---
title: "Dynamic characteristics: rise time, delay, response"
subject: "Instrumentation and Measurement"
chapter: 4
key: "semester_4/instrumentation_and_measurement:dynamic-characteristics-rise-time-delay"
source: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/dynamic-characteristics-rise-time-delay
site: BCT Engineering
---

# Dynamic characteristics: rise time, delay, response

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

- Asked in: year not given, 5 marks
- Video: [Time Domain Specifications](https://www.youtube.com/watch?v=MzrgBc4s-jk) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/dynamic-characteristics-rise-time-delay
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_and_measurement/chapter-4

## Question

Define and explain amplitude response, frequency response, phase response, rise time, delay time, settling time and dynamic error of a measurement system.

## Short answer

Amplitude response is output/input magnitude ratio M(ω). Phase response is the output lag φ(ω). Rise time is the 10%-90% step-response time. Delay time is time to 50% of final value. Settling time is time to remain within a tolerance band (e.g. 2%). Dynamic error is the instantaneous difference between indicated and true value for time-varying inputs.

## Answer

Dynamic characteristics describe how a measurement system responds to inputs that change with time.

- **Amplitude response (magnitude ratio):** the ratio of output amplitude to input amplitude, $M(\omega) = |q_o|/(K|q_i|)$, for a sinusoidal input of frequency $\omega$. An ideal instrument has $M = 1$ over its working range.
- **Frequency response:** the variation of $M$ and phase with frequency. The range over which $M$ stays within a stated tolerance (e.g. $\pm 5\%$) is the **bandwidth** or useful frequency range.
- **Phase response:** the phase angle $\phi(\omega)$ between output and input. For distortion-free measurement of complex signals, $\phi$ should be proportional to frequency (a constant time delay).
- **Delay time ($t_d$):** the time for the output to reach 50% of its final value after a step input.
- **Rise time ($t_r$):** the time for the step response to rise from 10% to 90% (sometimes 0 to 100%) of the final value. For a first-order system $t_r = 2.2\tau$.
- **Settling time ($t_s$):** the time after which the output stays within a band (2% or 5%) of the final value.
- **Dynamic error:** the difference between the indicated value and the true value of a time-varying input, excluding static error. For a sinusoidal input, the amplitude error is $(1 - M)$ times the input amplitude, and the output also lags by $\phi$.

```text
 y
 1.0 |           _.-~~~~~~~~~
 0.9 |--------.-'
 0.5 |------.'        <- delay time t_d
 0.1 |---.-'
     |  |<--- t_r --->|
     +--------------------> t
```

## Check yourself

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

1. Delay time is the time for a step response to reach
   A. 10% of final value
   B. 50% of final value
   C. 90% of final value
   D. within 2% band
   Answer: B. Delay time td is the time to reach 50% of the final value. (Key point: Delay time)

2. Rise time is commonly defined as the time to go from
   A. 0 to 50%
   B. 10% to 90% of final value
   C. 50% to 100%
   D. 63% to 99%
   Answer: B. Sometimes 0 to 100% is used; for first order tr = 2.2 tau. (Key point: Rise time)

3. Settling time is
   A. Time to first reach final value
   B. Time after which output stays within a band (2% or 5%) of final value
   C. Time to reach 63.2%
   D. Time of the first peak
   Answer: B. It is when the output stays within the tolerance band. (Key point: Settling time)

4. For distortion-free measurement of complex signals the phase angle should be
   A. Zero at all frequencies only
   B. Proportional to frequency (constant time delay)
   C. Constant 90°
   D. Random
   Answer: B. Proportional phase means all harmonics are delayed equally in time. (Key point: Phase response ideal)

## Related in Instrumentation and Measurement

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