1. Meters and Measurements
ECE Circuit Analysis
Lecture Set #4
Meters and Measurements
Dr. Dave Shattuck
Associate Professor, ECE Dept.

2. Meters

3. Overview Meters In this part, we will cover the following topics:
Voltmeters
Ammeters
Ohmmeters
You can click on the links to jump to the subject that you want to learn about now.

4. Textbook Coverage
This material is in your textbook in the following sections:
Electric Circuits 8th Ed. by Nilsson and Riedel: Sections 3.5 & 3.6
You should also read these sections in your text. This material is intended to complement your textbook coverage, not replace it.

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 don’t 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 1[MW] 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[kW/V], will have a resistance of 40[kW] if used on a 2[V] 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 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.

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 user’s 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 slide shows 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.
Let’s add a voltmeter with a resistance of 50[kW] 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.
{This is a good place for an animated graphic. The vX, along with the two labeled nodes A and B, should move from its position in the left hand figure to its position in the right hand figure. Ideally, it should wait a couple of seconds, and then begin moving slowly to the left, and then stopping in the final position for a short time. If this is done, the text needs to be adjusted accordingly.}

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 1[MW], 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.

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.
Extended Range Voltmeters – Proportional Scales
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.
Extended Range Voltmeters – Terminology
The 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 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.
Extended Range Voltmeters – Terminology
The 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.

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 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.

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 user’s 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 slide shows 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.
Let’s add an ammeter with a resistance of 50[W] 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.
{This is a good place for an animated graphic. The iX should move from its position in the left hand figure to its position in the right hand figure. Ideally, it should wait a couple of seconds, and then begin moving slowly to the left, and then stopping in the final position for a short time. If this is done, the text needs to be adjusted accordingly.}

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 0.5[W], 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.

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.
Extended Range Ammeters – Proportional Scales
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.
Extended Range Ammeters – Terminology
The 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.

41. Extended Range Ammeters – Terminology
Go back to Overview slide.
Extended Range Ammeters – Terminology
The 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.

42. Definitions for Meters – 1
This table is available on the course web page.
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

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
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,

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 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.

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.
Simple Ohmmeter Circuit – More Notes
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 8 The 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
You can click on the links to jump to the subject that you want to learn about now.

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
You should also read these sections in your text. This material is intended to complement your textbook coverage, not replace it.

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 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.

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 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 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.

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. 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.

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.
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,

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
Let’s 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.
The Wheatstone Bridge – Operating Notes
Let’s 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. 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.

69. Example Problem #1 This problem is taken from Quiz #2, Fall 2002.
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].
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.