ASE 369K Measurements and Instrumentation Performance Characteristics of Measuring Systems Measuring instruments must respond “faithfully” to changes in the signal and produce appropriate output. Why is this important? Consider the example: an instrument display such as a tachometer in an automobile; as the engine RPM changes, the needle deflects to show the current value of the RPM as closely as possible. There are three kinds of response of interest – amplitude response, frequency response and phase response. Let and q represent the time dependent input to and the output from the system respectively as indicated in Figure 1. )(tqi)(to )(tqi)(tqo System Figure 1. Amplitude response addresses the issue of linearity of the input-output relationship. Figure 2 shows a sketch of how the ratio of the output to input might vary with the magnitude of the input. For small values of the input, the output is proportional to the input, but as the input amplitude increases, the output is unable to keep up with the proportional increase and the output drops off. Note that the input and output do not always have to be linearly related, but the relationship should be known. We will encounter an example when we talk about a displacement measuring transducer; the value of the displacement at which the output begins to drop-off defines the useful amplitude range of the measuring instrument. )()(tqtqio)(tqiFigure 2 It is also necessary to determine the response of the system to variations in the frequency content of the input signal in cases where the interest is in measuring transient signals. Typical frequency response for a measuring system is shown in Figure 3. This is what would be desirable; the amplitude ratio should be independent of the input frequency, but of course the output cuts off after a certain frequency; this frequency defines the useful frequency range of the measuring instrument. )()(tqtqioωFigure 3 Professor K. Ravi-Chandar 1ASE 369K Measurements and Instrumentation The output signal q should track the time variations of the input ; time delays in the tracking are called phase lags; a corresponding phase response can be defined. )(to)(tqi In general, the relationship between the input and the output can be written in the form of a differential equation: )()(01222tqtqadtdadtdaio=++ (1) The constants are the parameters that characterize the system and must be known; we will see examples of these later. The output response of the system is written as a second order partial differential equation; higher order equations can be written, but most measuring systems do not need such a description. Analogies to electrical and mechanical systems are described in the boxes below. Let us divide through by and rearrange the differential equation: ia0a )()(121222tKqtqdtddtdionn=++ωζω (2) where 20aan=ω is the natural frequency of the system, ()2012 aaa=ζ is the damping ratio, 01 aK = is the static sensitivity. There is no reason to study the dynamic performance characteristics of a zeroth order system, so here we will look into first and second order system response. We want to evaluate the response of these systems first to a step input and then to a sinusoidal input. Professor K. Ravi-Chandar 2ASE 369K Measurements and Instrumentation )(tiv)(tov)()(122ttdtdRCdtdLCiovv =++)()(121222ttdtddtdionnvv =++ωζωRCLCRLCnn====ωζτζω22,12RC=τnωζElectrical Analogy to Second Order Systems The relationship between the input voltage and output voltage for an RLC circuit shown below is given by a 2nd order differential equation: R L + C - 1. If L and R are zero, we get a zeroth order system. The voltage across the capacitor is equal to the stored voltage. 2. If L is zero, we get a first order system, with a time constant, . 3. For an arbitrary combination of RLC, we have a second order system, with defining the system natural frequency and defining the damping ratio; the time constant is not relevant in this case. Professor K. Ravi-Chandar 3ASE 369K Measurements and Instrumentation ktFtxdtddtdnn)()(121222=++ωζωkckmcmknn====ωζτζω22,2kc /=τnωζ)()(22tFtxkdtdcdtdm =++Mechanical Analogy to Second Order Systems The relationship between the applied force F(t) and displacement x(t) for a sping-mass-damper system shown below is given by a 2nd order differential equation: c kmx(t) F(t)1. If m and c are zero, we get a zeroth order system. The deflection across the spring is proportional to the applied force. 2. If m is zero, we get a first order system, with a time constant, . 3. For an arbitrary combination of m, c, and k, we have a second order system, with defining the system natural frequency and defining the damping ratio; the time constant is not relevant in this case. Professor K. Ravi-Chandar 4ASE 369K Measurements and Instrumentation First order system We eliminate the second derivative term in Eq. (1) and end up with a first order equation: )()(1 tKqtqdtdio=+τ (3) where 01 aK = is the static sensitivity and 01aa=τ is the time constant of the first order system. These two parameters characterize the system; the importance these terms will be clear in examining solutions to the first order system. Consider a step input function q, where is the steady state amplitude and is the Heaviside step function or unit function defined as follows: )()( tHqtisi=isq)(tH (4) ≥≤=0for10for0)(tttH The general solution to the differential equation is: ([)]τtKqtqiso−−= exp1)( (5) A sketch of this solution is shown in Figure 4. For a step input, the output of the first order system displays a gradual response, reaching 63% of the steady-state value at time τ=t, 95% at τ3=, and % at t99.3τ5=t. The im tion of this result is that a first order system will not respond instantaneously to changes in the input, but will respond with a delay; if you consider a 5% error in the output response to be a tolerable precision error, then the first order system can provide adequate response for plicaτ3>t; or at shorter times will be larger. /(Kqis)00.511.5012345t/τ qoFigure 4the err Professor K. Ravi-Chandar 5ASE 369K Measurements and Instrumentation Now consider a signal that varies sinusoidally at a frequency ω: )sin()( tqtqisiω=. The solution to the