24.6
Tracé et calibrage du lieu des racines
Les lieux des racines divergent souvent lorsque les pôles du système passent de l'axe réel au plan complexe. Les points clés de cette transition sont…
Les loci racines divergent généralement lorsque les pôles du système passent du plan réel au plan complexe.
Les points de rupture et de rodage signalent où le locus quitte et rejoint l’axe réel. Les branches du locus racinaire forment un angle de 180/n degrés avec l’axe réel.
Le gain culmine au point de rupture entre les pôles en boucle ouverte sur l’axe réel, tandis que le gain minimum se produit au point de rupture entre deux zéros.
L’augmentation du gain peut pousser certains pôles du système dans le demi-plan droit, indiquant une instabilité potentielle. Les croisements de l’axe jω marquent la frontière entre le fonctionnement stable et instable du système.
L’analyse du locus racine consiste à localiser des points spécifiques et à calculer leur gain associé.
Pour connaître les coordonnées exactes du locus racine lorsqu’il traverse une certaine ligne de rapport d’amortissement, plusieurs points de test le long de la ligne sont sélectionnés et leur somme angulaire est évaluée.
Le locus racine existe lorsque la somme des angles totaux est égale à un multiple impair de 180 degrés.
Le gain en ce point précis est calculé en divisant le produit des longueurs des pôles par le produit des longueurs nulles jusqu’à ce point.
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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.