dave shattuck ece 2300 -...
TRANSCRIPT
Dave Shattuck
University of Houston
© University of Houston ECE 2300
Circuit Analysis
Dr. Dave Shattuck
Associate Professor, ECE Dept.
Lecture Set #4
Meters and Measurements
713 743-4422
W326-D3
Part 7 Meters
Dave ShattuckUniversity of Houston
© University of Houston Overview of this Part
Meters
In this part, we will cover the following
topics:
• Voltmeters
• Ammeters
• Ohmmeters
Dave ShattuckUniversity of Houston
© University of Houston
Textbook Coverage
This material is in your textbook in the following
sections:
• Electric Circuits 7th Ed. by Nilsson and Riedel:
Sections 3.5 & 3.6
Dave ShattuckUniversity of Houston
© University of Houston
Meters –
Making Measurements
The subject of this part is meters. We will
consider devices to measure voltage, current, and
resistance. We have two primary goals in this study:
1. Learning how to
connect and use these
devices.
2. Understanding the
limitations of the
measurements.
Dave ShattuckUniversity of Houston
© University of Houston
Voltmeters –
Fundamental ConceptsA voltmeter is a device that measures voltage. There are
a few things we should know about voltmeters:
1. Voltmeters must be placed in parallel with the voltage they are to measure. Generally, this means that the two terminals, or probes, of the voltmeter are connected or touched to the two points between which the voltage is to be measured.
2. Voltmeters can be modeled as resistances. That is to say, from the standpoint of circuit analysis, a voltmeter behaves the same way as a resistor. The value of this resistance may, or may not, be very important.
3. The addition of a voltmeter to a circuit adds a resistance to the circuit, and thus can change the circuit behavior. This change may, or may not, be significant.
Dave ShattuckUniversity of Houston
© University of Houston
Voltmeters –
Fundamental Concept #1
Voltmeters must be placed in parallel with the voltage they are to measure. Generally, this means that the two terminals, or probes, of the voltmeter are connected or touched to the two points between which the voltage is to be measured.
We usually say that we don’t have to break any connections to connect a voltmeter to a circuit.
Dave ShattuckUniversity of Houston
© University of Houston
Voltmeters –
Fundamental Concept #2Voltmeters can be modeled as resistances. That is to say, from the
standpoint of circuit analysis, a voltmeter behaves in the same way as a
resistor. The value of this resistance may, or may not, be very important.
Generally, we will know the resistance of the voltmeter. For most
digital voltmeters, this value is 1[MΩ] or higher, and constant for each
range of measurement. For most analog voltmeters, this value is lower,
and depends on the voltage range being measured. The larger the
resistance, the better, since this will cause a smaller change in the circuit
it is connected to.
For analog voltmeters, the sensitivity of the
meter is the resistance of the voltmeter per [Volt]
on the full-scale range being used. A meter with
a sensitivity of 20[kΩ/V], will have a resistance
of 40[kΩ] if used on a 2[V] scale.
Dave ShattuckUniversity of Houston
© University of Houston
Voltmeters –
Fundamental Concept #3
The addition of a voltmeter to a circuit adds a resistance to the circuit, and thus can change the circuit behavior. This change may, or may not, be significant.
Of course, we would like to know if it is going to be significant.
There are ways to determine whether it will be significant, such as by comparing the resistance to the Thevenin resistance of the circuit being measured. However, we have not yet covered Thevenin’s Theorem. Therefore, for now, we will solve the circuit, with and without the resistance of the meter included, and look at the difference.
Dave ShattuckUniversity of Houston
© University of Houston
Voltmeter Errors
Two kinds of errors are possible with voltmeter measurements.
1. One error is that the meter does not measure the voltage across it accurately. This is a function of how the meter is made, and perhaps the user’s reading of the scale.
2. The other error is that from the addition of a resistance to the circuit. This added resistance is the resistance of the meter. This can change the circuit behavior.
In a circuits course, the primary concern is with the second kind of error, since it relates to circuit concepts. Generally, we assume for circuits problems that the first type of error is zero. That is, we will assume that the voltmeter accurately measures the voltage across it; the error occurs from the change in the circuit caused by the resistance added to the circuit by the voltmeter. The next slideshows an example of what we mean by this.
Dave ShattuckUniversity of Houston
© University of Houston
Voltmeter Error Example
Here is an example on voltmeter errors. We will assume that the voltmeter accurately measures the voltage across it; the error occurs from the change in the circuit caused by the resistance added to the circuit by the voltmeter.
Let’s add a voltmeter with a resistance of 50[kΩ] to terminals A and B in the circuit shown here. The goal would be to measure the voltage across R2, labeled here as vX. We will calculate the voltage it is intended to measure, and then the voltage it actually measures. The difference between these values is the error.
+
-
vS=
4[V]
R1=
83[kΩ]
R2=
33[kΩ]
A
B
+
-
vX
Dave ShattuckUniversity of Houston
© University of Houston
Voltmeter Error Example –
Intended MeasurementThe voltage without the
voltmeter in place is the voltage
that we intend to measure. Stated
another way, this is the voltage
that would be measured with an
ideal voltmeter, with a resistance
that is infinite. Performing the
circuit analysis, we can say that
without the voltmeter in place, the
voltage vX can be found from the
Voltage Divider Rule,
+
-
vS=
4[V]
R1=
83[kΩ]
R2=
33[kΩ]
A
B
+
-
vX
2
2 1
33[k ]4[V] 1.14[V].
33[k ] 83[k ]SX
vR
vR R
Ω= = =
+ Ω + Ω
Dave ShattuckUniversity of Houston
© University of Houston
Voltmeter Error Example –
Actual MeasurementNext, we want to find the voltage vX again, this time with the voltmeter
in place. We have shown the voltmeter in its place to measure the
voltage across R2. Notice that the circuit does not have to be broken to
make the measurement. The next step is to convert this to a circuit that
we can solve; this means that we will replace the voltmeter with its
equivalent resistance.
+
-
vS=
4[V]
R1=
83[kΩ]
R2=
33[kΩ]
A
B
+
-
vX
VoltmeterV
The standard
voltmeter schematic
symbol is shown here.
You will sometimes
see other symbols for
the voltmeter, or
variations on this
symbol.
Dave ShattuckUniversity of Houston
© University of Houston
Voltmeter Error Example –
Actual MeasurementNext, we want to find the voltage vX again, this time with the voltmeter
in place. We have shown the voltmeter in its place to measure the
voltage across R2. Notice that the circuit does not have to be broken to
make the measurement. The next step is to convert this to a circuit that
we can solve; this means that we will replace the voltmeter with its
equivalent resistance.
+
-
vS=
4[V]
R1=
83[kΩ]
R2=
33[kΩ]
A
B
+
-
vX
Voltmeter
A non-standard,
alternative voltmeter
schematic symbol is
shown here. It has an
arrow at an angle to
the connection wires,
implying a
measurement. The
same symbol is often
used with ammeters.
Dave ShattuckUniversity of Houston
© University of Houston
Voltmeter Error Example –
Solving the Circuit We have replaced the voltmeter with its equivalent resistance, RM,
and now we can solve the circuit. We may be tempted to use the voltage divider rule using R1 and R2 again, but this will not work since R1 and R2
are no longer in series.
However, if we combine RM and R2 to an equivalent resistance in parallel, this parallel combination will indeed be in series with R1. We can do this, and still solve for vX, since vX can be identified outside the equivalent parallel combination. This is shown by identifying vX in the diagram at right, showing the voltage between two other points on the same nodes.
+
-
vS=
4[V]
R1=
83[kΩ]
R2=
33[kΩ]
A
B
+
-
vX
RM=
50[kΩ]
+
-
vS=
4[V]
R1=
83[kΩ]
R2=
33[kΩ]
A
B
+
-
vX
RM=
50[kΩ]
Dave ShattuckUniversity of Houston
© University of HoustonVoltmeter Error Example –
The Resulting ErrorWe have replaced the parallel combination of RM and R2
with an equivalent resistance, called RP. Now, RP is in series
with R1, and we can use the voltage divider rule to find vX.
We get
+
-
vS=
4[V]
R1=
83[kΩ]
RP=
20[kΩ]
A
B
+
-
vX
20[k ]4[V]
20[k ] 83[k ]
0.78[V].
X
X
v
v
Ω= =
Ω + Ω
=
As we can see, in this case, the resistance of the voltmeter was too low
to make a very accurate measurement. Repeat this problem, with RM equal to
1[MΩ], and you will see that the
measured voltage will then be 1.11[V], which is much closer to the voltage we
intend to measure (1.14[V]) for this circuit.
Dave ShattuckUniversity of Houston
© University of Houston
Extended Range and
Multirange Voltmeters
A voltmeter with a certain full scale reading, can be made to measure even larger voltages by placing a resistor in series with it. The resistor and the voltmeter combination can then be viewed as a new voltmeter, with a larger range. The measurement requires that the meter resistance be known. This can be used to calculate a multiplying factor for what the voltmeter reads. Once done, this can be repeated for other resistance values, to get a voltmeter with multiple ranges. This allows for simple and inexpensive analog multiple range voltmeters.
Dave ShattuckUniversity of Houston
© University of Houston Extended Range Voltmeters A voltmeter with a certain full scale reading, can be made to measure
even larger voltages by placing a resistor, RV, in series with it. The resistor
and the voltmeter can then be viewed as a new voltmeter, with a larger
range. This is shown here.
+
-
vT
RV
Existing
Voltmeter
+
-
vM
Extended Range Voltmeter
+
-
vT
RV
+
-
vM
Extended Range Voltmeter
RMV
By using the Voltage
Divider Rule, we can
find the multiplying
factor to use to find the
reading for the new
extended range
voltmeter. We replace
the voltmeter with its
equivalent resistance,
RM, and then write the
expression relating vTand vM,
.M
M
M T
V
R
Rv v
R=
+
Dave ShattuckUniversity of Houston
© University of Houston
Multiplying Factor for
Extended Range Voltmeters
A voltmeter with a certain full scale reading, can be made to measure even larger voltages by placing a resistor, RV, in series with it. The resistor and the voltmeter can then be viewed as a new voltmeter, with a larger range.
We solve the VDR
equation we wrote on
the last slide for vT and
we get the multiplying
factor, which is the sum
of the resistances over
the meter resistance.
.
M
M V
M V
T
M
M
T
M
v
v
R
R R
Rv
R
v
R
= ⇒+
+=
+
-
vT
RV
Existing
Voltmeter
+
-
vM
Extended Range Voltmeter
+
-
vT
RV
+
-
vM
Extended Range Voltmeter
RMV
Dave ShattuckUniversity of Houston
© University of Houston
Extended Range Voltmeters -
- NotesThe new Extended Range Voltmeter can now be used to read larger voltages.
The reading of the Existing Voltmeter is multiplied by the sum of the resistances divided by the meter resistance. Thus, the Extended Range Voltmeter can read larger voltages, and in addition has a larger effective meter resistance, which is the sum of the resistances.
By choosing different values of RV, we can also obtain a multirange voltmeter. Inexpensive multirange analog voltmeters are built by using a switch, or a series of connection points, to connect different series resistances to a single analog meter.
.M V
M
MTv
R Rv
R
+=
+
-
vT
RV
Existing
Voltmeter
+
-
vM
Extended Range Voltmeter
+
-
vT
RV
+
-
vM
Extended Range Voltmeter
RMV
Dave ShattuckUniversity of Houston
© University of Houston
Extended Range Voltmeters
– Proportional ScalesThe new Extended Range Voltmeter can now be used to read larger voltages. The
reading of the Existing Voltmeter is multiplied by the sum of the resistances divided by the meter resistance. Thus, the Extended Range Voltmeter can read larger voltages, and in addition has a larger effective meter resistance, which is the sum of the resistances.
By choosing different values of RV, we can also obtain a multirange voltmeter. Inexpensive multirange analog voltmeters are built by using a switch, or a series of connection points, to connect different series resistances to a single analog meter.
.M V
M
MTv
R Rv
R
+=
Since the scale on
an analog voltmeter
is linear, several
scales can be easily
labeled on the same
meter, each
proportional to the
other.
Go back to
Overview
slide.
+
-
vT
RV
Existing
Voltmeter
+
-
vM
Extended Range Voltmeter
+
-
vT
RV
+
-
vM
Extended Range Voltmeter
RMV
Dave ShattuckUniversity of Houston
© University of Houston
Extended Range Voltmeters
– TerminologyThe new Extended Range Voltmeter is referred to with
some common terminology. The Existing Voltmeter is often an analog meter called a d’Arsonval meter movement. The voltage at full scale across the d’Arsonval meter movement is called vd’A,rated. The current at full scale through the d’Arsonval meter movement is called id’A,rated.
.M V
M
MTv
R Rv
R
+=
The full-scale values are
used to characterize
meters. Remember that
all of the full-scale
characteristics occur at the same time.
Go back to
Overview
slide.
+
-
vT
RV
Existing
Voltmeter
+
-
vM
Extended Range Voltmeter
+
-
vT
RV
+
-
vM
Extended Range Voltmeter
RMV
Dave ShattuckUniversity of Houston
© University of Houston
Extended Range Voltmeters
– TerminologyThe new Extended Range Voltmeter is referred to with
some common terminology. The Existing Voltmeter is often an analog meter called a d’Arsonval meter movement. The voltage at full scale across the d’Arsonval meter movement is called vd’A,rated. The current at full scale through the d’Arsonval meter movement is called id’A,rated.
.M V
M
MTv
R Rv
R
+=
The ratio of vd’A,rated to id’A,ratedwill be the resistance of the
d’Arsonval meter movement.
Remember, the d’Arsonval
meter movement is simply a meter, and can be modeled
with a resistance.
Go back to
Overview
slide.
+
-
vT
RV
Existing
Voltmeter
+
-
vM
Extended Range Voltmeter
+
-
vT
RV
+
-
vM
Extended Range Voltmeter
RMV
Dave ShattuckUniversity of Houston
© University of Houston
Ammeters –
Fundamental Concepts
An ammeter is a device that measures current. There are a few things we should know about ammeters:
1. Ammeters must be placed in series with the current they are to measure. Generally, this means that the circuit is broken, and then the two terminals, or probes, of the ammeter are connected or touched to the two points where the break was made.
2. Ammeters can be modeled as resistances. That is to say, from the standpoint of circuit analysis, an ammeter behaves the same way as a resistor. The value of this resistance may, or may not, be very important.
3. The addition of an ammeter to a circuit adds a resistance to the circuit, and thus can change the circuit behavior. This change may, or may not, be significant.
Dave ShattuckUniversity of Houston
© University of Houston
Ammeters –
Fundamental Concept #1
Ammeters must be placed in series with the current they are to measure. Generally, this means that the circuit is broken, and then the two terminals, or probes, of the ammeter are connected or touched to the two points where the break was made.
We usually say that we have to break a connection to connect a ammeter to a circuit.
Dave ShattuckUniversity of Houston
© University of Houston
Ammeters –
Fundamental Concept #2Ammeters can be modeled as resistances. That is to
say, from the standpoint of circuit analysis, an ammeter
behaves in the same way as a resistor. The value of this
resistance may, or may not, be very important.
Generally, we will know the resistance of the ammeter.
The smaller the resistance, the better, since this will cause a
smaller change in the circuit it is connected to.
Dave ShattuckUniversity of Houston
© University of Houston
Ammeters –
Fundamental Concept #3
The addition of an ammeter to a circuit adds a resistance to the circuit, and thus can change the circuit behavior. This change may, or may not, be significant.
Of course, we would like to know if it is going to be significant.
There are ways to determine whether it will be significant, such as by comparing the resistance to the Thevenin resistance of the circuit being measured. However, we have not yet covered Thevenin’s Theorem. Therefore, for now, we will solve the circuit, with and without the resistance of the meter included, and look at the difference.
Dave ShattuckUniversity of Houston
© University of Houston Ammeter Errors
Two kinds of errors are possible with ammeter measurements.
1. One error is that the meter does not measure the current through it accurately. This is a function of how the meter is made, and perhaps the user’s reading of the scale.
2. The other error is that from the addition of a resistance to the circuit. This added resistance is the resistance of the meter. This can change the circuit behavior.
In a circuits course, the primary concern is with the second kind of error, since it relates to circuit concepts. Generally, we assume for circuits problems that the first type of error is zero. That is, we will assume that the ammeter accurately measures the current through it; the error occurs from the change in the circuit caused by the resistance added to the circuit by the ammeter. The next slideshows an example of what we mean by this.
Dave ShattuckUniversity of Houston
© University of Houston
Ammeter Error Example Here is an example on
ammeter errors. We will assume that the ammeter accurately measures the current through it; the error occurs from the change in the circuit caused by the resistance added to the circuit by the ammeter.
Let’s add an ammeter with a resistance of 50[Ω] to terminals A and B in the circuit shown here. The goal would be to measure the current through R2, labeled here as iX. We will calculate the current it is intended to measure, and then the current it actually measures. The difference between these values is the error.
R1=
150[Ω]
A B
R2=
330[Ω]
iS=
2[A]
iX
Dave ShattuckUniversity of Houston
© University of Houston
Ammeter Error Example –
Intended MeasurementThe current without the
ammeter in place is the current
that we intend to measure. Stated
another way, this is the current
that would be measured with an
ideal ammeter, with a resistance
that is zero. Performing the circuit
analysis, we can say that without
the ammeter in place, the current
iX can be found from the Current
Divider Rule,
1
2 1
150[ ]2[A] 0.63[A].
150[ ] 330[ ]SX
iR
iR R
Ω= = =
+ Ω + Ω
R1=
150[Ω]
A B
R2=
330[Ω]
iS=
2[A]
iX
Dave ShattuckUniversity of Houston
© University of Houston
Ammeter Error Example –
Actual MeasurementNext, we want to find the current iX again, this time with the ammeter
in place. We have shown the ammeter in its place to measure the current
through R2. Notice that the circuit had to be broken to make the
measurement. The next step is to convert this to a circuit that we can
solve; this means that we will replace the ammeter with its equivalent
resistance.
R1=
150[Ω]
A B
R2=
330[Ω]
iS=
2[A]
iX
Ammeter
A
The standard
ammeter schematic
symbol is shown here.
You will sometimes
see other symbols for
the ammeter, or
variations on this
symbol.
Dave ShattuckUniversity of Houston
© University of Houston
Ammeter Error Example –
Actual MeasurementNext, we want to find the current iX again, this time with the ammeter
in place. We have shown the ammeter in its place to measure the current
through R2. Notice that the circuit had to be broken to make the
measurement. The next step is to convert this to a circuit that we can
solve; this means that we will replace the ammeter with its equivalent
resistance.
A non-standard
alternative ammeter
schematic symbol is
shown here. It has an
arrow at an angle to
the connection wires,
implying a
measurement. The
same symbol is often
used with voltmeters.
R1=
150[Ω]
A B
R2=
330[Ω]
iS=
2[A]
iX
Ammeter
Dave ShattuckUniversity of Houston
© University of Houston
Ammeter Error Example –
Solving the Circuit We have replaced the ammeter with its equivalent resistance, RM,
and now we can solve the circuit. We may be tempted to use the current divider rule using R1 and R2 again, but this will not work since R1 and R2
are no longer in parallel.
However, if we combine RM and R2 to an equivalent resistance in series, this series combination will indeed be in parallel with R1. We can do this, and still solve for iX, since iX can be identified outside the equivalent series combination. This is shown by identifying iX in the diagram at right, showing the current entering the same combination.
R1=
150[Ω]
A B
R2=
330[Ω]
iS=
2[A]
iX
RM=
50[Ω]
R1=
150[Ω]
A B
R2=
330[Ω]
iS=
2[A]
iX
RM=
50[Ω]
Dave ShattuckUniversity of Houston
© University of HoustonAmmeter Error Example –
The Resulting ErrorWe have replaced the series combination of RM and R2
with an equivalent resistance, called RS. Now, RS is in
parallel with R1, and we can use the current divider rule to
find iX. We get
150[ ]2[A]
150[ ] 380[ ]
0.57[A].
X
X
i
i
Ω= =
Ω + Ω
=
As we can see, in this case, the resistance of the ammeter was too large
to make a very accurate measurement. Repeat this problem, with RM equal to
0.5[Ω], and you will see that the
measured current will then be 0.62[A], which is much closer to the current we
intend to measure (0.63[A]) for this circuit.
R1=
150[Ω]
RS=
380[Ω]
iS=
2[A]
iX
Dave ShattuckUniversity of Houston
© University of Houston
Extended Range and
Multirange Ammeters
An ammeter with a certain full scale reading, can be made to measure even larger currents by placing a resistor in parallel with it. The resistor and the ammeter combination can then be viewed as a new ammeter, with a larger range. The measurement requires that the meter resistance be known. This can be used to calculate a multiplying factor for what the ammeter reads. Once done, this can be repeated for other resistance values, to get an ammeter with multiple ranges. This allows for simple and inexpensive analog multiple range ammeters.
Dave ShattuckUniversity of Houston
© University of Houston Extended Range Ammeters An ammeter with a certain full scale reading, can be made to measure
even larger currents by placing a resistor, RA, in parallel with it. The
resistor and the ammeter can then be viewed as a new ammeter, with a
larger range. This is shown here.
By using the Current Divider
Rule, we can find the
multiplying factor to use to find
the reading for the new
extended range ammeter. We
replace the ammeter with its
equivalent resistance, RM, and
then write the expression
relating iT and iM,
.A
M
M T
A
R
Ri i
R=
+
Existing
Ammeter
Extended Range Ammeter
RA
iT
iM
Extended Range Ammeter
RA
iT
iM
RM
A
Dave ShattuckUniversity of Houston
© University of Houston
Multiplying Factor for
Extended Range Ammeters An ammeter with a certain full scale reading, can be made to measure
even larger currents by placing a resistor, RA, in parallel with it. The resistor and the ammeter can then be viewed as a new ammeter, with a larger range.
We solve the CDR
equation we wrote on
the last slide for iT and
we get the multiplying
factor, which is the sum
of the resistances over
the parallel resistance.
.
A
A M
M A
T
M
A
T
M
i
i
R
R R
Ri
R
i
R
= ⇒+
+=
Existing
Ammeter
Extended Range Ammeter
RA
iT
iM
Extended Range Ammeter
RA
iT
iM
RM
A
Dave ShattuckUniversity of Houston
© University of Houston
Extended Range Ammeters --
Notes
The new Extended Range Ammeter can now be used to read larger currents. The reading of the Existing Ammeter is multiplied by the sum of the resistances divided by the parallel resistance. Thus, the Extended Range Ammeter can read larger currents, and in addition has a smaller effective meter resistance, which is the parallel combination of the resistances.
By choosing different values of RA, we can also obtain a multirange ammeter. Inexpensive multirange analog ammeters are built by using a switch, or a series of connection points, to connect different parallel resistances to a single analog meter.
.M A
A
MTi
R Ri
R
+=
Existing
Ammeter
Extended Range Ammeter
RA
iT
iM
Extended Range Ammeter
RA
iT
iM
RM
A
Dave ShattuckUniversity of Houston
© University of Houston
Extended Range Ammeters –
Proportional ScalesThe new Extended Range Ammeter can now be used to read larger currents. The
reading of the Existing Ammeter is multiplied by the sum of the resistances divided by the parallel resistance. Thus, the Extended Range Ammeter can read larger currents, and in addition has a smaller effective meter resistance, which is the parallel combination of the resistances.
By choosing different values of RA, we can also obtain a multirange ammeter. Inexpensive multirange analog ammeters are built by using a switch, or a series of connection points, to connect different parallel resistances to a single meter.
Since the scale on an analog ammeter
is linear, several scales can be easily
labeled on the same meter, each
proportional to the other.
.M A
A
MTi
R Ri
R
+=
Go back to
Overview
slide.
Existing
Ammeter
Extended Range Ammeter
RA
iT
iM
Extended Range Ammeter
RA
iT
iM
RM
A
Dave ShattuckUniversity of Houston
© University of Houston
Extended Range Ammeters –
TerminologyThe new Extended Range
Ammeter is referred to with some common terminology. The Existing Ammeter is often an analog meter called a d’Arsonval meter movement. The voltage at full scale across the d’Arsonval meter movement is called vd’A,rated. The current at full scale through the d’Arsonval meter movement is called id’A,rated.
The full-scale values are used to
characterize meters. Remember
that all of the full-scale
characteristics occur at the same
time.
Go back to
Overview
slide.
.M A
A
MTi
R Ri
R
+=
Existing
Ammeter
Extended Range Ammeter
RA
iT
iM
Extended Range Ammeter
RA
iT
iM
RM
A
Dave ShattuckUniversity of Houston
© University of Houston
Extended Range Ammeters –
TerminologyThe new Extended Range
Voltmeter is referred to with some common terminology. The Existing Voltmeter is often an analog meter called a d’Arsonval meter movement. The voltage at full scale across the d’Arsonval meter movement is called vd’A,rated. The current at full scale through the d’Arsonval meter movement is called id’A,rated.
The ratio of vd’A,rated to id’A,rated will be the
resistance of the d’Arsonval meter
movement. Remember, the d’Arsonval
meter movement is simply a meter, and
can be modeled with a resistance.
Go back to
Overview
slide.
.M A
A
MTi
R Ri
R
+=
Existing
Ammeter
Extended Range Ammeter
RA
iT
iM
Extended Range Ammeter
RA
iT
iM
RM
A
Dave ShattuckUniversity of Houston
© University of HoustonDefinitions for Meters – 1
Term or Variable Definition in words
d’Arsonval meter
movement
A common version of an analog meter. The
deflection of the meter is proportional to the
current through it, and to the voltage across it. It
can be modeled as a resistance.
Rated value for
d’Arsonval meter
movement
Full scale value for a d’Arsonval meter
movement
id’A rated Full scale current for a d’Arsonval meter
movement, which is typically used to produce an
ammeter or a voltmeter by adding resistors
vd’A rated Full scale voltage for a d’Arsonval meter
movement, which is typically used to produce an
ammeter or a voltmeter by adding resistors
This table is available on the course web page.
Dave ShattuckUniversity of Houston
© University of HoustonDefinitions for Meters – 2
Term or Variable Definition in words
imeter, fullscale or iFS Full scale current for an extended range meter
vmeter, fullscale or vFS Full scale voltage for an extended range meter
d’Arsonval based
voltmeter
Extended range voltmeter built with a d’Arsonval
meter movement
d’Arsonval based
ammeter
Extended range ammeter built with a d’Arsonval
meter movement
Rd’A The resistance of a d’Arsonval meter
movement. As with any meter, this resistance
can be found from the full scale voltage divided
by the full scale current. Thus,
This table is available on the course web page.
' '
'
.d A RATEDd A
d A RATED
vR
i=
' '
'
.d A RATED
d Ad A RATED
vR
i=
Dave ShattuckUniversity of Houston
© University of Houston
Ohmmeters –
Fundamental Concepts
An ohmmeter is a device that measures
resistance. There are a few things we should know
about ohmmeters:
1. Ohmmeters must have a source in them.
2. An ohmmeter measures the ratio of the voltage at
its terminals, to the current through its terminals,
and reports the ratio as a resistance.
3. An analog ohmmeter is often characterized by its
half-scale reading.
Dave ShattuckUniversity of Houston
© University of Houston
Ohmmeters –
Fundamental Concept #1Ohmmeters must have a source in
them.
The voltmeters and ammeters we discussed earlier may or may not have a source within them; they may use the voltage or current that they are measuring to power the measurement. However, a resistor does not provide power, and a source must be present to provide this.
Thus, while an analog voltmeter or ammeter may work without a battery, it is not possible for an ohmmeter to work without a battery or other source of power.
Dave ShattuckUniversity of Houston
© University of Houston
Ohmmeters –
Fundamental Concept #2An ohmmeter measures the ratio of the
voltage at its terminals, to the current through its terminals, and reports the ratio as a resistance.
This is a key idea about ohmmeters. We could say that an ohmmeter assumes that everything is a resistor. If we connect the ohmmeter to something other than a resistor, such as a battery, it will report the ratio of the voltage to the current at its terminals, even though this may be a meaningless number.
Electrical-Engineer General’s Warning: It is
important to remove a resistor from its circuit
before measuring it with an ohmmeter. If we do
not, the measurement we obtain may not have any
meaning.
Dave ShattuckUniversity of Houston
© University of Houston
Ohmmeters –
Fundamental Concept #3
An analog ohmmeter is
often characterized by its
half-scale reading.An analog ohmmeter will
have a scale which has zero on
one end, and infinity on the other
end. This is true no matter what
the “range” it is set to. To
understand this, it is useful to
look at the internal circuit of the
ohmmeter. A typical circuit for a
simple analog ohmmeter is
shown here.
Ohmmeter Circuit
+
-
vB
(Battery
voltage)
Meter
Adjustable
Resistor RO
Unknown
Resistor RX
Dave ShattuckUniversity of Houston
© University of Houston
Simple Ohmmeter Circuit Notes
We may note several things about this circuit. 1. If the resistor RX is infinity (an open circuit), the current through the meter will be zero. The meter will be at one end of its scale.2. If the resistor RX is zero (a short circuit), the resistor RO is adjusted to make the meter read full scale.
Ohmmeter Circuit
+
-
vB
(Battery
voltage)
Meter
Adjustable
Resistor RO
Unknown
Resistor RX
Dave ShattuckUniversity of Houston
© University of Houston
Simple Ohmmeter Circuit –
More NotesThus, the value of the
resistor RO is adjusted to make the meter read full scale when RX is zero. Thus, the full-scale current must be equal to vB divided by the series combinations of the meter resistance and RO. It follows that half the full-scale current will result when RX equals this series combination.
A potentially useful bit of information is this: the half-scale reading of an
analog ohmmeter is equal to the internal resistance of the meter. Ohmmeter Circuit
+
-
vB
(Battery
voltage)
Meter
Adjustable
Resistor RO
Unknown
Resistor RX
Go back to
Overview
slide.
Dave ShattuckUniversity of Houston
© University of Houston What is the Point of
Considering Analog Meters?
• This is a good question, considering how accurate, inexpensive, and easy to use digital meters have become.
• The answer is two fold: First, there are still several applications for analog meters, and it is important to understand them. The benefits are made more important since the meters themselves are relatively simple and easy to understand.
• Second, an understanding of these meter concepts allow digital meters to be understood, from an applications standpoint. For example, we can extend the operating range of a digital voltmeter by adding a series resistor, just as we did with analog voltmeters. Go back to
Overview
slide.
Part 8
The Wheatstone Bridge
Dave ShattuckUniversity of Houston
© University of Houston Overview of this Part
The Wheatstone Bridge
In this part, we will cover the following
topics:
• Null Measurement Techniques
• Wheatstone Bridge Derivation
• Wheatstone Bridge Measurements
Dave ShattuckUniversity of Houston
© University of Houston
Textbook Coverage
This material is covered in your textbook in the
following section:
• Electric Circuits 7th Ed. by Nilsson and Riedel:
Section 3.6
Dave ShattuckUniversity of Houston
© University of Houston
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 ShattuckUniversity of Houston
© University of HoustonNull-Measurement
Techniques – Note 1Null-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 ShattuckUniversity of Houston
© University of HoustonNull-Measurement
Techniques – Note 2The 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 ShattuckUniversity of Houston
© University of HoustonNull-Measurement
Techniques – Note 3We 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 ShattuckUniversity of Houston
© University of Houston 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 R1and R3, and R2 and RX. The meter is said to “bridge” the
midpoints of these two pairs of resistors, which is
where the name
comes from.
A source (vS) is
used to power the
entire combination.
See the diagram here.
+
-
vS
R1
RX
R2
R3
Meter
Dave ShattuckUniversity of Houston
© University of Houston
The Wheatstone Bridge – Notes
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 might be considered to be a voltmeter. However, it should be noted that a meter is a resistor from a circuits viewpoint, so that when the voltage is zero the current is also zero.
+
-
vS
R1
RX
R2
R3
Meter
Go back to
Overview
slide.
Dave ShattuckUniversity of Houston
© University of Houston
The Wheatstone Bridge –
The Nulling StepTo 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.
+
-
vS
R1
RX
R2
R3
Meter
+ vM -
iM
Dave ShattuckUniversity of Houston
© University of Houston The Wheatstone Bridge –
Derivation Step 1Using 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.
+
-
vS
R1
RX
R2
R3
Meter
+ vM -
iM
Dave ShattuckUniversity of Houston
© University of Houston
The Wheatstone Bridge –
Derivation Step 2Second, 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,
+
-
vS
R1
RX
R2
R3
Meter
+ vM -
iM +
vX
-
+
v3
-
3
33
1
2
, and
.X
S
X
S
X
RvR R
Rv
R
v
Rv
=+
=+
Dave ShattuckUniversity of Houston
© University of Houston
The Wheatstone Bridge –
Derivation Step 3Third, 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,
+
-
vS
R1
RX
R2
R3
Meter
+ vM -
iM +
vX
-
+
v3
-
3
3 1 2
.X
S S
X
R Rv vR R R R
=+ +
Dave ShattuckUniversity of Houston
© University of Houston
The Wheatstone Bridge –
Derivation Step 4Fourth, 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,
+
-
vS
R1
RX
R2
R3
Meter
+ vM -
iM +
vX
-
+
v3
-
3
3 1 2
23
1
.
This can be solved
for ,
.
X
X
X
X
R R
R R R R
R
RR R
R
=+ +
=
Go back to
Overview
slide.
Dave ShattuckUniversity of Houston
© University of Houston
The Wheatstone Bridge –
EquationSo, 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.
+
-
vS
R1
RX
R2
R3
Meter
+ vM -
iM +
vX
-
+
v3
-
23
1
X
RR R
R=
Dave ShattuckUniversity of Houston
© University of Houston The Wheatstone Bridge –
MeasurementsLet’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.
+
-
vS
R1
RX
R2
R3
Meter
+ vM -
iM +
vX
-
+
v3
-
23
1
X
RR R
R=
Dave ShattuckUniversity of Houston
© University of Houston The Wheatstone Bridge –
Operating NotesLet’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. The voltage vS only needs to be large enough to deflect the meter when the bridge is not “balanced”. +
-
vS
R1
RX
R2
R3
Meter
+ vM -
iM +
vX
-
+
v3
-
23
1
X
RR R
R=
Go back to
Overview
slide.
Dave ShattuckUniversity of Houston
© University of Houston What’s So Special About
Null-Measurement Techniques?
• 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.
Dave ShattuckUniversity of Houston
© University of HoustonExample Problem #1
The extended-range ammeter shown in
Figure 1 uses an internal ammeter with
a 5[mA] full-scale current, and three
resistors. The internal ammeter has a
full-scale voltage of 100[mV].
Extended Range Ammeter
7[Ω]
Internal Ammeter
100[mV]
5[mA]
10[Ω] 5[Ω]a
b
Figure 1
3[Ω]c
d
Figure 2
12[Ω]50[mV]
+
-
a) Find the full-scale current of the extended range ammeter.
b) The circuit shown in Figure 2 was connected to the extended-
range ammeter, connecting terminal a to terminal c, and terminal b to
terminal d. Find the reading of the extended-range ammeter for this
situation.
This
problem
is taken
from
Quiz #2, Fall
2002.
Dave ShattuckUniversity of Houston
© University of HoustonExample Problem #2
This problem is
taken from Problem
3.44 in the Nilsson
and Riedel text.