ECE 2300 Circuit Analysis

Shattuck@uh.edu 713 743-4422 W326-D3 ECE 2300 Circuit Analysis Lecture Set #4 Meters and Measurements Dr. Dave Shattuck Associate Professor, ECE Dept.

Part 7 Meters

Overview of this PartMeters In this part, we will cover the following topics: • Voltmeters • Ammeters • Ohmmeters

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

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.

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 #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.

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.

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 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[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.

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,

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.

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.

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.

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.

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.

Extended Range Voltmeters 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, 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.

Multiplying Factor for Extended Range Voltmeters 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. 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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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[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.

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,

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.

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.

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.

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.

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.

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,

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.

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.

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.

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.

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.

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

Definitions for Meters – 2 This table is available on the course web page.

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.

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.

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.

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.

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.

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.

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.

ECE 2300 Circuit Analysis