ME 101 Measurement Demonstration (MD 2)

Linear Systems and Simple Resistance Measurements

Measurement Demonstration (MD2) - Instructions & Homework.doc

Objective: To learn about linear systems and linear transducers from an engineering standpoint and to conduct measurements on a simple example: a resistive circuit.

Technical Outcomes: Upon completion of this experiment, the student will be able to:

− Understand the concept of a linear system and determine linearity − Understand the concept of resistance and conductance − Conduct simple experiments with the power supply (PS) and digital multimeter (DMM) − Become acquainted with breadboards and build simple resistive series and parallel circuits

using the solderless breadboard − Experimentally measure resistance, voltage and current

Equipment: The following is a list of equipment used in the experiment:

− Tektronix TX3 handheld digital multimeter (DMM) − Tektronix PS280 Triple output Direct Current (DC) power supply − Breadboard − Assorted resistors − Jumper Kit

Glossary of Basic Physical Quantities Used in this Experiment In this experiment, you will be asked to measure voltage, current, and resistance in an electric circuit. It is very important to define these quantities before describing the experiment in detail. Charge (Q) – Charge is the property that certain particles have which produce either forces of attraction or repulsion between the particles. Charge has polarity (+/-) and is measured in Coulombs. A single electron has a charge of -1.6 × 10-19 Coulombs. The symbol for charge is Q. Current (I) - Electrical current is simply the flow or rate of change of charge with respect to time. If the rate of change of charge is constant, then direct current (DC) results. Current has direction indicated by an arrow and is measured in amperes which is simply a Coulomb/second. Voltage (V) – Voltage is simply the measure of potential energy available to move or drive a current from one point to another. Voltage has polarity as indicated by a plus (+) and minus (-) sign and it is measured in Volts which is simply joules/Coulomb and is represented by the symbol, V. Typical voltage sources are batteries and power supplies. Circuit – A connection of basic circuit components which has at least one complete path through which electrons can flow from the negative (-) terminal of a voltage source to the positive terminal. Circuit Element - A basic component of a circuit. Examples include resistors, capacitors, voltage sources, diodes, and inductors. Resistance (R) - Resistance is a measure of how much a circuit element opposes the flow of electric current. Resistance is measured in Ohms and is represented by the symbol, R.

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Measurement Demonstration (MD2) Introduction: A favorite analogy is the comparision of engineers to carpenters. A master carpenter can design and fashion extraordinary pieces of fine furniture but is powerless to do so without a set of tools. With a limited set of rudimentary tools, the carpenter can probably practice his/her trade, but the final quality and amount of time required to complete a project will certainly be impaired. Similarly, an engineer can design systems and devices to resolve many complex problems. But, like the carpenter, an engineer’s productivity is severely hampered without mastery of the critical tools of the trade. In lieu of a saw, hammer, and file an engineer uses math, science and computers to craft solutions. Perhaps the most important tool an engineer has is the ability to conduct experiments. In fact, the Accreditation Board for Engineering and Technology (ABET) demands that every graduate of an accredited program have “an ability to design and conduct experiments, as well as to analyze and interpret data.” The purpose of the measurement labs in ME 101 is to do precisely that. Although this experiment has already been designed, the student will have to analyze and interpret the resulting data. Additionally, this experiment will give a glimpse into laboratory experimentation as it pertains to all fields of engineering and to impart certain basic competencies needed in future curricula and as an engineering professional. This laboratory is divided into two parts. The first part is to acquaint you with the use of the basic equipment in this experiment and the second part is devoted to conducting actual experiments on linear systems and simple resistive networks.

Equipment used in Experiment 2 In this experiment, you will learn how to use the Tektronix TX3 handheld digital multimeter (DMM), the Tektronix PS280 Triple output Direct Current (DC) power supply, and how to construct simple circuits using a solderless breadboard. Tektronix TX3 DMM A handheld Digital Multimeter (DMM) is very much like a calculator for the engineer in the field. With a DMM, an engineer can quickly measure voltage (both DC and AC), current, resistance, capacitance, frequency, and circuit continuity. One might make the mistake of thinking a DMM is only useful to Electrical and Computer Engineers. Yet, with the addition of special transducers available for most DMMs, the engineer can also measure temperature, pressure, force and other “non-electrical engineering” entities. In this experiment, the DMM will be used to measure resistance, voltage, and current. A picture of the Tektronix TX3 DMM is shown in Figure 2. There are three main areas of operation of the DMM: the terminal connections, the rotary dial, and the soft (menu) buttons. The DMM uses probes or leads (red and black wires) to connect to a circuit and measure different quantities. These leads insert into different terminals of the DMM depending upon the quantity you wish to measure. Similarly, the rotary dial and menu buttons must be configured differently for measuring current, voltage, or resistance. The following lists the procedures for configuring the DMM correctly: To measure DC Voltage drop across a circuit element with the DMM:

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− Insert the red probe into the rightmost red DMM terminal marked V¬°C (V for Volts)

− Insert the black probe into the middle black DMM terminal marked COM − Turn the rotary dial to the V position − Press the second menu key from the left until DC appears on the LCD Display − Touch the other end of the red/black probes to either side of the circuit element and the

display will show the DC voltage drop across this circuit element. NOTE: Inserting the probes across a circuit element like this is called a parallel connection. Two circuit elements in parallel always have the same voltage across them. In this case, the DMM is in parallel with the circuit element and hence the voltage measured by the DMM is the voltage across the circuit element. Figure 1 shows how to use the DMM to measure voltage.

Figure 1 – Using the DMM to measure the voltage across a circuit element.

1.234 V DC

OFF

V

RANGE

M/M/A

1MS

A

HOLD

ME A COM V oC

5A 1000V

+

V = ? -

V = 1.234 Volts

Circuit Element

Red Probe

Black Probe

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Figure 2 - A Picture of the Tektronix TX3 Digital Multimeter (courtesy of Tektronix). To measure DC current through a circuit element with the DMM:

LCD Display (Measuring 120.03 Volts at 60 Hertz on Auto range)

Menu Button #2 (Measuring AC Volts)

Function Dial (Set to Measure Volts)

3 Probe Connection Terminals A = Amperage

COM = Common V = Voltage

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− Insert the red probe into the leftmost red DMM terminal marked A (A for Amperes)

− Insert the black probe into the middle black DMM terminal marked COM − Turn the rotary dial to the A position − Press the second menu key from the left until DC appears on the LCD Display − Disconnect one end of the circuit element from the circuit then touch the black probe to the

end of the circuit element just disconnected. Next, touch the red probe to the circuit at the point where the element was disconnected. Note that the DMM is now being used to reconnect the circuit element to the circuit and the display will show the DC current (amperage) flowing through this circuit element.

NOTE: Inserting the probes to re-connect a circuit element into the circuit like this is called a series connection. Two circuit elements in series always have the same current flowing through them. In this case, the DMM is in series with the circuit element since the same current runs through the DMM and the circuit element. Thus, the current measured through the DMM is the same current through the circuit element. Figure 3 depicts how to use the DMM to measure current.

To measure the current flowing through a circuit element, disconnect one end of the circuit element from the circuit then insert the DMM in series as shown in Figure 3.

Current = ?

Circuit Element

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Figure 3 – Using the DMM to measure current flowing through a circuit element

Current = 1.234 Amps

1.234 A DC

OFF

A Δ

HOLD

MEM

RANGE

M/M/A

1MS

A COM V oC

5A 1000V

Red Probe

Black Probe

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To measure resistance using the DMM:

− Insert the red probe into the rightmost red DMM terminal marked V¬°C (V for volts)

− Insert the black probe into the middle black DMM terminal marked COM − Turn the rotary dial to the Ω/ position − Touch the other end of the red/black probes to the points in the circuit across which the

resistance is to be measured. The LCD the display will show the value of the resistance in Ohms across these two points in the circuit.

NOTE: The value you read using the above method will not necessarily be the value of the resistance of the circuit element connected across the two points of measurement. If the circuit is active or if other elements are connect across the same points, the value measured will not be the resistance of the circuit element. To determine the value of the resistance of a particular circuit element, the element must be completely removed from the circuit in question prior to measurement. Figure 4 shows how to use the DMM to measure the resistance of a circuit element.

Power Supply

1.234 ΩΩ DC

OFF

ΩΩ RANGE

M/M/A

1MS

A

HOLD

MEM

A COM V oC

5A 1000V

ΩΩ = ?

Circuit Element

Red Probe

Black Probe

ΩΩ = 1.234 Ohms

Figure 4 – Using the DMM to measure resistance of a circuit element.

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This experiment will also incorporate the Tektronix PS280 triple output DC power supply. Direct Current or DC simply means that the voltage (or current) eminating from the supply is a constant and does not vary with time. This is different from the power available from a wall outlet which varies sinusoidally with time. The PS280 contains a total of 3 different power supplies: two 0-30 Volt variable supplies and one fixed 5 Volt supply. See Figure 5 for a picture of the PS280 and its functions.

Figure 5 – Picture of front of Tektronix PS280 Power Supply (courtesy of Tektronix). Basic Circuit Elements A circuit is a interconnection of elementary circuit elements that contains at least one closed-path for current to flow. For this lab, we will deal with just two simple circuits and two simple current elements, an independent voltage source (represented by the PS280 power supply) and a resistor. An independent voltage source is depicted in Figure 6 and is characterized by supplying a constant voltage regardless of the amount of current passing through it. Also, note that plus and minus signs are used to indicate the polarity of the voltage. In the example, the independent voltage source has a voltage of 1.234 volts with plus polarity indicated at the top. The diagram for a resistor is also shown in Figure 6. The resistors we will use in this experiment are made of carbon and are used to oppose the flow of current in a circuit. There are many other types of circuit elements, diodes, capacitors, inductors, transistors, current sources, etc. The independent voltage source and resistor will more than suffice for this first experiment.

Internal Voltmeter Displays

Current Limiter Knobs (2) Voltage Adjust Knobs (2)

+/-/GND Output Terminals (two 30 Volt variable, one 5 Volt fixed)

Power On/Off Button

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Figure 6 – Basic Circuit Symbols for an independent voltage source and a resistor. Solderless Breadboard A solderless breadboard will be used in this experiment to rapidly build circuits. The breadboard is made of white plastic with many holes on the top and a metal plate on the bottom. Running down the length of the breadboard is a trough or channel which is used for inserting analog and digital dual-inline package DIP chips. We wont be using DIP chips in this experiment, but it is important to know how the holes are electrically connected to correctly construct a circuit. If we could peel the plastic cover from the bread board, under each row of five holes we would reveal a number of five fingered metal clips looking like this:

Figure 7 – Internal construction of a breadboard.

Breadboard channel electrically isolates the two rows of 5 holes

Metal clip underneath breadboard electrically connects this set of 5 holes

+ _ 1.234 Volts

Symbol for Independent Voltage Source

567 Ohms

Symbol for Resistor

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Thus, any wires or circuit elements connected in this row of five holes are electrically connected together. The channel electrically isolates a single row (i.e., there is no internal connection across the channel). On the outside of the breadboard, there are four columns of holes with long metal clips electrically connecting each of the holes in a single column together. Thus, from an electrical standpoint, the breadboard looks like:

Figure 8 - Schematic of a Breadboard Layout

Spend a few minutes at this juncture to familiarize yourself with the concepts of circuits and circuit elements as well as the function of the DMM, the power supply, and the breadboard.

Concept of Linear Engineering Systems and Resistance Engineering systems are omnipresent in every field of engineering. Examples of engineering systems include an aircraft and flight control system, an automated coal conveyor, a GPS controlled grain harvester, a vibration suppression device for bridges, an oil refinery, an active suspension for an automobile, a computer or data network, or even a computer program. A system could have very few components (e.g., the circuit we will build) or many millions (e.g., the Pentium 4 chip). Whether simple or complex, engineers spend a great deal of their time and effort analyzing and designing systems. Often times, an engineer cannot see “inside” a given system and knowledge of this system must be determined by making external measurements. Such systems are often referred to as “black boxes” meaning that the internal workings of the system are hidden from direct examination. Almost all electronic and compurer systems are “black boxes” in that unless we designed the system or have a complete schematic, only certain input and output points or nodes are accessible to us. Anyone who has ever looked at a computer

This column of holes is electrically connected as are the other 3 simliar columns on the outside of the breadboard. However, the 4 columns are electrically isolated from each other

Row of 5 holes is electrically connected as are the other simliar rows of 5 holes. However, each row of 5 holes is electrically isolated from each other row of 5 (including the opposite row across the channel).

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motherboard knows that the vast majority of this system structure is shrouded behind charcoal grey electronic chips. Black box systems as well as many other engineering systems are characterized by external inputs and output variables or signals. An input to a system is simply a signal or variable which causes the system to react in some way. Examples of inputs include a voltage to an electric motor (electrical), the profile of a road to an automobile suspension (mechanical), or the position of a valve on an input pipe for a water treatment plant (civil). Changing these variables causes the system to react. An output of a system is simply a signal or variable which is externally measureable by an engineer and characterizes the performance of the system. Examples of outputs include the speed of our electric motor, the displacement of our automobile chasis, the level of pollutants in the final tank of our water treatment plant. Engineering systems can be classified into many different categories: continuous versus discrete, digital versus analog, time varying versus time-invariant, dertiministic versus stochastic. Perhaps the most frequently used classification of a system is whether it is linear or nonlinear. The most elementary definition of a linear system is that the principle of superposition holds. Superposition simply means if you apply the sum of two different inputs to a linear system, then the output produced will be the sum of the outputs produced by each of the two inputs, individually. Thus, if you double a given input to a linear system the output will double, as well. In this ME 101 experiment, we will build two very simple linear systems comprised of resistors and independent voltage sources. Other engineering examples of simple linear systems include springs (governed by Hooke’s law) and accelerometers (convert acceleration into voltage). Both of these system we will study at at later time in ME 101. In fact, the two resistive electrical circuits we will construct have analogous or “dual” mechanical circuits comprised of simple dashpots or viscous dampers! Resistance and Ohm’s Law In the previous section, the circuit symbol for a resistor was given and as well as an explanation that a resistor is generally made of carbon and serves to oppose current flow in a circuit. The higher the resistance, the more voltage that is needed to force the same amount of current to flow. The equation that governs the relationship between current, voltage, and resistance is called Ohm’s law in honor of Georg Simon Ohm, the german physicist who is credited for first discovering it. Quite simply stated, Ohm’s law says that voltage equals current times resistance or V = I x R. Obviously, the unit on resistance must be Volts/Ampere, which is true. However, a volt/ampere is also called an ohm and symbolized with a capital omega, Ω. Since voltage has polarity and current has direction, ohm’s law is only valid if the current direction enters the resistor at the positive (plus) side as shown below:

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If this convention (called the passive sign convention) is not true, Ohm’s law is off by a minus sign. Two other concepts we will employ in this first experiment are the concept of series and parallel. Circuit elements are said to be in series when the same current flows through them. If the same voltage exists across two or more circuit elements, they are said to be in parallel (see below for illustration)

+ V (volts) _

I (Amperes)

R (Ohms)

Ohm’s Law: V = I ⋅ R

Circuit Elements in Series (same current flows through each)

Circuit Elements in Parallel (same voltage appears across each)

+ _

+ _

+ _

ME 101 Measurement Demonstration (MD 2)

Linear Systems and Simple Resistance Measurements

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Lab 2 - In Lab Experimental Procedures: Name:______________________________ Section:____________Team:____________ Resistors in Series

1. Your lab teaching assistant (TA) will give you three resistors: R1, R2, and R3 . Use the DMM to measure the resisistance (in ohms) of each. Record the values Don’t forget units.:

R1 = ___________ (Brown-Black-Red-Gold color bars) R2 = ___________ (Red-Black-Red-Gold color bars) R3 = ___________ (Brown-Black-Orange-Gold color bars)

2. Using the jumper kit and solderless breadboard, connect the following series circuit. Use one of the two 0-30 volt outputs set to 2.0 volts on the PS280 power supply as the independent voltage source.

R2

R3

-2V +

I R1

+ V1 -

+ V3 -

+ V2 -

Once the circuit is in place, use the DMM to measure and record the quantities. Don’t forget units.: I = ___________ V = ___________ V1 = ___________ V2 = ___________ V3 = ___________ Req = ___________

3. Change the voltage on the PS280 supply to 4.0 Volts and measure the current, I, again.

I4v = __________

Note Remainder of this Section is Homework: Given what you now know about linear systems, does your answer seem reasonable? Explain why?

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4. It is possible to replace the three resistors you used with only one resistor that would give the same current I that you measured in step 2 (see figure below). This resistance is called the equivalent resistance.

R1

R2

R3

2V

I

+- 2V -

I

+ Requivalent=

a) What is the value of the equivalent resistance Req? Show your work. (hint: use Ohm’s

Law). Req = _________

b) Now add the three resistor values you used in step 2.

Rsum = R1 + R2 + R3 = _________

c) Do you see the relationship between Req and Rsum? Come up with a general relationship that you could use to combine resistors in series for any circuit.

Req =__________

5. Find the equivalent resistance of the following circuit if: R1 = R2 = 100 Ohms and R3 = R4 = R5 = 1000 Ohms.

R3

R4

R2

R5

R1

Req = _________

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6. Now consider a generic series circuit as a system with the input voltage of 1V from the power supply and the output of the system being the current, I. a) Is this a linear system (i.e., if you double your input, the output doubles)? b) If yes, what current would you expect if we changed the input voltage from the power supply

to be 9 volts?

7. Sum the voltages V1, V2, and V3 together and compare your answer to V. Can you see a relationship between the sum of voltage drops and the power supply voltage? What is that relationship?

8. Consider the following black box system with input/output measurements taken by the DMM.

X input value Y output value 10 volt 0.25 amperes 30 volts 0.75 amperes 1 volt 0.025 amperes

a) Is it linear? Show how you came to that conclusion.

b) What would the equivalent resistance be for this system?

Req = ________

c) Come up with a series circuit, with at least 2 resistors, that would behave like this system.

Draw the circuit and label all the required elements (voltage, current, and resistances).

V (input voltage) I (output current) System (linear?)

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Resistors in Parallel 1. Reconfigure the resistors on the breadboard so that the are now in parallel and reconnect the

PS280 power supply so that you form the following circuit:

Use the DMM to measure and record the quantities: V = ___________ I1 = ___________ I2 = ___________ I3 = ___________ I = ___________ Req = ___________

Note Remainder of this Section is Homework:

2. Is the parallel circuit you built also a linear system? If yes, what value of the current I would result if the input voltage were changed to 6.4 volts?

I6.4V = ________

3. It is possible to replace the three resistors you used with only one resistor that would give the

same current I that you measured in step 1 (see figure below). This resistance is called the equivalent resistance.

=R21V +- R3

I

R1 1V Requivalent+-

I

a) What is the value of the equivalent resistance Req? Show your work. (hint: use Ohm’s Law).

+ -

+ V _

R1 R2 R3 1.0 Volt

I2 I1 I3

I

5

Req = _________

b) The relationship between the equivalent resistance and resitance in parallel is as follow:

...31

21

11

1

+++=

RRR

Req

Using your values of R1, R2, R3, and the calculated value or Req in part a), show that this relationship is correct.

4. What should be the equivalent resistance for the following parallel combination if R1=R2= R3=R4=R5=400 ohms.

R1 R2 R3 R4 R5

Req = __________

5. Sum the currents I1, I2, and I3 together and compare your answer to I. Can you see a relationship

between the sum of current in the branches and the current coming out of the power supply? What is that relationship?

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Extra Credit Now that you know how to reduce resistors in series and resistors in parallel, can you reduce the following circuit into a single equivalent resistor using repeated applications of your series and parallel rules if R1=R2=R3=R4=R5=100 ohms?

R1

R2

R3R4

R5

You must show your work! Do not work with other teams on this problem! Req = ________