SPECTRUM TRANSFORMATION OF AN AMPLITUDE-MODULATED SIGNAL ON AN OHMIC NONLINEAR ELEMENT

2020. Т. 20, No 1. С. 71–78 7


Introduction
In many physical processes, the spectrum of the modulated signal is transferred to the lowfrequency region, for example, in the occurrence of radio sound in active dielectrics [1] or the appearance of an acoustic signal at the beats frequency during the interaction of signals from two ultrasonic sources [2]. In the general case, the analysis of the spectrum transformation process is a very difficult task related to solving a system of nonlinear differential equations. And in this case, the principle of superposition is not applicable. Nevertheless, research and analysis can be carried out using relatively simple methods if the form of the current-voltage characteristic (CVC) for a nonlinear element (NE) in the dynamic mode is the same as for a constant voltage (or in static mode). Such a NE is called a resistive inertialess element [3,4]. Physically inertialess NE means the appearance of a response in the form of a function of time from the current i(t) after the input action as a function of time from the voltage u(t). Formally, there are practically not inertialless NE. At the same time, many modern circuit elements are perfect in their frequency parameters and can be idealized from the point of view of their inertialess. Such elements are called not only resistive, but also active or ohmic [5].
In order to effectively transmit signals in any environ, it is necessary to transfer the spectrum of these signals from the low-frequency region to the region of sufficiently high frequencies. This procedure is called modulation in communication technology. The frequency ɷ for the physical information carrier is selected taking into account the peculiarities of the propagation of oscillations in communication lines or in the environ of radio communication. But in any case, the frequency ɷ is much higher than the highest frequency Ω of the primary signal, which performs modulation [5,6].
Under these conditions, the parameters of the modulated oscillation change slowly compared to the rate of change of the carrier oscillation. In one period of the modulating signal T F = 1/F = 2π/Ω usually contain hundreds, thousands and more periods of high-frequency oscillations. Consequently, during several periods of the last T f = 1/f = 2π/ɷ, only slight changes in the parameters of the modulating signal occurs [7,8].

The concept of current-voltage characteristics (CVC)
CVC -a special case of the transfer characteristics that determine the dependence (function) of the output quantity on the input for a given specific device or circuit. CVC is a graph of the current through a two-terminal circuit versus the voltage at this two-terminal circuit. CVC characteristic describes the behavior of a two-terminal circuit in static mode. Most often, the analysis of nonlinear elements by CVC degree of nonlinearity, which is determined by the coefficient of nonlinearity = . For linear elements, CVC is a straight line and is not of particular interest [7,8].
While analyzing the spectral transformation under the influence of harmonic voltage, for example, a power-law approximation in the form of a polynomial with trigonometric functions is used: ( ) = + cos ω + cos ω + cos ω , (1) where we limit ourselves to a polynomial of the third degree. Here b i -dimensional parameters; constant component is not of interest in the work. Example 1.
1. If t = 0 c, R = 1 Ohm, then based on Ohm's law from (1) we obtain: where CVC (and below in the example) is represented in dimensionless form.
(6) CVC for example 1 in dimensionless form are presented in Fig. 1.

Spectrum transformation for a mono signal
Consider the analysis of an example of a current spectrum under supplying a harmonic voltage. When the element is linear, then receive harmonic current (one component). When the element is nonlinear, then receive a lot of components [6].
For the spectrum concept, it is important to find the amplitude spectral components and initial phases. The frequencies of all components will be multiples of the fundamental frequency or the frequency of exposure [7][8][9][10][11][12].
The best choice of approximation method depends on the type of nonlinear characteristic, as well as on the mode of operation of the nonlinear element. One of the most common methods is power polynomial approximation [13][14][15].
In the analysis of the spectral transformation under the influence of harmonic voltage, for example, a power approximation in the form of a polynomial with trigonometric functions is used: ( ) = + cos ω + cos ω + cos ω + ⋯ + cos ω , When using the cosines of all the initial phase is zero. In this paper, we mainly limit ourselves to the polynomial of the third degree (1).

Spectrum transformation for amplitude-modulated signal
Under AM, the spectrum of the modulating signal is transmitted to the region of the carrier frequency, forming the upper and lower side components of the spectrum. Since such a transformation creates new frequencies, the modulation procedure is a nonlinear transformation. But since the AM spectrum of the modulating signal does not change, but is transmitted only to the high-frequency region, AM is considered a type of linear modulation. In many cases, the spectrum of the AM signal is relatively simple and can be determined from the spectrum of the modulating signal, which is noticeably simpler than its direct calculation. The basic relationships required for this can be relatively easily obtained using the example of the AM signal when the AM signal is performed by a harmonic signal [9][10][11][12][13].
Carrier frequency, frequency of harmonic oscillations subjected to modulation by signals for transmitting information. Low-frequency oscillations are sometimes called carrier oscillations. The oscillations with the LF themselves do not contain information, they only "carry" it. The spectrum of modulated oscillations contains, in addition to low frequencies, side frequencies, which contain the transmitted information [4].
Next, we use the following formulas to reduce the degrees: For current ( ) because of substitution, we obtain: Here and below, the low-frequency components of the spectrum are of interest. Example 3. Given: = 1, = 0, = 1, = 1, = 1, = 1. Determine the low frequency current.
Solution. From (13) we get the current LF: There is a constant component and there are two harmonics. For Fig. 3 and 4 the results are presented as reamers of the signal and its spectrum. Solution. If b 2 = 0, then (14) results in the absence of a low-frequency current and a constant component.

Beat Spectrum transformation
Beats occur due to the fact that one of the two signals is linear in time lags the other in phase, and at those moments when the oscillations coincide in phase, the total signal is the maximum, and at those moments when the two signals are not in phase, they mutually suppress each other. These moments periodically replace each other as the lag increases [4].
If at the same time two tuning-forks with slightly different frequencies are slightly excited, the resulting sound periodically oscillates and decays. These modulations are called beats; their frequency is equal to the difference in frequency of the initial tones. Beats are obtained when electrical signals from the outputs of two generators are mixed and fed to the speaker. On the other hand, these same signals can be simultaneously applied to two different dynamics and also hear beats [5].
Beats relate to amplitude modulation, but without a carrier frequency. Consider the addition of two high-frequency oscillations: where the difference frequency Δω = 2Ω.
(16) We use the formulas for lowering the degrees (11) and (12) Fig. 5 and 6 shows the results in the form of a sweep of the signal and its spectrum.

Conclusion
The transformation for the current spectrum under supplying a modulated voltage to an ohmic nonlinear element is considered. Transformation for any type of amplitude modulation to the low-frequency region is observed when there is a quadratic nonlinearity of the CVC. Similar transformations can be observed in active dielectrics, for example, in piezoceramics. In the presented examples, the transformation process has nothing to do with signal detection.