can write: U1 = I1R1 and U2 = I2R2, but since U1 = U2, then I1R1 = I2R2.
Applying the rule of proportion to this expression, we obtain I1 / I2 = U2 / U1 i.e. the current in the first branch will be as many times greater (or less) than the current in the second branch, how many times the resistance of the first branch is less (or greater) than the resistance of the second branches.
So we have come to the important conclusion that When resistances are connected in parallel, the total current of the circuit branches into currents that are inversely proportional to the resistance values of the parallel branches. In other words, the greater the resistance of a branch, the less current will flow through it, and, conversely, the less resistance of a branch, the greater the current will flow through this branch.
Let us verify the correctness of this dependence using the following example. Let's assemble a circuit consisting of two parallel-connected resistances R1 and R2 connected to a current source. Let R1 = 10 ohms, R2 = 20 ohms and U = 3 V.
Let's first calculate what the ammeter included in each branch will show us:
I1 = U / R1 = 3 / 10 = 0.3 A = 300 mA
I 2 = U / R 2 = 3 / 20 = 0.15 A = 150 mA
Total current in the circuit I = I1 + I2 = 300 + 150 = 450 mA
Our calculation confirms that when resistances are connected in parallel, the current in the circuit branches out in inverse proportion to the resistances.
Indeed, R1 == 10 Ohm is half as much as R 2 = 20 Ohm, while I1 = 300 mA is twice as much as I2 = 150 mA. The total current in the circuit I = 450 mA branched into two parts so that most of it (I1 = 300 mA) went through a smaller resistance (R1 = 10 Ohms), and a smaller part (R2 = 150 mA) went through a larger resistance (R 2 = 20 Ohm).
This branching of current in parallel branches is similar to the flow of liquid through pipes. Imagine pipe A, which at some point branches into two pipes B and C of different diameters (Fig. 4). Since the diameter of pipe B is larger than the diameter of pipes C, more water will pass through pipe B at the same time than through pipe B, which offers greater resistance to the flow of water.
Rice. 4
Let us now consider what the total resistance of an external circuit consisting of two parallel-connected resistances will be equal to.
Underneath this The total resistance of the external circuit must be understood as a resistance that could replace both parallel-connected resistances at a given circuit voltage, without changing the current before branching. This resistance is called equivalent resistance.
Let's return to the circuit shown in Fig. 3, and let’s see what the equivalent resistance of two parallel-connected resistances will be. Applying Ohm's law to this circuit, we can write: I = U/R, where I is the current in the external circuit (up to the branch point), U is the voltage of the external circuit, R is the resistance of the external circuit, i.e. equivalent resistance.
Similarly, for each branch I1 = U1 / R1, I2 = U2 / R2, where I1 and I 2 are the currents in the branches; U1 and U2 - voltage on branches; R1 and R2 - branch resistances.
According to the branched chain law: I = I1 + I2
Substituting the current values, we get U / R = U1 / R1 + U2 / R2
Since in a parallel connection U = U1 = U2, we can write U / R = U / R1 + U / R2
Taking U on the right side of the equality out of brackets, we get U / R = U (1 / R1 + 1 / R2)
Now dividing both sides of the equality by U, we will finally have 1 / R = 1 / R1 + 1 / R2
Remembering that conductivity is the reciprocal of resistance, we can say that in the resulting formula 1/R is the conductivity of the external circuit; 1 / R1 conductivity of the first branch; 1/R2 is the conductivity of the second branch.
Based on this formula we conclude: with a parallel connection, the conductivity of the external circuit is equal to the sum of the conductivities of the individual branches.
Hence, to determine the equivalent resistance of resistances connected in parallel, it is necessary to determine the conductivity of the circuit and take its reciprocal value.
It also follows from the formula that the conductivity of the circuit is greater than the conductivity of each branch, which means that the equivalent resistance of the external circuit is less than the smallest of the resistances connected in parallel.
Considering the case of parallel connection of resistances, we took the simplest circuit, consisting of two branches. However, in practice there may be cases when the chain consists of three or more parallel branches. What to do in these cases?
It turns out that all the relationships we obtained remain valid for a circuit consisting of any number of parallel-connected resistances.
To see this, consider the following example.
Let's take three resistances R1 = 10 Ohms, R2 = 20 Ohms and R3 = 60 Ohms and connect them in parallel. Let's determine the equivalent resistance of the circuit (Fig. 5).
Rice. 5. Circuit with three resistances connected in parallel
Applying the formula 1 / R = 1 / R1 + 1 / R2 for this circuit, we can write 1 / R = 1 / R1 + 1 / R2 + 1 / R3 and, substituting known values, we get 1 / R = 1 / 10 + 1 /20 + 1/60
Let's add these fractions: 1/R = 10/60 = 1/6, i.e. the conductivity of the circuit is 1/R = 1/6 Therefore, equivalent resistance R = 6 Ohm.
Thus, equivalent resistance is less than the smallest of the resistances connected in parallel in the circuit, i.e. less than resistance R1.
Let's now see whether this resistance is really equivalent, that is, one that could replace resistances of 10, 20 and 60 Ohms connected in parallel, without changing the current strength before branching the circuit.
Let us assume that the voltage of the external circuit, and therefore the voltage across the resistances R1, R2, R3, is 12 V. Then the current strength in the branches will be: I1 = U/R1 = 12 / 10 = 1.2 A I 2 = U/R 2 = 12 / 20 = 1.6 A I 3 = U/R1 = 12 / 60 = 0.2 A
We obtain the total current in the circuit using the formula I = I1 + I2 + I3 = 1.2 + 0.6 + 0.2 = 2 A.
Let's check, using the formula of Ohm's law, whether a current of 2 A will be obtained in the circuit if, instead of three parallel-connected resistances known to us, one equivalent resistance of 6 Ohms is connected.
I = U / R = 12 / 6 = 2 A
As we can see, the resistance R = 6 Ohm we found is indeed equivalent for this circuit.
You can also verify this using measuring instruments if you assemble a circuit with the resistances we took, measure the current in the external circuit (before branching), then replace the parallel-connected resistances with one 6 Ohm resistance and measure the current again. The ammeter readings in both cases will be approximately the same.
In practice, there may also be parallel connections for which it is possible to calculate the equivalent resistance more simply, i.e., without first determining the conductivities, you can immediately find the resistance.
For example, if two resistances R1 and R2 are connected in parallel, then the formula 1 / R = 1 / R1 + 1 / R2 can be transformed as follows: 1/R = (R2 + R1) / R1 R2 and, solving the equality with respect to R, obtain R = R1 x R2 / (R1 + R2), i.e. When two resistances are connected in parallel, the equivalent resistance of the circuit is equal to the product of the resistances connected in parallel divided by their sum.
In the previous summary, it was established that the current strength in a conductor depends on the voltage at its ends. If you change conductors in an experiment, leaving the voltage on them unchanged, then you can show that at a constant voltage at the ends of the conductor, the current strength is inversely proportional to its resistance. Combining the dependence of current on voltage and its dependence on conductor resistance, we can write: I = U/R . This law, established experimentally, is called Ohm's law(for a section of chain).
Ohm's law for a circuit section: The current strength in a conductor is directly proportional to the voltage applied to its ends and inversely proportional to the resistance of the conductor. First of all, the law is always true for solid and liquid metal conductors. And also for some other substances (usually solid or liquid).
Consumers of electrical energy (light bulbs, resistors, etc.) can be connected to each other in different ways in an electrical circuit. Dva main types of conductor connections : serial and parallel. And there are also two more connections that are rare: mixed and bridge.
Series connection of conductors
When connecting conductors in series, the end of one conductor will connect to the beginning of another conductor, and its end to the beginning of a third, etc. For example, connecting light bulbs in a Christmas tree garland. When the conductors are connected in series, current passes through all the bulbs. In this case, the same charge passes through the cross section of each conductor per unit time. That is, the charge does not accumulate in any part of the conductor.
Therefore, when connecting conductors in series The current strength in any part of the circuit is the same:I 1 = I 2 = I .
The total resistance of series-connected conductors is equal to the sum of their resistances: R1 + R2 = R . Because when conductors are connected in series, their total length increases. It is greater than the length of each individual conductor, and the resistance of the conductors increases accordingly.
According to Ohm's law, the voltage on each conductor is equal to: U 1 = I* R 1 ,U 2 = I*R 2 . In this case, the total voltage is U = I( R1+ R 2) . Since the current strength in all conductors is the same, and the total resistance is equal to the sum of the resistances of the conductors, then the total voltage on series-connected conductors is equal to the sum of the voltages on each conductor: U = U 1 + U 2 .
From the above equalities it follows that a series connection of conductors is used if the voltage for which the electrical energy consumers are designed is less than the total voltage in the circuit.
For series connection of conductors, the following laws apply: :
1) the current strength in all conductors is the same; 2) the voltage across the entire connection is equal to the sum of the voltages on the individual conductors; 3) the resistance of the entire connection is equal to the sum of the resistances of the individual conductors.
Parallel connection of conductors
Example parallel connection conductors serve to connect electrical energy consumers in the apartment. Thus, light bulbs, a kettle, an iron, etc. are switched on in parallel.
When connecting conductors in parallel, all conductors at one end are connected to one point in the circuit. And the second end to another point in the chain. A voltmeter connected to these points will show the voltage on both conductor 1 and conductor 2. In this case, the voltage at the ends of all parallel-connected conductors is the same: U 1 = U 2 = U .
When conductors are connected in parallel, the electrical circuit branches out. Therefore, part of the total charge passes through one conductor, and part through the other. Therefore, when connecting conductors in parallel, the current strength in the unbranched part of the circuit is equal to the sum of the current strength in the individual conductors: I = I 1 + I 2 .
According to Ohm's law I = U/R, I 1 = U 1 /R 1, I 2 = U 2 /R 2 . It follows from this: U/R = U 1 /R 1 + U 2 /R 2, U = U 1 = U 2, 1/R = 1/R 1 + 1/R 2 The reciprocal of the total resistance of parallel-connected conductors is equal to the sum of the reciprocals of the resistance of each conductor.
When conductors are connected in parallel, their total resistance is less than the resistance of each conductor. Indeed, if two conductors having the same resistance are connected in parallel G, then their total resistance is equal to: R = g/2. This is explained by the fact that when connecting conductors in parallel, their cross-sectional area increases. As a result, resistance decreases.
From the above formulas it is clear why consumers of electrical energy are connected in parallel. They are all designed for a certain identical voltage, which in apartments is 220 V. Knowing the resistance of each consumer, you can calculate the current strength in each of them. And also the correspondence of the total current strength to the maximum permissible current strength.
For parallel connection of conductors, the following laws apply:
1) the voltage on all conductors is the same; 2) the current strength at the junction of the conductors is equal to the sum of the currents in the individual conductors; 3) the reciprocal value of the resistance of the entire connection is equal to the sum of the reciprocal values of the resistance of individual conductors.
« Physics - 10th grade"
What does the dependence of the current in a conductor on the voltage across it look like?
What does the dependence of the current strength in a conductor on its resistance look like?
From a current source, energy can be transmitted through wires to devices that consume energy: an electric lamp, a radio receiver, etc. For this, they are composed electrical circuits of varying complexity.
The simplest and most common conductor connections include series and parallel connections.
Series connection of conductors.
With a series connection, the electrical circuit has no branches. All conductors are connected to the circuit one after another. Figure (15.5, a) shows a series connection of two conductors 1 and 2, having resistances R 1 and R 2. These can be two lamps, two electric motor windings, etc.
The current strength in both conductors is the same, i.e.
I 1 = I 2 = I. (15.5)
In conductors, electric charge does not accumulate in the case of direct current, and the same charge passes through any cross section of the conductor over a certain time.
The voltage at the ends of the circuit section under consideration is the sum of the voltages on the first and second conductors:
Applying Ohm's law for the entire section as a whole and for sections with conductor resistances R1 and R2, it can be proven that the total resistance of the entire section of the circuit when connected in series is equal to:
R = R 1 + R 2. (15.6)
This rule can be applied to any number of conductors connected in series.
The voltages on the conductors and their resistances in a series connection are related by the relation
Parallel connection of conductors.
Figure (15.5 b) shows a parallel connection of two conductors 1 and 2 with resistances R 1 and R 2. In this case, the electric current I branches into two parts. We denote the current strength in the first and second conductors by I 1 and I 2.
Since at point a - the branching of the conductors (such a point is called a node) - the electric charge does not accumulate, the charge entering the node per unit time is equal to the charge leaving the node during the same time. Hence,
I = I 1 + I 2. (15.8)
The voltage U at the ends of conductors connected in parallel is the same, since they are connected to the same points in the circuit.
The lighting network usually maintains a voltage of 220 V. Devices that consume electrical energy are designed for this voltage. Therefore, parallel connection is the most common way to connect different consumers. In this case, the failure of one device does not affect the operation of the others, whereas with a series connection, the failure of one device opens the circuit. Applying Ohm's law for the entire section as a whole and for sections of conductors with resistances R 1 and R 2, it can be proven that the reciprocal of the total resistance of the section ab is equal to the sum of the reciprocals of the resistances of individual conductors:
It follows that for two conductors
The voltages on parallel-connected conductors are equal: I 1 R 1 = I 2 R 2. Hence,
Let us pay attention to the fact that if in some section of the circuit through which direct current flows, a capacitor is connected in parallel to one of the resistors, then the current will not flow through the capacitor, the circuit in the section with the capacitor will be open. However, between the plates of the capacitor there will be a voltage equal to the voltage across the resistor, and a charge q = CU will accumulate on the plates.
Let's consider a chain of resistances R - 2R, called a matrix (Fig. 15.6).
On the last (right) link of the matrix, the voltage is divided in half due to the equality of resistances; on the previous link, the voltage is also divided in half, since it is distributed between a resistor with resistance R and two parallel resistors with resistance 2R, etc. This idea - voltage division - lies in the basis of converting binary code into direct voltage, which is necessary for the operation of computers.
