MAE 334 - INTRODUCTION TO COMPUTERS AND INSTRUMENTATION MeasurementSystemBehaviorNotes.docx 1 of 12 9/11/2009 Scott H Woodward 3 Dynamic Behavior of Measurement Systems Order of a Dynamic Measurement System Every measurement system responds to inputs in a unique way. For example, your ability to hear high frequency sounds will probably degrade as you age and will never be as keen as most dogs hearing. Sound pressure waves are a dynamic signal and the sensing of these pressure waves by a flexible membrane (like your ear drum) can be mathematically modeled and therefore simulated. Our goal in this section is to apply our understanding of the physics involved in sensing a signal and build a mathematical model that could be used to describe the response of the measurement system to a dynamic signal. In prior sections we described the response of a measurement system to a static signal and built a mathematical model which described that response. The process of characterizing that response is referred to as a static calibration and the resulting mathematical model is called the static calibration curve. In the first lab you will perform both a static and dynamic calibration of a temperature sensor and determine the corresponding static and dynamic models which describe the sensor response. In the case of a signal that is changing with time (dynamic) a sensor that can keep up, or is fast enough, is needed to accurately detect the change. In the case of the temperature sensors used in the first lab both the sensor and the environment being sensed must be at the same temperature to make an accurate measurement. If the sensor is initially at a different temperature then some amount of time is required for the sensor and the environment to become the same temperature. There has been a dynamic change in the sensor temperature in response to a dynamic change in the input temperature signal. In this example we understand that heat must be transferred from the environment to the sensor. The physics of that heat transfer might be modeled based on our understanding of conduction, convection, radiation or possibly some combination thereof. In general we could reason that the temperature sensor performs some mathematical operation on the input signal and outputs the result. In fact most measurement systems can be modeled using a differential equation that describes the relationship between the input signal and the output signal. In the first lab you will find the linear equation that describes the response to a static input (a static calibration) and the first order differential equation that describes the conductive heat transfer to and from the sensor (a dynamic calibration).MAE 334 - INTRODUCTION TO COMPUTERS AND INSTRUMENTATION MeasurementSystemBehaviorNotes.docx 2 of 12 9/11/2009 Scott H Woodward Figure 3.2 Measurement system operation on an input signal, F(t), provides the output signal, y(t). Measurement System Model If the measurement system operation performed on the input signal, F(t), in figure 3.2 is an nth-order linear differential equation then the output signal, y(t), can be represented with the equation: 11 1 01()nnnnnnd y d y dya a a a y F tdt dt dt (3.1) where the coefficients, a0, a1, a2, …, an represent the physical system parameters whose properties and values will depend on the measurement system itself. The forcing function, F(t), can also be generalized into an mth-order equation of the form: 11 1 01( ) mmmmmmd x d x dxF t b b b b x m ndt dt dt where b0, b1,…, bm also represent physical system parameters. The nature of these equations should reflect the governing equations of the pertinent fundamental physical laws of nature that are relevant to the measurement system. Zero-Order System If all the derivatives in Equation 3.1 are zero then the most basic model of a measurement system is obtained, the zero-order differential equation: 0()a y F t From this equation it is easy to see that any input, F(t), is instantly reflected in the output y with only a factor, a0, modification. If the input is a dynamically varying signal b0x then y = b0/a0x or y = Kx. The factor K isMAE 334 - INTRODUCTION TO COMPUTERS AND INSTRUMENTATION MeasurementSystemBehaviorNotes.docx 3 of 12 9/11/2009 Scott H Woodward often times referred to as the static sensitivity found during a static calibration. First-Order System A linear time-invariant (LTI) first-order system contains a single mode of energy storage. A simple Resister-Capacitor circuit is a first order system. Here the underlying physics is described by the equation outout indVRC V Vdt This circuit is called a single pole low-pass RC filter and will be discussed in greater detail in subsequent sections on signal conditioning and filters. Systems with thermal capacity like a bulb thermometer or thermocouple require heat transfer, Q, from their environment to effect a sensor temperature change. The change in energy, E, with respect to time is described by the first-order equation. ( ) ( )v s sdE dTQ mC hA T t T tdt dt where m is the sensor’s mass, Cv is the sensor’s specific heat, h is the convective heat transfer coefficient, As is the surface area of the sensor, T is the temperature of the surrounding material and Ts is the temperature of the sensor. This can be rearranged as ( ) ( )( ) ( )v s s sv s s sdTmC hAT t hAT tdtdTmC hAT t hA F tdt This can obviously be represented as a first-order differential equation in the form of equation 3.1 as 10()dya a y F tdt To help clarify the underlying physics the equation can be recast by dividing through by a0 and setting y dy dt.MAE 334 - INTRODUCTION TO COMPUTERS AND INSTRUMENTATION MeasurementSystemBehaviorNotes.docx 4 of 12 9/11/2009 Scott H Woodward ()y y KF t where 10aa. The parameter is called the time constant of the system. Reflecting back it is easy to see that the time constant of a single-pole low-pass RC filter is 1/RC and that of a temperature sensor is based on the mass, specific heat, heat transfer coefficient, and the surface area of the sensor, vsmC hA. It is essential that you grasp the insight that the time constant of such systems (LTI) or sensors is based on properties that do not change (under normal operating conditions). I.e. a bulb thermometer does not change in mass when subjected to a temperature change nor does its specific heat, surface area or heat transfer coefficient change therefore its time constant remains constant.