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Angle and magnitude criteria for k/[(s+1)(s+3)]

Short answer

Yes, the point lies on the locus because the angle criterion gives -180° (sd is on the vertical line σ=-2). The required gain from the magnitude criterion is k = |sd+1|·|sd+3| = 13.25.

For the unity feedback system with open loop transfer function (OLTF) G(s) = k/[(s+1)(s+3)], use angle criteria to check whether the root locus passes from point sd = −2 + j3.5. If yes, use magnitude criteria to select the appropriate value of gain parameter.

G(s)=k(s+1)(s+3),sd=−2+j3.5G(s) = \frac{k}{(s+1)(s+3)}, \qquad s_d = -2 + j3.5

Angle criterion

The locus passes through sds_d only if ∠G(sd)=±180∘(2q+1)\angle G(s_d) = \pm 180^\circ(2q+1).

VectorValueAngle
sd+1s_d + 1−1+j3.5-1 + j3.5180∘−tan⁡−1(3.5)=105.95∘180^\circ - \tan^{-1}(3.5) = 105.95^\circ
sd+3s_d + 31+j3.51 + j3.5tan⁡−1(3.5)=74.05∘\tan^{-1}(3.5) = 74.05^\circ
∠G(sd)=−(105.95∘+74.05∘)=−180∘\angle G(s_d) = -(105.95^\circ + 74.05^\circ) = -180^\circ

The angle criterion is satisfied, so the root locus passes through −2+j3.5-2 + j3.5. This is expected: for two real poles, the complex part of the locus is the vertical line through their midpoint, σ=−2\sigma = -2.

Magnitude criterion

k=∣sd+1∣ ∣sd+3∣=1+3.52×1+3.52=3.640×3.640=13.25\begin{aligned} k &= |s_d + 1|\,|s_d + 3| \\ &= \sqrt{1 + 3.5^2} \times \sqrt{1 + 3.5^2} \\ &= 3.640 \times 3.640 = 13.25 \end{aligned}

Check

With k=13.25k = 13.25: s2+4s+3+13.25=s2+4s+16.25=0s^2 + 4s + 3 + 13.25 = s^2 + 4s + 16.25 = 0, so s=−2±j3.5s = -2 \pm j3.5. ✓

Answer: Yes, the point lies on the root locus. The required gain is k = 13.25.

✓ Answer checked by Claude Opus 5.5 (high effort)

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