Transcript DPKC_Mod02_Part04_v05

Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Dynamic Presentation of Key
Concepts
Module 2 – Part 4
The Wheatstone Bridge
Filename: DPKC_Mod02_Part04.ppt
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Overview of this Part
The Wheatstone Bridge
In this part of Module 2, we will cover the
following topics:
• Null Measurement Techniques
• Wheatstone Bridge Derivation
• Wheatstone Bridge Measurements
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Textbook Coverage
This material is introduced in different ways in different
textbooks. Approximately this same material is covered in
your textbook in the following sections:
• Circuits by Carlson: Section 3.5
• Electric Circuits 6th Ed. by Nilsson and Riedel: Section 3.6
• Basic Engineering Circuit Analysis 6th Ed. by Irwin and
Wu: Section 2.8
• Fundamentals of Electric Circuits by Alexander and
Sadiku: Section 4.10.2
• Introduction to Electric Circuits 2nd Ed. by Dorf: Not
covered
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
The Wheatstone Bridge –
A Null-Measurement Technique
The subject of this part of Module 2 is the Wheatstone
Bridge, a null-measurement technique for measuring
resistance. There are also null-measurement techniques
for measurements of things like voltage, but we will just
consider this one example to illustrate the principle.
These techniques have the following properties:
1. They use a standard meter,
such as an ammeter or voltmeter.
2. The measurement occurs when
the reading on this ammeter or
voltmeter is zero.
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Null-Measurement
Techniques – Note 1
Null-measurement techniques use a standard meter,
such as an ammeter or voltmeter. Typically, they use an
analog meter, such as the D’Arsonval meter movement,
which is described in many circuits textbooks. Such
meters are sometimes thought of as ammeters, since their
response is due to the magnetic field in a coil, caused by a
current. However, since these meters can be modeled as
resistances, which means that
the current through them is
proportional to the voltage
across them, the distinction
is not really important.
In this sense, all of these meters
are both voltmeters and ammeters.
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Null-Measurement
Techniques – Note 2
The null-measurement occurs when the reading on this ammeter
or voltmeter is zero. This is a huge practical benefit. Making a
meter which is precisely linear, with an accurate scale, and
negligible resistance, is a challenge. None of these issue matter in a
null measurement, since the purpose of the meter to determine the
presence or absence of current or voltage. It does not need to be
linear; it is only important to detect the zero value. The resistance
does not matter, since there is no current through the meter at the
point of measurement.
The only concern is that the
meter be able to detect fairly small
currents, during the nulling step.
This makes the design much easier.
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Null-Measurement
Techniques – Note 3
We will consider the particular null-measurement technique known as the
Wheatstone Bridge. This is a very accurate resistance measurement technique,
which also has applications in measurement devices such as strain gauges.
There are other null-measurement techniques. One such technique is called
the Potentiometric Voltage Measurement System. This is discussed in the
textbook Circuits, by A. Bruce Carlson, on pages 121 and 122. A diagram from
the text is included here. While interesting, we will concentrate on the
Wheatstone Bridge in this module.
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
The Wheatstone Bridge
The Wheatstone Bridge is a resistance measuring technique that
uses a meter to detect when the voltage across that meter is zero.
The meter is placed across the middle of two resistor pairs. The
resistor pairs in the circuit here are R1 and R3, and R2 and RX. The
meter is said to “bridge” the midpoints of these two pairs of
resistors, which is
where the name
R1
R2
comes from.
A source (vS) is
Meter
used to power the
+
vS
entire combination.
See the diagram here.
R3
RX
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
The Wheatstone Bridge – Notes
Go back to
Overview
slide.
The resistor RX is an unknown resistor, that is, the resistor whose resistance is
being measured. The other three resistors are known values. The resistor R3 is a
variable resistor, calibrated so that as it is varied its value is known. The meter is
conceptually a voltmeter. However, it should be noted that a meter is a resistor
from a circuits standpoint, so that when the voltage is zero the current is also zero.
R1
R2
Meter
+
vS
R3
RX
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
The Wheatstone Bridge –
The Nulling Step
To make the measurement, the resistor R3 is a varied so that the
voltmeter reads zero. Thus, when R3 is the proper value, then vM and
iM are both zero.
R1
R2
Meter
+
vS
+ vM
iM
R3
RX
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
The Wheatstone Bridge –
Derivation Step 1
Using the fact that vM and iM are both zero, we can derive the
operating equation for the Wheatstone Bridge. Let’s take this
derivation one step at a time.
First, since iM is zero, we can say that R1 and R3 are in series, and
R2 and RX are in series.
R1
R2
Meter
+
vS
+ vM
iM
R3
RX
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
The Wheatstone Bridge –
Derivation Step 2
Second, since R1 and R3 are in series, and R2 and RX are in series,
we can write expressions for v3 and vX using the voltage divider rule,
R3
v3  vS
, and
R3  R1
RX
v X  vS
.
RX  R2
R1
R2
Meter
+
+ vM
vS
-
iM
+
v3
R3
-
+
vX
RX
-
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
The Wheatstone Bridge –
Derivation Step 3
Third, since vM is zero, we can write KVL around the loop and
show that v3 is equal to vX. Thus, we can set the expressions for
these two voltages equal,
R1
R3
RX
vS
 vS
.
R3  R1
RX  R2
R2
Meter
+
+ vM
vS
-
iM
+
v3
R3
-
+
vX
RX
-
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Go back to
Overview
slide.
The Wheatstone Bridge –
Derivation Step 4
Fourth, we can divide through by vS. This is important, since it
means that the exact value of vS does not matter. For example, the
source could be a battery, and if the battery runs down a little, it does
not change the measurement. We get,
R3
RX

.
R3  R1 RX  R2
This can be solved
for RX ,
R2
RX  R3 .
R1
R1
R2
Meter
+
+ vM
vS
-
iM
+
v3
R3
-
+
vX
RX
-
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
The Wheatstone Bridge –
Equation
So, we have shown that when R3 is adjusted so that meter reads
zero, this results in the equation below. Since R1, R2, and R3 are
known, we now know RX.
R2
RX  R3
R1
R1
R2
Meter
+
+ vM
vS
-
iM
+
v3
R3
-
+
vX
RX
-
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
The Wheatstone Bridge –
Measurements
Let’s review the basics of the Wheatstone Bridge.
1. The resistors R1, R2, and R3 are known, and R3 is variable.
2. The resistor R3 is varied until the meter reads zero.
3. Because the meter reads zero, the current through it is zero, leaving two series
resistor pairs.
4. Because the meter reads zero, the voltage across it is zero, making the voltage
divider rule voltages equal.
5. Setting these voltages equal and solving yields the equation below.
R1
R2
RX  R3
R1
R2
Meter
+
+ vM
vS
-
iM
+
v3
R3
-
+
vX
RX
-
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Go back to
Overview
slide.
The Wheatstone Bridge –
Operating Notes
Let’s review the advantages of the Wheatstone Bridge.
1. The accuracy of the measurement is determined almost entirely by the
accuracy of the values of the resistors R1, R2, and R3. Typically, it is relatively
easy to have these resistances accurately known.
2. The meter reads zero during the measurement, so the linearity, accuracy and
resistance of the meter do not matter. The meter only needs to detect the point at
which the voltage across it is zero. At this point the bridge is said to be
“balanced”.
3. The source voltage term cancels, so if
vS changes, the accuracy of the
measurement is not seriously affected.
R1
R2
The voltage vS only needs to be large
enough to deflect the meter when the
Meter
bridge is not “balanced”.
+
v
R2
RX  R3
R1
+
vS
-
iM
+
v3
R3
-
-
M
+
vX
RX
-
Dave Shattuck
University of Houston
What’s So Special About
Null-Measurement Techniques?
© Brooks/Cole Publishing Co.
• Null-Measurement Techniques are a clever way of
using the strengths of meters, particularly analog
meters, while minimizing their weaknesses. As
such, they are a good example of problem-solving
approaches.
• In addition, these techniques allow us to exercise
the concepts covered earlier in the module, such as
series resistors and the voltage divider rule.
Go back to
Overview
slide.