Harmonic analysis of periodic and non-periodic signals. Spectral (harmonic) analysis of signals. Mathematical notation of harmonic oscillations. Amplitude and phase spectra of a periodic signal. Spectrum of a periodic sequence directly

With one of the arms with capacitive resistance to alternating current.

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Electrical circuits (part 1)

Lecture 27

Lecture 29

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We spent a lot of time discussing electrostatic fields and the potential of the charge, or the potential energy of a stationary charge. Now let's see what happens if we let the charge move. And it will be much more interesting, because you will learn how most of the modern world around us. So, suppose there is a voltage source. How would I draw it? So be it. I'll take yellow. This is the source of voltage, also known to us as a battery. Here is a positive contact, here is a negative one. The principle of battery operation is a topic for a separate video, which I will definitely record. All that needs to be said is that no matter how much charge - I'll explain everything to you in a second - so, no matter how much charge flows from one side of the battery to the other, somehow the voltage remains constant. And this is not a completely clear thing, because we have already studied capacitors, and we will learn even more about them in the context of circuits, but what we already know about capacitors is that if you remove part of the charge from one of its ends, then the total voltage across the capacitor will decrease. But the battery is a magical thing. It seems that Volta invented it, and therefore we measure the voltage in volts. But even when one side of the magic battery loses charge, the voltage, or potential between the two poles, remains constant. This is the nature of the battery. So let's suppose that there is this magical tool. You probably have a battery in a calculator or phone. Let's see what happens if we let the charge move from one pole to the other. Let's say I have a conductor. Ideal conductor. It needs to be portrayed straight line which, unfortunately, does not work for me. Well, that's about it. What did I do? In the process of connecting the positive to the negative, I show you the standard notation for engineers, electricians, and so on. So take note, you might find it useful someday. These lines are wires. They do not have to be drawn at right angles. I do this just for the sake of clarity. It is assumed that this wire is an ideal conductor, through which the charge flows freely, without encountering obstacles. These zigzags are a resistor, and it will just be an obstacle to charging. It will not allow the charge to move at maximum speed. And behind him, of course, again our ideal guide. So, in which direction will the charge flow? As I said before, electrons flow in electrical circuits. Electrons are such small particles that revolve very quickly around the nucleus of an atom. And they have a fluidity that allows them to move through the conductor. The very movement of objects, if electrons can be called objects at all - some will argue that electrons are just a set of equations - but their very movement is from a negative contact to a positive one. The people who originally designed electronic circuit diagrams, electrical engineering pioneers, electricians or whatever, decided, and I think solely to confuse everyone, that current flows from positive to negative. Exactly. Therefore, the direction of the current is usually indicated in this direction, and the current is denoted by the Latin letter I. So, what is the current? The current is… Wait a minute. Before I tell you what current is, remember that most textbooks, especially if you become an engineer, will state that current flows from positive to negative, but the actual flow of particles goes from negative to positive. Large and heavy protons and neutrons will not be able to move in this direction. Just compare the sizes of a proton and an electron and you will see how crazy this is. These are electrons, small super-fast particles that move through the conductor from the negative terminal. Therefore, the voltage can be represented as the absence of a flow of electrons in this direction. I don't want to confuse you. But be that as it may, just remember that this is the generally accepted standard. But the reality is, to some extent, the opposite of it. So what is a resistor? When the current flows - and I want to depict this as close to reality as possible so that you can clearly see what is happening. When the electrons are flowing - there are these little electrons here, going through the wire - we think this wire is so amazing that they never collide with its atoms. But when the electrons get to the resistor, they start crashing into the particles. They start colliding with other electrons in that environment. This is the resistor. They start colliding with other electrons in matter, colliding with atoms and molecules. And because of this, electrons slow down when colliding with particles. Therefore, the more particles they have in their path, or the less space for them, the more the material slows down the movement of electrons. And as we will see later, the longer it is, the more chance the electron has to crash into something. This is the resistor, it resists and determines the speed of the current. "resistance" is English word denoting resistance. So the current, although assumed to flow from positive to negative, is simply the flow of charge per second. Let's write it down. We're getting off topic a bit, but I think you'll get the idea. Current is the flow of charge, or the change in charge per second, or rather per change over time. What is tension? Voltage is how much charge is attracted to a contact. Therefore, if there is a high voltage between these two contacts, then the electrons are strongly attracted to the other contact. And if the voltage is even higher, then the electrons are attracted even more strongly. Therefore, before it became clear that voltage is just a potential difference, it was called electromotive force. But now we know that this is not power. This is a potential difference, we can even call it electrical pressure, and earlier voltage was called that - electrical pressure. How strongly are the electrons attracted to the other contact? As soon as we open the way for the electrons through the circuit, they will start moving. And, since we consider these wires to be ideal, having no resistance, the electrons will be able to move as fast as possible. But when they get to the resistor, they will start colliding with particles, and this will limit their speed. Since this object limits the speed of the electrons, no matter how fast they move after, the resistor was the limiter. I think you understand. So, although the electrons here can move very fast, they will have to slow down here, and even if they speed up afterwards, the electrons at the beginning will not be able to move faster than through the resistor. Why is this happening? If these electrons are slower, then the current is less here, because the current is the speed at which the charge moves. So if the current is lower here and higher here, then excess charge will start to form around here while the current waits to pass through the resistor. And we know that this does not happen, all electrons move through the circuit at the same speed. And I'm going against the conventional wisdom that suggests that positive particles somehow move in that direction. But I want you to understand what's going on in the chain, because then the complex tasks won't seem so... So intimidating, or something. We know that current, or current strength, is proportional to the voltage of the entire circuit, and this is called Ohm's Law. Ohm's law. So, we know that the voltage is proportional to the current strength in the entire circuit. Voltage equals current times resistance, or otherwise voltage divided by resistance equals current. This is Ohm's law, and it always works if the temperature remains constant. Later we will study this in more detail, and we will find out that when the resistor heats up, the atoms and molecules move faster, the kinetic energy increases. And then the electrons collide with them more often, so the resistance increases with temperature. But, if we assume that for some material the temperature is constant, and later we learn that different materials different resistance coefficients. But for a particular material at a constant temperature for a given shape, the voltage across the resistor divided by its resistance equals the current flowing through it. The resistance of an object is measured in ohms and is denoted Greek letter Omega. A simple example: let's say it's a 16 volt battery that has 16 volts of potential difference between positive and negative. So, a 16-volt battery. Let's assume that the resistance of the resistor is 8 ohms. What is the current strength? I continue to ignore the generally accepted standard though, let's get back to it. What is the current strength in the circuit? Everything is quite obvious here. You just need to apply Ohm's law. Its formula is: V = IR. So the voltage is 16 volts, and it equals the current times the resistance, 8 ohms. That is, the current strength is equal to 16 volts divided by 8 ohms, which equals 2.2 amperes. Amps are denoted by a capital letter A, and they measure the strength of the current. But, as we know, current is the amount of charge over some time, that is, two coulombs per second. So, 2 coulombs per second. Okay, it's been over 11 minutes. Need to stop. You learned the basics of Ohm's law and, perhaps, began to understand what is happening in the circuit. See you in following form about. Subtitles by the Amara.org community

Integrating RC chain

If the input signal is applied to V in , and the output is taken from V c (see figure), then such a circuit is called an integrating circuit.

Response of an integrating type circuit to a single step impact with amplitude V is defined by the following formula:

U c (t) = U 0 (1 − e − t / R C) . (\displaystyle \,\!U_(c)(t)=U_(0)\left(1-e^(-t/RC)\right).)

Thus, the time constant τ of this aperiodic process will be equal to

τ = R C . (\displaystyle \tau=RC.)

Integrating circuits pass the constant component of the signal, cutting off high frequencies, that is, they are low-frequency filters. Moreover, the higher the time constant τ (\displaystyle \tau ), the lower the cutoff frequency. Only the constant component will pass in the limit. This property is used in secondary power supplies in which it is necessary to filter the AC component of the mains voltage. A cable made of a pair of wires has integrating properties, since any wire is a resistor, having its own resistance, and a pair of wires going side by side also form a capacitor, albeit with a small capacitance. When signals pass through such a cable, their high-frequency component may be lost, and the stronger, the longer the cable length.

Differentiating RC chain

A differentiating RC circuit is obtained by interchanging the resistor R and capacitor C in the integrating circuit. In this case, the input signal goes to the capacitor, and the output signal is taken from the resistor. For a DC voltage, the capacitor represents a break in the circuit, that is, the DC component of the signal in the differentiating type circuit will be cut off. Such circuits are high-pass filters. And the cutoff frequency in them is determined by the same time constant τ (\displaystyle \tau ). The more τ (\displaystyle \tau ), the lower the frequency that can be passed through the circuit without change.

Differentiating circuits have another feature. At the output of such a circuit, one signal is converted into two successive voltage jumps up and down relative to the base with an amplitude equal to the input voltage. The base is either the positive source terminal or ground, depending on where the resistor is connected. When the resistor is connected to the source, the amplitude of the positive output pulse will be twice the supply voltage. This is used to multiply the voltage, as well as, in the case of connecting a resistor to the "ground", to form a bipolar voltage from an existing unipolar one.

DIFFERENTIATION CIRCUIT- a device designed for differentiation in time electric. signals. Output reaction D. c. u out ( t) is related to the input action u in ( t) ratio , where - post. a quantity that has the dimension of time. There are passive and active D. c. Passive D. c. used in pulse and digital devices to shorten pulses. Active D. c. used as differentiators in analog computing. devices. The simplest passive D. c. shown in fig. one, a. The current through the capacitance is proportional to the derivative of the voltage applied to it. If the parameters of D. c. are chosen thus,

what u c =u vh, then , a . Condition u c =u input is performed if at the highest frequency of the spectrum of the input signal Option passive D. c. shown in fig. one, b. Under the condition we have and

Rice. 1. Schemes of passive differentiating circuits: a- capacitive RC; b- inductive RL.

Therefore, at the given parameters D. of c. differentiation is the more accurate, the lower the frequencies on which the energy of the input signal is concentrated. However, the more accurate the differentiation, the lower the coefficient. transfer circuit and hence the output level. This contradiction is eliminated in active D. c., where the process of differentiation is combined with the process of amplification. In active D. c. use operational amplifiers(OS) covered by negative feedback (Fig. 2). Input voltage u in ( t) is differentiated by a chain formed by a succession. container connection FROM and R eq - the equivalent resistance of the circuit between the terminals 2-2 ", and then the op-amp is amplified. If you apply voltage to the inverting input of the op-amp, then, provided that its gain, , we get

Rice. 2. Scheme of an active differentiating circuit.

Rice. 3. Passage of an impulse through a differentiating circuit RC: a- input impulse, u in = E at ; b- voltage on the capacitance u c (t); in- output voltage .

For compare. assessments of active and passive D. c. ceteris paribus, you can use the ratio . When passing through D. c. pulse signals there is a decrease in their duration, hence the concept of D. c. as about shortening. Time diagrams illustrating the passage of a rectangular pulse through a passive D. c. are shown in fig. 3. It is assumed that the input voltage source is characterized by zero ext. resistance, and D. c. - the absence of parasitic capacitances. The presence of internal resistance leads to a decrease in the voltage amplitude at the input terminals and, consequently, to a decrease in the amplitudes of the output pulses; the presence of parasitic capacitances - to delaying the processes of rise and fall of output pulses. Active D. of c also have a similar shortening effect.

Complex electronic devices are made up of simple circuits. Consider a circuit consisting of a resistor and a capacitor connected in series with an ideal voltage generator, shown in Fig. 3.3.

Fig.3.3. Differentiating Circuit

If the output voltage is removed from the resistor, then the circuit is called differentiating, if from the capacitor - integrating. These linear circuits are characterized by stationary and transient characteristics. This is due to the fact that a change in the magnitude of the voltage acting in the circuit leads to the fact that currents and voltages in different parts of the circuit acquire new values. The change in the state of the circuit does not occur instantly, but over a certain period of time. Therefore, a distinction is made between a steady state and a transitional state of an electric circuit.

Electrical processes are considered to be steady (stationary) if the law of change of all voltages and currents coincides, up to constant values, with the law of change of the voltage acting in the circuit from an external source. Otherwise, the circuit is considered to be in a transient (non-stationary) state.

The stationary characteristics include the amplitude-frequency and phase characteristics of a linear circuit.

The non-stationary state of a linear circuit is described by a transient response.

We assume that an ideal voltage generator is connected to the input of the circuit. Based on Kirchhoff's second law, for a differentiating circuit, one can write differential equation, linking voltage and current in the branches of the circuit:

(3.2)

Since the voltage at the output of the circuit, then:

(3.3)

Substituting the value of the current into the integral, we get:

(3.4)

Differentiate the left and right sides of the last equation with respect to time:

(3.5)

Let's rewrite this equation in the following form:

, (3.6)

Where = is a circuit parameter called the circuit time constant.

Depending on the value of the time constant, two different relationships are possible between the first and second terms on the right side of the equation.

If the time constant is large compared to the period of harmonic signals >> Or with the pulse duration >> that can be applied to the input of this circuit, then

And the voltage at the output of the circuit repeats the input voltage with slight distortion:

If the time constant is small compared to the period of harmonic signals<<Или с длительностью импульсов <<, то

Hence the output voltage is:

Thus, depending on the value of the time constant, such a circuit can either transmit the input signal to the output with certain distortions, or differentiate it with a certain degree of accuracy. In this case, the shape of the output signal will be different. Below in fig. 3.4 shows the input voltage, voltage across the resistor and capacitor for cases where the time constant is large and the time constant is small.

BUT B

Rice. 3.4. Voltages on the elements of the differentiating circuit at ( BUT) and ( B)

At the initial moment of time, a voltage jump appears on the resistor, equal to the amplitude of the input signal, and then the capacitor begins to charge, during which the voltage across the resistor will decrease.

When the time constant is , the capacitor does not have time to charge up to the amplitude of the input pulse and the circuit transmits the input signal to the output with slight distortion. At<< конденсатор успеет полностью зарядиться до амплитуды входного напряжения за время действия первого импульса, а за время паузы между импульсами – полностью разрядиться. При этом на выходе цепи появляются укороченные импульсы, приблизительно соответствующие производной от входного сигнала. Считается, что когда Цепочка дифференцирует входной сигнал.

Now let's determine the transfer coefficient of the differentiating circuit. The complex transfer coefficient of the differentiating circuit when a harmonic signal is applied to the input is:

. (3.11)

Denote the relation , where is the cutoff frequency of the passband of the differentiating circuit.

The expression for the transfer coefficient will take the form:

The modulus of the transfer coefficient is equal to:

. (3.13)

- the cutoff frequency of the passband, at which the reactance module becomes equal to the value of the active resistance, and the transfer coefficient of the circuit is equal to . The dependence of the transmission coefficient modulus on frequency is called the amplitude-frequency characteristic (AFC).

The dependence of the phase angle between the output and input voltages on frequency is called the phase response (PFC). Phase response:

Below in fig. 3.5 shows the frequency response and phase response of the differentiating circuit:

Rice. 3.5. Amplitude-frequency and phase characteristics

Differentiating circuit

From the amplitude-frequency characteristic, it can be seen that the passage of signals through the differentiating circuit is accompanied by a decrease in the amplitudes of the low-frequency components of its spectrum. The differentiator circuit is a high pass filter.

It can be seen from the phase response that the phases of the low-frequency components are shifted by a larger angle than the phases of the high-frequency components.

The transient response of a differentiating circuit can be obtained by applying a voltage in the form of a single jump to the input. The complex gain is

We have every right to move on to the consideration of chains consisting of these elements 🙂 This is what we will do today.

And the first circuit, the work of which we will consider - differentiating RC circuit.

Differentiating RC circuit.

From the name of the circuit, in principle, it is already clear what elements are included in its composition - this is a capacitor and a resistor 🙂 And it looks like this:

This scheme is based on the fact that current flowing through a capacitor, is directly proportional to the rate of change of the voltage applied to it:

The voltages in the circuit are related as follows (according to the Kirchhoff law):

At the same time, according to Ohm's law, we can write:

We express from the first expression and substitute into the second:

Provided that (that is, the rate of change of voltage is low), we get an approximate dependence for the output voltage:

Thus, the circuit fully justifies its name, because the output voltage is differential input signal.

But another case is also possible, when title="(!LANG:Rendered by QuickLaTeX.com" height="22" width="134" style="vertical-align: -6px;"> (быстрое изменение напряжения). При выполнении этого равенства мы получаем такую ситуацию:!}

That is: .

It can be seen that the condition will be better satisfied for small values ​​of the product , which is called circuit time constant:

Let's see what meaning this characteristic of the chain carries 🙂

The charge and discharge of the capacitor occurs according to the exponential law:

Here, is the voltage across the charged capacitor at the initial moment of time. Let's see what the voltage value will be after time:

The voltage on the capacitor will decrease to 37% of the original.

It turns out that is the time for which the capacitor:

• when charged - will charge up to 63%
• when discharged - discharged by 63% (discharged up to 37%)

With the time constant of the circuit we figured out, let's get back to differentiating RC circuit 🙂

We have analyzed the theoretical aspects of the functioning of the circuit, so let's see how it works in practice. And for this, let's try to apply some kind of signal to the input and see what happens at the output. As an example, let's apply a sequence of rectangular pulses to the input:

And here is how the waveform of the output signal looks like (the second channel is blue):

What do we see here?

Most of the time, the input voltage is constant, which means its differential is 0 (derivative of a constant = 0). This is exactly what we see on the graph, which means that the chain performs its differentiating function. And what are the bursts on the output oscillogram connected with? It's simple - when the input signal is “turned on”, the capacitor is charging, that is, the charging current passes through the circuit and the output voltage is maximum. And then, as the charging process proceeds, the current decreases exponentially to zero, and with it the output voltage decreases, because it is equal to . Let's zoom in on the waveform and then we will get a clear illustration of the charging process:

When the signal is “turned off” at the input of the differentiating circuit, a similar transient occurs, but it is only caused not by charging, but by discharging the capacitor:

In this case, we have a small circuit time constant, so the circuit differentiates the input signal well. According to our theoretical calculations, the more we increase the time constant, the more the output signal will be similar to the input. Let's put it to the test 🙂

We will increase the resistance of the resistor, which will lead to an increase:

You don’t even need to comment on anything here - the result is obvious 🙂 We have confirmed the theoretical calculations by conducting practical experiments, so let's move on to the next question - to integrating RC circuits.

We write expressions for calculating the current and voltage of this circuit:

At the same time, we can determine the current from Ohm's Law:

Equate these expressions and get:

We integrate the right and left sides of the equality:

As in the case with differentiating RC chain two cases are possible here:

In order to make sure that the circuit is working, let's apply to its input exactly the same signal that we used when analyzing the operation of the differentiating circuit, that is, a sequence of rectangular pulses. For small values, the output signal will be very similar to the input signal, and for large values ​​of the circuit time constant, we will see a signal at the output that is approximately equal to the integral of the input. What will be the signal? The sequence of pulses is a section of equal voltage, and the integral of the constant is a linear function (). Thus, at the output, we should see a sawtooth voltage. Let's check the theoretical calculations in practice:

The yellow color here shows the signal at the input, and the blue, respectively, the output signals at different values ​​of the time constant of the circuit. As you can see, we got exactly the result that we expected to see 🙂

This is where we end today's article, but we do not finish studying electronics, so see you in new articles! 🙂