24.6
Trazado y calibración del lugar de las raíces
Los lugares de las raíces suelen divergir a medida que los polos del sistema se desplazan del eje real al plano complejo. Los puntos clave en esta tra…
Los loci de las raíces comúnmente divergen cuando los polos del sistema pasan del plano real al complejo.
Los puntos de ruptura y ruptura señalan dónde sale el lugar geométrico y se vuelve a unir con el eje real. Las ramas del locus de la raíz forman un ángulo de 180/n grados con el eje real.
La ganancia alcanza su punto máximo en el punto de ruptura entre los polos de bucle abierto en el eje real, mientras que la ganancia mínima se produce en el punto de ruptura entre dos ceros.
El aumento de la ganancia puede empujar algunos polos del sistema hacia el medio plano derecho, lo que indica una posible inestabilidad. Los cruces del eje jω marcan el límite entre las operaciones estables e inestables del sistema.
El análisis del lugar geométrico de la raíz implica la localización de puntos específicos y el cálculo de su ganancia relacionada.
Para conocer las coordenadas exactas del lugar geométrico de la raíz a medida que cruza una determinada línea de relación de amortiguamiento, se seleccionan varios puntos de prueba a lo largo de la línea y se evalúa su suma angular.
El lugar geométrico de la raíz existe cuando la suma de los ángulos totales es igual a un múltiplo impar de 180 grados.
La ganancia en ese punto específico se calcula dividiendo el producto de las longitudes de los polos por el producto de las longitudes cero hasta ese punto.
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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.