Aerospace Engineering Laboratory I  Basics for Physical Quantities and Measurement  Physical Quantity  Measured Quantity VS Derived Quantity  Some Terminology  Measurement / Measure  Measurand / Measured Variable  Instrument / Measuring Instrument / Measurement System  Physical Principle/Relation of An Instrument and Sensor and Sensing Function f s  Measurement System Model  Input-Output relation: y = f ( x ; …)  Important: Identify MS, input x, and output y clearly

2  Theoretical Input-Output Relation and Theoretical Sensitivity  Linear Instrument VS Non-Linear Instrument  Examples:  Example 1: Define MS, Input x, Output y Clearly  Example 2:Redefine our measurement system for convenience  Example 3: Find the theoretical input-output relation and the theoretical sensitivity  Example 4: Theoretical sensitivity and how to increase sensitivity in the design of the instrument

3  Some Common Mechanical Measurements  Calibration  Static Calibration  Calibration Points and Calibration Curve  Calibration Process VS Measurement Process  Some Basic Instrument Parameters  Range and Span  Static Sensitivity K  Resolution  Some Common Practice in Indicating Instrument Errors

4 Basics for Physical Quantities and Measurement

5 Physical Quantity Describing A Physical Quantity  In an experiment, we want to determine the numerical values of various physical quantities. Physical quantity  A quantifiable/measurable attribute we assign to a particular characteristic of nature that we observe. Describing a physical quantity q 1.Dimension 2.Numerical value with respect to the unit of measure 3.Unit of measure

6 Measured Quantity VS Derived Quantity The Determination of The Numerical Value of A Physical Quantity q must be either through  Measurement with an instrument  Measured quantity or  Derived through a physical relation  Derived quantity (and by no other means) Because of existing physical relations/laws, we don’t want anybody to make up any number for a physical quantity.

7 Some Terminology Measurement / Measure  The process of  quantifying, or  assigning a specific numerical value corresponding to a specific unit (of measure) to a physical quantity q of interest in a (real) physical system. Measurand / Measured Variable  The physical quantity q that we want to measure, e.g., velocity, pressure, etc. Instrument / Measuring Instrument / Measurement System  The physical tool that we use for quantifying the measurand, e.g., thermometer, manometer, etc.

8  Often, our desired physical quantity x – measurand – cannot be measured directly (in its own dimension and unit). Physical Principle/Relation of An Instrument ( f s ) and Sensor and Sensing Function f s

9 What is the pressure difference ( p a – p b )? Do we measure the pressure difference ( p a – p b ) directly in the unit of pa with a U-tube manometer? p a pressure at surface a p b pressure at surface b hh mm Class Discussion

10 What is the pressure difference ( p a – p b )?  We do not measure ( p a – p b ) directly.  Instead, we measure  h.  Then, determine the desired measurand ( p a – p b ) from the physical principle/relation (from static fluid) p a pressure at surface a p b pressure at surface b hh mm Measurement System (MS) y = f s ( x ; …) (sensor stage) Input measurand x Output y ( p a – p b ) hh

11  Often, our desired physical quantity x – the measurand – cannot be measured directly (in its own dimension and unit).  We need to determine/derive its numerical value from  another physical quantity y, which is more easily measured, and  a physical relation/principle. Measurement System (MS) (sensor stage) Input measurand x ( p a – p b ) Output y hh x ( p a – p b ) y hh

12 Physical Principle/Relation of An Instrument [and Sensing Function f s ] Physical Principle of An Instrument [and Sensing Function f s ]  The physical principle that allows us to determine the desired measurand x with dimension [ x ] in terms of another physical quantity y s with different dimension [ y s ].  We refer to the underlying physical relation as sensing function f s. p a pressure at surface a p b pressure at surface b hh mm The physical principle of a U-tube manometer is static fluid (fluid in static equilibrium).

13 Measurement System (MS) Input measurand x Output y ( p a – p b ) hh Principle: Fluid Static papa pbpb hh mm Physical Principle: Fluid Statics f s is the sensing function.

14 Measurement System Model

15 Measurement System Model Input x Measurand q in a physical system Output y (Numerical value ) Output Stage Sensor stage (Sensing element) s f Signal Modification Stage m f Physical Principle of The Instrument and Sensing Function ( f s ) We shall refer to  the physical principle that allows the sensor to sense the desired measurand x with dimension [ x ] in terms of another physical quantity y s with different dimension [ y s ] as the physical principle of the instrument, and  the corresponding underlying physical relation as the sensing function f s.

16 Input - Output Relation:  We are then interested in the input-output relation Measurement System (MS) Input measurand x (physical quantity) Output y (physical quantity) How to find the output-input relation - Theoretical - Actual  Static Calibration

17 Important: Identify MS, input x, and output y clearly When considering measurement system (or subsystem) characteristics 1.Measurement System:Identify the measurement system (MS) clearly (physically as well as functionally), from input x to output y. 2.Input Measurand x :Identify the physical quantity that is the input measurand x and its dimension/unit. 3.Output y :Identify the physical quantity that is the output y and its dimension/unit. 4.Calibration Curve:Find and draw the calibration curve ( y VS x ) for the system Measurement System (MS) Input measurand x Output y Note: 1.It helps to identify the dimensions of the input and output physical quantities clearly. Is it length, pressure, velocity, or voltage, etc? 2.Recognize that if there is no output indicator, we cannot yet know the numerical value.  For example, the output of the pressure transducer is voltage output, but without a voltmeter or an output indicator, we cannot yet know the numerical value of this voltage output.  In this regard, e.g., when perform uncertainty analysis, the output indicator must be accounted for as part of the measurement system.

18 Theoretical Input-Output Relation and Theoretical Sensitivity

19 The Theoretical Input-Output Relation and Sensitivity p a – p b (input measurand)  h (output) The input-output relation: The Theoretical Input-Output Relation and Theoretical Sensitivity for U-Tube Manometer MS: U-Tube Manometer Input measurand x ? Output y ? (p a – p b ) [pressure]  h [Length] Define The Measurement System MS (Define the input and the output quantities clearly.) p a pressure at surface a p b pressure at surface b hh mm

20 Theoretical Input-Output Relation and Graph, and Linear VS Non-Linear Instrument slope Linear Instrument  Output y is a linear function of input measurand x.  The slope K is constant throughout the range General Non-Linear Instrument  Output y is not a linear function of input measurand x.  The slope K is not constant throughout the range

21 Sensitivity K Sensitivity Measurement System (MS) Input measurand x Output y If K is large, small change in input produces large change in output.  The instrument can detect small change in input measurand more easily.

22 Example 1: Define MS, Input x, Output y Clearly Note: Here, we define a measurement system in a more general term, based on the interested functional relation.

23  m = density of manometer fluid  a = density of fluid a  b = density of fluid b p a = static pressure at center a p b = static pressure at center b p 1 = static pressure at 1 p 2 = static pressure at 2 h a = elevation at center a h b = elevation at center b h 1 = elevation at free surface 1 h 2 = elevation at free surface 2  h = h 2 – h

24 Example 1 Determine the theoretical input-output relations for the two measurement systems [See Appendix A for the derivation] U-Tube Manometer Input measurand x ? Output y ? p a – p b [pressure]  h [Length] + + Measurement System 1 (MS1) U-Tube Manometer Input measurand x ? Output y ? p 1 – p 2 [pressure]  h [Length] + + Measurement System 2 (MS2)

25 Example 2: Redefine our measurement system for convenience

26 Example 2 Redefine our measurement system for convenience U-Tube Manometer p a – p b [pressure]  h [Length] Measurement System 1 (MS1)  MS1: It is not convenient to measure the change from the two interfaces. We may redefine our output/system.  MS2: Here, it is more convenient to measure the change with respect to one stationary reference point. Measurement System 2 (MS2) Equilibrium position U-Tube Manometer p a – p b [pressure] h [Length]

27 Example 3: Find the theoretical input-output relation and the theoretical sensitivity

28 Example 3 Find the theoretical input-output relation and the theoretical sensitivity The theoretical input-output relation is Measurement System (MS) (p a - p b ) hh

29 Example 4: Theoretical sensitivity and how to increase sensitivity in the design of an instrument

30 Example 4 How can we increase the sensitivity of the manometer? Differential Pressure Measurement - Inclined Manometer Fox et al, 2010, Example Problem 3.2 pp Principle: Static fluid Appropriate sensitivity K can be chosen by changing d/D and sin , e.g.,  Smaller   Higher K

31 Some Common Mechanical Measurements

32 Some Common Mechanical Measurements  Temperature  Pressure  Velocity  Volume Flowrate  Displacement  Velocity  Acceleration  Force  Torque

33 Example 5:Differential Pressure Measurement Inclined Manometer From Dwyer Principle: Fluid Static

34 From Dwyer Inclined Manometer Principle: Fluid Static Input measurand x ? Output y ? p a – p b [pressure] (at the free surfaces) L [Length, mmW] Balance position: apply p a = p b L = 0 Measure position: apply p a > p b L = 0 L (mmW)

35 Example 6:Differential Pressure Measurement Pressure Transducer: Capacitance From Omega: Principle: Capacitance

36 Pressure Transducer (alone) From Omega: Pressure Transducer Principle: Capacitance Input measurand x ? Output y ? p a – p b [pressure] (at the ports) V [Voltage, Vdc]  Without output stage such as voltmeter or output indicator, however, we cannot yet know the numerical value of the output (voltage).  This is not yet a complete measurement system – no output stage. of the pressure transducer alone

37 Pressure Transducer + Output Indicator From Omega: Pressure Transducer + Output Indicator Principle: Capacitance Input measurand x ? Output y ? p a – p b [pressure] (at the ports) V [Voltage, Vdc] Specification/Characteristics (of the measurement system, e.g., accuracy, etc.)  Need to take into account the characteristics (e.g., accuracy, etc.) of the output stage – i.e., output indicator – also.

38 Output Indicator (alone) From Omega

39 Differential Pressure Measurement Inclined Manometer Principle: Fluid Static (Fluid in Equilibrium) Pressure Transducer Principle: Capacitance

40 Calibration  Static Calibration  Calibration Points and Calibration Curve  Some Basic Instrument Parameters

41 Calibration Static Calibration Calibration is the act of applying a known/reference value of input to a measurement system. Objectives of Calibration Process 1.Determine the actual input-output relation of the instrument. 2.Quantify various performance parameters of the instrument, e.g., range, span, linearity, accuracy, etc. The known value used for the calibration is called the reference/standard. Static Calibration: A calibration procedure in which the values of the variables involved remain constant. That is, they do not change with time.

42 Calibration points: y M (x) Static calibration curve (fit): y C (x) If it is linear, y C (x) = Kx + b. Reference: Known value Calibration Points and Calibration Curve  Calibration Points:We first have a set of calibration points.  Calibration Curve:For convenience in usage, we fit the curve through calibration points and use the fitted equation in measurement.

43 Calibration Process VS Measurement Process Reference: Known value Calibration process Measurement process Static calibration curve (fit): y C (x) Reference: Known value

44 Range: Input range: Output range: Span: Input span: Output span: Some Basic Instrument Parameters Range and Span r o = y max - y min y max y min x min x max r i = x max - x min Calibration Full - scale operating range (FSO) = Output span =

45 Static Sensitivity: is the slope of the static calibration curve y C (x) at that point. In general, K = K(x). If the calibration curve is linear, K = constant over the range. Some Basic Instrument Parameters Static Sensitivity K Calibration points: y M (x) Static calibration curve (fit): y C (x) If it is linear, y C (x) = Kx + b.

46 [Output] Resolution (  R y ) is the smallest physically indicated output division that the instrument displays or is marked. Note that while the input may be continuous (e.g., temperature in the room), the indicated output displays are finite/digit (e.g., ‘digital’ or numbered scale on the output display). Some Basic Instrument Parameters Resolution (  R y ) RyRy Indicated output displays are finite/digit Continuous t

47 Some Basic Instrument Parameters Resolution and Static Sensitivity K The smallest change in input that an instrument can indicate. Linear static calibration curve: y C (x) = Kx + b.  R x: Input Resolution Output Resolution:  R y Indicated output displays are finite/digit, the input resolution is correspondingly considered finite.

48 Some Common Practice in Indicating Instrument Errors

49 Some Common Practice in Indicating Instrument Errors  Error e in % FSO:  Error e in % Reading:  Error e in [unit output]/[unit input]:

50 Example: the input range of a pressure transducer is 0 – 10 bar, the output range is V, and the current reading is 3 V (or 6 bar), then  Error: 1% FSO This error is considered fixed and applied to all reading values. That means the % reading error at lower reading values are more than this value.  Error: 1% Reading  Error: 5 mV / bar

51 Appendix A

52 U-Tube Manometer Relation Principle: Static fluid (fluid in static equilibrium) U-Tube Manometer Input measurand x ? p a – p b [pressure] Output y ?  h [Length]

If  b =  b =  f h a = h b Then Eq. (2) becomes If the fluid is gas (while manometer fluid is liquid),