Total resistance in series connection. Parallel and serial connection. Series and parallel connections of conductors
Let's check the validity of the formulas shown here using a simple experiment.
Let's 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 with a digital multimeter. As we can see, it is equal to the sum of the resistances of the resistors included in this chain.
Measuring 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. This leads to another well-known rule that can be applied in 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 is taken into account. For example, we need to select a replacement resistor for 100 Ohm and power 1 W. Let's take two resistors of 50 ohms each and connect them in series. How much power dissipation should these two resistors be rated for?
Since the same direct current flows through series-connected resistors (for example 0.1 A), and the resistance of each of them is equal 50 ohm, then the dissipation power of each of them must be at least 0.5 W. As a result, each of them will have 0.5 W power. In total this will be the same 1 W.
This example is quite crude. Therefore, if in doubt, you should take resistors with a power reserve.
Read more about resistor power dissipation.
Secondly, when connecting, you should use resistors of the same type, for example, the MLT series. Of course, there is nothing wrong with taking different ones. This is just a recommendation.
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 charges cannot accumulate at current branching points (nodes A and B) in a DC circuit. 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 circuits 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, several resistors can be connected to one point (electrical node). With this connection, a separate current will flow through each resistor. The strength of this current will be inversely proportional to the resistance of the resistor.
Thus, when connecting resistors with different resistances in parallel, the total resistance will always be less than the value of the smallest individual 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 is also a faster way to calculate 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 the simplest example of a mixed resistor connection. 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 with a mixed connection of 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, calculating the resistance of two parallel resistors is much more convenient. Parallel connection of resistors is often used in cases where higher power resistance is needed. To do this, as a rule, resistors with the same power and the same resistance are used.
Total resistance Rtot
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 a circuit, it can be proven that the total resistance in a series connection is equal to the sum of the resistances of the individual conductors.
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 the physical verification itself.
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 kind of religion that controls 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.
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. Equivalent are those transformations of a part of an electrical circuit circuit in which the currents and voltages in the non-transformed part 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 striking example of a series connection is an 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 a significant disadvantage of a 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.
Parallel and series connection of conductors are methods of switching an electrical circuit. Electrical circuits of any complexity can be represented using these abstractions.
Definitions
There are two ways to connect conductors; it becomes possible to simplify the calculation of a circuit of arbitrary complexity:
- The end of the previous conductor is connected directly to the beginning of the next one - the connection is called serial. A chain is formed. To turn on the next link, you need to break the electrical circuit by inserting a new conductor there.
- The beginnings of the conductors are connected by one point, the ends by another, the connection is called parallel. A ligament is usually called a branch. Each individual conductor forms a branch. Common points are called electrical network nodes.
In practice, a mixed connection of conductors is more common, some are connected in series, some in parallel. You need to break the chain into simple segments and solve the problem for each separately. An arbitrarily complex electrical circuit can be described by a parallel, series connection of conductors. This is how it is done in practice.
Using parallel and series connection of conductors
Terms applied to electrical circuits
Theory serves as the basis for the formation of solid knowledge; few people know how voltage (potential difference) differs from voltage drop. In physics terms, the internal circuit is the current source; the one located outside is called the external circuit. The demarcation helps to correctly describe the distribution of the field. The current does work. In the simplest case, heat generation follows the Joule-Lenz law. Charged particles, moving towards a lower potential, collide with the crystal lattice and release energy. The resistances heat up.
To ensure movement, it is necessary to maintain a potential difference at the ends of the conductor. This is called circuit section voltage. If you simply place a conductor in a field along the power lines, the current will flow and will be very short-lived. The process will end with the onset of equilibrium. The external field will be balanced by the own field of charges, in the opposite direction. The current will stop. For the process to become continuous, an external force is needed.
The current source acts as such a drive for the movement of the electrical circuit. To maintain potential, work is done inside. Chemical reaction, as in a galvanic cell, mechanical forces - a hydroelectric power station generator. The charges inside the source move in the direction opposite to the field. The work of outside forces is being done on this. You can paraphrase the above formulations and say:
- The outer part of the circuit, where the charges move, carried away by the field.
- The interior of a circuit where charges move against the voltage.
The generator (current source) is equipped with two poles. The one with less potential is called negative, the other is called positive. In the case of alternating current, the poles continuously change places. The direction of movement of charges is not constant. Current flows from the positive pole to the negative pole. The movement of positive charges goes in the direction of decreasing potential. According to this fact, the concept of potential drop is introduced:
The potential drop of a section of a circuit is the decrease in potential within the section. Formally, this is tension. For branches of a parallel circuit it is the same.
Voltage drop also means something else. The value characterizing heat losses is numerically equal to the product of the current and the active resistance of the section. Ohm's and Kirchhoff's laws, discussed below, are formulated for this case. In electric motors and transformers, the potential difference can differ significantly from the voltage drop. The latter characterizes losses in active resistance, while the former takes into account the full operation of the current source.
When solving physical problems, for simplification, the motor can include an EMF, the direction of action of which is opposite to the effect of the power source. The fact of energy loss through the reactive part of the impedance is taken into account. School and university physics courses are distinguished by their isolation from reality. That is why students listen with open mouths about the phenomena taking place in electrical engineering. In the period preceding the era of the industrial revolution, the main laws were discovered; a scientist must combine the role of a theorist and a talented experimenter. The prefaces to Kirchhoff's works openly speak about this (Georg Ohm's works have not been translated into Russian). The teachers literally attracted people with additional lectures, flavored with visual, amazing experiments.
Ohm's and Kirchhoff's laws as applied to series and parallel connection of conductors
Ohm's and Kirchhoff's laws are used to solve real problems. The first deduced equality purely empirically - experimentally - the second began with a mathematical analysis of the problem, then tested his guesses with practice. Here is some information to help solve the problem:
Calculate the resistance of elements in series and parallel connection
The algorithm for calculating real circuits is simple. Here are some points regarding the topic under consideration:
- When connected in series, the resistances are summed up; when connected in parallel, the conductivities are summed up:
- For resistors, the law is rewritten in unchanged form. With a parallel connection, the final resistance is equal to the product of the original ones divided by the total amount. In case of sequential, the denominations are summed up.
- Inductance acts as reactance (j*ω*L) and behaves like a regular resistor. In terms of writing the formula, it is no different. The nuance, for any purely imaginary impedance, is that you need to multiply the result by the operator j, the circular frequency ω (2*Pi*f). When inductors are connected in series, the values are summed up; when inductors are connected in parallel, the reciprocal values are added up.
- The imaginary resistance of the capacitance is written as: -j/ω*С. It’s easy to notice: adding up the values of a series connection, we get a formula exactly as it was for resistors and inductances in a parallel connection. For capacitors the opposite is true. When connected in parallel, the values are added; when connected in series, the reciprocal values are added.
The theses can easily be extended to arbitrary cases. The voltage drop across two open silicon diodes is equal to the sum. In practice it is 1 volt, the exact value depends on the type of semiconductor element and characteristics. Power supplies are considered in a similar way: when connected in series, the ratings are added up. Parallel is often found in substations where transformers are placed side by side. The voltage will be the same (controlled by equipment), divided between the branches. The transformation coefficient is strictly equal, blocking the occurrence of negative effects.
Some people find it difficult: two batteries of different ratings are connected in parallel. The case is described by Kirchhoff's second law; physics cannot imagine any complexity. If the ratings of two sources are unequal, the arithmetic mean is taken, if the internal resistance of both is neglected. Otherwise, the Kirchhoff equations are solved for all contours. The unknown currents will be (three in total), the total number of which is equal to the number of equations. For complete understanding, a drawing has been provided.
An example of solving Kirchhoff's equations
Let's look at the image: according to the conditions of the problem, source E1 is stronger than E2. We take the direction of the currents in the circuit from common sense. But if they had entered it incorrectly, after solving the problem, one would have turned out with a negative sign. Then it was necessary to change direction. Obviously, current flows in the external circuit as shown in the figure. We compose the Kirchhoff equations for three circuits, this is what follows:
- The work of the first (strong) source is spent on creating a current in the external circuit, overcoming the weakness of the neighbor (current I2).
- The second source does not perform useful work on the load and fights with the first. There's no other way to say it.
Connecting batteries of different ratings in parallel is certainly harmful. What is observed at a substation when using transformers with different transmission ratios. Equalizing currents do no useful work. Different batteries connected in parallel will begin to function effectively when the strong one drops to the level of the weak one.
