2. 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:

3. 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.

4. 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

5. 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.

6. 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.

7. In-Class Example RTD: aRTD = 0.0005 / ºC; R = 25 W at 20.0 ºC
Figure 4.18
RTD: aRTD = / 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

8. In-Class Example

9. 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.

10. 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.