AVOIDING PITFALLS IN MEASUREMENT UNCERTAINTY ANALYSIS

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Denis Taylor
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1 VOIDING ITFLLS IN ESREENT NERTINTY NLYSIS Benny R. Sith Inchwor Solutions Santa Rosa, Suary: itfalls, both subtle and obvious, await the new or casual practitioner of easureent uncertainty analysis. This paper describes the ore coon istakes that are ade and gives a prescription for staying out of trouble. worked exaple illustrates the preferred technique. ISO-G ROESS The ISO Guide to the Expression of ncertainty in easureent (G prescribes six steps for the analysis of a easureent: odel the easureent Develop the easureent Equation Derive the ncertainty Equation Deterine the Standard ncertainties alculate the obined Standard ncertainty Deterine the Expanded easureent ncertainty While these steps are designed to help analysts focus on the core of a easureent and to proote consistency in uncertainty analysis, each step offers opportunities to go astray. istake in one step ay cascade, perhaps aplified, to subsequent steps. ESREENT EXLE bsolute plitude ccuracy of a Spectru nalyzer Specification: 0. 7 db, for aplitude levels between 10 db and 50 db. To verify this specification, a two-part procedure is followed. The equipent connections for each part are shown below. In art 1, an accurate reference power level is established with a power sensor. The spectru analyzer easures this reference level. In art, several aplitude-test levels are established with a precision step-attenuator. t each test level, the spectru analyzer

2 again easures the absolute aplitude. The difference between the true power available fro the network and the power indicated by the spectru analyzer is defined as the absolute aplitude accuracy of the spectru analyzer. Figure 1: easure the reference power with a power eter. Figure : easure the reference and other power levels with the spectru analyzer.

3 The easureent can be odeled as: ODEL THE ESREENT Gen. able tt. able db dapter D..T. Each piece of equipent used in the easureent is odeled as either a two-port network or a 11 s, s,, and s one-port network. The two-ports can be represented by s-paraeters: 1 1. The signal generator can be represented as an active one-port with a reflection coefficient. The spectru analyzer and the power sensor can be represented as passive one-ports with only a reflection coefficient. ITFLL: Oitting fro the odel any piece of equipent that sees insignificant. It is unwise to eliinate anything fro the odel which is present in the physical network, including cables and adapters, until a easureent equation has been created and an uncertainty equation is derived. ncertainty ters can always be set to zero later in order to check the liits of changes to the network. ESREENT EQTION ITFLL: Not writing a easureent equation at all. Do not be tepted to bypass the writing of a easureent equation and go straight to an assessent of the uncertainty. Without a easureent equation you risk: Not identifying all of the sources of uncertainty in the easureent. Not knowing the sensitivity of the total uncertainty to the individual contributors. s ITFLL: Treating uncertainty as a rando variable separate fro the associated paraeter and including the uncertainty in the easureent equation. EXLE: [ ] [ ] SORE SORE ES Reeber that each variable in the easureent equation is considered to be a rando variable with a known (or assued probability distribution, characterized at the least by a ean value and a standard deviation. dding extra ters to represent the uncertainty (standard deviation of the variable is redundant and will lead to trouble. ES

4 ITFLL: Writing the easureent equation in the wrong doain (log or linear. If the perforance specification being verified is expressed in db units, then the easureent equation ust be written in log ters. If it is expressed in linear units (including percent, the easureent equation ust be written in linear ters. Once the easureent equation is written, it can be transfored fro one doain to the other (see exaple below in order to expand the original equation into a ore detailed version that accounts for all of the influence paraeters (cable loss, attenuator settings, reflection coefficients, etc. NOTE: There are two good reasons why the easureent equation should be transfored to the linear doain (if it is not written there in the first place: In the linear doain, the easureent equation will often have the for will allow writing the uncertainty equation by inspection. (See exaple below. Y B, which D E F The ters in the uncertainty equation will have the for ( or %. any instruent specifications are quoted in the for of db or %., which has the diensions of db ITFLL: Incoplete or incorrect easureent equations. Once the atheatical odel (cascade of two-ports, in this exaple is coplete, the developent of the easureent equation, while essy, should be straightforward. EXLE: Specification: plitude 50 Hz: 50 db. 0.7 db, for input signals fro -10 db to - Since the plitude ccuracy specification is given in db, the easureent equation will be expressed in db. db EQN. 1 V where: is the absolute aplitude accuracy V is the absolute power available fro the effective source is the power indicated by the spectru analyzer This equation siply states that aplitude accuracy is the difference between the true power available at the spectru analyzer s input and the power indicated by the spectru analyzer.

5 The expression for where: V [ ] gz0 G V is given by a signal-flow-graph analysis of Figure EQN. [ gz0 ] G is the power available fro the signal generator., are the gains of the external attenuators., 1, are the gains of the two Type N cables and the adapter. 1 1 Γ s is the isatch between the source and the first Type N cable. g 11a 1 Γ s a attenuator. 11b is the isatch between the first Type N cable and the coposite 1 b 11c cable. Γ s is the isatch between the coposite attenuator and the second Type N Γ s c 1 11d is the isatch between the second Type N cable and the db attenuator. 1 Γd s11 e is the isatch between the db attenuator and the adapter. Notice that the expression for V is in the linear doain, whereas the easureent equation is in the log doain. Rather than convert the expression for V to the log doain, the easureent equation will be converted to the linear doain: V LIN EQN. 3 (Subtraction in the log doain is equivalent to division in the linear doain. For the reference aplitude level of this easureent (-10 db, the power sensor deterines V, the available power. The spectru analyzer akes its own reading of the reference power.

6 Therefore, the easureent equation at the reference aplitude is: ] S [ ] gz0 S V EQN. 4 where [ 0 is the power that the network would deliver to a atched load, as estiated by gz the power sensor/power eter. In ters of the signal source and the intervening network, the easureent equation for the reference level easureent is: t the test power levels: [ ] gz0 G T V 1 1 Since V [ gz0 ] and V [ gz ] T V V T G T 0 G EQN. 5 where: T 1 1 and T 1 1 Therefore: V EQN. NOTE: Since the final coaxial adapter has been deleted for the test connection to the spectru analyzer, the ters and do not appear in EQN.. (If an adapter is needed, the ters can be re-inserted into T.

7 , and 1 Since do not change throughout the test, V 1 1 EQN. 7 Define: [ ] [ ] [ 1 ] [ 1 ] Then: [ ] V S EQN. 8 The classic ISO-G uncertainty equation is: NERTINTY EQTION ( S ( V S ( T ( ( ( ( R T R where: T,, etc. ITFLL: artial differentiation in deterining sensitivity coefficients. It is possible to avoid partial differentiation if the easureent equation has the for: Y B. D E F The uncertainty equation can be written as: ( ( ( ( ( ( ( ( ( ( ( ( (

8 This equation contains relative uncertainty ters, each of which is diensionless. This special for of the general ISO-G uncertainty equation is very handy for any uncertainty analysis situations. STNDRD NERTINTY ITFLL: onfusing Type and Type B uncertainty. Type evaluation of easureent uncertainty requires the taking of data fro repeated observations of the subject paraeter. For exaple, the D voltage accuracy of a particular odel of DV could be established by easuring a voltage standard repeatedly, under the sae circustances, with several DV units. The ean and standard deviation of the resulting data would provide a noinal value (ean and an uncertainty (standard deviation of that value. Type B evaluation of easureent uncertainty is obtained by applying scientific judgent to all of the available inforation about the paraeter in question, including anufacturer s data sheets, previous easureent data, etc. NOTE: In coon verification testing using off-the-shelf test equipent and standards, Type B evaluation of uncertainty is prevalent. EXLE: Type Evaluation of uncertainty Fro the uncertainty equation, evaluate the standard relative uncertainty: ( (. The relative uncertainty of the attenuation value of the coposite attenuator is taken fro the calibration report for the attenuator. The exaple data shown below is typical of that obtained fro a standards laboratory. ssue that the calibration data has a gaussian probability distribution and that the values given represent the s (or 95% confidence liits. recision ttenuator ccuracy Data at 50 Hz oposite tten. Setting (db tten. fro al. Report (db tten. al. ncertainty (± db

9 σ al. nc. ( db db ± ± 0. L 05 onverting to the linear doain (since the uncertainty equation is written in the linear doain: ( ± ITFLL: How to deterine the underlying distribution and the confidence level for a perforance paraeter value taken fro an instruent s data sheet. EXLE: Type B Evaluation of uncertainty The true power delivered to the spectru analyzer is estiated by the power eter. The equation that describes the available power interns of the power eter and power sensor paraeters is: db V S IKL where is the power indicated by the power eter. The relative uncertainty of this paraeter is: ( The power eter reading is a digital value, delivered via the front-panel display or via the GIB. The resolution of this reading is selectable. ssue for this exaple, that the resolution is 0.1 db. ny power reading that falls within one of the 0.1 db intervals will be given the sae reported value. Therefore, a reported value could be due to any value within the interval. This situation is best odeled by a unifor distribution. For a unifor distribution of width 0.1 db, the standard deviation is: σ onverting to linear ters: 0.1dB 1 SLE 0.03dB. ( ± ± NOTE: For ost specifications given in anufacturer s data sheets, assue that the value given represents at least 95% of the population of values. 95% is equivalent to s for a gaussian (noral distribution and 1.45s for a unifor distribution. hoose an assued distribution based on the nature of the specification. ost specifications can be treated as if they are taken fro a gaussian distribution.

10 ITFLL: Do data sheet values represent ( or (? ( will always have diensional units: volts, ohs, hertz, etc. ( will always be diensionless. It will be specified as %, db, or as a ratio. OBINED STNDRD NERTINTY Once a value is obtained for each standard uncertainty represented in the uncertainty equation, the cobined standard uncertainty can be found. NOTE: The uncertainty equation has the for: σ σ σ σ σ... EQN.9 B D Each of the squared ters in this equation represents a statistical variance. Standard deviation (a.k.a, standard uncertainty is defined as the square-root of the variance, regardless of the statistical distribution involved. EQN. 9 states that the cobined variance of the response paraeter (absolute aplitude accuracy, in this exaple is equal to the su of the individual variances of the contributing paraeters. Recalling the uncertainty equation: ( ( ( ( ( ( ( ( ( ( ( ( ( ITFLL: The units of each standard-uncertainty ter are not diensionless, or are not consistent with one another. Since the uncertainty equation is written in the linear doain, each standard-uncertainty ter (e.g., ( will be expressed as a decial nuber or as a %. learly, the ters ust all have the sae units: either decial or %. The cobined standard uncertainty, (, represents the standard deviation of the ( coposite rando variable,. If there are three or ore contributors to the resulting distribution will be approxiately gaussian, regardless of the distributions of the contributors., then

11 ( EXNDED NERTINTY is a linear quantity. The original specification for bsolute plitude ccuracy is a logarithic quantity (db. onverting EXLE: ( ± 0.01 ( ( back to the log doain gives ( ( db 10 log 1 ± 10 log ± 05 ( σ ± 0. db db 05 ( 1± db ITFLL: hoosing the correct coverage factor for the final uncertainty value. in db. Now that the cobined standard uncertainty is back in the log doain, it is a siple atter to deterine a coverage factor that, when ultiplied by the cobined standard uncertainty, will give the desired degree of confidence that the true value of bsolute plitude ccuracy is contained within the liits ( ±. db For a gaussian distribution, a coverage factor of will provide 95% coverage (i.e., σ. The expanded uncertainty becoes: ( db. db ( ± 0. db EX db 10

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