What happens to power when connected in parallel. Parallel connection of resistors
Series connection of resistances
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 is connected 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 alternating. Of course, the current in such a circuit will be the same at all its points.
Rice 1 . Series connection of resistances
How to find the total resistance of a circuit if we already know all the resistances included in it one by one? 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 have R = R1 + R2 + R3
Thus, we concluded that when resistances are alternately connected, 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 one by one (Fig. 2) and connect them to a current source whose EMF is 60 V ( internal resistance current source is neglected).
Rice. 2. Example of alternate connection of 3 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 using Ohm's law: 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 got it in such a way 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 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. By then connecting points a and b with a current source, we obtain a closed electronic 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, 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 by I1 and I 2, respectively. Any of these currents will follow 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 resistances are connected in parallel, a branched circuit results. Let's see what the relationship between the currents in the circuit we have created 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.
Means current in the circuit before it branches(up to point a) is equal to current strength 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 strength in the first branch I1, and in the 2nd branch - I 2. By adding these two ammeter readings, we get the total current, in magnitude equal to current I until the branching (to point a).
Properly, 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 because it is connected, as can be seen from Fig. 3, specifically to the terminals of the current source. Secondly, the voltmeter will show the voltage drops U1 and U2 across resistances R1 and R2, because it is connected to the beginning and end of each resistance.
As follows, when connecting resistances 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 you can write: U1 = I1R1 and U2 = I2R2, but because 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 2nd branch, how many times the resistance of the first branch is less (or greater) resistance of the 2nd branch.
So, we have come to the fundamental 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 the branch, the less current will flow through it, and, conversely, the lower the resistance of the branch, the higher current will flow through this branch.
Let us verify the correctness of this dependence in 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 back in 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 the smallest resistance (R1 = 10 Ohms), and the smallest part (R2 = 150 mA) went through the larger resistance (R 2 = 20 Ohm).
This branching of 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 B, then through pipe B to the same time will pass more water than through pipe B, which provides 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 this The total resistance of the external circuit must be understood to be such a resistance that could be used to change 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 2 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 branching point), U is the voltage of the external circuit, R is the resistance of the external circuit, i.e. equivalent resistance.
In the same way, 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
Because with 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)
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 external 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 resistances connected in parallel, you need to find the conductivity of the circuit and take the 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 smaller of the resistances connected in parallel.
Considering the case of parallel connection of resistances, we took a more ordinary circuit consisting of two branches. But in practice, there may be cases when the chain consists of 3 or more parallel branches. What to do 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 see this, let's look at 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). R = 1/6 As follows, equivalent resistance R = 6 Ohm.
In this way, equivalent resistance is less than the smaller of the resistances connected in parallel in the circuit, i.e. less than resistance R1.
Let's see now whether this resistance is really equivalent, that is, one that could change 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 as follows, 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 3 parallel-connected resistances we recognize, one equivalent resistance of 6 Ohms is connected.
I = U / R = 12 / 6 = 2 A
As we see, the resistance we found R = 6 Ohms is indeed equivalent for this circuit.
You can also verify this using measuring devices 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 similar.
In practice, you may also encounter parallel connections, for which it is easier to calculate the equivalent resistance, i.e., without first determining the conductivity, 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 converted as follows: 1/R = (R2 + R1) / R1 R2 and, solving the equality with respect to R, get 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.
Content:All electrical circuits use resistors, which are elements with exactly set value resistance. Thanks 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 work of almost all electronic devices and equipment. So, the voltage in parallel and serial connection resistors will be different. Therefore, each type of connection can only be used under certain conditions, so that one or another electrical diagram could fully perform its functions.
Series voltage
In a series connection, two or more resistors are connected into a common circuit in such a way that each of them has contact with another 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, etc.
A feature of this circuit is that the same value passes through all connected resistors electric current. As the number of elements in the section of the circuit under consideration increases, the flow of electric current becomes more and more difficult. This occurs due to an increase in the total resistance of the resistors when they are connected in series. This property reflected by the formula: Rtot = 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 = I Rn x R n. Thus, as the resistance of the resistor increases, the voltage dropped across it also increases.
Parallel voltage
In a parallel connection, resistors are included in the electrical circuit in such a way that all resistance elements are connected to each other by both contacts at once. One point, representing an electrical node, can connect several resistors simultaneously.
This connection involves the flow of a separate current in each resistor. The strength of this current is inversely proportional. As a result, there is an increase in the overall conductivity of a given section of the circuit, with a general decrease in resistance. In the case of 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 diagram shown, the voltage between points A and B represents not only the total voltage for the entire section, but also the voltage supplied to each individual resistor. Thus, in case of parallel connection, the voltage applied to all resistors will be the same.
As a result, the voltage between parallel and series connections will be different in each case. Thanks to this property, there is a real opportunity to adjust this value at any part of the chain.
1. Find the equivalent resistance of sections of the circuit with parallel connection of resistors. Figure 2. Series connection of resistors. To calculate the resistance of such connections, the entire circuit is divided into simple sections, consisting of resistors connected in parallel or in series.
This result follows from the fact that at the current branching points (nodes A and B) in the circuit direct current charges cannot accumulate. This result is valid for any number of conductors connected in parallel.
In Fig. 1.9.3 shows an example of such a complex circuit and indicates the sequence of calculations. It should be noted that not all complex chains, consisting of conductors with different resistances, can be calculated using formulas for series and parallel connections.
When conductors are connected in series, the current in all conductors is the same. In a parallel connection, the voltage drop between the two nodes connecting the elements of the circuit is the same for all elements.
That is, the greater the resistance of the resistor, the greater the voltage drops across it. As a result, to one point ( electrical unit) several resistors can be connected. With this connection, a separate current will flow through each resistor. Force given current will be inversely proportional to the resistance of the resistor.
Thus, when parallel connection resistors with different resistances, the total resistance will always be less than value the smallest single resistor. The voltage between points A and B is both the total voltage for the entire circuit section and the voltage across each resistor individually. A mixed connection is a section of a circuit where some resistors are connected in series and some in parallel.
The circuit is divided into sections with only parallel or only serial connections. The total resistance is calculated for each individual section. Calculate the total resistance for the entire mixed connection circuit. There are also more quick way calculating the total resistance for a mixed connection. If the resistors are connected in series, add them together.
That is, with a series connection, the resistors will be connected one after another. Figure 4 shows simplest example mixed connection of resistors. After calculating the equivalent resistances of the resistors, the circuit is redrawn. Usually a circuit of equivalent resistances connected in series is obtained.4. Figure 5. Calculation of the resistance of a circuit section at mixed connection resistors.
As a result, you will learn from scratch not only how to develop your own devices, but also how to interface various peripherals with them! A node is a branching point in a circuit at which at least three conductors are connected. Series connection of resistors is used to increase resistance.
Parallel voltage
As you can see, calculate the resistance of two parallel resistors much more convenient. Parallel connection of resistors is often used in cases where resistance with more power. To do this, as a rule, resistors with the same power and the same resistance are used.
Total resistance Rtotal
This connection of resistances is called series. We thus obtained that U = 60 V, i.e. the non-existent equality of the emf of the current source and its voltage. We will now turn on the ammeter in turn in each branch of the circuit, remembering the readings of the device. 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 branching of current in parallel branches is similar to the flow of liquid through pipes. Let us now consider what the total resistance of an external circuit consisting of two parallel-connected resistances will be equal to.
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. Similarly, for each branch I1 = U1 / R1, I2 = U2 / R2, where I1 and I2 are the currents in the branches; U1 and U2 - voltage on branches; R1 and R2 - branch resistances.
This means that the total resistance of the circuit will always be lower than any resistor connected in parallel. 2. If these sections include resistors connected in series, then first calculate their resistance. By applying Ohm's law to a section of the circuit, it can be proven that impedance when connected in series, it is equal to the sum of the resistances of the individual conductors.
When solving problems, it is customary to transform the circuit so that it is as simple as possible. To do this, equivalent transformations are used. Such transformations of part of the circuit are called equivalent electrical circuit, at which currents and voltages in the non-transformed part of it remain unchanged.
There are four main types of conductor connections: series, parallel, mixed and bridge.
Serial connection
Serial connection- this is a connection in which the current strength throughout the entire circuit is the same. A prime example of a serial connection is the old Christmas tree garland. There the light bulbs are connected in series, one after another. Now imagine one light bulb burns out, the circuit is broken and the rest of the light bulbs go out. The failure of one element leads to the shutdown of all the others, this is significant drawback serial connection.
When connected in series, the resistances of the elements are summed up.
Parallel connection
Parallel connection- this is a connection in which the voltage at the ends of the circuit section is the same. Parallel connection is the most common, mainly because all the elements are under the same voltage, the current is distributed differently and when one of the elements exits, all the others continue to work.
In a parallel connection, the equivalent resistance is found as:
In the case of two parallel connected resistors 
In the case of three resistors connected in parallel:
Mixed compound
Mixed compound– a connection, which is a collection of serial and parallel connections. To find the equivalent resistance, you need to “collapse” the circuit by alternately transforming parallel and serial sections of the circuit.
First, let's find the equivalent resistance for the parallel section of the circuit, and then add to it the remaining resistance R 3 . It should be understood that after the conversion, the equivalent resistance R 1 R 2 and resistor R 3 are connected in series.
So, that leaves the most interesting and most complex connection of conductors.
Bridge circuit
The bridge connection diagram is shown in the figure below.
In order to collapse the bridge circuit, one of the bridge triangles is replaced with an equivalent star.
And find the resistances R 1, R 2 and R 3.
Did you know,
What is a thought experiment, gedanken experiment?
This is a non-existent practice, an otherworldly experience, an imagination of something that does not actually exist. Thought experiments are like waking dreams. They give birth to monsters. Unlike a physical experiment, which is an experimental test of hypotheses, a “thought experiment” magically replaces experimental testing with desired conclusions that have not been tested in practice, manipulating logical constructions that actually violate logic itself by using unproven premises as proven ones, that is, by substitution. Thus, the main task of the applicants of “thought experiments” is to deceive the listener or reader by replacing a real physical experiment with its “doll” - fictitious reasoning on parole without any physical check.
Filling physics with imaginary, “thought experiments” has led to the emergence of an absurd, surreal, confused picture of the world. A real researcher must distinguish such “candy wrappers” from real values.
Relativists and positivists argue that “thought experiments” are a very useful tool for testing theories (also arising in our minds) for consistency. In this they deceive people, since any verification can only be carried out by a source independent of the object of verification. The applicant of the hypothesis himself cannot be a test of his own statement, since the reason for this statement itself is the absence of contradictions in the statement visible to the applicant.
We see this in the example of SRT and GTR, which have turned into a unique type of religion that governs science and public opinion. No amount of facts that contradict them can overcome Einstein’s formula: “If a fact does not correspond to the theory, change the fact” (In another version, “Does the fact not correspond to the theory? - So much the worse for the fact”).
The maximum that a “thought experiment” can claim is only the internal consistency of the hypothesis within the framework of the applicant’s own, often by no means true, logic. This does not check compliance with practice. Real verification can only take place in an actual physical experiment.
An experiment is an experiment because it is not a refinement of thought, but a test of thought. A thought that is self-consistent cannot verify itself. This was proven by Kurt Gödel.