EXPERIMENT 5
PHYSICS 250
WHEATSTONE BRIDGE
Apparatus
Power supply
Precision multimeter
Precision resistors
Wheatstone bridge
Temperature bath
Strain gauge
Introduction
During this laboratory period you will study the Wheatstone bridge, a special technique used
in high-precision electrical DC measurements. The Wheatstone bridge is very important in its own
right, but it also is typical of many techniques in which changes in a value can be measured much
better than the value itself. Two preliminary comments provide helpful background for this type of
study:
It is important to remember the difference between the words “precision” and “accuracy.”
Precision involves the ability to measure with great sensitivity and to measure repeatedly a very small
difference. Accuracy further implies that the measured value can be related to an absolute scale.
Note that high precision is required before high accuracy can be considered, but it is more difficult
to obtain accuracy.
In high-precision measurements you must exercise great care to control or eliminate spurious
or random effects. It is foolish to talk about the measurement of a quantity to within one part in 106
if the quantity is fluctuating in time with a variation of a few parts in 104 or if there is a large spatial
variation in the quantity. Stability and uniqueness are axiomatic to high-precision measurements. For
example, it would be meaningless to talk about measuring the temperature of an ordinary beaker of
water to a precision of 0.001º C. The temperature varies throughout the volume by as much as 1º
C, and at one specific point the temperature will fluctuate in time by almost as much.
In electrical circuits assembled in the laboratory, there are three major sources of error:
changes in the resistance of electrical connections due to poor contacts, variations of electrical
parameters with temperature, and noise pickup in unshielded wires. In some cases these effects may
be eliminated, and in other situations they must simply be controlled. For instance, in this lab you will
be provided with special wires with “spade” ends to be clamped into binding posts rather than
“banana plug” ends to ensure good contact between the wires and the binding posts. In some
situations you may need to use shielded wires to avoid noise pickup or you may use some method
to be sure that the temperature of the devices being investigated remains constant.
It is almost invariably true that you can make comparisons with much greater precision than
you can make an independent measurement. For example, you will see it is relatively easy to
5-1
determine that an electrical resistance is 0.001% larger than a similar resistance, but a direct
measurement of current and voltage to determine R to this accuracy would be very difficult. The
secret of a comparison technique is the use of a known or assumed reference. The great advantage
comes because you have to measure only the difference rather than the total quantity. Often you may
wish to study a small change in a quantity (a resistance, for instance) due to a change in its
environment, and the reference is the quantity itself before the change is made. In such a case the
question of accuracy is not involved.
If you require absolute accuracy, you must employ a standard. The accuracy of the
measurement is then limited only by the accuracy of the standard and by the sensitivity with which
you can make the comparison with the standard. The most accurate standards will generally be
referred to in the USA as “NIST-traceable” or “NBS-traceable” which means that the calibration of
that instrument can be traced directly to a standard maintained by the National Institute of Standards
and Technology (NIST; NBS is the acronym for the National Bureau of Standards, a previous
incarnation of the same group).
The development in the late 1970's of the digital electronic comparator meters significantly
changed the field of precision electrical measurements. The operation of this meter itself is based
upon the principle of the comparison-type measurements discussed above. This type of meter has
an internal fixed-voltage standard, and requires the use of a technique to obtain a variable-standard
voltage for comparison with measured voltages by taking simple ratios. The best meters available
in this laboratory have four significant digits, and you can use them as voltage-reference meters to
this accuracy. (Actually, the reference drifts slightly in time and requires recalibration at intervals of
several months if you desire the ultimate accuracy.) You can use this meter for any measurements
requiring accuracy of four significant figures or fewer. This meter has a sensitivity of 10 V on the
200-mV range and is thus also useful as a null detector to demonstrate that two voltages are equal
to within 10 V.
As a prime example of a very high-precision electrical measurement made by using the
comparison technique, you will study the Wheatstone bridge for resistance measurements. This
technique has been used for over one hundred years for precision resistance measurements greater
than a few percent. It is still used broadly today when precision greater than one part in 104 is
required. Using this technique, you will be able to observe changes in resistance with a precision of
a few parts per million.
All the actual measurements made during this laboratory period will consist of simple
measurements of electrical resistance. When using the Wheatstone bridge you will use either
temperature or strain to vary the resistance slightly.
5-2
Wheatstone Bridge
The basic idea of the simple Wheatstone resistance bridge is illustrated in Fig. 1. The
resistance Rs is generally a standard resistance that is variable and made with great precision so that
its value is well known, and Ru is an unknown resistance. When Rs is adjusted until the voltage
difference V in Fig. 1 becomes zero, the two resistance ratios Rs/Ru and R1/R2 are equal, and the
bridge is said to be in balance. Thus at balance
Ru = R s
R2 .
R1
The accuracy of the instrument in
determining Ru is limited by the accuracy of the
known resistances Rs, R1, and R2, and the precision
(or sensitivity) is determined by the sensitivity of the
null indicator that measures V.
(1)
Voltage
Source
R1
Rs
V
Historically, the Wheatstone bridge was used
with a standard resistance that was also variable over
its total range so that unknown resistances of any
R2
Ru
size could be measured. For measurements with a
6
sensitivity of a few parts in 10 , this setup requires a
very costly set of standard resistances. The Figure 1. The Wheatstone bridge.
dominant use of the Wheatstone bridge in modern
science is in special situations where only very small
changes are being studied, and specific bridges are created for specific uses. This situation is the one
you will set up in the laboratory.
The Wheatstone bridge is most sensitive
when R1 = R2 and Rs = Ru. Consider the
R0
R
specialized bridge shown in Fig. 2, which
Voltage
Source
utilizes a small variable standard Rvs and thus
will measure very accurately only a resistance
Ru near the value R0. The standard variable
or
r
V
Rvs
resistance Rvs is placed in either the Rs branch
(Rs = R0 + Rvs) or in the Ru branch (Ru = Rx +
Rvs), depending upon whether Rx is greater or
less than R0. In a typical application, Rvs may
R
Rx
vary between 0 and 0.01R0, so the total possible
variation of Rx from R0 would be one percent.
In the bridge used in this laboratory, R0 = 5,000 Figure 2. A specialized Wheatstone bridge.
, and Rvs can vary from 0 to 1000 . Thus,
Rvs < 0.2 R0, so Rx can differ from R0 by as much as twenty percent. The small variable resistance r
is also included as shown to allow an independent zero adjustment of the bridge to account for any
small difference in the two values R and any small changes in the values of R and R0 over a period of
time. The total variation of r is only 1.0 while R is approximately 5,000 . You can set the r zero
adjustment by placing a standard 5,000 resistance in the position Rx when you balance the bridge
5-3
by using r. This procedure represents a calibration of the bridge. After you have completed this
calibration then any subsequent measurements will really be a comparison of the unknown with the
value of the standard resistor.
Before leaving the analysis of the Wheatstone bridge, note that it is not necessary to have a
standard variable resistance with extremely small divisions. A more thorough analysis of the circuit
in Fig. 1 where the bridge is not balanced shows that if the bridge is near balance ( V is small) and
if V is measured with a voltmeter that has an internal resistance much greater than Rs, Ru, R2, or R1,
then
Ru = R s +
4 ∆V
Rs
E
(2)
Note that V can be either positive or negative so that Ru can be either larger or smaller than Rs. You
must be careful to note the relationship between the sign of V and the effect on Ru.
Because of this simple relationship near balance, a linear interpolation between the smallest
divisions on the Rvs adjustment is possible. The resolution of the bridge in resistance (the smallest
detectable resistance change) is determined by the smallest detectable voltage difference V, which
is 1 x 10-5 volts for the available 4 ½-digit electronic meters. Note that if E = 20 volts, this resolution
is
4 ( ∆V) R s 4 (1x 10 -5 ) (5000)
=
Ω = 0.01 Ω .
E
20
(3)
If you measure a resistance near 5,000 , this value represents two parts in 106, which is not bad for
such a simple bridge.
If the change of the unknown resistance, Rx, is sufficiently small (a fraction of a percent) it is not
necessary to change the value of Rvs during the course of the measurement. And, in fact, if you do
not change the value of Rvs during the course of the measurement you can actually achieve higher
accuracy. And if you are looking at resistance changes, R2 - R1, then the (unchanged) value of Rvs
will cancel entirely.
Objective: To become acquainted with comparison-type, high-precision, electrical measurements
and to gain an appreciation for the care required to make such measurements.
Procedure
During this laboratory period you will use the Wheatstone bridge to measure with very high
precision a change in resistance that is so small that using the 4 ½-digit electronic meter as an
ohmmeter you can barely observe the change. In each case you should contrast resistance
measurements made with the electronic multimeter with measurements made with the more precise
comparison-type techniques discussed above.
Select one of the following two situations for study. The comments given with each situation
will aid your understanding of the measurement you will make.
5-4
1. A strain gauge is mounted on a sheet of steel. When a force is applied to the steel, the steel
bends, causing a change in the resistance of the strain gauge. You are to measure the change
in resistance as a function of the force applied to the steel plate. The strain gauge is
composed of a number of high resistance “wires” deposited on the surface of an insulating
backing. As the backing is stretched, the wires both elongate and decrease in cross-sectional
area, thereby causing the resistance to increase. The change in resistance is proportional to
the applied force over a fairly wide range of forces.
2. Precision resistors used as references in electronic equipment change only slightly with
temperature. The precision resistances mentioned are made of very special metal alloys, and
change only very slightly with temperature, and are thus very stable and usable when you need
a standard resistance of known value. The resistances used in radio, TV, etc. are made of
carbon, and their resistances vary significantly with temperature. Pure metals also have
sizeable temperature coefficients of resistance. The resistor available for evaluation in this
experiment is a carbon resistor. An oil bath is available to heat the resistor and a thermometer
is available with which to measure the oil temperature. Be careful with this bath – the oil can
become very hot. When you are through with the oil be sure to pour it into the can labeled
“hot” so that others won’t be surprised (or injured) when they pour hot oil from the can
labeled as “cold” oil.
Before setting up your Wheatstone bridge, think about what source voltage you should use.
Why is 20 volts a convenient voltage? If E = 20 volts, you should be able to determine each value
of R with a sensitivity of 0.01 using the electronic meter to interpolate.
REMEMBER TO DISCONNECT ALL WIRES AND TO TURN OFF ALL EQUIPMENT
EXCEPT THE COMPUTER!
5-5