All internal lining in series connection electrified through influence. Their charges are equal

in magnitude, but opposite in sign (1 + q1 = 1-q1 = q, Figure 12).

Consequently, the charges on all the capacitors are successively connected, and the potentials are added up,

Dj = j 1 - j 2 = Dj 1 + Dj 2 + ... + Dj n,

where .

Consequently, . (17)

Parallel connection of capacitors

Fig. 13

With parallel connection, all capacitors have a constant potential difference

j 1 - j 2 = const. The full charge of the capacitor bank (Figure 1.31): q = q 1 + q 2 + ... + q n

By definition, the capacity of the capacitor bank ,

Consequently,

C = C 1 + C 2 + ... + C n. (18)

Electric field energy

The interaction energy of electric charges

It is known that dW 12 = - dA 12. For a system of three charges

dW = -d (W 12 + W 13 + W 23) = -dA,

W = W 12 + W 13 + W 23. (19)

This situation remains valid for an arbitrary system of point charges. To find the interaction energy of a system N of point charges, the formula (19) can be represented in the form

  , where W ij = W ji.

Consequently, ,

where W i is the interaction energy of the i-th charge with the remaining charges.

It is known that W i = q i j i, where q i is the i-th charge of the system; j i is the resultant potential created by all other charges of the system together with the charge q i. In this way,

. (20)

The total energy of the system of charges

If the charges are distributed over the volume with the space charge density r, then the charge system can be represented as a set of elementary charges dq = rdV, ie dW = j dq = j rdV.

Taking this into account, formula (20) after integration becomes

, (21)

where j is the potential created by all charges in the elementary volume dV.

If the charges are distributed with the surface charge density s, then

. (22)

Formulas (21) and (22) allow us to find the total energy of the system, and formula (20) - only the self-energy of the charge. Indeed, according to (21), W = W 1 + W 2 + W 12, where W 1, W 2 are the eigenenergy of the charge q 1 and q 2; W 12 is the interaction energy of these charges.

The energy of a system of charged conductors

Using formula (21), we find the energy of an isolated (solitary) conductor. If a conductor has a charge q and a potential j = const at all points where the charge is distributed, then

. (23)

Since for a flat capacitor (two charged conductors)

, (24)

where k1 + q1 = k1-q1 = q; Dj is the potential difference between positively and negatively charged capacitor plates; W is the total interaction energy of not only charges of one electrode with charges of the other, but also the energy of charge interaction inside each of the plates.

Formula (24) remains valid even in the presence of a dielectric between the plates of the capacitor.

If we use capacitive coefficients, then

. (25)

Electric field energy

To find the energy, we used only charges and potentials. The main characteristic of the electric field is the tension vector. Then the energy of the electric field between the plates of the flat capacitor can be found by transforming the formula (23), taking into account the fact that Dj = Ed; .

After the substitution, we obtain

. (26)

Taking into account the dielectric between the plates of the capacitor

. (27)

It is known that an electric field is a special case of an electromagnetic field, which can exist separately from field sources, i.e. The propagation of electromagnetic waves in space is associated with the transfer of energy.

Consequently, the electrostatic field has the energy distributed in it with the bulk density w el.

In the case of a homogeneous electric field

If the electric field is not homogeneous, then

where .

In this case, the volume density of the energy of the electric field

. (29)

Consequently, the total energy of the electric field

. (30)

Thus, unlike the gravitational field, the electrostatic (electromagnetic) field is characterized by the bulk density of energy, and we can talk about the localization of electrical energy in space.

In many cases, in order to obtain the required electrical capacitance, the capacitors must be connected to a group called battery.

Consistent   This is the connection of capacitors in which the negatively charged lining of the previous capacitor is connected to the positively charged plate of the subsequent one (Figure 15.31). In a series connection, all capacitor plates will have the same charge q.   Since the charges on the capacitor are in equilibrium, the potentials of the plates, connected by conductors, will be the same.

Given these circumstances, we derive a formula for calculating the electrical capacity of a battery of series-connected capacitors.   From Fig. 15.31 that the voltage across the battery U 6 is equal to the sum of the voltages on the series-connected capacitors. Really:

(φ 1 - φ 2) + (φ 2 - φ 3) + ... + (φ n-1 - φ n) = φ 1 - φ n

U 1 + U 2 + ... + U n = U 6

Using relationsq =CU, we obtain:

After a reduction by q, we have:

From (15.21) it can be seen that, in a series connection, the electric capacity of the battery is less than the smallest of the capacitances of the individual capacitors.

Parallel   is the connection of capacitors, in which all positively charged plates are connected to one wire, and negatively charged ones to the other (Figure 15.32). In this case, the voltages on all the capacitors are the same and equal to U, and the charge on the battery q b is equal to the sum of the charges on the individual capacitors:

q b = q 1 + q 2 + ... = q n

C b U = C 1 U + C 2 U + ... + C n U

After a reduction to u, we obtain a formula for calculating the electrical capacity of the battery parallel connected capacitors:

C b = C 1 + C 2 + ... + C n (15.22)

It is seen from (15.22) that when the connection is parallel, the electric capacity of the battery is greater than the largest of the capacitances of the individual capacitors.   When manufacturing capacitors of large electrical capacitance, use the parallel connection shown in Fig. 15.33. This method of connection provides savings in the material, since the charges are located on both sides of the capacitor plates (except for the two outer plates).

In Fig. 15.33 is connected in parallel to 6 capacitors, and the plates are made 7. Consequently, in this case the parallel-connected capacitors are one less, the number of metal sheets n in the battery of capacitors, i.e.,

C б = Ɛ C S (n - 1) / d (15.23)

Coupling of capacitors

In electrical circuits and circuits, various methods of connecting capacitors are used. The connection of capacitors can be serial, parallel and series-parallel (mixed connection of capacitors).

If the connection of the capacitors to the battery is carried out in the form of a chain and only the first and last capacitors are connected to the connection points in the circuit, this connection is called consistent.

When the capacitors are connected in series, they are charged with the same amount of electricity, although only the two end plates are charged from the current source, and the remaining plates are charged through the influence of the electric field. In this case, the charge of plate 2 will be equal to the nominal, but opposite to the sign of the charge of plate 1, the charge of plate 3 will be equal to the charge of plate 2, but will also be of opposite polarity, and so on.

But more precisely, the voltages on different capacitive elements will differ, since for a charge of the same amount of electricity at different nominal capacities, different voltages are always needed. The lower the capacity of the capacitor, the higher the level of voltage required to charge the radio component with the required amount of electricity, and vice versa.

Thus, when a group of capacitors connected in series are charged, the capacitors of low capacitance will have higher voltages and lower capacitance on high-capacity cells.

Consider the entire group of capacitors connected in series, as one equivalent capacitance, between plates of which there is some level of voltage equal to the sum of the voltages on all elements of the group, and the charge of which is equal to the charge of any component of the given group.

If we take a closer look at the smallest capacity rating in the group, then it should have the highest voltage level. But in fact, the voltage level on it is only part of the total voltage value, from the total group. The voltage across the entire group is always higher than the voltage on the capacitor having the smallest capacitance value.   And so we can say that the total capacitance of a group of capacitors connected in series is smaller than the capacitance of the smallest capacitor in the group.

To calculate the total capacity of a group, in this example, we use the following formula:

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

For the particular case of two consecutively connected elements, the formula takes the form:

C total = C 1 × C 2 / C 1 + C 2

If a group of capacitive elements is included in the circuit in such a way that all the components of the circuit are connected to the direct connection points, then this connection is called a parallel connection of the capacitors.

When you charge a group of capacitors connected in parallel, between the plates of all the elements will be the same voltage, since they are all charged from a single power source. The total amount of electricity on all the elements will be equal to the sum of the quantities of electricity that are placed on each tank separately, since the charge of each of them is independent of the charge of the other components of this circuit. Proceeding from this, the whole system can be considered as one common equivalent capacitor. Then the total capacitance at parallel connection of capacitors is equal to the sum of the capacitances   all connected elements.

Denote the total capacity of the cells connected to the battery by the symbol With total, then we can write the formula:

C total = C 1 + C 2 + C 3

A series-parallel connection of capacitors is a circuit or circuit having in its composition parts, both with parallel and with a serial connection of radio components.

When calculating the total capacity of such a scheme with serial-parallel connection type   this section (as in case c) is divided into elementary sections consisting of simple groups with a serial or parallel connection of the tanks. Further the algorithm of calculations takes the form:

1. Calculate the equivalent capacitance of segments with a series connection of capacitors
2. If these sections consist of series-connected capacitors, then first calculate their capacitance.
3. After calculating the equivalent capacities, the scheme is redrawn. Usually, a circuit is obtained from the series-connected equivalent capacitors.
4. Calculate the total capacity of the resulting circuit.

Serial connection

In many cases, to obtain the desired electrical capacitance, the capacitors are combined into a group called a battery. The capacitance of the capacitor bank depends on the connection circuit of its constituent capacitors. There are two types of connection: serial and parallel. It is also possible to mix capacitors in a battery.

Fig. 2.14. Sequential connection of two capacitors

Parallel connection

The capacitor is a very common radio component, which is found in all the circuit diagrams. It is two conductors separated by a dielectric (depending on the type of capacitors, various types of capacitors are used), that is, physically this is a break in the circuit, but a dielectric can accumulate a charge. The main characteristic of any capacitor is the ability to accumulate charge - capacity, and this charge. Electrolytic capacitors have a polarity and are characterized by a large capacitance and a wide range of voltages, paper can withstand high voltage, but have a small capacity. There are also devices with varying capacities, but each type has its own application.

Often radio amateurs are faced with the problem of selecting capacitors for capacitance or voltage. Professionals know: in the absence of the right one, it is possible to assemble a combination of several devices, a battery of them. In batteries, a combination capacitor is allowed.

By connecting devices in parallel, you can achieve an increase in capacity. The total in such a battery will be equal to the sum of all the capacitances (Сэкв. = С1 + С2 + ...), the voltage on each element will be equal. This means that the minimum voltage of the capacitor applied in the connection is the maximum allowable for the entire battery.

A series connection of capacitors is used in cases where it is necessary to increase the voltage capable of withstanding devices or reducing their capacitance. In this variant, the elements are joined according to the following scheme: the beginning of one with the end of the other, that is, the "plus" of one with the minus of the other. The capacitance of an equivalent capacitor in this case is calculated by the following formula: 1 / Секв. = 1 / С1 + 1 / С2 + ... It follows from this that for two capacitors Сэкв = С1 * С2 / (С1 + С2), which means that the battery capacity will be less than the minimum capacity used in it.

A capacitor bank often provides a combined (mixed)
compound. To calculate the capacitance of such a device, in which a parallel and series connection of capacitors is applied, the circuit is divided into sections, and then the capacity of each of them is alternately calculated. Thus, the capacitance C12 = C1 + C2 is calculated, and then Сэкв = С12 * С3 / (С12 + С3).

Thanks to the development of capacitor banks with different configurations and circuitry
connection, you can pick up any capacity for any voltage of interest. capacitors, as well as combined, is used in many ready-made radio amateur circuits. It must be taken into account that each capacitor has a very important individual parameter - the leakage current, it can unbalance the voltage at parallel connection and the capacitance in the series. It is very important to select the necessary shunt resistance.

When working with capacitors and electronics, do not forget the rules of personal safety and the risk of electric shock.