Page 206 - The Mechatronics Handbook

0066_Frame_C11 Page 12 Wednesday, January 9, 2002 4:14 PM

FIGURE 11.15 Wheatstone bridge circuits.

The objective is to determine the unknown resistance R x .

1. Find the value of the voltage v ad = v ad – v bd in terms of the four resistances and the source voltage,
v S . Note that since the reference point d is the same for both voltages, we can also write v ab = v a – v b .
2. If R 1 = R 2 = R 3 = 1 kΩ, v S = 12 V, and v ab = 12 mV, what is the value of R x ?

Solution
1. First, we observe that the circuit consists of the parallel combination of three subcircuits: the
voltage source, the series combination of R 1 and R 2 , and the series combination of R 3 and R x . Since
these three subcircuits are in parallel, the same voltage will appear across each of them, namely,
the source voltage, v S .
Thus, the source voltage divides between each resistor pair, R 1 -R 2 and R 3 -R x , according to the
voltage divider rule: v a is the fraction of the source voltage appearing across R 2 , while v b is the
voltage appearing across R x :

R x
v a = ----------------- and v b = -----------------
R 2
R 1 + R 2 v S R 3 + R x
v S
Finally, the voltage difference between points a and b is given by

v ab = v a – v b = v S ----------------- – ------------------  
R 2
R x

R 3 +
R 1 +
R 2
R x
This result is very useful and quite general, and it finds application in numerous practical circuits.
2. In order to solve for the unknown resistance, we substitute the numerical values in the preceding
equation to obtain
1000

0.012 = 12 ----------- – ------------------------  
R x

1000 +
2000
R x
which may be solved for R x to yield
R x = 996 Ω
Practical Voltage and Current Sources
Idealized models of voltage and current sources fail to take into consideration the finite-energy nature
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. As the load resistance (R) decreases,
the source is required to provide increasing amounts of current to maintain the voltage v S (t) across

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