Biomathematical Analysis on Impedance Measurement During COVID-19 Pandemic

Introduction

In the industrial field, impedance is an important parameter, and the measurement and analysis of impedance helps us to understand the changes of morphological characteristics of the object under test [1]. For example, in the oil and gas pipeline transmission system, the impedance of the pipeline corrosion protection layer needs to be measured to grasp its service life [2] in the field of biomedicine, the clinical application of bioimpedance technology has a great front [3] in the field of corrosion monitoring, the monitoring of aircraft coatings, which is an effective means to prevent corrosion of the base metal[4] these belong to the category of high impedance measurement, with an impedance of up to 108Ω. Therefore, in the processing of these weak signals processing, the idea of sinusoidal fitting is used to process to obtain more information during COVID-19 pandemic [5].

Differential Amplifier Circuit Mathematical Model

The excitation signal is formed by bucking the sine signal generated by filtering, assuming that the bucking scale factor is k1 and the amplitude of the sine signal before bucking is 𝐴, the excitation signal is expressed as [6,7]:

If the differential amplification is K₂, the output V_o of the differential amplifier circuit and the input V_i satisfy the following vector relationship [8,9]:

Translated into a specific functional form of time it can be expressed as [10-12]:

Z₀: Reference impedance, standard resistance is generally used in measurement circuits.
𝛗: Difference between Zx and the impedance angle of Z_x + Z₀.

If the presence of the input capacitance _C0 of the amplified input system is not considered, Z_x is the measured coating complex impedance Z_c, and if the presence of the input capacitance is considered, it is the combined impedance of the coating complex impedance in parallel with the capacitance, satisfying the following relationship [13-15].

The complex impedance of the coating under test is:

Understanding of Sine Fitting

Simple moving average method: First, we use the simple moving average method to feel the meaning of “moving”. There is a sequence as follows (Table 1).

Table 1.Sequence table.

If a moving average with a window of 5 is used, the element with a sequence of 11 and a value of 2 should be replaced with [16,17].

Then the element with a sequence of 12 and a value of 1 should be replaced with

For fast calculation, the following formula can be used for recursion

Simple moving average is defined as: Simple moving average of data P1, P2,…,PM with window n:

The iteration form is:

In MATLAB, there is a corresponding smooth function available. We can use 𝑚𝑎𝑡𝑙𝑎𝑏 to verify smooth (y, span). The following figure shows the result of data2= smooth (data1,5) (Figure 1).

Figure 1:Smooth validation chart.

Digital Sine Fitting: We can draw the following discrete curve by inputting the following commands in the 𝑚𝑎𝑡𝑙𝑎𝑏 command line window [18].

Figure 2: Discrete curve chart.

Since the known curve is drawn, the above parameters can be easily obtained, and the error of each point is 0. However, for a relatively disordered sequence, there will always be a certain deviation from the ideal curve, so the error will not be 0, but our purpose is to find a curve that is closest to it, and the evaluation standard is the sum of squares of errors, which is consistent with the least square method. (Figure 3) The following diagram shows a discrete sequence of approximate cosine curves [19].

Figure 3: Discrete sequences.

Similarly, we use the following expression

For subsequent operation processing, another form is used to express

Among,

So, we think that x_i = acos Δ_i + bsinΔ_i + c is the function that can best represent the discrete curve in the above figure, but a, B and C are temporarily unknown. Define the sum of squares of errors as

In order to find the A, B and C that minimize the above formula, the partial derivatives of a, b and c in the above formula are obtained, and the partial derivatives are zero, then

We generally use the whole period fitting. For the whole period

After getting a and b, of course, you can also know the parameter A and 0

Moving Sine Fitting

For ease of understanding, only the whole cycle is described here. Using the method in 2.2, perform sine fitting on the interval Zi (corresponding to the moving average window, which contains exactly one cycle, assuming that the number of cycle points is n) where the element Xi is located. a, b, and c can be obtained, and only ai, bi, ci of element xi are obtained. Can also get Ai, θI (Figure 4).

Figure 4: Four-period discrete series.

So, Then, the interval Z_i+1 of element X_i+1 is sinusoidally fitted to obtain a_i+1, b_i+1, c_i+1 of element x_i+1, and also A_i+1 and θ_{i +}1.

In fact, for the convenience of calculation, the calculation formulas of a, b and c in 2.2 are combined. It can be inferred that [20].

Where [n/2]is the downward rounding of n/2, that is, when n=5, its value is.

The moving sine has a faster running speed in the microprocessor environment because of the recursive idea of replacing old and new elements, which is similar to the moving average. It is also more suitable for the case with unstable amplitude as shown in the figure below. It can accurately calculate a, b, c, amplitude A and initial phase of each pointθ (Figure 5).

Figure 5: Ultrasonic signals.

For an ultrasonic echo signal, after the amplitude A of each point is obtained by moving fitting, the amplitude envelope can be obtained, as shown in the dotted envelope in the below (Figure 6).

Figure 6: Ultrasound echo signal.

Results

In large impedance measurements, the signal is extremely weak, so a differential circuit is introduced to change it into complex impedance form, and the simplicity of sine fitting in amplitude and phase calculation is used to assist in calculating the impedance magnitude, which plays an important role in subsequent large impedance measurements.

Conflict of Interest

We have no conflict of interests to disclose, and the manuscript has been read and approved by all named authors.

Acknowledgments

This work was supported by the Philosophical and Social Sciences Research Project of Hubei Education Department (19Y049), and the Staring Research Foundation for the Ph.D. of Hubei University of Technology (BSQD2019054), Hubei Province, China.

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