Dave Shattuck ECE 2300 - libvolume3.xyzlibvolume3.xyz/.../digitalinstruments/digitalinstrumentspresentation2.pdf · Dave Shattuck University of Houston ... Thevenin’s Theorem. Therefore,

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

[email protected]

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


© 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



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.



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.



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.



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.



Voltmeters –


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.



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.



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



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




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.




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.




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]


© 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.



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.


© 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


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=

+



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


+

-

vT

RV

+

-

vM


RMV



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


+

-

vT

RV

+

-

vM


RMV




– 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


+

-

vT

RV

+

-

vM


RMV




– 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


+

-

vT

RV

+

-

vM


RMV




– 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


+

-

vT

RV

+

-

vM


RMV



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.



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.

We usually say that we have to break a connection to connect a ammeter to a circuit.



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.



Ammeters –


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.


© 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.



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



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




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.




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




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]


© 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



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.


© 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


RA

iT

iM

RM

A



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


RA

iT

iM


RA

iT

iM

RM

A



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


RA

iT

iM


RA

iT

iM

RM

A



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


RA

iT

iM


RA

iT

iM

RM

A




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


RA

iT

iM


RA

iT

iM

RM

A




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


RA

iT

iM


RA

iT

iM

RM

A


© 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.


© 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=



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.



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.



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.



Ohmmeters –


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



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



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.


© 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


© 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



Textbook Coverage

This material is covered in your textbook in the

following section:

• Electric Circuits 7th Ed. by Nilsson and Riedel:

Section 3.6



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.


© 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.



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.



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.


© 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



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.




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


© 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




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

=+

=+




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

=+ +




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.




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=



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=



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.


© 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.


© 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].


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.


© University of HoustonExample Problem #2

This problem is

taken from Problem

3.44 in the Nilsson

and Riedel text.

Documents

Dave Shattuck ECE 2300 - libvolume3.xyzlibvolume3.xyz/.../digitalinstruments/digitalinstrumentspresentation2.pdf · Dave Shattuck University of Houston ... Thevenin’s Theorem. Therefore,