Capacitors in electronic circuits

In previous articles, we briefly talked about the operation of capacitors in AC circuits, how and why capacitors pass AC current (see - AC Capacitors) In this case, the capacitors do not heat up, power is not allocated to them: in one half-wave of the sinusoid, the capacitor charges, and in the other, it naturally discharges, while transferring the stored energy back to the current source.

This method of passing current allows you to call the capacitor a free resistance, and that is why the capacitor connected to the outlet does not make the counter spin. And all this is because the current in the capacitor is ahead of exactly 1/4 of the time the voltage applied to it.

But this phase advance allows not only to “trick” the counter, but makes it possible to create various circuits, for example, generators of sinusoidal and rectangular signals, time delays and various frequency filters.

In the process of this story, it will be necessary to recall sometimes what has already been said before, so to speak, to summarize. This will help not to return to previous articles in order to recall a simple formula, or simply, “what is it?”

Parallel and series connection of capacitors

With a parallel connection of capacitors, the total capacity is simply the arithmetic sum of the capacities. Naturally, with this inclusion, the total capacitance will be greater than the capacity of the largest capacitor. Ctotal = C1 + C2 + C3 + ... + Cn.

In the case of a series connection, the total capacity is less than that of the smallest one.

1 / Ctotal = 1 / C1 + 1 / C2 + 1 / C3 + ... + 1 / Cn.

When two identical capacitors are connected in series, the total capacitance will be equal to half of the capacitance of one: for example, when connecting two capacitors of 1 µF each, the total capacitance will be 0.5 µF.

Capacitance Xc

Here, everything, as when connecting resistors, is only the exact opposite: a series connection reduces the total capacitance, while a parallel one increases it. This circumstance should not be forgotten when connecting capacitors, since an increase in capacitance leads to a decrease in capacitance Xc

Xc = 1/2 * π * f * C.

From the point of view of mathematics, this is quite natural, because the capacity C is in the denominator of the fraction. By the way, the frequency f is in the same place, so an increase in the frequency also leads to a decrease in capacitance Xc. The physical meaning of this is that through the same capacitor it is better, more unhindered, that high frequencies pass. This will be discussed a little later, when it comes to the low-pass and high-pass filters.

If we take a capacitor with a capacity of 1 μF, then for a frequency of 60 Hz its Xc will be 2653 Ohms, and for a frequency of 400 Hz the same capacitor has an Xc of only 398 Ohms. Those who wish can check these results by the formula, substituting π = 3.14, the frequency in hertz, and the capacitance in farads. Then the result will be in ohms. Everything must comply with the SI system!

But capacitors are used not only as free-damping damping resistances or in rectifier filters. Without their participation, circuits for low and high frequency generators, various waveform converters, differentiating and integrating circuits, amplifiers and other schemes.

Next, various electrical signals that capacitors have to work with will be considered. First of all, these are periodic signals suitable for observation with oscilloscope.

Period and frequency of oscillations

Periodic oscillation is therefore called periodic, which, without ceasing, repeats the same form, for example, one sinusoidal oscillation. The duration of this full swing is precisely called the period T, and is measured in seconds, milliseconds, microseconds.Modern electronics even deals with nanoseconds (a billionth of a second).

The number of periods per second is called the frequency (how often) of the oscillations f, and is expressed in hertz. 1Hz is the frequency at which one oscillation, one full period, is performed in 1 second. The ratio of the period and frequency is expressed by the simple formula T = 1 / f.

Accordingly, knowing the oscillation period, it is very simple to calculate the frequency f = 1 / T.

This is how the frequency is calculated when measuring with an oscilloscope: the number of cells in a period is calculated, multiplied by the duration of one cell, and the period is obtained, for example, in microseconds. And to find out the frequency, they simply used the last formula.

Normal electronic oscilloscope Allows you to observe only periodic signals that can be synchronized with the sweep frequency in order to obtain a still image suitable for research. If you send a signal to a music program to the input of the oscilloscope, you won’t be able to stop the image for anything. To observe such signals, storage oscilloscopes are used.

When a period is measured in milliseconds, the frequency is obtained in kilohertz, for a period measured in microseconds, the frequency is already expressed in megahertz. This is if you do not follow the requirements of the SI system: period in seconds, frequency in hertz.

Non-sinusoidal vibrations

As mentioned earlier, a sine wave is the most common and suitable for study and practical use of the periodic curve. In industrial conditions, it is obtained using electric generators, for example, in hydroelectric power plants. In electronic devices, vibrations of the most various shapes are used.

Basically, these are three forms: sinusoidal, rectangular and triangular, as shown in Figure 1. Both current and voltage can have such a shape, therefore, the figure shows only the time axis, the ordinate axis is left without a name.

Such oscillations are generated by special electronic circuits. Rectangular and triangular signals are often called pulsed. However, there are a lot of electronic circuits that perform signal conversion: for example, a rectangle, a triangle can be made from a sinusoid.

Picture 1.

For all three signals, the figure shows two periods, all signals have the same frequency.

Spectrum of non-sinusoidal signals

Any electrical signal can be represented as a measurement of the amplitude at some point in time. The frequency of these samples is called the sampling frequency, and at least two times higher than the upper frequency of the measured signal. Then from these samples, you can restore the original signal. This method is used, for example, in digital sound recording. This method is also called time analysis.

Another method assumes that any signal, even a rectangular one, can be represented as the algebraic sum of sinusoids with different frequencies and phases. This method is called frequency analysis. But, what was said “with different frequencies” is not entirely true: the constituent sinusoids are called harmonics and their frequencies obey certain laws.

A sine wave whose frequency is equal to the frequency of a square wave is called the fundamental or first harmonic. Even harmonics are obtained by multiplying the fundamental frequency by an even number, and odd harmonics, respectively, by odd.

Thus, if the first harmonic has a frequency of 1000 Hz, then the second is 2000 Hz, the fourth is 4000 Hz, etc. Odd harmonics will have frequencies of 3000Hz, 5000Hz. Moreover, each harmonic is smaller in amplitude than the main one: the higher the harmonic, the smaller the amplitude.

In music, harmonics are called overtones. They form the timbre of sound, allow you to distinguish the violin from the piano, and the guitar from the saxophone. They do not allow to confuse the male and female voice or to distinguish Petrov from Ivanov. And only the sinusoid itself can no longer be decomposed or assembled from any signals.

Figure 2 shows the construction of a rectangular pulse.

Figure 2

The first and third harmonics are shown in the upper part of the figure. It is easy to see that in one period of the first harmonic three periods of the third pass. In this case, the amplitude of the third harmonic is one third of the first. The sum of the first and third harmonics is also shown here.

Below, together with the sum of 1 and 3 harmonics, another 5 harmonics are shown: for a period of a rectangular signal it manages to do exactly five periods. In this case, its amplitude is even smaller, more precisely, exactly 1/5 of the main (first) one. But one should not think that everything ends at the fifth harmonic: it simply cannot be shown in the figure, in fact there are much more.

The formation of sawtooth and triangular signals, shown in Figure 3, is somewhat more complicated. If in the previous case only odd harmonics took part, then even harmonics come into play.

Figure 3

Thus, we can state the fact that with the help of many harmonics a signal of any shape is synthesized, and the number and type of harmonics depend on the waveform, as shown in Figures 2 and 3.

When repairing and setting up electronic equipment, an oscilloscope is used to study electrical signals. It allows you to consider the form of periodic signals, their amplitude, measure the repetition period. But the harmonics shown in Figures 2 and 3 cannot be seen.

Even if you connect, for example, an electric guitar to an oscilloscope, pull one string, a sinusoid appears on the screen, it is also the first harmonic. In this case, there can be no talk of any overtones. The same sinusoid will result if you blow into the pipe or flute in front of the microphone.

How to get rectangular impulses

After getting acquainted with electrical signals, we need to recall the capacitors with which the article began. First of all, you should get acquainted with one of the classical electronics circuits - multivibrator, (Figure 4) it is he who generates rectangular pulses. The circuit is so classic that it starts working right away without requiring any settings or adjustments.

Figure 4

The multivibrator is a two-stage amplifier, covered by positive feedback. If the collector load resistors R1 = R4, the base resistors R2 = R3 and the capacitors C1 = C2 are equal, the multivibrator is called symmetrical and generates square wave pulses of the meander type - the pulse duration is equal to the pause duration.

The duty cycle of such pulses (the ratio of the period to the pulse duration) is equal to two. In English-language schemes, everything is exactly the opposite: they call it duty cycle. It is calculated as the ratio of the pulse duration to the period of its succession and is expressed as a percentage. Thus, for the meander, the duty cycle is 50%.

Is the computer correct?

The name multivibrator was proposed by the Dutch physicist van der Pol, because the spectrum of a rectangular signal contains many harmonics. You can verify this if you can place a radio receiver operating in the medium wave range near a multivibrator that works even at an audio frequency: howls will come from the speaker. This suggests that in addition to sound frequency, the multivibrator also emits high-frequency oscillations.

To determine the generation frequency, one can use the formula f = 700 / (C1 * R2).

With this form of the formula, the capacitance of the capacitor in microfarads (μF), the resistance in kilo-ohms (KΩ), the result in hertz (Hz). Thus, the frequency is determined by the time constant of the C1 * R2 circuit; collector loads do not affect the frequency. If we take C1 = 0.02 μF, R2 = 39 KΩ, then we get f = 700 / (0.02 * 39) = 897.4 Hz.

Multivibrator in the age of computers and microcontrollers According to this scheme, it is almost never used, although it may well be suitable for various experiments. First of all, using computers. This is how the multivibrator circuit assembled in the Multisim program looks like. The connection of the oscilloscope is also shown here.

Figure 5

In this circuit, capacitors and resistors are installed as in the previous example. The task is to check the calculation according to the formula whether the same frequency will be obtained. To do this, measure the period of the pulses, and then recalculate them in frequency. The result of the Multisim oscilloscope is shown in Figure 6.

Figure 6

Some clarifications to Figure 6.

On the oscilloscope screen, the red pulse shows the pulses on the transistor collector, and the blue on the bases. Below the screen in a large white window, the numbers show the measurement results. We are interested in the column "Time". Time is measured by indicators T1 and T2 (red and blue triangles above the screen).

Thus, the pulse repetition period T2-T1 = 1.107ms is shown quite accurately. It remains only to calculate the frequency f = 1 / T = 1 / 1.107 * 1000 = 903Hz.

The result is almost the same as in the calculation according to the formula, which is given a little higher.

Capacitors can be used not only separately: in combination with resistors, they allow you to easily create various filters or create phase shift circuits. But this will be discussed in the next article.

Continuation of the article: Capacitors in electronic circuits. Part 2

Boris Aladyshkin