---
title: "Second-order transducer frequency and step response"
subject: "Instrumentation and Measurement"
chapter: 4
key: "semester_4/instrumentation_and_measurement:second-order-transducer-numerical"
source: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/second-order-transducer-numerical
site: BCT Engineering
---

# Second-order transducer frequency and step response

Instrumentation and Measurement · Chapter 4: Dynamic Response of Measurement System

- Asked in: year not given, 8 marks
- Video: [Intro to Control - 9.3 Second Order System: Damping & Natural Frequency](https://www.youtube.com/watch?v=pzRTyxf31fw) — Engineering with Prof. Kim
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/second-order-transducer-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_and_measurement/chapter-4

## Question

A second-order pressure transducer has a natural frequency of 1000 rad/s and a damping ratio of 0.7. Find (a) the amplitude ratio and phase lag for a sinusoidal input of 400 rad/s, (b) the percentage overshoot for a step input, and (c) the damped frequency, peak time, rise time and 2% settling time.

## Short answer

Amplitude ratio at 400 rad/s is 0.9905, phase lag 33.7°. Step overshoot is 4.60%. Damped frequency 714.1 rad/s, peak time 4.40 ms, rise time 3.29 ms, 2% settling time 5.71 ms.

## Answer

Given: $\omega_n = 1000$ rad/s, $\zeta = 0.7$.

#### (a) Frequency response at 400 rad/s
Frequency ratio $r = \omega/\omega_n = 0.4$.

$$
M = \frac{1}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}} = \frac{1}{\sqrt{(0.84)^2 + (0.56)^2}} = \frac{1}{\sqrt{0.7056 + 0.3136}} = 0.9905
$$

$$
\phi = -\tan^{-1}\left(\frac{2\zeta r}{1 - r^2}\right) = -\tan^{-1}\left(\frac{0.56}{0.84}\right) = -33.7^\circ
$$

The amplitude error is only 0.95%.

#### (b) Overshoot
$$
M_p = \exp\left(\frac{-\pi\zeta}{\sqrt{1-\zeta^2}}\right) = \exp\left(\frac{-\pi(0.7)}{0.7141}\right) = \exp(-3.079) = 0.0460 = 4.60\%
$$

#### (c) Time-domain parameters
Damped frequency:

$$
\omega_d = \omega_n\sqrt{1-\zeta^2} = 1000(0.7141) = 714.1\ \text{rad/s}
$$

Peak time: $t_p = \pi/\omega_d = 4.40$ ms.

Rise time (0 to 100%): $t_r = \dfrac{\pi - \cos^{-1}\zeta}{\omega_d} = \dfrac{3.1416 - 0.7954}{714.1} = 3.29$ ms.

2% settling time: $t_s \approx \dfrac{4}{\zeta\omega_n} = \dfrac{4}{700} = 5.71$ ms.

| Quantity | Value |
|---|---|
| Amplitude ratio at 400 rad/s | 0.9905 |
| Phase lag | $33.7^\circ$ |
| Overshoot | 4.60% |
| $\omega_d$ | 714.1 rad/s |
| $t_p$ | 4.40 ms |
| $t_r$ | 3.29 ms |
| $t_s$ (2%) | 5.71 ms |

**Answer: $M = 0.9905$, lag $= 33.7^\circ$, overshoot $= 4.6\%$, $\omega_d = 714$ rad/s, $t_p = 4.40$ ms, $t_r = 3.29$ ms, $t_s = 5.71$ ms.**

## Check yourself

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

1. For r = 0.4 and ζ = 0.7, the amplitude ratio is
   A. 0.9905
   B. 0.84
   C. 1.0095
   D. 0.56
   Answer: A. M = 1/sqrt(0.84^2 + 0.56^2) = 1/sqrt(1.0192) = 0.9905. (Key point: Amplitude ratio)

2. The phase lag at r = 0.4, ζ = 0.7 is
   A. 56°
   B. 33.7°
   C. 45°
   D. 71.5°
   Answer: B. φ = tan^-1(0.56/0.84) = 33.7°. (Key point: Phase lag)

3. Percent overshoot for ζ = 0.7 is
   A. 0.46%
   B. 4.6%
   C. 46%
   D. 7%
   Answer: B. exp(-π·0.7/0.7141) = 0.046 = 4.6%. (Key point: Overshoot)

4. Peak time with ωn = 1000 rad/s and ζ = 0.7 is
   A. 3.14 ms
   B. 4.40 ms
   C. 5.71 ms
   D. 7.14 ms
   Answer: B. ωd = 714.1 rad/s and tp = π/ωd = 4.40 ms. (Key point: Peak time)

## Related in Instrumentation and Measurement

- [Second-order step response and damping](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Asecond-order-step-response-damping) — Instrumentation and Measurement, chapter 4
- [Thermocouple time constant and frequency response](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Athermocouple-first-order-response-numerical) — Instrumentation and Measurement, chapter 4
- [Step response of a first-order system](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Afirst-order-step-response-time-constant) — Instrumentation and Measurement, chapter 4
- [Models of zero, first and second order systems](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Azero-first-second-order-models) — Instrumentation and Measurement, chapter 4
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