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Strain Gages • Electrical resistance in material changes when the material is deformed R A R – Resistance ρ – Resistivity l – Length A – Cross-sectional area log R log log A Taking the differential dR d d A R A For linear deformations R Ss R Change in resistance is from change in shape as well as change in resistivity ε – strain Ss – sensitivity or gage factor (2-6 for metals and 40 – 200 for semiconductor) • The change in resistance is measured using an electrical circuit • Many variables can be measured – displacement, acceleration, pressure, temperature, liquid level, stress, force and torque • Some variables (stress, force, torque) can be determined by measuring the strain directly • Other variables can be measured by converting the measurand into stress using a front-end device Housing Output vo Strain Gage m Seismic Mass Strain Member Cantilever Base Mounting Threads Strain gage accelerometer Direction of Sensitivity (Acceleration) Strain gages are manufactured as metallic foil (copper-nickel alloy – constantan) Direction of Sensitivity Foil Grid Single Element Two-Element Rosette Backing Film Solder Tabs (For Leads) Three-Element Rosettes Semiconductor (silicon with impurity) Doped Silicon Crystal (P or N Type) Phenolic Glass Backing Plate Welded Gold Leads Nickle-Plated Copper Ribbons Potentiometer or Ballast Circuit + vo Output Strain Gage vref R - (Supply) Rc R vo vref R Rc • Ambient temperature changes will introduce error • Variations in supply voltage will affect the output • Electrical loading effect will be significant • Change in voltage due to strain is a very small percentage of the output Question: Show that errors due to ambient temperature changes will cancel if the temperature coefficients of R and Rc are the same Wheatstone Bridge Circuit A Small i + R1 R2 RL R4 R3 vo Load (High) - B - vref + (Constant Voltage) vo R1vref (R1 R2 ) R3vref (R3 R4 ) (R1R4 R2 R3 ) vref (R1 R2 )(R3 R4 ) When the bridge is balanced R1 R3 R2 R4 True for any RL Null Balance Method • When the stain gage in the bridge deforms, the balance is upset. • Balance is restored by changing a variable resistor • The amount of change corresponds to the change in stain • Time consuming – servo balancing can be used Direct Measurement of Output Voltage • Measure the output voltage resulting from the imbalance • Determine the calibration constant • Bridge sensitivity vo R2 R1 R1R2 R4 R3 R3 R4 2 v ref R1 R2 R3 R4 2 To compensate for temperature changes, temperature coefficients of adjacent pairs should be the same The Bridge Constant • More than one resistor in the bridge can be active • If all four resistors are active, best sensitivity can be obtained • R1 and R4 in tension and R2 and R3 in compression gives the largest sensitivity • The bridge sensitivity can be expressed as vo vref Bridge Constant k k R 4R bridge output in the general case bridge output if only one strain gage is active Example 4.4 A strain gage load cell (force sensor) consists of four identical strain gages, forming a Wheatstone bridge, that are mounted on a rod that has square crosssection. One opposite pair of strain gages is mounted axially and the other pair is mounted in the transverse direction, as shown below. To maximize the bridge sensitivity, the strain gages are connected to the bridge as shown. Determine the bridge constant k in terms of Poisson’s ratio v of the rod material. Axial Gage 2 1 1 2 3 Cross Section Of Sensing Member + vo Transverse Gage 4 − 3 4 − Transverse strain = (-v) x longitudinal strain vref + Calibration Constant vo C v ref k C Ss 4 k – Bridge Constant Ss – Sensitivity or gage factor R Ss R vo vref k R 4R Example 4.5 A schematic diagram of a strain gage accelerometer is shown below. A point mass of weight W is used as the acceleration sensing element, and a light cantilever with rectangular cross-section, mounted inside the accelerometer casing, converts the inertia force of the mass into a strain. The maximum bending strain at the root of the cantilever is measured using four identical active semiconductor strain gages. Two of the strain gages (A and B) are mounted axially on the top surface of the cantilever, and the remaining two (C and D) are mounted on the bottom surface. In order to maximize the sensitivity of the accelerometer, indicate the manner in which the four strain gages A, B, C, and D should be connected to a Wheatstone bridge circuit. What is the bridge constant of the resulting circuit? A C Strain Gages A, B + δvo W − C, D D B l b h A B C D − vref + Obtain an expression relating applied acceleration a (in units of g) to bridge output (bridge balanced at zero acceleration) in terms of the following parameters: W = Mg = weight of the seismic mass at the free end of the cantilever element E = Young’s modulus of the cantilever l = length of the cantilever b = cross-section width of the cantilever h = cross-section height of the cantilever Ss = gage factor (sensitivity) of each strain gage vref = supply voltage to the bridge. • If M = 5 gm, E = 5x1010 N/m2, l = 1 cm, b = 1 mm, h = 0.5 mm, Ss = 200, and vref = 20 V, determine the sensitivity of the accelerometer in mV/g. • If the yield strength of the cantilever element is 5xl07 N/m2, what is the maximum acceleration that could be measured using the accelerometer? • If the ADC which reads the strain signal into a process computer has the range 0 to 10 V, how much amplification (bridge amplifier gain) would be needed at the bridge output so that this maximum acceleration corresponds to the upper limit of the ADC (10 V)? • Is the cross-sensitivity (i.e., the sensitivity in the two directions orthogonal to the direction of sensitivity small with this arrangement? Explain. • Hint: For a cantilever subjected to force F at the free end, the maximum stress at the root is given by 6 F bh 2 MEMS Accelerometer Signal Conditioning Mechanical Structure Applications: Airbag Deployment Data Acquisition Dynamic Strain AC Bridge Amplifier Oscillator Power Supply Demodulator And Filter Calibration Constant • Supply frequency ~ 1kHz • Output Voltage ~ few micro volts – 1 mV • Advantages – Stability (less drift), low power consumption • Foil gages - 50Ω – kΩ • Power consumption decreases with resistance • Resolutions on the order of 1 m/m Strain Reading Semiconductor Strain Gages Conductor Ribbons Single Crystal of Semiconductor Gold Leads Phenolic Glass Backing Plate • Gage factor – 40 – 200 • Resitivity is higher – reduced power consumption • Resistance – 5kΩ • Smaller and lighter Properties of common strain gage material Material Composition Gage Factor (Sensitivity) Temperature Coefficient of Resistance (10-6/C) Constantan 45% Ni, 55% Cu 2.0 15 Isoelastic 36% Ni, 52% Fe, 8% Cr, 4% (Mn, Si, Mo) 3.5 200 Karma 74% Ni, 20% Cr, 3% Fe, 3% Al 2.3 20 Monel 67% Ni, 33% Cu 1.9 2000 Silicon p-type 100 to 170 70 to 700 Silicon n-type -140 to –100 70 to 700 Disadvantages of Semiconductor Strain Gages • The strain-resistance relationship is nonlinear • They are brittle and difficult to mount on curved surfaces. • The maximum strain that can be measured is an order of magnitude smaller 0.003 m/m (typically, less than 0.01 m/m) • They are more costly • They have a much larger temperature sensitivity. Resistance Change R R Resistance Change P-type R 0.4 0.4 = 1 Microstrain 0.3 = Strain of 0.2 0.3 1×10-6 0.2 0.1 0.1 −3 −2 N-type R −1 1 −0.1 2 Strain 3 ×103 −3 −2 −1 1 −0.1 −0.2 −0.2 −0.3 −0.3 2 3 ×103 Strain For semiconductor strain gages R S1 S2 2 R • S1 – linear sensitivity • Positive for p-type gages • Negative for n-type gages • Magnitude is larger for p-type • S2 – nonlinearity • Positive for both types • Magnitude is smaller for p-type Linear Approximation R R Ss L R R Change in Resistance Quadratic Curve Linear Approximation −max 0 Strain max Error e R R S1 S2 2 Ss R R L S1 Ss S2 2 Quadratic Error J J 0. Minimize Error Ss max (2 ) S1 S s S 2 2 d = 0 max e d max 2 S1 Ss S2 max max 2 Maximum Error max S1 S s emax S2 2max 2 2 d Range – change in resistance R S1 max S2 2max S1 max S2 2max R 2 S1 max Percentage nonlinearity error 2 S2 max max error Np 100% 100% range 2S1 max N p 50 S 2 max S1 % Temperature coefficients (per °F) Temperature Compensation α = Temperature Coefficient of Resistance β = Temperature Coefficient of Gage Factor 3 α 2 Compensation Feasible (−β) 1 Compensation Not Feasible Compensation Feasible 0 Concentration of Trace Material (Atoms/cc) Sensitivity change due to temperature R Ro 1 .T Ss Sso 1 .T Resistance change due to temperature Self Compensation with a Resistor R1 R R2 R + δvo − R R + − − vi R4 R3 vi + Compensating Resistor Rc vref vi R vref R Rc v o kSs R v ref R Rc 4 Possible only for certain ranges − vref Rc + For self compensation the output after the temperature change must be the same Ro 1 .T Ro Sso Sso 1 .T R R Ro 1 .T Rc o c Ro Rc ( ) Ro Rc T Rc Ro