Properties of the Root Locus

The root locus method helps analyze higher-order systems without requiring the factorization of the denominator of the transfer function.

A pole is identified when the characteristic polynomial in the transfer function's denominator equals zero.

The criteria for a point to be on the root locus are that the total angle contribution from all zeros minus the total angle contribution from all poles, is equal to an odd multiple of 180 degrees.

The gain at any point on the root locus is determined by dividing the product of the pole lengths by the product of the zero lengths.

Consider a transfer function for a unity feedback system. The angle at a specific point is the algebraic sum of angles of vectors drawn from the system's zeros and poles of the transfer function to that point.

Since the angle equals an odd multiple of 180 degrees, the point is on the root locus.

The gain can be computed by dividing the pole lengths product by the zero lengths product.