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Comments: Systematic methods for analyzing arbitrary circuit con gurations are
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The Wheatstone bridge is a resistive circuit that is frequently encountered in a variety of measurement circuits The general form of the bridge circuit is shown in Figure 236(a), where R1 , R2 , and R3 are known while Rx is an unknown resistance, to be determined The circuit may also be redrawn as shown in Figure 236(b) The latter circuit will be used to demonstrate the use of the voltage divider rule in a mixed series-parallel circuit The objective is to determine the unknown resistance, Rx
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c R1 vS + _ a R2 d (a) c R1 vS
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R3 va vb Rx b
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1 Find the value of the voltage vab = vad vbd in terms of the four resistances and the source voltage, vS Note that since the reference point d is the same for both voltages, we can also write vab = va vb 2 If R1 = R2 = R3 = 1 k , vS = 12 V, and vab = 12 mV, what is the value of Rx
Solution
Known Quantities: Source voltage, resistance values, bridge voltage
R3 va vb b Rx
Find: Unknown resistance Rx Schematics, Diagrams, Circuits, and Given Data: See Figure 236
a R2
R1 = R2 = R3 = 1 k ; vS = 12 V; vab = 12 mV
Analysis:
d (b)
Figure 236 Wheatstone bridge circuits
1 First, we observe that the circuit consists of the parallel combination of three subcircuits: the voltage source, the series combination of R1 and R2 , and the series combination of R3 and Rx Since these three subcircuits are in parallel, the same voltage will appear across each of them, namely, the source voltage, vS Thus, the source voltage divides between each resistor pair, R1 R2 and R3 Rx , according to the voltage divider rule: va is the fraction of the source voltage appearing across R2 , while vb is the voltage appearing across Rx : v a = vS R2 R1 + R 2 and vb = vS Rx R3 + R x
Finally, the voltage difference between points a and b is given by: vab = va vb = vS Rx R2 R1 + R2 R3 + R x
This result is very useful and quite general 2 In order to solve for the unknown resistance, we substitute the numerical values in the preceding equation to obtain 0012 = 12 1,000 Rx 2,000 1,000 + Rx
Part I
Circuits
which may be solved for Rx to yield Rx = 996
Comments: The Wheatstone bridge nds application in many measurement circuits and
instruments
Focus on Computer-Aided Tools: Virtual Lab You will nd a Virtual Lab version of the
circuit of Figure 236 in the electronic les that accompany this book If you have practiced building some simple circuits using Electronics Workbench, you should by now be convinced that this is an invaluable tool in validating numerical solutions to problems, and in exploring more advanced concepts
The Wheatstone Bridge and Force Measurements
Strain gauges, which were introduced in a Focus on Measurements section earlier in this chapter, are frequently employed in the measurement of force One of the simplest applications of strain gauges is in the measurement of the force applied to a cantilever beam, as illustrated in Figure 237 Four strain gauges are employed in this case, of which two are bonded to the upper surface of the beam at a distance L from the point where the external force, F , is applied and two are bonded on the lower surface, also at a distance L Under the in uence of the external force, the beam deforms and causes the upper gauges to extend and the lower gauges to compress Thus, the resistance of the upper gauges will increase by an amount R, and that of the lower gauges will decrease by an equal amount, assuming that the gauges are symmetrically placed Let R1 and R4 be the upper gauges and R2 and R3 the lower gauges Thus, under the in uence of the external force, we have: R1 = R4 = R0 + R2 = R3 = R0 R R
FOCUS ON MEASUREMENTS
where R0 is the zero strain resistance of the gauges It can be shown from elementary statics that the relationship between the strain and a force F
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