Capacitance unit of measurement is in si. Capacitance in an AC circuit

1 Real and ideal sources of electricity. energy. Equivalent circuits. Any source electrical energy converts other types of energy (mechanical, light, chemical, etc.) into electrical energy. The current in the source of electrical energy is directed from negative terminal to positive due to outside forces, determined by the type of energy that the source converts into electrical energy. A real source of electrical energy when analyzing electrical circuits can be represented either in the form voltage source, or in the form of a current source. This is shown below using an example of an ordinary battery.

Methods of representing a real source of electrical energy differ from each other in equivalent circuits (calculation circuits). In Fig. 15 the real source is represented (replaced) by a voltage source circuit, and in Fig. 16, the real source is represented (replaced) by a current source circuit.

Period(T) is the time (s) during which the variable makes a complete oscillation. Frequency- number of periods per second. The unit of frequency is Hertz (abbreviated Hz), 1 Hz is equal to one oscillation per second. Period and frequency are related T = 1/f. Changing over time, the sinusoidal quantity (voltage, current, emf) takes on different values. Value in at the moment time is called instantaneous. Amplitude - highest value sinusoidal value. The amplitudes of current, voltage and EMF are denoted in capital letters with an index: I m, U m, E m, and their instantaneous values ​​are indicated in lowercase letters i, u, e. The instantaneous value of a sinusoidal quantity, for example current, is determined by the formula i = I m sin(ωt + ψ), where ωt + ψ is the phase-angle that determines the value of the sinusoidal quantity at a given time; ψ is the initial phase, i.e., the angle that determines the value of the quantity at the initial moment of time. Sinusoidal quantities having the same frequency but different initial phases are called phase-shifted.

3 In Fig. Figure 2 shows graphs of sinusoidal quantities (current, voltage) shifted in phase. When the initial phases of two quantities are equal ψ i = ψ u, then the difference ψ i − ψ u = 0 and, therefore, there is no phase shift φ = 0 (Fig. 3). The effectiveness of the mechanical and thermal action of alternating current is assessed by its effective value. The effective value of the alternating current is equal to the value of the direct current, which, in a time equal to one period of the alternating current, will release in the same resistance the same amount of heat as the alternating current. The effective value is indicated in capital letters without an index: I, U, E. Rice. 2 Graphs of sinusoidal current and voltage, phase shifted. Rice. 3 Graphs of sinusoidal current and voltage that are in phase

For sinusoidal quantities, the effective and amplitude values ​​are related by the relations:

I=I M /√2; U=U M /√2; E=E M √2. The effective values ​​of current and voltage are measured with ammeters and alternating current voltmeters, and the average power value is measured with wattmeters.

4 RMS (effective) valuestrengthAC They call the amount of direct current, the action of which will produce the same work (thermal or electrodynamic effect) as the alternating current in question during one period. In modern literature it is more often used mathematical definition this value is the root mean square value of the alternating current. In other words, the effective current value can be determined by the formula:

.

For harmonic current oscillations

5Formula of inductive reactance:

where L is inductance.

Capacitance formula:

where C is capacity.

We propose to consider an alternating current circuit, which includes one active resistance, and draw it in your notebooks. After checking the drawing, I tell you that in electrical circuit(Fig. 1, a) under the influence of alternating voltage, alternating current flows, the change of which depends on the change in voltage. If the voltage increases, the current in the circuit increases, and when the voltage is zero, there is no current in the circuit. A change in its direction will also coincide with a change in the direction of voltage

(Fig. 1, c).

Fig. 1. AC circuit with active resistance: a – diagram; b – vector diagram; c – wave diagram

I graphically depict on the board sinusoids of current and voltage that are in phase, explaining that although it is possible to determine the period and frequency of oscillations, as well as the maximum and effective values ​​from a sinusoid, it is nevertheless quite difficult to construct a sinusoid. A simpler way to represent current and voltage values ​​is vector. For this, the voltage vector (to scale) should be plotted to the right from an arbitrarily chosen point. The teacher invites students to plot the current vector themselves, reminding them that voltage and current are in phase. After constructing a vector diagram (Fig. 1, b), it should be shown that the angle between the voltage and current vectors is zero, i.e. = 0. The current strength in such a circuit will be determined by Ohm’s law: Question 2. AC circuit with inductive reactance Consider the AC electrical circuit (Fig. 2, a), which includes inductive reactance. Such resistance is a coil with a small number of turns of large cross-section wire, in which the active resistance is usually considered equal to 0.

Rice. 2. AC circuit with inductive reactance

Around the turns of the coil, when current passes, an alternating magnetic field will be created, inducing a self-induction emf in the turns. According to Lenz's rule, the effect of induction always counteracts the cause that causes it. And since self-induction is caused by changes in alternating current, it prevents its passage. The resistance caused by self-induction is called inductive and is denoted by the letter x L. The inductive reactance of the coil depends on the rate of change of current in the coil and its inductance L: where X L is the inductive reactance, Ohm; – angular frequency of alternating current, rad/s; L is the inductance of the coil, G.

Angular frequency == ,

hence, .

Capacitance in an alternating current circuit. Before starting the explanation, it should be recalled that there are a number of cases when in electrical circuits, in addition to active and inductive resistance, there is also capacitive reactance. A device designed to accumulate electric charges, is called a capacitor. The simplest capacitor is two wires separated by a layer of insulation. Therefore, multi-core wires, cables, electric motor windings, etc. have capacitance. The explanation is accompanied by a display of the capacitor various types and capacitances with their connection into an electrical circuit. I propose to consider the case when one capacitive reactance predominates in the electrical circuit, and active and inductive reactance can be neglected due to their small values ​​(Fig. 6, a). If a capacitor is connected to a DC circuit, then no current will flow through the circuit, since there is a dielectric between the capacitor plates. If capacitance is connected to an alternating current circuit, then current / will flow through the circuit, caused by recharging the capacitor. Overcharging occurs because the alternating voltage changes its direction and hence if we connect an ammeter in this circuit, then it will indicate the charging and discharging current of the capacitor. No current passes through the capacitor in this case either. The strength of the current passing in a circuit with capacitive reactance depends on the capacitance of the capacitor Xc and is determined by Ohm's law

where U is the voltage of the emf source, V; Xc – capacitance, Ohm; / – current strength, A.

Rice. 3. AC circuit with capacitance

Capacitance, in turn, is determined by the formula

where C is the capacitance of the capacitor, F. I invite students to construct a vector diagram of current and voltage in a circuit with capacitance. Let me remind you that when studying processes in an electrical circuit with capacitive reactance, it was found that the current leads the voltage by an angle φ = 90°. This phase shift of current and voltage should be shown on a wave diagram. I graphically depict a voltage sinusoid on the board (Fig. 3, b) and instruct students to independently draw a current sinusoid leading the voltage by an angle of 90°.

Electric current in conductors is continuously associated with magnetic and electric fields. Elements that characterize the conversion of electromagnetic energy into heat are called active resistances (denoted R). Typical representatives of active resistances are resistors, incandescent lamps, electric ovens, etc.

Inductive reactance. Formula of inductive reactance.

Elements associated with the presence of only a magnetic field are called inductances. Coils, windings and etc. have inductance. Inductive reactance formula:

where L is inductance.

Capacitance. Capacitance formula.

Elements associated with the presence of an electric field are called capacitances. Capacitors, long power lines, etc. have capacitance. Capacitance formula:

where C is capacity.

Total resistance. Total resistance formulas.

Real consumers of electrical energy may also have a complex value of resistance. In the presence of active R and inductive L resistances, the value of the total resistance Z is calculated using the formula:

Similarly, the total resistance Z is calculated for the circuit of active R and capacitive resistance C:

Consumers with active R, inductive L and capacitive resistance C have a total resistance:

In a DC circuit, a capacitor represents an infinitely greater resistance: D.C. does not pass through the dielectric separating the capacitor plates. The capacitor does not break the alternating current circuit: by alternately charging and discharging, it ensures the movement of electrical charges, i.e., it maintains alternating current in the external circuit. Based on Maxwell's electromagnetic theory (see § 105), we can say that the alternating conduction current is closed inside the capacitor by a displacement current. Thus, for alternating current, the capacitor is a finite resistance called capacitance.

Experience and theory show that the strength of alternating current in a wire depends significantly on the shape that is given to this wire. The current strength will be greatest in the case of a straight wire. If the wire is coiled in the form of a coil with a large number turns, then current strength it will decrease significantly: a particularly sharp decrease in current occurs when a ferromagnetic core is introduced into this coil. This means that for alternating current the conductor, in addition to ohmic resistance, also has additional resistance, which depends on the inductance of the conductor and is therefore called inductive reactance. The physical meaning of inductive reactance is as follows. Under the influence of changes in current in a conductor with inductance, an electromotive force of self-induction arises, preventing these changes, i.e., reducing the amplitude of the current and, consequently, the effective current. A decrease in the effective current in a conductor is equivalent to an increase in the resistance of the conductor, i.e., equivalent to the appearance of additional ( inductive) resistance.

Let us now obtain expressions for capacitive and inductive reactances.

1. Capacitance. Let an alternating sinusoidal voltage be applied to a capacitor with capacitance C (Fig. 258)

Neglecting the voltage drop across the low ohmic resistance of the supply wires, we will assume that the voltage on the capacitor plates is equal to the applied voltage:

At any moment of time, the charge of the capacitor is equal to the product of the capacitance of the capacitor C and the voltage (see § 83):

If over a short period of time the charge of the capacitor changes by an amount, this means that a current equal to

Since the amplitude of this current

then we finally get it

Let us write formula (37) in the form

The last relationship expresses Ohm's law; the quantity that plays the role of resistance is the resistance of the capacitor for alternating current, i.e. capacitance

Thus, capacitance is inversely proportional to the circular frequency of the current and the magnitude of the capacitance. The physical meaning of this dependence is not difficult to understand. The greater the capacitance of the capacitor and the more often the direction of the current changes (i.e., the greater the circular frequency, the greater the charge passes per unit time through the cross-section of the supply wires. Consequently,). But current and resistance are inversely proportional to each other.

Therefore, resistance

Let's calculate the capacitance of a capacitor with a capacitance connected to an alternating current circuit with a frequency of Hz:

At a frequency of Hz, the capacitance of the same capacitor will drop to approximately 3 ohms.

From a comparison of formulas (36) and (38) it is clear that changes in current and voltage occur in different phases: the current phase is greater than the voltage phase. This means that the current maximum occurs a quarter of a period earlier than the voltage maximum (Fig. 259).

So, across capacitance, the current leads the voltage by a quarter of a period (in time) or by 90° (in phase).

The physical meaning of this important phenomenon can be explained as follows. At the initial moment of time, the capacitor is not yet charged. Therefore, even a very small external voltage easily moves charges to the capacitor plates, creating a current (see Fig. 258). As the capacitor charges, the voltage on its plates increases, preventing further inflow of charges. In this regard, the current in the circuit decreases, despite the continuing increase in external voltage

Consequently, at the initial moment of time, the current had a maximum value ( When and along with it reaches a maximum (which will happen after a quarter of the period), the capacitor will be fully charged and the current in the circuit will stop. So, at the initial moment of time, the current in the circuit is maximum, and the voltage is minimum and only begins to increase; after a quarter of the period, the voltage reaches its maximum, and the current already has time to decrease to zero. Thus, the current actually leads the voltage by a quarter of the period.

2. Inductive reactance. Let an alternating sinusoidal current flow through the self-induction coil with inductance

caused by alternating voltage applied to the coil

Neglecting the voltage drop across the low ohmic resistance of the supply wires and the coil itself (which is quite acceptable if the coil is made, for example, of thick copper wire), we will assume that the applied voltage is balanced by the electromotive force of self-induction (equal to it in magnitude and opposite in direction):

Then, taking into account formulas (40) and (41), we can write:

Since the amplitude of the applied voltage

then we finally get it

Let us write formula (42) in the form

The last relationship expresses Ohm's law; the value that plays the role of resistance is the inductive resistance of the self-induction coil:

Thus, inductive reactance is proportional to the circular frequency of the current and the magnitude of the inductance. This kind of dependence is explained by the fact that, as noted in the previous paragraph, inductive reactance is caused by the action of the electromotive force of self-induction, which reduces the effective current and, therefore, increases the resistance.

The magnitude of this electromotive force (and, therefore, resistance) is proportional to the inductance of the coil and the rate of change of current, i.e. circular frequency

Let's calculate the inductive reactance of a coil with inductance connected to an alternating current circuit with a frequency of Hz:

At a frequency of Hz, the inductive reactance of the same coil increases to 31,400 ohms.

We emphasize that the ohmic resistance of a coil (with an iron core) having inductance is usually only a few ohms.

From a comparison of formulas (40) and (43) it is clear that changes in current and voltage occur in different phases, and the current phase is less than the voltage phase. This means that the current maximum occurs a quarter of a period (774) later than the voltage maximum (Fig. 261).

So, in inductive reactance the current lags behind the voltage by a quarter of a period (in time), or by 90° (in phase). The phase shift is due to the braking effect of the electromotive force of self-induction: it prevents both the increase and decrease of the current in the circuit, so the maximum current occurs later than the maximum voltage.

If inductive and capacitive reactances are connected in series in an alternating current circuit, then the voltage across the inductive reactance will obviously lead the voltage across the capacitive reactance by half a cycle (in time), or by 180° (in phase).

As already mentioned, both capacitive and inductive reactances are common name reactance. On D active resistance no electricity is consumed; in this way it differs significantly from active resistance. The fact is that the energy periodically consumed to create an electric field in the capacitor (during its charging), in the same quantity and with the same frequency, is returned to the circuit when this field is eliminated (during the discharge of the capacitor). In the same way, the energy periodically consumed to create the magnetic field of the self-induction coil (during the current increase) is returned in the same amount and with the same frequency to the circuit when this field is eliminated (during the current decrease).

In AC technology, instead of rheostats (ohmic resistance), which always heat up and waste energy, chokes (inductive resistance) are often used. The choke is a self-induction coil with an iron core. Providing significant resistance to alternating current, the inductor practically does not heat up and does not consume electricity.

Capacitance is the resistance to alternating current that electrical capacitance provides. The current in a circuit with a capacitor leads the voltage in phase by 90 degrees. Capacitive reactance is reactive, that is, energy loss does not occur in it, as, for example, in active resistance. Capacitance is inversely proportional to the frequency of alternating current.

Let's conduct an experiment, for this we will need. A capacitor is an incandescent lamp and two voltage sources, one DC and the other AC. First, let's build a circuit consisting of a source DC voltage, lamp and capacitor are all connected in series.

Figure 1 - capacitor in a DC circuit

When the current is turned on, the lamp will flash short time, and then goes out. Since for direct current the capacitor has a large electrical resistance. This is understandable, because between the plates of the capacitor there is a dielectric through which direct current is not able to pass. And the lamp will flash because at the moment the constant voltage source is turned on, a short-term current pulse occurs, charging the capacitor. And since the current flows, the lamp glows.

Now in this circuit we will replace the DC voltage source with an AC generator. When such a circuit is turned on, we will find that the lamp will glow continuously. This happens because the capacitor in the alternating current circuit is charged in a quarter of the period. When the voltage across it reaches the amplitude value, the voltage across it begins to decrease, and it will discharge for the next quarter of the period. In the next half of the period the process will repeat again, but this time the voltage will be negative.

Thus, current flows continuously in the circuit, although it changes its direction twice per period. But charges do not pass through the dielectric of the capacitor. How does this happen?

Let's imagine a capacitor connected to a constant voltage source. When turned on, the source removes electrons from one plate, thereby creating a positive charge on it. And on the second plate it adds electrons, thereby creating a negative charge of equal magnitude but opposite sign. At the moment of redistribution of charges in the circuit, a capacitor charging current flows. Although the electrons do not move through the dielectric of the capacitor.

Figure 2 - capacitor charge

If you now remove the capacitor from the circuit, the lamp will shine brighter. This suggests that the capacitance creates resistance, limiting the current flow. This happens due to the fact that at a given current frequency the value of the capacitance is small and it does not have time to accumulate enough energy in the form of charges on its plates. And during a discharge, a current will flow less than the current source is capable of developing.

Reactance– electrical resistance to alternating current due to energy transfer magnetic field in inductances or by an electric field in capacitors.

Elements that have reactance are called reactive.

Reactance of the inductor.

When AC current flows I in a coil, a magnetic field creates an EMF in its turns, which prevents the current from changing.
When the current increases, the EMF is negative and prevents the current from increasing; when it decreases, it is positive and prevents its decrease, thus resisting the change in current throughout the entire period.

As a result of the created counteraction, a voltage is formed at the terminals of the inductor in antiphase U, suppressing EMF, equal to it in amplitude and opposite in sign.

When the current passes through zero, the amplitude of the emf reaches maximum value, which forms a discrepancy in time between current and voltage of 1/4 of the period.

If you apply voltage to the terminals of the inductor U, the current cannot start instantly due to the counter-emf equal to -U, therefore, the current in the inductance will always lag behind the voltage by an angle of 90°. The shift at the lagging current is called positive.

Let us write down the expression for the instantaneous voltage value u based on EMF ( ε ), which is proportional to the inductance L and the rate of change of current: u = -ε = L(di/dt).
From here we express the sinusoidal current.

Integral of a function sin(t) will -cos(t), or an equal function sin(t-π/2).
Differential dt functions sin(ωt) will leave the integral sign with a factor of 1 /ω .
As a result, we obtain the expression for the instantaneous current value with a shift from the stress function by an angle π/2(90°).
For RMS values U And I in this case we can write .

As a result, we have a dependence of sinusoidal current on voltage according to Ohm’s Law, where in the denominator instead of R expression ωL, which is the reactance:

The reactance of inductors is called inductive.

Capacitor reactance.

The electric current in a capacitor is a part or a set of processes of its charge and discharge - the accumulation and release of energy by the electric field between its plates.

In an AC circuit, the capacitor will charge to a certain maximum value until the current reverses direction. Consequently, at the moments of the amplitude value of the voltage on the capacitor, the current in it will be equal to zero. Thus, the voltage across the capacitor and the current will always have a time difference of a quarter period.

As a result, the current in the circuit will be limited by the voltage drop across the capacitor, which creates an alternating current reactance that is inversely proportional to the rate of change of current (frequency) and the capacitance of the capacitor.

If you apply voltage to a capacitor U, the current will instantly start from the maximum value, then decrease to zero. At this time, the voltage at its terminals will increase from zero to maximum. Consequently, the voltage on the capacitor plates lags the current in phase by an angle of 90 °. This phase shift is called negative.

The current in a capacitor is a derivative function of its charge i = dQ/dt = C(du/dt).
Derivative of sin(t) will cos(t) or an equal function sin(t+π/2).
Then for sinusoidal voltage u = U amp sin(ωt) Let's write the expression for the instantaneous current value as follows:

i = U amp ωCsin(ωt+π/2).

From here we express the ratio of the root-mean-square values .

Ohm's law dictates that 1 /ωC is nothing more than reactance for a sinusoidal current:

The reactance of a capacitor in technical literature is often called capacitive. It can be used, for example, in organizing capacitive dividers in alternating current circuits.

Online reactance calculator

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