ECE 2300 Circuit Analysis

1
ECE 2300 Circuit Analysis
Lecture Set 4 Meters and Measurements
Dr. Dave Shattuck Associate Professor, ECE Dept.
2
Part 7 Meters
3
Overview of this Part Meters

• In this part, we will cover the following topics
• Voltmeters
• Ammeters
• Ohmmeters

4
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

5
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
• Learning how to connect and use these devices.
• Understanding the limitations of the
  measurements.

6
Voltmeters Fundamental Concepts

• A voltmeter is a device that measures voltage.
  There are a few things we should know about
  voltmeters
• 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.
• 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.
• 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.

7
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 dont have to break any
  connections to connect a voltmeter to a circuit.

8
Voltmeters Fundamental Concept 2

• Voltmeters 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 1MW 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 20kW/V, will have
  a resistance of 40kW if used on a 2V scale.

9
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
  Thevenins Theorem. Therefore, for now, we will
  solve the circuit, with and without the
  resistance of the meter included, and look at the
  difference.

10
Voltmeter Errors

• Two kinds of errors are possible with voltmeter
  measurements.
• 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 users
  reading of the scale.
• 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.

11
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.
• Lets add a voltmeter with a resistance of 50kW
  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.

12
Voltmeter Error Example Intended Measurement

• The 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,

13
Voltmeter Error Example Actual Measurement

• Next, 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.

The standard voltmeter schematic symbol is shown
here. You will sometimes see other symbols for
the voltmeter, or variations on this symbol.
14
Voltmeter Error Example Actual Measurement

• Next, 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.

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.
15
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.

16
Voltmeter Error Example The Resulting Error

• We 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

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 1MW, and you will see that the measured
voltage will then be 1.11V, which is much
closer to the voltage we intend to measure
(1.14V) for this circuit.
17
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.

18
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.

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 vT and vM,

19
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.
20
Extended Range Voltmeters -- Notes

• The 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.

21
Extended Range Voltmeters Proportional Scales
Go back to Overview slide.

• The 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.

Since the scale on an analog voltmeter is linear,
several scales can be easily labeled on the same
meter, each proportional to the other.
22
Extended Range Voltmeters Terminology
Go back to Overview slide.

• The new Extended Range Voltmeter is referred to
  with some common terminology. The Existing
  Voltmeter is often an analog meter called a
  dArsonval meter movement. The voltage at full
  scale across the dArsonval meter movement is
  called vdA,rated. The current at full scale
  through the dArsonval meter movement is called
  idA,rated.

The full-scale values are used to characterize
meters. Remember that all of the full-scale
characteristics occur at the same time.
23
Extended Range Voltmeters Terminology
Go back to Overview slide.

• The new Extended Range Voltmeter is referred to
  with some common terminology. The Existing
  Voltmeter is often an analog meter called a
  dArsonval meter movement. The voltage at full
  scale across the dArsonval meter movement is
  called vdA,rated. The current at full scale
  through the dArsonval meter movement is called
  idA,rated.

The ratio of vdA,rated to idA,rated will be the
resistance of the dArsonval meter movement.
Remember, the dArsonval meter movement is simply
a meter, and can be modeled with a resistance.
24
Ammeters Fundamental Concepts

• An ammeter is a device that measures current.
  There are a few things we should know about
  ammeters
• 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.
• 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.
• 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.

25
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.

26
Ammeters Fundamental Concept 2

• Ammeters 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.

27
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
  Thevenins Theorem. Therefore, for now, we will
  solve the circuit, with and without the
  resistance of the meter included, and look at the
  difference.

28
Ammeter Errors

• Two kinds of errors are possible with ammeter
  measurements.
• 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 users reading of the scale.
• 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.

29
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.
• Lets add an ammeter with a resistance of 50W
  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.

30
Ammeter Error Example Intended Measurement

• The 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,

31
Ammeter Error Example Actual Measurement

• Next, 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.

The standard ammeter schematic symbol is shown
here. You will sometimes see other symbols for
the ammeter, or variations on this symbol.
32
Ammeter Error Example Actual Measurement

• Next, 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.
33
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.

34
Ammeter Error Example The Resulting Error

• We 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

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 0.5W, and you will see that the measured
current will then be 0.62A, which is much
closer to the current we intend to measure
(0.63A) for this circuit.
35
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.

36
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,

37
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.
38
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.

39
Extended Range Ammeters Proportional Scales
Go back to Overview slide.

• 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 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.
40
Extended Range Ammeters Terminology
Go back to Overview slide.

• The new Extended Range Ammeter is referred to
  with some common terminology. The Existing
  Ammeter is often an analog meter called a
  dArsonval meter movement. The voltage at full
  scale across the dArsonval meter movement is
  called vdA,rated. The current at full scale
  through the dArsonval meter movement is called
  idA,rated.

The full-scale values are used to characterize
meters. Remember that all of the full-scale
characteristics occur at the same time.
41
Extended Range Ammeters Terminology
Go back to Overview slide.

• The new Extended Range Voltmeter is referred to
  with some common terminology. The Existing
  Voltmeter is often an analog meter called a
  dArsonval meter movement. The voltage at full
  scale across the dArsonval meter movement is
  called vdA,rated. The current at full scale
  through the dArsonval meter movement is called
  idA,rated.

The ratio of vdA,rated to idA,rated will be the
resistance of the dArsonval meter movement.
Remember, the dArsonval meter movement is simply
a meter, and can be modeled with a resistance.
42
Definitions for Meters 1
This table is available on the course web page.
Term or Variable Definition in words
dArsonval 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 dArsonval meter movement Full scale value for a dArsonval meter movement
idA rated Full scale current for a dArsonval meter movement, which is typically used to produce an ammeter or a voltmeter by adding resistors
vdA rated Full scale voltage for a dArsonval meter movement, which is typically used to produce an ammeter or a voltmeter by adding resistors
43
Definitions for Meters 2
This table is available on the course web page.
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
dArsonval based voltmeter Extended range voltmeter built with a dArsonval meter movement
dArsonval based ammeter Extended range ammeter built with a dArsonval meter movement
RdA The resistance of a dArsonval meter movement. As with any meter, this resistance can be found from the full scale voltage divided by the full scale current. Thus,
44
Ohmmeters Fundamental Concepts

• An ohmmeter is a device that measures resistance.
  There are a few things we should know about
  ohmmeters
• Ohmmeters must have a source in them.
• An ohmmeter measures the ratio of the voltage at
  its terminals, to the current through its
  terminals, and reports the ratio as a resistance.
  
• An analog ohmmeter is often characterized by its
  half-scale reading.

45
Ohmmeters Fundamental Concept 1

• Ohmmeters 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.

46
Ohmmeters Fundamental Concept 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.
  
• 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 Generals 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.
47
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.

48
Simple Ohmmeter Circuit Notes

• We may note several things about this circuit.
• 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.
• If the resistor RX is zero (a short circuit), the
  resistor RO is adjusted to make the meter read
  full scale.

49
Simple Ohmmeter Circuit More Notes
Go back to Overview slide.

• Thus, 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.
50
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.
51
Part 8The Wheatstone Bridge
52
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

53
Textbook Coverage

• This material is covered in your textbook in the
  following section
• Electric Circuits 7th Ed. by Nilsson and Riedel
  Section 3.6

54
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
• They use a standard meter, such as an ammeter or
  voltmeter.
• The measurement occurs when the reading on this
  ammeter or voltmeter is zero.

55
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 DArsonval 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.

56
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.

57
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.

58
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 comes from.
• A source (vS) is used to power the entire
  combination. See the diagram here.

59
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 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.

60
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.

61
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. Lets 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.

62
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,

63
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,

64
The Wheatstone Bridge Derivation Step 4
Go back to Overview slide.

• 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,

65
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.

66
The Wheatstone Bridge Measurements

• Lets review the basics of the Wheatstone Bridge.
• The resistors R1, R2, and R3 are known, and R3 is
  variable.
• The resistor R3 is varied until the meter reads
  zero.
• Because the meter reads zero, the current through
  it is zero, leaving two series resistor pairs.
• Because the meter reads zero, the voltage across
  it is zero, making the voltage divider rule
  voltages equal.
• Setting these voltages equal and solving yields
  the equation below.

67
The Wheatstone Bridge Operating Notes
Go back to Overview slide.

• Lets review the advantages of the Wheatstone
  Bridge.
• 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.
• 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.
• 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.

68
Whats 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.
69
Example Problem 1
The extended-range ammeter shown in Figure 1 uses
an internal ammeter with a 5mA full-scale
current, and three resistors. The internal
ammeter has a full-scale voltage of 100mV.
This problem is taken from Quiz 2, Fall 2002.
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.
70
Example Problem 2
This problem is taken from Problem 3.44 in the
Nilsson and Riedel text.