Models of zero, first and second order systems

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