Frequency response analysis. The concept of amplitude-frequency and phase-frequency characteristics of the system. Methods for experimental reading of frequency response and phase response Frequency response analysis

I bought Motorola Pulse Escape Bluetooth headphones. Overall I liked the sound, but one thing remained unclear. According to the instructions, they have an equalizer switch. Presumably, the headphones have several built-in settings that switch in a circle. Unfortunately, I could not determine by ear what settings there were and how many there were, so I decided to find out by measuring.

So, we want to measure the amplitude-frequency response (AFC) of headphones - this is a graph that shows which frequencies are reproduced louder and which ones are quieter. It turns out that such measurements can be made “on the knee”, without special equipment.

We will need a computer with Windows (I used a laptop), a microphone, and also a sound source - some kind of player with bluetooth (I took a smartphone). Well, the headphones themselves, of course.

(There are a lot of pictures under the cut).

Preparation

I found this microphone among my old gadgets. The microphone is cheap, for conversations, not intended for recording music, much less for measurements.

Of course, such a microphone has its own frequency response (and, looking ahead, directional pattern), so it will greatly distort the measurement results, but it is suitable for the task at hand, because we are interested not so much in the absolute characteristics of the headphones, but in how they change when the equalizer is switched.

The laptop had only one combined audio jack. We connect our microphone there:

Windows asks what kind of device we connected. We answer that this is a microphone:

Windows is German, sorry. I promised to use improvised materials.

Thus, the only audio connector is occupied, which is why an additional sound source is needed. We download a special test audio signal to the smartphone - the so-called pink noise. Pink noise is a sound that contains the entire spectrum of frequencies, and equal power over the entire range. (Do not confuse it with white noise! White noise has a different power distribution, so it cannot be used for measurements, as this may damage the speakers).

Adjust the microphone sensitivity level. Right-click on the speaker icon in Windows and select adjust recording devices:

Find our microphone (I called it Jack Mic):

Select it as a recording device (bird in a green circle). We set its sensitivity level closer to the maximum:

Microphone Boost (if present) is removed! This is automatic sensitivity adjustment. It’s good for the voice, but during measurements it will only interfere.

We install the measuring program on the laptop. I love TrueRTA for the ability to see many charts on one screen at once. (RTA - frequency response in English). In the free demo version, the program measures the frequency response in octave steps (that is, adjacent measurement points differ in frequency by a factor of 2). This, of course, is very crude, but for our purposes it will do.

Using tape, secure the microphone near the edge of the table so that it can be covered with an earphone:

It is important to fix the microphone so that it does not move during the measurement process. We connect the headphones with a wire to the smartphone and place one earphone on top of the microphone, so as to tightly close it on top - something like how the earphone covers the human ear:

The second earphone hangs freely under the table, from which we will hear the test signal turned on. We make sure that the headphones are stable and cannot be moved during the measurement process. We can begin.

Measurements

We launch the TrueRTA program and see:

The main part of the window is the field for graphs. To the left of it are the buttons for the signal generator; we don’t need it, because we have an external signal source, a smartphone. On the right are settings for graphs and measurements. At the top are some more settings and controls. Set the field color to white to better see the graphs (menu View → Background Color → White).

We set the measurement limit to 20 Hz and the number of measurements, say, 100. The program will automatically make the specified number of measurements in a row and average the result; this is necessary for a noise signal. Turn off the display of bar charts, let graphs be drawn instead (the button at the top with the image of bars is marked in the next screenshot).

Having made the settings, we make the first measurement - this will be the measurement of silence. We close the windows and doors, ask the children to be silent and press Go:

If everything is done correctly, a graph will begin to appear in the field. Let’s wait until it stabilizes (stops “dancing” back and forth) and click Stop:

We see that the “volume of silence” (background noise) does not exceed -40dBu, and we set (the dB Bottom control on the right side of the window) the lower display limit to -40dBu in order to remove background noise from the screen and see the graph of the signal we are interested in in a larger view.

Now we will measure the real test signal. Turn on the player on your smartphone, starting with low volume.

We start the measurement in TrueRTA with the Go button and gradually turn up the volume on the smartphone. A hissing noise begins to come from the free earphone, and a graph appears on the screen. Add volume until the graph reaches a height of approximately -10...0dBu:

After waiting for the graph to stabilize, we stop the measurement using the Stop button in the program. We also stop the player for now. So what do we see on the graph? Good bass (except for the deepest ones), some roll-off towards the mid-range frequencies and a sharp roll-off towards the high frequencies. Let me remind you that this is not the real frequency response of headphones; the microphone makes its contribution.

We will take this graph as a reference. The headphones received a signal via wire, in this mode they work as passive speakers without any equalizers, their buttons do not work. Let’s save the graph into memory number 1 (via the menu View → Save to Memory → Save to Memory 1 or by pressing Alt+1). You can save graphs in memory cells, and use the Mem1..Mem20 buttons at the top of the window to enable or disable the display of these graphs on the screen.

Now we disconnect the wire (both from the headphones and from the smartphone) and connect the headphones to the smartphone via bluetooth, being careful not to move them on the table.

We turn on the player again, start the measurement with the Go button and, by adjusting the volume on the smartphone, bring the new graph in level to the reference one. The reference chart is shown in green, and the new chart is shown in blue:

We stop the measurement (you don’t have to turn off the player if the hiss from a free earphone doesn’t irritate you) and are glad that via Bluetooth the headphones produce the same frequency response as via wire. We save the graph into memory number 2 (Alt+2) so that it does not leave the screen.

Now we switch the equalizer using the headphone buttons. The headphones report in a cheerful female voice “EQ changed.” We turn on the measurement and, after waiting for the graph to stabilize, we see:

Hm. In some places there are differences of 1 decibel, but this is somehow not serious. More like measurement errors. We put this graph into memory, switch the equalizer again and after the measurement we see another graph (if you look closely):

Well, you already understand. No matter how much I switched the equalizer on the headphones, it made no difference!

On this, in principle, we can finish the work and draw the following conclusion: These headphones do not have a working equalizer. (Now it’s clear why he couldn’t be heard).

However, the fact that we did not see any changes in the results is disappointing and even raises doubts about the correctness of the methodology. Maybe we measured something wrong?

Bonus dimensions

To make sure that we measured the frequency response, and not the weather on the Moon, let's turn the equalizer in another place. We have a player in our smartphone! Let's use its equalizer:

Amplitude-frequency response of headphones (abbreviated frequency response, also “frequency response of the system”, in English - frequency response) is the dependence of the oscillation amplitude (volume) at the headphone output on the frequency of the reproduced harmonic signal. The amplitude-frequency response shows the tonal balance. From the amplitude-frequency response, a frequency response is obtained, which is also called the frequency range, indicated on the boxes or in the documentation for the headphones.

The frequency range is divided into low, medium and high frequencies; the picture above shows how the frequency grid and the names of the frequency ranges relate. Below are the meanings of each range. As you can see, frequencies are perceived in a logarithmic representation - through doubling the frequency. The frequency range in which the upper frequency is twice the lower frequency is called an octave. For example, octaves are frequency ranges: 20 ~ 40 Hz, 250 ~ 500 Hz, 3 ~ 6 kHz.

Common names of frequency ranges
20 - 40 Hz	Low Bass	Sub bass
40 - 80 Hz	Mid Bass	Midbass
80 - 160 Hz	Upper Bass	Upper Bass
160 - 320 Hz	Lower Midrange	Lower middle
320 - 640 Hz	Middle Midrange	Center mid range
640 Hz - 1.28 kHz	Upper Midrange	Upper middle
1.28 - 2.56 kHz	Lower Treble	Bottom high
2.56 - 5.12 kHz	Middele Treble	Mid high
5.12 - 10.2 kHz	Upper Treble	Upper high
10.2 - 20.4 kHz	Top Octave	Upper octave

To evaluate the sound of instruments and different sounds, we offer the following diagram for your reference:

Green color - the main sound range (gray-green - non-dominant low frequencies), orange - aftertones, overtones, additional. harmonic series, (gray-orange - upper non-dominant range).

The vertical axis of the graph indicates the volume level, usually expressed in decibels (dB). A double change in sound pressure corresponds to 6 dB. Subjective perception of loudness depends on many factors (equal loudness curves, spectral composition, etc.), but in general cases we can roughly estimate that a double change in sound pressure will correspond to a double change in loudness.

The values can be relative or absolute in SPL (Sound pressure level). The SPL level can be used to determine the sensitivity of the headphones.

This example shows the frequency response of two headphones, A and B. Headphone A reproduces low and high frequencies quieter than earphone B, but at the same time reproduces mid frequencies louder.

This shows the deviation between headphones and shows more clearly that earphone A is up to 6 dB quieter at low frequencies, and up to 6 dB quieter at the highest frequencies (upper octave). But on average it’s louder by almost 6 dB. In other words, Earphone A plays the low and highest frequencies twice as quiet, and, conversely, the mid frequencies are almost twice as loud.

To assess sound evenness, we offer several general graphs.

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General types of frequency response for open, closed and on-ear headphones. Peculiarities.

Here you can observe several characteristic types of frequency response. The green graph is a subjectively flat frequency response; at the highest frequencies you can see a decline; it is perceived smoothly due to the fact that we are accustomed to perceiving the smooth frequency response that is reproduced by acoustic systems located in front of the listener. In relation to the ear, it is at 60 degrees. If acoustics with a direct frequency response are placed on the side, at 0 degrees, then an excess of the highest frequencies will be perceived. Therefore, thanks to the smooth roll-off, a subjectively smooth sound is achieved. The yellow graph is usually audiophile headphones with accentuated low and high frequencies.

Such headphones are especially in demand among those who listen to recordings of live music, in which the lowest and highest frequencies are minimal. The blue graph is headphones with an emphasis on the upper mid frequencies; usually this graph is found on monitor headphones for musicians for whom it is important to hear their voice as clearly and intelligibly as possible. This can also be found in audiophile headphones for those who prefer listening to vocals. The red graph is a special dip that can serve as a solution against sibilance or other sound emphasis. which does not suit listeners when listening to certain genres. Having decided for what tasks you want to purchase headphones, you can select a number of models based on their characteristic frequency response characteristics.

In the high frequency region, unevenness can usually be observed. It is not worth calculating the exact frequencies and heights of peaks and dips, because they depend on how the headphones are worn. At our special stand there are much fewer variations to put on headphones than at simpler stands, and the stand is closest to reality. However, if your auricle is different, and you wear headphones slightly differently, then this unevenness will only be approximate. Also, depending on the volume level, this unevenness will be subjectively perceived a little differently, as follows from studies of equal volume curves.

The graph line may have some irregularity. Unevenness in the frequency response can appear either from resonances that decay for a long time, or from the interference of sound waves (which is typical for headphones with complex profiles of protective grilles). In the first case, this indicates worse sound, in the second case it does not affect the sound. For a complete picture, you need to look at diagrams of the cumulative spectrum (which is a three-dimensional sonogram) or the attenuation of resonances depending on periods at specific frequencies.

A number of dips are caused by wave interference. On graphs without smoothing, they represent dips in a narrow frequency range. Such dips are not significant and strongly depend on the fit of the headphones.

General types of frequency response for in-ear headphones (plugs). Peculiarities.

Here you can observe several characteristic types of frequency response. The green graph is a subjectively flat frequency response; at the highest frequencies you can see a decline; it is perceived as flat due to the fact that we are accustomed to perceiving the smooth frequency response that speaker systems reproduce when they are in front of the listener. In relation to the ear, it is at 60 degrees. If acoustics with a direct frequency response are placed on the side, at 0 degrees, then an excess of the highest frequencies will be perceived. Therefore, thanks to the smooth roll-off, a subjectively smooth sound is achieved.

The orange graph shows headphones with increased output at low frequencies; such headphones are preferred mainly for portable use when listening to music from a mobile phone or player. Many players and phones have a roll-off in the low frequencies (for example, to save batteries) and more bassy headphone models can correct this deficiency. The blue graph is headphones with an emphasis on the upper mids; this graph is usually found on monitor headphones for musicians for whom it is important to hear their voice as clearly and intelligibly as possible. This can also be found in audiophile headphones for those who prefer listening to vocals. Having decided for what tasks you want to purchase headphones, you can select a number of models based on their characteristic frequency response characteristics.

Unevenness above 10 kHz is highly dependent on the fit of the earphone in the ear canal, and a shift of half a millimeter completely changes the graph. For this reason, it is worth evaluating the graph as the average value in this area.

On the frequency response one can observe from one to several resonances, depending on the landing depth. The frequency of such a resonance is purely individual for each listener, so this resonance is excluded on the graph, but the typical resonance value is shown in dim color. Ideally, it is better to choose headphones that have little or no such resonances.

Dependence of frequency response on amplifier and headphone impedance

The type of frequency response depends on the impedance of the headphones and the impedance of the amplifier (output impedance). As a rule, the frequency response of headphones remains unchanged when the output impedance of the amplifier is close to zero, as well as when the impedance of the headphones has a minimal deviation close to resistive in nature. The higher the output impedance of the amplifier and the more the Rz curve fluctuates, the more the frequency response of the headphones changes.

When measuring the amplifier output in RMAA with an active load, where the load is headphones, you can see the frequency response with a hump in the low-frequency region. In this case, it shows how the frequency response of headphones changes against an amplifier with zero resistance. The error of such a graph depends on the input resistance of the sound card, and the higher it is, the lower the error.

In the example, we will consider the dependence of the frequency response on amplifiers with different output impedances. In our example, the headphones have an impedance of 20 ohms with a maximum value of 60 ohms at 60 Hz.

In the Rz graph, the resistance changes to 60 ohms at low frequencies. Along the horizontal axis of frequency, along the vertical axis - resistance in a logarithmic scale.

When connected to amplifiers with different output impedances, you can see how the frequency response changes. You can see that when you connect headphones to an amplifier with an output impedance of 300 Ohms, the frequency response at 60 Hz changes to 7 dB.

The frequency range indicated on headphone boxes does not show the amplitude-frequency response, but only shows the extreme frequencies after which a decline is expected. For amplifiers that usually have a flat frequency response, limits are indicated in dB, for example -1 dB, -3 dB or another number. For example, 20Hz - 20kHz - 3dB, will mean that already at 20Hz and 20kHz the signal amplitude is 3dB lower than at frequencies around 1kHz.

Amplitude Frequency Response (AFC)

Amplitude-frequency response - (abbreviated as frequency response, in English - frequency response) - amplitude dependence fluctuations (volume) at the output from frequency reproduced harmonic signal.
The term “ amplitude-frequency response” applies only for signal processing devices and sensors- i.e. for devices through which the signal passes. When talking about devices designed to generate signals (generator, musical instruments, etc.), it is more correct to use the term “frequency range”.
Let's start from far away.
Sound is a special type of mechanical vibrations of an elastic medium that can cause auditory sensations.
The basis for the processes of creation, propagation and perception of sound are mechanical vibrations of elastic bodies:
- creation of sound - determined by vibrations of strings, plates, membranes, columns of air and other elements of musical instruments, as well as diaphragms of loudspeakers and other elastic bodies;
- sound propagation - depends on mechanical vibrations of particles of the medium (air, water, wood, metal, etc.);
- sound perception - begins with mechanical vibrations of the eardrum in the hearing aid, and only after this a complex process of information processing occurs in various parts of the auditory system.
Therefore, in order to understand the nature of sound, we must first consider mechanical vibrations.
Oscillations are called repeating processes of changing any parameters of the system (for example, temperature changes, heartbeat, movement of the Moon, etc.).
Mechanical vibrations- these are repeated movements of various bodies (rotation of the Earth and planets, oscillations of pendulums, tuning forks, strings, etc.).
Mechanical vibrations are primarily the movements of bodies. Mechanical motion of a body is called “a change in its position over time in relation to other bodies.”
All movements are described using concepts such as displacement, velocity and acceleration.
Bias is the path (distance) traveled by a body during its movement from some reference point. Any movement of a body can be described as a change in its position in time (t) and space (x, y, z). Graphically, this can be represented (for example, for bodies that are displaced in one direction) as a line on the x (t) plane - in a two-dimensional coordinate system. Displacement is measured in meters (m).
If for each equal period of time a body moves an equal distance, then this is uniform motion. Uniform motion is motion at a constant speed.
Speed is the path traveled by the body per unit time.
It is defined as “the ratio of the length of a path to the period of time during which this path is traveled”
Speed is measured in meters per second (m/s).
If the displacement of a body over equal periods of time is unequal, then the body makes uneven motion. At the same time, its speed changes all the time, i.e. it is a movement with variable speed.
Acceleration is the ratio of the change in speed to the period of time during which this change occurred.
If a body moves at a constant speed, then the acceleration is zero. If the speed changes uniformly (uniformly accelerated motion), then the acceleration is constant: a = const. If the speed changes unevenly, then the acceleration is defined as the first derivative of the speed (or the second derivative of the displacement): a = dv I dt = drx I dt2.
Acceleration is measured in meters per second squared (m/s2).
Simple harmonic oscillations (amplitude, frequency, phase).

In order for the movement to be oscillatory (i.e., repeating), a restoring force must act on the body, directed in the direction opposite to the displacement (it must return the body back). If the magnitude of this force is proportional to the displacement and directed in the opposite direction, i.e. F = - kx, then under the influence of such a force the body makes repeated movements, returning at regular intervals to the equilibrium position. This motion of a body is called simple harmonic oscillation. This type of movement underlies the creation of complex musical sounds, since it is the strings, membranes, and soundboards of musical instruments that vibrate under the action of elastic restoring forces.
An example of simple harmonic oscillations is oscillations of a mass (load) on a spring.
Amplitude of oscillations (A) is called the maximum displacement of the body from the equilibrium position (with steady oscillations it is constant).
Oscillation period (T) is called the shortest period of time after which the oscillations are repeated. For example, if a pendulum goes through a full cycle of oscillations (in one direction and the other) in 0.01 s, then its period of oscillation is equal to this value: T = 0.01 s. For a simple harmonic oscillation, the period does not depend on the amplitude of the oscillations.
Oscillation frequency (f) is determined by the number of oscillations (cycles) per second. Its unit of measurement is equal to one oscillation per second and is called hertz (Hz).
The oscillation frequency is the reciprocal of the period: f = 1/T.
w- angular (circular) frequency. The angular frequency is related to the oscillation frequency according to the formula с = 2Пf, where the number П = 3.14. It is measured in radians per second (rad/s). For example, if the frequency is f = 100 Hz, then co = 628 rad/s.
f0 - initial phase. The initial phase determines the position of the body from which the oscillation began. It is measured in degrees.
For example, if a pendulum begins to oscillate from an equilibrium position, then its initial phase is zero. If the pendulum is first deflected to the extreme right and then pushed, it will begin to oscillate with an initial phase of 90°. If two pendulums (or two strings, membranes, etc.) begin to oscillate with a time delay, then a phase shift will form between them
If the time delay is equal to one quarter of a period, then the phase shift is 90°, if half a period is -180°, three quarters of a period is 270°, one period is 360°.
At the moment of passing the equilibrium position, the body has maximum speed, and at these moments the kinetic energy is maximum and the potential energy is zero. If this sum were always constant, then any body removed from an equilibrium position would oscillate forever, and the result would be a “perpetual motion machine.” However, in a real environment, part of the energy is spent on overcoming friction in the air, friction in supports, etc. (for example, a pendulum in a viscous medium would oscillate for a very short period of time), so the amplitude of oscillations becomes less and less and gradually the body (string, pendulum, the tuning fork) stops - the oscillations die out.
A damped oscillation can be graphically represented as oscillations with a gradually decreasing amplitude.
In electroacoustics, radio engineering and musical acoustics, a quantity called quality factor systems - Q.
Quality factor(Q) is defined as the reciprocal of the attenuation coefficient:
that is, the lower the quality factor, the faster the oscillations decay.
Free vibrations of complex systems. Spectrum

The oscillatory systems described above, for example a pendulum or a load on a spring, are characterized by the fact that they have one mass (weight) and one stiffness (springs or threads) and move (oscillate) in one direction. Such systems are called systems with one degree of freedom.
Real oscillating bodies (strings, plates, membranes, etc.) that create sound in musical instruments are much more complex devices.
Let us consider the oscillations of systems with two degrees of freedom, consisting of two masses on springs.
When a string is actually excited, the first few natural frequencies are usually excited in it; the vibration amplitudes at other frequencies are very small and do not have a significant effect on the overall shape of the vibrations.

The set of natural frequencies and amplitudes of vibrations that are excited in a given body when exposed to an external force (blow, pinch, bow, etc.) is called amplitude spectrum .
If a set of oscillation phases is presented at these frequencies, then such a spectrum is called a phase spectrum.
An example of the vibration shape of a violin string excited by a bow and its spectrum are shown in the figure.
The basic terms that are used to describe the spectrum of an oscillating body are as follows:
the first fundamental (lowest) natural frequency is called fundamental frequency(sometimes called fundamental frequency).
All natural frequencies above the first are called overtones, for example, in the figure, the fundamental frequency is 100 Hz, the first overtone is 110 Hz, the second overtone is 180 Hz, etc. Overtones whose frequencies are in integer ratios with the fundamental frequency are called harmonics(in this case, the fundamental frequency is called first harmonic). For example, in the figure, the third overtone is the second harmonic because its frequency is 200 Hz, i.e., it has a 2:1 ratio to the fundamental frequency.
To be continued... .
To the question: “Why so far away?” I'll answer right away. That the frequency response graph is not as simple as many people imagine it to be. The main thing is to understand how it is formed and what it will tell us.
It just so happens that the average human ear can distinguish signals in the range from 20 to 20,000 Hz (or 20 kHz). This rather substantial range, in turn, is usually divided into 10 octaves (it can be divided into any other number, but 10 is accepted).
In general octave is a frequency range whose boundaries are calculated by doubling or halving the frequency. The lower limit of the next octave is obtained by doubling the lower limit of the previous octave.
Actually, why do you need knowledge of octaves? It is necessary in order to stop the confusion about what should be called lower, middle or some other bass and the like. The generally accepted set of octaves clearly determines who is who to the nearest hertz.
The last line is not numbered. This is due to the fact that it is not included in the standard ten octaves. Pay attention to the column "Title 2". This contains the names of the octaves that are highlighted by musicians. These "strange" people have no concept of deep bass, but they have one octave above - from 20480 Hz. Therefore, there is such a discrepancy in numbering and names.
Now we can talk more specifically about the frequency range of speaker systems. We should start with some unpleasant news: there is no deep bass in multimedia acoustics. The vast majority of music lovers have simply never heard 20 Hz at a level of -3 dB. And now the news is pleasant and unexpected. There are no such frequencies in a real signal either (with some exceptions, of course). An exception is, for example, a recording from an IASCA Competition judge's disc. The song is called "The Viking". There, even 10 Hz are recorded with a decent amplitude. This track was recorded in a special room on a huge organ. The judges will decorate the system that wins over the Vikings with awards, like a Christmas tree with toys. But with a real signal everything is simpler: bass drum – from 40 Hz. Hefty Chinese drums also start from 40 Hz (among them, however, there is one megadrum. So it starts playing as early as 30 Hz). Live double bass – generally from 60 Hz. As you can see, 20 Hz is not mentioned here. Therefore, you don’t have to worry about the absence of such low components. They are not needed to listen to real music.
Here is another quite informative page where you can visually (using the mouse), in more detail, see this sign
Knowing the alphabet of octaves and music, you can begin to understand the frequency response.
Frequency response (amplitude-frequency response) – dependence of the oscillation amplitude at the device output on the frequency of the input harmonic signal. That is, the system is supplied with a signal at the input, the level of which is taken as 0 dB. From this signal, speakers with an amplification path do what they can. What they usually end up with is not a straight line at 0 dB, but a somewhat broken line. The most interesting thing, by the way, is that everyone (from audio enthusiasts to audio manufacturers) strives for a perfectly flat frequency response, but they are afraid to “strive.”
Actually, what is the benefit of the frequency response and why do they constantly try to measure this curve? The fact is that it can be used to establish real frequency range boundaries, and not those whispered by the “evil marketing spirit” to the manufacturer. It is customary to indicate at what signal drop the boundary frequencies are still played. If not specified, it is assumed that the standard -3 dB was taken. This is where the catch lies. It is enough not to indicate at what drop the boundary values were taken, and you can absolutely honestly indicate at least 20 Hz - 20 kHz, although, indeed, these 20 Hz are achievable at a signal level that is very different from the prescribed -3.
Also, the benefit of the frequency response is expressed in the fact that from it, although approximately, you can understand what problems the selected system will have. Moreover, the system as a whole. The frequency response suffers from all elements of the path. To understand how the system will sound according to the schedule, you need to know the elements of psychoacoustics. In short, the situation is like this: a person speaks within medium frequencies. That’s why he perceives them best. And at the corresponding octaves the graph should be the most even, since distortions in this area put a lot of pressure on the ears. The presence of tall narrow peaks is also undesirable. The general rule here is that peaks are heard better than valleys, and a sharp peak is heard better than a flat one.
The abscissa scale (blue) shows frequencies in hertz (Hz)
The ordinate scale (red) shows the sensitivity level (dB)
Green - the frequency response itself
When carrying out frequency response measurements, not a sine wave is used as a test signal, but a special signal called “pink noise”.
Pink noise is a pseudo-random broadband signal in which the total power at all frequencies within any octave is equal to the total power at all frequencies within any other octave. It sounds very much like a waterfall.
Loudspeakers are directional devices, i.e. they focus the emitted sound in a specific direction. As you move away from the main axis of the loudspeaker, the sound level may decrease and its frequency response becomes less linear.
Volume
Often the terms “loudness” and “sound pressure level” are used interchangeably, but this is incorrect, since the term “loudness” has its own specific meaning. The sound pressure level in dB is determined using sound level meters.
Equal Loudness Curves and Backgrounds
Will listeners perceive noise-like or sine-wave test signals with linear frequency response across the entire audio frequency range, directed to a linear frequency response power amplifier and then into a linear frequency response loudspeaker, equally loud at all frequencies? The fact is that the sensitivity of human hearing is nonlinear, and therefore listeners will perceive sounds of equal loudness at different frequencies as sounds with different sound pressure.
This phenomenon is described by the so-called “equal loudness curves” (figure), which show what sound pressure is required to be created at different frequencies so that for listeners the loudness of these sounds is equal to the loudness of a sound with a frequency of 1 kHz. In order for us to perceive higher and lower frequency sounds to be as loud as a 1 kHz sound, they must have a greater sound pressure. And the lower the sound level, the less sensitive our ear is to low frequencies.
The sound pressure level of the reference sound is set at a frequency of 1000 Hz (for example, 40 dB), then the subject is asked to listen to the signal at a different frequency (for example, 100 Hz), and adjust its level so that it seems equally loud to the reference one. Signals can be presented via telephones or loudspeakers. If you do this for different frequencies, and set aside the resulting values of the sound pressure level, which are required for signals of different frequencies so that they are equally loud with the reference signal, you will get one of the curves in the figure.
For example, for a 100 Hz sound to appear as loud as a 1000 Hz sound at 40 dB, its level must be higher, about 50 dB. If a sound is supplied with a frequency of 50 Hz, then in order to make it equally loud as the reference one, you need to raise its level to 65 dB, etc. If we now increase the reference sound level to 60 dB and repeat all the experiments, we will get an equal loudness curve corresponding to a level of 60 dB...
A family of such curves for various levels of 0, 10, 20...110 dB is shown in the figure. These curves are called curves of equal volume. They were obtained by scientists Fletcher and Manson as a result of processing data from a large number of experiments they conducted among several hundred visitors to the 1931 World's Fair in New York.
Currently, the international standard ISO 226 (1987) accepts updated measurement data obtained in 1956. It is the data from the ISO standard that is presented in the figure, while the measurements were carried out under free-field conditions, that is, in an anechoic chamber, the sound source was located frontally and the sound was supplied through loudspeakers. New results have now been accumulated, and it is expected that these data will be refined in the near future. Each of the presented curves is called an isophone and characterizes the volume level of sounds of different frequencies.
If we analyze these curves, we can see that at low sound pressure levels, the estimate of the loudness level is very dependent on frequency - hearing is less sensitive to low and high frequencies, and it is necessary to create much higher sound pressure levels in order for the sound to sound equally loud with the reference sound 1000 Hz At high levels, the isophones are leveled out, the rise at low frequencies becomes less steep - the volume of low-frequency sounds increases more quickly than that of medium and high frequencies. Thus, at higher levels, low, mid, and high sounds are graded more evenly in loudness level.
So. We have the sound pressure level measured using measuring equipment and the volume that is physically perceived by a person.

This raises a question! By measuring the frequency response of a speaker using measuring equipment, what do we get? What does OUR ear hear? Or what readings does the microphone take with its sensitive element of the measuring equipment? And what conclusion can be drawn from these testimonies?
This raises a question! By measuring the frequency response of a speaker using measuring equipment, what do we get? What does OUR ear hear? Or what readings does the microphone take with its sensitive element of the measuring equipment? And what conclusion can be drawn from these testimonies?

Frequency analysis. frequency response

15. Save the text from the output file in the report template, having previously removed empty lines from it. Highlight in the text the results of calculating the small-signal transfer function in the analysis mode for direct current, input and output resistance (Fig. 13).

** Profile: "SCHEMATIC1-post" [ C:\OrCAD_Data\test-

* pspicefiles\schematic1\post.sim ]

****JOB STATISTICS SUMMARY

Total job time (using Solver 1) = .02

Rice. 13. Output file fragment

The text interface of the PSpise A/D program, working with *.cir and *.out files, and modeling directives are described in more detail in .

Frequency analysis. frequency response

16. Transform the diagram in accordance with paragraph 3 of the laboratory assignment. Instead of the input source, put a VAC or IAC source (in accordance with the option), set the amplitude of the variable component arbitrarily, but not equal to zero. Other sources should be excluded from the diagram.

The current source has infinite internal resistance (open circuit), and the voltage source has zero (jumper).

Since the circuit is linear, and it is necessary to remove the frequency response and phase response, the amplitude of the input influence does not play a role (within the limits of values permissible in

PSpice, for voltages and currents - 10 10 volts or amperes).

VAC and IAC are harmonic signal sources for frequency analysis and can be used for DC analysis.

17. Create a new modeling profile. 3

18. Select analysis type AC Sweep – analysis of a circuit in the frequency domain. Set the initial analysis parameters as shown in Fig. 14.

Selecting a frequency step: Linear – linear, Logarithmic – logarithmic. For a linear step, the total number of points per scale (Total Points) is indicated, for a logarithmic step the number of points per decade or octa-

wu (Points/Decade (Octave)).Start Frequency – initial frequency of analysis, cannot be equal to 0.End Frequency – final frequency of analysis.

Laboratory work No. 1. Static, frequency and timing analysis of a passive RLC circuit

Rice. 14. Simulation settings window. Setting up AC Sweep Analysis

19. Run the simulation. 2

20. Open output file ( Output File)4 find and copy the section with analysis directives to the report template (Analysis directives).

Frequency domain analysis is specified by the .AC directive.

21. Construct frequency response graphs.

The frequency response is the dependence of the modulus of the complex coefficient

The frequency transfer ratio can be defined as the ratio of the amplitudes of the input and output signals.

21.a. Open the Add Traces window. In PSpice A/D, the command Trace>Add Trace..., the Insert key or a button on the toolbar (Fig. 15).

In OrCAD 16, you can also add a graph through the context menu, called by right-clicking on an empty plot area.

Rice. 15. Calling the Add Traces window

The functions of constructing graphs and post-processing of simulation results are performed directly by a graphical post-processor

Probe built into PSpice A/D.

Laboratory work No. 1. Static, frequency and timing analysis of a passive RLC circuit Customizing the appearance of the plotting area and graphs

21.b. In the Add Traces window, using the keyboard or mouse, enter into the Trace Expression line expressions for the frequency response of all outputs (Fig. 16), as the ratio of output, input voltages (even version) or currents (odd version).

The left side of the Add Traces window lists all the currents and potentials of the nodes in your circuit. On the right side is a list of mathematical functions and connectors that Probe can apply to individual graphs.

Rice. 16. Entering graph expressions in the Add Traces window

IN result of analysis AC Sweep nodal voltages are calculated

And branch currents, which are complex quantities. In mode AC Sweep Probe supports calculations with complex numbers. Entering expressions for complex values into the Trace Expression line of the Add Traces window without using any mathematical functions or Probe operators displays the result module. If an expression is entered for a real value, for example the phase of the complex transmission coefficient, then the result may be negative. If the expression is complex, for example, the complex voltage transfer coefficient V(N1)/V(N4) - defined as the ratio of the potentials of nodes N1 and N4, then its module is displayed, which is always non-negative.

To access the real and imaginary parts of the calculated quantities, the R and IMG functions are used, respectively.

IN The Probe program also uses the ABS (absolute value) function - absolute value and similar to it M (magnitude) - module, corresponding

valid expressions: V(N1)/V(N4), M(V(N1)/V(N4)), ABS(V(N1)/V(N4)) and SQRT(PWR(R(V(N1)/ V(N4)),2)+PWR(IMG(V(N1)/V(N4)),2)) – completely equivalent

valence. The SQRT function is the square root, and the PWR function is the exponentiation, in the example given, the square.

Laboratory work No. 1. Static, frequency and timing analysis of a passive RLC circuit Customizing the appearance of the plotting area and graphs

21st century Analyze the form of the obtained frequency response, open the simulation profile settings window (Simulation Settings) and change, if necessary, the limiting frequencies of the analysis, the type of frequency step, the number of points so that the graphs take on the most informative form.

You can call the Simulation Settings window and change the simulation directives directly from the PSpice A/D program by clicking the corresponding toolbar icon (Fig. 17) or using the command Simulation>Edit Profile….

21. In the Simulation Settings window, on the Probe Windows tab check the box Last plot in the Show group (Fig. 18 ) – displays graphs for the last entered expressions.

21.d. If the simulation directive has been changed, run the simulation again.

You can start the simulation directly from the PSpice A/D program by clicking the corresponding button on the toolbar (Fig. 17) or using the command

Simulation>Run.

Rice. 17. Calling the Simulation Settings window (Edit Profile command)

and starting the simulation (Run command) from the PSpice A/D program

Rice. 18. Simulation Settings window.

Probe Window tab – setting up the display of simulation results

Laboratory work No. 1. Static, frequency and timing analysis of a passive RLC circuit Customizing the appearance of the plotting area and graphs

After each simulation, information about the expressions entered in the Trace Expression line is reset; the Show Last plot option allows you not to enter expressions again.

Customizing the appearance of the plotting area and graphs

21.e. If necessary, change the display scale along the axes (linear or logarithmic) (Fig. 19).

Rice. 19. Changes the display scale along the axes.

Opening the Axis Settings window

21.g. Remove intermediate grid lines.

Open the window for setting up grid and axes parameters (Axis Settings). Command Plot>Axis Settings..., or double-click the left mouse button in the value area of one of the axes, or select the context menu item available by right-clicking on the grid line (Settings... item) (Fig. 19).

In the Axis Settings window on the X Grid and Y Grid tabs in the Minor Grids section check the box None (Fig. 20).

21.z. Configure the display of graphs.

Open the chart properties window (Trace Properties). Right-click the graph line or icon in the line with the graph legends, X axis (Fig. 21). In the context menu that appears, select Properties….

In the Trace Properties window, change the graph display parameters: increase the thickness of graph lines, change the color and type of lines.

Repeat the steps for all graphs.

The display settings for frame and grid lines can be configured in the same way.

Laboratory work No. 1. Static, frequency and timing analysis of a passive RLC circuit Frequency analysis. FCHH

The thickness of the lines affects the quality of printing and perception. Line colors should be selected that, when printed in black and white, provide acceptable clarity and contrast against a white background.

Rice. 20. Axis Settings window. Setting the display of intermediate grid lines

Rice. 21. Setting the appearance of graphs

21.i. Save frequency response graphs. Command Window>Copy to Clipboard (save to clipboard), in the window that opens, in the Foreground section, check the box change white to black (change white with black), click OK (Fig. 22). Paste the picture from the clipboard into the report template (Ctrl+V

or Shift+Ins).

The construction area, including axes, grid, graphs, axes labels, legend and text notes is copied to the buffer (Fig. 23). The size of the image in the buffer depends on the actual size of the construction area at the time of copying.