1. The problem asks to determine if each root locus sketch in Figure P8.1 can be a valid root locus and explain why or why not. 2. Root locus rules to remember: - Root loci start at poles (marked with crosses) and end at zeros (marked with circles) or go to infinity if zeros are fewer than poles. - The number of branches equals the number of poles. - Root loci cannot cross each other. - Root loci on the real axis exist to the left of an odd number of poles and zeros. - Angles of departure and arrival must satisfy angle conditions. 3. Analyze each sketch: (a) Two poles left of two zeros on real axis with a half-circle above real axis: Valid root locus because loci start at poles and end at zeros, and the half-circle is a typical locus shape. (b) One zero and two poles on real axis with a full circle passing through them: Invalid because root loci cannot form closed loops; they start at poles and end at zeros or infinity, not forming closed circles. (c) Two zeros labeled "Double pole" with a half-circle above real axis: Invalid because zeros cannot be labeled as poles; also, root loci start at poles, not zeros. (d) Two poles on real axis with no loci shown: Incomplete sketch, but possible root locus if loci are drawn; no loci means no root locus shown. (e) Two zeros and one pole with lines emanating upward forming an angle: Invalid because number of poles must be equal or greater than zeros; also, loci must start at poles. (f) One zero and one pole on real axis with arc curving below axis starting at zero: Invalid because loci start at poles, not zeros. (g) Two poles flanking one zero on real axis: Valid root locus; loci start at poles and end at zero. (h) Two poles on real axis with curved locus crossing real axis between them: Valid root locus; loci can cross real axis between poles. Final answers: (a) Valid (b) Invalid - root loci cannot form closed loops (c) Invalid - zeros cannot be poles (d) Incomplete - no loci shown (e) Invalid - number of poles less than zeros (f) Invalid - loci start at poles, not zeros (g) Valid (h) Valid