Electrical resistance of two resistors connected in series. Conductor current in parallel and series connection
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 total resistance the resulting chain digital multimeter. As we can see, it is equal to the sum of the resistances of the resistors included in this chain.
Measurement of total resistance at serial 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 current flows through resistors connected in series D.C.(let's say 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, on each of them there will be 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.
Consistent This 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 explained by the fact that charges cannot accumulate in the nodes of the circuit. Their accumulation would lead to a change in the electric field strength, and consequently to a change in the current strength. That's why
Ammeter A measures the current in a circuit and has a small internal resistance(R A 0).
The switched on voltmeters V 1 and V 2 measure the voltage U 1 and U 2 across resistances R 1 and R 2 . Voltmeter V measures the voltage U supplied to terminals M and N. Voltmeters show that with a series connection, voltage U is equal to the sum of the voltages in individual sections of the circuit:
Applying Ohm's law for each section of the circuit, we obtain:
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 individual resistors are equal to each other, i.e. R 1 = R 2 = ... = R n, then the total resistance of these resistors when connected in series is n times greater than the resistance of one resistor: R = nR 1.
When connecting resistors in series, the following relation is true:
those. The voltages across the resistors are directly proportional to the resistances.
Parallel This connection of resistors is called when some ends of all resistors are connected into one node, the other ends into another node (Fig. 2). A node is a point in a branched circuit where 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 strength I in the unbranched part of the circuit is equal to the sum of the current strengths I 1 and I 2 in parallel connected conductors R 1 and R 2:
This also follows from the law of conservation of electric charge. Let's apply Ohm's law for individual areas circuit and the entire circuit with total resistance R:
Substituting I, I 1 and I 2 into formula (2), we get.
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 when connecting resistors in parallel and in series 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, turning on the resistors in electrical circuit is performed in such a way that all resistance elements are connected to each other by both contacts at once. One point representing electrical unit, 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.
Moreover, these can be not only conductors, but also capacitors. It is important here not to get confused about what each of them looks like on the diagram. And only then apply specific formulas. By the way, you need to remember them by heart.
How can you differentiate between these two compounds?
Look carefully at the diagram. If you imagine the wires as a road, then the cars on it will play the role of resistors. On a straight road without any branches, cars drive one after another, in a chain. The series connection of conductors looks the same. In this case, the road can have an unlimited number of turns, but not a single intersection. No matter how the road (wires) twist, the machines (resistors) will always be located one after another, in one chain.
It is a completely different matter if one considers parallel connection. Then the resistors can be compared to athletes at the start line. They each stand on their own path, but their direction of movement is the same, and the finish line is in the same place. The same goes for resistors - each of them has its own wire, but they are all connected at some point.
Formulas for current strength
Always about her we're talking about in the topic "Electricity". Parallel and series connections have different effects on the value in resistors. Formulas have been derived for them that can be remembered. But it’s enough just to remember the meaning that is put into them.
So, the current when connecting conductors in series is always the same. That is, in each of them the current value is not different. An analogy can be drawn by comparing a wire with a pipe. The water always flows in it the same way. And all obstacles in her path will be swept away with the same force. Same with current strength. Therefore, the formula for the total current in a circuit with resistors connected in series looks like this:
I total = I 1 = I 2
Here the letter I denotes the current strength. This is a common designation, so you need to remember it.
The current in a parallel connection will no longer be a constant value. Using the same analogy with a pipe, it turns out that water will split into two streams if the main pipe has a branch. The same phenomenon is observed with current when a branching wire appears in its path. Formula for total current at:
I total = I 1 + I 2
If the branching is made up of more than two wires, then in the above formula there will be more terms by the same number.
Formulas for voltage
When we consider a circuit in which the conductors are connected in series, the voltage across the entire section is determined by the sum of these values on each specific resistor. You can compare this situation with plates. One person can easily hold one of them; he can also take the second one nearby, but with difficulty. One person will no longer be able to hold three plates in their hands next to each other; the help of a second person will be required. And so on. People's efforts add up.
The formula for the total voltage of a circuit section with a series connection of conductors looks like this:
U total = U 1 + U 2, where U is the designation adopted for
A different situation arises when considering When the plates are stacked on top of each other, they can still be held by one person. Therefore, there is no need to fold anything. The same analogy is observed when connecting conductors in parallel. The voltage on each of them is the same and equal to that on all of them at once. The formula for total voltage is:
U total = U 1 = U 2
Formulas for electrical resistance
You no longer need to memorize them, but know the formula of Ohm’s law and derive the necessary one from it. From this law it follows that voltage is equal to the product of current and resistance. That is, U = I * R, where R is resistance.
Then the formula you need to work with depends on how the conductors are connected:
- sequentially, which means we need equality for the voltage - I total * R total = I 1 * R 1 + I 2 * R 2;
- in parallel, it is necessary to use the formula for current strength - Utot / Rtot = U 1 / R 1 + U 2 / R 2 .
What follows are simple transformations, which are based on the fact that in the first equality all currents have same value, and in the second - the voltages are equal. This means they can be reduced. That is, the following expressions are obtained:
- R total = R 1 + R 2 (for series connection of conductors).
- 1 / R total = 1 / R 1 + 1 / R 2 (for parallel connection).
As the number of resistors that are connected to the network increases, the number of terms in these expressions changes.
It is worth noting that parallel and series connections of conductors have different effects on the total resistance. The first of them reduces the resistance of the circuit section. Moreover, it turns out to be smaller than the smallest of the resistors used. With a serial connection, everything is logical: the values are added, so the total number will always be the largest.
Current work
The previous three quantities make up the laws of parallel connection and series arrangement of conductors in a circuit. Therefore, it is imperative to know them. You just need to remember about work and power basic formula. It is written like this: A = I * U * t, where A is the work done by the current, t is the time it passes through the conductor.
In order to determine general work When connecting in series, you need to replace the voltage in the original expression. The result is the equality: A = I * (U 1 + U 2) * t, opening the brackets in which it turns out that the work on the entire section is equal to their sum on each specific current consumer.
The reasoning is similar if a parallel connection scheme is considered. Only the current strength must be replaced. But the result will be the same: A = A 1 + A 2.
Current power
When deriving the formula for the power (designation “P”) of a section of the circuit, you again need to use one formula: P = U * I. After similar reasoning, it turns out that parallel and serial connections are described by the following formula for power: P = P 1 + P 2.
That is, no matter how the circuits are drawn up, the total power will be the sum of those involved in the work. This explains the fact that you cannot connect many powerful devices to your apartment’s network at the same time. She simply cannot withstand such a load.
How does the connection of conductors affect the repair of a New Year's garland?
Immediately after one of the bulbs burns out, it will become clear how they were connected. When connected in series, none of them will light up. This is explained by the fact that a lamp that has become unusable creates a break in the circuit. Therefore, you need to check everything to determine which one is burned out, replace it - and the garland will start working.
If it uses a parallel connection, it does not stop working if one of the bulbs fails. After all, the chain will not be completely broken, but only one parallel part. To repair such a garland, you do not need to check all the elements of the circuit, but only those that do not light up.
What happens to a circuit if it includes capacitors rather than resistors?
When they are connected in series, the following situation is observed: charges from the pluses of the power source are supplied only to the outer plates of the outer capacitors. Those that are between them simply transfer this charge along the chain. This explains the fact that identical charges appear on all plates, but with different signs. That's why electric charge each capacitor connected in series can be written as follows:
q total = q 1 = q 2.
In order to determine the voltage on each capacitor, you will need to know the formula: U = q / C. In it, C is the capacitance of the capacitor.
The total voltage obeys the same law that is valid for resistors. Therefore, replacing the voltage with the sum in the capacitance formula, we get that the total capacitance of the devices must be calculated using the formula:
C = q / (U 1 + U 2).
You can simplify this formula by reversing the fractions and replacing the voltage-to-charge ratio with capacitance. We get the following equality: 1 / C = 1 / C 1 + 1 / C 2 .
The situation looks somewhat different when the capacitors are connected in parallel. Then the total charge is determined by the sum of all charges that accumulate on the plates of all devices. And the voltage value is still determined according to general laws. Therefore, the formula for the total capacitance of parallel-connected capacitors looks like this:
C = (q 1 + q 2) / U.
That is, this value is calculated as the sum of each of the devices used in the connection:
C = C 1 + C 2.
How to determine the total resistance of an arbitrary connection of conductors?
That is, one in which successive sections replace parallel ones, and vice versa. All the laws described are still valid for them. You just need to apply them step by step.
First, you need to mentally unfold the diagram. If it’s difficult to imagine, then you need to draw what you get. The explanation will become clearer if we consider it in specific example(see picture).
It is convenient to start drawing it from points B and C. They need to be placed at some distance from each other and from the edges of the sheet. One wire approaches point B from the left, and two are already directed to the right. Point B, on the contrary, on the left has two branches, and after it there is one wire.
Now you need to fill the space between these points. Along the top wire you need to place three resistors with coefficients 2, 3 and 4, and the one with the index equal to 5 will go below. The first three are connected in series. They are parallel with the fifth resistor.
The remaining two resistors (the first and sixth) are connected in series with the considered section of the BV. Therefore, the drawing can simply be supplemented with two rectangles on either side of the selected points. It remains to apply the formulas to calculate the resistance:
- first the one given for the serial connection;
- then for parallel;
- and again for consistency.
In this way, you can deploy any, even very complex, scheme.
Problem on serial connection of conductors
Condition. Two lamps and a resistor are connected in a circuit one behind the other. The total voltage is 110 V and the current is 12 A. What is the value of the resistor if each lamp is rated at 40 V?
Solution. Since a series connection is considered, the formulas of its laws are known. You just need to apply them correctly. Start by finding out the voltage across the resistor. To do this, you need to subtract the voltage of one lamp twice from the total. It turns out 30 V.
Now that two quantities are known, U and I (the second of them is given in the condition, since total current equal to current in each series consumer), you can calculate the resistance of the resistor using Ohm’s law. It turns out to be equal to 2.5 ohms.
Answer. The resistor's resistance is 2.5 ohms.
Parallel and serial problem
Condition. There are three capacitors with capacities of 20, 25 and 30 μF. Determine their total capacitance when connected in series and in parallel.
Solution. It's easier to start with In this situation, all three values just need to be added. Thus, the total capacitance is equal to 75 µF.
The calculations will be somewhat more complicated when these capacitors are connected in series. After all, you first need to find the ratio of one to each of these containers, and then add them to each other. It turns out that one divided by the total capacity is equal to 37/300. Then the desired value is approximately 8 µF.
Answer. The total capacitance for a series connection is 8 µF, for a parallel connection - 75 µF.
Details Category: Articles Created: 09/06/2017 19:48How to connect several lamps in a dollhouse
When you think about how to make lighting in a dollhouse or roombox, where there is not one, but several lamps, the question arises of how to connect them and network them. There are two types of connections: serial and parallel, which we heard about from school. We will consider them in this article.
I'll try to keep it simple accessible language, so that everything is clear even to the most humanists who are not familiar with electrical intricacies.
Note: In this article we will only consider a circuit with incandescent light bulbs. Lighting with diodes is more complex and will be discussed in another article.
For understanding, each diagram will be accompanied by a drawing and an electrical wiring diagram next to the drawing.
Let's first consider symbols on electrical diagrams.
| Item name | Symbol on the diagram | Image |
| battery/battery | ||
| switch | ||
| the wire | ||
| crossing of wires (without connection) | ||
| connecting wires (soldering, twisting) | ||
| incandescent lamp | ||
| faulty lamp | ||
| broken lamp | ||
| burning lamp |
As already mentioned, there are two main types of connections: serial and parallel. There is also a third, mixed: series-parallel, combining both. Let's start with the sequential one, as it is simpler.
Serial connection
It looks like this.
The light bulbs are placed one after the other, as if holding hands in a round dance. Old Soviet garlands were made according to this principle.
Advantages- ease of connection.
Flaws- if at least one light bulb burns out, the entire circuit will not work.
You will have to go through and check each light bulb to find the faulty one. This can be tedious when large quantities light bulbs Also, the light bulbs must be of the same type: voltage, power.
With this type of connection, the voltages of the light bulbs are added. The voltage is indicated by the letter U, measured in volts V. The voltage of the power supply must be equal to the sum of the voltages of all the bulbs in the circuit.
Example No. 1: you want to connect 3 1.5V light bulbs in a series circuit. The power source voltage required for the operation of such a circuit is 1.5+1.5+1.5=4.5V.
For ordinary AA batteries voltage 1.5V. To get a voltage of 4.5V from them, they also need to be connected in a series circuit, their voltages will add up.
Read more about how to choose a power source in this article.
Example #2: you want to connect 6V bulbs to a 12V power source. 6+6=12v. You can connect 2 of these bulbs.
Example #3: you want to connect 2 3V light bulbs in a circuit. 3+3=6V. A 6 V power supply is required.
To summarize: serial connection is easy to manufacture; you need bulbs of the same type. Disadvantages: if one bulb fails, not all of them light up. You can only turn the circuit on and off as a whole.
Based on this, to illuminate a dollhouse, it is advisable to connect no more than 2-3 light bulbs in series. For example, in sconces. To connect a larger number of light bulbs, you need to use a different type of connection - parallel.
Read also articles on the topic:
- Review of miniature incandescent lamps
- Diodes or incandescent lamps
Parallel connection of light bulbs
This is what it looks like parallel connection light bulbs
In this type of connection, all bulbs and the power supply have the same voltage. That is, with a 12v power source, each of the bulbs must also have a voltage of 12V. And the number of light bulbs may vary. And if, for example, you have 6V light bulbs, then you need to take a 6V power source.
When one bulb fails, the others continue to burn.
The light bulbs can be turned on independently of each other. To do this, each one needs to have its own switch.
Electrical appliances in our city apartments are connected according to this principle. All devices have the same voltage 220V, they can be turned on and off independently of each other, the power of electrical devices can be different.
Conclusion: When there are many lamps in a dollhouse, parallel connection is optimal, although it is a little more complicated than serial connection.
Let's consider another type of connection, combining serial and parallel.
Combined connection
An example of a combined connection.
Three series circuits connected in parallel
Here's another option:
Three parallel circuits connected in series.
Sections of such a circuit connected in series behave like a series connection. And parallel sections are like a parallel connection.
Example
With such a scheme, the burnout of one light bulb will disable the entire section connected in series, and the other two series circuits will remain operational.
Accordingly, sections can be turned on and off independently of each other. For this, everyone series circuit You need to install your own switch.
But you can't turn on just one light bulb.
With a parallel-series connection, if one light bulb fails, the circuit will behave like this:
And if there is a violation in a sequential section like this:
Example:
There are 6 3V bulbs connected in 3 series circuits of 2 bulbs each. The circuits, in turn, are connected in parallel. We divide it into 3 consecutive sections and calculate this section.
In the series section, the voltages of the light bulbs add up, 3v+3V=6V. Each series circuit has a voltage of 6V. Since the circuits are connected in parallel, their voltage does not add up, which means we need a 6V power source.
Example
We have 6 6V bulbs. The light bulbs are connected in groups of 3 in a parallel circuit, and the circuits, in turn, are connected in series. We divide the system into three parallel circuits.
One parallel circuit the voltage of each light bulb is 6V, since the voltage does not add up, then the voltage of the entire circuit is 6V. And the circuits themselves are already connected in series and their voltages are already added up. It turns out 6V+6V=12V. This means you need a 12V power source.
Example
For dollhouses, you can use this mixed connection.
Let's say there is one lamp in each room, all lamps are connected in parallel. But the lamps themselves have a different number of light bulbs: two have one light bulb each, there is a two-arm sconce made of two light bulbs and a three-arm chandelier. In a chandelier and sconce, the light bulbs are connected in series.
Each lamp has its own switch. Power supply 12V voltage. Single light bulbs connected in parallel must have a voltage of 12V. And for those connected in series, the voltage is added to the section of the circuit
. Accordingly, for a sconce section of two light bulbs, divide 12V (total voltage) by 2 (the number of light bulbs), we get 6V (voltage of one light bulb).
For the chandelier section 12V:3=4V (voltage of one chandelier bulb).
You should not connect more than three light bulbs in one lamp in series.
Now you have learned all the tricks of connecting incandescent light bulbs different ways. And I think that it won’t be difficult to make lighting in a dollhouse with many light bulbs, of any complexity. If something else is difficult for you, read the article about the simplest way to make light in a dollhouse, the most basic principles. Good luck!