Parallel connected capacitors formula. Connections of capacitors when connected in series

a) in parallel with a large capacitor, connect exactly the same capacitor, but with a small capacitance;

b) instead of one large capacitor, include two or three smaller capacitors of the same type;

c) instead of one large capacitor, include many small capacitors.

Naturally, it must be switched on in parallel, in which case the capacities are summed up, and the total capacity in all these cases is the same. Let's look into this issue (all the necessary information is in Table 1 and Fig. 47).

Option a). They say that a small capacitor will help a large one work.

The maximum operating frequency of a capacitor can be considered the frequency at which its resistance is minimal. Further, as the frequency increases, the total resistance of the capacitor begins to increase - this affects the inductance of the capacitor design. In this case, the inductive reactance outweighs the capacitive reactance, and the capacitor behaves like an inductor. That is, it is no longer a capacitor.

For a small-capacity capacitor, the minimum resistance actually occurs at a higher frequency, but its resistance is still greater than that of a large-capacity capacitor (the properties of which already deteriorate at this frequency). But the main task of a capacitor at these frequencies is to pass the load current through itself, influencing it as little as possible. Therefore, the lower the capacitor's resistance, the better. And a small capacitor will not really help a “large” capacitor, its resistance is too high. Only at point A the resistances of both capacitors become equal, and at a higher frequency the small capacitor has less resistance than the “large” capacitor. But look - at this point even a small capacitor does not work well! In reality, these graphs are shown in Fig. 47, where numbers 1...5 indicate capacitors of smaller capacity, and numbers 8...12 indicate capacitors of larger capacity.

But if there is a ceramic or film capacitor in the system, then it works well both at this frequency and at higher frequencies (Fig. 48). Only its capacity must be large enough,

so that at the desired frequencies it has low resistance.

Conclusion: parallel connection of a small-capacity electrolytic capacitor will not bring any noticeable benefit (although it will not harm); it is much more profitable to bypass a large-capacity electrolyte with a good film capacitor, which is probably much more high-frequency.

This begs the question: why do they do this? And even in industrial equipment? Well, firstly, sometimes you can actually find conditions where a “small” capacitor will help a little. And most importantly

– why not install such a capacitor, since buyers believe in it? Moreover, it is very cheap.

Option b). Instead of one large capacitor, we include two smaller capacitors of the same type. Let's consider this situation for the capacitors given in the last two rows of Table 1. Let's say we install two 4700 µF capacitors instead of one 10000 µF. Then their resistance will be 0.071/2 = 0.0355 Ohm, and the permissible current is 3-2 = 6 amperes. It turns out that in terms of ESR it’s about the same, and in terms of current it’s even better than a single capacitor. You just need to remember that capacitors have quite a wide spread, so you can put two bad ones instead of one good one. Or vice versa. Longer wires connecting two capacitors will have greater resistance than a single capacitor. And the charge currents of the capacitors will be slightly different. As a result, this small benefit from doubling the capacitors will most likely be “eaten up” by the imperfections of the remaining elements of the circuit.

So in this case, these options for choosing capacitors can be considered equivalent. And choose one or another option based on some other considerations. For example, which capacitors will fit in your case. Or what capacitors are sold in your city.

Option c). We install 10 1000 µF capacitors instead of one 10,000 µF. What the math says: ESR = 0.199/10 = 0.0199 Ohm (compared to 0.033 Ohm for a 10000 µF capacitor), maximum current = 10-1.4 = 14A (compared to 5 A for a 10000 µF capacitor). It seems that the gain in resistance is 1.5 times, and in current almost 3 times. Judging by the figures obtained, many capacitors are better than one.

Have you ever heard how theorists are scolded, saying that in practice everything turns out completely different from their theory? This is about those would-be theorists who simply multiply and divide numbers and don’t think about the other factors influencing the situation. Look at fig. 49. Inductances and resistors are the resistance and inductance of the conductors connecting this whole bunch of capacitors. Since there are now many capacitors, the length of the wires increases significantly, and the inductance-resistance also increases. This is where all the benefits that we calculated using the formulas are lost! No, the formulas are correct! Only they do not take into account these elements - after all, we wrote these formulas without taking them into account, without thinking about them.

As a result, the total resistance can be even greater than that of a single capacitor.

the capacitance is low, and the current is distributed very unevenly. For example, when charging capacitors, the charge starts from the leftmost one according to circuit C1, and at the very first moment of time the entire maximum current flows into it (current will flow into C2 only after C1 has already charged a little), and the capacitor is designed for only 1.4 amps! Therefore, it may happen that this capacitor will be overloaded with charging current, which means it will not last long. In the same way, the rightmost capacitor SY is discharged first, and it will be overloaded with the discharge current.

In general, all the benefits are usually obtained only on paper. This is exactly the situation when “too good is also not good.” Everything should always be within reasonable limits, but here we are beyond them. Actually, “many small” capacitors will not always be worse than “one large” one, but it will not always be better. A good professional will be able to benefit from such inclusion (when it is justified), but a beginner will most likely ruin everything.

In fact, there is a case when connecting two or three capacitors in parallel will be beneficial. For example, when a filter capacitor is installed near a hot diode and it is not possible to move it away. Then, with several capacitors, only one of them will heat up.

And further. For any set of electrolytes, connecting a film capacitor is welcome.

Content:

Circuits in electrical engineering consist of electrical elements in which the methods of connecting capacitors can be different. You need to understand how to properly connect a capacitor. Individual sections of the circuit with connected capacitors can be replaced with one equivalent element. It will replace a series of capacitors, but a mandatory condition must be met: when the voltage supplied to the plates of an equivalent capacitor is equal to the voltage at the input and output of the group of capacitors being replaced, then the charge on the capacitor will be the same as on the group of capacitors. To understand the question of how to connect a capacitor in any circuit, let's consider the types of its connection.

Parallel connection of capacitors in a circuit

Parallel connection of capacitors is when all the plates are connected to the switching points of the circuit, forming a bank of capacitors.

The potential difference on the plates of the capacitance storage devices will be the same, since they are all charged from the same current source. In this case, each charging capacitor has its own charge with the same amount of energy supplied to them.

Parallel capacitors, a general parameter for the amount of charge of the resulting storage battery, are calculated as the sum of all the charges placed on each capacitor, because each capacitor charge does not depend on the charge of another capacitor included in the group of capacitors connected in parallel to the circuit.

When capacitors are connected in parallel, the capacitance is equal to:

From the presented formula we can conclude that the entire group of drives can be considered as one capacitor equivalent to them.

Capacitors connected in parallel have a voltage:

Series connection of capacitors in a circuit

When a series connection of capacitors is made in a circuit, it looks like a chain of capacitive storage devices, where the plate of the first and last capacitive storage device (capacitor) is connected to a current source.

Series connection of a capacitor:

When capacitors are connected in series, all devices in this section take the same amount of electricity, because the first and last plate of the storage devices are involved in the process, and plates 2, 3 and others up to N are charged through influence. For this reason, the charge of plate 2 of the capacitance storage device is equal in value to the charge of plate 1, but has the opposite sign. The charge of drive plate 3 is equal to the charge value of plate 2, but also with the opposite sign; all subsequent drives have a similar charge system.

Formula for finding charge on a capacitor, capacitor connection diagram:

When capacitors are connected in series, the voltage on each capacitance storage device will be different, since different capacitances are involved in charging with the same amount of electrical energy. The dependence of capacitance on voltage is as follows: the smaller it is, the greater the voltage must be applied to the drive plates to charge it. And the inverse value: the higher the storage capacity, the less voltage is required to charge it. We can conclude that the capacitance of series-connected drives matters for the voltage on the plates - the lower it is, the more voltage is required, and also high-capacity drives require less voltage.

The main difference between the series connection of capacitance storage devices is that electricity flows in only one direction, which means that in each capacitance storage device of the stacked battery the current will be the same. This type of capacitor connections ensures uniform energy storage regardless of the storage capacity.

A group of capacitance storage devices can also be considered in the diagram as an equivalent storage device, the plates of which are supplied with a voltage determined by the formula:

The charge of the common (equivalent) storage device of a group of capacitive storage devices in a serial connection is equal to:

The general value of the capacitance of series-connected capacitors corresponds to the expression:

Mixed inclusion of capacitive storage devices in a circuit

The parallel and series connection of capacitors in one of the sections of the circuit circuit is called a mixed connection by specialists.

Section of the circuit of mixed-connected capacity storage devices:

The mixed connection of capacitors in the circuit is calculated in a certain order, which can be represented as follows:

the circuit is divided into sections that are easy to calculate; this is a series and parallel connection of capacitors;
we calculate the equivalent capacitance for a group of capacitors connected in series in a parallel connection section;
we find the equivalent capacity in a parallel section;
when the equivalent storage capacities are determined, it is recommended to redraw the diagram;
The capacity of the resulting electrical energy storage devices after sequential switching on is calculated.

Capacitance storage devices (double-terminal networks) are connected in different ways to the circuit; this provides several advantages in solving electrical problems compared to traditional methods of connecting capacitors:

Use for connecting electric motors and other equipment in workshops, in radio engineering devices.
Simplifying the calculation of electrical circuit values. Installation is carried out in separate sections.
The technical properties of all elements do not change when the current strength and magnetic field change; this is used to turn on different storage devices. It is characterized by a constant value of capacitance and voltage, and the charge is proportional to the potential.

Conclusion

Various types of inclusion of capacitors in a circuit are used to solve electrical problems, in particular, to obtain polar storage devices from several non-polar two-terminal networks. In this case, the solution would be to connect a group of single-pole capacitance storage devices using an anti-parallel method (triangle). In this circuit, minus is connected to minus, and plus is connected to plus. The storage capacity increases, and the operation of the two-terminal network changes.

The following entries are not displayed: serial parallel and mixed connection of capacitors, series and parallel connection of capacitors, and capacitance when connecting capacitors in parallel.

In electrical engineering, there are various options for connecting electrical elements. In particular, there is a series, parallel or mixed connection of capacitors, depending on the needs of the circuit. Let's look at them.

Parallel connection

A parallel connection is characterized by the fact that all the plates of electrical capacitors are connected to the switching points and form batteries. In this case, when charging the capacitors, each of them will have a different number of electrical charges with the same amount of energy supplied

Parallel mounting scheme

The capacitance for parallel installation is calculated based on the capacitances of all capacitors in the circuit. In this case, the amount of electrical energy supplied to all individual two-pole elements of the circuit can be calculated by summing the amount of energy placed in each capacitor. The entire circuit connected in this way is calculated as one two-terminal network.

Ctot = C 1 + C 2 + C 3

Diagram - voltage on drives

Unlike a star connection, the plates of all capacitors receive the same voltage. For example, in the diagram above we see that:

V AB = V C1 = V C2 = V C3 = 20 Volts

Serial connection

Here, only the contacts of the first and last capacitor are connected to the switching points.

Diagram – serial connection diagram

The main feature of the circuit is that electrical energy will flow in only one direction, which means that the current in each of the capacitors will be the same. In such a circuit, for each storage device, regardless of its capacity, equal accumulation of passing energy will be ensured. You need to understand that each of them is in sequential contact with the next and previous ones, which means that the capacity of the sequential type can be reproduced by the energy of the neighboring storage device.

The formula that reflects the dependence of the current on the connection of capacitors is as follows:

i = i c 1 = i c 2 = i c 3 = i c 4, that is, the currents passing through each capacitor are equal to each other.

Consequently, not only the current strength will be the same, but also the electric charge. According to the formula, this is defined as:

Q total = Q 1 = Q 2 = Q 3

And this is how the total total capacitance of capacitors in a series connection is determined:

1/C total = 1/C 1 + 1/C 2 + 1/C 3

Video: how to connect capacitors in parallel and series method

Mixed connection

But, it is worth considering that to connect different capacitors, it is necessary to take into account the network voltage. For each semiconductor, this indicator will differ depending on the capacitance of the element. It follows that individual groups of small-capacity semiconductor biterminals will become larger when charging, and vice versa, a large-sized electrical capacitance will require less charging.

Scheme: mixed connection of capacitors

There is also a mixed connection of two or more capacitors. Here, electrical energy is distributed simultaneously by connecting electrolytic cells in parallel and in series in a circuit. This circuit has several sections with different connections of condensing two-terminal networks. In other words, on one the circuit is connected in parallel, on the other - in series. This electrical circuit has a number of advantages compared to traditional ones:

Can be used for any purpose: connecting an electric motor, machine equipment, radio equipment;
Simple calculation. For installation, the entire circuit is divided into separate sections of the circuit, which are calculated separately;
The properties of the components do not change regardless of changes in the electromagnetic field or current strength. This is very important when working with opposite two-terminal networks. The capacitance is constant at a constant voltage, but the potential is proportional to the charge;
If you need to assemble several non-polar semiconductor two-terminal devices from polar ones, then you need to take several single-pole two-terminal networks and connect them in a back-to-back (triangle) manner. Minus to minus, and plus to plus. Thus, by increasing the capacitance, the operating principle of a bipolar semiconductor changes.

Any electronics in the house can fail. However, you shouldn’t immediately run to the service center - even a novice radio amateur can diagnose and repair the simplest devices. For example, a burnt capacitor is visible to the naked eye. But what if you don’t have a part of a suitable value at hand? Of course, connect 2 or more in a chain. Today we’ll talk about concepts such as parallel and series connection of capacitors, we’ll figure out how to do it, learn about connection methods, and the rules for doing it.

Read in the article:

There is no capacitor of the required value: what to do

Very often, novice home craftsmen, having discovered a breakdown of the device, try to independently discover the cause. Having seen a burnt part, they try to find a similar one, and if this fails, they take the device for repair. In fact, it is not necessary that the indicators coincide. You can use smaller capacitors by connecting them in a circuit. The main thing is to do it right. In this case, 3 goals are achieved at once - the breakdown is eliminated, experience is gained, and family budget funds are saved.

Let's try to figure out what connection methods exist and what tasks the series and parallel connection of capacitors are designed for.

Connecting capacitors into a battery: methods of execution

There are 3 connection methods, each of which has its own specific purpose:

Parallel– is performed if it is necessary to increase the capacity while leaving the voltage at the same level.
Sequential– the opposite effect. The voltage increases, the capacitance decreases.
Mixed– both capacitance and voltage increase.

Now let's look at each of the methods in more detail.

Parallel connection: diagrams, rules

It's actually quite simple. With a parallel connection, the calculation of the total capacitance can be calculated by simply adding all the capacitors. The final formula will look like this: C total = C₁ + C₂ + C₃ + … + C n . In this case, the voltage on each of their elements will remain unchanged: V total = V₁ = V₂ = V₃ = … = V n .

The connection with this connection will look like this:

It turns out that such an installation involves connecting all the capacitor plates to the power points. This method is the most common. But a situation may arise where it is important to increase the voltage. Let's figure out how to do this.

Serial connection: less commonly used method

When using the method of connecting capacitors in series, the voltage in the circuit increases. It consists of the voltage of all elements and looks like this: V total = V₁ + V₂ + V₃ +…+ V n . In this case, the capacity changes in inverse proportion: 1/С total = 1/С₁ + 1/С₂ + 1/С₃ + … + 1/С n . Let's look at changes in capacitance and voltage when connected in series using an example.

Given: 3 capacitors with a voltage of 150 V and a capacity of 300 μF. Connecting them in series, we get:

voltage: 150 + 150 + 150 = 450 V;
capacity: 1/300 + 1/300 + 1/300 = 1/C = 299 uF.

Externally, such a connection of plates (plates) will look like this:

This connection is made if there is a risk of breakdown of the dielectric of the capacitor when voltage is applied to the circuit. But there is another way of installation.

Good to know! Series and parallel connections of resistors and capacitors are also used. This is done in order to reduce the voltage supplied to the capacitor and prevent its breakdown. However, it should be borne in mind that the voltage must be sufficient to operate the device itself.

Mixed connection of capacitors: diagram, reasons for the need for use

This connection (also called series-parallel) is used if it is necessary to increase both capacity and voltage. Here, calculating the general parameters is a little more complicated, but not so much that it would be impossible for a novice radio amateur to figure it out. First, let's see what such a scheme looks like.

Let's create a calculation algorithm.

the entire circuit needs to be divided into separate parts, the parameters of which are easy to calculate;
calculate denominations;
We calculate the general indicators, as with sequential switching.

Such an algorithm looks like this:

The advantage of mixed inclusion of capacitors in a circuit compared to series or parallel

Mixed connection of capacitors solves problems that parallel and series circuits cannot do. It can be used when connecting electric motors or other equipment; its installation is possible in separate sections. Its installation is much simpler due to the possibility of performing it in separate parts.

Interesting to know! Many radio amateurs consider this method simpler and more acceptable than the previous two. In fact, this is true if you fully understand the algorithm of actions and learn to use it correctly.

Mixed, parallel and series connection of capacitors: what to look for when doing it

When connecting capacitors, especially electrolytic ones, pay attention to strict polarity. Parallel connection implies a minus/minus connection, and serial connection means a plus/minus connection. All elements must be of the same type - film, ceramic, mica or metal paper.

Good to know! Failure of capacitors often occurs due to the fault of the manufacturer, which skimps on parts (usually these are Chinese-made devices). Therefore, correctly calculated and assembled elements in the circuit will work much longer. Of course, provided that there is no short circuit in the circuit, in which the operation of capacitors is impossible in principle.

Capacitance calculator for series connection of capacitors

What to do if the required capacity is unknown? Not everyone wants to independently calculate the required capacitor capacity manually, and some simply don’t have the time for this. For the convenience of carrying out such actions, the editors of the site invite our dear reader to use an online calculator for calculating capacitors in series connection or calculating capacitance. It is extremely simple to work with. The user only needs to enter the required data in the fields and then click the “Calculate” button. Programs that contain all the algorithms and formulas for connecting capacitors in series, as well as calculating the required capacity, will instantly produce the required result.

Send the result to me by email

How to calculate the energy of a charged capacitor: we derive the final formula

The first thing you need to do for this is to calculate the force with which the plates are attracted to each other. This can be done using the formula F = q₀ × E, where q₀ is an indicator of the magnitude of the charge, and E – tension of the plates. Next, we need an indicator of the tension of the plates, which can be calculated using the formula E = q / (2ε₀S) , Where q – charge, ε₀ – constant value, S – area of the coverings. In this case, we obtain a general formula for calculating the force of attraction between two plates: F = q₂ / (2ε₀S) .

The result of our conclusions will be the derivation of the expression for the energy of a charged capacitor, as W=A=Fd . However, this is not the final formula we need. We follow further: taking into account the previous information, we have: W = dq₂ / (2ε₀S) . With the capacitance of the capacitor expressed as C = d / (ε₀S) we get the result W = q₂ / (2С) . Applying the formula q = CU , we get the result: W = CU² /2.

Of course, for a novice radio amateur, all these calculations may seem complex and incomprehensible, but with desire and some perseverance, you can figure them out. Having delved into the meaning, he will be amazed at how simply all these calculations are made.

Why do you need to know the energy indicator of a capacitor?

In fact, energy calculations are rarely used, but there are areas in which it is necessary to know this. For example, a camera flash - here calculating the energy indicator is very important. It accumulates over a certain time (several seconds), but is issued instantly. It turns out that a capacitor is comparable to a battery - the only difference is in capacity.

Summarizing

Sometimes you can’t do without connecting capacitors, because it’s not always possible to choose the right ones according to their ratings. Therefore, knowing how to do this can help out when household appliances or electronics break down, which will significantly save on labor costs for a repair specialist. As Dear Reader has probably already understood, this is not difficult to do and even novice home craftsmen can do it. This means it’s worth spending a little of your precious time and understanding the algorithm of actions and the rules for their implementation.

Details 03 July 2017

Gentlemen, one wonderful summer day I took my laptop and left the house to go to my summer cottage. There, sitting in a rocking chair in the shade of apple trees, I decided to write this article. The breeze rustled in the branches of the trees, swaying them from side to side, and in the air there was that very atmosphere conducive to the flow of thoughts, which is so sometimes necessary...

However, enough of the lyrics, it’s time to move directly to the essence of the issue indicated in the title of the article.

So, parallel connection of capacitors... What is parallel connection anyway? Those who have read my past articles will certainly remember the meaning of this definition. We came across it when we were talking about parallel connection of resistors. In the case of capacitors, the definition will have exactly the same form. So, a parallel connection of capacitors is a connection when some ends of all capacitors are connected to one node, and the other to another.

Of course, it is better to see once than to hear a hundred times, so in Figure 1 I showed an image of three capacitors that are connected in parallel. Let the capacity of the first be C1, the second - C2, and the third - C3.

Figure 1 - Parallel connection of capacitors

In this article we will look at the laws by which currents, voltages and AC resistance when connecting capacitors in parallel, and what will be the total capacitance of such a design. Well, of course, let’s talk about why such a connection might be needed at all.

I suggest starting with tension, because everything is very clear with it. Gentlemen, it should be quite obvious that When capacitors are connected in parallel, the voltages across them are equal to each other. That is, the voltage on the first capacitor is exactly the same as on the second and third

Why, exactly, is this so? Yes, very simple! The voltage across a capacitor is calculated as the potential difference between the two legs of the capacitor. And with a parallel connection, the “left” legs of all capacitors converge into one node, and the “right” legs into another. Thus, the “left” legs everyone capacitors have one potential, and the “right” ones have another. That is, the potential difference between the “left” and “right” legs will be the same for any capacitor, and this just means that all capacitors have the same voltage. You can see a slightly more rigorous conclusion to this statement in this article. In it we presented it for parallel connection of resistors, but here it will sound absolutely the same.

So, we found out that the voltage on all parallel-connected capacitors is the same. This, by the way, is true for any type of voltage- both for constant and variable. You can connect a battery to three capacitors connected in parallel 1.5 V. And all of them will have permanent 1.5 V. Or you can connect to them a sinusoidal voltage generator with a frequency 50 Hz and amplitude 310 V. And each capacitor will have a sinusoidal voltage with a frequency 50 Hz and amplitude 310 V. It is important to remember that parallel-connected capacitors will have the same not only the amplitude, but also the frequency and phase of the voltage.

And if with voltage everything is so simple, then with current the situation is more complicated. When we talk about current through a capacitor, we usually mean alternating current. You remember that direct currents do not flow through capacitors? A capacitor for DC is like an open circuit (there is actually some capacitor leakage resistance, but it is usually neglected because it is so large). Alternating currents flow quite well through capacitors, and can have very, very large amplitudes. It is obvious that these alternating currents are caused by some alternating voltages applied to the capacitors. So, let us still have three parallel-connected capacitors with capacitances C1, C2 and C3. Some alternating voltage is applied to them complex amplitude. Due to this applied voltage, some alternating currents with complex amplitudes will flow through the capacitors. For clarity, let's draw a picture in which all these quantities will appear. It is presented in Figure 2.

Figure 2 - Looking for currents through capacitors

First of all, you need to understand how the currents are related to the total source current. And they are connected, gentlemen, all in the same way Kirchhoff's first law, which we have already met in a separate article. Yes, then we looked at it in the context of direct current. But it turns out that Kirchhoff’s first law remains true in the case of alternating current! It’s just that in this case it is necessary to use complex current amplitudes. So, the total current of three capacitors connected in parallel is related to the total current like this

That is the total current is actually simply divided between the three capacitors, while its total value remains the same. It is important to remember one more important thing - the frequency of the current and its phase will be the same for all three capacitors. The total current will have exactly the same frequency and phase I. Thus, they will differ only in amplitude, which will be different for each capacitor. How to find these same current amplitudes? Very simple! In the article about capacitor resistance we connected the current through the capacitor and the voltage across the capacitor through the resistance of the capacitor. We can easily calculate the resistance of a capacitor, knowing its capacity and the frequency of the current flowing through it (remember that for different frequencies a capacitor has different resistance) using the general formula:

Using this wonderful formula, we can find the resistance of each capacitor:

Using this formula, we can easily find the current through each of three parallel-connected capacitors:

The total current in the circuit, which flows into node A and then flows out of node B, is obviously equal to

Just in case, let me remind you once again that this happened on the basis Kirchhoff's first law. Please note, gentlemen, one important fact - The larger the capacitor's capacitance, the lower its resistance and the more current will flow through it.

Let's imagine the total current through three parallel-connected capacitors as the ratio of the voltage applied to them and some equivalent total resistance Z c∑ (which we do not yet know, but which we will find later) of the three parallel-connected capacitors:

Reducing the left and right sides by U, we get

Thus, we get an important conclusion: when connecting capacitors in parallel, the reverse equivalent resistance is equal to the sum of the reverse resistances of the individual capacitors. If you remember, we received exactly the same conclusion when parallel connection of resistors .

What happens to the capacity? What is the total capacitance of a system of three capacitors connected in parallel? Is it possible to find this somehow? Of course you can! And what's more, we almost did it. Let's substitute the decoding of capacitor resistances into our last formula. Then we will get something like this:

After elementary mathematical transformations, accessible even to a fifth grader, we obtain that

This is our next extremely important conclusion: the total capacitance of a system of several parallel-connected capacitors is equal to the sum of the capacitances of the individual capacitors.

So, we have looked at the main points regarding parallel connection of capacitors. Let's summarize them all in a concise manner:

The voltage on all three parallel-connected capacitors is the same (in amplitude, phase and frequency);
The amplitude of the current in a circuit containing parallel-connected capacitors is equal to the sum of the amplitudes of the currents through the individual capacitors. The greater the capacitance of the capacitor, the greater the amplitude of the current through it. The phases and frequencies of the currents on all capacitors are the same;
When connecting capacitors in parallel, the reverse equivalent resistance is equal to the sum of the reverse resistances of the individual capacitors;
The total capacitance of parallel-connected capacitors is equal to the sum of the capacitances of all capacitors.

Gentlemen, if you remember and understand these four points, then, one can say, I wrote the article not in vain.

Now let’s try to consolidate the material solve some problem for parallel connection of capacitors. Because, very likely, if you have not heard anything before about parallel connection of capacitors, then everything written above can be perceived simply as a set of abstract letters that are not very clear how to apply in practice. Therefore, in my opinion, the presence of tasks close to practice is an integral part of the educational process. So, the task.

Let's say we have three parallel-connected capacitors with capacitances C1=1 µF, C2=4.7 µF And C3=22 μ F. An alternating sinusoidal voltage with an amplitude is applied to them U max =50 V and frequency f=1 kHz. Need to determine

a) voltage on each capacitor;

b) the current through each capacitor and the total current in the circuit;

c) the resistance of each capacitor to alternating current and the total resistance;

d) the total capacity of such a system.

Let's start with tension. We remember that We have the same voltage on all capacitors- that is, sinusoidal with a frequency f = 1 kHz and amplitude U max = 50 V. Let us assume that it changes according to a sinusoidal law. Then we can write the following

So we have answered the first question of the problem. The voltage oscillogram on our capacitors is shown in Figure 3.

Figure 3 - Voltage oscillogram on capacitors

Yes, we see that our resistances are not only complex, but also with a minus sign. However, this should not bother you, gentlemen. This only means that the current through the capacitor and the voltage across the capacitor are out of phase relative to each other, with the current leading the voltage. Yes, the imaginary unit here shows only a phase shift and nothing more. To calculate the current amplitude, we only need the modulus of this complex number. All this has already been discussed in the past two articles (one and two). Perhaps this is not entirely obvious and some kind of visual illustration of this matter is required. This can be done on a trigonometric circle and, hopefully, a little later, I will prepare a separate article dedicated to this, or you can figure out how to show it visually yourself, using data from my article about complex numbers in electrical engineering.
Now nothing prevents you from finding the inverse total resistance:

Find the total resistance of our three parallel connected capacitors

It should be remembered that this is resistance true only for 1 kHz frequency. For other frequencies the resistance value will obviously be different.

The next step is to calculate the amplitudes of the currents through each capacitor. In the calculation we will use resistance modules (discard the imaginary unit), remembering that the phase shift between current and voltage will be 90 degrees (that is, if our voltage changes according to the sine law, then the current will change according to the cosine law). You can also carry out calculations with complex numbers, using complex amplitudes of current and voltage, but, in my opinion, in this problem it is easier to simply take into account the phase relationships. So, the amplitudes of the currents are equal

The total amplitude of the current in the circuit is obviously equal to

We can afford to add the signal amplitudes like this, because all currents through parallel-connected capacitors have the same frequency and phase. If this requirement is not met, you cannot simply take it and fold it.

Now, remembering the phase relationships, no one is stopping us from writing down the laws of current change through each capacitor

And the total current in the circuit

Oscillograms of currents through capacitors are shown in Figure 4.

Figure 4 - Oscillograms of currents through capacitors

Well, to complete the task, the simplest thing is to find the total capacity of the system as the sum of the capacities:

By the way, this capacitance can be used to calculate the total resistance of three parallel-connected capacitors. As an exercise, the reader is invited to see this for himself.

In conclusion, I would like to clarify one, perhaps the most important question: a why is it necessary to connect capacitors in parallel in practice?? What does this give? What opportunities does it open to us? Below, point by point, I outlined the main points:

Well, we end here, gentlemen. Thank you for your attention and see you again!

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