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1 University of Waterloo Electrical and Computer Engineering Department Physics of Electrical Engineering 2 ECE-106 Lab manual LS 3: Resistance & Resistivity Winter 2010 Last Updated March 10 th 2010 © Electrical and Computer Engineering University of Waterloo Waterloo, Ontario N2L 3G1, Canada

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Page 1: University of Waterloo Electrical and Computer Engineering

1

University of Waterloo

Electrical and Computer Engineering Department

Physics of Electrical Engineering 2

ECE-106 Lab manual

LS 3: Resistance & Resistivity

Winter 2010

Last Updated March 10th 2010

© Electrical and Computer Engineering

University of Waterloo

Waterloo, Ontario N2L 3G1, Canada

Page 2: University of Waterloo Electrical and Computer Engineering

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1. Objective The purpose of this laboratory exercise is to

Investigate the resistivity of different materials

Study resistance as a function of material, physical construction and temperature

Learn how to make resistance measurements using the Four Point Probe method as well as the

Wheatstone bridge

Observe and measure the conversion of electrical energy to heat energy

2. Background

2.1. Resistivity and Resistance At a given temperature: Resistance is related to resistivity by the Equation

𝑅 = 𝜌𝐿

𝐴 (2.1)

Where R = resistance of a material

A = cross- sectional area = 𝑊 × 𝜏 (see Figure 1)

L = length (see Figure 1)

𝜌 = 1

𝜎 = resistivity (Ω∙m)

W

L

Figure 1 Film with Dimensions

The material parameter 𝜎 is known as conductivity and this parameter is temperature dependant.

Thus resistivity, 𝜌, is also temperature dependent. Note: current is always in the “L” dimension.

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2.2. Resistance by Squares If we examine films, material that are rectangular with a constant thickness, one can obtain the

relationship between the resistance of an object to its length and width by using Eq. (2.1) 𝑅 =𝜌𝐿𝐴

Substituting 𝐴 = 𝑊 ∙ 𝜏 in Eq (2.1) gives

𝑅 = 𝜌𝐿

𝜏 ∙ 𝑊 (2.2)

Since in most real world situations thickness (𝜏) is held constant, and since resistivity (𝜌) is also a constant (for a given temperature) we can rewrite Eq. (2.2) as

𝑅 =𝜌

𝜏∙ 𝐿

𝑊 (2.3)

As can be seen from Eq. (2.3) resistance of a film will vary as the length and width are varied.

Resistance will be at a minimum when the length and width are equal, i.e. when the ratio 𝐿

𝑊 goes to

unity. Therefore what is important is the ratio of length to width not their absolute values (for

determining resistance). When 𝐿

𝑊 is at unity, the film is square and Eq. (2.3) reduces to

𝑅𝑠 =𝜌

𝜏 (2.4)

Where 𝑅𝑠 is called the sheet resistance of the material (for a given thickness). Since most practical films are rectangles, they can be seen as a finite number of squares added to together. Therefore one can calculate the resistance of an object by multiplying 𝑅𝑠 by the number of squares in the object, remember that current is parallel to “L”. Many technologies use films or thin films to construct electrical circuits and components. A film is a material layer of uniform thickness usually deposited on a substrate. A microprocessor is an example of a product that uses thin uniform layers of material (e.g. aluminium) to build circuits. Also most circuit boards built today are not only built using films but most use surface mount components (see Figure 1), many of which use films in their construction.

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Figure 2. SMT vs. Through Hole Components

Remember: Current in a conductor results from the potential difference across the conductor. The

current in a resistor is related to the potential difference across the resistor by

∆𝑉 = 𝐼𝑅 𝑜𝑟 𝐼 = ∆𝑉

𝑅

2.3. Resistivity as a function of temperature The resistance of a pure metal will vary almost linearly over a broad temperature range to a good

approximation;

R = Roα(T-To) + Ro

where R is the resistance of the material at temperature T and Ro is the resistance at some reference

temperature To . The quantity α in the above expression is called the temperature co-efficient of

resistance, and is characteristic of the material. We will use our data to determine α for a sample of

copper wire.

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2.4. Four Point Probe Resistance Measurement The four-point probe method apparatus, illustrated in Figure 3, provides a convenient way to obtain

resistance. Four equally spaced and collinear probes are connected onto the surface of the material.

A current I is passed through the outer probes while the inner probes measure a voltage V. With

known current and voltage the resistance can be calculated.

Figure 3. Four-point probe method

2.5. The Wheatstone Bridge The Wheatstone bridge (Figure 4) is an electrical circuit used to measure resistance. It is also a circuit

that is used for the comparison of resistances.

Recall that Kirchhoff’s voltage law (KVL) states that the algebraic sum of voltages around a closed

loop must equal zero. If we look at the Basic Wheatstone Bridge (Figure 4A) we see that VTest = VAB +

VBC = VAD + VDC. Note V1 is a voltmeter. If the circuit resistors are chosen such that VAB = VAD and VBC =

VDC, then VBD=0 volts. When VBD=0 volts the bridge is ‘balanced’, also any one resistor (Rstandard,RX, RL,

or RR) is called an arm of the bridge.

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.

Figure 4. A) The Basic Wheatstone Bridge B) The Slide Wire Wheatstone Bridge

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3. Pre-Lab 1) Define : Resistance, resistivity, conductivity, contact potential, ohms, Siemens,

2) Explain: the Resistor color code, the resistor power rating / power dissipation, % tolerance, axial

vs SMT resistors, and the circuit symbols ( i.e. fixed, variable, potentiometer),

3) Why do we want to measure the voltage at a different point than the current in the four point

probe method?

4) Prove that: Rstandard/RX = RL/RR for the Wheatstone Bridge.

5) What do we mean by #26 AWG copper wire , and what is its resistance/meter, its diameter, is it

solid or stranded wire?

6) What Power is generated in the heated Test Wire by the test current in the #26AWG wire

(section 4.4)?

7) Name and describe two different types of resistors (e.g. an SMT resistor).

8) What do we mean by 3 1/2 digit DMM and 4 1/2 digit DMM.

9) The heated copper test wire used in 4. 1 has a thin coating of insulating material on its surface,

could this wire be used to build an Inductor? Why?

10) What is the normal AWG wire used in your house wiring circuits, i.e. the wire connected to wall

power outlets, and what is the maximum AC current that you can receive from a wall outlet?

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4. In-Lab Procedure Remember electricity is dangerous, only 100mA passing through your heart will cause Ventricular

Fibrillation which, if continued, will be fatal. Always turn the DC Power Supply off when rewiring

circuits.

In this lab you will be using resistors as heaters, they will become hot enough to burn you – be

careful.

You will be using DMMs as voltmeters and ammeters, if you do not hook them up in the correct

orientation you will damage the DMM, double check your work.

We will look at two different ways to measure resistance: 1) The four point probe method (or Ohms Law

for linear resistances) and 2) the Wheatstone bridge (an electrical circuit used to measure resistance).

4.1. Four Point Probe Technique for Measuring Film Resistance Use the four-point probe method illustrated in Figure 5 to calculate the resistance of simple Aluminium film resistors of varying physical size. By applying a known current through the Aluminium resistors and measuring the voltage drop across a known length (L) and width (W) the resistance can be calculated.

WL

V

AVtest

Figure 5. Four-point probe for resistance measurement setup

1) Using the DC power supply and a resistor pass a current through each of the Aluminium resistors. Adjust the Vtest to get approximately 100mA, use an ammeter (DMM) to measure the current. Connect the 4.5digit voltmeter (DMM) to the resistor, record both the current and voltage drop obtained. Record results in Table 1

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2) What correlation (if any) do the varying sizes of resistors have to one another? (i.e. Compare resistance vs. width and # of Squares)

3) Given the resistivity of aluminium calculate the thickness of the aluminium foil? Is this result reasonable?

Table 1. Film Resistor Measurement

Resistor Dimensions

(cm) I (mA) V (mV) R (Ω)

2 X 2

2 X 4

2 X 6

4 X 4

4 X 8

4 X 12

6 X 6

6 X 12

6 X 18

4.2. The Four Point Probe Method for Measuring Wire Resistance: To accurately determine the resistance of wire we need to apply a constant current through the wire

and measure the voltage drop across a known length of the wire. This is necessary to eliminate

contact resistance and the fact that test resistance may change over time.

1) Wire up the circuit in Figure 6. Note: V1 is a voltmeter, RS = 56 ohms, A1 is an ammeter, MgR

and MgB are minigrabber red and black respectively, The Fuse is contained in the yellow

cable.

2) A constant current supply of Itest = 100mA is created by adding a resistor RS in series with a

DC voltage source. The resistor RS must be much larger than the resistance being measured.

This will ensure that the current in the circuit will be a function of the series resistor and

practically independent of the wire under test.

3) Make voltage measurements between various points along the slide wire with a voltmeter

and the minigrabbers connecting leads.

4) Using the DMM to measure wire #3. Measure the voltages between meter stick readings at:

0cm, 50cm, 100cm, and between the Banana Plugs R3Red and R3Blue. Fill the data in Table 2.

Connect MgB at the 100 cm position to start.

i. How does the resistance vary with length of wire?

ii. Calculate the resistance of two meters for slide wire R3 using your data for current and

voltage.

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iii. Repeat the above procedure for wires R1, R2, R4, and R5. Make sure you also measure

between the ends of the banana plugs, i.e. the full length for each wire.

iv. Wires #1, 2, 4, and 5 are copper with a Tin covering what are they’re diameter and gauge?

v. What are there AWG gage and diameter?

vi. How does the resistance vary with gauge of wire?

The length of wire between the meter stick and the binding posts is 6cm at each end. The wires

numbered 1, 2, 4, and 5 are thus 1.12 meters long. We can call this 6 cm a “dead zone”.

vii. What is the resistance of the dead end zone near the end of these wires?

Figure 6. The Circuit used to find the Resistance of the test board wire numbered R3

Table 2. Place the black minigrabber at the 100cm end of the wire. Then move the red minigrabber to different meter stick positions and make readings.

Resistor 0 cm voltage

50 cm

voltage 100 cm

voltage full length

voltage

Ohm/m I Test

V Test Gage #

R1, R2

R3, R4,

R5,

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4.3. The Wheatstone Bridge with Slide Wire Using the Wheatstone Slide Wire Bridge found in Figure 7 find the resistance of the “Unknown

Resistors.” Our Unknown Resistors will be constructed from different lengths and diameters of wire

of different materials. Note each of the five wires on the Wire Board, once measured, can be used as

a known standard resistance for other measurements.

4.3.1. 500mH Inductor: Using the Wheatstone Bridge how do we find the value of the Unknown Resistance? Refer to Figure 7; when the voltage of V1 = 0 volts, then the ratio Rstandard/Rx is equal to the ratio RL/RR . The Unknown Resistance Rx can be computed from the known resistance Rstandard and the known ratio RLeft/RRight of wire #3. The error in the computed Rx is not affected by the test voltage Vtest or meter loading. The voltage divider concept is an important concept to understanding how this bridge is used.

1) Rstandard = 1.5 K ohms. Note: C1 and C2 are short cables 15 cm long

2) Apply a test voltage VTest and adjust it to give a current of 100mA

3) What is the test current reading from the power supply in the circuit?

4) Connect MgR to the DMM and move it along wire #3 measuring the voltages at: 10 cm,

20cm, 40cm, 60cm, 80cm, 100cm, and the node connecting RX to RStandard. Record this

voltage as V1. Fill in Table 3

5) Adjust the placement of MgR along the slide wire so that V1= 0 volts and record the meter

stick reading. This is the bridge balanced position. Now calculate RX

6) What is the resistance value of Rx?

7) What is the wire length of Rx if it has a gauge of #30?

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Figure 7. Wheatstone Bridge Circuit used to measure Rx = 500mH inductor

Table 3. Results for 500mH Inductor

Meter : cm V1 = reading , mV VTest ITest

V1 = 0 V

4.3.2. 1mH Inductor: How does the resistance vary with length and gauge of wire?

i. Wire up Figure 8, Note: C1 is a short 15 cm cable.

ii. Adjust the test voltage to get a test current of 100mA.

iii. What is the test current reading from the power supply in the circuit?

iv. Connect the Minigrabbers to the DMM. Move the MgR along wire #1. Measure the

voltages at: 0.1cm, 20cm, 40cm, 60cm, 80 cm, 100cm and the node connecting RX to

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RStandard. Record this as V1 in Table 4. Adjust MgR so that V1= 0 volts and record the meter

stick reading. This is the bridge balanced position. Now calculate RX.

v. What is the length of the wire in the inductor given the measured resistance, and the

diameter of the wire if it has a gage of #22? Show your calculations.

Figure 8. The Wheatstone Bridge for Unknown Rx = 1mH (inductor)

Table 4. Results for 1mH Inductor

Meter : cm V1 = reading , mV VTest Itest

V1 = 0 V

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4.4. Copper wire heating circuit: How does the copper resistance vary with

temperature? The Heater Resistor consists of two 100 ohm, 10 watt resistors connected in series, See Figure 9.

1) What is their total resistance?

2) When a Heater supply voltage of VHeater = 25 Volts is applied to the resistors, what should the

current through these resistors be?

3) Construct the circuit to test the heated copper wire, see Figure 1. Use Rstandard = 1 ohm @ 1%.

Note: C1 and C2 are short cables 15 cm long.

4) Turn on VTEST so that the test current is 100mA. Balance the bridge; it should balance around

the centre of the R3 wire.

5) Turn on VHEATER , the Heater current should be 115 mA maximum. Use the Fluke IR

Temperature sensor to make sure the copper wire sample temperature gives a reasonable

value before proceeding. Take a number of readings as the wire heats up. The maximum

temperature should be around 65oC. When the copper wire sample has obtained a

temperature of at least 62oC, stop the acquisition.

6) Fill in the data in Table 5. Record the copper wire temperature data at about 100C intervals.

At the same time as you record the temperature adjust V1 to be 0 Volts and record the

Meter-Stick cm, balanced reading. From the meter-stick balanced readings calculate how the

copper wire resistance is changing with temperature.

7) Plot the sample resistance as a function of temperature and comment on your results. From

the graph, determine the temperature co-efficient of resistance α for copper. Compare this

to the accepted value for copper which is (4.3 x 10-3 OC-1) . Suggest some possible reasons for

any discrepancy between your results and the accepted value. The wire is #26 gauge how

long is it? Calculate this resistivity of copper at a temperature of 25oC.

Important Testing Instructions

Hold the Temperature “gun” with the laser pointer just above the heater, be consistent for all

readings.

Reset the “gun” before repeating each reading by measuring temperature of the floor (room

temperature).

Record the maximum value of temperature displayed on the “gun.” You should repeat the

measurement a few times to get a consistent value (don’t forget to reset before making each

measurement).

Make sure the EMS setting is at “medium”. You can change the setting by pressing EMS and

then the up/down arrow buttons if needed

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DANGER :

Be Carefull. Do not point the laser into anyone’s eye. Ensure the laser is pointed

in a safe direction at all times

Figure 9. Wheatstone Bridge and Heated copper wire (#26 inductor wire).

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Table 5. Slide Wire Wheatstone Bridge data

Unknown R

RSstandard Wire No. Meter Stick cm , balanced

Calculated R Vtest Itest mA

Heater

ImA

Wire vs. room Temp.

around 23oC

Wire vs. Temp.

1

Wire vs. Temp.

2

Wire vs. Temp.

3

Wire vs. Temp.

4

All the temperature values (Temp 1 to 4) should be around values at 10 interval between room temp

and the last value (62oC).

4.5. Strain Gauge circuit: How does the material resistance vary with

strain? Construct the circuit in Figure 10. Adjust Vtest so that Itest = 2mA and set RA = RB = 1.5 K ohms.

The gauge Load is one Master Lock = 88grams.With no load and MgR at the 50 cm position to

start, adjust RDECADE so that V1 = 0 Volts. This is the no-load balanced position. What is the

resistance value of RDECADE?

Then add the weight, which is a metal Lock. Now with the weight added adjust the slide wire

(MgR) so that V1 = 0 Volts. Record the distance change caused by adding the weight. This is the

full-load balanced position. Calculate the change in resistance in the Strain Gauge. What

change in resistance did you see in your Gauge? Fill in Table 6

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Figure 10.Wheatstone Bridge and Strain Gauge.

Table 6 Results for Strain Gauge

Weight Meter , cm V1 = reading , mV VTest Itest

0

Lock, grams

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Equipment List 1) Aluminium film resistor box

2) A Dual DC Power Supply , Agilent E3620A

3) One 4 ½ digit DMM ; Fluke 8050A or Keithley 179

4) One 3 ½ digit DMM ; Fluke 8010A

5) Fluke IR Thermometer

6) Slide – Wire Wheatstone Bridge Board / Five Wire Resistance Board,

Slide – Wire Potentiometer , about 1meter long,

7) Strain Gauge chassis, gauge resistance: some with positive change, some with negative change

with load.

8) Heater Board chassis

9) Strain Gauge bridge box

10) Aluminum sheets box

11) 500 mH inductor

12) 1 mH inductor

13) Two Fuse in cables , 1/8 A Buss type

14) 1 ohm resistor

15) 56 ohm resistor

16) Two 1.5 k ohm resistors

17) Two Minigrabber/Banana plug cables , red/black, Pomona

18) Two short Banana leads, red/black

19) One bench lock , as a test weight

20) One 6 inch ruler