24.6
Plotando e Calibrando o Lugar das Raízes
Os lugares das raízes geralmente divergem conforme os pólos do sistema mudam do eixo real para o plano complexo. Os pontos-chave nessa transição são o…
Os loci raiz geralmente divergem quando os pólos do sistema fazem a transição do plano real para o complexo.
Os pontos de ruptura e ruptura sinalizam onde o locus sai e se junta ao eixo real. Os ramos do locus radicular formam um ângulo de 180 / n graus com o eixo real.
O ganho atinge o pico no ponto de ruptura entre os pólos de malha aberta no eixo real, enquanto o ganho mínimo ocorre no ponto de ruptura entre dois zeros.
O aumento do ganho pode empurrar alguns pólos do sistema para o meio-plano direito, indicando instabilidade potencial. Os cruzamentos do eixo jω marcam a fronteira entre as operações estáveis e instáveis do sistema.
A análise do locus raiz envolve a localização de pontos específicos e o cálculo de seu ganho relacionado.
Para saber as coordenadas exatas do locus raiz à medida que cruza uma determinada linha de taxa de amortecimento, vários pontos de teste ao longo da linha são selecionados e sua soma angular é avaliada.
O locus raiz existe onde a soma dos ângulos totais é igual a um múltiplo ímpar de 180 graus.
O ganho nesse ponto específico é calculado dividindo o produto dos comprimentos dos pólos pelo produto dos comprimentos zero até aquele ponto.
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Q1: What are breakaway and break-in points in root locus analysis?
Breakaway points are where the root locus leaves the real axis as system poles transition to the complex plane, occurring between open-loop poles where gain peaks. Break-in points are where the locus rejoins the real axis between two zeros, where minimum gain occurs. These critical points signal pole migration and are essential for understanding system behavior.
Q2: How do you determine if a point lies on the root locus?
A point lies on the root locus where the sum of total angles from all poles and zeros to that point equals an odd multiple of 180 degrees. Test points along a damping ratio line are selected and their angular sum is evaluated. Once a point satisfies this angle condition, the gain at that location can be calculated.
Q3: How is gain calculated at a specific point on the root locus?
Gain is calculated by dividing the product of distances from all poles to the point by the product of distances from all zeros to that point. This ratio determines the system gain required for the poles to occupy that specific location. The calculation uses vector representation complex numbers to measure these distances accurately.
Q4: What does it mean when poles cross into the right half-plane?
When increasing gain pushes system poles into the right half-plane, it signals potential instability. The jω-axis serves as the boundary between stable and unstable operations. Crossing this boundary indicates the system may become unstable, requiring careful gain selection during control system design.
Q5: What angle do root locus branches form with the real axis?
Root locus branches form an angle of 180/n degrees with the real axis, where n represents the number of branches at a breakaway or break-in point. This angular relationship is a fundamental property that helps predict how poles diverge as they transition from the real axis to the complex plane.
Q6: How do you find the exact coordinates where a root locus crosses a damping ratio line?
Multiple test points are selected along the damping ratio line, and the angular sum from poles and zeros to each point is evaluated. When the total angle equals an odd multiple of 180 degrees, that point lies on the root locus. The gain at the intersection is then calculated using the pole and zero distance ratio.
Q7: Why is the jω-axis crossing important in root locus analysis?
The jω-axis crossing marks the critical boundary between stable and unstable system operations. This crossing point indicates the gain value at which the system transitions from stability to instability. Identifying this crossing is essential for determining safe operating ranges and ensuring robust control system design.