Novel Method of Fitting a Nonlinear Function to Data of Measurement Based on Linearization by Change Variables, Examples and Uncertainty
Abstract
:1. Introduction
2. Main Idea of Proposed Method
- The simplest form with the determined dependent variable y, i.e., y = f (x).
- On each side of the equal sign are nonlinear functions of only y or x, .
- The implicit form of the nonlinear function H(x, y) = 0.
3. Examples of Nonlinear Functions Fitted by Change Variables for Linearization
3.1. Transformations of Single Coordinate x (Type I)
Parabola Fitting—Method I
3.2. Transformation of Both Separatable Coordinates (Type II)
3.3. Fitting the Implicit Dependence (Type III)
- standard uncertainties:
- expanded uncertainties:
4. Examples of Measurement Circuits with Nonlinear Functions
4.1. Matching the Characteristics of a MOSFET
4.2. Measurement of a Series Connection of a Diode D and Resistor r Representing Internal Resistance of This Circuit (Type III)
4.3. Measuring the Thickness of a Plane-Parallel Plate Using the Law of Refraction
4.4. Calculation of the Measured Value d and Its Uncertainty
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
y | 9.18 | 16.50 | 23.31 | 28.29 | 33.67 | 37.13 | 41.59 | 44.00 | 47.84 | 49.50 |
No. | Coordinates of Measurement Points | Matching Errors to Nominal Parabolic Characteristic Δy [V] | |
---|---|---|---|
xi [V] | [V] | ||
1 | 2.79 | 10.00 | −1.00 |
2 | 7.13 | 17.50 | 0.50 |
3 | 13.47 | 35.00 | −4.00 |
4 | 21.81 | 46.00 | 2.00 |
5 | 32.15 | 71.00 | −2.00 |
6 | 44.49 | 97.00 | −3.00 |
7 | 58.83 | 122.00 | 1.00 |
8 | 75.17 | 158.00 | −2.00 |
9 | 93.51 | 194.00 | −1.00 |
10 | 113.85 | 232.00 | 2.00 |
(mA) | 0.25 | 0.5 | 0.75 | 1 | 1.25 | 1.5 | 1.75 | 2 | 3 | 4 |
4.07 | 8.06 | 11.41 | 14.93 | 17.87 | 20.48 | 22.96 | 24.98 | 29.74 | 30.02 |
I (mA) | 1.10 | 1.37 | 3.29 | 6.80 | 21.98 | 69.09 | 148.75 | 317.59 | 970.12 | 1975.35 |
U (mV) | 120 | 130 | 150 | 170 | 200 | 230 | 250 | 270 | 300 | 320 |
α [deg] | 0 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 | 39 | 42 | 45 |
b [μm] | 0 | 247.9 | 609.7 | 824.1 | 1132.3 | 1494.1 | 1855.9 | 2284.7 | 2726.9 | 3088.7 | 3571.1 | 4013.3 | 4602.9 | 5219.3 | 5849.1 | 6545.9 |
α [rad] | 0.05 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 | 0.35 | 0.40 | 0.45 | 0.50 |
b (mm) | 0.51 | 1.02 | 1.53 | 2.06 | 2.59 | 3.15 | 3.72 | 4.31 | 4.92 | 5.56 |
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Warsza, Z.L.; Puchalski, J.; Więcek, T. Novel Method of Fitting a Nonlinear Function to Data of Measurement Based on Linearization by Change Variables, Examples and Uncertainty. Metrology 2024, 4, 718-735. https://doi.org/10.3390/metrology4040042
Warsza ZL, Puchalski J, Więcek T. Novel Method of Fitting a Nonlinear Function to Data of Measurement Based on Linearization by Change Variables, Examples and Uncertainty. Metrology. 2024; 4(4):718-735. https://doi.org/10.3390/metrology4040042
Chicago/Turabian StyleWarsza, Zygmunt L., Jacek Puchalski, and Tomasz Więcek. 2024. "Novel Method of Fitting a Nonlinear Function to Data of Measurement Based on Linearization by Change Variables, Examples and Uncertainty" Metrology 4, no. 4: 718-735. https://doi.org/10.3390/metrology4040042
APA StyleWarsza, Z. L., Puchalski, J., & Więcek, T. (2024). Novel Method of Fitting a Nonlinear Function to Data of Measurement Based on Linearization by Change Variables, Examples and Uncertainty. Metrology, 4(4), 718-735. https://doi.org/10.3390/metrology4040042