---
title: "Models of zero, first and second order systems"
subject: "Instrumentation and Measurement"
chapter: 4
key: "semester_4/instrumentation_and_measurement:zero-first-second-order-models"
source: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/zero-first-second-order-models
site: BCT Engineering
---

# Models of zero, first and second order systems

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

- Asked in: year not given, 8 marks
- Video: [Order of Instruments | Zero Order | First Order | Second Order | Fundamentals of Instrumentation](https://www.youtube.com/watch?v=AvrZHJmR_dw) — MissionMotivateMillionMinds-Prof.K.NagabhushanRaju
- 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-first-second-order-models
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_and_measurement/chapter-4

## Question

Write the differential equations of zero-order, first-order and second-order measurement systems. Define the parameters (static sensitivity, time constant, natural frequency, damping ratio) and give one physical example of each type.

## Answer

A measurement system with input $q_i(t)$ and output $q_o(t)$ is modelled by a linear ordinary differential equation. Its order is the highest derivative of the output.

#### Zero-order system
$$
a_0 q_o = b_0 q_i \quad\Rightarrow\quad q_o = K q_i, \qquad K = \frac{b_0}{a_0}
$$

$K$ is the **static sensitivity**. The output follows the input instantly, with no lag or distortion. Example: linear potentiometer, resistance strain gauge at low frequency.

#### First-order system
$$
a_1\frac{dq_o}{dt} + a_0 q_o = b_0 q_i \quad\Rightarrow\quad \tau\frac{dq_o}{dt} + q_o = K q_i
$$

with $\tau = a_1/a_0$ the **time constant** and $K = b_0/a_0$. It has one energy-storing element. Examples: a thermometer or thermocouple in a fluid (heat capacity and film resistance), RC low-pass filter, a pressure gauge with a long tube.

#### Second-order system
$$
a_2\frac{d^2q_o}{dt^2} + a_1\frac{dq_o}{dt} + a_0 q_o = b_0 q_i
$$

$$
\frac{1}{\omega_n^2}\frac{d^2q_o}{dt^2} + \frac{2\zeta}{\omega_n}\frac{dq_o}{dt} + q_o = K q_i
$$

where

$$
\omega_n = \sqrt{\frac{a_0}{a_2}}, \qquad \zeta = \frac{a_1}{2\sqrt{a_0 a_2}}, \qquad K = \frac{b_0}{a_0}
$$

$\omega_n$ is the undamped **natural frequency** and $\zeta$ the **damping ratio**. Two energy-storing elements exchange energy. Examples: spring-mass-damper accelerometer, bourdon-tube pressure gauge with fluid, U-tube manometer, seismic transducer, RLC circuit.

#### Summary

| Order | Equation | Parameters | Example |
|---|---|---|---|
| Zero | $q_o = Kq_i$ | $K$ | Potentiometer |
| First | $\tau \dot q_o + q_o = Kq_i$ | $K, \tau$ | Thermocouple |
| Second | $\ddot q_o/\omega_n^2 + 2\zeta\dot q_o/\omega_n + q_o = Kq_i$ | $K, \omega_n, \zeta$ | Accelerometer |

## Check yourself

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

1. A zero-order system is described by
   A. tau dqo/dt + qo = K qi
   B. qo = K qi with no lag
   C. a second-order derivative equation
   D. qo = K dqi/dt
   Answer: B. The output follows the input instantly, scaled by static sensitivity K = b0/a0. (Key point: Zero-order)

2. In a first-order system tau dqo/dt + qo = K qi, the time constant is
   A. a0/a1
   B. b0/a0
   C. a1/a0
   D. sqrt(a0/a1)
   Answer: C. tau = a1/a0 and K = b0/a0. (Key point: Time constant)

3. For a second-order system, the natural frequency ωn is
   A. sqrt(a0/a2)
   B. a1/(2 sqrt(a0 a2))
   C. a0/a2
   D. sqrt(a2/a0)
   Answer: A. ωn = sqrt(a0/a2) and ζ = a1/(2 sqrt(a0 a2)). (Key point: Natural frequency)

4. Which pairing of system order and example is correct?
   A. Potentiometer - second order
   B. Accelerometer - first order
   C. Thermocouple in fluid - first order
   D. RC filter - zero order
   Answer: C. A thermocouple in a fluid has one energy-storing element (first order); a potentiometer is zero order and an accelerometer second order. (Key point: Examples)

## Related in Instrumentation and Measurement

- [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
- [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
- [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
- [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
- [Zero drift and sensitivity drift](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Azero-drift-sensitivity-drift) — Instrumentation and Measurement, chapter 3
- [Characteristics and response of zero-order system](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Azero-order-system-characteristics) — Instrumentation and Measurement, chapter 4

## Related in other subjects

- [First and second order finite difference schemes](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Afd-first-second-order-schemes) — Computational Techniques in Civil Engineering, chapter 5, asked 2×
- [Mapping constant damping ratio line to z-plane](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aconstant-damping-ratio-mapping) — Digital Control System, chapter 3
- [Derive maximum overshoot of second order system](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Amp-expression-derivation) — Control Systems, chapter 3
- [Second order constant delay filter](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aconstant-delay-second-order) — Filter Design, chapter 2, asked 4×