Series connection of a resistor, capacitor and inductor. Parallel and series connection of resistors, capacitors and inductors Capacitor coil connected in series capacitance

According to the equations of the elements

. (15.1)

We found a current complex. Along the way, in the denominator we received the complex resistance of the two-terminal network , active resistance of a two-terminal network and reactance of a two-terminal network .

Phase resonance A two-terminal network is a mode in which the current and voltage of the two-terminal network are in phase: . In this case, the reactance and reactive conductivity of the two-terminal network are equal to zero.

Voltage resonance A two-terminal network is a mode in which the voltages of the circuit elements are maximally compensated. The impedance of the two-terminal network is minimal.

Resonance of currents A two-terminal circuit is called a mode in which the currents of the circuit elements are maximally compensated. The total resistance of the two-terminal network is maximum.

For a series connection of a resistor, inductor and capacitor, the phase resonance coincides with the voltage resonance. The resonant frequency is determined by the formula

which is derived from the equality to zero reactance: .

Dependence of effective voltage values ​​on frequency for a series connection R, L, C shown in Fig. 15.3. Expressions for calculating these voltages are obtained by multiplying the effective current value (formula 15.2) by the impedances of the elements: , , (see paragraph 12).

Let's construct a vector diagram of current and voltage (Fig. 15.4, the case is shown here U L > U C). The easiest way to do this is if the initial phase of the current is zero: . Then the vector representing the current complex will be directed at an angle to the real axis of the complex plane. The voltage across the resistor is in phase with the current, so the vector representing the voltage complex across the resistor will be directed in the same direction as the vector representing the current complex.

Rice. 15.3.

Rice. 15.4.

Rice. 15.5.

The voltage on the inductor is ahead of the current in phase by an angle , so the vector representing the voltage complex on the inductor will be directed at an angle to the vector representing the current complex. The voltage on the capacitor lags in phase from the current by an angle , so the vector representing the voltage complex on the capacitor will be directed at an angle – to the vector representing the current complex. The vector representing the complex of the applied voltage will be equal to the sum of the vectors representing the complex voltages on the resistor, capacitor and coil. The lengths of all vectors are proportional to the effective values ​​of the corresponding quantities. That is, in order to draw vectors, you need to set the scale, for example: 1 centimeter is 20 volts, 1 centimeter is 5 amperes.

The vector diagram for the resonance mode is shown in Fig. 15.5.

Let's calculate the ratio of the effective voltage values ​​on the inductor and on the capacitor to the effective value of the source voltage in resonance mode.

Let us take into account that during resonance, the voltages on the coil and on the capacitor completely compensate each other (voltage resonance), and therefore the source voltage is equal to the voltage on the resistor: (Fig. 15.5). We use the relationship between the effective values ​​of current and voltage for the resistor, coil and capacitor, as well as the formula for the resonant frequency. We get:

where .

The quantity is called wave impedance oscillatory circuit and is designated by the letter r. The relation is denoted by the letter Q and is called quality factor oscillatory circuit. It determines the amplification properties of the circuit at the resonant frequency. In good circuits, the quality factor can be on the order of several hundred, that is, in resonance mode, the voltage on the coil and capacitor can be hundreds of times greater than that applied to the two-terminal network.

Resonance is often used in electrical engineering and electronics to amplify sinusoidal voltages and currents, as well as to separate oscillations of certain frequencies from complex oscillations. However, unwanted resonance in information electrical circuits leads to the emergence and intensification of interference, and in power circuits it can lead to dangerously high voltages and currents.

Every electrical circuit is characterized by active resistance, inductance and capacitance. Components with these properties can be connected to each other in various ways. Depending on the connection method, the values ​​of active and reactive resistances are considered. In conclusion, we describe the phenomenon of resonance, which plays a vital role in radio engineering.

My dear friends, you have become familiar with passive components. This is the name given to resistors, inductors, and capacitors, in contrast to the active components: vacuum tubes and transistors, which you will study shortly.

Coexistence of R, L and C

Everything that you, Lyuboznaykin, explained to your friend is absolutely correct. However, I must add that in reality any of the components has more than just the property that determines its name. Thus, even a simple conductor from a straight piece of wire simultaneously has resistance, inductance and capacitance. In fact, no matter how good its conductivity, it still has some active resistance.

You remember that when an electric current passes through a conductor, it creates a magnetic field around it. And if the flowing current is variable, then this field is variable; it induces currents in the conductor that counteract the main current flowing through the conductor. Therefore, here we observe the phenomenon of self-induction.

And finally, like any conductor, our piece of wire is capable of holding some electrical charge - both negative and positive. This means that it also has some capacity.

Everything that is characteristic of a simple straight piece of wire is, of course, also characteristic of a coil: in addition to its basic property of inductance, it also has some active resistance and some capacitance.

The capacitor, in turn, in addition to the capacitance that characterizes it, has some, usually very small, active resistance. In fact, passing through the plates of the capacitor, electric charges cross a certain mass of the plates, which has a small active resistance. And these small movements of charges also give rise to induction.

Thus you see that none of these three characteristics, denoted by the letters R, L and C, can exist separately without the presence of the other two. However, we will not take into account these side effects, since they are immeasurably less than the main property of the component.

Serial connection

We need to study the connection of homogeneous and heterogeneous components. We will analyze what value is obtained as a result and what resistance the components connected to each other have to the passage of current.

The components can be connected in series or in parallel (Fig. 31). A series connection is when the end of one component is connected to the beginning of another, etc.

In this case, the current alternately passes through all the components forming the chain. In a parallel connection, pins of the same name are connected to each other. Here the current, branching, simultaneously passes through all components connected in this way.

You can easily understand that resistors connected in series add up. Let's take resistors with a resistance of 100, 500 and 1000 Ohms. Let's connect them in series; the resulting chain will have resistance

Let us now take the inductors and connect them in series. provided that there is no mutual induction between them, their inductances must add up.

Let's take coils with inductances of 0.5 and 1.25 G, respectively, and connect them in series, placing them far enough apart to avoid mutual influence. The inductance of the circuit will be:

It all seems very simple. Will it be just as easy when connecting capacitors in series?

Rice. 31. Serial (a) and parallel (b) connections of components.

Rice. 32. Series connection of capacitors. The total capacity is less than the capacity of each.

We said that with such a connection, the resistances of the components add up. And capacitors add capacitance. Let's consider the case with two capacitors having capacitances, respectively, through which current flows with frequency (Fig. 32). The capacitances of these capacitors add up and make up the total capacitance:

Considering the capacitance of the entire chain as corresponding to capacitance C, we can write:

Multiplying all terms of this equality by , we get:

The transformations carried out allow us to conclude that when connecting capacitors in series, we need to add the reciprocal values ​​of their capacitances in order to obtain the reciprocal value of the capacitance of the entire chain.

In the case we have considered, i.e. the case of a series connection of two capacitors, from the last formula we can, without much mathematical effort, derive a formula for calculating the capacitance of the entire chain:

Parallel connection

Let's now move on to studying components connected in parallel. This connection method facilitates the passage of current. In fact, the conductivities of the components are added up here. This is the name given to the reciprocal of resistance.

Let's consider the case of parallel connection of active resistances (Fig. 33). Their conductivities add up. When two resistors are connected in parallel, the conductivity of the entire chain is equal to the sum of the conductivities of the connected resistors:

As you can see, there is an analogy with a series connection of capacitors, and you can easily calculate the total circuit resistance R of two parallel-connected resistors:

Now, if my reasoning has not yet bored you, consider the case of a parallel connection of two coils between which there is no mutual induction (Fig. 34). The inductive reactances of the coils are proportional to their inductance. Therefore, they will behave similarly to active resistances.

So, we will not be mistaken if we say that two coils connected in parallel have a common inductance, which is calculated by the formula

And finally, consider the case of two capacitors connected in parallel (Fig. 35). Here you need to add up the conductivities, which are the reciprocals of capacitance. But the capacitances themselves, as you remember, are inversely proportional to the capacitances. This means that the conductivities of capacitors are directly proportional to their capacitances.

Rice. 33. When resistors are connected in parallel, the total resistance decreases.

Rice. 34. Parallel connection of inductors.

Rice. 35. Parallel connection of capacitors.

Therefore, being connected in parallel, the containers add up:

However, by analyzing the physical phenomena that occur when capacitors are charged, you could easily come to this conclusion.

Try to remember, dear Neznaykin, that when components are connected in series, their resistances are added up, and when connected in parallel, conductances are added up, i.e., the reciprocal of the resistance.

Combined connection

Everything I just said applies only to circuits consisting of homogeneous components. But the situation becomes much more complicated if we connect active resistances, inductors and capacitors together.

Here I should have used the term impedance, which, as the word “impedance” itself shows, means a complex resistance consisting of active and reactive resistance. In contrast to the active resistance inherent in a particular conductor material, inductive and capacitive resistance are called reactance.

Impedance is denoted by the letter Z, and its reciprocal is called admittance.

I don't want to bore you with all the possible combinations. We will limit ourselves only to those that are found in all electronic devices (Table 2).

Let us first consider the series connection of an inductor with a capacitor (Fig. 36). Their reactances add up, but this does not give us reason to write a formula with a plus sign. In fact, inductive and capacitive reactances have seemingly opposite properties.

Inductance, as you know, delays the appearance of current when an alternating voltage is connected to it. This is called a phase shift, and in this case the current lags behind the voltage.

The opposite phenomenon occurs in a capacitor, where the current is ahead of the voltage in phase. Indeed, as the charge of the capacitor increases, the voltage on its plates increases, but as it approaches saturation, the current decreases. Therefore, it will not surprise you that when adding inductive reactance and capacitive reactance, I will put a minus sign in front of the latter:

Rice. 36. A coil and a capacitor connected in series. The total resistance of the circuit is equal to the difference between the inductive and capacitive reactances.

Rice. 37. The relationship between the hypotenuse and the legs of a right triangle.

The active resistance in this case is very small, and therefore it is not taken into account in the above formula. But if the value R of active resistance is significant, then our formula takes on a more complex form:

As you can see, you need to take the square root of the sum of the squares of the active and reactive resistance to get the total resistance.

Table 2

Does this remind you of anything, Neznaykin, from the field of geometry? Isn’t this how the length of the hypotenuse is calculated (Fig. 37), taking the square root of the sum of the squares of the legs?

Series connection of resistors

A series connection of resistors is a connection where resistors are connected in series one after another. In this case, the same current will flow through all resistors.

To calculate the total resistance of all series-connected resistors, use the formula:

Rtotal = R1 + R2 + R3 + … + Rn.

Parallel connection of resistors

A parallel connection of resistors is when one of the contacts of all resistors is connected to one common point, and the other contact of all resistors is connected to another common point. In this case, each individual resistor flows its own specific current.

If you need to determine the resistance of two parallel connected resistors, you can use the following formula:

Rtot= (R1*R2)/(R1+R2)

If two resistors connected in parallel have the same resistance, then their total resistance will be equal to half the resistance of one of them:

Rtot=(R1)/2 if R1=R2

Capacitors

Parallel connection of capacitors

A parallel connection of capacitors is when one of the contacts of all capacitors is connected to one common point, and the other contact of all capacitors is connected to another common point. In this case, there will be the same potential difference between the plates of each capacitor, since they are all charged from a common source.

For two capacitors connected in series, the total capacitance is determined by the following formula:

Commun = (C1*C2)/(C1+C2)

Inductors

Series connection of inductors

When connecting inductors in series, the total inductance is equal to the sum of the inductance of all coils, but provided that, when connecting inductors in series, their magnetic fields do not affect each other.

Ltot=L1+L2+L3+…+Ln

Parallel connection of inductors

When inductors are connected in parallel, the total inductance (provided that the magnetic fields of the inductors do not affect each other) is determined by the formula:

Ltot=1/(1/L1+1/L2+1/L3+1/Ln)

The inductance of two coils connected in parallel is determined by the following formula:

Ltotal= (L1*L2)/(L1+L2)

• Related articles

Using the results obtained above, you can find the relationship between current and voltage fluctuations in any circuit. Let's consider a series connection of a resistor, capacitor and inductor (Fig. 8.).

Assume as before that the current in the circuit varies according to the law

,

and calculate the voltage between the ends of the circuit u. Since when the conductors are connected in series, the voltages are added, the desired voltage u is the sum of three voltages: across the resistance , on the container and on inductance , and each of these voltages, as we have seen, changes over time according to the cosine law:

, (5)

, (6)

To add these three oscillations, we will use a vector voltage diagram. Voltage fluctuations across the resistance are represented by a vector
, directed along the current axis and having a length
, voltage fluctuations across capacitance and inductance are vectors
And
, perpendicular to the current axis, with lengths ( I m / C) And ( I m L) (Fig. 9.). Let's imagine that these vectors rotate counterclockwise around a common origin with angular velocity . Then the projections onto the axis of the vector currents
,
And
, will be described respectively by formulas (5)-(7). Obviously, the projection onto the current axis of the total vector

equal to the sum
, that is, equal to the total voltage in the circuit section. The maximum value of this voltage is equal to the vector modulus
. This value is easily determined geometrically. First, it is advisable to find the magnitude of the vector
:

,

and then according to the Pythagorean theorem:

. (8)

It is also clear from the figure that

. (9)

For the voltage on a section of the circuit, we can write

where the voltage amplitude and the phase shift between current and voltage are determined by formulas (8), (9). If
, then the voltage leads the current in phase, otherwise the voltage lags behind the phase.

Formula (8) is similar to Ohm's law in the sense that the voltage amplitude is proportional to the current amplitude. Therefore, it is sometimes called Ohm's law for alternating current. However, it must be remembered that this formula applies only to amplitudes, but not to instantaneous values
And
. Size

is called the circuit resistance for alternating current, the value

is called the reactance of the circuit, and the value R- active resistance.

The resulting formulas are also valid for a closed circuit that includes an alternating voltage generator, if under R, C And L understand their meanings for the entire chain (for example R represents the total active resistance of the circuit, including the internal resistance of the generator). In this case, all formulas should be replaced u on the emf of the generator. Indeed, for all our reasoning it was indifferent where exactly the capacitance, inductance and resistance are concentrated, therefore in a closed circuit (Fig. 8) we can assume that represents the total active resistance of the circuit, including the internal resistance of the generator, and And - capacitance and inductance of the circuit, and replace the real generator with an imaginary one, whose internal resistance is zero. In this case, the voltage u between points a And b will be equal to the emf of the generator . It follows that formulas (8), (9) are also valid for a closed alternating current circuit, if under ,, And understand their meanings for the entire chain and replace them in all formulas u on the EMF of the generator .