L R4 R1
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c ia + vS R2 h R4 R1 va ib R3 vb
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R2, R3 bonded to bottom surface
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Beam cross section
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Figure 237 A force-measuring instrument
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Fundamentals of Electric Circuits
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applied at a distance L for a cantilever beam is: 6LF wh2 Y where h and w are as de ned in Figure 237 and Y is the beam s modulus of elasticity In the circuit of Figure 237, the currents ia and ib are given by vS vS ia = and ib = R1 + R 2 R3 + R 4 = The bridge output voltage is de ned by vo = vb va and may be found from the following expression: vo = ib R4 ia R2 = = vS = vS v S R4 v S R2 R3 + R 4 R1 + R 2 R vS R0 + R0 R R + R0 R
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R0 + R R + R0
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R = vS G R0
where the expression for R/R0 was obtained in Focus on Measurements: Resistance Strain Gauges section Thus, it is possible to obtain a relationship between the output voltage of the bridge circuit and the force, F , as follows: 6LF 6vS GL = F = kF 2Y wh wh2 Y where k is the calibration constant for this force transducer vo = vS G = vS G
Comments Strain gauge bridges are commonly used in
mechanical, chemical, aerospace, biomedical, and civil engineering applications (and wherever measurements of force, pressure, torque, stress, or strain are sought)
Check Your Understanding
+ Vbattery + _ Unknown element i R
24 Repeat Example 28 by reversing the reference direction of the current, to show that the same result is obtained 25 The circuit in the accompanying illustration contains a battery, a resistor, and an unknown circuit element
1 2 If the voltage Vbattery is 145 V and i = 5 mA, nd power supplied to or by the battery Repeat part 1 if i = 2 mA
26 The battery in the accompanying circuit supplies power to the resistors R1 , R2 , and R3 Use KCL to determine the current iB , and nd the power supplied by the battery if Vbattery = 3 V
iB + Vbattery + _
R1 i1 = 02 mA
R2 i2 = 04 mA
R3 i3 = 12 mA
27 Use the results of part 1 of Example 211 to nd the condition for which the voltage vab = va vb is equal to zero (this is called the balanced condition for the bridge) Does this result necessarily require that all four resistors be identical Why 28 Verify that KCL is satis ed by the current divider rule and that the source current iS divides in inverse proportion to the parallel resistors R1 , R2 , and R3 in the circuit of Figure 233 (This should not be a surprise, since we would expect to see more current ow through the smaller resistance) 29 Compute the full-scale (ie, largest) output voltage for the force-measuring apparatus of Focus on Measurements: The Wheatstone Bridge and Force Measurements Assume that the strain gauge bridge is to measure forces ranging from 0 to 500 N, L = 03 m, w = 005 m, h = 001 m, G = 2, and the modulus of elasticity for the beam is 69 109 N/m2 (aluminum) The source voltage is 12 V What is the calibration constant of this force transducer 210 Repeat the derivation of the current divider law by using conductance elements that is, by replacing each resistance with its equivalent conductance, G = 1/R
PRACTICAL VOLTAGE AND CURRENT SOURCES
The idealized models of voltage and current sources we discussed in Section 23 fail to consider the internal resistance of practical voltage and current sources The objective of this section is to extend the ideal models to models that are capable of describing the physical limitations of the voltage and current sources used in practice Consider, for example, the model of an ideal voltage source shown in Figure 29 As the load resistance (R) decreases, the source is required to provide increasing amounts of current to maintain the voltage vS (t) across its terminals: i(t) = vS (t) R (224)
This circuit suggests that the ideal voltage source is required to provide an in nite amount of current to the load, in the limit as the load resistance approaches zero Naturally, you can see that this is impossible; for example, think about the ratings of a conventional car battery: 12 V, 450 A-h (ampere-hours) This implies that there is a limit (albeit a large one) to the amount of current a practical source can deliver to a load Fortunately, it will not be necessary to delve too deeply into the physical nature of each type of source in order to describe the behavior of a practical voltage source: The limitations of practical sources can be approximated quite simply by exploiting the notion of the internal resistance of a source Although the models