ELE427 - Testing Linear Sensors. Linear Regression, Accuracy, and Resolution.

Transcription

1 ELE47 - Testng Lnear Sensors Lnear Regresson, Accurac, and Resoluton. Introducton: In the frst three la eperents we wll e concerned wth the characterstcs of lnear sensors. The asc functon of these sensors s to convert a echancal propert to an electrcal sgnal. For nstance, the potentoeter wll take the angular poston of ts shaft as an nput and convert ths to voltage at ts output. Lnear sensors are those devces for whch a plot of the output versus the nput fors a straght lne. Thus, for an deal potentoeter, knowng the voltage output ou could easl calculate the poston nput. Ths would ake the potentoeter a ver good sensor. Unfortunatel, real sensors suffer fro a nuer of real proles that cause the nput/output relatonshp to e less than deal. Electronc nose, frcton, endng of phscal structures, and looseness n couplers and earngs all contrute to the non-deal ehavor oserved n the la. When ou desgn a product that ncludes a sensor, ou wll need soe wa to know f the sensor ou choose s gong to e good enough. If ou choose a cheap sensor, t a degrade the perforance of our entre product. Conversel, f ou choose a hgh qualt sensor, the cost a ake our product less copettve n the arketplace. Accurac s a quanttatve easure of how closel a real sensor approaches the deal lnear odel. The IEEE, ISA, and SAMA organzatons defne accurac as havng three coponents: lneart, repeatalt, and hsteress. Manufacturers wll usuall nclude soe or all of these easureents n ther descrptons of a sensor ut t s wse to otan a saple sensor and test t ourself. The three ccle test s the accepted ethod of easurng accurac. Ths test requres easurng the nputs and outputs of the sensor over the entre operatng range of the devce. Start wth the nput at one etree and var t untl the other etree s reached takng 0 data ponts along the wa. Then return the nput to the orgnal poston takng data at the sae 0 ponts as ou go. Ths akes one ccle of data, one end to the other and ack. Repeat the sae procedure to collect two ore ccles of data. Use the sae nput ponts for all three ccles. It s useful to ake a tale for recordng our data. Lael the coluns so ou wll know whch data was taken when the nput was ncreasng, and whch data was taken when the nput was decreasng. Lnear Regresson: Snce we have assued that the data we collected s fro a lnear sensor, t sees logcal to ask, What s the one lne that ths fts data est?" More atheatcall, we are lookng for a lne,, that passes as close as possle to the data ponts we collected, (, ),,,.... We know, and. We want to fnd the slope,, and the -ntercept,. In order to fnd these unknowns, we need to e ore precse aout what s eant as close as possle. Consderng an nput, the lne requres that the output should e, ut the actual data pont s. Thus the vertcal dstance etween the deal lne and the actual pont s,

2 dst ( ) To get a easure of the dstance etween all the data ponts and the deal lne, we use the su of the squares of the ndvdual dstances. E [( ) The square s used to prevent negatve dfferences fro cancelng postve dfferences gvng an unwarranted presson of accurac. In ths equaton E s referred to as ean square error etween the data ponts and the deal lne. Eaple: 3 Ccle Test Data Input Up Down Up Down Up Down Input-Encoder counts, Output-volts Our goal s to fnd the paraeters and that nze the ean square error. To do ths we wll use the fact that the nu of a functon occurs when the frst dervatves equal zero. Because E depends on oth and, we wll take partal dervatves wth respect to and. ] δe δ [( ) ] δe δ [( ) ] Settng the dervatves equal to zero and epandng, we otan:

3 0 ) ( 0 ) ( Reorderng the ters, we recognze two sultaneous equatons n two unknowns: or, n atr forat: Usng deternants (Craer' s rule) ou can solve for and. Although ths looks ver ess, the calculatons are not that ad (especall f ou use MATLAB). Lets consder the eaple data set shown n the prevous tale ote that the su of s not just the su of the nput values shown on the tale. It s s tes the su of the tale values ecause each value shown actuall corresponds to s values. Usng these values we fnd that, 0.03 and Ths est-ft lne and the actual la data are shown n Fgure.

4 Lneart: Once we have found the est-ft lne, t s now possle to consder a quanttatve easure of how closel the sensor approaches the deal lnear ehavor. Specfcall, we want to otan a sngle nuer that characterzes how far the data les fro the lne. There are several was that ths s coonl done. One ethod s called the worst case lneart. In ths case we are nterested n the data pont that les furthest fro the deal lne. Frst, ou need to calculate where the deal lne s for each of the nput data ponts that ou used when ou took the data. Then fnd the dfference etween these deal outputs and the actual la easureents. The worst case lneart s just the largest (agntude) of these dfferences. In our eaple the worst case lneart occurs on the thrd easureent of the frst ccle when the nput s 60 counts and the output s.35 volts. The deal lne passes through (60) , and so the dfference s volts. However, lneart s never reported n unts of volts. Instead, t s reported as a percentage of the full-scale operatng range of the sensor. The etrees of the deal lne are at volts and volts ( ) so the full-scale s 9.84 ( ) volts. Thus the worst case lneart s / or 9.8%. A second coon ethod of calculatng lneart s the ean square lneart. As the nae suggests, ths nuer s otaned squarng the ndvdual dfferences etween the deal lne and the data ponts, and then averagng the squares. Agan, the result s reported as a percentage of the full-scale output of the sensor. Calculatng ths quantt for the eaple data set we fnd that the ean of the squares s and so the ean square lneart s 0.339/ or.36%.

5 Repeatalt: Recall that the second part of accurac s repeatalt. Ths refers to the alt of the sensor to eld the sae output an te t receves the sae nput. One of the reasons for takng three ccles of data s to test whether ou get the sae output each te ou appl the sae nput. Slar to lneart there are several ethods whch repeatalt s calculated. Worst case repeatalt refers to the largest dfference etween two outputs taken wth the sae nput and gong n the sae drecton. Ths eans ou are nterested n the largest dfference etween the three outputs ou otaned for at each nput pont as the nput was ncreased or decreased. We are not nterested n the dfference etween outputs that were taken wth the nput gong n dfferent drectons, nor do we care where the deal lne les. Ths s just a easure of whether the sensor wll repeat ts prevous outputs when ou gve t the sae nputs. For our eaple data set the largest dfference etween two outputs wth the sae nput n the sae drecton occurs etween the frst and second ccle n the decreasng drecton when the nput was 30. Durng ccle the output was 5.6 volts, durng ccle the output was 3.53 volts for a dfference of.63 volts. Lke lneart, repeatalt s reported as a percentage of the fullscale output of the sensor. Thus the worst case repeatalt s.63/ or 6.6%. A ean square repeatalt could e found takng the dfferences etween all the ponts otaned wth the sae nput n the sae drecton, squarng each ndvdual dfference, and then averagng all the squares. Usng ths ethod, our data set has an average of squares of whch elds a ean square repeatalt of.58%. Hsteress: The thrd easure of accurac has to do wth the ehavor of the sensor when the sae nput s approached fro opposte drectons. Here we are nterested n the dfference etween outputs that occurred as the nput was ncreasng and outputs that occurred as the nput was decreasng. The locaton of the deal lne doesn't atter. Worst case hsteress s the largest dfference etween two outputs taken wth the sae nput and gong n opposte drectons. Ths occurs etween the frst ccle ncreasng and the frst ccle decreasng when the nput was 30 counts. The outputs were 3.79 volts and 5.6 volts respectvel for a dfference of.37 volts or 3.9% of full-scale. Lke the other coponents of accurac, hsteress s alwas reported as a percentage of the full-scale output of the sensor. We could calculate a ean square hsteress the sae ethod used to get ean square repeatalt. A ore llunatng procedure ght e to fnd est ft lnes. One usng onl data taken wth the nput ncreasng, and the other usng onl data taken wth the nput decreasng. Perforng ths calculaton on our eaple data set we fnd that, up 0.04, up 0.363, whle, down 0.0, down Fgure shows the actual data and these two est ft lnes. It can e seen that there s a dstnct dfference n the outputs dependng on whch drecton the nput was ovng.

6 Resoluton: Although a sensor ght e ver accurate wth respect to lneart, repeatalt, and hsteress, t a stll not e sutale for soe applcatons. Resoluton has to do wth the alt of a sensor to dscern fne detal. We defne resoluton as the sallest change n nput that can alwas e detected at the output. Consder an optcal encoder that has 000 wndows. Such an encoder wll generate 000 counts ever te t turns 360 degrees. Ths eans that f ou turn the encoder 360/ degrees or ore ou wll alwas generate a count at the output. So, we would sa that the resoluton of ths encoder s 0.36 degrees. otce that resoluton s an asolute quantt, not a relatve percentage. It wll alwas have the sae unts as the nput to the sensor. A potentoeter that s set up as a voltage dvder s another coon stuaton. In ths case, an oveent of the nput shaft wll change the output voltage. Ths devce sees to have an nfnte resoluton! However, for the potentoeter to e of an use the output voltage ust e easured. So the lt of resoluton for the potentoeter sste s defned how eactl ou can easure the output voltage. If our volteter can easure a one llvolt change, and the potentoeter voltage changes one volt per 3 degrees, then we would sa that the resoluton of the potentoeter/volteter sste s (0-3 volts)(36 degrees/volt)0.036 degrees. Last update: C. Turtle, August, 000