Wheatstone Bridge Calibration for Strains

Transcription

1 M8 Wheatstone Brdge albraton Wheatstone Brdge albraton for Strans Background The stran lab uses two stran gages ounted on the top and botto surface of a bea n bendng to easure the surface stran due to bendng. In the process of calbratng the brdge a calbraton resstor s used to relate a known resstance change to an equvalent stran. Ths docuent descrbes the theoretcal background behnd ths calbraton process. Wheatstone Brdge The Wheatstone brdge conssts of resstors connected n a confguraton as shown on the rght. cross two opposte connecton ponts ( and B a voltage s appled (also called the exctaton voltage. We can ake the followng assupton: (hgh pedance volteter B o Usng Krchhoff s nd law, whch states that the su of the voltage n a closed loop has to be zero, we can wrte for the loop, whch goes fro the exctaton voltage through resstors and :, therefore ( or the loop gong through resstors and we get D - Wheatstone Brdge - ; therefore ( ro these equatons the unknown currents have been deterned. However, we are nterested n the voltage dfference between ponts and D ( the output voltage of the brdge,, whch we want to relate to resstance changes n the brdge. Ths can be found fro a rd voltage loop across resstors and and. Solvng for the output voltage and substtutng the earler results for the currents gves an equaton for the brdge output voltage as a functon of the resstors and the nput exctaton voltage. or Ths equaton can be used to deterne, how a resstance change n the brdge changes the output voltage. HD Page /5

2 M8 Wheatstone Brdge albraton balanced brdge has zero output voltage. Balancng s acheved by choosng resstors that ake the expresson go to zero. Ths s equvalent to or Wheatstone Brdge wth two stran gages The Wheatstone brdge s now odfed so that resstor and are stran gages. eeber that a stran gage produces a resstance change proportonal to stran. The proportonalty factor G s called the gage factor and depends on the stran gage ateral. G L L G G G We can therefore wrte for the resstance change n the top and botto stran gage G G - G G ( G ( G G G where G s the nonal (unstraned gage resstance. Substtuton of these results nto the equaton for the brdge output voltage yelds G ( G G ( G where s the nonal value of and. Ths equaton s non-lnear n the stran. However, for sall strans, we can lnearze ths equaton around usng a Taylor Seres. ( G G G B G G D - o where we only kept the ter lnear n. Hgher order ters are sall because of the sall strans. If we choose the resstors n the Wheatstone brdge to have the sae nonal resstance as the stran gages, G, - Wheatstone Brdge wth two stran gages the expresson splfes even ore to G ( for G HD Page /5

3 M8 Wheatstone Brdge albraton Therefore for a stran gage brdge wth nonal resstors beng dentcal on all ars, the rato of output to exctaton voltage wll be sply the stran * gage factor /. Note that n the lab the ars do not have the sae resstance. Note that snce typcal strans are sall and exctaton voltages have to be kept low to anage current flow and power dsspaton n the crcut, very sall voltages have to be easured relably. Lab Brdge confguraton In the lab a coercal brdge aplfer s used. It has adjustable resstors to balance the brdge, calbraton resstors and an aplfer to ncrease the agntude of the brdge output. The wrng dagra of the brdge crcut s shown on the rght. In order to deterne the equvalent resstance n ths crcut for branch - and - D, we have to add the resstors shown. or a balanced brdge wth unstraned stran gages we know that ½ of the adjustable kω resstor s on ether sde of the brdge crcut ars - and -D. ollowng the rules for resstor addton we get.5, 5, The stran gage has a nonal value of G.5Ω and a gage factor G.9.Ω Wheatstone Brdge onfguraton n Lab? kω 5kΩ.5Ω esstor Detal ( to Usng the results fro the prevous secton we have G ( G Ω.5Ω (.5Ω Ω.9 G.77 The above equaton establshes the relatonshp between stran, brdge exctaton voltage and easured output voltage. The adjustable brdge aplfer n the BM further aplfes wth an undeterned gan factor. What s eventually easured wth the data acquston s DQ, where DQ G pl, where G pl s the aplfer gan. HD Page /5

4 M8 Wheatstone Brdge albraton Lab Brdge confguraton wth albraton esstor When the calbraton (L button s pressed, an addtonal resstor s added to the - branch of the Wheatstone brdge. The wrng dagra of the brdge crcut s shown on the rght but now wth the L button pressed. Ths changes the equvalent resstance n the - branch crcut only but not n the -D branch. The balance n the brdge between - and -D wll produce an output voltage. ro before we know that. Ω ollowng the rules for resstor addton for the - branch we get gan we ake use of the fact that the calbraton resstor wll be consderably larger than. Therefore we can do a lnear Taylor Seres expanson to approxate. Wheatstone Brdge onfguraton n Lab? kω 5kΩ.5Ω esstor Detal ( to wth c The equaton for the output voltage becoes G G The calbraton resstor s large copared to the brdge resstance, therefore we can assue that <<. s a shortcut we wrte ρ. The Taylor seres approxaton for the brdge output n ters of the noralzed (sall calbraton resstor rato ρ then becoes G ρ ( ( G G G Ths equaton can be used to deterne the output voltage for known nonal brdge and gage resstance G and known calbraton resstor. HD Page /5

5 M8 Wheatstone Brdge albraton oparson of Brdge Output due to stran vs calbraton resstor So far we found that the brdge output voltage vares wth stran accordng to G G brdge output to stran ( G and f no stran s appled but a calbraton resstor s nserted nto the - branch of the brdge G brdge output to calbraton resstor ( G The next queston we ask s, what knd of stran cal would be requred to produce the sae output that we get by addng the calbraton resstor to the crcut. Note that n the lab we also have an aplfer wth adjustable gan, but the aplfcaton s the sae whether the brdge voltage occurs due to stran or a calbraton resstor. Therefore we can copare the voltages ether before or after aplfcaton. We set Stran alesstor whch (snce the exctaton voltage s constant s equvalent to Stran alesstor Therefore G G G al ( G ( G Solvng for the stran as a functon of the calbraton resstor we get al G Ths stran, cal, s the stran whch s equvalent to an appled calbraton resstor of agntude. In the lab the nonal cal resstor ( s MΩ, G.9 and was deterned to be Ω. Therefore the equvalent stran s al Ω Ω 6 n n The table n the lab notebook lsts a value of , whch ore closely corresponds to.5ω. But ths s a nor dfference (.%. or dfferent settngs of the L swtch (, dfferent cal resstors can be engaged. The general expresson for the BM s MΩ whch can be used to deterne the equvalent stran based on the settng of the swtch. correspondng table for dfferent values of s gven n the lab anual. HD Page 5/5