RU2536833C2 - Method to measure resonant frequency - Google Patents

Abstract

FIELD: measurement equipment.

SUBSTANCE: invention relates to measurement equipment and is designed for the high-precision detection of a resonant frequency using digital methods of signal processing, and also to detect values, which are functionally connected to the resonant frequency of resonators included within radio frequency sensors and used in various fields of engineering and scientific research. The method to measure the resonant frequency is based on serial excitation of various frequencies in a resonator from the specified set and fixation of these frequencies and their appropriate values of the resonator output signal amplitude. Besides, for each of the excitation frequencies they fix amplitude of the output signal after attenuation of a transition process in a resonator, approximate the obtained discrete dependence of the amplitude on the frequency with a continuous function and the resonant frequency of the resonator is determined by the maximum of this function.

EFFECT: reduced error of measurement specified by discreteness of frequencies of the resonator excitation and impact of transition processes in this resonator, as well as availability of noise in the output signal of the resonator.

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Description

The invention relates to measuring technique and can be used for high-precision determination of the resonant frequency of electromagnetic resonators, as well as quantities that are functionally related to the resonant frequency of the resonators that are part of radio frequency sensors and are used in various fields of technology and scientific research. In particular, it can be used in radio-frequency resonance sensors for the moisture content of oil products, concentration of solutions and resonant level gauges of various media.

Various methods for measuring a resonant frequency are known. The method implemented in the device (SU 1758510 A1, 08/30/1992), which is based on an extreme controller, was widely used. In this device, a voltage-controlled high-frequency generator (VCO) is automatically tuned to the resonant frequency of the radio frequency sensor and monitors all its changes. In this case, the frequency of the high-frequency generator oscillates around the maximum amplitude-frequency characteristic (AFC) of this sensor and the average value of this frequency is considered the resonant frequency of this sensor. The specified method has the following disadvantages:

1) The frequency response of the resonator should be with only one maximum, since in the presence of other local maxima, false response and capture of a local maximum are possible instead of the main one;

2) since the resonant frequency is determined as the average value of the oscillating frequency of the VCO, to obtain the exact value of the resonant frequency, it is necessary to measure the frequency of the VCO for a long time;

3) when measuring the resonant frequency using tunable generators, a dynamic resonance curve is observed at the resonator output, the maximum of which is shifted relative to the maximum of the static resonance curve, which leads to an additional error in the measurement of the resonance frequency. This shift is explained by the presence of a transition process in the resonator during the tuning of the excitation frequency, and the magnitude of this shift s is determined by the formula (Kharkevich A. A. Spectra and analysis. M: State publishing house of technical and theoretical literature. 1957)

, $$s=8\lambda /\left({\omega }_r^2{d}^2\right)$$

,

where λ is the rate of change of the frequency of the generator;

ω _r is the resonant frequency;

d is the attenuation of the resonator.

Closest to the proposed method in technical essence is the method of measuring the resonant frequency described in the device (RU 2427805 C1, 08.27.2011) and adopted as a prototype. According to this method, using a frequency synthesizer controlled by a microprocessor, oscillations of various frequencies from a given range are sequentially excited in a resonator (radio frequency sensor). At the resonator output, the amplitude of these oscillations is converted into digital form using an analog-to-digital converter (ADC), and then this amplitude and its corresponding frequency are recorded using a microprocessor. After that, the microprocessor determines the maximum voltage at the resonator output and the frequency corresponding to this maximum, which is taken equal to the resonant frequency.

The disadvantage of the prototype method implemented in the specified device is the presence of a significant error in the measurement of the resonant frequency, due to both the discrete frequency tuning of the synthesizer and the influence of interference caused by transients when the resonator is excited by a series of radio pulses, which, in essence, are the signals at the output of the synthesizer.

As a result of this, oscillations with a frequency Ω are superimposed on the envelope of the signal at the resonator output, which are the result of adding damped natural oscillations of the resonator with frequency ω ₀ and oscillations of the synthesizer with frequency ω _i .

$$\Omega =|\; |\; |{\omega }_0-{\omega }_i|\; |\; |.(\mathrm{one})$$

These oscillations distort the shape of the frequency response obtained in this way, and therefore the maximum of this frequency response does not correspond to the resonant frequency.

The objective of the present invention is to expand the arsenal of technical means.

The technical result of the invention is to reduce the measurement error of the resonant frequency due to the discreteness of the excitation frequencies and the influence of transients and random noise, as well as reducing the measurement time of the specified resonant frequency without compromising the accuracy of the measurement.

The technical result is achieved in that the proposed method for measuring the resonant frequency, based on the sequential excitation in the resonator of oscillations of various frequencies from a given set and fixing these frequencies and the corresponding values of the amplitude of the output signal of the resonator, differs in that for each of the excitation frequencies the amplitude of the output signal is fixed after the transient attenuation in the resonator, the obtained discrete dependence of the amplitude on the frequency is approximated by a continuous function and ansnuyu resonator frequency is determined by the maximum of this function.

The invention is illustrated by the following figures.

Figure 1 - view of the signals at the characteristic points of the circuit that implements the proposed method, where 1 is the dependence of the frequency of the output signal of the synthesizer ω _i on time t (figa); 2 - signal at the output of the detector, obtained with a discrete change in the excitation frequency (discrete frequency response); 3 is a graph of a function approximating the obtained discrete frequency response of the resonator (Fig. 1b).

Figure 2 - structural diagram of a device that implements the proposed method, where 4 is a digital signal processor (DSP), 5 is a frequency synthesizer, 6 is a low-pass filter (LPF), 7 is a resonator, 8 is a detector, 9 is an analog-to-digital converter (ADC).

Figure 3 - experimental curves of the dependence of the error in determining the resonant frequency on the relative frequency range in which the approximation is made, for various methods for determining the resonant frequency (curves 10 - approximation of the discrete frequency response by a fourth degree polynomial and determination of the resonant frequency from the maximum of the polynomial and curves 11 - determination of the resonant frequency directly from the maximum of the discrete frequency response) and for various values of the signal-to-noise ratio N.

Figure 4 - view of the signal with noise at the output of the detector (curve 12) and a polynomial of the 4th degree approximating this signal (curve 13).

The essence of the invention lies in the fact that the predetermined measuring range of the resonant frequency is divided into N intervals, each of which corresponds to the frequency ω _i . A signal with a frequency ω _i - at a time t _i is supplied to the resonator. The response of this resonator is the signal u (t), which, as shown in (Kharkevich A.A. Fundamentals of Radio Engineering. M.: State Publishing House of Literature on Communications and Radio. 1962), has the form

$$u\left(t-t_i\right)\approx \frac{U_m}{\sqrt{{\epsilon }_i^2+d^2}}\left[\sin \left({\omega }_i\left(t-t_i\right)+\varphi \right)-e^{-\alpha \left(t-t_i\right)}\sin \left({\omega }_0\left(t-t_i\right)+\varphi \right)\right],(2)$$

- относительная расстройка;Where $${\epsilon }_i=\mathrm{one}-{\mathrm{<figure-callout\; id="2"\; label="\omega \; i"\; filenames="00000013.png"\; state="\{\{state\}\}">\omega \; </figure-callout>}}_i^{}/{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="00000014.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_0^2$$

- relative detuning;

α = dω 0/2 _- cavity attenuation constant;

U _m is the amplitude of the signal at the input of the resonator;

t is the current time;

φ is the initial phase;

d is the attenuation of the resonator;

ω ₀ is the natural frequency of the resonator.

The envelope of this signal at the output of the resonator U _i (tt _i ) is expressed by the formula

$$u\left(t-t_i\right)\approx \frac{U_m}{\sqrt{{\epsilon }_i^2+{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="00000013.png"\; state="\{\{state\}\}">d</figure-callout>}}^2}}\sqrt{\mathrm{one}+e^{-\alpha \left(t-t_i\right)}-2e^{-\alpha \left(t-t_i\right)}\cos \left(\Omega \left(t-t_i\right)\right)},(3)$$

From expression (3), a formula can be obtained for the duration of the transient process in the resonator τ, provided that

e ^-ατ = 0.1.

From here

$$\tau =2,3/\alpha =\mathrm{four},6L/r=\mathrm{four},6Q/{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="00000014.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_0=\mathrm{four},6/\left({\omega }_{0}d\right)(\mathrm{four})$$

It follows from the foregoing that when a sinusoidal signal 1 is supplied to the resonator input with a stepwise changing frequency (Fig. (figb).

From figb it is seen that, using discrete excitation of the resonator at time t _{i by a} signal with a frequency ω _i - it is necessary to take into account the presence of transients that occur at each of the moments t _i . Therefore, to obtain more accurate values of the discrete frequency response, the authors propose to measure the amplitude of the signal U _i at the resonator output in the interval Δt _i only after the transient attenuation caused by the action of a single oscillation with a frequency ω _i on the resonator _. In this case, the interval Δt _i must be not less than the sum of the duration of the transient process in the resonator τ and the duration of the digital processing of the analog signal taken from the output of the resonator, τ _p .

$$\Delta t_i\ge \tau +{\tau }_p.(5)$$

In this case, the value of the minimum frequency tuning step of the exciting signal Δω _s can be calculated by the formula

, $$\Delta {\omega }_s\ge \left({\omega }_{\max }-{\omega }_{\min }\right)\Delta t_i/t_{sk}(6)$$

,

where ω _max , ω _min - the boundaries of the resonant frequency range of the investigated resonator;

t _sk is the scan duration of a given range of resonant frequencies;

Δt _i - is the time interval during which the resonator is excited by a single oscillation with a frequency ω _i .

To reduce the measurement error of the resonant frequency due to the final value of the tuning step Δω _s and the presence of transients in the resonator, the invention proposes to approximate the discrete frequency response of the resonator by a continuous function. As such a function, various analytical dependencies can be taken that are widely used to smooth experimental data, for example, an nth degree polynomial, splines, the resonance curve equation, etc. Recommendations for choosing the type of approximating function are as follows.

If the analytical expression for the frequency response of the resonator under study is known, then as an approximating function, one should choose an expression that most accurately describes the specified frequency response, for example, the equation of the resonance curve, otherwise polynomials or splines should be preferred.

The use of polynomials and splines is facilitated by the fact that various computer approximation programs have already been developed and are available for them. If, for example, an nth degree polynomial is used for approximation, then to determine the coefficients of this polynomial it is necessary to solve a system of n linear equations. Obviously, these equations can be obtained with only n pairs of values of U ₁ , ω _I, however, in practice, for the most accurate approximation, it is necessary to use the maximum possible number of pairs U _i , ω _i , or, taking into account the final value of the tuning step Δω _i , increase the frequency range in which the processing of the values of A _i , ω _i . This is due to the fact that various types of noise and interference are superimposed on the envelope of the signal at the output of the resonator. Therefore, processing the maximum number of pairs A _i , ω _i , for example, by the least squares method leads to smoothing of this envelope, which ultimately reduces the effect of these noises and interference. This can also be explained by the fact that the application of the least squares method for approximating the frequency response of the resonator and the subsequent determination of the resonant frequency is equivalent to averaging the measurement data of the resonant frequency. However, it must be borne in mind that any approximation most accurately describes the frequency response of the resonator only in a narrow frequency band, the central frequency of which coincides with the resonant frequency, and with an increase in this band, the approximation error also increases. In addition, an increase in the approximation range leads to a decrease in the signal-to-noise ratio at the resonator output. Therefore, there is an optimal value of the frequency band at which the total error, that is, the error due to the influence of noise, plus the approximation error is minimized. This optimal value, as studies have shown, depends on the quality factor of the resonator, noise level, frequency tuning step, degree of approximating polynomial, but always lies within the resonator passband Δω defined at 0.707 (Δω = ω ₀ / Q), which is shown by the curves 11 the dependence of the error in determining the resonant frequency from the relative range of approximation frequencies, presented in figure 3.

It should be noted that there is also an optimal value of the degree of the approximating polynomial at which the error in determining the resonance frequency is minimal, and this value is in the range of 3–5. This is due to the fact that, in order to obtain high approximation accuracy, it is necessary to maximize the degree of the polynomial and, accordingly, the required number of equations, however, the number of independent systems that can be composed of these equations is reduced, which leads to a decrease in the averaging efficiency.

The algorithm for determining the resonant frequency when approximating the discrete frequency response of the resonator by a continuous function in the general case is as follows.

Using the fixed pairs of values of U _i , ω _i , the parameters of the approximating function are determined by the least square method (for the polynomial these are the coefficients of the nth degree polynomial P (ω); for the resonance curve equation, this is the natural frequency ω ₀ and attenuation d. Next, using numerical methods, find the maximum of this polynomial and the frequency corresponding to this maximum, which is taken to be equal to the resonant frequency, or the roots of the equation P '(ω) = 0 are found by the analytical method, one of which is taken to be equal to the resonant frequency. AANII as a function of the equation approximating the resonance curve of the resonance frequency ω _r is determined by the known formula

$${\omega }_r={\omega }_0\sqrt{\left(\mathrm{one}-{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="00000013.png"\; state="\{\{state\}\}">d</figure-callout>}}^2/2\right)}.$$

Since at any approximation, the maximum number of pairs of values of U _i , ω _{i are} processed within a one-time scan, the measurement time without deterioration of accuracy can be reduced to the duration of this scan t _sk , which is also a significant advantage of the proposed method.

The proposed method can be implemented according to the scheme of the device shown in figure 2.

The specified device operates as follows. DSP 4 generates sampling intervals Δt _i and the corresponding codes for controlling the frequency of synthesizer 5 and ADC 9. These codes are periodically fed through the intervals Δt _i to the input of synthesizer 5 and, according to the stepwise law, change its frequency in the range of resonant frequencies (Fig. 1a). At the same time, these codes are remembered by the DSP 4. Codes from the DSP 4 also receive codes for the ADC control input to obtain samples at certain points in time that correspond to the steady state in the cavity under study. Next, the high-frequency signal from the output of the synthesizer 5 with a frequency ω _{i is} fed to the input of the low-pass filter 6, which is designed to suppress the harmonics and other components of the spectrum present in the output signal of the synthesizer 5. From the output of the low-pass filter 6, the high-frequency signal is fed to the input of the resonator 7 and then to the detector 8, which isolates the envelope voltage of this high-frequency signal U _i (Fig. 1b, curve 2). The voltage U _{i is} supplied to the input of the ADC 9 and at specified times is converted into a digital code. From the ADC 9, the codes of the voltage samples U _i are supplied to the DSP 4, where they are accumulated together with the corresponding frequency codes ω _i . At the end of the scan, the DSP 4 performs logical and computational operations related to the selection of the values of U _i , ω _i that are within the optimal frequency range, calculation of the coefficients of the approximating polynomial (or parameters ω ₀ and d) and determination of the resonant frequency ω ₀ according to the selected algorithm .

The effectiveness of the proposed method was confirmed by an experiment with a virtual resonator model, the discrete frequency response of which is described by the following expression

$$A_i=\mathrm{one}/\sqrt{{\left(\mathrm{one}-{\left({\omega }_i/{\omega }_0\right)}^2\right)}^2+{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="00000013.png"\; state="\{\{state\}\}">d</figure-callout>}}^2{\omega }_i^2/{\mathrm{<figure-callout\; id="0"\; label="\omega "\; filenames="00000014.png"\; state="\{\{state\}\}">\omega </figure-callout>}}_0^2}.(7)$$

The exact value of the resonant frequency ω _rs for a given resonator model is calculated by the formula

$${\omega }_{rs}={\omega }_0\sqrt{\left(\mathrm{one}-{\mathrm{<figure-callout\; id="2"\; label="d"\; filenames="00000013.png"\; state="\{\{state\}\}">d</figure-callout>}}^2/2\right)},$$

and this value was taken as a reference.

The model was performed in the Matlab environment and had the following parameters: ω ₀ = 3 · 10 ⁸ rad / s; ω _max = 1.2 · 10 ⁸ rad / s; ω _min = 0.8 · 10 ⁸ rad / sec. Scanning time t _sk = 100 ms. For the quality factor, varying in the interval Q = 20 ÷ 100, the minimum step of tuning the frequency of the synthesizer taking into account formulas (4) (5) and (6) Δω _s = 10 ³ rad / sec. For the data given by formula (7), a discrete frequency response was calculated, on which a random function of the Gaussian noise type was superimposed to simulate interference. Figure 4 presents a view of this discrete frequency response with a signal-to-noise ratio of N = 80dB and Q = 20 (curve 12) and the results of approximation of this discrete frequency response by a fourth-degree polynomial for the relative approximation range Δω _ar = 0.3 (curve 13) which was defined as

Δω _ar = Δω _a / Δω,

where Δω _a is the frequency range in which the discrete frequency response was approximated;

Δω is the passband of the investigated resonator.

When studying the influence of the relative approximation range on the error in determining the resonance frequency, the signal-to-noise ratio varied in the interval N = 40dB ÷ ∞. The resonance frequency was determined for Q = 20 by the following methods:

1) The method proposed by the authors for determining the resonant frequency from the maximum of a polynomial of the 4th degree approximating a discrete frequency response with noise.

2) By the method of determining the resonant frequency directly from the maximum of the discrete frequency response with interference (as in the prototype).

Then, using the obtained experimental values of the resonance frequency ω _re and the reference value ω _rs calculated by the formula (8), for each of the methods, the relative error in determining the resonance frequency ω _{r was} determined in accordance with the formula

δω _r = (ω _re- ω _rs ) / ω _rs .

Figure 3 presents graphs of the dependence of the error δω _r on the relative approximation range Δω _ar when using the proposed method (family of curves 10) and when using the prototype method (family of curves 11).

The results of the experiment, presented in figure 3, clearly show that the error of the method proposed by the authors is at least an order of magnitude smaller than the error of the prototype method in a wide range of signal-to-noise ratios.

Claims (1)

A method for measuring a resonant frequency, characterized in that oscillations of various frequencies from a given set are sequentially excited in the resonator and these frequencies and the corresponding values of the amplitude of the output signal of the resonator are fixed, characterized in that for each of the excitation frequencies the amplitude of the output signal is fixed after the transient attenuation in resonator, approximate the obtained discrete dependence of the amplitude on the frequency of the continuous function and the resonant frequency of the resonator is determined by max mind this feature.

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