ECE 2120 Electrical Engineering Laboratory II

A Companion Course to ECE 2620 - Electrical Circuits II

By Dr. Apoorva Kapadia (Undergraduate Laboratory Coordinator)

and

Dr. Afshin Ahmadi (Former Laboratory Teaching Assistant)

Updated on August 17, 2020 The Holcombe Department of Electrical & Computer Engineering Clemson University Clemson, SC - 29634.

2 Contents

Course Information i Introduction ...... i Student Responsibilities ...... i Laboratory Teaching Assistant Responsibilities ...... ii Faculty Coordinator Responsibilities ...... ii Lab Policy and Grading ...... ii Course Goals and Objectives ...... iii Use of Laboratory Instruments ...... iv Instrument Protection Rules ...... iv Data Recording and Reports ...... v The Laboratory Notebook ...... v The Lab Report ...... v

Lab 1 - Orientation 1

Lab 2 - Average and RMS Values 2

Lab 3 - Capacitors and Series RC Circuits 9

Lab 4 - Inductors and Series RL Circuits 18

Lab 5 - Parallel RC and RL Circuits 25

Lab 6 - Circuit Resonance 33

Lab 7 -Filters: High-pass, Low-pass, Bandpass, and Notch 42

Lab 8 - Transformers 52

Lab 9 - Two-Port Network Characterization 61

Lab 10 - Final Exam 70

Appendix A - Safety 72

Appendix B - Instruments for Electrical Measurements 78

Appendix C - Operating Instructions for a Typical Oscilloscope 82

Appendix D - LT SPICE AC Circuit Simulation 88 Course Information

1 Introduction

This course is intended to enhance the learning experience of the student in topics encountered in ECE 2620. In this lab, students are expected to gain experience in using the basic measuring devices used in electrical engineering and in interpreting the results of measurement operations in terms of the concepts introduced in the second electrical circuits course. How the student performs in the lab depends on his/her preparation, participation, and teamwork. Each team member must participate in all aspects of the lab to insure a thorough understanding of the equipment and concepts. The student, lab teaching assistant, and faculty coordinator all have certain responsibilities toward successful completion of the lab’s goals and objectives.

1.1 Student Responsibilities

The student is expected to be prepared for each lab. Lab preparation includes reading the lab experiment and related textbook material. If you have questions or problems with the preparation, contact your Laboratory Teaching Assistant (LTA), but in a timely manner. Do not wait until an hour or two before the lab and then expect the LTA to be immediately available.

Active participation by each student in lab activities is expected. The student is expected to ask the LTA any questions they may have. Do not make costly mistakes because you did not ask a question before proceeding.

A large portion of the student’s grade is determined in the comprehensive ﬁnal exam, resulting in a requirement of understanding the concepts and procedure of each lab experiment for the successful completion of the lab class. The student should remain alert and use common sense while performing a lab experiment. They are also responsible for keeping a professional and accurate record of the lab experiments in the lab manual wherever tables are provided. Students should report any errors in the lab manual to the teaching assistant.

i 1.2 Laboratory Teaching Assistant Responsibilities

The LTA shall be completely familiar with each lab prior to class. The LTA shall provide the students with a syllabus and safety review during the ﬁrst class. The syllabus shall include the LTA’s oﬃce hours, telephone number, and the name of the faculty coordinator. The LTA is responsible for ensuring that all the necessary equipment and/or preparations for the lab are available and in working condition. Lab experiments should be checked in advance to make sure everything is in working order. The LTA should fully answer any questions posed by the students and supervise the students performing the lab experiments. The LTA is expected to grade the lab notebooks and reports in a fair and timely manner. The reports should be returned to the students in the next lab period following submission. The LTA should report any errors in the lab manual to the faculty coordinator.

1.3 Faculty Coordinator Responsibilities

The faculty coordinator should ensure that the laboratory is properly equipped, i.e., that the teaching assistants receive any equipment necessary to perform the experiments. The coordinator is responsible for supervising the teaching assistants and resolving any questions or problems that are identiﬁed by the teaching assistants or the students. The coordinator may supervise the format of the ﬁnal exam for the lab. They are also responsible for making any necessary corrections to this manual and ensuring that it is continually updated and available.

1.4 Lab Policy and Grading

The student should understand the following policy:

ATTENDANCE: Attendance is mandatory and any absence must be for a valid excuse and must be documented. If the instructor is more than 15 minutes late, students may consider lab for the day cancelled.

LAB RECORDS: The student must:

1. Perform the PreLab assignment before the beginning of each lab,

2. Keep all work in preparation of and obtained during lab; and

3. Prepare a lab report on experiments selected by the LTA.

GRADING POLICY: The ﬁnal grade of this course is determined using the criterion detailed in the syllabus.

PRE-REQUISITES AND CO-REQUISITES: The lab course is to be taken during the same semester as ECE 2620, but receives a separate grade. If ECE 2620 is dropped, then ECE 2120 must be dropped as well. Students are required to have completed both ECE 202, MTHSC 206 and PHYS 221 with a C or better grade in each. Students are also assumed to have completed a programming class and be familiar with the use of a computer-based word processor.

ii Note that the instructor reserves the right to alter any part of this information at their discretion. Any changes will be announced in class and distributed in writing to the students prior to the changes taking eﬀect.

1.5 Course Goals and Objectives

The Electrical Circuits Laboratory II is designed to provide the student with the knowledge to use basic measuring instruments and techniques with proﬁciency. These techniques are designed to complement the concepts introduced in ECE 2620. In addition, the student should learn how to record experimental results eﬀectively and present these results in a written report. More explicitly, the class objectives are:

1. To gain proﬁciency in the use of common measuring instruments.

2. To enhance understanding of advanced electric circuit analysis concepts including:

• Inductance, Capacitance, and Reactance,

• AC voltage and current addition. Phasors,

• AC power (real and reactive, instantaneous and average),

• Series and parallel resonant circuit behavior,

• Passive Filters,

• Transfer functions,

• Transformers,

• Two-port network analysis;

3. To develop communication skills through:

• Maintenance of succinct but complete laboratory notebooks as permanent, written descriptions of procedures, results, and analyses.

• Verbal interchanges with the laboratory instructor and other students.

• Preparation of succinct but complete laboratory reports.

4. To compare theoretical predictions with experimental results and to determine the source of any apparent diﬀerences.

iii 2 Use of Laboratory Instruments

One of the major goals of this lab is to familiarize the student with the proper equipment and techniques for making electrical measurements. Some understanding of the lab instruments is necessary to avoid personal or equipment damage. By understanding the device’s purpose and following a few simple rules, costly mistakes can be avoided. You have already, in ECE 211, learned these rules, but they are repeated for convenience and emphasis below. Most of the instrumentation used in this laboratory is implemented through National Analog Discovery 2, breadboard, and circuit analysis system.

In general, all devices have physical limits. These limits are speciﬁed by the device manufacturer and are referred to as the device rating. The ratings are usually expressed in terms of voltage limits, current limits, or power limits. It is up to the engineer to make sure that in device operation, these ratings (limit values) are not exceeded. The following rules provide a guideline for instrument protection.

2.1 Instrument Protection Rules

1. Set instrument scales to the highest range before turning on the power/source.

2. Be sure instrument grounds are connected properly. Avoid accidental grounding of ”hot” leads, i.e., those that are above ground potential.

3. Check polarity markings and connections of instruments carefully before connecting power.4)Never connect an ammeter across a voltage source. Only connect ammeters in series with loads.

4. Never connect an ammeter across a voltage source. Only connect ammeters in series with loads. An ammeter is a low-resistance device that, if connected in parallel, will short out most components and usually destroy the ammeter or its protecting fuse.

5. Do not exceed the voltage and current ratings of instruments or other circuit elements. This particularly applies to wattmeters since the current or voltage rating may be exceeded with the needle still on the scale.

6. Be sure the fuse and circuit breakers are of suitable value. When connecting electrical elements to make up a network in the laboratory, it is easy to lose track of various points in the network and accidentally connect a wire to the wrong place. A procedure to follow that helps to avoid this is to connect the main series part of the network ﬁrst, then go back and add the elements in parallel. As an element is added, place a small check by it on your circuit diagram. Then go back and verify all connections before turning on the power. One day someone’s life may depend upon your making sure that all has been done correctly.

iv 3 Data Recording and Reports

3.1 The Laboratory Notebook

Students must record their experimental values in the provided tables in this laboratory manual and reproduce them in the lab reports. Reports are integral to recording the methodology and results of an experiment. In engineering practice, the laboratory notebook serves as an invaluable reference to the technique used in the lab and is essential when trying to duplicate a result or write a report. Therefore, it is important to learn to keep accurate data. Make plots of data and sketches when these are appropriate in the recording and analysis of observations. Note that the data collected will be an accurate and permanent record of the data obtained during the experiment and the analysis of the results. You will need this record when you are ready to prepare a lab report.

3.2 The Lab Report

Reports are the primary means of communicating your experience and conclusions to other profes- sionals. In this course you will use the lab report to inform your LTA about what you did and what you have learned from the experience. Engineering results are meaningless unless they can be communicated to others. You will be directed by your LTA to prepare a lab report on a few selected lab experiments during the semester. Your assignment might be diﬀerent from your lab partner’s assignment.

Your laboratory report should be clear and concise. The lab report shall be typed on a word processor. As a guide, use the format on the next page. Use tables, diagrams, sketches, and plots, as necessary to show what you did, what was observed, and what conclusions you can draw from this. Even though you will work with one or more lab partners, your report will be the result of your individual eﬀort in order to provide you with practice in technical communication.

3.2.1 Formatting and Style

• The lab report shall be typed in a word processor.

• All page margins must be 1.25 inches. All content (including text, ﬁgures, tables, etc.) must ﬁt within the margins.

• Body text should be double-spaced.

• Basic text should be in 12-point size in a commonly used text font.

• Set your main text justiﬁed (with even left/right margins).

• The ﬁrst line of each paragraph should have a left indent.

• All the tables should have titles and should be numbered. Tables should be labelled numerically as Table 1, Table 2, etc. Table captions appear above the table. The column headings should be labeled with the units speciﬁed.

v • Graphs should be presented as ﬁgures. All the ﬁgures should have titles and should be numbered. Figure captions appear below the ﬁgure. Graphs should have labeled axes and clearly show the scales and units of the axes.

• All the ﬁgures and tables must be centered on the page.

• All the ﬁgures and tables in your report must be referenced in your discussion. References to ﬁgures in the main body of the text are usually written in abbreviated form (e.g. ‘see Fig. 1”).

• Use MS-Word equation (under Insert → Equation menu), MathType, or a similar tool to type formulas.

• If you need to copy a schematic or ﬁgure from the lab manual to your report, use Copy and Paste function or take a screenshot by using Snipping Tool in MS-Windows.

• Do not place screenshots of your lab notebook in the report! Diagrams, tables, calculations, etc. must be generated using the existing tools in the word processor.

3.2.2 Order of Lab Report Components

COVER PAGE - Cover page must include lab name and number, your name, your lab partner’s name, and the date the lab was performed.

OBJECTIVE - Clearly state the experiment objective in your own words.

EQUIPMENT USED - Indicate which equipment was used in performing the experiment.

FOR EACH PART OF THE LAB:

• Write the lab’s part number and title in bold font.

• Firstly, describe the problem that you studied in this part, give an introduction of the theory, and explain why you did this experiment. Do not lift the text from the lab manual; use your own words.

• Secondly, describe the experimental setup and procedures. Do not follow the lab manual in listing out individual pieces of equipment and assembly instructions. That is not relevant information in a lab report! Instead, describe the circuit as a whole (preferably with diagram), and explain how it works. Your description should take the form of a narrative, and include information not present in the manual, such as descriptions of what happened during intermediate steps of the experiment.

• Thirdly, explain your ﬁndings. This is the most important part of your report, because here, you show that you understand the experiment beyond the simple level of completing it. Explain (compare expected results with those obtained). Analyse (analyze experimental error). Interpret (explain your results in terms of theoretical issues and relate to your experimental objectives). This part includes tables, graphs, and sample calculations. When showing calculations, it is usual to show the general equation, and one worked example. All the results should be presented even

vi if there is any inconsistency with the theory. It should be possible to understand what is going on by just reading through the text paragraphs, without looking at the ﬁgures. Every ﬁgure/table must be referenced and discussed somewhere in the text.

• Finally, provide a summary of what was learned from this part of the laboratory experiment. If the results seem unexpected or unreliable, discuss them and give possible explanations.

CONCLUSIONS - The conclusion section should provide a take-home message summing up what has been learned from the experiment:

• Brieﬂy restate the purpose of the experiment (the question it was seeking to answer)

• Identify the main ﬁndings (answer to the research question)

• Note the main limitations that are relevant to the interpretation of the results

• Summarise what the experiment has contributed to your understanding of the problem.

PROBING FURTHER QUESTIONS - Questions pertaining to this lab must be answered at the end of laboratory report.

vii

Lab 1 - Orientation

Introduction

In the ﬁrst lab period, the students should become familiar with the location of equipment and components in the lab, the course requirements, and the teaching instructor. Students should also make sure that they have all of the co-requisites and pre-requisites for the course at this time.

Objective

To familiarize the students with the lab facilities, equipment, standard operating procedures, lab safety, and the course requirements.

PreLab

Read the Introduction and Appendix A, of this manual. Download and install the “WaveForms” software on your personal computer, available here.

Equipment Needed

ECE 2120 lab manual.

Procedure

1. During the ﬁrst laboratory period, the instructor will provide the students with a general idea of what is expected from them in this course. Each student will receive a copy of the syllabus, stating the instructor’s contact information. In addition, the instructor will review the safety concepts of the course. 2. During this period, the instructor will brieﬂy review the equipment which will be used throughout the semester. The location of instruments, equipment, and components (e.g. resistors, capacitors, connecting wiring) will be indicated. The guidelines for instrument use will be reviewed.

1 Lab 2 - Average and RMS Values

Introduction

Waveforms of voltage and current that vary periodically with time may be characterized by their average value or their root mean square (rms) value. The latter is used to determine the power supplied, dissipated, or stored by a circuit element. Some of the measuring instruments you will use respond to average values of voltage or current, while others respond to rms values.

Objective

By the end of this lab, the student should learn how to determine the values of rms voltage for three types of waveforms: a sinusoid, a square wave, and a triangular wave. Also, the student should understand the diﬀerence between a true-rms and a conventional multimeter.

PreLab

Read Appendix B and Appendix C of this manual, paying particular attention to the methods of using measurement instruments. Prior to coming to lab class, complete Part 0 of the Procedure.

Equipment

Analog Discovery 2 Instrument. Digital Multimeter. Resistance Box.

2 Background

Measurements of AC signal:

• Amplitude is a non-negative scalar measure of a wave’s maximum magnitude of oscillation. In electrical engineering it may be thought of as the maximum absolute value reached by a voltage or current waveform as measured from the center of the oscillation. An amplitude measurement may be reported as peak, peak-to-peak, average, or RMS. • Peak amplitude is the height of an AC waveform as measured from the center of the oscillation to the highest positive or lowest negative point on a graph. Also known as the crest amplitude of a wave. • Peak-to-peak amplitude is the total height of an AC waveform as measured from maximum positive to minimum negative peaks (the highest peak to the lowest valley) on a graph of the waveform. Often abbreviated as “P-P”, e.g., Vp−p or Vpp. • Average value is the arithmetic “mean” of a waveform’s values over one cycle. The average value of any waveform with equal-area portions above and below the “zero” line on a graph is zero. However, often as a practical measure of amplitude, a waveform may be characterized by its average absolute value, calculated as the arithmetic mean of the absolute values of the waveform. • “RMS” stands for Root Mean Square, and is a way of expressing an AC quantity of voltage or current in terms functionally equivalent to DC. For example, 10 volts AC RMS is the amount of AC voltage that would produce the same amount of heat dissipation across a resistor of given value as a 10 volt DC power supply. Also known as the “equivalent” or “DC equivalent” value of an AC voltage or current. • Analog, electromechanical meter movements respond proportionally to the average absolute value of an AC voltage or current. When RMS indication is desired, the meter’s calibration must be adjusted accordingly, usually by applying a constant multiplicative factor assumed for a sinusoidal wave. This means that the accuracy of an electromechanical meter’s RMS indication is dependent on the purity of the waveform and whether it is the same wave shape as the waveform used in calibrating.

Figure 2.1: Measurements of AC Signal

3 In this experiment, we will work with periodic waveforms having period T and amplitude Vm. Speciﬁcally, we will work with the following three types of waveforms:

1. A sinusoidal wave voltage: 2Π v(t) = V cos( t) (2.1) m T 2. A square-wave voltage: T 3T v(t) = Vm 0 ≤ t ≤ ; T ≤ t ≤ ; etc 2 2 (2.2) T 3T v(t) = −Vm 2 ≤ t ≤ T ; 2 ≤ t ≤ 2T ; etc

3. A triangular-wave voltage: (t−nT ) 1 1 v(t) = 4Vm[ T ](n − 4 )T ≤ t ≤ (n + 4 )T ) 1 (2.3) (t−(n+ 4 )T ) 1 3 v(t) = Vm[ T ](n + 4 )T ≤ t ≤ (n + 4 )T ) where n = 0,1,2,3,...

AC voltage and current waveforms are further depicted in a number of ways. Two of the most common and useful are the average and the root-mean-square (rms) values.

The average value of a time-varying waveform x(t) with period T is given by:

1 Z T Xavg = x(t)dt (2.4) T 0

The root-mean-square value, useful for power calculations, is deﬁned by: s Z T 1 2 Xrms = x (t)dt (2.5) T 0

Since the sine, square, and triangular waveforms are symmetrical about the time axis, they all have mathematical average voltages of zero. However, each waveform will have an rms value, and a summary of the calculation steps for relating the voltage magnitude to the rms value for each waveform is shown below:

Sinusoidal Voltage: s s s Z T Z T 2 Z T cos(2 2π t) 1 2 1 2π 1 1 T Xrms = v (t)dt = Vm sin( t) dt = Vm − dt T 0 T 0 T T 0 2 2 s 1 1 Vm = Vm T + 0 = √ (2.6) T 2 2

4 Square-Wave Voltage: s v s Z T u "Z T/2 Z T # 1 2 u 1 2 2 1 2 T 2 T Xrms = v (t)dt = t (V m) dt + (−V m) dt = (V m) + (−V m) T 0 T 0 T/2 T 2 2 r 1 = [(V )2T ] = V (2.7) T m m

Triangular-Wave Voltage: s v Z T u "Z T/4 Z 3T/4 Z T # 1 2 u 1 4V m 2 4V m 2 4V m 2 Xrms = v (t)dt = t ( t) dt + (2V m − t) dt + (−4V m + t) dt T 0 T 0 T T/4 T 3T/4 T s 1 V 2T 2V 2T V 2T V = m + m + m = √m (2.8) T 12 12 12 3

If we use what is called a “true rms” voltage meter, then the relationship between the magnitude of the waveform and the measured value would be given by the three equations given above. However, many of the meters available are not “true rms” meters, and as a result are only designed for measurements in circuits with either DC voltages or sinusoidal AC voltages. For obtaining the AC voltage, most digital meters eﬀectively perform a full-wave rectiﬁcation of the waveform and compute the average absolute value. A constant factor is then applied to compute an RMS value. Often the constant factor is chosen to give a correct result for a sinusoidal waveform.

For a sinusoidal waveform, the average value for the rectiﬁed voltage is calculated as follows: Z T Z T "Z T/2 Z T # 1 1 2π Vm 2π 2π Xavg−rect = |v(t)|dt = |Vm sin( t)|dt = Vm sin( t)dt + −Vm sin( t)dt T 0 T 0 T T 0 T T/2 T Z T/2 2Vm 2π 2Vm = Vm sin( t)dt = (2.9) T 0 T π √ Since the rms value for a sinusoidal waveform should be related to the amplitude by 2, we need to apply a conversion factor to get the correct rms value on the meter readout. In this case the conversion factor would be: 1 π Vrms = √ Vavg−rectified = 1.111 ∗ Vavg−rectified (2.10) 2 2

Now what happens when we measure a square wave with 50% duty cycle? To ﬁnd out, we compute the average of the rectiﬁed square-wave waveform: " # 1 Z T 1 Z T/2 Z T Vavg−rect = |v(t)|dt = |Vm|dt + | − Vm|dt = Vm (2.11) T 0 T 0 T/2

The non-true-rms meter will apply the same conversion factor it applied to the sine wave. Hence, what we will see on the meter readout for a square wave is:

Vmeter = 1.111 ∗ Vm (2.12)

5 This meter will report a rms value that is 11% higher than the actual rms value we should have for a square wave.

Finally, for a triangular waveform, the average rectiﬁed voltage is:

1 Z T Vavg−rect = |v(t)|dt T 0 "Z T/4 Z T/2 Z 3T/4 Z T # 1 4V m 2 4V m 2 4V m 2 4V m 2 = ( t) dt + (2V m − t) dt + (−2V m + t) dt + (4V m − t) dt T 0 T T/4 T T/2 T 3T/4 T V = m (2.13) 2

So for a triangular waveform the rms voltage indicated on the non-true-rms meter will be: V m V = 1.111 ∗ = 0.555 ∗ V (2.14) meter 2 m

√ while it should be registering Vm/ 3 = 0.577 ∗ Vm. This meter gives a reading only 96% of what we should actually get, and so will under-report the rms voltage for a triangular wave.

In practice you will not get the exact results predicted by the equations, due to a number of errors, such as inability to set the peak voltage to the exact value, slight errors in the meter reading, and inaccuracies in the shape of the waveform produced by the function generator. Also, we have assumed that the duty cycle of the square wave is exactly 50% in these calculations, which might not actually be the case for your waveform generator.

Procedure

Part (0) After answering the following questions, review the laboratory exercise procedures and plan how you will use the experience gained in these calculations to ﬁnd the values sought.

1. Using the information provided in the Background section, calculate the average voltage, the average absolute voltage, and the rms voltage values for the symmetrical sine, square, and triangular waveforms, assuming that the peak value of each waveform is 2V.

Table 2.1: Calculated Voltage Values

Average Voltage Average Absolute Voltage RMS Voltage

Sine Square Triangular

6 2. Compute the RMS voltage value that would be reported by non-true rms voltmeters, assuming that the peak value of each waveform is 2V. You may use the equations derived in the Background section for these calculations.

Table 2.2: Reported Voltage Value by non-true RMS voltmeters

Reported RMS Voltage

Sine Square Triangular

Part (1) Setup the circuit in Figure 2.2, and connect the CH1 of the oscilloscope across the resistor. Use the “W1” and Ground terminals of the Analog Discovery for the Function Generator, and “1+” and “1-” pins for the Oscilloscope.

+

2Vpeak 1kΩ

−

Figure 2.2: A Simple Resistive Circuit

1. Open the WaveForms software. Use the settings under the “Wavegen” menu to output a 1kHz sine wave with amplitude = 2 V-Peak and DC oﬀset = 0V. “RUN” the FGEN.

(a) Open the Scope menu and display the voltage across the resistor. You may need to adjust the oscilloscope’s Range and Base settings to obtain a good display of the waveforms. (b) Use the “Measurements” function to display the AC RMS value (available under the “View” menu). (c) Record the RMS value in Table 2.3. (d) Disconnect the oscilloscope and connect the Digital Multimeter (DMM) across the resistor (VΩ and COM) to measure the RMS voltage. Record the value in Table 2.3.

2. Repeat the steps from 1.a to 1.d using a triangular wave output from the Function Generator. Record your values in Table 2.3.

3. Repeat the steps from 1.a to 1.d using a square wave output from the Function Generator. Record your values in Table 2.3.

7 Table 2.3: Calculated and Measured Voltage Values

Calculate RMS Voltage Measured RMS Voltage Measured RMS Voltage (Table 2.1) (Oscilloscope) (DMM)

Sine Triangular Square

4. Adjust the DC oﬀset control of FGEN to add a 2 Volt DC oﬀset to the original 1 kHz sine wave. Now the waveform should vary from 0 to 4 Volts. Use the DMM to measure the AC RMS voltage and the DC voltage across the resistance. Record the values. Repeat this same step for triangular wave and square wave.

Table 2.4: Measured Voltage Values with 2V DC Oﬀset

AC RMS Voltage DC Voltage (DMM) (DMM)

Sine Square Triangulare

Probing Further Questions

Q1. Compare the RMS voltage value calculations you made for the PreLab to the AC voltage measure- ments made in 1th, 2th, 3th, and 4th steps in the Procedure. Do your measured values agree with the calculated values in all cases? If not, why?

Q2. Summarize your comparisons: Does the Fluke digital multimeter provide true RMS measurements of voltage for all three waveforms? Justify your conclusion.

8 Lab 3 - Capacitors and Series RC Circuits

Introduction

Linear circuit elements — resistors, capacitors, and inductors — are the backbone of electrical and electronic circuits. These three types of elements respond to electrical voltages in diﬀerent ways, variously consuming, storing, or supplying electrical energy. Understanding these behaviors and learning to calculate the result of combining elements is critical for designing and working with electric circuits. While a resistor consumes electrical energy, converting it to heat, capacitors and inductors vary their responses according to the frequency of the voltage or current applied to them. This laboratory will explore those responses for series-connected capacitors.

Objectives

Educational

• Learn to measure capacitive reactance.

• Learn to measure phase angles between voltages.

• Learn to draw impedance and voltage phasor diagrams for resistors and capacitors in series.

• Understand how impedance and voltage phasors add (i.e., like vectors).

• Learn to simulate AC series circuit in LT SPICE.

Experimental

• Conﬁrm how capacitances add when two capacitors are connected in parallel; in series.

• Determine the reactance of a capacitor in a series RC circuit by measuring voltages.

• Draw impedance and voltage phasor diagrams for a series RC circuit.

• Explain the eﬀect of frequency on the impedance and voltage phasors for a series RC circuit.

9 PreLab

Reading

• Read and study the Background section of this Laboratory. • Read Appendix C, especially Avoiding Grounding Errors with Oscilloscope, Voltage Measurement, and Phase-Angle Measurement for an oscilloscope with two vertical inputs.

Written

• Prior to coming to lab class, complete Part 0 of the Procedure.

Equipment

Analog Discovery 2 Instrument. Digital Multimeter. Resistance Substitution Box. Capacitor: 2 x 0.01 µF.

Background

A capacitor is formed whenever two conductors are separated by an insulating material. Consider the simple example of two parallel conducting plates separated by a small gap that is ﬁlled with an insulating material (vacuum, air, glass, or other dielectric). If a potential diﬀerence exists between the two plates, then an electric ﬁeld exists between them, and opposite electric charges will be attracted to the two plates. The ability to store that electric charge is a fundamental property of capacitors. The larger the plates, the more charge can be stored. The closer the plates, the more charge can be stored. . . at least until the charges leap the gap and the dielectric breaks down.

If a voltage source is connected across a capacitor, charge will ﬂow in the external circuit until the voltage across the capacitor is equal to the applied voltage. The charge that ﬂows is proportional to the size of the capacitor (its “capacitance”) and to the applied voltage. The relationship is given by the equation: Q = CV (3.1) where Q is the charge in coulombs, C is the capacitance in farads, and V is the applied voltage in volts.

Capacitors in Series

Electric current I is the amount of charge that ﬂows per unit time; that is, I = Q/T. Thus, the total charge that ﬂows through a circuit (or a capacitor) is Q = IT. So, if two capacitors are connected in series and a voltage is applied across the pair, the same current, and therefore the same charge, must ﬂow through both capacitors, and the total voltage VT must be divided across both capacitors: Q Q 1 1 Q VT = V1 + V2 = + = Q + = (3.2) C1 C2 C1 C2 CT

10 where V1 and V2 are the voltages across the capacitors with capacitances C1 and C2. Thus, CT , the total capacitance of two capacitors in series, is found by:

1 1 1 C1C2 = + or CT = (3.3) CT C1 C2 C1 + C2

Capacitors in Parallel

Connecting capacitors in parallel is eﬀectively the same as making a single capacitor’s plates larger, and therefore able to hold more charge for a given applied voltage. This simple view is borne out if one analyzes the ﬂow of charge through a parallel array of capacitors connected to a voltage source. The result of such analysis is that capacitances in parallel add directly:

CT = C1 + C2 + C3 + ... (3.4)

Time Constant

If a voltage V0 is applied to a capacitor C connected in series with a resistor R, the voltage across the capacitor gradually increases. The rate at which the capacitor’s voltage changes is characterized by a “time constant”, τ: τ = RC (3.5) where τ is the time required for the voltage on the capacitor to rise from 0 to 0.632 V0. τ is also the time required for the voltage of a fully charged capacitor to fall from V0 to 0.368 V0. The number 0.368 = e−1 and the number 0.632 = (1 – e−1 ).

Capacitive Reactance

Reactance is a characteristic exhibited by capacitors and inductors in circuits with time-varying voltages and currents, such as common sinusoidal AC circuits. Like resistance, reactance opposes the ﬂow of electric current and is measured in ohms. Capacitive reactance XC can be found by the equation: 1 X = (3.6) c 2πfc where f is the frequency of the applied voltage or current and C is the capacitance in farads. As with resistance, reactance obeys Ohm’s law:

Vc Vc = IcXc or Xc = (3.7) Ic

If a sinusoidal voltage is applied across a resistor, the current through the resistor is in phase with the voltage. That is not true for a capacitor. If we connect a capacitor across a sinusoidal voltage, the maximum current ﬂows through the capacitor when the voltage’s rate of change is maximum (i.e., at V=0), and diminishes as the voltage on the capacitor increases, until ﬁnally the current is zero when the voltage is at maximum and its derivative is zero. At that instant, the maximum possible charge for the applied voltage is stored in the capacitor, and so the ﬂow of charge (i.e., the current) stops. The current and the voltage have exactly the same frequency, but the current through the capacitor is 1 ◦ leading the voltage by 4 cycle — 90 or π/2 radians. Figure 3.1 illustrates this relationship.

11 Figure 3.1: Resistive, Inductive, and Capacitive Loads

Phasors

When a sinusoidal voltage at frequency f drives a circuit that contains only linear elements, the waveforms throughout the circuit are also sinusoidal and at the same frequency. To understand the relationships among the sinusoidal voltages, currents, and impedances, we represent the various wave- forms as two-dimensional vectors called phasors. A phasor is a complex number used to represent a sinusoidal wave, taking into account both its amplitude and phase angle. As a complex number, a phasor has “real” and “imaginary” components, but like any two-dimensional vector, it can be drawn simply on ordinary XY axes, with the “real” axis in the usual X direction and the “imaginary” axis in the usual Y direction. Such phasor drawings are very helpful in analyzing circuits and understanding the relationships of the various voltages and currents. The algebra of complex numbers can then be used to perform arithmetic operations on the sinusoidal waves. Make no mistake: adding voltages or currents in an AC circuit without taking account of phase angles will lead to confusing and wrong results.

A series RC circuit is illustrated in Figure 3.2(a) and a phasor diagram of its impedances is shown in Figure 3.2(b). The vector for resistance R is shown along the Real (X) axis, while the reactance XC is shown in the negative Imaginary (Y) axis, since its voltage trails its current by 90◦. The vector sum of R and XC is labeled Z and has magnitude 5kΩ. Therefore, the magnitude of the current through the circuit is VS / Z = 1.0 mA.

Notice that the current phasor is in the same as the direction of the voltage across the resistor, because voltage and current are in phase for a resistor. The current is the same throughout the series circuit, including the capacitor, and the voltage across the capacitor trails the current by 90◦. The source voltage is the vector sum of the voltages across the resistor and the capacitor, as is illustrated

12 in the phasor diagram, Figure 3.2(c).

Figure 3.2: Phasor Diagram

Safety Precautions

1. Always turn oﬀ power to the circuit when changing the circuit.

2. Failure to turn oﬀ power when making circuit changes is a major reason for blowing fuses in the equipment, thereby rendering the equipment unusable and wasting your time and that of others. Please carefully check circuit wiring, resistor settings, and voltage settings before applying power to the circuits.

3. Only reapply power after verifying that the circuit is properly wired and that the voltage to be applied is at or below the required value.

Procedure

Part (0) Prior to coming to the lab, calculate theoretical voltage and current values in Figure 3.3 and record them in Table 3.1. Calculate all voltage and current values as peak-to-peak.

Table 3.1: Calculated Values of RC Circuit

ZT otal i1 (p-to-p) VR (p-to-p) VC (p-to-p) Freq. (Hz) Xc (Ω) R (Ω) X (Ω) mag (mA) angle mag (V) angle mag (V) angle

500 1000 2000 4000 8000

13 Part (1) Use the digital multimeter (DMM) to complete the following steps. Record your values in Table 3.2.

1. Set 10 kΩ on the resistor decade box and measure its resistance using the DMM.

2. Select two 0.01 µF capacitors. Identify them as C1 and C2. 3. Measure the capacitances using the DMM by connecting the capacitor leads to the “VΩ” and “COM” terminals.

4. Connect the capacitors in parallel and measure the combined capacitance using the DMM. Wait for the measurement to stabilize before recording. Compare the theoretical and measured values. Does the measurement tend to conﬁrm the theoretical prediction? Explain.

5. Connect the capacitors in series and measure the combined capacitance using the DMM. Wait for the measurement to stabilize before recording. Compare the theoretical and measured values. Does the measurement tend to conﬁrm the theoretical prediction? Explain.

6. Select capacitor C1. Compute its theoretical reactance XC1 at a frequency f = 500 Hz. Record the value in Table 3.2.

Table 3.2: Calculated and Measured Values

Nominal or Calculated Value Measured Value

R

C1

C2 C1 C2 (Parallel) C1 C2 (Series)

XC1 (500 Hz) -

Part (2) Set up the series RC circuit using the resistor R and capacitor C1 as shown in Figure 3.3. Connect Channel 1 of the oscilloscope across the function generator (through the bread board). Use the “W1” and Ground pins of the Analog Discovery for the Function Generator, “1+” and “1-” pins for Channel 1 of the Oscilloscope, and “2+” and “2-” pins for Channel 2 of the Oscilloscope.

14 For steps 1 through 4, record the results of the measurements in Table 3.3. Use peak-to-peak readings for all voltage and current measurements in this experiment.

10kΩ

i1 +

2Vp−p 0.01µF

−

Figure 3.3: Series RC Circuit

1. Set the function generator to give a sinusoidal wave output with amplitude of 1V, oﬀset = 0V, and frequency = 500 Hz. Check the voltage (VS) and frequency values on CH1 of the oscilloscope and record them in Table 3.3. Keep CH1 across FGEN throughout the experiment.

2. Connect CH2 of the oscilloscope across capacitor C1 and measure VC .

3. Connect CH2 of the oscilloscope across resistor R and measure VR. Do not remove CH2 leads.

4. Compute the peak-to-peak current Ipp from Ipp = VR/R. Remember, the current is the same throughout the circuit, so this current also ﬂows through the capacitor.

5. Measure the phase angle φ between VR and VS by using the cursors on the oscilloscope display.

6. Compute the capacitor’s reactance XC1 from XC1 = VC1/IPP . Compute C1 from the measured XC1 and compare to your earlier measurement.

7. Compute the total impedance ZT otal by applying Ohm’s law to the circuit (Z = Vs∠0 / I∠φ). Use the supply voltage set in step 1 and the current found in step 4. Remember, the impedance has both a magnitude and a phase angle (measured relative to the resistor). Convert the value to rectangular form and record it in the table. 8. Repeat steps 1 to 7 (resetting VS if necessary) for the following frequencies: 1000, 2000, 4000, 8000 Hz. Table 3.3: Recorded Values for Part 2

Freq. Freq. (Hz) VS (V) VC1 (V) VR (V) IPP (mA) φ (deg) XC1 (Ω) C1 (µF) ZT otal 500 (Hz) 1000 (Hz) 2000 (Hz) 4000 (Hz) 8000 (Hz)

15 9. Draw impedance and voltage phasors (as in Figure 3.2) for frequency f = 1000 Hz.

10. Draw impedance and voltage phasors for frequency f = 4000 Hz.

11. The phasor diagrams at various frequencies show how the impedances, and therefore the voltages, change with frequency. To better see the net eﬀect on the circuit, graph VC1 and VR versus frequency for the values in Table 3.3. Label the curves.

Probing Further Questions

Q1. Describe what happens to the current in this RC series circuit as the frequency increases. Explain in general terms why the observed change should occur.

16 Q2. Simulate the circuit in LT SPICE and graph VC1 and VR versus frequency over the range of values in Table 3.3. Compare to your manually drawn curves. [Instructions for setting up the necessary AC simulation in LT SPICE are shown in Appendix D.]

Q3. In this experiment it was shown that the voltage phasor diagram can be obtained by multiplying each of the impedance phasors by the current in the circuit. If each of the voltage phasors in the voltage phasor diagram is again multiplied by the current, the resulting diagram is the power phasor diagram. Using the data in Table 3.3 convert the current I and source voltage VS to RMS values. Then draw a plot of the power phasor diagrams at a frequency of 1000 Hz and another at a frequency of 4000 Hz. Determine the real power, the reactive power, and the apparent power in the RC circuit at those frequencies.

17 Lab 4 - Inductors and Series RL Circuits

Introduction

This laboratory continues the study of series linear circuits, this time looking at the eﬀect of inductors in series linear circuits. Besides studying the behavior of inductors, you will use measurements of magnitude and phase to construct phasor diagrams for sinusoidal voltages in a series circuit, and thereby validate Kirchhoﬀ’s Voltage Law even when the current varies with time.

Objectives

Educational

• Learn to predict and to measure inductive reactance.

• Learn to apply Ohm’s law to reactances of impedances.

• Learn to measure phase angles between voltages.

• Learn to draw impedance and voltage phasor diagrams for resistors and inductors in series.

• Gain experience in the construction and use of phasor diagrams.

Experimental

• Determine the reactance of an inductor in a series RL circuit by measuring voltages.

• Draw impedance and voltage phasor diagrams for a series RL circuit.

• Determine the real, reactive, and apparent power for a series RL circuit.

• Explain the eﬀect of frequency on the impedance and voltage phasors for a series RL circuit.

18 PreLab

Reading

• Read and study the Background section of this Laboratory.

• Read Appendix C, Voltage Measurement, and Phase-Angle Measurement for an oscilloscope with two vertical inputs.

Written

• Prior to coming to lab class, complete Part 0 of the Procedure.

Equipment

Analog Discovery 2 Instrument. Digital Multimeter. Resistance Substitution Box. Inductor: 50 mH.

Background

When a current passes through a wire, a magnetic ﬁeld is generated around the wire. The magnetic ﬁeld results from the movement of electric charge and is proportional to the magnitude of the current. Turning the wire into a coil concentrates that magnetic ﬁeld, with the ﬁeld of one turn of the coil reinforcing another. Such a coil is called an inductor. From an electrical point of view the especially interesting fact is that if the current in the wire changes, the magnetic ﬁeld will react to try to keep the current constant. This property of inductors is described by Lenz’s law.

An inductor’s response to changes of current is called inductance. Inductance opposes changes in current, just as capacitance opposes changes in voltage. Inductance is the electric current equivalent of inertia in mechanical systems.

Inductance is measured in henries. One henry is the amount of inductance present when one volt is generated as a result of a current changing at the rate of one ampere per second.

When inductors are connected in series, the total inductance is the sum of the individual induc- tors. This is similar to the way resistors in series add. When inductors are connected in parallel, the inductances add in the same way that parallel resistors add. However, an additional eﬀect can appear in inductance circuits that is not present with resistors. This eﬀect is called mutual inductance and is caused by the interaction of the magnetic ﬁelds of neighboring inductors. Mutual inductance can either increase or decrease the total inductance, depending on the orientation of the interacting inductors.

19 Time Constant

Inductance circuits have a time constant associated with them, just as do capacitance circuits, except that the curve of interest for inductors is the current, rather than the voltage. The time constant τ for inductors is: L τ = (4.1) R where τ is in seconds; L is in henries; R is in ohms.

The voltage induced across the inductor is a maximum when the change in current is a maximum. When a sinusoidal current is applied to an inductor, the largest induced voltage appears across the inductor when the current is passing through zero. At the peaks of the applied current, the slope is zero and the current is not changing, so the induced voltage is zero. Therefore, the voltage that appears across an inductor leads the current in the inductor by 1/4 cycle. Figure 4.1 illustrates this relationship.

Figure 4.1: Resistive, Inductive, and Capacitive Loads

Inductive Reactance

As the frequency of the sine wave increases, the rate of change of the current also increases, and so the induced (reacting) voltage across the inductor increases. As a result, the net current through the inductor decreases. That means, the inductor’s reactance increases with frequency. The inductive reactance is given by: XL = 2πfL (4.2)

20 As with capacitors and resistors, Ohm’s law can be applied to inductive circuits:

VL XL = (4.3) IL

Series RL circuits

When a sine wave is applied to a series circuit of linear components (resistors, capacitors, and inductors), the phase relationships between current and voltage depend on the types of components. The current and voltage are always in phase across an ideal resistor. The current through a capacitor leads the voltage across the capacitor by 90◦. The voltage across an ideal inductor leads the current through the inductor by 90◦.

A common memory aid for these relationships is “ELI the ICE man”, where E represents voltage (E is short for “electromotive force”, which is another term for voltage), I represents current, L represents inductance, and C represents capacitance.

From Kirchhoﬀ’s current law (KCL), we know that the current is the same throughout a series circuit. Since the current and voltage are in phase for a resistor, we can determine the phase of the current by measuring the phase of the voltage across the resistor. This is commonly done by using an oscilloscope to compare the voltage from the source to the voltage across the resistor.

Consider the series RL circuit shown in Figure 4.2(a). Graphical representations for the impedance phasors are shown in Figure 4.2(b). As with the series RC circuit, the total impedance of the series RL circuit is obtained by the vector sum of the impedance phasors. (Because the vectors are in the complex plane, this summation can also be done equivalently by summing the complex numbers.)

Figure 4.2: Phasor Diagram

In this example, 5V are applied and the total impedance is 5 kΩ, so the total current is 1.0 mA. Because the current is the same in all components of the series circuit, we use its direction through the resistor for the phase reference direction. Multiplying the impedance phasors by the current gives the voltage phasors, as shown in Figure 4.2(c).

In this experiment you will use an oscilloscope to measure the phase angles in the circuit. Because of the resistance of the wires in an inductor, actual inductors may have enough resistance to aﬀect the phase angle. To minimize that eﬀect in this experiment, we will use a series resistor that is large compared to the inductor’s resistance.

21 Safety Precautions

1. Always turn oﬀ power to the circuit when changing the circuit.

2. Only reapply power after verifying that the circuit is properly wired and that the voltage to be applied is at or below the required value.

3. Failure to turn oﬀ power when making circuit changes is a major reason for blowing fuses in the equipment, thereby rendering the equipment unusable and wasting your time and that of others. Please carefully check circuit wiring, resistor settings, and voltage settings before applying power to the circuits.

Procedure

Part (0) Prior to coming to the lab, calculate theoretical voltage and current values in Figure 4.3 and record them in Table 4.1. Calculate all voltages and currents as peak-to-peak values.

Table 4.1: Calculated Values of RC Circuit

ZT otal i1 (p-to-p) VR (p-to-p) VL (p-to-p) Freq. (Hz) XL (Ω) R (Ω) X (Ω) mag (mA) angle mag (V) angle mag (V) angle

10000

Part (1) Complete the following steps and record your values in Table 4.2.

1. Set 5 kΩ on the resistor decade box and measure its resistance using the DMM.

2. Select a 50 mH inductor or set that value on the inductor decade box. Measure its inductance L. To measure inductance requires connecting the leads to the Impedance Analyzer (the device that comes with AD2 kit) and using the Impedance tool of the WaveForms application.

3. Inductors are made of wires, so they have some internal resistance. Measure the winding resistance RW using the DMM by connecting the inductor leads to the “VΩ” and “COM” terminals.

Table 4.2: Measured Values

Nominal Value Measured Value

R 5kΩ L 50mH

RW 0

22 Part (2) Construct the series RL circuit in Figure 4.3. Connect Channel 1 of the oscilloscope across the function generator (through the bread board). Use the “W1” and Ground pins of the Analog Discovery for the Function Generator, “1+” and “1-” pins for Channel 1 of the Oscilloscope, and “2+” and “2-” pins for Channel 2 of the Oscilloscope.

For steps 1 through 4, record the results of the measurements in Table 4.3. Use peak-to-peak readings for all voltage and current measurements in this experiment.

50mH

i1 +

4Vp−p 5kΩ

−

Figure 4.3: Series RC Circuit

1. Set the function generator to give a sinusoidal wave output with amplitude of 2V, oﬀset = 0V, and frequency = 10 kHz. Check the voltage (VS) and frequency values on CH1 of the oscilloscope and record them in Table 4.3. Keep CH1 across FGEN throughout the experiment.

2. Connect CH2 of the oscilloscope across inductor L and measure VL.

3. Connect CH2 of the oscilloscope across resistor R and measure VR. Do not remove CH2 leads.

4. Compute the peak-to-peak current Ipp in the circuit by applying the Ohm’s law to the resistor. That is Ipp = VR/R.

5. Measure the phase angle φ between VR and VS by using the cursors on the oscilloscope display. Press STOP to improve accuracy when taking cursor measurements. Record this value as φmeas in the table.

6. Compute the inductive reactance XL by applying Ohm’s law to the inductor. That is XL = VL/IPP .

7. Compute the total impedance ZT otal by applying Ohm’s law to the circuit (Z = Vs∠0 / I∠φ). Use the supply voltage set in step 1 and the current found in step 4. Remember, the impedance has both a magnitude and a phase angle (measured relative to the resistor). Convert the value to rectangular form and record it in the table.

8. Compute the phase angle φ between VS and VR. Recall that φ = arctan(VL/VR). Record this value as φcalc in the table, and compare it to the value measured on the oscilloscope.

23 Table 4.3: Recorded Values for Part 2

Freq. (Hz) VS (V) VL (V) VR (V) IPP (mA) φmeas XL (Ω) ZT otal (Ω) φcalc

9. Using the values listed in the table, draw a diagram of the impedance phasors and a second diagram of the voltage phasors, as illustrated in Figure 4.2 of the Background section of this laboratory experiment.

10. Using the data in the table, convert the current I and source voltage VS to RMS values. Then draw plots of the power phasor diagrams (power triangle) at the frequency of 10 kHz. Determine the real power, the reactive power, and the apparent power in the RL circuit at that frequency.

Probing Further Questions

Q1. What you think would happen to the current in this RL series circuit if the frequency were decreased, say, to 2000 Hz? Why?

Q2. Simulate the circuit in LT SPICE and graph VL and VR versus frequency from 1000 Hz to 20,000 Hz. Does the simulation support your prediction in question 1?

24 Lab 5 - Parallel RC and RL Circuits

Introduction

This laboratory explores the behavior of parallel RC and RL circuits, and the application of Kirch- hoﬀ’s current and voltage laws to such circuits. For series RC and RL circuits, we saw that Kirchhoﬀ’s voltage law applies, but that the voltages must be added as phasors. Similarly, in parallel circuits, Kirchhoﬀ’s current law applies to any junction, but again, the currents must be added as phasors.

Objective

Educational

• Learn to apply Kirchhoﬀ’s voltage law (KVL) and Kirchhoﬀ’s current law (KCL) in parallel circuits.

• Learn to draw current phasor diagrams for parallel circuits.

• Gain experience in the construction and use of phasor diagrams.

• Gain experience in calculating real, reactive, and apparent power.

Experimental

• Conﬁrm Kirchhoﬀ’s current law (KCL) in parallel circuits.

• Draw current phasor diagrams for parallel circuits.

• Determine the real, reactive, and apparent power for a parallel RC circuit.

25 PreLab

Reading

• Study the Background section of this Laboratory.

Written

• Prior to coming to lab class, complete Part 0 of this lab.

• Conﬁrm Kirchhoﬀ’s current law (KCL) in parallel circuits.

Equipment

Analog Discovery 2 Instrument. Digital Multimeter. Resistors: 10kΩ, 2.2kΩ, 2 x 100Ω, 2 x 22Ω Inductor: 100 mH Capacitor: 0.01 µF Components may be discrete or via decade substitution boxes, unless otherwise indicated.

Background

As was seen in prior experiments, in a series circuit the same current is in all components, and so current is generally used as a reference in series circuits. However, in parallel circuits, the same voltage is across all components, so voltage is the logical and appropriate reference. The current in each branch then evolves from the circuit voltage.

For series RC and RL circuits, we saw that Kirchhoﬀ’s voltage law applies, but that the voltages must be added as phasors. Similarly, in parallel circuits, Kirchhoﬀ’s current law applies to any junction, but again, the currents must be added as phasors. The current entering a junction is always equal to the current leaving the junction.

In a parallel circuit, if the impedance of each branch is known, then the current in that branch can be determined directly from the applied voltage and Ohm’s law. The current phasor diagram can then be constructed, and the total current can be found as the phasor sum of the currents in each branch.

Consider the current phasor diagram for the parallel RC circuit shown in Figure 5.1. The current in the capacitor is shown at +90◦ from the voltage reference because the current leads the voltage in a capacitor. The current in the resistor is along the x-axis because current and voltage are in phase in a resistor.

26 Figure 5.1: Parallel RC Circuit

Figure 5.2 shows a parallel RL circuit and its current phasor diagram. Here we have assumed all components are “ideal”. The current in an ideal inductor is at -90◦ from the voltage reference, because the current lags the voltage in an inductor. However, practical inductors contain resistance that often is large enough to aﬀect the phasor. The eﬀect of the inductor’s resistance on the phasor diagram would be to reduce the angle between the IL and IR. In a practical circuit, this angle will be slightly less than the -90◦ of a pure inductor. This experiment will illustrate this diﬀerence between the approximation of circuit performance based on ideal components and the actual measured values.

Figure 5.2: Parallel RL Circuit

For both RC and RL circuits, the Pythagorean theorem and ordinary vector addition can be applied to the current phasors to determine the magnitude of the total current, IT . q q 2 2 2 2 IT = IR + IC and IT = IR + IL (5.1)

Recall that in series circuits, the phase angle was measured between the source voltage VS and the resistor voltage VR using the oscilloscope. The oscilloscope is a voltage-sensitive device, so examining those voltages and phase angles is straightforward. But in parallel circuits, the phase angle of interest usually is between the total current, IT , and one of the branch currents. To use the oscilloscope to measure the phase angle in a parallel circuit, we must convert the current to a voltage. This is commonly done by inserting a small resistor (a “sense” resistor) in the branch with the current to be measured. Such a sense resistor makes it easy to determine the magnitude and the phase of the current in that branch, but the resistor must be small enough not to have a signiﬁcant eﬀect on the values to be measured.

27 Part (0) Before coming to the lab, determine the voltages across and currents through each resistor in the circuits of Figure 5.3 and Figure 5.4. Record all of your calculated results in Tables 5.2 and 5.4. Record the calculated values in polar form.

Part (1) - Parallel RC Circuit

This part of the experiment will give you experience making the measurements with the digital multimeter (DMM). Because the electrodes of this device are isolated from the circuit ground and the grounds in AD2, you may make voltage measurements directly across any of the components, whether or not they are grounded.

itotal i2 + i1 C : 0.01µF

2VRMS R1 : 10kΩ

RS2 : 100Ω − RS1 : 100Ω itotal

Figure 5.3: Parallel RC Circuit

1. Before constructing the circuit in Figure 5.3, measure circuit components using the DMM: one 10kΩ resistor, two 100Ω sense resistors, and one 0.01 µF capacitor. (Be sure to wait until the capacitance measurement stabilizes.) Record the measured values in Table 5.1.

Table 5.1: Measured Values

Nominal Value Measured Value

R1 10kΩ

RS1 100Ω

RS2 100Ω C 0.01µF

2. Construct the circuit shown in Figure 5.3. Set the function generator to provide a sine wave with a voltage of 2.0 Vrms at 1.0 kHz. Verify the voltage and frequency with your oscilloscope while the circuit is connected and operating; adjust if necessary.

3. Using the DMM voltmeter in AC mode [V∼], measure the voltage drop across each resistor. The voltage drops are small, so measure as accurately as possible and keep three signiﬁcant ﬁgures in

28 your measurement (use mV mode of DMM if the displayed voltage is zero). Record the voltage drops in Table 5.2.

Table 5.2: Measured and Calculated Values for Figure 5.3

VS (RMS) VR1 (RMS) VRS1 (RMS) VRS2 (RMS) I1 (mA) I2 (mA) Itotal (mA)

Calculated 2∠0 Measured

4. Compute the current in each resistor using Ohm’s law, and record the calculated currents in Table 5.2.

5. Draw the current phasors I1, I2, and the total current Itotal in a plot similar to that of Figure 5.1. Ignore the small eﬀect of the sense resistors on the phasor diagram. Note carefully the direction of the phasors. Label each of the current phasors.

6. Compute XC for the 1.0 kHz frequency. Using this value and that of the sense resistor, calculate the expected current, IC (i2), through the capacitor. How does this value compare to that found from the sense resistor (Table 5.2)?

1 XC = −j 2πfc =

VS IC = = −jXC +RS2

7. Using the value of −jXC for the 1.0 kHz frequency and the measured resistance of R1, ﬁnd the total impedance, Ztotal (in polar form), of the circuit. Remember that these impedances add like parallel resistors. Ignore the sense resistors for this calculation. Show your work.

8. Using Ztotal and the applied voltage, VS, compute the total current, Itotal. Show your work. The total current should reasonably agree with the value determined in step 4.

29 Probing Further Questions

Q1. Ignoring the eﬀect of the sense resistors, compute the real, reactive, and apparent power of the circuit in Figure 5.3 at 1.0 kHz.

Q3. If the capacitor C were made smaller, what would happen to the current phasor diagrams?

Part (2) - Parallel RL Circuit

This part of the experiment will give you experience making similar measurements using the oscillo- scope. In this part of the experiment, record all voltages and currents as peak-to-peak values.

itotal i2 + i1 L : 100mH

4VPP R1 : 2.2kΩ

RS2 : 22Ω − RS1 : 22Ω itotal

Figure 5.4: Parallel RL Circuit

1. Before constructing the circuit in Figure 5.4, measure circuit components using the DMM and Impedance Analyzer. Record the measured values in Table 5.3.

30 Table 5.3: Measured Values

Nominal Value Measured Value

R1 2.2kΩ

RS1 22Ω

RS2 22Ω L 100mH

Table 5.4: Measured and Calculated Values for Figure 5.4

VS (P-P) VR1 (P-P) VRS1 (P-P) VRS2 (P-P) I1 (mA) I2 (mA) Itotal (mA)

Calculated 4∠0 Measured

2. Construct the circuit shown in Figure 5.4. Using the oscilloscope’s CHANNEL 1 to monitor the function generator, set the source voltage to a sine wave with a voltage of 4.0 VPP at 4.0 kHz. Verify both the voltage and the frequency with the oscilloscope. Record the measured values in Table 5.4.

3. Using oscilloscope CHANNEL 2, measure the peak-to-peak voltage across RS1. Apply Ohm’s law to calculate the peak-to-peak value of the total current, Itotal. Record the measured values in Table 5.4.

4. Because RS1 is small compared to R1, the source voltage VS is very nearly the voltage across the resistor R1. Furthermore, since only resistors are in that loop, the phase of the voltage across R1 — and current through R1 — is the same as the phase of the source voltage VS.

With CHANNEL 1 of the oscilloscope displaying the voltage across the function generator (VS), and CHANNEL 2 displaying the voltage across RS1, measure the phase angle between the generator voltage, VS, and the generator current, Itotal, (i.e., the current ﬂowing through RS1). This is equivalently the phase angle between IR1 and Itotal. You may press STOP to improve accuracy when taking cursor measurements. Record the measured phase angle between IR1 and Itotal.

φ =

5. Replace the sense resistor RS1 with a jumper wire (short it). CHANNEL 1 is still measuring the output voltage of the function generator, which now is also the voltage across R1. Connect CHANNEL 2 of the oscilloscope across sense resistor RS2. Record the voltages across R1 and across RS2 in Table 5.4. Apply Ohm’s law to calculate the current in each branch of the circuit, and record the currents. 6. Using the computed peak-to-peak currents from Table 5.4, draw the current phasor diagram for the circuit. Ignore the eﬀects of the sense resistors. Your diagram should look somewhat similar to Figure 5.2(b).

31 7. The phasor diagram depicts visually the relationship between the total current and the currents in each branch of the circuit. From the currents in the phasor diagram, compute the phase angle between the total current (Itotal) and the current in R1 (IR1). Then compute the phase angle between the total current (Itotal), and the current in L (IL). Ideally, what should be the sum of these angles?

8. On the oscilloscope, measure the angle between IL and IR1. (The oscilloscope’s leads were already connected to do this as a result of step 5). Ideally, this measurement should be 90◦, but resistance in the inductor, may reduce the angle. If necessary, adjust both scope channels to have the same apparent height on the oscilloscope screen in order to make the measurement.

9. In step 4 you measured the phase angle φRT between IR1 and Itotal. In step 8 you measured the phase angle φRL between IR1 and Itotal. Compute the phase angle φTL between Itotal and IL by subtracting φRT from φRL. That is,

φTL = φRL - φRT .

10. Compare the measured phase angles versus the computed phase angles in Table 5.4. Discuss likely causes for any discrepancies.

Probing Further Questions

◦ Q1. The currents IR1 and IL were measured in step 5. If those currents are 90 apart, we can calculate the total current Itotal using the Pythagorean theorem.

a. Compare this calculated total current to the total current measured in step 3.

b. What factors might cause any discrepancies observed between the values?

Q2. What eﬀect does the inductor’s coil resistance have on the phase angle between the currents in the resistor and the inductor?

32 Lab 6 - Circuit Resonance

Introduction

The response of a circuit containing both inductors and capacitors in series or in parallel depends on the frequency of the driving voltage or current. This laboratory will explore one of the more dramatic eﬀects of the interplay of capacitance and inductance, namely, resonance, when the inductive and ca- pacitive reactances cancel each other. Resonance is the fundamental principle upon which most ﬁlters are based — ﬁlters that allow us to tune radios, televisions, cell phones, and a myriad of other devices deemed essential for modern living.

Objective

Educational

• Learn the deﬁnition of resonance in AC circuits.

• Learn to calculate resonant frequencies, band widths, and quality factors for series and parallel resonant circuits.

• Learn to use AD2’s Bode plot function to view circuit response.

Experimental

• Predict resonant frequencies, band widths, and quality factors for series resonant circuits.

• Measure resonant frequencies, band widths, and quality factors for series resonant circuits, and compare them to predicted value.

• Determine what eﬀect changing the series resistance has on the resonance.

33 PreLab

Reading

• Study the Background section of this Laboratory exercise.

Written

• Prior to coming to lab class, complete Part 0 of the Procedure.

Equipment

Analog Discovery 2 Instrument. Digital Multimeter. Resistors: 10Ω, 100Ω Inductor: 100 mH Capacitor: 0.01 µF Components may be discrete or via decade substitution boxes, unless otherwise indicated.

Background

The reactance of inductors increases with frequency:

XL = 2πfL (6.1) The reactance of capacitors decreases with frequency: 1 X = (6.2) C 2πfC

In an LC circuit, whether series or parallel, there is some frequency at which the magnitudes of these two reactances are equal. That point is called resonance. Setting XL = XC , and solving for f, we ﬁnd that the resonant frequency fo of an LC circuit is: 1 fo = √ (6.3) 2π LC

The frequency f has units cycles/second or sec−1. The frequency may also be expressed as angular frequency, ω, where ω = 2πfC and has units radians/sec. Thus, the resonant frequency may also be written as: 1 ωo = 2πfo = √ (6.4) LC

The resonant frequency is generally the highest point of a peak (or the deepest point of a valley) with bandwidth BW (cycles/sec) or β (radians/sec). The resonant frequency is also called the center frequency, because it is at the mid-point of the peak frequency response.

34 The lowest frequency (f1 or ω1) and the highest frequency (f2 or ω2) of the band are the “half-power 1 points” at which the power is 2 that at the peak frequency. Since power goes like the square of the current, the current at the half-power points is √1 = 0.707 times the current at the maximum. Thus, 2 the bandwidth of a resonant circuit is the frequency range over which the current is at least 70.7% of the maximum. BW = f2 − f1 or β = ω2 − ω1 (6.5)

As the bandwidth narrows, the circuit becomes more highly selective, responding to a narrow range of frequencies close to the center frequency. The sharpness (narrowness) of that resonant peak is measured by the quality factor Q. The quality factor is a unitless quantity that is deﬁned as: maximum energy stored Q ≡ 2π (6.6) energy dissipated per cycle In more practical terms, f Q = o (6.7) BW

Figure 6.1: Circuit Bandwidth

Series Resonance

For a series LC circuit, the current is the same throughout. What about the voltages? To visualize the concept of resonance, consider the simple series RLC circuit in Figure 6.2 operating at resonance, and its associated reactance diagram.

Figure 6.2: Series RLC Circuit

35 The phase shift caused by the capacitor is directly opposite the phase shift caused by the inductor; that is, they are 180◦ out of phase. Therefore, in the reactance phasor diagram (b) for the circuit, the two phasors point in opposite directions. At resonance, the magnitudes of the capacitor reactance and the inductor reactance are equal, so the sum of the two phasors is zero, and the only remaining impedance is due to the resistor. Notice in the voltage phasor diagram (c) that the voltage drop across the inductor and the capacitor may be quite large — bigger even than the source voltage — but those voltages are opposite in phase and so cancel each other out as voltages are summed around the circuit. Kirchhoﬀ’s voltage law remains valid, and the generator’s voltage output is dropped entirely over the resistor R.

Since at resonance the only impedance is the resistance R, the impedance of the series circuit is at a minimum, and so the current is a maximum. That current is VS/R. The source voltage and the current are in phase with each other, so the power factor = 1, and maximum power is delivered to the resistor.

But what happens at neighboring frequencies? At lower frequencies, the inductor’s reactance de- creases, and the capacitor has greater eﬀect. At higher frequencies, the inductor dominates, and the circuit will take on inductive characteristics. How sharply deﬁned is the resonance? How selective is it? We have said that for a resonant circuit, the quality factor Q is the ratio of the resonant frequency to the bandwidth. Thus, Q gives a measure of the bandwidth normalized to the frequency, thereby describing the shape of the circuit’s response independent of the actual resonant frequency.

We list here two other useful relationships for Q in a series resonant circuit. The ﬁrst relates Q to the circuit’s capacitance, inductance, and total series resistance:

r 1 L Q = (6.8) R C

The value of R in this equation is the total equivalent series resistance in the circuit. This form of the equation makes it easy to see ways to optimize the Q for the desired circuit. Decreasing R, increasing inductance, or decreasing capacitance will all tend to make Q larger and increase the circuit’s selectivity.

The second useful relationship for Q can be derived from the previous equation. Recall that XL = 1 2πfL and XC = 2πfC .

1 p Q = X .X (6.9) R L C

Since at resonance the inductive and capacitive reactances are equal, this equation can be reduced to:

X X Q = L or Q = C (6.10) R R

where R is again the total equivalent series resistance of the circuit. Usually the XL form is used because the resistance of the inductor frequently is the dominant resistance in the circuit. An equivalent

36 form of this last equation is:

2πf L 1 Q = o or Q = (6.11) R 2πfoCR

Parallel Resonance

For a parallel RLC circuit, the same voltage is applied across all the branches. The current in each branch is determined by the voltage applied to the branch and the impedance in that branch. A parallel RLC circuit with ideal components is illustrated below, along with its current phasor diagram. Notice that the total source current at resonance is the current in the resistor. The currents in the inductor and capacitor cancel out because of their opposite phase shifts. The net impedance of the circuit at resonance is solely determined by R, since the inductor and the capacitor appear to be open.

Figure 6.3: Parallel RLC Circuit

In an ideal two-branch LC circuit including an inductor and capacitor (i.e., R is removed or equiva- lently R=∞), the source current would be zero. In a practical two-branch LC circuit, the only signiﬁcant resistance is the inductor’s winding resistance, so that resistance often plays an important role in the circuit’s behavior.

In series resonant circuits, we deﬁned the bandwidth by observing the response of the current. In parallel resonant circuits, the corresponding response to observe is the impedance. The bandwidth of a parallel resonant circuit is the frequency range over which the circuit impedance is at least 70.7% of the maximum impedance. The sharpness of the response is again measured as Q. Note that the Q of the circuit will be diﬀerent from the Q of the inductor if other resistance is in parallel with L and C. If there is no additional resistance (i.e., R=∞), then the circuit Q is the same as the inductor’s Q.

37 Table 6.1: Summary of the Characteristics of Resonant RLC Circuits

Characteristic Series Circuit Parallel Circuit

Resonant Frequency, f √1 √1 o 2π LC 2π LC Quality factor, Q 2πfoL or 1 R or 2πf RC R 2πfoRC 2πfoL o

fo fo Bandwidth, BW Q Q r 2 r 2 1 fo 1 fo Half-power frequencies f1, f2 fo 1 + 2Q ± 2Q fo 1 + 2Q ± 2Q

BW BW For Q ≥ 10, f1, f2 fo ± 2 fo ± 2

Procedure

Part (0) Before coming to the lab, calculate fo, Q, BW, f1, and f2 for the circuit shown in Figure 6.4 when R = 10Ω and 100Ω. Don’t forget to include the impedance of the function generator (RF GEN ≈ 50Ω) as part of the total resistance in the circuit. Record the results in the “predicted” column of Table 6.3.

Part (1) Using the DMM and the Impedance Analyzer, measure the values of the components in Table 6.2. Also, measure the winding resistance RW of the 100 mH inductor.

Table 6.2: Measured Values

Nominal Value Measured Value

R1 100Ω

R2 10Ω

RW - L 100mH C 0.01µF

Part (2) Complete the following steps and record your values in Table 6.3.

1. Construct the circuit shown in Figure 6.4 using a 100Ω resistor for R. Adjust the function generator to generate a sine wave with voltage 1.0 VPP . Initially set the frequency to 5000 Hz. 2. Connect oscilloscope CHANNEL 1 across the function generator (FGEN and GND) and conﬁrm that the voltage is 1.0 VPP .

38 0.01 µF

50Ω 100 mH +

1.0VPP − R = 100,10 Ω

Figure 6.4: Series RLC Circuit

3. Connect oscilloscope CHANNEL 2 across the resistor R and observe the voltage. We use VR instead of directly measuring the source current, since the current through the circuit and resistor R is proportional to the voltage across R.

4. Using your predicted values as a guide, adjust the frequency of the function generator to tune for resonance, as observed on CHANNEL 2 of the oscilloscope. Remember that resonance frequency is the frequency where the current in the circuit is at its maximum (or the VR is at its max). Measure the resonant frequency fo on the oscilloscope, and record the value in Table 6.3. Export and save the oscilloscope screen as an image ﬁle. Additionally, record the VR at this frequency:

VR−PP =

5. Reduce the frequency on the function generator until the voltage across R becomes 70.7% of the voltage value at resonance frequency (step 4). This is the lower half-power point f1. Record the measured frequency f1 in Table 6.3. 6. Increase the frequency through resonance and continue to increase it until the voltage across R is 70.7% of the value at resonance (step 4). This is the upper half-power point f2. Record the measured frequency f2 in Table 6.3.

7. Calculate the bandwidth BW = f2 – f1. Record the result in Table 6.3. 8. Stop the function generator. Change the value of resistor box to 10Ω.

9. Start the function generator and repeat steps 4 through 7. Record the measured values in Table 6.3.

10. Complete step 10 for both R = 10,100 Ω.

39 Table 6.3: Experimental Values

R = 100 Ω R = 10 Ω Predicted Measured Predicted Measured

fo Q BW

f1

f2

11. When building circuits, we often need to determine how the circuit will respond given certain inputs. This is especially true when we use energy storage elements like capacitors and inductors. How do these parts interact with and change the input signal to give us the output signal? By using the Bode Diagram, you can get a better idea of how your circuit responds to various types of inputs. In this step, Use the Bode analyzer of the AD2 device to view the plot of the last circuit’s response. To do so:

• Close all the active windows in the WaveForms application. • Connect CH1 across the FGEN (reference signal) and CH2 across the resistor R (output). • Once you have everything set up, get it all connected and open WaveForms. Click on the “Network” button near the bottom left. This will open the Network Analyzer window. • Before we get started, let’s look at what we see in the “Network” window. You’ll see two graphs, the top shows the magnitude of our signal and has a dB scale from 10 to -90 dB on the left. The lower graph shows phase and has a degrees scale from 180◦ to -180◦. (The vertical axis labels are on the far right of the plot windows.) The x-axis along the very bottom of the window lists frequencies on a logarithmic scale, with major divisions at each decade of frequency. (By default, 1kHz, 10kHz, 100kHz, and 1MHz. Logarithmic scales can be a bit tricky to understand at ﬁrst, but it converts the logarithmic behavior of the circuit to a linear representation, which can be easier to understand visually. At the far right you will see adjustments for the waveform generator, y-axis adjustments for each plot, and the channel adjustments. We will look more at those in a bit. • Set the settings to scan from 2.5 kHz to 8.5 kHz. Set the “Samples” to at least 300, although you may need to increase this value to as much as 500 to get accurate readings of cutoﬀ frequency and phase angle. Verify that the graph scale is “Logarithmic”. Set the “amplitude” to 1.0 V (that is, 1.0 VPP ). Press the RUN button.

• Turn on CURSORS to be able to locate speciﬁc values. Find fo, f1, f2 values on the phase graph and calculate the Bandwidth using this numbers.

• Save the Bode plot as an image (with the cursors showing the values for f1, f2). • Do you see any diﬀerence between the values in the Table 6.3 and those that you found in the previous step? Explain.

40 Probing Further Questions

Q1. From the measured L, C, fo, and BW, compute the total series resistance of the circuit. Suggest and explain likely causes for any discrepancy; what might not have been taken into account for the predictions?

Q2. Discuss the eﬀect of changing the resistor R from 100 Ω to 10 Ω. How dramatic was the impact?

Q3. What would happen to the resonant frequency if the inductance were doubled and the capacitance cut in half? What would happen to the bandwidth? What would happen to the quality factor?

Q4. Simulate the second circuit (with R=10Ω) in LT SPICE, using the AC Frequency Sweep (see Appendix D). In the simulation, set up a voltmeter across R and an ammeter to measure the total current. Be sure to have LT SPICE display both graphs and a table of the results. Select an appropriate range of frequencies to sweep and sample at least 100 steps per interval. When the graph is displayed, right-click the graph and toggle the X axis to be logarithmic. Do you see a sharp peak in the magnitude (dB) plots of voltage and current at the resonant frequency you predicted? Notice that the phase plot for the circuit rapidly changes from positive (capacitive) to negative (inductive) as the frequency passes through resonance. Record the resonant frequency indicated by the graph, and compare it to your calculated and measured values.

41 Lab 7 -Filters: High-pass, Low-pass, Band- pass, and Notch

Introduction

This laboratory studies the use of passive components to create ﬁlters to separate portions of time- dependant waveforms. Filters are an essential tool in our complex world of mixed signals — both electronic and otherwise. Passive components (resistors, capacitors, and inductors) have long served as ﬁlter components for everything from selecting radio stations to ﬁltering out electrical noise.

Objective

Educational

• Learn the four general ﬁlter types: High-pass, Low-pass, Bandpass, and Notch.

• Learn to alter ﬁlter type by changing contacts for output voltage.Learn to alter ﬁlter type by changing contacts for output voltage.

• Learn phase angle at cutoﬀ for simple RC and RL ﬁlters.

• Design simple ﬁlter.

• Gain experience with Bode Analyzer and Bode Plots.

Experimental

• Calculate and measure cutoﬀ frequency for series RC and RL ﬁlters.

• Design simple RL low-pass ﬁlter.

• Generate and interpret Bode plots for series ﬁlters.

• Observe behavior of a single circuit used as a Bandpass ﬁlter and as a notch ﬁlter.

42 PreLab

Reading

• Study the Background section of this Laboratory exercise.

• Study textbook regarding origin and meaning of “decibels”.

Written

• Prior to coming to lab class, complete Part 0 of the Procedure.

Equipment

Analog Discovery 2 Instrument. Digital Multimeter. Suitable Decade Substitution boxes may be used for the following components: Resistors: 10kΩ, 100Ω, and a resistor to be determined Inductor: 100 mH Capacitor: 0.005 µF, 0.01 µF

Background

In many circuits, a wide range of diﬀerent frequencies are present, some of which are desired, while others are not. The frequency response of capacitors and inductors allows us to construct ﬁlters that will pass or reject certain ranges of the electrical frequencies that are applied to them. “Passive ﬁlters” created from “passive” components (inductors, capacitors, and resistors) have served us well for a long time for such purposes as selecting radio and television stations and ﬁltering noise out of various signals. Indeed, much of the electronics we take for granted today would not be possible without the use of such ﬁlters.

The four typical types of ﬁlter behaviors are illustrated in Figure 7.1, along with schematics of simple ﬁlters that exhibit the indicated behavior.

The ﬁlter types are low-pass, high-pass, bandpass, and notch (or band-reject) ﬁlters. In Figure 7.1, the grayed area is the passband, that is, the part of the signal that is passed to the output of the ﬁlter. The rejected portions are called the stopband. The frequency that separates the passband from the stopband is called the cutoﬀ frequency. The cutoﬀ frequency is equivalent to the half-power points discussed in Laboratory 6. The cutoﬀ frequency is also sometimes called the corner frequency.

43 Figure 7.1: Types of Passive Filters

A low-pass ﬁlter would allow extracting a low frequency, such as an audio signal, that is mixed with a high frequency radio wave. A high-pass ﬁlter would do the opposite. A resonant circuit can be tuned as a bandpass ﬁlter to retain signals in a narrow range of frequencies, while rejecting frequencies outside that range. Such is the case with a radio tuner. A notch ﬁlter generally keeps all frequencies except those in a narrow band. Notch ﬁlters are widely used to block interfering signals from noise sources. Bandpass and notch ﬁlters require resonant circuits, studied in Lab 6.

Notice that the components making the low-pass and high-pass ﬁlters in Figure 7.1 are the same. Whether the circuit is low-pass or high-pass depends only upon which voltage we look at: the voltage across the capacitor or the voltage across the resistor. (Equivalent circuits could have been made using an inductor and a resistor.) Similarly, the notch ﬁlter is identical to the RLC series resonant circuit we looked at in Lab 6, however in Lab 6 we looked at the voltage across the resistor, and so saw a bandpass ﬁlter. Caution: While one may be able to obtain the opposite response from the ﬁlter simply by putting the output terminals across a diﬀerent ﬁlter component, one must be sure to stay within the power and current limitations of the circuit and its components.

RC and RL ﬁlters are simple, inexpensive, and often used eﬀectively as ﬁlters. Their major problem is their generally slow (in frequency) transition from passband to stopband. The few simple components in ﬁlter “stages” can increase the transition rate, giving the ﬁlter a sharper cutoﬀ.

The ratio of an output response to an input signal is referred to as a transfer function. The input signal and the output response do not need to be the same entity type. For example, a transfer function

44 may prescribe an output voltage resulting from an input current. Transfer functions are often used as a tool to characterize the eﬀect of a ﬁlter regardless of the details of the ﬁlter’s structure. It can make the analysis of complex circuits easier. In this lab, however, we will mostly be studying the ﬁlter itself.

Cutoﬀ Frequency for series RC and RL circuits

As mentioned, the cutoﬀ frequency, sometimes called the corner frequency, is equivalent to the half- power points discussed in Laboratory 6. Since the power√ is half that at the peak, the voltage (or current) will be the peak voltage (or current) multiplied by 1/ 2 = 0.707. For a simple 2-component RC or RL circuit, the half-power point will occur when half the power is dropped on the resistor and half on the capacitor or inductor. Thus, the cutoﬀ frequency will occur when the reactance of the capacitor or inductor equals the total series resistance in the circuit. That is,

1 XC = and XL = 2πfoL (7.1) 2πfoC And so,

1 R f = and f = (7.2) o 2πRC o 2πL

Resonant frequencies for RLC circuits were discussed in Laboratory 6. decibels (dB)

As discussed in your textbook, the decibel (dB) is commonly used for the magnitude of voltages and currents in making Bode plots. Keep in mind that a decibel is a unit created to measure the transfer function for power gain (or loss) through a circuit module or stage:

power out Number of decibels = 20 log (7.3) power in

Since power is proportional to the square of the voltage or the current, we have equivalently,

V I Number of decibels = 20 log out and Number of decibels = 20 log out (7.4) Vin Iin

How many decibels correspond to a doubling of the power? To a halving of the power?

Procedure

Part (0) Perform the design calculations for the ﬁlter in Part 2.

Note: In all cases below, you may use substitute a decade substitution box of the correct type and value for the speciﬁed resistor, capacitor, or inductor.

45 Part (1) Low-pass ﬁlter - complete the following steps and record your values.

10 kΩ

+

50Ω 0.005 µF Vout +

2.0VPP − −

Figure 7.2: Low-pass Filter

1. Obtain a 10 kΩ resistor and a 0.005 µF capacitor. Measure and record the actual values of the components.

2. Set up the circuit as shown in Figure 7.2. Use the function generator FGEN for the supply voltage.

3. Calculate the cutoﬀ frequency for the circuit, assuming the output is at Vo. At the cutoﬀ frequency, what, theoretically, will be the voltage Vo?

4. Connect CHANNEL 1 of the oscilloscope to measure the Vin (i.e., FGEN).

5. Connect CHANNEL 2 of the oscilloscope to measure the ﬁlter’s output voltage Vo. On the oscil- loscope, turn on the measurement functions for CH1 and CH2. Vary the frequency from 500 Hz to 10 kHz in steps indicated in Table 7.1, and record the indicated values. You may press STOP to freeze the display when taking cursor measurements. For the Vin and Vo measurements, be sure the time scale is suﬃcient to show at least a few cycles, or the instrument may not properly calculate the Peak-to-Peak and RMS values.

6. Use the oscilloscope’s cursors to measure the phase shift ∆t and then calculate the phase angle φ between Vin and Vo.

46 Table 7.1: Measured Values

Freq. Vin Vin Vo Vo ∆t φ Gain KhZ RMS PP RMS PP µs degrees db

0.500 2.000 3.000 4.000 6.000 12.000 20.000 Cutoﬀ

7. Using the data of Table 7.1, sketch a Bode plot of the of the ﬁlter’s output voltage.

Part (2) Design a low-pass inductive ﬁlter.

1. Design a low-pass ﬁlter using a 100 mH inductor and a single resistor R to obtain a cutoﬀ fre- quency of 7500 Hz ± 2%. In your notebook, show your design procedure, including your design calculations.

47 2. Measure the actual value of the 100 mH inductor and record its value.

3. Using the value of R that you calculated, construct an RL circuit designed to be a low-pass ﬁlter with the speciﬁed cutoﬀ frequency (similar to the RC circuit used in Part 1 but, since we are using an inductor, the order of components in the circuit should be L-R).

4. Connect CH1 across the function generator and CH2 across the ﬁlter output (resistor R).

5. Use the WaveForms application to generate the Bode plot for this circuit (similar to the last part of Lab 6). Scan from 1 kHz to 40 kHz. Set the “Samples” to at least 300, although you may need to increase this value to as much as 500 to get accurate readings of cutoﬀ frequency and phase angle. Verify that the graph scale is “Logarithmic”. Set the “amplitude” to 1.0 V (that is, 2.0 VPP ). Run the Bode analyzer. 6. Change the “Top” and “Bottom” values to ﬁt the plot.

7. When the analyzer has ﬁnished, use the cursors to locate the cutoﬀ frequency by ﬁnding the half- power point (What identiﬁes the half-power point?). Record the cutoﬀ frequency and the phase angle. Save the Bode plot as an image.

8. Compare the experimentally determined cutoﬀ frequency to the desired value.

9. If the cutoﬀ frequency is oﬀ by more than 5%, change the size of R as necessary to obtain the speciﬁed cutoﬀ frequency, and test again for the cutoﬀ frequency. Record the ﬁnal R, cutoﬀ frequency, and phase angle φ.

10. Draw a circuit diagram of the ﬁnal circuit that accomplishes the design objective. Explain any diﬀerences between the ﬁnal value of R and your originally calculated value.

Part (3) RLC Circuit 1.

1. Set up the series RLC circuit shown in Figure 7.3, using the function generator to provide the sinusoidal input voltage.

2. Calculate the resonant frequency f0 of the circuit. (See Laboratory 6.)

48 0.01 µF 100 mH

+

50Ω 100 Ω Vout +

2.0VPP − −

Figure 7.3: Series RLC Circuit

3. As in part 2, use the WaveForms application to generate the Bode plot for this circuit (similar to the last part of Lab 6). Scan from 1 kHz to 40 kHz. Set the “Samples” to at least 300, although you may need to increase this value to as much as 500 to get accurate readings of cutoﬀ frequency and phase angle. Verify that the graph scale is “Logarithmic”. Set the “amplitude” to 1.0 V (that is, 2.0 VPP ). Run the Bode analyzer. 4. Change the “Top” and “Bottom” values to ﬁt the plot.

5. What type of ﬁlter is this if the output voltage is Vo?

6. What is the measured resonant frequency? What is the measured bandwidth? What is the phase angle at resonance? To measure these values from the Bode plot, you may need to increase the “samples” or interpolate between points. Save the Bode plot as an image.

Part (4) RLC Circuit 2.

1. Switch the positions of the resistor with inductor and capacitor to get the series RLC circuit shown in Figure 7.4.

2. Calculate the resonant frequency fo of the circuit. (See Laboratory 6.)

49 100 Ω

+ 0.01 µF 50Ω Vout + 100 mH 2.0VPP − −

Figure 7.4: Modiﬁed Series RLC Resonant Circuit

3. As in part 2, use the WaveForms application to generate the Bode plot for this circuit (similar to the last part of Lab 6). Scan from 1 kHz to 40 kHz. Set the “Samples” to at least 300, although you may need to increase this value to as much as 500 to get accurate readings of cutoﬀ frequency and phase angle. Verify that the graph scale is “Logarithmic”. Set the “amplitude” to 1.0 V (that is, 2.0 VPP ). Run the Bode analyzer. 4. Change the “Top” and “Bottom” values to ﬁt the plot.

5. What type of ﬁlter is this if the output voltage is Vo?

6. From the Bode plots, what is the measured resonant frequency fc? What are the lower and upper half-power points, f1 and f2? What is the measured bandwidth? What is the phase angle φ at resonance? To measure these values from the Bode plot, you may need to increase the “samples” or interpolate between points. Save the Bode plot as an image.

Probing Further Questions

Q1. In Procedure parts 1 and 2 you measured the phase angle φ between Vin and Vo at the cutoﬀ frequency. What values did you observe? What value would you expect? Why?

50 Q2. In Procedure parts 3 and 4 you measured the resonant frequencies and the bandwidths using the Bode Analyzer. Construct a table comparing the theoretical values (see Laboratory 6) and the values for f0, f1, f2, bandwidth measured in Procedure parts 3 and 4. Calculate the Q for these circuits.

51 Lab 8 - Transformers

Introduction

Transformers are magnetic circuit elements used in AC circuits for a variety of applications. Trans- formers are often used to convert voltages from high to low values, or from low to high values. They are often used to isolate one AC line from another, for safety or for equipment isolation. And they are often used to connect two components with mismatched impedances, such as a high-impedance stereo ampliﬁer and a low-impedance speaker. This laboratory exercise will introduce transformer modeling concepts and explore the characteristics of a common power transformer.

Objective

Educational

• To teach the properties of common low-power transformers.

Experimental

• Characterize the behavior of common power transformers, including concepts such as turns-ratios, impedance matching, and loading.

PreLab

Reading

• Review the sections of your ECE-2620 textbook regarding transformer modeling. • Study the Background section below.

Written

• Prior to coming to lab class, complete Part 0 of the Procedure.

52 Equipment

Analog Discovery 2 Instrument. Digital Multimeter. Transformer Board with mounted 120V, center-tapped 12.6V transformer. Resistance Decade Substitution box.

Background

In its simplest form, a transformer consists of two closely coupled wire coils that share a common magnetic ﬁeld. If a voltage is applied to one of the coils (called the primary), then a voltage will be induced on the second coil (called the secondary).

Figure 8.1: Model for an Ideal Transformer

The actual voltage induced in the secondary coil is proportional to the ratio of the number of turns in the two coils. That ratio is called the turns ratio, for which we will use the symbol a. If the number of turns in the primary is N1 and the number of turns in the secondary is N2, the turns ratio α is: N α = 2 (8.1) N1 and so, N2 V2 I1 α = = = and V2 = αV1 (8.2) N1 V1 I2

The two coils (also called windings) may be wound around air (an air core) or a dielectric, but more often they are wound around a ferromagnetic material, such as steel (an iron core), in order to concentrate the magnetic ﬁeld and improve the coupling between the coils. Transformers are most often used for three principal reasons: to change AC voltages; for electrical isolation; and for impedance matching.

Often we can assume that a transformer is ”ideal”, meaning that it has no internal power loss and all magnetic ﬂux lines of the primary coil also pass through the secondary coil. The ideal model often ﬁts large power transformers, where the losses are very small compared to the power transferred. An ideal transformer delivers all of the applied power to the load. That is,

P ower = I1V1 = I2V2 (8.3)

53 Rearranging this equation reveals an important relationship: N V I α = 2 = 2 = 1 (8.4) N1 V1 I2

So, for example, if the secondary voltage is 1/10 the primary voltage, then the primary current will be 1/10 the secondary current. The equation further indicates that if there is no load current (I2=0), then the primary current is also zero.

Impedance-Matching

Besides converting voltages, transformers are often used to match impedances between two electrical components, such as an audio ampliﬁer (typically having 600Ω output impedance) and a speaker system (typically 4Ω or 8Ω impedance). The right transformer can make the load appear to have the same impedance as the source, and thereby maximize the power transferred to the load — and minimize the power lost in the source. The transformer that can perform such magical matching will have the following property: r Z α = load (8.5) Zsource where α is the turns ratio, Zload is the load impedance, and Zsource is the source impedance. Equation 8.5 is a direct result of Equation 8.3 after substituting V1 = I1Z1 and V2 = I2Z2.

In essence, Zsource is what the load impedance Zload will look like from the primary side of the transformer. Expressed in other terms, Zsource is the eﬀective load impedance reﬂected to the primary side of the transformer. That is, 1 Z = Z (8.6) source α2 load

Usually, matching transformers are designed diﬀerently from power transformers, because matching transformers must meet other requirements. For example, audio transformers must respond uniformly over a rather wide band of frequencies, and do so with very low distortion.

Non-Ideal Transformer Model

Real transformers do, in fact, have losses and do draw current even with no load connected. Real transformers have coil resistance; they require magnetizing currents to maintain the magnetic coupling between the coils; and there are eddy currents and other issues. And so, even though the ideal model works well most of the time, when required, we use other models suitable to the circ*mstances to represent non-ideal transformers. One such model is illustrated in Figure 8.2.

Figure 8.2: Circuit Model for non-ideal Transformer

54 For steady-state sinusoidal operation, the following two equations show the voltage sums in this circuit model: V = (R + jωL )I − jωMI S 1 1 1 2 (8.7) 0 = −jωMI1 + (R2 + jωL2 + Rload + jωLload)I2 where R1 and R2 are the winding resistances and M is the mutual inductance of the two windings. These equations can be used to compute the inductances of the transformer coils, as well as the mutual inductance between them.

Procedure

Part (0) Prior to coming to the lab, calculate the nominal turns ratio αnom for the transformer in Part (1) based on the nominal rating of the transformer’s primary and secondary voltages. Record this result in Table 8.1.

Part (1) For these experiments, use a 120V / 12.6V center-tapped transformer mounted on Plexiglas.

Figure 8.3: Testing Voltage Relationships

Note: Some of the resistances and voltages are small. The cables connecting the AD2 to the trans- former must have a good connection at each end or the measurements will be unstable and signiﬁcantly wrong.

1. Resistance: Using the DMM’s ohmmeter function, measure the transformer’s primary and sec- ondary resistances. Record the values in Table 8.1.

2. Measured Turns Ratio:

• Connect the function generator to the primary winding of the transformer, as illustrated in Figure 8.3. Select a sine wave output and set the generator’s voltage to 3VPP . Set the frequency at 60 Hz, since that is the operating frequency for which this transformer is designed.

55 Table 8.1: Recorded Values

Value

Primary resistance RP

Secondary resistance RS

Nominal turns ratio, αnom

X13 turns ratio, αmeas Percentage diﬀerence

X12 turns ratio

X23 turns ratio

VPP −P rimary

VPP −X1−X3

VPP −X1−X2

VPP −X2−X3

• Connect the oscilloscope across the transformer as indicated in Figure 8.3. Use the oscillo- scope to measure the transformer voltages indicated below and record the results in Table 8.1.

– VPP across the primary (H1 to H2).

– VPP across the secondary (X1 to X3).

– VPP from the center tap (CT) to one side of the secondary (X1 to X2)

– VPP from the center tap (CT) to the other side of the secondary (X2 to X3) • Do you see why the connection X2 is referred to as the center tap?

• Calculate the turns ratio αmeasured between the primary winding and the secondary winding X1–X3. Record the result in Table 8.1.

Calculate the % Diﬀerence between αnominal and αmeasured and record the result in Table 8.1: α − α P ercentage Difference = measured nominal . 100% (8.8) αnominal

Why are αnominal and αmeasured diﬀerent? Based on your measurements, estimate what the actual output voltage of the transformer (X1–X3) would be if the input voltage were 120V.

• Calculate the turns ratio αmeasured between the primary winding and the secondary winding X1–X2. Record the result in Table 8.1.

• Calculate the turns ratio αmeasured between the primary winding and the secondary winding X2–X3. Record the result in Table 8.1.

56 Part (2) Phase relationship between the primary and secondary voltages.

1. Connect the oscilloscope across the transformer, as indicated in Figure 8.3: CHANNEL 1 across the transformer’s primary (H1–H2) with H2 grounded; and CHANNEL 2 across the transformer’s secondary (X1–X3) with X3 grounded.

2. Start the function generator (3 VPP at 60 Hz) and compare the phases of the two waveforms. Are the voltages in phase or out of phase? In your notebook, sketch the waveforms and record the phase shift, if any. Save the oscilloscope output as an image.

3. Reverse the scope leads on the secondary: that is, connect CHANNEL 2’s signal lead to X3 and the ground lead to X1. Describe the result and sketch the waveforms in your notebook. Record the phase shift, if any. Save the oscilloscope output as an image.

4. Connect CHANNEL 2 of the scope to observe the secondary-side voltage from X2 (signal) to X1 (ground). What is the diﬀerence from what you measured just previously in previous part?

Part (3) Secondary center-tap voltages.

Figure 8.4: Testing Secondary Center Tap

57 1. Connect the oscilloscope across the transformer, as indicated in Figure 8.4: CHANNEL 1 across X1–X2; and CHANNEL 2 across X3–X2. In both cases, X2 must be connected to the scope’s ground lead.

2. Start the function generator (3 VPP at 60 Hz). View the signals on each side of the center tap at the same time. Sketch the waveforms, showing the measured voltages. Save the oscilloscope output as an image. Describe your observations regarding voltages, phases, and anything else you observe about the signal.

Notice that in Parts 1, 2, and 3 we could ground any terminal on the secondary. That is a demonstration of the third major use of the transformer: to isolate two circuits (especially the grounds) from one another. Attempting the same thing on the primary side (e.g., connecting the oscilloscope’s ground to H1) could be disastrous because of the internal grounds on the function generator and the oscilloscope.

Part (4) The eﬀect of loading.

Figure 8.5: Testing Secondary Center Tap

1. Construct the circuit shown in Figure 8.5. Use a resistor substitution box for the resistor R, and initially set the value to 1000 Ω.

2. Connect the oscilloscope across the transformer, as indicated in Figure 8.5: CHANNEL 1 across H1–H2; and CHANNEL 2 across X1–X3.

3. Set the function generator to 6 VPP at 60 Hz, and start the function generator and the oscilloscope.

4. For each of the resistor values in Table 8.2, record in turn the voltages V1 and V2, and record the values in Table 8.2.

58 Table 8.2: Measured and Calculated Values

R (Ω) V1 V2 α = V2/V1 1000 100 50 20 5

5. Calculate the apparent turns ratio for each of the loads and record the results in Table 8.2.

6. Graph V1, V2, and a versus R to visualize the relationships. You may create the graph by hand or in a spreadsheet.

7. Compare these turns ratios with the αnominal and αmeasured calculated in Parts 2 and 3. How might these new values explain the diﬀerence between those earlier values?

Probing Further Questions

Q1. Using the data from Procedure part 1, compute the resistance ratio between the secondary and primary windings by dividing the secondary resistance by the primary resistance. Record the result and compare it to the turns ratio. What factors might cause the resistance ratio to diﬀer from the turns ratio?

59 Q2. Suppose that you had two of these 120 V transformers and you needed to convert 120 V to about 25 V. Show a sketch of how you could connect these two transformers to get this transformation. Make sure to show the winding connection labels for each transformer.

Q3. If an ideal transformer with 120 V across the primary and draws a current of 200 mA, what power is dissipated in the load? If the transformer’s secondary is 30 V, what is the secondary current? What is the turns ratio?

Q4. Using the speciﬁed Thevenin impedance of the function generator (50 Ω) and the speciﬁed impedance of an acoustic speaker (8 Ω), calculate the turns ratio required to match the speaker with the function generator.

60 Lab 9 - Two-Port Network Characterization

Introduction

You have now had experience with transfer functions and the concept of relating output quantities to input quantities. Many circuits can be treated as modules where an input is presented to two terminals (the ”input port”) and an output is taken from two terminals (the ”output port”). Such circuits are termed two-port networks, and because they can be treated as a unit — a block — they allow great simpliﬁcation of large-scale designs. The last chapter in your textbook is an introduction to two-port circuits and to the methodology for developing the sets of parameters that may be used to relate the output variables of voltage and current, typically designated as V2 and I2, respectively, to the input variables of voltage and current, typically designated as V1 and I1, respectively. This approach is used to characterize a variety of components and circuit element combinations from ﬁlters, through transistors, to microwave circuits.

Objective

Educational

• The objectives of this laboratory exercise are to gain familiarity with three alternative two-port network parameter sets, to learn to measure the parameter sets, and to demonstrate the operational deﬁnition of these parameters.

Experimental

• To determine the impedance, admittance, and hybrid parameter sets for an unknown two-port network.

PreLab

Reading

• Study the Background section below.

61 • Read the chapter in your textbook on two-port networks. Pay particular attention to the way in which the various parameters in each set (i.e., the z parameters, the y parameters, the h parameters, etc.) are deﬁned.

Written

• Review the steps in the procedure below and plan how you will make your measurements to determine the required parameter values.

Equipment

Analog Discovery 2 Instrument. Digital Multimeter. Two-port Network Box. Resistance Decade Substitution box.

Background

A port consists of a pair of terminals; current enters through one of the terminals and the same current leaves through the other terminal. A resistor is a one-port network. In this lab we will study two-port networks with one input port and one output port. Such networks are often treated as “black boxes” or modules that may be plugged into a circuit to accomplish some task, such as ﬁltering the signal or providing a controlled voltage. Engineers need a way to characterize the behavior of such a network and have developed several sets of parameters to do that. Each of these parameter sets relates the input (side 1) and output (side 2) voltages and currents.

In this lab we will use impedance, admittance, and hybrid parameter sets to characterize a simple circuit. Impedance and admittance parameters are commonly used to characterize ﬁlters, and are often useful in designing and characterizing impedance matching and power distribution networks. The term immittance is often applied to the use of either impedance or admittance parameters.

Be aware that because the voltages and currents are phasors with magnitude and phase angle, the parameters also have magnitude and phase angle. A simple RMS measurement will not suﬃce.

The impedance parameters (z parameters) relate the input and output voltages to the input and output currents by the following two equations:

V = z I + z I 1 11 1 12 2 (9.1) V2 = z21I1 + z22I2 or in matrix notation: V1 z11 z12 I1 = (9.2) V2 z21 z22 I2

62 The z parameters have units of ohms and are most easily found by applying a set of open-circuit tests on the circuit. When we apply a voltage to the input with the output open-circuited, we can measure the input current and output voltage and ﬁnd the ﬁrst two z parameters as follows:

V1 V2 z11 = and z21 = (9.3) I I 1 I2 = 0 1 I2 = 0

We can determine the other two z parameters by applying a similar test to the output with the input open-circuited:

V1 V2 z12 = and z22 = (9.4) I I 2 I1 = 0 2 I1 = 0

Sometimes the impedance parameters do not exist because the voltages cannot be described by equation 9.1. Therefore, we need alternatives, such as the admittance parameters.

The admittance parameters (y parameters) relate the input and output currents to the input and output voltages by the following two equations:

I = y V + y V 1 11 1 12 2 (9.5) I2 = y21V1 + y22V2 or in matrix notation: I1 y11 y12 V1 = (9.6) I2 y21 y22 V2

The y parameters have units of siemens (or mhos) and are most easily found by applying a set of short-circuit tests on the circuit. When we apply a voltage to the inputs with the output short circuited, we can measure the input current and output current to ﬁnd the ﬁrst two y parameters:

I1 I2 y11 = and y21 = (9.7) V V 1 V2 = 0 1 V2 = 0

We can determine the other two y parameters by applying a similar test to the output with the input short-circuited:

I1 I2 y12 = and y22 = (9.8) V V 2 V1 = 0 2 V1 = 0

There are occasions where neither the impedance nor the admittance parameters exist, so there is need for still another set of parameters.

63 The hybrid parameters (h parameters) are based on making V1 and I2 the dependent variables, and relating them to cross-variables V2 and I1. The h parameters satisfy the following equations:

V = h I + h V 1 11 1 12 2 (9.9) I2 = h21I1 + h22V2 or in matrix notation: V1 h11 h12 I1 = (9.10) I2 h21 h22 V2

The h parameters are found using a mix of short- and open-circuit tests as follows:

Short-circuit tests:

V1 I2 h11 = and h21 = (9.11) I I 1 V2 = 0 1 V2 = 0

Open-circuit tests:

V1 I2 h12 = and h22 = (9.12) V V 2 I1 = 0 2 I1 = 0

Procedure

Warning: Unless you are careful in your planning, in carrying out the experiment, and in recording your data, you are very likely to make errors. Here in this last laboratory you must ﬁgure out how to do most steps on your own.

Hint: Performing this lab is actually very straightforward, if you take it a step at a time. Think about what you have learned during this semester. If you need to measure the phase angle of a voltage or current, you will need to use the oscilloscope. When necessary, insert a sense resistor of about 10 ohms and measure the voltage across it, including its phase shift, then calculate the magnitude of the current ﬂowing through the sense resistor. Be sure that you have accurately measured the resistor value, though, and not just relied on the nominal value. Use only one resistor decade box (it will make your life easier), and be careful where you put the resistor, so you don’t make a grounding error on the oscilloscope. Carefully record each of the measurements and calculations in tables that you have constructed for that purpose.

General Instructions:

1. Use the function generator to apply a 1 kHz sine wave of about 10 VPP to the input terminals of the Two-port Network Box given to you by your instructor.

64 2. Make the current and voltage measurements necessary to calculate the z,y,h-parameters. Note that you must determine not only the magnitude but also the phase of the voltages and currents involved. How can you do that with the instruments at hand?

Important Notes:

• CH1 of the oscilloscope must always be connected across the FGEN. • Phase angles can be positive (lead) or negative (lag). • Record the RMS values. • Place the sense resistor on the Ground side.

Part (1) - Calculating z11 and z21

Figure 9.1: Circuit Diagram for Open-circuit Test

Table 9.1: Measured and Calculated Values

Measured Values

Output Terminal Open (I2 = 0) Magnitude Angle

V1 0

I1

V2

I2 0 0 Calculated Values

Magnitude Angle

z11

z21

65 Part (2) - Calculating y11, y21, h11, and h21

Figure 9.2: Circuit Diagram for Short-circuit Test

Table 9.2: Measured and Calculated Values

Measured Values

Output Terminal Shorted (V2 = 0) Magnitude Angle

V1 0

I1

I2

V2 0 0 Calculated Values

Magnitude Angle

y11

y21

h11

h21

66 Part (3) - Calculating z12, z22, h12, and h22

Figure 9.3: Circuit Diagram for Open-circuit Test

Table 9.3: Measured and Calculated Values

Measured Values

Input Terminal Open (I1 = 0) Magnitude Angle

V2 0

I2

V1

I1 0 0 Calculated Values

Magnitude Angle

z12

z22

h12

h22

67 Part (4) - Calculating y12, and y22

Figure 9.4: Circuit Diagram for Short-circuit Test

Table 9.4: Measured and Calculated Values

Measured Values

Input Terminal Shorted (V1 = 0) Magnitude Angle

V2 0

I2

I1

V1 0 0 Calculated Values

Magnitude Angle

y12

y22

68 Probing Further Questions

Q1. Are the parameter values, which you have calculated for the various two-port network representa- tions in this exercise valid for, say, 10 kHz? Why or why not?

Q2. The z and y parameters should be related by a matrix inversion. Invert the 2x2 z parameter matrix and see if it matches with the y parameters you calculated in the lab.

69 Lab 10 - Final Exam

Introduction

This has been your ﬁrst engineering laboratory. Although you have been provided with a “cookbook” (this manual), hopefully you have not just blindly followed instructions in order to get a good grade. If you have, you have cheated only yourself. The theory which you have learned from your textbook and lectures is a way of looking at reality and thinking about how to organize experience, i.e., dealing with real, physical things. It is only through combining practical experience with theory that you can begin to develop the necessary analytical skills to aid you in “taking things apart and putting them together in new ways” that is the essence of the practice of real engineering.

Objective

This examination is designed to help you and your lab TA determine how much you have developed your knowledge, skills, and self conﬁdence.

PreLab

Review your lab manual. Think carefully of the procedures which you have followed and what you have learned from them. How do you measure voltage, current, resistance, frequency, etc.? How accurate are your measurements? How are your measurements aﬀected by the frequency response of your measuring instruments? Can you develop linear, semi-log, or log-log plots of your data to facilitate interpretation?

Equipment

• Calculator

• AD2 Device

• A Single Page Formula Sheet

• Laptop

70 Procedure

The ﬁnal exam will consist of a written exam and a practical exam. The practical exam involves performing a lab procedure to obtain a desired result. The written exam will cover the application of theory to understand circuit behavior, based upon your laboratory experiences in this course.

Probing Further Questions

Q1. This has been your second in a series of four (4) laboratory courses. What have you done to develop your expertise and skills? What will you do diﬀerently in the next course? What have you learned from your experience?

71 Appendix A - Safety

Electricity, when improperly used, is very dangerous to people and to equipment. This is especially true in an industrial environment where large amounts of power is used, and where high voltages are present [1]; in environments where people are especially susceptible to electric shock such as maintenance of a high voltage system (while in operation) or in hospitals where electrical equipment is used to test or control physiological functions [2, 3]; and in an experimental or teaching laboratory where inexperienced personnel may use electrical equipment in experimental or nonstandard conﬁguration.

Engineers play a vital role in eliminating or alleviating the danger in all three types of environments mentioned above. For conditions where standard equipment is used in standard conﬁgurations, govern- mental agencies and insurance underwriters impose strict laws and regulations on the operation and use of electrical equipment including switchgear, power lines, safety devices, etc. As a result, corporations and other organizations in turn impose strict rules and methods of operation on their employees and contractors. Engineers who are involved in using electrical equipment, in supervising others who use it, and in designing such systems, have a great responsibility to learn safety rules and practices, to observe them, and to see that a safe environment is maintained for those they supervise. In any working environment there is always pressure to “get the job done” and take short cuts. The engineer, as one who is capable of recognizing hazardous conditions, is in a responsible position both as an engineer and as a supervisor or manager and must maintain conditions to protect personnel and avoid damage to equipment.

Because of their non-standard activities, experimental laboratories are exempt from many of these rules and regulations. This puts more responsibility on the engineer in this environment to know and enforce the safest working procedures.

The knowledge and habit-forming experience to work safely around electrical equipment and the ability to design safe electrical equipment begins with the ﬁrst student laboratory experience and con- tinues through life. This includes learning the types of electrical injuries and damage, how they can be prevented, the physiology of electrical injuries, and steps to take when accidents.

Physiology of Electrical Injuries

There are three main types of electrical injuries: electrical shock, electrical burns, and falls caused by electrical shock. A fourth type, ’sunburned’ eyes from looking at electric arcs, such as arc-welding,

72 is very painful and may cause loss of work time but is usually of a temporary nature. Other injuries may be indirectly caused by electrical accidents, e.g., burns from exploding oil-immersed switch gear or transformers.

Although electric shock is normally associated with high-voltage AC contact, under some circum- stances death can occur from voltages from substantially less than the nominal 120 Volts AC found in residential systems. Electric shock is caused by an electric current passing through a part of the human body. The human body normally has a high resistance to electric currents so that a high voltage is usually required to cause lethal currents. This resistance is almost all in the skin, but when the skin is wet its resistance is much lower. When a person is hot and sweaty or is standing in water, contact with 120 Volts or less is likely to cause a fatal shock.

Electric shock is not a single phenomenon but is a disturbance of the nerves that is caused by electric current. A current through a part of the body such as the arm or leg will cause pain and muscle contraction. If a victim receives an electric shock from grasping a live conductor, a current of greater than 15 to 30 mA through the arm will cause muscle contractions so severe that the victim cannot let go. Similar currents through leg muscles may cause sudden contractions causing the victim to jump or fall, resulting in possible injuries or death. It is also possible for a prolonged period of contact of more than a minute or so to cause chest muscles to be contracted, preventing breathing and resulting in suﬀocation or brain damage from lack of oxygen.

The predominant cause of death by electric shock is generally attributed to ventricular ﬁbrillation, which is an uncontrolled twitching or beating of the heart that produces no pumping action and therefore no blood circulation. Unless corrective action is taken, death follows quickly from lack of oxygen to the brain. While the amount of current that will cause ﬁbrillation depends on several variables, 0.5 to 5A through the body will normally cause the very small current through the heart that causes ﬁbrillation in most people. Larger currents than this through the heart causes contraction or clamping of the heart muscle and resulting death unless corrective action is taken. Prolonged contact of more than a minute or so may cause chest muscles to contract, preventing breathing and resulting in suﬀocation or brain damage from lack of oxygen.

Death by electric shock is most often attributed to ventricular ﬁbrillation, which is an uncontrolled twitching or beating of the heart that produces no pumping action and therefore no blood circulation. Unless corrective action is taken, death follows quickly from lack of oxygen to the brain. While the amount of current that will cause ﬁbrillation depends on several variables, 0.5 to 5 amperes through the body will normally cause the very small current (approximately 1 mA) through the heart that is suﬃcient to cause ﬁbrillation in most people. Larger currents than this through the heart cause contraction or clamping of the heart muscle, resulting in death unless corrective action is taken.

Electric burns may be caused by electric currents ﬂowing in or near parts of the body. Such burns are similar to burns from ordinary heat sources, except that those caused by high-frequency currents are generally deeper and take longer to heal the other burns. Electrocution will often leave severe burns at the points where the current entered and left the body.

73 Source of Electric Shock

Since electric shock is caused by an electric current through a part of the body, it is prevented by not allowing the body to become part of any electric circuit. From this viewpoint, electric circuits may be classiﬁed as either grounded or ungrounded.

Electric circuits may be classiﬁed as either grounded or ungrounded. Grounded circuits are safer for most conditions, since they result in known voltages at other points in the circuit and provide easier and better protection against faulty conditions in the circuit. The disadvantage is that a person standing on a non-insulated ﬂoor can receive a shock by touching only one conductor.

Almost all electric power generation, transmission, and distribution systems are grounded to pro- tect people and equipment against fall conditions caused by windstorms, lightning, etc. Residential, commercial, and industrial systems such as lighting and heating are always grounded for greater safety. Communication, computer, and similar systems are grounded for safety reasons and to prevent or reduce noise, crosstalk, static, etc. Many electronic equipment or instruments are grounded for safety and noise prevention, also. Common examples are DC power supplies, oscilloscopes, oscillators, and analog and digital multimeters.

Ungrounded circuits are used in systems where isolation from other systems is necessary, where low voltages and low power are used, and in other instances where obtaining a ground connection is diﬃcult or impractical. In the ungrounded circuit, contact with two points in the circuit that are at diﬀerent potentials is required to produce an electrical shock. The hazard is that with no known ground, a hidden fault can occur, causing some unknown point to be grounded, in which case, touching a supposedly safe conductor while standing on the ground could result in an electric shock.

Protecting People and Equipment in the Laboratory

Prevention of electric shock to individuals and damage to equipment in the laboratory can be done by strict adherence to several common-sense rules summarized below:

Protecting People

1. When hooking up a circuit, connect to the power source last, while power is oﬀ.

2. Before making changes in a circuit, turn oﬀ or disconnect the power ﬁrst, if possible.

3. Never work alone where the potential of electric shock exists.

4. When changing an energized connection, use only one hand. Never touch two points in the circuit that are at diﬀerent potentials.

5. Know that the circuit and connections are correct before applying power to the circuit.

6. Avoid touching capacitors that may have a residual charge. The stored energy can cause a severe shock even after a long period of time.

7. Insulate yourself from ground by standing on an insulating mat where available.

74 The above rules and the additional rules given below also serve to protect instruments and other circuits from damage.

Protecting Equipment

1. Set the scales of measurement instrument to the highest range before applying power.

2. Before making changes in a circuit, turn oﬀ or disconnect the power ﬁrst, if possible.

3. When using an oscilloscope, do not leave a bright spot or trace on the screen for long periods of time. Doing so can burn the image into the screen.

4. Be sure instrument grounds are connected properly. Avoid ground loops and accidental grounding of “hot” leads.

5. Check polarity markings and connections of instruments carefully before connecting power.

6. Never connect an ammeter across a voltage source, but only in series with a load.

7. Do not exceed the voltage or current ratings of circuit elements or instruments. This particularly applies to wattmeters, since the current or voltage rating may be exceeded with the needle still reading on the scale.

8. Be sure any fuses and circuit breakers are of suitable value.

When connecting electrical elements to make up a network in the laboratory, it easy to lose track of various points in the network and accidentally connect a wire to the wrong place. One procedure to help avoid this problem is to connect ﬁrst the main series loop of the circuit, then go back and add the elements in parallel.

Types of Equipment Damage

Excessive currents and voltages can damage instruments and other circuit elements. A large over- current for a short time or a smaller over-current for a longer time will cause overheating, resulting in insulation scorching and equipment failure.

Blown fuses are the most common equipment failure mode in this laboratory. The principal causes for these failures include:

• incorrectly wired circuits;

• accidental shorts;

• switching resistance settings while power is applied to the circuit;

• changing the circuit while power is applied;

• using the wrong scale on ammeter;

75 • connecting an ammeter across a voltage source; • using a low-power resistor box (limit 1/2 amp) when high power is required; • turning on an auto-transformer at too high a setting.

All of these causes are the result of carelessness by the experimenter.

Some type of insulating material, such as paper, cloth, plastic, or ceramic, separates conductors that are at diﬀerent potentials in electrical devices. The voltage diﬀerence that this material can withstand is determined by design (type, thickness, moisture content, temperature, etc.). Exceeding the voltage rating of a device by an appreciable amount can cause arcing or corona, resulting insulation breakdown, and failure.

Some electrical devices can also be damaged mechanically by excessive currents. An example is the D’Arsonval meter, the indicator in most analog metering instruments. A large pulse of over current will provide mechanical torque that can cause the needle to wrap around the pin at the top of the scale, thereby causing permanent damage even though the current may not have been on long enough to cause failure due to overheating.

After Accident Action

Since accidents do happen despite all eﬀorts to prevent them, plans for appropriate reaction to an accident can save time and lives. Such a plan should include immediate availability of ﬁrst aid material suitable for minor injuries or for injuries that are likely because of the nature of the work. Knowledge of how to obtain trained assistance such as Emergency Medical Services (EMS) should be readily available for everyone.

Treating victims for electrical shock includes four basic steps that should be taken immediately. Step two requires qualiﬁcation in CPR and step three requires knowledge of mouth-to-mouth resuscitation. Everyone who works around voltages that can cause dangerous electrical shock should take advantage of the many opportunities available to become qualiﬁed in CPR and artiﬁcial respiration.

Immediate Steps After Electric Shock

1. Shut oﬀ all power and remove victim from the electric circuit. If the power cannot be shut oﬀ immediately,use an insulator of some sort, such as a wooden pole, to remove victim from the circuit. Attempts to pull the victim from the circuit with your hands will almost always result in your joining the victim in the electric shock. 2. If you are qualiﬁed in CPR, check for ventricular ﬁbrillation or cardiac arrest. If either is detected, external cardiac massage should be started at once. Whether you are qualiﬁed in CPR or not, notify EMS and the ECE Department at once, using the telephone numbers listed below. 3. Check for respiratory failure and take appropriate action. This may have resulted from physical paralysis of respiratory muscles or from a head injury. Sometimes many hours pass before normal respiration returns. Artiﬁcial respiration should be continued until trained EMS assistance arrives.

76 4. Check for and treat other injuries such as fractures from a fall or burns from current entry and exit sites. Investigations are always after accidents. As an engineer you will be involved as a part of the investigating team or in providing information to an investigator. Information obtained and notes written immediately after the emergency will aid this investigation and assist in preventing future accidents of a similar nature.

Investigations are always made after accidents. As an engineer, you will be involved as a part of the investigating team or in providing information to an investigator. Information obtained and notes written immediately after the emergency will aid the investigation and assist in preventing future accidents of a similar nature.

Emergency Numbers

Fire / EMS: 911 or (864) 656-2222 Student Health Center: (864) 656-2233 ECE Department Oﬃce: (864) 656-5650

Appendix A References

[1] W.F. Cooper, Electrical Safety Engineering, Newnes-Butterworth, London-Boston, 1978.

[2] W.H. Buschsbaum and B. Goldsmith, Electrical Safety in the Hospital, Medical Economics Company, Oradel, NJ, 1975.

[3] J.G. Wester, editor, Medical Instrumentation Application and Design, Houghton Miﬄin Company, Boston, 1978.

77 Appendix B - Instruments for Electrical Mea- surements

Electrical engineers measure and use a wide variety of electrical circuit variables, such as volt- age,current, frequency, power, and energy, as well as electrical circuit parameters, such as resistance, capacitance, and inductance. Many instruments can be used to make such measurements, but the proper use of the instruments and interpretation of the measurements depend on a fundamental understanding of how the instruments work, their capabilities, and their limitations.

This appendix provides a brief overview of the fundamentals of the electrical equipment and instru- ments that you will use in this and other laboratory courses. As you encounter more and varied types of electrical equipment and instruments in this and subsequent courses, you will ﬁnd several books, in addition to your textbook, useful in developing your understanding and measurement skills. In addition, many commercial instrument manufacturers publish handbooks and application notes that provide more information on speciﬁc measurement techniques.

Measurement of Current and Voltage

The basic electrical circuit variables of current and voltage are measured with ammeters (for cur- rent) and voltmeters (for voltage). These instruments may use either analog (continuous) or digital (numerical) indicators (“readouts”) to report the measurement results.

Basic facts you should keep in mind include the following:

• An ammeter is a low resistance instrument to measure current. It is inserted in series with the with circuit branch of interest. If connected in parallel with a component, it will likely short out the component and blow the fuse or burn out the circuit or the instrument.

• A voltmeter is a high resistance instrument for measuring voltage. It is connected in parallel with the component(s) of interest. Voltage is also called “electromotive force” or EMF.

78 Analog Meter Instruments

Analog meter instruments were developed early in the history of electrical science and technology. Most are based on the d’Arsonval galvanometer movement. A brief description of this meter movement and its use in ammeters and voltmeters is given in the textbook Electric Circuits by J. W. Nilsson and S.A. Riedel.

In the d’Arsonval galvanometer, current through a coil of ﬁne wire develops a magnetic ﬁeld that opposes the ﬁeld of a permanent magnet, and so rotates a needle across a scale that is marked oﬀ in units of the measured variable. This type of movement is used extensively in DC analog instruments. For AC measurement, however, the d’Arsonval movement is not suﬃcient by itself, but must be adapted in some way:

• it must be used in conjunction with a rectiﬁer diode that converts the AC into a waveform with a DC level to which the meter can respond, or • it must use an electromagnet instead of the permanent magnet in the standard d’Arsonval move- ment, or • it must use iron vanes.

Figure 12.1 (a) shows an equivalent circuit representation of a dual-coil electro-dynamometer wattmeter often used to measure 60 Hz AC power. It employs the electromagnet form of the galvanometer. Because the meter deﬂection is proportional to the product of the current through the current-sensing coil (in series with the load) and the current through the voltage sensing coil (across the load), the response is proportional to the product of the load current and the load voltage drop. The size of the resistance in series with the voltage coil determines the voltage range of the instrument.

Figure 12.1 (b) shows the equivalent circuit representation of the analog voltmeter. The size of the resistor determines the voltmeter range.

Figure 12.1 (c) shows the equivalent circuit of the analog ammeter. The size of the shunt (parallel) resistor determines the range of the ammeter.

Figure 12.1: (a) Equivalent circuit for analog wattmeter. (b) Equivalent circuit for analog voltmeter. (c) Equivalent circuit for analog ammeter.

79 Digital Multimeter

A multimeter is an electronic device that measures a multitude of electrical values, usually including at least AC and DC voltage and current, as well as resistance. Analog multimeters have an analog display and digital multimeter (DMM) have a digital display. The DMM in this laboratory is used to measure voltage (DC and AC), current (DC and AC), resistance, capacitance. Additionally it may be used for diode tests and audible continuity tests. For capacitance and inductance measurements you must make connections to the DMM/Impedance Analyzer on the prototyping board. For all other measurements make connections to the DMM banana jacks on the workstation.

Dual-Beam Oscilloscope

The oscilloscope is a tool to allow engineers to look at the shape of an electrical voltage versus time or versus a second signal. Until relatively recently, oscilloscopes used a cathode ray tube (CRT) to draw the waveforms onto a screen, just like an image on a television. Televisions and computer monitors also used cathode ray tubes until the advent of the new ﬂat screens. Most engineers refer to the instrument as a “scope”.

The Dual-Beam Oscilloscope has two vertical (“y”) input channels and one horizontal (“x”) channel. The horizontal channel can be connected either to an external AC voltage signal or to an internal time- base generator (x = time). You should become familiar with the scale options on the y input channels, the x input channel, and the time base, since you will be using these to obtain values for voltage and time. The y inputs can be either direct (1X) or through a 10X probe. Figure 12.2 (a) shows the equivalent input circuit for a direct input and Figure 12.2 (b) shows an equivalent circuit for the 10X probe input. Note that in both conﬁgurations one side of the input is grounded, which means that care must be used in connecting the ground clip of the probe or connector used to assure that these are not connected to a “hot” (|V | > 0) part of the circuit. (See the section “Oscilloscope Grounding Errors”.) The calibrated time base is useful when measuring the phase diﬀerence between two waveforms (on the y1 and y2 inputs) by carefully lining up the zero levels for both y inputs and then using in the ac-couple mode to observe the time diﬀerence between zero crossings of the two waveforms.

Figure 12.2: (a) Equivalent circuit for oscilloscope input with 1X probe (direct input). (b) Equivalent circuit for oscilloscope input with 10X probe.

Digital Storage Oscilloscope

80 The digital storage oscilloscope (DSO) is now the preferred type of oscilloscope for most industrial applications, although analog oscilloscopes are still widely used. The DSO uses digital memory to store data as long as required without degradation. The digital storage allows use of an enormous array of sophisticated digital signal processing tools for the analysis of complex waveforms in today’s circuitry.

The digital storage oscilloscopes of the Analog Discovery 2 are dual-beam oscilloscopes with two vertical inputs, as described above. The vertical input on the oscilloscope, instead of driving a vertical ampliﬁer, is digitized by an analog-to-digital (A-to-D) converter to create a data set that is stored in the memory of a microprocessor. The data set is processed and then sent to the display. The data set can be written to a ﬂash drive or sent over a LAN or a WAN for processing or archiving. The screen image can be directly recorded on paper by means of an attached printer or plotter, without the need for an oscilloscope camera. The scope’s own signal analysis software can extract many useful time- domain features (e.g. rise time, pulse width, amplitude), frequency spectra, histograms and statistics, persistence maps, and a large number of parameters meaningful to engineers in specialized ﬁelds such as telecommunications, disk drive analysis, and power electronics.

Digital oscilloscopes are limited principally by the performance of the analog input circuitry and the sampling frequency. In general, the sampling frequency should be at least the Nyquist rate – double the frequency of the highest-frequency component of the observed signal – to avoid aliasing.

81 Appendix C - Operating Instructions for a Typical Oscilloscope

The oscilloscope is an instrument for the analysis of electrical circuits by observation of voltage and current waves. It may be used to study frequency, phase angle, and time, and to compare the relation between two variables directly on the display screen. Perhaps the greatest advantage of the oscilloscope is its ability to display the periodic waveforms being studied.

Until recently, oscilloscopes used a cathode-ray tube to display the signals of interest. A cathoderay tube (CRT) contains an electron gun that directs a high-velocity beam of electrons onto a ﬂuorescent screen. The beam is controlled by a pair of horizontal and a pair of vertical deﬂecting plates. When the voltage on the deﬂection plates is equal to zero, the beam produces a spot of light in the center of the screen. Any potential applied to the plates creates an electric ﬁeld that deﬂects the electron beam proportionally to the applied voltage.

The basic components of the traditional oscilloscope are cathode-ray tube, ampliﬁers, sweep or tim- ing oscillator, and power supply. A voltage to be observed is applied to the vertical deﬂection plates. This signal may be ampliﬁed by the vertical ampliﬁer in order to obtain a satisfactory vertical deﬂec- tion. Meanwhile, a sweep oscillator moves the beam horizontally at a uniform rate. The simultaneous horizontal and vertical sweep of the beam across the CRT screen displays the waveform of the voltage applied to the vertical plates. The sweep oscillator blanks the CRT electron gun during its reverse sweep across the screen to switch oﬀ the electron beam.

If several voltage waveforms are to be studied and must maintain their relative phase positions, the sweep generator must be synchronized to the same voltage during the entire test. In this case, one voltage is applied to the oscilloscope as an external trigger.

An independent voltage may be applied to the horizontal input in place of the sweep oscillator voltage. In this case, two independent input voltages are displayed against one another. If the horizontal frequency is a submultiple of the vertical frequency, the trace will form a stationary pattern on the screen.

oday the traditional CRT oscilloscopes are rapidly being replaced with digital oscilloscopes that have ﬂat-panel liquid crystal displays (LCDs), some with color displays. Instead of directly applying the incoming voltages to deﬂection plates, the digital oscilloscopes capture the voltage information and store it in computer memory as digital signals, which are then analyzed and displayed on the LCD. While the

82 new digital scopes handle the incoming signal diﬀerently from the CRT-based scopes, the basic purpose and many of the operational controls remain the same. Therefore, the discussion that follows applies, for the most part, to both types of oscilloscopes.

Avoiding Grounding Errors with Oscilloscope

The shield or ground wire on the oscilloscope’s signal input is connected to the oscilloscope’s chassis ground, and therefore to the ground lead on the instrument’s electrical power connection. This fact creates the possibility of grounding errors when making connections to circuits, especially when trying to measure voltages across ungrounded components.

The problem is illustrated in the following diagram. Suppose one wants to measure the voltage across resistor R1. Because the circuit and the oscilloscope both have the same ground, connecting the oscilloscope’s input leads directly across R1, as indicated, will create a short across R2. Besides giving an incorrect reading, such a connection might damage either the oscilloscope or the circuit under test.

Figure 13.1: Grounding Error

There are two solutions to the problem.

The ﬁrst solution is simply to swap (interchange) the positions of R1 and R2, so that one end of R1 is at ground. Then there is no problem directly connecting the oscilloscope across R1, because both grounds are connected together.

The second solution is sometimes called the two-channel diﬀerence method, and is illustrated in the next ﬁgure. It requires an oscilloscope with two input channels and with the ability to subtract the signal on one channel from the other channel. As indicated in the illustration, the ground leads of both oscilloscope channels are connected to ground, so there is no grounding error. The signal leads are placed on both sides of R1. Channel 1 will show the output voltage from the signal generator. Channel 2 will show the voltage across R2. If the oscilloscope’s diﬀerence function is engaged so that the signal of Channel 2 is subtracted from Channel 1, the scope will display the voltage across R1.

83 Figure 13.2: Avoiding Grounding Error

Fortunately, Analog Discovery has a diﬀerence function and there is no need for the ﬁrst solution. However, if you ever come across an oscilloscope that does not have a diﬀerence function, you may apply the ﬁrst solution; namely, to swap the positions of R1 and R2, and then measure the voltage across the now-grounded R1.

Figure 13.3: Avoiding Grounding Error

Preliminary Adjustment To Obtain a Trace

To operate the oscilloscope, ﬁrst turn on the power switch and allow the unit to warm up and initialize. Place the HORIZONTAL CONTROL in the sweep position. Adjust the INTENSITY and FOCUS controls on a CRT scope until the desired brightness and line width are obtained. Do not leave the spot stationary on the screen, as doing so may leave a permanent burn on the screen coating.

Once a trace is obtained, some oscilloscopes must be checked for calibration and balance. For the HP 120B CRT oscilloscope (a typical example), the procedure is as follows: Place the TRIGGER LEVEL on AUTO and the TRIGGER SOURCE to INT. Turn the horizontal and vertical VERNIERS to CAL, the VERTICAL SENSITIVITY to CAL, and the HORIZONTAL DISPLAY to 1 msec/cm. With the controls in these positions, an internal calibrating signal is produced on the screen. Turn the CAL adjustment until the upper and lower peaks of the square wave are 6 cm apart.

Digital oscilloscopes often have automated setup and calibration upon power-up.

84 Waveform Observation

After the initial adjustments are made, the oscilloscope is ready for operation. To observe the waveform of any periodic signal, apply the signal to the vertical input terminals. Use DC coupling if the input is DC or very low in frequency or if you want to capture any DC oﬀset. Now the signal from an oscillator, signal generator, or some component of an electrical circuit may be observed on the screen. The best resolution of the waveform is obtained when the time scale is adjusted so one or two cycles appear on the screen and when the vertical scale is adjusted so the amplitude occupies most of the graticule. If the waveform will not stabilize, adjust the SYNC or TRIGGER just enough to cause the pattern to stop. Whenever possible, connect the oscilloscope ground to the common ground of the circuit. Exercise great care when making measurements with both terminals above ground potential, as there may be a diﬀerence in potential between two instrument cases, causing ground loop currents, faulty readings, and damaged equipment.

Voltage Measurement (AC and DC)

The oscilloscope has advantages as a voltmeter: a very high input impedance compared to an analog voltmeter; the ability to measure voltages over a very wide frequency range; and the ability to indicate magnitude regardless of waveform. Also, scopes measure peak-to-peak values of AC voltages, whereas standard AC voltmeters measure rms values of sine wave voltages. However, the oscilloscope only has an accuracy of 2% to 5%, while the AC voltmeter’s accuracy will be from 0.25% to 2%.

To use the oscilloscope as an AC voltmeter, apply the signal to the vertical input terminals, and adjust the calibrated VERTICAL SENSITIVITY (or VOLTS/DIV) so the amplitude is of suitable magnitude on the graticule. The peak-to-peak value is then the distance indicated multiplied by the vertical calibration. For example, assume that a sine wave generator is set to 1000 Hz and adjusted for maximum output voltage. A peak-to-peak value of 60V is observed on the oscilloscope. The output of the generator at 1000 Hz, therefore, is approximately 60V peak-to-peak, and 21.2 Vrms.

Figure 13.4: Voltage Measurement

For DC measurements, apply the voltage to the vertical input terminals, again suitably adjusting the VERTICAL SENSITIVITY. A straight line is produced with the horizontal sweep functioning. With no horizontal voltage applied, a spot will appear on the screen. In measuring DC voltages, it is necessary to remember where the trace was with 0V applied to the vertical input.

85 Frequency Measurement

The frequency of an unknown signal may be calculated from the oscilloscope very easily. The period of the waveform is the product of the distance along the x-axis covered by one cycle and the horizontal sweep setting. As an example, a sine-wave generator is set to 1000 Hz with the voltage applied to the oscilloscope vertical. One cycle covers 9.95 cm, with a sweep speed of 100 µsec/cm. The period is T = (9.95)·(100x10−6 )sec. The measured frequency is f = 1/T = 1005 Hz.

Figure 13.5: Frequency Measurement

Phase-Angle Measurement

The diﬀerence in phase angle between two waveforms may be measured directly on the oscilloscope with little diﬃculty.

For an oscilloscope with only one vertical input:

One wave is chosen as the reverence and applied to the vertical input terminals. This same wave is applied to the external trigger input of the scope. Next, a convenient point on the wave is selected as a time reference, such as where the wave is zero and about to swing positive. Then, this waveform is removed from the vertical input and a second waveform is applied. The voltage of this wave at the time reference is observed. The ratio of voltage at the time reference to the maximum voltage is equal to the sine of the phase diﬀerence between the two waves. This relation is shown below.

Figure 13.6: Phase-Angle Measurement

86 For an oscilloscope with two vertical inputs:

Connect the reference voltage to Channel 1 of the oscilloscope and connect the second voltage to Channel 2. Adjust the amplitudes so the overlapping signals look something like the ﬁgure below, where the solid curve is Channel 1 (the reference) and the dashed curve is Channel 2. (In this ﬁgure, the dashed curve is lagging the solid curve; if the dashed curve were shifted to the left so it “started” before the solid curve, then the dashed curve would be leading the solid curve.) The phase shift φ in degrees can be calculated by the following formula:

φ = ∆t.f.360o (13.1)

where φ is the phase shift, f is the frequency, and ∆t is the time diﬀerence between the two waveforms. Many new digital scopes have cursors that allow directly marking, calculating, and displaying the time diﬀerence and perhaps even the phase shift.

Figure 13.7: Frequency Measurement

87 Appendix D - LT SPICE AC Circuit Simu- lation

LT SPICE is a powerful simulation program, but sometimes it is a little diﬃcult to ﬁgure out how to get it to do what you want it to. A case in point is reporting voltages and currents in a simple AC circuit. To get you started, this appendix describes how to set up a simple series circuit and run LT SPICE to report the current and voltages in that circuit.

Drawing circuits in LT SPICE is straightforward, and becomes easy with a little practice. Start by opening LT SPICE and creating the circuit shown below.

Figure 14.1: LT-SPICE Circuit

For this example, the power supply, V1, is a function generator providing a 10 VPP sinusoidal output. R1 is a 1.0kΩ resistor, and C1 is a 0.1µF capacitor. It is asked that we calculate the current, the voltages, and the phase angles for frequencies from 500 Hz to 5000 Hz.

Once the circuit is drawn, insert ammeters in any branches of interest and apply voltmeters across the devices of interest. For now, don’t worry about the values indicated in the meters.

If necessary, change the values for the resistor and capacitor by double-clicking on the component and making the necessary changes to the name or the value. You must double-click on the component, not on the text describing it.

88 Figure 14.2: LT-SPICE Circuit

Set the power supply voltage by double-clicking on the round icon representing the voltage source. Change the part title to “V1” if necessary. Set the [DC VALUE] to 0.

Select the [SMALL SIGNAL AC AND DISTORTION] tab. Under [AC PROPERTIES FOR AC SMALL SIGNAL ANALYSIS ONLY], enter magnitude 10 V and phase 0. Check [USE]. Click [OK].

Set the range of frequencies to be simulated as follows: Click in the circuit window to be sure it is selected. Click on the menu item [SIMULATION] and then choose [SET UP SIMULATIONS]. Click on the button [AC (AC FREQUENCY SWEEP)]. Make the following entries:

Start Value: 500 (the frequency at which you want to begin) Stop Value: 5000 (the frequency at which you want to stop) Number of Each Steps per Interval: 10 (the number of frequencies at which you want calculations, beginning at the Start Value and ending at the Stop Value.) Stepping interval: Linear

Check (i.e., select) [SHOW APTITUDE PLOTS] and [SHOW PHASE PLOTS]. Select [USE DEGREES FOR PLOTS] and [USE MAGNITUDE FOR PLOTS]. Check [DISPLAY GRAPH]. Check [DISPLAY TABLE]. Click [OK]. Be sure that [AC (AC FREQUENCY SWEEP)] is enabled. Then click the button marked [RUN NOW...].

Immediately a table of voltages, currents, and phase angles will be displayed from 500 Hz to 5000 Hz. The phase angles may be a bit misleading for our purposes, because they force the supply voltage to have a phase angle of zero (0), while our usual procedure is to require the voltage and current of the resistor to point along the real (X) axis in our phasor diagrams. Nonetheless, the simulation data show the relative phases of the voltages, and as expected, the current through the capacitor leads the voltage across the capacitor by 90◦.

LT SPICE will create a graph of the data, if you checked DISPLAY GRAPH, but the voltage and current scales are usually so small that they require magniﬁcation. One can overcome the display

89 problem in the simulation by multiplying the simulation’s supply voltage by some constant so that it is near 100V. Be sure to correct for the multiplicative constant if you try to extract magnitude values from the graph later.

To graph the table of data in Excel, EXPORT the table to a text ﬁle (through the File menu), and then read that text ﬁle into Excel. Once the data is in Excel, use the search and replace function to change units multipliers: k to E3 m to E-3 u to E-6 n to E-9 meg to E6

Excel recognizes the E# as equivalent to 10∧#, where # is an appropriate number. Be sure you do not put any blank spaces in the substitution, or Excel will think it is all text.

After the substitutions are completed, use the standard Excel commands to graph the columns of interest. In this case you would probably choose an XY scatterplot VC and VR magnitudes versus frequency.

Warning: NEVER copy the simulated values and pretend that they are your calculations or your measurements. That is cheating. Whenever you report values from a simulation, be sure to state that they are from the simulation and what program you used to do that simulation. To do otherwise is to be dishonest.

Oscilloscope-like Display

To have LT SPICE show an oscilloscope-like display of the circuit’s voltages and currents, requires only a few changes. Set up the circuit as before, and add meters for anything you want measured.

Double-click on the voltage source. In the [TRANSIENT PROPERTIES] section select [SINU- SOIDAL]. To the right, click on the [PEAK AMPLITUDE OF VOLTAGE] and in the [VALUE] box, enter “10” and click [ACCEPT NEW VALUE]. Then click on [FREQUENCY] and enter “500” in the [VALUE] box and click [ACCEPT NEW VALUE]. Click [OK].

Setting the simulation parameters

Click on the circuit drawing to be sure it is selected. From the menu click on [Simulation] and choose [Set up Simulations]. Enable [Transient and Fourier Analyses] and then click that button. Input a reasonable STOP TIME that will show a few cycles. In this case the frequency is 500 Hz, and so a stop time of 0.01 seconds will display ﬁve cycles. Enable [Display Graph]. Click [OK].

Click the button marked [RUN NOW...]. Immediately LT SPICE will generate a graph of whatever voltages and currents you had meters for on the circuit layout. If [AC (AC FREQUENCY SWEEP)] was also still enabled, then LT SPICE would also generate that graph and table of data. The best of both worlds.

90