---
title: "Characteristics and response of zero-order system"
subject: "Instrumentation and Measurement"
chapter: 4
key: "semester_4/instrumentation_and_measurement:zero-order-system-characteristics"
source: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/zero-order-system-characteristics
site: BCT Engineering
---

# Characteristics and response of zero-order system

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

- Asked in: year not given, 4 marks
- Video: [Time Respone Analysis of a Zero Order System | Basic Concept](https://www.youtube.com/watch?v=33sTOQKAd5A) — ENGINEERING TUTORIAL
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/zero-order-system-characteristics
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_and_measurement/chapter-4

## Question

Explain the characteristics and response of a zero-order measurement system. Why is a linear potentiometer considered a zero-order instrument, and what are the limits of this assumption?

## Short answer

A zero-order system output is proportional to input instantly, with no lag or phase shift; amplitude ratio = 1, phase = 0 at all frequencies. A linear potentiometer gives e_o ∝ x with K = E/L, behaving as zero-order only when input changes are slow compared to its natural frequency, as real pots have mass, inductance and loading effects that introduce higher-order dynamics.

## Answer

A zero-order system has the model $a_0 q_o = b_0 q_i$, so $q_o = K q_i$ with static sensitivity $K = b_0/a_0$.

#### Characteristics
- The output is proportional to the input at every instant. There is no lag, delay, phase shift or distortion.
- Amplitude ratio $M = 1$ (relative to $K$) and phase angle $\phi = 0$ at all frequencies.
- The only parameter is $K$, found by static calibration.
- No energy storage (no mass, spring or capacitance) in the model.

#### Response
For any input, a step, ramp or sine, the output has the same shape scaled by $K$.

```text
 q_i |  ____        q_o |  ____   (scaled by K)
     | |    |           | |    |
     +-+----+--> t      +-+----+--> t
```

#### Potentiometer as example
A linear potentiometer of length $L$ and supply $E$ gives $e_o = (E/L)\,x$, so $K = E/L$ V/m. The output follows the wiper position with no dynamic lag.

#### Limits
A zero-order model is an idealization valid only when the input changes slowly compared with the instrument's natural frequency. In a real potentiometer, wiper mass and inertia, wire-wound resolution, friction, stray capacitance and inductance, and the loading error of a finite load give time-dependent behaviour at high frequency, so the true response is of higher order.

## Check yourself

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

1. For an ideal zero-order system, amplitude ratio and phase angle are
   A. M = 1 and φ = 0 at all frequencies
   B. M falls with frequency
   C. φ = -90° always
   D. M = 0.707 at ωn
   Answer: A. There is no lag or distortion at any frequency. (Key point: Frequency behaviour)

2. For a linear potentiometer with supply E and length L, K equals
   A. L/E
   B. E/L
   C. E×L
   D. E
   Answer: B. eo = (E/L)x so K = E/L V/m. (Key point: Potentiometer K)

3. Why does the zero-order model have no dynamics?
   A. It contains no mass, spring or capacitance (no energy storage)
   B. It has high damping
   C. It has a very large time constant
   D. It has two energy stores
   Answer: A. Without energy-storing elements the output follows the input instantly. (Key point: Energy storage)

4. When does the zero-order assumption for a pot break down?
   A. When the input changes slowly
   B. At high frequency because of wiper inertia, stray C and L, and friction
   C. At DC only
   D. Never
   Answer: B. It is valid only when the input varies slowly compared with the natural frequency. (Key point: Limits)

## Related in Instrumentation and Measurement

- [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
- [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
- [Potentiometer transducer and loading error](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Apotentiometer-transducer-loading-error) — Instrumentation and Measurement, chapter 5
- [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
- [Dynamic characteristics: rise time, delay, response](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Adynamic-characteristics-rise-time-delay) — Instrumentation and Measurement, chapter 4
- [Second-order transducer frequency and step response](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Asecond-order-transducer-numerical) — Instrumentation and Measurement, chapter 4

## Related in other subjects

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- [First-order system impulse and step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afirst-order-system-response) — Signals and Systems, chapter 5
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- [Magnitude response of second order system](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Amagnitude-response-0-5-0-3) — Digital Signal Analysis and Processing, chapter 3, asked 2×