Capacitive resistance of a capacitor: influence on alternating and direct current, formulas for calculation

A capacitor is used in circuits to separate the AC and DC voltage components, while it conducts high-frequency signals well, and poorly conducts low-frequency signals. Being in a DC circuit, its impedance is assumed to be infinitely large. For alternating current, the capacitance of the capacitor is not constant. Therefore, the calculation of this value is extremely important when designing various electronic devices.

general description

Physically, an electronic device - a capacitor - consists of two plates made of a conductive material, between which there is a dielectric layer. Two electrodes are brought out from the surface of the plates, intended for connection to an electric circuit. Structurally, the device can be of various sizes and shapes, but its structure remains unchanged, that is, there is always an alternation of conductive and dielectric layers.

The word "capacitor" comes from the Latin "condensatio" - "accumulation". The scientific definition says that an electrical storage device is a two-terminal device, characterized by constant and variable capacitance values ​​and high resistance. It is designed to store energy and charge. Farad (F) is taken as the unit of measurement of capacity.

In the diagrams, the capacitor is depicted in the form of two straight lines corresponding to the conducting plates of the device, and perpendicular to their midpoints by drawn segments - the terminals of the device.

The principle of operation of the capacitor is as follows: when the device is connected to an electrical circuit, the voltage in it will have zero value. At this moment, the device begins to receive and accumulate a charge. The electrical current flowing into the circuit will be as high as possible. After a while, positive charges will begin to accumulate on one of the electrodes of the device, and negative charges on the other.

The duration of this process depends on the capacity of the device and the active resistance. A dielectric located between the leads interferes with the movement of particles between the plates. But this will only happen until the potential difference of the power source and the voltage at the capacitor terminals are equal. At this moment, the capacity will become the maximum possible, and the electric current - the minimum.

If the voltage is no longer applied to the element, then when the load is connected, the capacitor begins to give its accumulated charge to it. Its capacity decreases, and the voltage and current levels in the circuit decrease. In other words, the storage device itself turns into a power source. Therefore, if the capacitor is connected to an alternating current, then it will begin to recharge periodically, that is, create a certain resistance in the circuit.

Instrument characteristics

The most important characteristic of a storage device is its capacity. The charging time depends on it when the device is connected to a current source. The discharge time is directly related to the value of the load resistance: the higher it is, the faster the process of returning the accumulated energy occurs. This capacity is determined by the following expression:

C = E * Eo * S / d, where E is the relative dielectric constant of the medium (reference value), S is the area of ​​the plates, d is the distance between them.

In addition to capacity a capacitor is characterized by a number of parameters, such as:

• specific capacity - determines the ratio of the capacity to the mass of the dielectric;
• operating voltage - the nominal value that the device can withstand when it is applied to the element plates;
• temperature stability - the interval in which the capacitance of the capacitor practically does not change;
• insulation resistance - characterized by self-discharge of the device and determined by the leakage current;
• equivalent resistance - consists of losses formed at the terminals of the device and the dielectric layer;
• absorption - the process of the emergence of a potential difference on the plates after the device is discharged to zero;
• capacitive resistance - a decrease in conductivity when an alternating current is applied;
• polarity - due to the physical properties of the material used in the manufacture, the capacitor can work correctly only if a potential with a certain sign is applied to the plates;
• equivalent inductance - a parasitic parameter that appears on the contacts of the device and turns the capacitor into an oscillatory circuit.

Element impedance

The total resistance of a capacitor (impedance) to an alternating signal consists of three components: capacitive, resistive and inductive resistance. All these values ​​must be taken into account when designing circuits containing a storage element. Otherwise, in an electrical circuit, with appropriate piping, the capacitor can behave like a choke and is in resonance. Of all the three quantities, the most significant is the capacitance of a capacitor, but under certain circumstances inductive also has an effect.

Often, in calculations, parasitic values ​​such as inductance or active resistance are assumed to be negligible, and the capacitor in this case is called ideal.

Element impedance expressed in the formula Z = (R2 + (Xl-Xc) ² ) ^½, where

• Xl - inductance;
• Xc - capacity;
• R is the active component.

The latter arises due to the appearance of the electromotive force (EMF) of self-induction. The inconstancy of the current leads to a change in the magnetic flux, which maintains the EMF current of self-induction constant. This value is determined by the inductance L and the frequency of the flowing charges W. Xl = wL = 2 * p * f * L. Xc is the capacitive resistance depending on the storage capacity C and the current frequency f. Xc = 1 / wC = ½ * p * f * C, where w is the angular frequency.

The difference between capacitive and inductive values ​​is called the reactance of the capacitor: X = Xl-Xc. By formulas it can be seen that with an increase in the frequency f of the signal, the inductive value begins to prevail, with a decrease - capacitive. Therefore, if:

• X> 0, the element exhibits inductive properties;
• X = 0, only the active value is present in the tank;
• X <0, capacitive resistance appears in the element.

Active resistance R is associated with power losses, the transformation of its electrical energy into heat. Reactive - with the exchange of energy between alternating current and an electromagnetic field. Thus, the impedance can be found using the formula Z = R + j * X, where j is the imaginary unit.

Capacitance

To understand the process, one should imagine a capacitor in an electrical circuit through which an alternating current flows. Moreover, there are no other elements in this chain. The value of the current passing through the capacitor and the voltage applied to its plates change over time. Knowing any of these values, you can find another.

Let the current change according to the sinusoidal dependence I (t) = Im * sin (w * t + f 0). Then the voltage can be described as U (t) = (Im / C * w) * sin (w * t + f 0 -p / 2). When taking into account the 90-degree phase shift between the signals in the formula, a complex value j is introduced, which is called the imaginary unit. Therefore, the formula for finding the current will look like I = U / (1 / j * w * C). But given that the complex number only denotes the offset of the voltage relative to the current, and does not affect their amplitude values, it can be removed from the formula, thereby significantly simplifying it.

Since, according to Ohm's law, the resistance is directly proportional to the voltage in the circuit section and inversely proportional to the current, then transforming the formulas, you can get the following expression:

• Xc = 1 / w * C = ½ * p * f * C. The unit of measurement is ohm.

It becomes clear that the capacitive resistance depends not only on the capacitance, but also on the frequency. Moreover, the greater this frequency, the less resistance the capacitor will provide to the current passed through it. In relation to capacity, this statement will be the opposite. That is why for a direct current, the frequency of which is equal to zero, the storage resistance will be infinitely large.

In practice, things are a little different. The closer the signal frequency approaches zero, the greater the resistance of the capacitor becomes, but at the same time, an open circuit still cannot occur. This is due to the leakage current. In the case when the frequency tends to infinity, the resistance of the capacitor should become zero, but this also does not happen - due to the presence of parasitic inductance and all the same current leaks.

Inductive component

When an alternating signal passes through a storage device, it can be represented as an inductor connected in series with a power supply. This coil is characterized by a higher resistance in the AC signal circuit than in the DC one. The value of the current at a certain point in time is found as I = I 0 * sinw.

Taking into account that the instantaneous value of the voltage U 0 is opposite in sign to the instantaneous value of the EMF self-induction E 0, as well as using Lenz's rule, you can get the expression E = L * I, where L - inductance.

Therefore: U = L * w * I 0 * cosw * t = U 0 * sin (wt + p / 2), and the current lags behind the voltage by p / 2. Using Ohm's law and assuming that the resistance of the coil is equal to w * L, we get a formula for a section of an electrical circuit that has only an inductive component: U 0 = I 0 / w * L.

Thus, the inductive reactance will be equal to Xl = w * L, it is also measured in ohms. From the obtained expression, it can be seen that the higher the signal frequency, the stronger the resistance to the passage of current will be.

Calculation example

Capacitive and inductive reactances are reactive, that is, those that do not consume power. Therefore, Ohm's law for a section of a circuit with a capacitance has the form I = U / Xc, where current and voltage denote rms values. It is because of this that capacitors are used in circuits to separate not only direct and alternating currents, but also low and high frequencies. In this case, the lower the capacity, the higher the frequency the current can pass. If an active resistance is connected in series with the capacitor, then the total impedance of the circuit is Z = (R ² + Xc ² ) ^½.

The practical application of formulas can be considered when solving a problem. Let there be an RC circuit consisting of a capacitance C = 1 μF and a resistance R = 5 kΩ. It is necessary to find the impedance of this section and the circuit current if the signal frequency is f = 50 Hz, and the amplitude is U = 50 V.

First of all, you need to determine the resistance of the capacitor in the AC circuit for a given frequency. Substituting the data into the formula, we get that for a frequency of 50 Hz, the resistance will be

Xc = 1 / (2 * p * F * C) = 1 / (2 * 3.14 * 50 * 1 * 10 ⁻⁶ ) = 3.2 kΩ.

According to Ohm's law, you can find the current: I = U / Xc = 50/3200 = 15.7 mA.

The voltage is taken to be variable according to the sine law, therefore: U (t) = U * sin (2 * p * f * t) = 50 * sin (314 * t). Accordingly, the current will be I (t) = 15.7 * 10 ⁻³ + sin (314 * t + p / 2). Using the results obtained, you can plot the current and voltage at this frequency. The total resistance of the circuit section is found as Z = (5000²+3200²) ½ = 5,936 Ω = 5.9 kΩ.

Thus, it is not difficult to calculate the impedance at any part of the circuit. In this case, you can also use the so-called online calculators, where initial data such as frequency and capacity are entered, and all calculations are performed automatically. This is convenient, since there is no need to memorize formulas and the probability of an error tends to zero.