---
title: "s-plane region for ζ>0.707 and ts<2 s"
subject: "Control Systems"
chapter: 3
key: "semester_3/control_systems:s-plane-region-zeta-ts"
source: https://www.bctengineer.com/q/semester_3/control_systems/s-plane-region-zeta-ts
site: BCT Engineering
---

# s-plane region for ζ>0.707 and ts<2 s

Control Systems · Chapter 3: System Transfer Function and Responses

- Asked in: 2073 Shrawan, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/s-plane-region-zeta-ts
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-3

## Question

Draw the region in s-plane that satisfies following requirements: i) ζ > 0.707 ii) ts < 2 s.

## Short answer

The region is left of the vertical line Re(s) = -2 and inside the sector bounded by lines at ±45° to the negative real axis (ζ > 0.707).

## Answer

Each specification puts a boundary on where the closed-loop poles $s = -\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2}$ may lie.

#### i) $\zeta > 0.707$
A pole makes angle $\theta$ with the negative real axis, where $\cos\theta = \zeta$.
$$
\zeta > 0.707 \;\Rightarrow\; \theta < \cos^{-1}(0.707) = 45^\circ
$$
So the poles must lie inside the wedge between the two lines at $\pm 45^\circ$ to the negative real axis.

#### ii) $t_s < 2$ s
Using the 2% criterion, $t_s = \dfrac{4}{\zeta\omega_n} = \dfrac{4}{\sigma}$:
$$
\frac{4}{\sigma} < 2 \;\Rightarrow\; \sigma = \zeta\omega_n > 2
$$
So the poles must lie to the left of the vertical line $\text{Re}(s) = -2$.

#### Required region
Both conditions together: the shaded region (marked `#`), left of $\sigma = -2$ and inside the $\pm 45^\circ$ lines.

```text
                          jw
  ##############\     :   |
  ################\   :   |
  ##################\ :   |
  ####################\   |
  ####################: \ |
----------------------+---+------ sigma
  ####################: / |
  ####################/   |
  ##################/ :   |
  ################/   :   |
  ##############/     :   |
                     -2   0
  # = allowed region; \ / = 45 deg lines
```
The region is bounded by the lines $\omega = \pm\sigma$ (45 degrees) and the line $\sigma = -2$; it extends to the left without limit. (With the 5% criterion, $t_s = 3/\sigma$, the vertical line is at $\sigma = -1.5$.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does zeta > 0.707 mean in the s-plane?
   A. Poles to the left of Re(s) = -0.707
   B. Poles inside a circle of radius 0.707
   C. Poles inside a wedge within 45 degrees of the negative real axis
   D. Poles above the real axis only
   Answer: C. cos(theta) = zeta, so zeta > 0.707 means theta < 45 degrees. (Key point: Constant zeta lines)

2. What does ts < 2 s (2% criterion) require?
   A. sigma < 2, poles right of Re(s) = -2
   B. sigma = zeta wn > 2, so poles lie left of Re(s) = -2
   C. wn > 2, poles outside a circle
   D. sigma > 4
   Answer: B. ts = 4/sigma < 2 gives sigma > 2. (Key point: Settling time boundary)

3. What is the final region?
   A. Left of the line s = -2 and inside the +/-45 degree lines
   B. Right of s = -2 and outside the 45 degree lines
   C. Only the area on the imaginary axis
   D. Inside the 45 degree wedge only
   Answer: A. Both constraints must hold, so it is their intersection. (Key point: Combined region)

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