Question

The output of a potentiometer is measured by a voltmeter having a resistance Rm, as shown in Figure 3.12. R, is the resistance of the total length X, of the potentiometer and R, is the resistance between the wiper and common point for a general wiper position X/. Show that the measurement error due to the resistance R of the measuring instrument is given by R (R-R;) Error = E R.(R;R + R„R, -R) Hence show that the maximum error occurs when X; is approximately equal to 2X,/3. (Hint-differentiate the error expression with respect to R; and set to 0. Note that maximum error does not occur exactly at X;= 2X,/3, but this value is very close to the position where the maximum error occurs). Figure 3.12

Answer 1

Solution 42 Error tre given by E = RPC RiRi) Rtfrit PL + Rim Re - Rit] RTRL Here from om the given circuit

t h ere from the Effres Circuit EG Rm. E measured = E Rill Rm+ RE-RE Ž Rm E measured . E. Rill Rom 1. Rillemt Re-Ri q ERI Eartual SERI

Now the error. E = Eactuali tmeasured Then we get 100 . Eri Erm Ri Rt R+Rm-R tre -R, 2 we have get this Equation by Cross multiplying ERR (R+Rm - R+Ri - R2) Rt (ERM Rig ERi (R+Rm+R+ Ri -Ri-Rm RB (RE) (R+RM-R TRI-Riz) Then ERP (R+ Ri) Rt (R+Rm - Rt Ri Ri - Hence proved for Enor d enor =0 then dri => Rt le+Rm -P+ Rin Ria) (Peri (R + _R.) + Erice) - ERI(RE-Ri) (-2R:RE)=0 ->(Rť" Rm-e** P; -Ri-)(ZERIRE- ZER: 2-fr. & ERI3R +(RE-R) = 0

->PREPRm Ri - Ri Rtº-2RiP RE-3R ;?RtRm +3RIER++3pill +2R; BR; 2 2R," Ri rally Rt = 2 Rt = 2 Ri