Formation of the Spectral Composition of the Output Voltage of Converters for Nuclear Magnetic Resonance

—In this paper, we consider the construction of an asynchronous control system for semiconductor converters for nuclear magnetic resonance. The relevance of research in this direction, main problems that arise during the construction of these systems are shown. Derived the mathematical model of the inverter, on the basis of which a control system with asynchronous pulse-width modulation is constructed. Proposed an equivalent converter circuit with a constant structure, constant parameters, and an equivalent EMF generator. Established correspondence between the equivalent circuit of the converter and its mathematical model in the form of differential equations. On the basis of mathematical equations, a structural diagram of the converter with a control system was developed and the principle of operation of the device was described according to it. To develop an algorithm for the control system, the dependence of the frequency change relative to the resonant frequency on the phase shift between the current through the filter and the voltage on the antenna circuit is determined. For which the model was built in Simulink and the corresponding simulation was carried out. Numerical dependences of reference signal frequency change and phase shift were obtained. The increase in efficiency of device for nuclear magnetic resonance is considered due to the use of multilevel inverters with tuning the frequency of operation to the frequency of the resonant circuit. Simulation of a three-level diode-clamped inverter showed that when the capacitance of the resonant circuit changes and, therefore, the resonant frequency of the circuit, the input current increases. The work obtained specific numerical dependencies. Time diagrams of voltages and currents on the main elements of the converter are given, which illustrate the implementation asynchronous pulse-width modulation in the control system. After working out the automatic frequency control algorithm, the increase in current consumption can be leveled off. The simulation results also show that it is possible to reduce the amplitude of the third harmonic. The disadvantages of the proposed system include the fact that the frequency of the converter is adjusted at each subsequent period of its operation. At the same time, the parameters of the filter can change again, which leads us to believe that adjusting the frequency will never give a 100% result, but will only allow to get as close as possible to the set parameters of the sounding signal. The work also indicates that it is possible to improve the spectral composition of the probing voltage generated by the converter by using more levels of a diode-clamped multi-level inverter. However, increasing the number of levels reduces the action speed of the system and complicates the control system itself. Therefore, the need to maintain a balance between the number of levels of the inverter and the complexity of the system is indicated.


I. INTRODUCTION
Maintaining the energy balance in the world is not only the most important, but also a very difficult task, because maintaining the currently achieved level of oil production, and even more so increasing it in the modern conditions of growing demand for energy resources, is also a non-trivial task.The success and effectiveness of the application of new oil production technologies in such conditions largely depends on the reliability and completeness of information about the physical properties of mineral deposits.One of the most effective and modern methods is nuclear magnetic resonance (NMR) [1]- [5].NMR logging has earned a special place among other research methods due to its unique accuracy and complexity [6]- [9].
When developing systems for NMR, strict requirements are put forward for the accuracy of the envelope voltage (5%) and harmonic composition of the output probing voltage.One of the ways to meet these requirements is the use of multi-level inverters, which gives advantages in the power load of circuit components and greater possibilities of formation a various form of the output probing voltage.At the same time, a compromise is necessary between the complexity of the circuit (the number of inverter levels) and the harmonic composition of the output voltage.
The maximum power of the probing voltage, as well as the reduction of the content of higher harmonics, is achieved by the use of asynchronous pulse width modulation (asPWM) in the control system [10]- [19].

II. METHODS OF OBTAINING A MATHEMATICAL MODEL OF THE CONVERTER
Let's consider the formation of the equations of the mathematical model of the converter of the logging system using the example of a three-level inverter [20]- [25], the diagram of which is given in Fig. 1.
A stepped bipolar voltage with zero pauses is formed at the output of the inverter.When transistors During the operation of the device for nuclear magnetic resonance, the parameters of the oscillating circuit and the emitter change, which, in turn, causes a change the Larmor resonance frequency.Therefore, it is necessary to adjust the frequency of the converter to ensure a quasi-resonant mode to reduce dynamic losses.
In order to implement the asPWM, it is necessary to obtain mathematical equations that will describe the currents and voltages on the elements.Let's establish the correspondence between the equivalent circuit of the converter and its mathematical model in the form of differential equations.The equivalent circuit of the converter with a constant structure, constant parameters and a generator of the equivalent EMF ( ) e t , which can take the values 2 U , -2 U and 0, is shown in Fig. 2.
According to Kirchhoff's first law, for the first node: Considering the current direction ( ) 0 i t and the direction of the first loop as indicated in Fig. 2, according to Kirchhoff's second law, write: For the second and third circuits, taking into account the direction of the currents ( )

I t and
( ) 33 I t , as well as the directions of the branch currents, as indicated in the diagram, we write: Let's express the current ( ) Considering ( 6), let's write equations ( 2) and (3) in the form of a system: It is obvious that for any structure of the passive part, the equivalent circuit of the converter with a constant structure, parameters, and generator of the equivalent EMF is described by a linear differential equation with constant coefficients and a piecewise continuous periodic function in the right-hand side.
Let's enter new variables: Than: Considering ( 8) and ( 9), we pass from the system of two differential equations of the second order (7) to the system of four differential equations of the first order with variables ( ) x t , ( ) After mathematical transformations and simplifications, we write system (10) in matrix form: where Let's write down the final mathematical model in matrix form:

III. RESULTS
To implement asynchronous pulse-width modulation when developing the structure of the control system, it is necessary to supplement the mathematical model of the converter (12) with equations for the feedback voltage fb U , the voltages of the output arms of ( ) ( ) where -the maximum value of the module of the probing voltage ( ) System (13) is the basis for the development of the control system of the converter with asPWM.
The structural diagram of the converter control system with asynchronous pulse width modulation is shown in Fig. 3.
The voltage from the battery AB is supplied to the inverter I.The multi-level inverter forms the sensing voltage according to the control pulses.The stepped voltage from the inverter is converted into an amplitudemodulated sine wave using resonance filters.To ensure the quasi-resonance mode of operation of the device, it is necessary to generate a probing voltage with an accuracy of 5% [3].For this, it is necessary to monitor the amplitude value of the modulated sinusoid on each half-cycle of the converter.The VS voltage sensor is a voltage divider, the voltage from which, after rectification, is fed to the AS amplitude sensor.DOI: 10.20535/2523-4455.mea.291566 the moment of the control pulse of the previous interval, the sawtooth voltage generator begins to unfold each subsequent period from the beginning of the previous control pulse.Thus, control pulses are formed at the output of the comparator, which are summed with the output pulses from the generator PG.The PD switching unit is designed to distribute control pulses across transistors.The BA amplifies the pulses to the required level to open the power transistors of the multi-level inverter.
In the event that, according to the SVG voltage, the duration of the pulse will exceed the duration of the period, its duration will be − δ, T where T -duration of the standard period, δ -some minimum time interval to ensure a pause before the start of the next control pulse.
Timing diagrams illustrating the implementation of asPWM in the control system (see Fig. 3) are shown in Fig. 5.
The use of such a control system, which works according to the obtained mathematical ratios, allows you to ensure the necessary THD (Total Harmonic Distortion) indicators [26]- [30].To check the performance of the system, a model of the converter with asPWM was developed in the MATLAB ® Simulink ® environment.
The value of the phase shift at a change the frequency of the reference signal within ±10% was obtained (Table 1).
Simulations have shown that changing the capacitance of the resonant circuit by 4% (and therefore the resonant frequency of the circuit changes by 2%, from 500 kHz to 490 GHz) leads to an increase input current by 9.09% from 1.21 A to 1.32 A. Automatic frequency adjustment algorithm allow to level the increase in current consumption.The amplitude of the third harmonic also decreases from 17% ( 3 ) of the amplitude of the fundamental (Table 1).

CONCLUSIONS
The obtained mathematical model of a multi-level inverter for nuclear magnetic resonance allows designing a control system of the converter based on asPWM.
The frequency adjustment allows to reduce current consumption by 9.09% and reduce third harmonic of the probe voltage from 17% to 6%.The obtained simulation results also show that it is possible to reduce the amplitude of the third harmonic when using a threelevel inverter and an algorithm with automatic frequency adjustment by 11 percent.

Fig. 4
Fig. 4 shows the time diagrams of the three-level inverter circuit with asPWM.The voltage fb U from the amplitude sensor is com- pared with the USVG on comparator C. Since in asPWM systems the control angle is counted each time from

Fig. 5
Fig.5The timing diagrams of the control system

TABLE 1 DEPENDENCE
OF THE FREQUENCY CHANGE FL ON THE PHASE SHIFTAND