The Wheatstone Bridge 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: Figure 4.10 I1 = I2 (at top node) I3 = I4 (at bottom node). Combining the above gives the Wheatstone Bridge (WB) equation:

bridge equation if another resistor, Rx, is added in What is the Wheatstone bridge equation if another resistor, 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. FLOW Rx

When Eo = 0, the WB is said to be ‘balanced’ → When the WB is balanced and 3 of the 4 resistances are known, the 4th (unknown) resistance can be found using the balanced WB equation. The is called the null method.

Now consider the case when all 4 resistors are the same initially and, then, one resistance, say R1, is changed by an amount dR. This is called the deflection method. Often, dR is associated with a change in a physical variable.

In-Class Example RTD: aRTD = 0.0005 / ºC; R = 25 W at 20.0 ºC Figure 4.18 RTD: aRTD = 0.0005 / ºC; R = 25 W at 20.0 ºC Wheatstone Bridge: R2 = R3 = R4 = 25 W; Ei = 5 V Amplifier: Gain = G Multimeter: 0 V to 10 V range (DC) Experimental Operating Range: 20 ºC to 80 ºC

In-Class Example

Cantilever Beam with Four Strain Gages From solid mechanics, for a cantilever beam, both the elongational strain, eL, and the compressive strain, eC, are directly proportional to the applied force, F. Figure 4.11 Figure 6.2 When F is applied as shown (downward), R1 and R4 increase by dR (due to elongation), and R2 and R3 decrease by dR (due to compression). Here, dR is directly proportional to the strains.

The Cantilever Beam with Four Gages The WB equation becomes which reduces to Eo = Ei (dR/R). Because dR ~ (eL or eC) and (eL or eC) ~ F, Eo = constant x F This instrumented cantilever beam system is the basis for many force measurement systems (like force balances and load cells.