Meteorological Instrument Performance Characteristics

### Dynamic Response » Transfer Functions and the Transfer-Function Diagram » First Order vs Second Order Systems

Sensors differ in their response characteristics, but many follow general patterns. A first-order system is one that can be described by a first-order differential equation. A second-order system requires a second-order differential equation for its description. The thermometer is an example of a first-order system in temperature that responds directly to the heat flow induced by the difference in temperature between the sensor and the bath. A wind-measuring system consisting of a wind vane and a cup anemometer, as shown in the next illustration, involves both first-order and second-order responses. The rotor in the cup anemometer has angular momentum that resists change when the wind changes, so it is basically a first-order system in velocity with no preferred orientation or position. An exponential change between the initial and final states describes the solution to a first-order system. The wind vane is a second-order system in position that will tend to oscillate about the equilibrium position when the wind direction changes. Thus, second-order solutions include sinusoidally-varying components that describe the oscillatory nature of the system, one which often includes damping, which is represented by a damping coefficient. Many other commonly used systems fall into one of these two categories. A wind measuring system consisting of a cup anemometer is an example of a first-order system with a response to changes in angular momentum, and a wind vane represents a second-order system where the kinetic energy is stored in the angular motion of the vane and potential energy stored in the position of the vane relative to the wind vector. Image from R. Clark.

#### Question

Drag each of the following to the type of system (first order or higher order) it represents.

Derivative of the signal
with respect to time
Exhibits no energy loss
Wind vane
Thermocouple,
or rotors on a cup anemometer
Second derivative of the
signal with respect to time
Exhibits energy loss

The correct answers are shown below.

First Order System

Derivative of the signal
with respect to time
Thermocouple,
or rotors on a cup anemometer
Exhibits energy loss

Second Order System

Exhibits no energy loss
Wind vane
Second derivative of the
signal with respect to time

#### Question

Which of the following is(are) true of dynamic performance characteristics for a first-order system?

The correct answers are a and b.

In a first-order system, only slowly varying input fluctuations will be passed through to the output, while rapidly varying input fluctuations will be dampened or attenuated.