Ganago Student Lab3

Lab 3: Thevenin Equivalent Circuit; Beyond Parallel and Series

LAB EXPERIMENTS USING

NI ELVIS II AND NI MULTISIM

Alexander Ganago Jason Lee Sleight

University of Michigan

Ann Arbor

Lab 3 Thevenin Equivalent Circuit; Beyond Parallel and Series

© 2010 A. Ganago Introduction Page 1 of 8


Goals for Lab 3

Learn about the Thevenin equivalent circuit:

• in the pre-lab, model a simple circuit with a source resistance and variable load resistance

• in the lab, build such circuits and do the measurements of voltage and current

• in the post-lab, compare the results of your modeling with your lab data; draw conclusions on the maximal power transfer in your circuit

Explore (for extra credit) whether the NI ELVIS power supply acts as an independent voltage source (without a resistance in series); if your measurements disprove it, calculate the source resistance.

Learn about circuits that do not reduce to parallel/series connection:

• model them in pre-lab

• build them on the NI ELVIS PB in the lab; measure the currents and voltages

• compare your lab data with the results of your modeling in the post-lab.

Explore (for extra credit) the relationship between the circuits you built in this lab with the Wheatstone bridge circuit, which you learned in Lab 2

Learn about potentiometers

Explore (for extra credit) how to build circuits with potentiometers

Explore (for extra credit) how the equivalent resistance of the circuits depends on the terminals, between which you measure it.



Introduction In this lab, you will be working with more circuits and more methods for their analysis; you will learn concepts of key importance for you future careers.

One of the reasons for learning new stuff is that the old stuff does not work so well.

Consider an independent voltage source shown on this diagram. It is called so because we expect that this source maintained voltage difference of VBSB between its terminals is independent of anything else in the circuit, for example, independent of the equivalenload resistance RBeqB, to which the source is connect

t ed.

ls

According to this circuit model, the current through

the load equa SV and the power absorbed by the load equa

eq

IR

= ls 2

2 SV thus

both the current and the power can grow infinitely high as the load resistance eqR decrea

eqeq

I RR

⋅ =

ses to zero.

If your voltage source is a model of an AAA battery, and your equivalent load resistance is that of many light bulbs connected in parallel, you can reach a conclusion that one battery could be enough to illuminate a large lecture hall, which is certainly absurd.

What’s not working so well in this example is the oversimplified circuit model for a voltage source.

A better model includes a source resistance as shown on the second diagram. Here the load is connected to the right of the terminals a and b, while to the left of these terminals are the source and its resistance. You as the user can alter the load resistance but have no control over the source resistance.

As you will observe in the pre-lab, this circuit model is free from absurd conclusions such as infinitely high current or power.

The current in this circuit and the voltage across the load resistance are easy to calculate using the formulas for series connection of resistors, voltage division, etc. You have probably noticed that subscripts on the second diagram changed from those on the first.

The new subscripts VBTB and RBTB remind us about the important Thevenin theorem, which serves as a powerful tool for circuit analysis.



According to this theorem, any linear circuit that contains sources and resistors can be reduced to its equivalent, which includes only one voltage source VBTB along with its resistance RBTB.

The Thevenin circuit is equivalent to the original one if they both exhibit the same load characteristics, which can be plotted as a voltage-current curve.

For the Thevenin circuit, the load characteristic is linear; it can be drawn on the plot as a straight line connecting the two points on the axes, as shown on this sketch. One of these points—on the voltage axis —corresponds to zero current and maximal voltage; it is called the open-circuit voltage and measured or calculated when nothing is connected to the terminals a and b. The other point—on the current axis—corresponds to zero voltage and maximal current; it is called the short-circuit current and could be measured when terminals a and b are connected with a wire of zero resistance. Note that in practice it can be dangerous to short-circuit the terminals of a voltage source thus we often obtain the load characteristics by plotting data measured with several different load resistances. For theoretical calculations, there is no danger in the assumption that terminals are short-circuited.

The power that can be transferred from the circuit to the load reaches its maximum if the load resistance equals the Thevenin resistance of the circuit; in this case

T T

T L T

22 T

L L, MAXT

V VIR R 2 R

VP I R4 R

= =+ ⋅

= ⋅ =⋅

On the other hand, from the measured voltage and current in the circuit with the known load resistance, you can calculate the Thevenin equivalent voltage and the Thevenin equivalent resistance (of course, in order to calculate both parameters, you should do the measurements with at least 2 load resistances).

LV I

TV

TR

Multisim presents a powerful tool for modeling the circuits and predicting results of lab measurements but its predicting power—or accuracy of simulations—is limited, in particular, by the circuit model used in simulations: for example, simulation results depend on whether you take into account or neglect the Thevenin equivalent resistance of the power supply used in the lab.



Many circuits can be reduced to parallel/series combinations of resistors thus solved using familiar formulas for voltage and current division, but many more circuits cannot be reduced and should be solved with more advanced methods such as node voltages and mesh currents.

Also, Multisim allows you to simulate circuits beyond parallel and series combinations of resistors, to calculate voltages and currents in such circuits, and to determine their parameters such as the equivalent resistance.

Resistance measurements To measure the resistance of a resistor, we simply connect it to the two terminals of an

ohmmeter, labeled HI and LO on the sketch below, and read the resistance from the display.

Remember: you must disconnect the

resistor from your circuit before measuring the resistance.

Otherwise, you will not obtain the correct reading of resistance.

Here is why.

The ohmmeter’s internal circuitry includes a voltage source VBSB (usually, ~1 V) and has a certain equivalent internal resistance RBS B.B



The component whose resistance you wish to measure is connected to the two terminals labeled HI and LO on the equivalent circuit diagram shown here.

From the voltage VBSB and the current I that flows through the terminals HI and LO (or the voltage drop between the terminals HI and LO), the instrument calculates the resistance R using the formula for voltage division, etc., and displays R in the units of ohms.

t.

r

ted and ce R will be calculated

low

noticeable), results of such resistance measurement will be worthless.

Note the important rule: NOTHING is connected to the resistor on the righIf this rule is broken, the relationship between the voltage VBSB applied by youohmmeter and the current through the terminals HI and LO will be distorthe resistanwrongly.

A typical beginner’s mistake is to hold theresistor between fingers while measuringits resistance: some current IB2B will fthrough the human skin and body.

Although, in resistance measurements, such currents are not harmful (and barely



Another typical mistake is to keep the resistor, whose resistance you measure, connected to other parts of your circuit as shown here.

Even if your circuit is not connected to a power supply, the current through the terminals HI and LO will now depend not only on the resistance R but also on the other parts of the circuit thus, from the relationship between this current and VBSB, the resistance R will be calculated wrongly.

The worst mistake is to measure resistance of a resistor, which is still connected to the circuit that has its power supply turned on.

In this case, the current through the terminals HI and LO will depend on the voltage VBS2B applied by the power supply; if VBS2B is high, it can even damage the ohmmeter.

Learn about these mistakes and avoid them in the lab.

The equivalent resistance of a circuit is defined as the ratio of the voltage applied to the circuit to the current that flows through that circuit; it does not matter whether the circuit is reducible to a series/parallel combination of resistors.

For example, the entire Circuit 5 (see the pre-lab) can be represented with its

equivalent resistance SEQ

Total

VRI

= .

This definition applies to simulations and measurements: in the pre-lab, you will simulate the circuit, determine

S TotalV and I and calculate EQR ; in the you will build the circuit and m

S TotalV and I ; in the post-lab you will compare th two values of .

lab easure

e EQR



Potentiometer A potentiometer, or pot in the EE jargon, is basically a resistor with three connectors: in addition to the end connectors A and B, there is a movable tap C. The resistance between the potentiometer’s ends is fixed RBABB = RBP B(the potentiometer’s resistance) while the resistance RBXB = RBACB between the end and the tap is variable.

Evidently, the resistance between the tap and the other end connector equals (RBPB – RBXB).

Thus the potentiometer is equivalent to two resistors as shown on the second diagram here. Sometimes, for circuit analysis, it may help you tredraw the diagram with a pot into that with tworesistors.

o

Potentiometers are used for user control, such as loudness of sound (volume control) of an audio system, and can control a wide variety of functions in electronic circuits.

In circuits, potentiometers are used as variable resistors and voltage dividers.

When a pot is used as a variable resistor, one of its end connectors is either left open (this is what you will do in this lab) or connected to the tap, as shown on two diagrams below. This device—potentiometer with two connectors—is sometimes called a rheostat.

Two equivalent circuit diagrams for a potentiometer. Node C is the movable tap.

Potentiometer as a variable resistor



Pre-Lab Problem 1 The purpose of this problem is to compare circuit models for an independent voltage source and introduce the concept of maximal power transfer to the load. If a similar problem is assigned in homework, this one can be skipped.

Part 1 Consider a C-type battery as an independent voltage source VBSB = 1.5 V, and equivalent load resistance that is varied from 0.01 Ω to 100 Ω. Use software, with which you are comfortable (EXCEL, MATLAB, MATHSCRIPT, etc.) to generate a plot that shows two variables as functions of the load resistance RBeqB (use a logarithmic axis for RBeq Band show at least 5 points per decade, such as between 0.01 Ω and 0.1 Ω or between 10 Ω and 100 Ω):

a. Current I in the circuit

b. Power absorbed by the load resistance

Part 2 Consider a C-type battery as a voltage source VBTB = 1.5 V with the source resistance equal RBTB = 1 Ω, and the load resistance RBLB that is varied from 0.01 Ω to 100 Ω. Use software, with which you are comfortable (EXCEL, MATLAB, MATHSCRIPT, etc.) to generate a plot (use a logarithmic axis for RBeqB, as in Part 1) that shows:

a. Current I in the circuit

b. Power absorbed by the load resistance

Part 3 Use your results obtained in Parts 1 and 2 to answer the following questions.

A. Does the source resistance limit the power absorbed by the load?

B. At what RBL, maxB does the power absorbed by the load reach its maximum in each of the circuits?

C. Which of the models—Part 1 or Part 2—seems more realistic to you and why?

D. In Part 2, how does load resistance RBL, maxB relate to the source resistance RBTB?

© 2010 A. Ganago Pre-Lab Page 1 of 6


Pre-Lab (continued) Problem 2 The purpose of this problem is to compare circuit models for an independent voltage source and introduce the concept of maximal power transfer to the load. If a similar problem is assigned in homework, this one can be skipped.

For the circuit used in Pre-Lab Problem 1 Part 2, plot the load characteristics and determine the open-circuit voltage and short-circuit current.

From theory, we expect that the maximal power absorbed by the load resistance

equals L, max OC SC1P V4

= ⋅ ⋅ I .

Calculate PBL, maxB from the load characteristics and measure it from your plot in Pre-Lab Problem 1 Part 2; calculate the percentage difference between the two values and discuss whether they agree.

Problem 3 In the lab, you will build a circuit, which includes a voltage source that has terminals a and b and two resistors—100 Ω that is fixed and RBLB, which is variable.

Using this circuit, you will measure the voltage across the load resistor RBLB and calculate the power transferred to the load (absorbed by RBLB).

When you begin this experiment, you do not know which circuit model (shown here as Circuit 1 and Circuit 2) to use for the source; in other words, whether you may neglect the source resistance RBTB or have to take it into account. Note that you will have access only to terminals a and b and the circuit to the right of them, but not to the “guts” of the voltage source.

(Continued on the next page…)



Pre-Lab (continued)

Problem 3 (continued) Use Multisim to model Circuits 1 and 2 shown on these diagrams. In both circuits, assume VBTB = 6 V. In Circuit 2, assume RBTB = 50 Ω (which is typical for many function generators). In both circuits, consider load resistances: 20 Ω, 50 Ω, 100 Ω, 200 Ω, and 500 Ω.

For each load resistance, obtain the voltage VBLB in each circuit and the power absorbed by the load. Fill the table below (use ohms, volts, and milliwatts).

Explain how you would choose between the two models for the voltage source.

Circuit 1 Circuit 2

R_Load Voltage V_L Power P_L Voltage V_L Voltage V_a Power P_L

20

50

100

200

500

Show your work on additional pages.



Pre-Lab (continued)

Problem 4

In the lab, you will build Circuits 3, 4, and 5 (shown on the next page) and measure currents through resistors, node voltages, and the equivalent resistances.

Use Multisim to model Circuits 3, 4, and 5 shown on the diagrams. In all circuits, assume VBSB = 6 V.

In all circuits, consider the following resistances:

RB1B = 50 ΩΩ

RB2B = 1000 Ω

RB3B = 500 Ω

RB4B = 150 Ω

RB5B = 200 Ω (circuit 5 only).

For each circuit, determine:

• the node voltages VBAB and VBBB in volts,

• the total current IBTotal Band the current through each resistor (in Circuit 4, also, the current through the wire connecting nodes A and B) in milliamps, and

• the circuit’s equivalent resistance in ohms.

Record your results in the following table:

Circuit VBAB VBBB IB1B IB2B IB3B IB4B IB5B/IBWireB IBTotalB RBeqB

3 NA

4

5



Pre-Lab (continued)

Problem 4 (continued)

Briefly explain why the equivalent resistances of the three circuits, which are built of the same components, are so different.

Problem 5 (for extra credit)

Circuit 6 shown here is built with the same resistors as Circuit 5 above, with the variable resistor RBXB added in series with RB4B.

Determine the value of RBXB such that the current IB5B vanishes.

Your result: RBXB = _____________ Ω.

Show your work.

Hint: Refer to Lab 2; note that the current IB5B vanishes if VBAB=VBBB, which means that the Wheatstone bridge built of RB1B, RB2B, RB3B, RB4B is balanced.

Use Multisim to verify your result.



Pre-Lab (continued)

Problem 6 (for extra credit)

Circuit 7 shown here is built with the same resistors as Circuit 5 above.

Calculate the equivalent resistance RBCDB in ohms between terminals C andD. Note that Circuit 7 is reducible parallel/series combination of resistors.

to

Your result: RBCDB = ______________ Ω.

Show your work.

Use Multisim to verify your result.

Briefly explain why the equivalent resistances of circuits 5 and 7, which are built of the same components, are so different.



In-Lab Work Part 1: Thevenin Equivalent Circuits SAFETY NOTE: for this part of the lab, the power consumed by each individual resistor will be relatively high. You should thus use ½ W resistors instead of the standard ¼ W resistors. Turn on the NI ELVIS II.

Build Circuit 1 from pre-lab, it is repeated here for your convenience:

Where RBLB = 100 Ω. Use the VPS to supply the input voltage.

Power on the PB.

Open the NI ELVISmx Instrument launcher and run the DMM and VPS.

On the VPS VI set the SUPPLY+ voltage to be 6 V.

Run the VPS VI.

On the DMM VI, ensure you are in DC Voltage mode.

Run the DMM VI.

Measure the voltage drop across RBLB and record it in the following table.

Power off the PB.

Repeat the measurements for each of the other values of RBLB.

© 2010 A. Ganago In-Lab Page 1 of 8


RBLB (Ω) Voltage across RBLB (V)

20

50

100

200

500



Part 2: Node Voltage, Mesh Current, and Equivalent Resistance In this part, you will build Circuits 3, 4, and 5 from the pre-lab (the diagrams are repeated here for your convenience), take data and record them in the table below. In the post-lab, you will analyze your lab data and compare them with your pre-lab results.



In all circuits use:

RB1B = 50 Ω,

RB2B = 1000 Ω,

RB3B = 500 Ω,

RB4B = 150 Ω,

RB5B = 200 Ω.

Use the VPS to supply VBSB.

On the VPS VI, set VBSB to be 6 V.

Take data on Circuit 3:

• use the DMM’s voltmeter to measure VBAB and VBBB

• use the DMM’s ammeter to measure IB5B/IBWireB and IBTotalB

• use the DMM’s ohmmeter to measure RBeqB directly (make sthe power supplydisconnected for this measuremen

ure is

t).

Record your results in the table on page 5.

Compare the results of your measurements with the results of your pre-lab simulations. If they differ by more than 5%, try to find your error but do not spend more than 5 minutes troubleshooting: ask your Lab instructor for help.



Repeat the measurements for Circuit 4.

Repeat the measurements for Circuit 5.

Circuit VBAB VBBB IB5B/IBWireB IBTotalB RBeqB

3 NA

4

5

This is the end of the required lab. If you are not going to continue with the explorations, power off the PB and NI ELVIS II and clean up your workstation.



Part 3 (optional): Exploration Build Circuit 6 from the pre-lab, it is repeated here for your convenience:

Where RB1B = 50 Ω,

RB2B = 1000 Ω,

RB3B = 500 Ω,

RB4B = 150 Ω,

RB5B = 200 Ω,

RBPB has a max value of 10 kΩ.

Use the VPS to supply VBSB.

On the VPS VI, set VBSB to be 6 V.

Use the DMM’s ammeter to measure the current through RB5B.

Power on the PB and run the DMM and VPS VIs.

Adjust RBXB until IB5B is 0 mA.

Continued on the next page…



After you adjusted RBXB so that IB5B is 0 mA, your next challenge is to measure RBXB.

This measurement can be done in at least three ways.

This diagram shows the dangerous way to measure RBXB: the current from VBSB can flow through your ohmmeter and might damage it. This diagram, which we will call Method 1, shows a safer way to measure RBXB, which can be erroneous because other resistors contribute to the measured value (see below).



The third way to measure RBX,B which we will call Method 2, is to disconnect the potentiometer from the prototyping board and measure the resistance between its two terminals. Two challenges here are:

• avoid moving the tap, and remember between which terminals you have to measure RBXB

Record the resistance of RBXB using both Method 1 and Method 2. The rationale for using Method 1 is to get convinced that it can indeed lead to errors.

Disconnect the power supply from the circuit. Measure RBXB according to Method 1 (see the diagram above).

Remove the potentiometer from the circuit and measure its resistance RBXB according to Method 2.

Record both results.

Method RBXB (Ω)

1

2

This is the end of the lab. Power off the PB and NI ELVIS II and clean up your workstation.



Post-Lab Problem 1

In pre-lab Problem 3, you calculated the power transferred to the load in two circuits shown below: the difference is that in Circuit 1 you neglected the source resistance RBTB, while in Circuit 2 you assumed that RBTB equals 50 Ω.

In the lab, you built a circuit, with a voltage source and two resistors—100 Ω (fixed) and RBL B(variable), in which you measured the voltage VBLB across the load resistor RBLB .

Use your lab data and calculate the power transferred to the load (absorbed by RBLB). Write your results in the table on the next page. Show your work.

When doing this experiment, you did not know which model (Circuit 1 or 2) to use for the source; in other words, whether to take RBTB into account. Now, from the known VBTB, RBLB, and VBLB, you can calculate RBTB in the circuit and conclude which of the models is correct.

Use your lab data for each load resistance and calculate RBTB in ohms. Write your results in the table on the next page. Show your work. If your results for various loads differ, calculate the average and standard deviation in ohms. Briefly explain which model (Circuit 1 or 2) is better for the NI ELVIS II power supply. If you choose Circuit 2, specify the value of RBTB in ohms.

Continued on the next page

© 2010 A. Ganago Post-Lab Page 1 of 6


Post-Lab (continued) Problem 1 (continued)

R_Load Voltage V_L

(your lab data)

Power P_L in mW

Source resistance RBTB

Comments

20

50

100

200

500

Average RBTB

Standard deviation for RBTB




TTPost-Lab (continued) Problem 2

In pre-lab Problem 4, you simulated Circuits 3, 4, and 5 (shown on the next page), obtained the currents through resistors, node voltages, and the equivalent resistances.

In the lab, you built each of these circuits and measured the voltages, currents, and the equivalent resistance.

Compare your lab data with your pre-lab simulations: for each of the parameters in the circuits, calculate the percentage difference Measured Simulated 100%

Simulated−

⋅

Record your results in the table on the next page.

Briefly explain whether your simulation results agree with your lab data.

In case of a serious disagreement (>10%), repeat the simulation using the actual resistances, which you measured in the lab.




TTPost-Lab (continued) Problem 2 (continued)

Circuit 3 VBAB VBBB IBTotalB RBeqB

Simulated

Measured

% difference

Circuit 4 VBA B(= VBBB) IBWireB IBTotalB RBeqB

Simulated

Measured

% difference

Circuit 5 VBAB VBBB IB5B IBTotalB RBeqB

Simulated

Measured

% difference




Post-Lab (continued) Problem 3 (for extra credit)

In pre-lab Problem 5 you simulated Circuit 6, which is built with the same resistors as Circuit 5 above, with the variable resistor RBXB added in series with RB4B, and determined the value of RBXB such that the current IB5B vanishes.

In the lab, you built this circuit and varied the value of RBXB until IB5B vanished.

Summarize your results:

Simulated: RBXB = _____________ Ω

Measured (according to the correct Method 2, see the circuit diagram below): RBXB = _____________ Ω

Calculate the percentage difference

Measured Simulated 100%Simulated

−⋅ =

Discuss the agreement/disagreement between theory and experiment.

In case of a serious disagreement (>10%), explain its possible cause.




Post-Lab (continued) Problem 4 (for extra credit)

In the lab, you measured RBXB using two methods: Method 1, which is erroneous because other resistors in the circuit contribute to the measured value,

and

Method 2, in which you disconnected the potentiometer from the prototyping board and measured the resistance between its two terminals.

In pre-lab Problem 6, you calculated the equivalent resistance RBCDB whose presence in Method 1 circuit makes the difference.

Use your pre-lab results to calculate the resistance measured according to the erroneous Method 1.

Calculated: _____________ Ω

Measured: _____________ Ω

Calculate the percentage difference

Measured Simulated 100%Simulated

−⋅ =

Discuss the agreement/disagreement between theory and experiment.

In case of a serious disagreement (>10%), explain its possible cause.


Documents

Ganago Student Lab3