where the −zi are the m zeros, the −pi are the m+n poles, and K is a scalar gain. Typically, a root locus diagram will indicate the transfer function's pole locations for varying values of K. A root locus plot will be all those points in the s-plane where G(s)H(s) = -1 for any value of K.
The factoring of K and the use of simple monomials means the evaluation of the rational polynomial can be done with vector techniques that add or subtract angles and multiply or divide magnitudes. The vector formulation arises from the fact that each monomial term in the factored G(s)H(s), (s−a) for example, represents the vector from a to s. The polynomial can be evaluated by considering the magnitudes and angles of each of these vectors. According to vector mathematics, the angle of the result is the sum of all the angles in the numerator add minus the sum of all the angles in the denominator. Similarly, the magnitude of the result is the product of all the magnitudes in the numerator divided by the product of all the magnitudes in the denominator. It turns out that the calculation of the magnitude is not needed because K varies; one of its values may result in a root. So to test whether a point in the s-plane is on the root locus, only the angles to all the open loop poles and zeros need be considered. A graphical method that uses a special protractor called a "Spirule" was once used to determine angles and draw the root loci.[3]
From the function T(s), it can be seen that the value of K does not affect the location of the zeros.[citation needed] The root locus only gives the location of closed loop poles as the gain K is varied. The zeros of a system do not move.
Using a few basic rules, the root locus method can plot the overall shape of the path (locus) traversed by the roots as the value of K varies. The plot of the root locus then gives an idea of the stability and dynamics of this feedback system for different values of K.
where the −zi are the m zeros, the −pi are the m+n poles, and K is a scalar gain. Typically, a root locus diagram will indicate the transfer function's pole locations for varying values of K. A root locus plot will be all those points in the s-plane where G(s)H(s) = -1 for any value of K.
The factoring of K and the use of simple monomials means the evaluation of the rational polynomial can be done with vector techniques that add or subtract angles and multiply or divide magnitudes. The vector formulation arises from the fact that each monomial term in the factored G(s)H(s), (s−a) for example, represents the vector from a to s. The polynomial can be evaluated by considering the magnitudes and angles of each of these vectors. According to vector mathematics, the angle of the result is the sum of all the angles in the numerator add minus the sum of all the angles in the denominator. Similarly, the magnitude of the result is the product of all the magnitudes in the numerator divided by the product of all the magnitudes in the denominator. It turns out that the calculation of the magnitude is not needed because K varies; one of its values may result in a root. So to test whether a point in the s-plane is on the root locus, only the angles to all the open loop poles and zeros need be considered. A graphical method that uses a special protractor called a "Spirule" was once used to determine angles and draw the root loci.[3]
From the function T(s), it can be seen that the value of K does not affect the location of the zeros.[citation needed] The root locus only gives the location of closed loop poles as the gain K is varied. The zeros of a system do not move.
Using a few basic rules, the root locus method can plot the overall shape of the path (locus) traversed by the roots as the value of K varies. The plot of the root locus then gives an idea of the stability and dynamics of this feedback system for different values of K.
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