ME-327 Instrumentation and Measurement Lecture 13 Variable Conversion Element Part C By Dr. Muhammad Usman Bhutta usmanbhutta@smme.nust.edu.pk +92-51-9085-6044 Office 215-F Contents 1. Variable Conversion Element 1. Variable Conversion Element • Non-linear relationship between output reading (voltage) and unknown physical quantity is not acceptable • PRIMARY REQUIREMENT OF INSTRUMENT: LINEAR RELATION BETWEEN PHYSICAL QUANTITY AND INSTRUMENT • Change in the unknown resistance 𝛿𝑅1 is typically small compared with the nominal value of 𝑅1 1. Variable Conversion Element • Equation of new voltage 𝑉 ′ 0 when the resistance R1 changes by an amount 𝛿𝑅1 𝑽′ 𝟎 𝑹𝟏 + 𝜹𝑹𝟏 𝑹𝟑 = 𝑽𝒊 − 𝑹𝟏 + 𝜹𝑹𝟏 + 𝑹𝟐 𝑹𝟑 + 𝑹𝟒 𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒗𝒐𝒍𝒕𝒂𝒈𝒆 𝒐𝒖𝒕𝒑𝒖𝒕 𝒊𝒔 𝒈𝒊𝒗𝒆𝒏 𝒃𝒚: 𝜹𝑽𝟎 = 𝑽′ 𝟎 − 𝑽𝟎 𝜹𝑽𝟎 = 𝑽𝒊𝜹𝑹𝟏 𝑹𝟏 + 𝜹𝑹𝟏 + 𝑹𝟐 𝑰𝒇 𝜹𝑹𝟏 << 𝑹𝟏 , 𝒇𝒐𝒍𝒍𝒘𝒊𝒏𝒈 𝒍𝒊𝒏𝒆𝒂𝒓 𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏𝒔𝒉𝒊𝒑 𝒊𝒔 𝒐𝒃𝒕𝒂𝒊𝒏𝒆𝒅: 𝜹𝑽𝟎 𝑽𝒊 = 𝜹𝑹𝟏 𝑹𝟏 + 𝑹𝟐 1. Variable Conversion Element • 𝜹𝑽𝟎 𝜹𝑹𝟏 = 𝑽𝒊 𝑹𝟏+𝑹𝟐 is valid for instruments where typical change in resistance is very small compared to nominal resistance EXAMPLE: Strain gauges • 𝜹𝑽𝟎 𝜹𝑹𝟏 = 𝑽𝒊 𝑹𝟏+𝑹𝟐 is not valid for instruments where change in resistance is comparable with nominal resistance EXAMPLE: Resistance Thermometers 1. Variable Conversion Element • REMEDY METHOD Make the values of the resistance 𝑹𝟐 and 𝑹𝟒 at least ten times those of 𝑹𝟏 and 𝑹𝟑 1. Variable Conversion Element EXAMPLE Consider a platinum resistance thermometer with a range of 𝟎°𝑪 to 𝟓𝟎°𝑪, whose resistance at 𝟎°𝑪 is 𝟓𝟎𝟎Ω and whose resistance varies with temperature at the rate of 𝟒Ω/°𝑪. Over this range of measurement, the output characteristic of the thermometer itself is nearly perfectly linear. Analyse linearity between temperature and Wheatstone bridge voltage output with and without Remedy Method. 1. Variable Conversion Element SOLUTION 𝑮𝒊𝒗𝒆𝒏, 𝑹𝟏 = 𝑹𝟐 = 𝑹𝟑 = 𝑹𝟒 = 𝟓𝟎𝟎Ω 𝑨𝒕 𝟐𝟓°𝑪 𝑹𝟏 = 𝟔𝟎𝟎Ω 𝑽𝒐 = 𝑽𝒊 𝑹𝟏 𝑹𝟑 − 𝑹𝟏 + 𝑹𝟐 𝑹𝟑 + 𝑹𝟒 𝑽𝒊 = 𝟏𝟎𝑽 𝑽𝒐 = 𝟎. 𝟒𝟓𝟓𝑽 𝑨𝒕 𝟎°𝑪 𝑨𝒕 𝟓𝟎°𝑪 𝑽𝟎 = 𝟎𝑽 𝑹𝟏 = 𝟕𝟎𝟎Ω 𝑽𝒐 = 𝑽𝒊 𝑹𝟏 𝑹𝟑 − 𝑹𝟏 + 𝑹𝟐 𝑹𝟑 + 𝑹𝟒 𝑽𝒐 = 𝟎. 𝟖𝟑𝟑𝑽 1. Variable Conversion Element SOLUTION 𝑻𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒄𝒉𝒂𝒏𝒈𝒆 𝒇𝒓𝒐𝒎 𝟎°𝑪 to 𝟐𝟓°𝑪 = 𝟐𝟓°𝑪 𝑽𝒐𝒍𝒕𝒂𝒈𝒆 𝒄𝒉𝒂𝒏𝒈𝒆 = 𝟎. 𝟒𝟓𝟓𝑽 𝑻𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒄𝒉𝒂𝒏𝒈𝒆 𝒇𝒓𝒐𝒎 𝟐𝟓°𝑪 to 𝟓𝟎°𝑪 = 𝟐𝟓°𝑪 𝑽𝒐𝒍𝒕𝒂𝒈𝒆 𝒄𝒉𝒂𝒏𝒈𝒆 = 𝟎. 𝟑𝟕𝟖𝑽 𝑵𝒐𝒕𝒆: 𝑁𝑜𝑛 − 𝑙𝑖𝑛𝑒𝑎𝑟 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑎𝑛𝑑 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 1. Variable Conversion Element SOLUTION USING REMEDY METHOD 𝑹𝟏 = 𝑹𝟑 = 𝟓𝟎𝟎Ω 𝑹𝟐 = 𝑹𝟒 = 𝟓𝟎𝟎𝟎Ω 𝑨𝒕 𝟐𝟓°𝑪 𝑹𝟏 = 𝟔𝟎𝟎Ω 𝑽𝒐 = 𝑽𝒊 𝑹𝟏 𝑹𝟑 − 𝑹𝟏 + 𝑹𝟐 𝑹𝟑 + 𝑹𝟒 𝑽𝒐 = 𝟎. 𝟒𝟐𝟒𝑽 𝑽𝒊 = 𝟐𝟔. 𝟏𝑽? 𝑨𝒕 𝟓𝟎°𝑪 𝑨𝒕 𝟎°𝑪 𝑹𝟏 = 𝟕𝟎𝟎Ω 𝑽𝟎 = 𝟎𝑽 𝑽𝒐 = 𝑽𝒊 𝑹𝟏 𝑹𝟑 − 𝑹𝟏 + 𝑹𝟐 𝑹𝟑 + 𝑹𝟒 𝑽𝒐 = 𝟎. 𝟖𝟑𝟑𝑽 EXAMPE A null-type Wheatstone bridge is used to accurately measure the resistance of a platinum resistance thermometer during a calibration procedure. Values of known fixed resistance values are R2 = 98.3 Ω and R4 = 102.2 Ω. The thermometer is inserted in the circuit as R1 and then the variable resistance box R3 is adjusted until the bridge output voltage V0 goes to zero. At this balance point, the value of R3 is 95.7 Ω. Calculate the resistance of the thermometer Solution At the balance point, the resistance values are related according to 𝑹𝟑 . 𝑹𝟐 𝑹𝟏 = 𝑹𝟒 Substituting the resistance values 𝑹𝟏 = 𝟗𝟐. 𝟎Ω 1. Variable Conversion Element • Contribution of component-value tolerances to total measurement system accuracy limits must be clearly understood • Steps for calculating maximum and minimum values of 𝑹𝟏 at null position STEP 1 Maximize 𝑹𝟏 by considering relevant tolerance of 𝑹𝟐, 𝑹𝟑 and 𝑹𝟒 STEP 2 Calculate percentage increase in the value of 𝑹𝟏 compared to its nominal value STEP 3 Minimize 𝑹𝟏 by considering relevant tolerance of 𝑹𝟐, 𝑹𝟑 and 𝑹𝟒 STEP 4 Calculate percentage decrease in the value of 𝑹𝟏 compared to its nominal value 1. Variable Conversion Element • Cumulative effect of errors in individual bridge circuit components is clearly seen • Maximum error in any one component = ±0.2% • Possible error in the measured value of Ru = ± 0.4% • Such magnitude of error is often not acceptable • Special measures are taken to overcome the introduction of error by component-value tolerances • One such practical measure - introduction of apex balancing 1. Variable Conversion Element APEX BALANCING • Place an additional variable resistor R5 at the junction C between the resistances R2 and R3 • Apply the excitation voltage Vi to the wiper of R5 • During calibration - Ru and Rv are replaced by two equal resistances whose values are accurately known - R5 is varied until the output voltage V0 is zero • If the portions of resistance on either side of the wiper on R5 are R6 and R7 (R5 = R6 + R7), we can write: R3 + R6 = R2 + R7 1. Variable Conversion Element APEX BALANCING • Any source of error due to the tolerance in the value of R2 and R3 eliminated • Error in the measured value of Ru depends only on the accuracy of one component - Rv Quiz-2 1. Variable Conversion Element A.C. Bridges • Bridges with A.C. excitation are used to measure unknown impedances 1. Variable Conversion Element A.C. Bridges - Typical null-type impedance bridge 1. Variable Conversion Element A.C. Bridges • Null point - conveniently detected by monitoring output with a pair of headphones connected via an operational amplifier across the points BD • Much cheaper method of null detection than the application of an expensive galvanometer that is required for a D.C. Wheatstone bridge 1. Variable Conversion Element A.C. Bridges • If Zu is capacitive - Zu = 1/jωCu - Zv must be variable capacitance • If Zu is inductive - Zu = Ru + jωLu - Zv must consist of a variableresistance and variable-inductance • Expression for Zu as an inductive impedance has a resistive term in it because it is impossible to realize a pure inductor • Inductor coil always has a resistive component, though this is made as small as possible by designing the coil to have a high Q factor (Q factor is the ratio inductance/resistance) 1. Variable Conversion Element A.C. Bridges • Variable inductance are not readily available - difficult and hence expensive to manufacture - set of fixed value inductors to make up a variable-inductance box • Alternative kind of null-type bridge circuit (Maxwell bridge) commonly used to measure unknown inductances 1. Variable Conversion Element A.C. Bridges • Maxwell Bridge Click to add text 1. Variable Conversion Element A.C. Bridges • Maxwell Bridge 1. Variable Conversion Element A.C. Bridges • Maxwell Bridge 1. Variable Conversion Element A.C. Bridges • Maxwell Bridge 1. Variable Conversion Element A.C. Bridges • Maxwell Bridge 1. Variable Conversion Element A.C. Bridges • Maxwell Bridge 1. Variable Conversion Element Deflection-type A.C. bridge 1. Variable Conversion Element Deflection-type A.C. bridge 1. Variable Conversion Element Deflection-type A.C. bridge 1. Variable Conversion Element Deflection-type A.C. bridge 1. Variable Conversion Element Deflection-type A.C. bridge 1. Variable Conversion Element Deflection-type A.C. bridge 1. Variable Conversion Element Deflection-type A.C. bridge
