2.1

Structure and density of calibration kit

The operating frequency of equipment based on microwave technology for measuring cigarette cutting positions is typically in the range of 2.0 to 3.0 GHz. Therefore, the design of the calibration kit must consider the material response within this frequency band. Table 1 shows the dielectric properties and density information for common plastic materials. Since the density of cigarettes is mostly in the range of 200 to 300 mg/cm³ (i.e., 0.2 to 0.3 g/cm³), which is significantly different from the densities of common plastics listed in Table 1, polypropylene materials with low density values (0.85 to 0.91 g/cm³) and moderate dielectric constants were selected. In addition to its stability and processability, polypropylene was chosen as the calibration material due to its suitable microwave response.

Table 1.

Electrical characteristics of plastics.

Based on the analysis of the differences in distribution position and density between the “dense-end part” and the “normal dense part” of cigarettes, the calibration kit structure shown in Figure 1 was designed. The white segment in the figure simulates the “normal dense part”, while the gray segment simulates the “dense-end part”. The length of the “dense-end part” varies between the left and right ends, mimicking the actual length difference of the “dense-end part” in real cigarette strips. In addition, this paper incorporates the density distribution characteristics of cigarettes, specifically the density ratio of the “dense-end part” to the “normal dense part” (8 ~ 15%). On this basis, and in accordance with the actual machining accuracy (± 0.1 mm), the processing was carried out.

Figure 1.

Structure of calibration kit.

Figure 2 shows the density distribution diagram of the calibration kits, where different colors represent the density distribution of the calibration kits with varying dense-end lengths. The density distribution within the range of 1 to 11 mm reflects the corresponding length differences, and the “normal dense part” in the middle section exhibits similar density and length. Additionally, the density ratio of the “dense-end part” to the “normal dense part” falls within the design range, indicating that the density distribution range of the calibration kits is reasonable and suitable for calibrating the cutting position measurement.

Figure 2.

Density of calibration kits.

2.2

Calibration of evaluation parameters

In this study, two key calibration techniques are elaborated in depth: density measurement calibration and cutting position calculation calibration. In the domain of microwave density measurement and calibration, a calibration kit made from polypropylene material is selected as the calibration reference. This calibration kit exhibits stable density characteristics at room temperature (25°C), ensuring the accuracy and repeatability of measurements with an error less than 3 mg/cm³, thus meeting the high standard requirements for calibration.

In the cutting position calibration procedure, this paper uses half the length difference of the “inner and outer row” dense-end parts as the key calibration index. Calibration kits with different specifications have varying dense-end part lengths, specifically 0 mm, 1 mm, 2 mm, 3 mm, and 4 mm, corresponding to sizes 0# to 4#. These length values establish the standard offset for the cutting position. During the calibration process, the cutting position of the calibration kit is precisely calculated and compared with the actual offset of the cutting position. The calibration result is considered acceptable when the deviation between the calculated outcome and the actual offset is less than 1 mm, and the repeatability of the calibration is less than 0.5 mm.

2.3

Calibration methods

3.1

Calibration of density measurement and evaluation of uncertainty

3.1.2

Uncertainty evaluation measurement model

Where, ΔD is the indication error of densimeter, in mg/cm3; D is the indication value of densimeter, in mg/cm3; DS is the constant density value of calibration kit at 25°C, in mg/cm3.

3.1.3

Uncertainty propagation rate

Sensitivity coefficients c1 and c2 are:

3.1.4

Source and evaluation of standard uncertainty

3.1.4.1

Standard uncertainty assessment of D

The standard uncertainty [11,12] u1(D) was introduced for repeated measurements of 0 # calibration kit by MW-T, with the density values were 380.72 mg/cm3, 381.81 mg/cm3, 382.41 mg/cm3, 378.72 mg/cm3, 378.75 mg/cm3, 379.38 mg/cm3, 380.94 mg/cm3, 382.25 mg/cm3, 380.81 mg/cm3, 380.64 mg/cm3 for 10 times. The standard deviation Sn of the density values was 1.26 mg/cm3. Referring to JJG 42-2011 Verification Regulation of Working Glass Hydrometers, by repeating the measurement 10 times, the standard uncertainty u₁(D) is calculated as:

In addition, the standard uncertainty u₂(D) is introduced from the MW-T resolution d (0.01 mg/cm3), which is calculated by the following formula:

3.2

Standard uncertainty evaluation of DS

The standard uncertainty of DS is primarily introduced by the density of 0# calibration kit. The density uncertainty of the calibration kit is not more than U = 1 mg/cm3, with a coverage factor K = 2, then:

3.2.1

Combined standard uncertainty

If the components of the standard uncertainty are not correlated, the combined standard uncertainty is:

Considering the inclusion factor K = 2, then:

Therefore, the calibration uncertainty of MW-T using 0# calibration kit is: U = 1.30 mg/cm3, K = 2.

3.3

Calibration and analysis of cigarettes cutting position

3.2

Calibration repeatability

According to the data in Table 4, the results of calibrating each calibration kit ten times were obtained. The standard deviations were all less than 0.5 mm. The coefficient of variation was less than 0.2 for all kits except for the relatively large 0# calibration kit, indicating that all indicators have met the evaluation criteria for calibration repeatability.

Table 4.

Repeatability of cutting position calibration.

Note: The average value in the table represents the average of calibrated results obtained from ten repetitions; the standard deviation is the standard deviation of calibrated results obtained from ten repetitions; the coefficient of variation is the ratio of standard deviation to the mean value.

3.3

Calibration stability

For the calibration kits, the MW-T was used to perform calibration measurements continuously for five months under the same ambient temperature and humidity (25 ± 2 °C, 40-70% RH), and the results were presented in Table 5. The results indicate that after long-term calibration tests, the standard deviation of calibrated offset of each calibration kit is less than 0.5 mm. This suggests that the calibration kits and the calibration method employed exhibit good stability and are suitable for long-term calibration applications.

Table 5.

Stability of cutting position calibration.

Note: The calibration results in the table represent the average of the calibration values obtained from three calibration runs per month; the standard deviation is the standard deviation of the average measurements over five months.

3.4

Calibration applicability

In order to verify the applicability of the calibration kit for cutting position calibration across multiple MW-T systems, four MW-T systems were selected as experimental equipment in this study. Firstly, each piece of equipment was calibrated for density and cutting position using the 0# calibration kit, following the steps outlined in Section 2.3. Then, the cutting position of the calibration kits was measured using each equipment. The optimal calibration parameters and average errors after calibration are presented in Table 6. According to the table, there is a certain degree of inter-device variability between different pieces of equipment. Even when the same group of calibration kits is used for calibration, the optimal calibration parameters for each equipment differ. However, the average measurement error of the calibration kit after calibration is less than 1 mm, which indicates that the method has good universal applicability for cutting position calibration across different equipment.

Table 6.

Multi-equipment cutting position measurements.