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In mathematics, the angle condition is a constraint that is satisfied by the locus of points in the s-plane on which closed-loop poles of a system reside. In combination with the magnitude condition, these two mathematical expressions fully determine the root locus. Let the characteristic equation of a system be , where . Rewriting the equation in polar form is useful. where are the only solutions to this equation. Rewriting in factored form, and representing each factor and by their vector equivalents, and , respectively, may be rewritten. Simplifying the characteristic equation, for ,

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  • Angle condition (en)
  • 角度條件 (zh)
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  • 角度條件（angle condition）是自動控制的根軌跡圖中，有關角度的限制條件，根軌跡圖中的點和閉迴路極點、零點組成向量的角度會滿足角度條件。角度條件和量值條件可以完全確定根軌跡圖。 令系統的特徵方程為，其中。若將控制方程改為極坐標表示 其中為方程式中的解。將改寫為各因式相乘的形式 再將因式及用向量的形式及表示，因此可以改寫如下。 簡化特徵方程式， 因此可以得到角度條件： 其中, 為極點1至n的角度，而 為零點1至m的角度。 也可以用類似的方式推導量值條件。 (zh)
  • In mathematics, the angle condition is a constraint that is satisfied by the locus of points in the s-plane on which closed-loop poles of a system reside. In combination with the magnitude condition, these two mathematical expressions fully determine the root locus. Let the characteristic equation of a system be , where . Rewriting the equation in polar form is useful. where are the only solutions to this equation. Rewriting in factored form, and representing each factor and by their vector equivalents, and , respectively, may be rewritten. Simplifying the characteristic equation, for , (en)
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  • In mathematics, the angle condition is a constraint that is satisfied by the locus of points in the s-plane on which closed-loop poles of a system reside. In combination with the magnitude condition, these two mathematical expressions fully determine the root locus. Let the characteristic equation of a system be , where . Rewriting the equation in polar form is useful. where are the only solutions to this equation. Rewriting in factored form, and representing each factor and by their vector equivalents, and , respectively, may be rewritten. Simplifying the characteristic equation, from which we derive the angle condition: for , are the angles of zeros 1 to n, and are the angles of poles 1 to m. The magnitude condition is derived similarly. (en)
  • 角度條件（angle condition）是自動控制的根軌跡圖中，有關角度的限制條件，根軌跡圖中的點和閉迴路極點、零點組成向量的角度會滿足角度條件。角度條件和量值條件可以完全確定根軌跡圖。 令系統的特徵方程為，其中。若將控制方程改為極坐標表示 其中為方程式中的解。將改寫為各因式相乘的形式 再將因式及用向量的形式及表示，因此可以改寫如下。 簡化特徵方程式， 因此可以得到角度條件： 其中, 為極點1至n的角度，而 為零點1至m的角度。 也可以用類似的方式推導量值條件。 (zh)
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