Existence and Stability of <i>nT</i>-Periodic Orbits in the Boost Converter
In high load conditions, the boost converter presents some phenomena, such as chattering, chaos, subharmonics, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>T</mi&g...
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MDPI AG
2022-09-01
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| Series: | Applied Sciences |
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| Online Access: | https://www.mdpi.com/2076-3417/12/19/9565 |
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| author | Simeón Casanova Trujillo John E. Candelo-Becerra Fredy E. Hoyos |
| author_facet | Simeón Casanova Trujillo John E. Candelo-Becerra Fredy E. Hoyos |
| author_sort | Simeón Casanova Trujillo |
| collection | DOAJ |
| description | In high load conditions, the boost converter presents some phenomena, such as chattering, chaos, subharmonics, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits, which require studying them with the aim of reducing the effects and improving the performance of these electronic devices. In this paper, sufficient conditions for the existence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits are analytically obtained and the system stability is evaluated using eigenvalues of the Jacobian matrix of the Poincaré application. It is demonstrated numerically that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits occur for a broad range of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> parameters. The research obtains a particular class of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits in the boost converter and a formula that provides sufficient conditions for the existence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits with and without saturation in the duty cycle. In addition, an analysis of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits is performed with a biparametric diagram. The system stability is computed using a variational equation that allows perturbation of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits. Moreover, an analytical calculation of the Floquet exponents is performed to determine the stability limit of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbit. Finally, the phenomena found in this research are described according to the behavior of real applications encountered in previous literature. |
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| spelling | doaj.art-e02ef91d9dbd47dc8a43c3db40d24f112023-11-23T19:41:45ZengMDPI AGApplied Sciences2076-34172022-09-011219956510.3390/app12199565Existence and Stability of <i>nT</i>-Periodic Orbits in the Boost ConverterSimeón Casanova Trujillo0John E. Candelo-Becerra1Fredy E. Hoyos2Grupo de Investigación Cálculo Científico y Modelamiento Matemático, Universidad Nacional de Colombia, Sede Manizales, Manizales 170003, ColombiaDepartamento de Energía Eléctrica y Automática, Facultad de Minas, Universidad Nacional de Colombia, Sede Medellín, Carrera 80 No. 65-223, Robledo, Medellín 050041, ColombiaDepartamento de Energía Eléctrica y Automática, Facultad de Minas, Universidad Nacional de Colombia, Sede Medellín, Carrera 80 No. 65-223, Robledo, Medellín 050041, ColombiaIn high load conditions, the boost converter presents some phenomena, such as chattering, chaos, subharmonics, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits, which require studying them with the aim of reducing the effects and improving the performance of these electronic devices. In this paper, sufficient conditions for the existence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits are analytically obtained and the system stability is evaluated using eigenvalues of the Jacobian matrix of the Poincaré application. It is demonstrated numerically that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits occur for a broad range of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> parameters. The research obtains a particular class of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits in the boost converter and a formula that provides sufficient conditions for the existence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits with and without saturation in the duty cycle. In addition, an analysis of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits is performed with a biparametric diagram. The system stability is computed using a variational equation that allows perturbation of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbits. Moreover, an analytical calculation of the Floquet exponents is performed to determine the stability limit of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mi>T</mi></mrow></semantics></math></inline-formula>-periodic orbit. Finally, the phenomena found in this research are described according to the behavior of real applications encountered in previous literature.https://www.mdpi.com/2076-3417/12/19/9565Poincaré mapperiodic orbitsboost convertersystem stabilitybiparametric diagram |
| spellingShingle | Simeón Casanova Trujillo John E. Candelo-Becerra Fredy E. Hoyos Existence and Stability of <i>nT</i>-Periodic Orbits in the Boost Converter Applied Sciences Poincaré map periodic orbits boost converter system stability biparametric diagram |
| title | Existence and Stability of <i>nT</i>-Periodic Orbits in the Boost Converter |
| title_full | Existence and Stability of <i>nT</i>-Periodic Orbits in the Boost Converter |
| title_fullStr | Existence and Stability of <i>nT</i>-Periodic Orbits in the Boost Converter |
| title_full_unstemmed | Existence and Stability of <i>nT</i>-Periodic Orbits in the Boost Converter |
| title_short | Existence and Stability of <i>nT</i>-Periodic Orbits in the Boost Converter |
| title_sort | existence and stability of i nt i periodic orbits in the boost converter |
| topic | Poincaré map periodic orbits boost converter system stability biparametric diagram |
| url | https://www.mdpi.com/2076-3417/12/19/9565 |
| work_keys_str_mv | AT simeoncasanovatrujillo existenceandstabilityofintiperiodicorbitsintheboostconverter AT johnecandelobecerra existenceandstabilityofintiperiodicorbitsintheboostconverter AT fredyehoyos existenceandstabilityofintiperiodicorbitsintheboostconverter |