The Wheatstone Bridge
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Transcript of The Wheatstone Bridge
The Wheatstone Bridge
Figure 4.10
Using Kirchhoff’s Voltage Law:
red loop: Ei = I1R1 + I2R2
green loop: Ei = I3R3 + I4R4
blue loop: Eo = I4R4 - I2R2
gold loop: Eo = -I3R3 + I1R1
Using Kirchhoff’s Current Law:
Combining the above gives the Wheatstone Bridge (WB) equation:
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RR
R
RR
REE io
I1 = I2 (at top node) I3 = I4 (at bottom node).
• What is the Wheatstonebridge equation if anotherresistor, Rx, is added in parallel with R1 ?
Rx
• 1/Rnew = 1/Rx+1/R1
• To solve, simply substitute
Rnew = R1Rx / (R1+Rx)
into the original WB equation for R1.
• One advantage of using two resistors in parallel in one leg of the WB is that the added resistor can be located remotely from the actual WB, such as in a flow.Here, that resistor can serve as a sensor.
Rx
FLOW
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R
RR
R
• When Eo = 0, the WB is said to be ‘balanced’ →
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4321 // RRRR →
• When the WB is balanced and 3 of the 4 resistances are known, the 4th (unknown) resistance can be foundusing the balanced WB equation. The is called the null method.
RR
R
RRR
RREE io )(
)(
• Now consider the case when all 4 resistors are thesame initially and, then, one resistance, say R1, is changedby an amount R. This is called the deflection method.
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RR
RREi /24
/
• Often, R is associated with a change in a physical variable.
In-Class Example
Figure 4.18
RTD: RTD = 0.0005 / ºC; R = 25 at 20.0 ºCWheatstone Bridge: R2 = R3 = R4 = 25 Ei =5 VAmplifier: Gain = GMultimeter: 0 V to 10 V range (DC)Experimental Operating Range: 20 ºC to 80 ºC
In-Class Example
Cantilever Beam with Four Strain Gages
• When F is applied as shown (downward), R1 and R4 increase by R (due to elongation), and R2 and R3 decrease by R (due to compression). Here, R is
directly proportional to the strains.
Figure 4.11
• From solid mechanics, for a cantilever beam, both the elongational strain, L, and the compressive strain, C, are directly proportional to the applied force, F.
Figure 6.2
The Cantilever Beam with Four Gages
• The WB equation
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RR
R
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)()(
)(
)()( RRRR
RR
RRRR
RREE io
• This instrumented cantilever beam system is the basis for many force measurement systems (like force balances and load cells.
• Because R ~ (L or C) and (L or C) ~ F, Eo = constant x F
becomes
which reduces to Eo = Ei (R/R).