An open-loop system has poles at s = 0, −3, and −6 with no finite zeros (n = 3, m = 0). As K increases from 0 to ∞, what are the asymptote angles and the centroid from which they radiate?
AAll three branches converge to the centroid at s = −3 as K → ∞
BThree branches go to infinity at 60°, 180°, and −60° from the centroid σ_a = (0 − 3 − 6)/3 = −3
CAll three branches go to infinity at 45°, 135°, and 225° because the system has three poles
DThe branches cannot be determined without numerically solving the characteristic equation
With n = 3 poles and m = 0 zeros, all three branches go to infinity (n − m = 3 asymptotes). The centroid is σ_a = (Σpoles − Σzeros)/(n − m) = (0 − 3 − 6)/3 = −3. Asymptote angles are θ = 180°(2k + 1)/(n − m) for k = 0, 1, 2: yielding 60°, 180°, and 300° (= −60°). The branches radiate to infinity in these three directions from s = −3, regardless of the exact pole spacing (which only affects the curve shape before it approaches the asymptotes).
Question 2 Multiple Choice
A system has real-axis open-loop poles at s = 0, −2, −5 and a real-axis zero at s = −3. Which real-axis segments belong to the root locus?
AThe entire negative real axis, because poles always outnumber zeros
BBetween s = 0 and s = −2 (one real pole to the right), and between s = −3 and s = −5 (three real critical values to the right: pole at 0, pole at −2, zero at −3)
CBetween s = −2 and s = −3 (two poles to the right), and to the left of s = −5 (four critical values)
DOnly to the left of s = −5 because the zero at −3 cancels one pole contribution
The real-axis rule: a segment belongs to the root locus when the total count of real-axis poles AND zeros to its right is odd. (1) Between 0 and −2: one pole at 0 is to the right → count = 1 (odd) → on locus. (2) Between −2 and −3: poles at 0 and −2 → count = 2 (even) → not on locus. (3) Between −3 and −5: poles at 0, −2 and zero at −3 → count = 3 (odd) → on locus. (4) Left of −5: all four elements → count = 4 (even) → not on locus.
Question 3 True / False
The centroid σ_a = (Σpoles − Σzeros)/(n − m) indicates where root-locus branches cross or intersect each other on the real axis.
TTrue
FFalse
Answer: False
The centroid determines only where the asymptotes RADIATE FROM — it is the geometric origin of the n − m lines along which branches escape to infinity. The centroid is not a point where branches intersect, and root-locus branches do not necessarily pass through it. Actual breakaway and break-in points on the real axis are found by solving dK/ds = 0, which can yield answers far from the centroid — especially when zeros are present. This is one of the most common misapplications of the asymptote rules.
Question 4 True / False
Departure angles from complex open-loop poles must be computed explicitly; without them, a hand-sketched root locus may be qualitatively wrong about whether branches initially move toward the right half-plane.
TTrue
FFalse
Answer: True
Departure angles are computed by applying the angle condition at a point infinitesimally close to a complex pole. The contributions from all other poles and zeros determine the net phase, and the departure angle is whatever value satisfies the ±180° requirement. This angle determines whether the branch initially moves left (toward stability) or right (toward instability). For systems with complex open-loop poles — common in underdamped or resonant plants — omitting this computation can produce a qualitatively wrong sketch that gives incorrect stability predictions at low gain.
Question 5 Short Answer
The real-axis rule states a segment belongs to the root locus only when there is an ODD count of real poles and zeros to its right. Why odd and not even?
Think about your answer, then reveal below.
Model answer: The root locus is defined by the angle condition: ∠G(s)H(s) = ±180°(2k+1). On the real axis, conjugate complex poles and zeros contribute phase in pairs that cancel exactly. Only real-axis poles and zeros contribute net phase: each real pole or zero to the RIGHT of a test point contributes ±180°. For the total to equal an odd multiple of 180° (the angle condition), the number of such contributions must be odd — an even count sums to a multiple of 360°, which fails the condition. The real-axis rule is therefore not a separate memorized fact but a direct consequence of the angle condition evaluated on the real axis.
This derivation shows why construction rules are not arbitrary mnemonics. Every rule — real axis, asymptotes, breakaway points, departure angles — is the angle condition applied in a different geometric context. Understanding this derivation means you can reconstruct the rules from first principles if you forget them, and you can correctly apply the real-axis rule even in unusual configurations with multiple zeros.