Continuous modulation techniques
Signal modulation techniques
Lecture number 7
In some cases, during telemetry, it is necessary to transmit information about a continuous process using continuous messages. And if at the same time it is necessary to obtain information about infinitely a large number gradation, then the signals by which continuous messages are transmitted must be continuous.
Continuous signal formed using continuous modulation techniques.
Modulation is the formation of a signal by changing the parameters of the carrier under the influence of a message.
At continuous methods modulation as a carrier is used HF - sinusoidal oscillation, or non-sinusoidal. Since a sinusoidal oscillation is characterized by such basic parameters as amplitude, frequency and phase, there are three main types of modulation: amplitude (AM), frequency (FM) and phase (PM). There are also varieties of these modulations, which will be discussed below, as well as oscillations of the main types of modulation, the so-called double modulation.
It is possible to transmit a continuous message directly without using an HF carrier, i.e. no modulation. However, modulation expands the possibilities of transmitting messages over following reasons:
a) the number of messages that can be transmitted over one communication line increases by using frequency division signals and subcarriers;
b) the reliability of the transmitted signals increases when using noise-immune types of modulation;
c) the efficiency of signal emission during transmission over the radio channel is increased. This is due to the fact that the size of the antenna must be at least 1/10 of the wavelength of the transmitted signal. So, when transmitting a message with a frequency of 10 kHz, having a wavelength of 30 km, an antenna with a length of 3 km would be required. If this message is transmitted to a carrier of 200 kHz, then this will reduce the antenna length by 20 times (150 m).
Amplitude modulation (AM) is the formation of a signal by changing the amplitude of a harmonic oscillation in proportion to the instantaneous value of the voltage or current of another electrical signal(messages).
We will consider the case amplitude modulation with which transmitted message is the simplest harmonic oscillation U c = UΩ cos Ω t(rice. a) where Ω is the frequency, and UΩ - vibration amplitude, HF - carrier, or carrier, U n = U w 0 = cos ω 0 t(rice. b), ω 0 is the carrier frequency, and Uω 0 - amplitude.
Under the influence of the message on the carrier amplitude, a new oscillation is formed, in which the amplitude changes, but the frequency ω 0 remains constant.
The carrier amplitude will change linearly.
U a m = Uω 0 + ku c = Uω 0 + k UΩ cos Ω t = Uω0 (1+ m cos Ω t).
where k Is the coefficient of proportionality, and
– (4-2)
- the relative change in carrier amplitude, called modulation rate or depth. Sometimes the modulation factor is expressed as a percentage. If the amplitude of the modulated waveform rises to twice the amplitude of the carrier, then the modulation depth is 100%.
Amplitude - modulation of the oscillation will have the form shown in Fig. c), and its instantaneous value will be determined by equality
Uam = Uω 0(1 + m cos Ω t) cos ω 0 t(4-3)
Expanding the brackets and using the fact that
cos Ω t cosω 0 t =+ cos [(? 0 + ? ) t + ? ]} —
— Um J 2 (Mchm) (sin [(? 0 — 2 ? ) t +j] + sin [(? 0 +2 ? ) t + ? ]}+
+ Um J 3 (Mchm) (cos [(? 0 — 3 ? ) t +j] + cos [(? 0 +3 ? ) t +? ]} —
— Um J 4 (Mchm) (sin [(? 0 — 4 ? ) t +j] + sin [(? 0 +4 ? ) t +? ]} — … (15)
where J k (Mpm) are proportionality coefficients.
J k (Mfm) are determined by the Bessel functions and depend on the frequency modulation index. Figure 8 shows a graph containing eight Bessel functions. To determine the amplitudes of the components of the FM signal spectrum, it is necessary to determine the value of the Bessel functions for a given index. And how
Figure 8 - Bessel functions
It can be seen from the figure that various functions have their origin in different values of MFM, and therefore, the number of components in the spectrum will be determined by MFM (as the index increases, the number of spectrum components also increases). For example, it is necessary to determine the coefficients J k (Mchm) at Mchm = 2. The graph shows that for a given index it is possible to determine the coefficients for five functions (J 0, J 1, J 2, J 3, J 4) Their value at a given index will be: J 0 = 0.21; J 1 = 0.58; J 2 = 0.36; J 3 = 0.12; J 4 = 0.02. All other functions start after the value Mhm = 2 and are equal to zero, respectively. For the given example, the number of components in the FM signal spectrum will be 9: one component of the carrier signal (Um J 0) and four components in each sideband (Um J 1; Um J 2; Um J 3; Um J 4).
Another important feature of the FM signal spectrum is that it is possible to achieve the absence of a carrier signal component or make its amplitude much less than the amplitudes of information components without additional technical complications of the modulator. To do this, it is necessary to select such a modulation index MFM, at which J 0 (MFM) will be equal to zero (at the intersection of the function J 0 with the MFM axis), for example, MFM = 2.4.
Since an increase in the components leads to an increase in the width of the FM signal spectrum, it means that the spectrum width depends on the FM signal (Figure 9). As can be seen from the figure, at Mpm ≥ 0.5, the width of the FM signal spectrum corresponds to the width of the AM signal spectrum, and in this case the frequency modulation is narrowband, with an increase in Mpm, the spectrum width increases, and the modulation in this case is broadband... For an FM signal, the spectrum width is determined
D? chm= 2 (1 + Mchm) ? (16)
The advantages of frequency modulation are:
- high noise immunity;
- more efficient use of transmitter power;
- comparative ease of obtaining modulated signals.
The main disadvantage of this modulation is the wide spectrum of the modulated signal.
Frequency modulation is used:
- in television broadcasting systems (for the transmission of sound signals);
- satellite TV and radio broadcasting systems;
- systems of high quality stereo broadcasting (FM range);
- radio relay lines (RRL);
- cellular telephone communication.
Figure 9 - Spectra of the FM signal with a harmonic modulating signal and with various indices Mchm: a) at Mchm = 0.5, b) at Mchm = 1, c) at Mchm = 5
Phase modulation
Phase modulation- the process of changing the phase of the carrier signal in accordance with instantaneous values modulating signal.
Consider the mathematical model phase-modulated(PM) signal with harmonic modulating signal. When exposed to a modulating signal
u(t) = Um u sin? t
on the carrier vibration
S(t) = Um sin(? 0 t+ ? )
there is a change in the instantaneous phase of the carrier signal according to the law:
? fm (t) =? 0 t +? + a fmUm u sin? t(17)
where a fm is the frequency modulation proportionality factor.
Substituting ? fm (t) in S (t) we obtain a mathematical model of the FM signal with a harmonic modulating signal:
Sfm (t) = Um sin (? 0 t +a fmUm u sin? t +? ) (18)
The product a fm Um u = Dj m is called phase modulation index or phase deviation.
Since a change in phase causes a change in frequency, then using (11) we determine the law of change in the frequency of the FM signal:
? fm(t)= d ? fm (t)/ dt= w 0 + a fmUm u? cos ? t (19)
The product a fm Um u ? =?? m is the deviation of the phase modulation frequency. Comparing the frequency deviation with frequency and phase modulation, it can be concluded that in both FM and FM, the frequency deviation depends on the proportionality coefficient and the amplitude of the modulating signal, but in FM, the frequency deviation also depends on the frequency of the modulating signal.
Timing diagrams explaining the process of forming the FM signal are shown in Figure 10.
Upon decomposition mathematical model The FM signal into harmonic components will result in the same series as in the case of frequency modulation (15), with the only difference that the coefficients J k will depend on the phase modulation index? ? m (J k (? ? m)). These coefficients will be determined in the same way as for FM, that is, according to the Bessel functions, with the only difference that on the abscissa axis it is necessary to replace MFM with? ? m. Since the spectrum of the FM signal is constructed similarly to the spectrum of the FM signal, it has the same conclusions as for the FM signal (paragraph 1.4).
Figure 10 - Formation of the FM signal
The spectrum width of the FM signal is determined by the expression:
? ? fm=2(1+ ? jm) ? (20).
The advantages of phase modulation are:
- high noise immunity;
- more efficient use of transmitter power.
- The disadvantages of phase modulation are:
- large spectrum width;
- comparative difficulty of obtaining modulated signals and their detection
Discrete Binary Modulation (Harmonic Carrier Keying)
Discrete Binary Modulation (Keying)- a special case of analog modulation, in which a harmonic carrier is used as a carrier signal, and a discrete binary signal is used as a modulating signal.
There are four types of manipulation:
- amplitude keying (AMn or AMT);
- frequency shift keying (FMN or TBI);
- phase shift keying (PMN or PMT);
- relative phase shift keying (OFMn or OFM).
Temporary and spectral diagrams modulated signals for various types of manipulation are shown in Figure 11.
At amplitude shift keying , as well as with any other modulating signal, the envelope S AMn (t) repeats the shape of the modulating signal (Figure 11, c).
At frequency shift keying are there two frequencies? 1 and? 2. Is a higher frequency used when there is a pulse in the baseband signal (burst)? 2, in the absence of a pulse (active pause), a lower frequency w 1 is used corresponding to the unmodulated carrier (Figure 11, d)). The spectrum of the frequency-shift keyed signal S FSK (t) has two bands near the frequencies? 1 and? 2.
At phase shift keying the phase of the carrier signal changes 180 ° when the amplitude of the modulating signal changes. If a series of several pulses follows, then the phase of the carrier signal does not change in this interval (Figure 11, e).
Figure 11 - Time and spectral diagrams of modulated signals of various types of discrete binary modulation
At phase-shift keying the phase of the carrier signal changes by 180 ° only at the moment the pulse is applied, i.e., when switching from an active pause to a message (0? 1) or from a message to a message (1? 1). With a decrease in the amplitude of the modulating signal, the phase of the carrier signal does not change (Figure 11, e). The signal spectra for PSK and SPM have the same form (Figure 9, f).
Comparing the spectra of all modulated signals, it can be noted that the FSK signal spectrum has the greatest width, the AMn, PMn, OFMn spectrum has the smallest, but the carrier signal component is absent in the PMN and PMN spectra of signals.
In view of the greater noise immunity, the most widespread are frequency, phase and relative-phase keying. Their various types are used in telegraphy, in data transmission, in mobile radio communication systems (telephone, trunking, paging).
Pulse modulation
Pulse modulation Is a modulation in which a periodic sequence of pulses is used as a carrier signal, and an analog or discrete signal can be used as a modulating signal.
Since a periodic sequence is characterized by four information parameters (amplitude, frequency, phase and pulse duration), there are four main types of pulse modulation:
- pulse-amplitude modulation (AIM); there is a change in the amplitude of the pulses of the carrier signal;
- pulse frequency modulation (PFM), there is a change in the pulse repetition rate of the carrier signal;
- phase-pulse modulation (PPM), there is a change in the phase of the pulses of the carrier signal;
- pulse width modulation (PWM), there is a change in the duration of the pulses of the carrier signal.
Timing diagrams of pulse-modulated signals are shown in Figure 12.
With AMP, the amplitude of the carrier signal S (t) changes in accordance with the instantaneous values of the modulating signal u (t), i.e. the pulse envelope repeats the shape of the modulating signal (Figure 12, c).
With PWM, the pulse duration S (t) changes in accordance with the instantaneous values of u (t) (Figure 12, d).
Figure 12 - Timing diagrams of signals with pulse modulation
With PFM, the period, and accordingly the frequency, of the carrier signal S (t) changes in accordance with the instantaneous values of u (t) (Figure 12, e).
With PPM, the pulses of the carrier signal are displaced relative to their clock (temporal) position in the unmodulated carrier (clock moments are indicated on the diagrams by points T, 2T, 3T, etc.). The PPM signal is shown in Figure 12, f.
Since in pulse modulation the carrier of the message is a periodic sequence of pulses, the spectrum of pulse-modulated signals is discrete and contains many spectral components. This spectrum is a spectrum of a periodic sequence of pulses in which the components of the modulating signal are located near each harmonic component of the carrier signal (Figure 13). The structure of the sidebands around each component of the carrier signal depends on the modulation type.
Figure 13 - Spectrum of a pulse-modulated signal
Also an important feature of the spectrum of pulse-modulated signals is that the width of the spectrum of the modulated signal, except for PWM, does not depend on the modulating signal. It is completely determined by the pulse width of the carrier signal. Since with PWM the pulse duration changes and depends on the modulating signal, then with this type of modulation and the width of the spectrum also depend on the modulating signal.
The pulse repetition rate of the carrier signal can be determined by the theorem of V.A.Kotelnikov as f 0 = 2Fmax. In this case, Fmax is the upper frequency of the spectrum of the modulating signal.
The transmission of pulse-modulated signals over high-frequency communication lines is impossible, since the spectrum of these signals contains low-frequency components. Therefore, for transmission, re-modulation... This is a modulation in which a pulse-modulated signal is used as a modulating signal, and a harmonic waveform is used as a carrier. With re-modulation, the spectrum of the pulse-modulated signal is transferred to the carrier frequency region. For re-modulation, any of the types of analog modulation can be used: AM, ChS, FM. The resulting modulation is indicated by two abbreviations: the first indicates the type of pulse modulation and the second indicates the type of analog modulation, for example, AIM-AM (Figure 14, a) or PWM-FM (Figure 14, b), etc.
Figure 14 - Timing diagrams of signals with pulse repetitive modulation
As you know, AM is a type of modulation in which the amplitude of the carrier signal changes according to the law of the modulating (information) signal. There are many sources with theoretical and practical descriptions of AM. The description is given primarily to show the frequency composition of the AM signal. A single-tone signal is usually considered as the modulating signal. This signal is given by a simple sine function. I was always asked, and I wondered how to describe AM in case an arbitrary signal is used as a modulating signal. It is an arbitrary signal, the frequency spectrum of which consists of many components, that is of interest, since AM is used in broadcasting to transmit sound.
Let's try to describe AM for the above case, taking into account that the modulating signal can be represented as a continuous sum of simple single-tone signals of different frequencies with different amplitudes and phases. Without going into the intricacies of mathematical analysis, this signal can be written as a continuous sum (integral) Fourier:
Where is the upper limit of the signal frequency (baseband signal bandwidth), is the integration variable responsible for the frequency, and. Functions and are the amplitude and phase of the signal component at the frequency.
The integrand of this formula is the so-called. trigonometric convolution into the amplitude-phase form of the Fourier series term, into which the signal can be decomposed. The integral in (1) can be called the Fourier integral, since, in fact, it is a continuous sum, i.e. a continuous Fourier series into which the original signal is decomposed. Decomposition of the signal in a similar series gives an idea of the frequency composition of this signal. Thus, the original modulating signal is presented as a continuous sum of sinusoids (in this case, for convenience -) of different frequencies from to, each of them has its own amplitude and phase shift. The function represents the frequency spectrum of the original signal.
It should be noted that the signal is considered for a limited period of time. Generally speaking, if we are talking about an audio signal, then, as a rule, it makes practical sense to consider the frequency spectrum for very short signal fragments. Obviously, the longer the duration of the signal, the more low-frequency (approaching zero) components will appear in the spectral composition, which cannot be compared with audio frequencies in the audible range.
In addition to the modulating signal, there is a tone signal, which is a carrier wave with frequency, amplitude and zero initial phase:
Moreover. Indeed, in broadcasting, the carrier frequency is many times greater than the bandwidth transmitted signal.
Now let's go directly to the amplitude modulation process.
It is known that the AM signal is the result of multiplying the carrier signal and the modulating signal, previously biased and "indexed" by the modulation index, i.e.
To avoid so called overmodulation.
Substitute the initial data (1) and (2) into expression (3), open the brackets, and add some factors independent of the integration variable under the integral:
We apply the well-known school trigonometric formula for transforming a product for integrands:
This formula is key in AM and emphasizes these same "two side" in the spectral composition of the AM signal.
Continuing the equality, we split the integral of the resulting sum into the sum of two integrals, expand the brackets and put out the required factors in the arguments of the functions:
The three resulting terms, respectively, represent, as can be seen from the equality, the carrier signal, the signals of the "lower" and "upper" side. Before giving concrete clarification, let's continue the equality by applying the variable replacement method in the following configuration:
Let's use this very replacement:
By changing the limits of integration in places in the first integral (as a result of which the sign in front of the integral will change to the opposite), two integrals can be combined into one. Moreover, the first term describing the carrier signal can also be added there. In this case, naturally, the integrands of the amplitude and phase must be generalized. This is all done conditionally and for more detailed clarity, without going into the intricacies of mathematical analysis. Thus, it will turn out:
Thus, new piecewise specified functions (4) and (5) were introduced, describing the change in amplitude and phase depending on frequency. Looking at the components of function (4), one can notice that the third component is obtained by parallel transfer of the function to, and the first one is also obtained with a preliminary mirror image. I do not take into account the constant multipliers in front of the functions that reduce the amplitude. That is, there are three components in the AM signal spectrum: carrier, upper side and lower side, which was reflected in (4).
In conclusion, it is worth noting that AM can be described using a more sophisticated approach based on complex signals and complex numbers. The common signal discussed in this article does not have an imaginary component. Taking into account the representation by means of vector diagrams on the complex plane, a signal without an imaginary component is added from two complex signals with both components. This is obvious if you represent a single-tone signal as the sum of two vectors that rotate in opposite directions symmetrically about the x-axis (Re). The rotation speed of these vectors is equivalent to the signal frequency, and the direction is equal to the sign of the frequency (positive or negative). It follows from this that the frequency spectrum of a signal without an imaginary component has not only a positive, but also a negative component. And, of course, it is symmetrical about zero. It is with this representation that it can be argued that in the process of amplitude modulation, the spectrum of the modulating signal is shifted along the frequency scale to the right from zero to the carrier frequency (and to the left too). In this case, the "lower side" does not arise, it already exists in the original modulating signal, although it is located in the negative frequency range. It sounds strange at first glance, since it would seem that negative frequencies do not exist in nature. But mathematics is full of surprises.
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Amplitude modulation is the process of forming an amplitude-modulated signal, i.e. signal, the amplitude of which changes according to the law of the modulating signal (transmitted message). This process is carried out by an amplitude modulator.
The amplitude modulator should form a high-frequency vibration, the analytical expression for which in the general case has the form
where is the envelope of the modulated oscillation, described by a function that characterizes the law of amplitude change;
Modulating signal;
And - the frequency and the initial phase of the high-frequency oscillation.
To obtain such a signal, it is necessary to multiply the high-frequency (carrier) oscillation and the low-frequency modulating signal in such a way that an envelope of the form is formed. The presence of a constant component in the envelope structure ensures unipolarity of its change, the coefficient excludes overmodulation, i.e. provides depth of modulation. It is clear that such a multiplication operation will be accompanied by a transformation of the spectrum, which makes it possible to consider amplitude modulation as an essentially nonlinear or parametric process.
The structure of the amplitude modulator in the case of using a nonlinear element is shown in Fig. 8.4.
Rice. 8.4. Amplitude modulator block diagram
The non-linear element converts the carrier wave and the modulating signal, as a result of which a current (or voltage) is formed, the spectrum of which contains components in the frequency band from to, and is the highest frequency in the spectrum of the modulating signal. A bandpass filter separates these components of the spectrum, forming an amplitude-modulated signal at the output.
Multiplication of two signals can be performed using a non-linear element, the characteristic of which is approximated by a polynomial containing a quadratic term. Due to this, the square of the sum of the two signals is formed, containing their product.
The essence of what has been said and the general idea of the formation of an amplitude-modulated oscillation are illustrated by rather simple mathematical transformations under the assumption that tonal (one frequency) modulation is carried out.
1. As a nonlinear element, we use a transistor, the I – V characteristic of which is approximated by a polynomial of the second degree 
.
2. The input of the nonlinear element is supplied with a voltage equal to the sum of two oscillations: carrier and modulating, i.e.
3. The spectral composition of the current is determined as follows:
In the expression obtained, the spectral components are arranged in ascending order of their frequencies. Among them there are components with frequencies, and, which form an amplitude-modulated oscillation, i.e.
Transmitting devices usually combine modulation and amplification processes, which ensures minimal distortion of modulated signals. For this purpose, amplitude modulators are built according to the scheme of resonant power amplifiers, in which a change in the amplitude of high-frequency oscillations is achieved by changing the position of the operating point according to the law of the modulating signal.
Amplitude modulator circuit and operating modes
The diagram of an amplitude modulator based on a resonant amplifier is shown in Fig. 8.5.
Rice. 8.5. Amplitude modulator circuit based on a resonant amplifier
To the input of a resonant amplifier operating in a nonlinear mode, the following are fed:
carrier oscillation from an auto-generator using high-frequency transformer connection of the input circuit circuit with the base of the transistor;
modulating signal using a low-frequency transformer.
Capacitors and - blocking, provide decoupling of the input circuits at the frequencies of the carrier oscillation and the modulating signal, i.e. decoupling at high and low frequencies. The oscillatory circuit in the collector circuit is tuned to the frequency of the carrier oscillation, the Q-factor of the circuit provides the bandwidth, where is the highest frequency in the spectrum of the modulating signal.
The operating mode of the modulator is determined by the choice of the operating point. Two modes are available: small signal mode and large signal mode.
a. Small input mode
This mode is set by choosing the operating point in the middle of the quadratic section of the I - V characteristic of the transistor. The choice of the amplitude of the carrier oscillation ensures the operation of the modulator within this section (Fig. 8.6).
Rice. 8.6. Amplitude modulator small input mode
The amplitude of the voltage on the oscillatory circuit, the resonant frequency of which is equal to the carrier frequency, is determined by the amplitude of the first harmonic of the current, i.e. , where is the resonant resistance of the circuit. Considering that the average slope of the I - V characteristic within the working section is equal to the ratio of the amplitude of the first harmonic to the amplitude of the carrier vibration, i.e. 
, you can write
.
Under the influence of the modulating voltage applied to the base of the transistor, the position of the operating point will change, which means that the average slope of the I – V characteristic will also change. Since the amplitude of the voltage on the oscillatory circuit is proportional to the average slope, to ensure the amplitude modulation of the carrier wave, it is necessary to provide a linear dependence of the slope on the modulating signal. Let us show that this is possible when using the working section of the I – V characteristic approximated by a polynomial of the second degree.
So, within the quadratic section of the I - V characteristic, described by a polynomial, there is an input voltage equal to the sum of two oscillations: the carrier and modulating, i.e.
The spectral composition of the collector current is determined as follows:
We select the first harmonic of the current:
Thus, the amplitude of the first harmonic is:
As can be seen from the obtained expression, the amplitude of the first harmonic of the current linearly depends on the modulating voltage. Therefore, the average slope will also be linear with the modulating voltage.
Then the voltage on the oscillatory circuit will be equal to:
Therefore, at the output of the modulator under consideration, an amplitude-modulated signal of the form is formed:
Here is the modulation depth coefficient;
- the amplitude of the high-frequency oscillation at the output of the modulator in the absence of modulation, i.e. at .
When designing transmission systems, an important requirement is the formation of high-power amplitude-modulated oscillations with sufficient efficiency. It is obvious that the considered mode of operation of the modulator cannot meet these requirements, especially the first of them. Therefore, the so-called large signal mode is most often used.
b. Large input mode
This mode is set by choosing the operating point on the I - V characteristic of the transistor, at which the amplifier operates with current cutoff. In turn, the choice of the amplitude of the carrier oscillation ensures the change in the amplitude of the collector current pulses according to the law of the modulating signal (Fig. 8.7). This leads to a similar change in the amplitude of the first harmonic of the collector current and, consequently, to a change in the voltage amplitude on the oscillatory circuit of the modulator, since
and .
Rice. 8.7. Amplitude modulator large input mode
The change in the amplitude of the input high-frequency voltage over time is accompanied by a change in the cutoff angle, and hence the coefficient. Consequently, the shape of the voltage envelope on the circuit may differ from the shape of the modulating signal, which is a disadvantage of the considered modulation method. To ensure minimal distortion, it is necessary to set certain limits for changing the cutoff angle and work with a not too high modulation factor.
In the amplitude modulator circuit shown in Fig. 8.8, the modulating signal is applied to the base of the constant current generator transistor. The value of this current is proportional to the input voltage. At small values of input voltages, the amplitude of the output voltage will depend on the modulating signal as follows
where are the proportionality coefficients.
Amplitude modulator characteristics
To select the operating mode of the modulator and assess the quality of its operation, various characteristics are used, the main of which are: static modulation characteristic, dynamic modulation characteristic and frequency response.
Rice. 8.8. Amplitude modulator circuit with current generator
a. Static modulation characteristic
Static modulation characteristic (CMX) is the dependence of the amplitude of the output voltage of the modulator on the bias voltage at a constant amplitude of the carrier voltage at the input, i.e. 
.
In the experimental determination of the static modulation characteristic, only the carrier frequency voltage is applied to the modulator input (the modulating signal is not supplied), the value changes (as if the change in the modulating signal in static is simulated), and the change in the amplitude of the carrier oscillation at the output is recorded. The type of characteristic (Fig. 8.9, a) is determined by the dynamics of the change in the average slope of the I – V characteristic when the bias voltage is changed. The linear increasing section of the CMX corresponds to the quadratic section of the I - V characteristic, since in this section, with an increase in the bias voltage, the average steepness increases. The horizontal section of the CMX corresponds to the linear section of the I - V characteristic, i.e. a section with a constant average steepness. When the transistor enters the saturation mode, a horizontal section of the I - V characteristic with zero slope appears, which is reflected by the decrease in the CMX
The static modulation characteristic allows you to determine the magnitude of the offset voltage and the acceptable range of the modulating signal in order to ensure its linear dependence on the output voltage. The modulator should operate within the linear section of the CMX. The value of the bias voltage should correspond to the middle of the linear section, and the maximum value of the modulating signal should not go beyond the limits of the linear section of the CMX. You can also define the maximum modulation factor at which there is no distortion yet. Its value is 
.
Rice. 8.9. Amplitude modulator characteristics
b. Dynamic modulation response
Dynamic modulation characteristic (DMX) is the dependence of the modulation factor on the amplitude of the modulating signal, i.e. ... This characteristic can be obtained experimentally, or by the static modulation characteristic. The DMX type is shown in Fig. 8.9, b. The linear section of the characteristic corresponds to the operation of the modulator within the linear section of the CMX.
v. Frequency response
Frequency response is the dependence of the modulation factor on the frequency of the modulating signal, i.e. ... The influence of the input transformer leads to a blockage of the characteristic on low frequencies ah (Figure 8.9, c). With an increase in the frequency of the modulating signal, the side components of the amplitude-modulated oscillation move away from the carrier frequency. This leads to their less amplification due to the selective properties of the oscillatory circuit, which causes a blockage of the characteristic by more high frequencies ah. If the bandwidth occupied by the modulating signal is within the horizontal section of the frequency response, then modulation distortion will be minimal.
Balanced amplitude modulator
For efficient use of transmitter power, balanced amplitude modulation is used. In this case, an amplitude-modulated signal is formed, in the spectrum of which there is no component at the carrier frequency.
A balanced modulator circuit (Figure 8.10) is a combination of two typical amplitude modulator circuits with specific connections to their inputs and outputs. The inputs on the frequency of the carrier are connected in parallel, and the outputs are connected with inversion relative to each other, forming a difference in output voltages. The modulating signal is applied to the modulators in antiphase. As a result, at the outputs of the modulators we have
And, and at the output of the balanced modulator
Rice. 8.10. Balanced amplitude modulator circuit
Thus, the spectrum of the output signal contains components with frequencies and. There is no component with the carrier frequency.
where is the carrier amplitude; - proportionality factor, chosen so that the amplitude is always positive. The frequency and phase of the carrier wave at AM remain unchanged.
For the mathematical description of the AM signal in (2.2), instead of the coefficient depending on the specific modulator circuit, the modulation index is introduced:
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,those. the ratio of the difference between the maximum and minimum values of the amplitudes of the AM signal to the sum of these values. For a balanced AM baseband signal, the signal is also balanced, i.e. 
... Then the modulation index is equal to the ratio of the maximum amplitude gain, to the carrier amplitude.
Amplitude modulation by harmonic oscillation. In the simplest case, the modulating signal is a harmonic vibration with frequency. Moreover, the expression
corresponds to the single-tone AM signal shown in Fig. 2.26.
A single-tone AM signal can be represented as a sum of three harmonic components with frequencies: - carrier; - upper side and - lower side:
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.The spectral diagram of a single-tone AM signal, built according to (2.7), is symmetric with respect to the carrier frequency (Fig. 2.2, c). The amplitudes of lateral vibrations with frequencies and are the same and even if they do not exceed half the amplitude of the carrier vibration.
Harmonic baseband signals and thus a single tone AM signal are rare in practice. In most cases, the modulating primary signals are complex functions of time (Figure 2.3, a). Any complex signal can be represented as a finite or infinite sum of harmonic components using the Fourier series or integral. Each harmonic component of a signal with frequency will result in two sidebands with frequencies in the AM signal.
Many harmonics in a baseband signal with frequencies 
will correspond to many side components with frequencies 
... For clarity, such a transformation of the spectrum at AM is shown in Fig. 2.3, b. The spectrum of a complex-modulated AM signal, in addition to the carrier oscillation with frequency, contains groups of upper and lower lateral oscillations, forming, respectively, the upper sideband and the lower side stripe AM signal.
In this case, the upper sideband of frequencies is a scale copy of the spectrum of the information signal, shifted to the high frequency region by an amount. The lower sideband also repeats the spectral diagram of the signal, but the frequencies in it are arranged in a mirror order relative to the carrier frequency.
The width of the spectrum of the AM signal is equal to twice the value of the highest frequency of the spectrum of the baseband signal, i.e.
The presence of two side bands leads to the expansion of the occupied frequency band by about two times in comparison with the spectrum of the information signal. The power per oscillation of the carrier frequency is constant. The power contained in the sidebands depends on the modulation index and increases with increasing modulation depth. However, even in the extreme case, when only the entire oscillation power falls on the two side bands.
