Meteorological Instrument Performance Characteristics

Dynamic Response » Transfer Functions and the Transfer-Function Diagram » Dynamic Responses

The transfer function diagram can be used for dynamic systems of higher order. An example is the simplified response of a wind vane to the direction of the wind. [Note that a wind vane is actually a much more complex system and the following simplification is not realistic except as an example of a second-order dynamic system. For a real wind vane, the restoring torque becomes nonlinear for any but very small deflection angles, varying wind speed changes the restoring torque, and there are aerodynamic contributions to the drag term and other complicating factors.]

In this simplification, two factors control how the vane will respond to the direction of the wind when the wind speed remains constant. First, a difference between the orientations of the vane and the wind will produce a torque proportional to that difference, with proportionality constant k. However, such a system will tend to oscillate about the correct orientation, like a weight suspended on a spring. Therefore a wind vane will have some resistance to motion built into its mounting structure, and this resistance provides damping of the oscillation.

A schematic representation of the transfer function of this system is shown in the figure, where the amplifier with gain D provides the damping effect of this feedback.

General schematic of a transfer function for a damped, forced harmonic oscillation. Dots over M represent the order of the time derivatives

General schematic of a transfer function for a damped, forced harmonic oscillation. Dots over M represent the order of the time derivatives. Image from NCAR/EOL.

This schematic shows a damped forced harmonic oscillator, where k is analogous to a spring constant but in this case is the proportionality constant between the restoring torque and the angle between the orientation of the wind and of the wind vane. The symbol m in the amplifier with gain 1 / m is analogous to mass for the loaded spring but here represents the moment of inertia of the wind vane. The response to changes in wind direction is determined by the relative magnitudes of the parameters {k, the spring constant; m,the mass; and D, the damping constant}. If the damping constant is very small, the vane tends to oscillate too much, while a large value of D causes the vane to respond too slowly. “Critical damping,” for which the vane just moves to a new equilibrium position without overshooting, occurs when D = 2√  mk , and the natural oscillation without damping has an angular frequency 𝜔0 equal to √ k/m .

In contrast to the first-order transfer function, which serves as a low-pass filter, the response of a second-order transfer function can lead to erroneous large-amplitude fluctuations near the natural frequency if the damping is insufficient. In situations with large damping, the transfer function again serves as a low-pass filter.

The governing differential equation can be determined from this and other similar transfer-function diagrams, as follows:

Start at the point in the diagram labeled as expression (or whichever term is the highest-order derivative).

Working backward, incorporate each contribution into the differential equation to obtain, in the case of the diagram,

equation

It is then possible to solve the resulting differential equations for specified values of the measurand as a function of time. Another benefit of characterizing the sensor with a transfer-function diagram is that it provides a structure easily incorporated, with initial values, into a numerical solution that gives the response to any input.

To demonstrate how solutions to these equations or the transfer-function diagrams relate to characteristics of instruments, the responses to some specific types of input will be discussed in the next pages.