Transmission Lines Impedance Fitting Using Analytical Impedance Equation and Frequency Response Analysis
Abstract
:1. Introduction
- Develop a fitting method for MTL impedance based on the analytical impedance equation. This will ensure that the proposed method behaves similarly to the transmission line impedance.
- Develop a fitting method for MTL impedance which is a function of transmission line length. This is to ensure that the proposed model will have an accurate representation of the transmission line impedance for different transmission line lengths.
- Develop a fitting method capable of fitting MTL impedance for a wide range of frequencies without modifying the model for higher frequencies.
2. Wave Characteristics on Finite Transmission Lines
- , attenuation constant, in Np/m.
- , phase constant, in rad/m.
- R, resistance per unit length, in /m.
- L, inductance per unit length, in H/m.
- G, conductance per unit length, in S/m.
- C, capacitance per unit length, in F/m.
- l, length of the transmission line, in m.
3. The Frequency Response of Transmission Lines Impedance and the Proposed Model of MTL
4. Parameters of MTLs Proposed Impedance Equation
4.1. Derivation of Resonance Equations
4.2. Model Parameters Calculation of
4.3. Model Parameters Calculation of
5. Developed Algorithm for an Accurate Fitting
Algorithm 1: Developed Algorithm for an accurate model. |
|
6. Simulation Results
6.1. Case 1: MTL Impedance Fitting for up to 10 kHz Frequency Range
6.2. Case 2: MTL Impedance Fitting for up to 100 kHz Frequency Range
6.3. Case 3: MTL Impedance Fitting for up to 1 MHz Frequency Range
6.4. Case 4: Changing the Transmission Line Length for 1 MHz Frequency Range
6.5. Computational Time Comparison between the Proposed Method and VF
6.6. Comparison between VF and the Proposed Method
7. Features of the Proposed Method
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Advantages | Disadvantages |
---|---|
VF model can be used in time domain and frequency domain. | The rational function approximation is not a function in the transmission line length. |
VF Capable of fitting the impedance to a high degree of accuracy. | The rational function approximation is a mathematical function that is not correlated to the impedance of the transmission line. Thus, it fits the data as-is. |
VF enforces passivity to obtain a passive model. | VF required high computation power to obtain the model. |
VF is a general mathematical model capable of fitting any physical quantity with an amplitude and angle (Vector). | The obtained model changes with the frequency range. |
l | ||
---|---|---|
Inductive | ||
Capacitive | ||
0 | ||
where |
Model | Bergeron (RLC Data Entry) |
---|---|
Line Length | 100 [Km] |
Transposition | Ideally Transposed |
Frequency | 60 [Hz] |
Ground Resistivity | 100 [- m] |
Number of Phases | 3 |
Positive Sequence Series Resistance | 0.018547 [/Km] |
Positive Sequence Series Ind. Reactance | 0.37661 [/Km] |
Positive Sequence Shunt Cap. Reactance | 0.22789 [Meaga× Km] |
Zero Sequence Series Resistance | 0.3618376[/Km] |
Zero Sequence Series Ind. Reactance | 1.227747 [/Km] |
Zero Sequence Shunt Cap. Reactance | 0.34513 [Meaga× Km] |
Model | Bergeron (RLC Data Entry) |
---|---|
Line Length | 100 [Km] |
Frequency | 60 [Hz] |
Ground Resistivity | 100 [-m] |
Number of Phases | 1 |
Positive Sequence Series Resistance | 0.018547 [/Km] |
Positive Sequence Series Ind. Reactance | 0.37661 [/Km] |
Positive Sequence Shunt Cap. Reactance | 0.22789 [Meaga × Km] |
Transmission Line Parameters | Calculation |
---|---|
12.070792 [] | |
0.108695 [H] | |
2.304245 [F] | |
1.226930 [] | |
0.066348 [H] | |
1.752583 [F] |
Transmission Line Parameters | Calculation |
---|---|
12.068966 [] | |
0.1086480 [H] | |
2.3052476 [F] | |
0.6259449 [] | |
0.0335024 [H] | |
3.4709012 [F] |
TL Parameters | Calculated | Fitted | Error % |
---|---|---|---|
12.070792 [] | 12.06 [] | 0.0895 | |
0.108695 [H] | 0.108547 [H] | 0.1371 | |
2.304245 [F] | 2.305941 [F] | 0.0735 | |
1.226930 [] | 1.237722 [] | 0.8719 | |
0.066348 [H] | 0.066593 [H] | 0.3678 | |
1.752583 [F] | 1.746129 [F] | 0.3696 |
TL Parameters | Calculated | Fitted | Error % |
---|---|---|---|
12.068966 [] | 12.06 [] | 0.0743 | |
0.1086480 [H] | 0.108546 [H] | 0.0944 | |
2.3052476 [F] | 2.305959 [F] | 0.0309 | |
0.6259449 [] | 0.618861 [] | 1.1447 | |
0.0335024 [H] | 0.033297 [H] | 0.6179 | |
3.4709012 [F] | 3.492238 [F] | 0.6110 |
TL Parameters | Calculated | Fitted | Error % |
---|---|---|---|
12.068966 [] | 12.06 [] | 0.0743 | |
0.1086480 [H] | 0.108546 [H] | 0.0944 | |
2.3052476 [F] | 2.305959 [F] | 0.0309 | |
0.6259449 [] | 0.618861 [] | 1.1447 | |
0.0335024 [H] | 0.033296 [H] | 0.6191 | |
3.4709012 [F] | 3.492282 [F] | 0.6122 |
Model | Bergeron (RLC Data Entry) |
---|---|
Line Length | 100 [Km] |
Transposition | Ideally Transposed |
Frequency | 60 [Hz] |
Ground Resistivity | 100 [-m] |
Number of Phases | 3 |
Positive Sequence Series Resistance | 0.018547 [/Km] |
Positive Sequence Series Ind. Reactance | 0.37661 [/Km] |
Positive Sequence Shunt Cap. Reactance | 0.22789 [Meaga × Km] |
Zero Sequence Series Resistance | 0.3618376[/Km] |
Zero Sequence Series Ind. Reactance | 1.227747 [/Km] |
Zero Sequence Shunt Cap. Reactance | 0.34513 [Meaga × Km] |
Computer | Specs |
---|---|
RAM | 32 GB, 3200 MHz |
CPU | i7-11700F |
Operating system | Windows 11 |
Method | Computational Time [S] |
---|---|
VF | 24,663.37 |
Proposed Model | 516.75 |
Frequency [kHz] | Number of Parameters | Computational Power | ||
---|---|---|---|---|
Vector Fitting | Proposed Model | Vector Fitting | Proposed Model | |
10 | 50 | 24 | Low | Low |
100 | 750 | 24 | Medium | Low |
1000 | NA | 24 | High | Low |
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Alharbi, H.; Khalid, M.; Abido, M. Transmission Lines Impedance Fitting Using Analytical Impedance Equation and Frequency Response Analysis. Mathematics 2022, 10, 2677. https://doi.org/10.3390/math10152677
Alharbi H, Khalid M, Abido M. Transmission Lines Impedance Fitting Using Analytical Impedance Equation and Frequency Response Analysis. Mathematics. 2022; 10(15):2677. https://doi.org/10.3390/math10152677
Chicago/Turabian StyleAlharbi, Hosam, Muhammad Khalid, and Mohammad Abido. 2022. "Transmission Lines Impedance Fitting Using Analytical Impedance Equation and Frequency Response Analysis" Mathematics 10, no. 15: 2677. https://doi.org/10.3390/math10152677
APA StyleAlharbi, H., Khalid, M., & Abido, M. (2022). Transmission Lines Impedance Fitting Using Analytical Impedance Equation and Frequency Response Analysis. Mathematics, 10(15), 2677. https://doi.org/10.3390/math10152677