eel 3123 networks and systems

LABORATORY MANUAL

EEL 3123 NETWORKS AND SYSTEMS

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

UNIVERSITY OF CENTRAL FLORIDA

Prepared by

Dr. PARVEEN WAHID Ms. YA SHEN

MAY 2013

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PREFACE

This lab manual for EEL 3123 - Networks and Systems is an updated version of the earlier manual. Changes have been made to keep it up to date with the recent curriculum changes in Electrical and Computer Engineering. The experiments have been re-written with a lot more detail to help the student conduct them in a straight forward manner and to help in the understanding of the material.

Every effort has been made to check for any errors and to make sure the experiments are outlined correctly. If you should find errors that need to be corrected, please contact Dr. Parveen Wahid at [email protected].

Dr. Parveen Wahid

Fall 2011

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TABLEOFCONTENTS

SAFETY RULES AND OPERATING PROCEDURES ........................................................... - 4 -

LABORATORY SAFETY INFORMATION ............................................................................. - 5 -

INTRODUCTION ................................................................................................................... - 7 -

TROUBLESHOOTING HINTS ............................................................................................... - 8 -

EXPERIMENT #1 DC MEASUREMENTS ........................................................................... - 9 -

EXPERIMENT #2 AC MEASUREMENTS ......................................................................... - 20 -

EXPERIMENT #3 NETWORK ANALYSIS METHODS ..................................................... - 28 -

EXPERIMENT #4 FIRST ORDER CIRCUITS ................................................................... - 34 -

EXPERIMENT #5 SECOND ORDER CIRCUITS .............................................................. - 41 -

EXPERIMENT #6 SINUSOIDAL STEADY STATE ............................................................ - 48 -

EXPERIMENT #7 SERIES AND PARALLEL RESONANCE ............................................. - 52 -

EXPERIMENT #8 TRANSFER FUNCTIONS .................................................................... - 58 -

EXPERIMENT #9 FREQUENCY RESPONSE .................................................................. - 64 -

APPENDIX I. STANDARD RESISTOR COLOR CODE .................................................. - 69 -

APPENDIX II. LIST OF AVAILABLE RESISTORS AND CAPACITORS .......................... - 71 -

APPENDIX III. GUIDELINES TO WRITE A FINAL LABORATORY REPORT .................. - 74 -

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SAFETYRULESANDOPERATINGPROCEDURES

1. Students are allowed in the laboratory only when the instructor is present.

2. Be aware of the location of the Emergency Disconnect (red button near the door) to shut off power in an emergency.

3. Open drinks and food are not allowed near the lab benches.

4. Report any broken equipment or defective parts to the lab instructor immediately. Do not open, remove the cover, or attempt to repair any equipment.

5. When the experiment is finished, all equipment, except for computers, must be turned off. Return substitution boxes (resistor boxes or capacitor boxes) to the designated location. Clean up any cables and components on the bench before leaving.

6. University property must not be taken out of the laboratory.

7. Do not move equipment from one lab station to another.

8. Do not tamper with or remove security straps, locks or other security devices.

9. ANYONE VIOLATING ANY RULES OR REGULATIONS MAY BE DENIED ACCESS TO THESE FACILITIES.

I have read and understand these rules and procedures. I agree to abide by these rules and procedures at all times while using these facilities. I understand that failure to follow these rules and procedures will result in my immediate dismissal from the laboratory and additional disciplinary action may be taken.

________________________________________ ________________________

Signature Date Lab name and section

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LABORATORYSAFETYINFORMATION

Introduction

The danger of injury or death from electrical shock, fire, or explosion is present while conducting experiments in this laboratory. To work safely, it is important that you understand the prudent practices necessary to minimize the risks and what to do if there is an accident.

ElectricalShock

Avoid contact with conductors in energized electrical circuits. Electrocution has been reported at DC voltages as low as 42 volts. Just 100 mA of current passing through the chest is usually fatal. Muscle contractions can prevent the person from moving away while being electrocuted.

Do not touch someone who is being shocked while still in contact with the electrical conductor or you may also be electrocuted. Instead, press the Emergency Disconnect (red button located near the door to the laboratory). This shuts off all power, except the lights.

Make sure your hands are dry. The resistance of dry, unbroken skin is relatively high and thus reduces the risk of shock. Skin that is broken, wet or damp with sweat has a low resistance.

When working with an energized circuit, work with only your right hand, keeping your left hand away from all conductive material. This reduces the likelihood of an accident that results in current passing through your heart.

Be cautious of rings, watches, and necklaces. Skin beneath a ring or watch is damp, lowering the skin resistance. Shoes covering the feet are much safer than sandals.

If the victim isn’t breathing, find someone certified in CPR. Be quick! Some of the staff in the Department Office are certified in CPR. If the victim is unconscious or needs an ambulance, call 911 and contact the Department Office for help. If able, the victim should go to the Student Health Services for examination and treatment.

Fire

Transistors and other components can become extremely hot and cause severe burns if touched. If resistor or other components on your breadboard catch fire, turn off the power supply and notify the instructor. If electronic instruments catch fire, press the Emergency Disconnect (red button). These small electrical fires extinguish quickly after the power is shut off. Avoid using fire extinguishers on electronic instruments.

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Explosion

When using electrolytic capacitors, be careful to observe proper polarity and do not exceed the voltage rating. Electrolytic capacitors can explode and cause injury. A first aid kit is located on the wall near the door. Proceed to Student Health Services, if needed.

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INTRODUCTION

The objective of this first laboratory in electrical engineering is to familiarize the student with operating basic laboratory instrumentation such as an oscilloscope, a function generator, a multimeter, etc. The student will learn correct laboratory techniques and procedures. In addition the student will get to work with the circuit simulator Multisim. Another goal of conducting the lab experiments is to re-enforce the theoretical knowledge learned in the classroom with practice and vice-versa.

Each of the experiments specifies the objective of the experiment, the equipment that will be needed to conduct that experiment and the measurements that need to be taken. All experiments have a brief write-up of the theory behind the experiment. Each experiment also has three sections to it (i) the theoretical calculation of the results (this is typically done as a per-lab effort) (ii) a simulation section using the circuit simulator and (iii) an experimental section. The pre-lab allows the student to understand the material presented in the classroom and to know what to expect during the lab experiment. The final lab report, that needs to be submitted for each experiment, should contain a comparison between the calculated, simulated and measured values. Any discrepancies obtained should be clearly explained. It is essential that the students learn how to write a detailed lab report and to address all the question listed within each experiment.

In the execution of the experiment, the highest benefit is gained if the student can distinguish between performing the experiment step-by-step following instructions and actually understanding the reasons behind the methodology and the process. To understand the experiments it is essential that both the theory of the circuit under test and the instruments used to test them are known clearly.

Most experiments are designed to be one week experiments. The pre-lab with the calculated and simulated results, is to be completed prior to coming to the lab to do the measurements.

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TROUBLESHOOTINGHINTS

1. Be sure that the power is turned on.

2. Be sure that the supply voltages are correct and the ground connections are common.

For some power supplies, the ‘ON’ button needs to be clicked in order to turn on the

output channel.

3. Be sure the circuit you built is identical to that in the diagram. Do a node-by-node check

if needed.

4. Be sure the resistors, capacitors and inductors used in your circuit all have the correct

values. You can get that information using a digital multimeter, color code or the

nominal value printed on the component.

5. Be sure that the equipment is set up correctly and you are measuring the correct

parameter.

6. If steps 1 through 5 are correct, you probably have used a component that doesn’t work.

It is also possible that the equipment does not work (although this is not probable) or the

breadboard you are using may have some unwanted paths between nodes. To find your

problem you much trace through the voltages in your circuit node by node and compare

the signal you have to the signal you expect to have. If they are different, use your

engineering judgment to decide what is causing the difference or ask you lab assistant.

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EXPERIMENT#1 DCMEASUREMENTS

OBJECTIVES

To understand the basics of DC (direct current) circuits. To use a digital multimeter (DMM) to measure DC voltage, current and resistance. To understand the valid measurement condition for a digital multimeter.

EQUIPMENT

Breadboard DC power supply Digital multimeter (DMM)

BACKGROUND

I. DCcircuitbasics

A DC circuit is an electrical circuit that consists of any combination of constant voltage sources, constant current sources and resistors. The voltages and currents in this circuit are invariant with time, in other words, constant. A DC circuit is usually powered by a DC voltage source or a DC current source.

There are basic concepts and laws that are fundamental to circuit analysis. These laws are Ohm’s law, KCL (Kirchhoff’s current law or Kirchhoff’s first law) and KVL (Kirchhoff’s voltage law or Kirchhoff’s second law). In addition, the voltage divider rule and the current divider rule are often applied to simplify the circuit analysis.

II. Breadboard

A breadboard is also referred to as a solderless breadboard or a plugboard. It is used to build temporary circuits for testing or to experiment new circuit ideas. It has many holes, which can be used to plug in resistors, capacitors, inductors, ICs, and etc. A typical breadboard is shown in Figure 1 - 1. The backside of the bread board, Figure 1 - 1 (b), has strips of metal connecting the holes on the front side. The holes connected by a same metal strip form one common node in a circuit. Different components at a given node are connected by pushing in a corresponding end of each component into holes connected to the same node. It is also noticed that some common nodes are longer than most of the five-hole nodes. They are

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typically used for power supply connections or for those nodes to which many components are connected. A jumper wire can also be used to combine two nodes into one.

The breadboard we are using for this lab also has four binding posts on one side of the board. They are used for DC or AC power supply connections. To connect a binding post onto the breadboard, a wire with long-enough metal exposed is inserted into the hole at the bottom of the post followed by tightening the plastic cap to ensure good connection. The other end of the wire is then plugged into one of the long common nodes on the breadboard.

Here are a few tips for using the breadboard.

1. Never build your circuit without a breadboard, even for the easiest circuit configurations. 2. Always use the binding posts and side-lines (long common nodes) for power supply

connections. 3. It is recommended to use black wires for ground and red (or other colors if there are

multiple voltages needed) for positive voltage (or other DC/AC voltages). 4. Keep the jumper wires short and flat on the board, so that the circuit doesn’t look

cluttered. 5. Route jumper wires around the chips, so that it makes it easy to change the chips. 6. You could trim or bend the resistor/capacitor/inductor lead, so that they will fit in snugly

and won’t get pulled out by accident. 7. A wire should be used to connect the probe of an oscilloscope onto the breadboard,

since the probe connection might loosen the existing connection of your components.

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III. DCpowersupply

A DC power supply is a device that supplies DC voltage and current to a circuit. The one we are going to use for this lab is an Agilent E3630A Triple Output DC Power Supply, as shown in Figure 1 - 2. It offers three output ratings: 0 to +20V (0 to 0.5A), 0 to -20V (0 to 0.5A) and 0 to 6V (0 to 2.5A), with a total maximum power of 35W. The +/-20V output have the ability to track

(a) (b) Figure 1 - 1 Breadboard. (a) front; (b) back.

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each other by adjusting the Tracking Ratio knob, while the 6V output is adjusted separately. When setting each output, the corresponding range needs to be selected on METER section in order to have that particular output displayed on the LED. On VOLTAGE ADJUST section, the +6V knob is used to adjust the 6V output. The +/-20V knob adjusts both the +20V and the -20V output at the same time. Therefore, when two voltages with the same absolute value but opposite polarities are needed, the Tracking Ratio knob needs to be set at Fixed position, which means the tracking ratio is 1. However, when two voltages with different absolute values and polarities are needed from the +20V and the -20V output, one of the two (usually the +20V) needs to be adjusted first using the +/-20V knob, then by varying the Tracking Ratio knob, the other voltage is then set to the desired output. The COM knob in OUTPUT section is used as the common ground for all three outputs. The LED will show both the voltage and current for the voltage range selected, and when the current exceeds the maximum current rating during measurement, the OVERLOAD indicator will turn on.

IV. Digitalmultimeter

A digital multimeter (DMM) is an electronic measuring instrument that combines several measurement functions, such as voltage, current and resistance measurement, in one unit, and displays its result digitally. The one we are going to use in this lab is a Tektronix DMM4050 6-1/2 Digit Precision Multimeter, as shown in Figure 1 - 3. This DMM is capable of measuring DC/AC voltages and currents, resistance, integrated frequency, period, capacitance and temperature measurement.

(a) (b) Figure 1 - 2 Agilent E3630A DC power supply. (a) front; (b) back.

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Here are a few tips for using the DMM.

1. Voltage is measured by placing the DMM in parallel with the device under test (DUT) on which the voltage is to be measured, as shown in Figure 1 - 4 (a). First connect the DMM input connector (red probe) to 1000V/600V input, and select DCV or ACV. The ground probe (black probe) is connected to the corresponding LO input. Then place the probes on two sides of the DUT. Preferably, the red probed should be connected to the side with higher potential. But sometimes it is hard to decide in a circuit with multiple DC voltage sources as for which end of the DUT has higher potential. It is fine to just randomly pick a side. If the reading ends up being negative, that means the red probe is connected to the lower-potential end.

2. Current is measured by inserting the DMM into the circuit and letting the current being measured go through the DMM, as shown in Figure 1 - 4 (b). First connect the DMM input connector (red probe) to 400 mA or 10 A input, and select DCI or ACI. The ground probe (black probe) is connected to the corresponding LO input. Then insert the probes into the branch being measured. Preferably, current should flow into the DMM from the red probed and flow out from the black probe. But sometimes it is hard to decide the

Figure 1 - 3 Tektronix DMM 4050

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current direction in a circuit with multiple DC voltage sources. It is fine to assume a direction. If the reading ends up being negative, that means the current flows in the opposite direction from your assumption.

3. Resistance measurement is done similar to the voltage measurement by placing the DMM across the resistor to be measured, as shown in Figure 1 - 4 (c) and use the Ω button. The ground probe (black probe) is connected to the corresponding LO input. Note that the resistor being measured should be disconnected from the rest of the circuit during this measurement.

4. A DMM usually has an internal resistance (typically of 10 MΩ), as shown in the circuit in Figure 1 - 6 (e). The measurement result will be inaccurate if the resistor for which the voltage or current being measured is comparable in value to the internal resistance.

V. Multisim

Multisim is an electronic schematic capture and simulation program used to analyze circuit behavior. The DC/AC voltage, DC/AC current, resistance, frequency, time-domain waveform, etc, can be determined using this software. An example circuit simulation measurement is shown in Figure 1 - 5. In this simulation, all the components are laid out in a way that is the same as the circuit diagram. Each DMM is connected in the same way that a physical DMM would be connected on the breadboard. Results are obtained by running the simulation and then double clicking on each piece of equipment (DMM, oscilloscope, etc) to read the desired output values..

(a) (b) (c)

Figure 1 - 4 Voltage, current and resistance measurement using DMM, (a) voltage; (b) current; (c) resistance.

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VI. Standardresistorcolorcode

Low-power resistors have a standard set of values. Color-band codes indicate the resistance value as well as a tolerance value. Refer to APPENDIX I on how to read the resistance using the color codes.

Figure 1 - 5 An example for using Multisim

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PREPARATION

For the circuits in Figure 1 - 6, use 9V as input voltage. Pick any resistor values between 1kΩ and 56 kΩ and assign the same values to all resistors with the same name and different values to those with different names. Refer to APPENDIX II for available resistors and capacitors.

1. For the circuit in Figure 1 - 6(a), calculate V1, V2 and I using Ohm’s law. Use voltage divider rule to calculate V1 and V2 again, and then compare them with the V1 and V2 calculated earlier. Are they the same? If yes, the voltage divider rule is verified. Add V1 and V2, is this value equal to the total voltage supplied by the source? If yes, KVL is verified.

2. For the circuit in Figure 1 - 6 (b), calculate I1, I2, I and VX using Ohm’s law. Use current divider rule to calculate I1 and I2 and then compare them with the I1 and I2 calculated earlier. Are they the same? If yes, current divider rule is verified. Add I1 and I2, is this value equal to the total current I? If yes, KCL is verified.

3. For the circuit in Figure 1 - 6 (c), calculate the equivalent resistance between points AB and points CD. Rearrange the resistors in the circuit or use Y-Δ transformation if needed.

4. For the circuit in Figure 1 - 6 (d), calculate VX. 5. Assume that a DMM with an internal resistance of 10 MΩ is used to measure VX in the

circuit in Figure 1 - 6 (e). Calculate the result if: a. R is between 1kΩ and 10kΩ. b. R is larger than 2MΩ.

SIMULATION

Build and simulate the circuits in Figure 1 - 6 using Multisim.

1. For circuit (a), use a DMM to read I, V1 and V2. 2. For circuit (b), use a DMM to read I, I1, I2 and VX. 3. For circuit (c), use a DMM to read RAB and RCD. 4. For circuit (d), use a DMM to read VX. 5. For circuit (e), use a DMM to read VX for both cases considered in PREPARATION

(Figure 1-6e).

Compare all the results with those calculated in PREPARATION.

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EXPERIMENT

1. Measure the resistance of each resistor used in this experiment using a DMM. Compare the nominal value with the one measured. Refer to ‘Digital multimeter’ in ‘BACKGROUND’ section for how to use a DMM to measure a resistor. The nominal value of a resistor can also be obtained from its color bands or color code. Build the circuits in Figure 1 - 6 on a breadboard. a. For circuit (a), use a DMM to measure I, V1 and V2. Refer to ‘Digital multimeter’ in

‘BACKGROUND’ section for how to use a DMM to measure the voltage and current. b. For circuit (b), use a DMM to measure I, I1, I2and VX. c. For circuit (c), use a DMM to measure RAB and RCD. d. For circuit (d), use a DMM to measure VX. e. For circuit (e), use a DMM to measure VX for both cases considered in

PREPARATION (Figure 1-6e).

Compare the results with those calculated in PREPARATION and those in SIMULATION.

REPORT

Prepare your report as per the guidelines given in APPENDIX III.

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(a) (b)

(e)

(d)

(c)

Figure 1 - 6 Circuits

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EXPERIMENT#2 ACMEASUREMENTS

OBJECTIVES

To understand the basics of AC (alternating current) circuits. To use an oscilloscope to display and record a waveform. To use an oscilloscope to measure frequency, period, voltage (magnitude, peak-to-

peak, maximum, minimum, and etc), DC offset, etc, of the waveform. To use a digital multimeter to measure AC voltage and current.

EQUIPMENT

Function generator Oscilloscope Digital multimeter (DMM)

BACKGROUND

I. ACcircuitbasics

An AC (alternating current) is a function of time. In AC circuits, the voltage and current sources are time-varying, with the most common being a sinusoidal variation. Other time-varying applications, depending on the application, include the square wave and the triangular wave.

A sinusoidal AC voltage or current is described by its amplitude, frequency and phase, for example, sin , where A is the magnitude, w the angular frequency and is the phase. Special attention must be paid when measuring the voltage or current of an AC circuit, as three different forms of voltage or current may be obtained from a measurement, namely Vmagnitude or Imagnitude, VRMS or IRMS or Vpeak-to-peak or Ipeak-to-peak. They can be converted from one form to another using the following equations:

For square waveforms,

For sine waveforms,

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√2

√2

For triangular waveforms,

√3

√3

II. Functiongenerator

A function generator is a piece of equipment used to generate electrical waveforms. It is widely used in development, testing and repair of electronic equipment, for example, as a signal source to test amplifiers, or to introduce an error signal into a control loop. The function generator we are going to use for this lab is a Tektronix AFG 3022B, as shown in Figure 2 - 1. It is able to provide 12 different standard waveforms, with dual-channel capability. Sine, square and triangle waveforms (3 most used waveforms in this lab) can be selected from Function section. Both frequency and amplitude adjustment buttons are in the same section below ‘Run Mode’ section, which can be set either by the numeric keypad or by the knob above that keypad. The two arrows right below the numeric keypad can change which digit is to be adjusted. The ‘On’ button will turn each channel on or off, and ‘CH1/CH2’ button decides if it is a single or a dual channel output. Detailed functions will be shown in EXPERIMENT section.

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III. Oscilloscope

An oscilloscope is a type of electronic test instrument that allows observation of constantly varying signal voltages. Usually a two-dimensional graph of one or more electrical potentials is displayed using the vertical or ‘Y’ axis with time along the horizontal or ‘X’ axis. In most instances, an oscilloscope shows events that repeat with either no change or very slow

Figure 2 - 1 Function generator

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changes. Thus, the exact wave shape of an electrical signal can be observed. In addition to the amplitude of the signal, an oscilloscope can show distortion, the time between two events (such as pulse width, period, rise time, and etc), and relative timing between two related signals. The oscilloscope we are going to use for this lab is a Tektronix DPO 4034 Digital Phosphor Oscilloscope, as shown in Figure 2 - 2. It has 4 analog and 16 digital channels for analyzing both analog and digital signals. Detailed functions will be shown in EXPERIMENT section.

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Figure 2 - 2 Oscilloscope

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PREPARATION

1. Calculate the RMS voltage of the following waveforms with 10 V peak-to-peak: a. Sine wave; b. Square wave; c. Triangle wave.

2. Calculate the period of a waveform with the frequency of: a. 100 Hz; b. 1 kHz; c. 100 kHz.

SIMULATION

Build a circuit in Multisim consisting of a function generator and a resistor. Set the frequency to 1 kHz and amplitude to 5V (peak-to-peak to 10 V). Use a multimeter to read the voltage (VRMS) over the resistor. Use an oscilloscope to display the waveform over the resistor. Peak voltage and period can also be measured using cursors. Verify the frequency of the output waveform by converting measured period to frequency.

EXPERIMENT

A. Waveformdisplayandmeasurement

1. Click ‘Default Settings’ button located at below the LCD screen on oscilloscope to clear all the settings from the previous user. For the same purpose, click the ‘Default’ button on the function generator, which is located on the left of the numeric keypad, and then click Enter on the right of the keypad.

2. On the function generator, click ‘Output Menu’ located on the right of the LCD screen, select ‘Load Impedance’ then ‘High Z’ option. Click the return arrow in the lower right corner of the screen until the default screen shows up.

3. Set the function generator output to sine wave. 4. Connect Ch.1 of the oscilloscope to the function generator output. Polarities (positive

and negative) of both probes need to be matched, i.e. positive to positive, negative to negative. Turn on Channel 1 output on function generator by clicking the ‘On’ button of channel 1. Click ‘Autoset’ button on the oscilloscope to automatically find the waveform on the LCD screen. How many cycles are displayed?

5. Click on the different ‘Function’ selection buttons on the function generator. What other waveforms do you observe on the screen? Set the waveform back to sinusoidal. Click the ‘Autoset’ button if needed.

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6. Click ‘Ch1’ button (yellow) on oscilloscope to bring up the setting options for Ch1. Click it again, what do you see on the screen? Click the button once again, did the waveform show up again? With the options for Channel 1 available on the bottom of the screen, set coupling to DC, and keep the rest at the default setting.

7. Various measurement options for all four channels can be set on the oscilloscope. To do this, click ‘Measure’ button in ‘Wave Inspector’ section and press ‘Add Measurement’ button to bring up the menu. Use the ‘Multipurpose a’ knob to select the source (Ch1, Ch2, Ch3 or Ch4, whichever is available) and ‘Multipurpose b’ knob to select the measurement type. Using the ‘Multipurpose b’ knob highlight the frequency, period, amplitude, mean and RMS, and click on ‘OK Add Measurement’ each time to bring up the readings on the screen. When finished, click on the ‘Menu Off’ button to turn off the menu. (Note: the amplitude definition might be different from that in your text book. Make sure to check the definition on the oscilloscope when selecting this measurement.)

8. Vary the frequency and then the amplitude on the function generator. It can be done by clicking on the ‘Frequency/Period’ button or the ‘Amplitude/High’ button, then either type in the number using the numeric keypad or adjust the knob above that. How does the waveform change on the oscilloscope screen? Do the readings on the bottom of the oscilloscope screen change as well? Use ‘Autoset’ button or manually adjust the scale vertically or horizontally when needed. (Note: the readings on the oscilloscope will not be accurate if the waveform is not properly displayed on the LCD screen, e.g. too few cycles or the waveform exceeds the screen in the vertical direction.)

9. Vary the channel scale knob in the ‘Vertical’ section and the time scale knob in the ‘Horizontal’ section on the oscilloscope. How does the waveform change on the screen? Do the readings on the bottom of the screen change as well?

10. Now, with all the measurement displayed on the screen, set the frequency of the input signal to 1 kHz and magnitude to 5 V. (Try to keep about 4 - 5 cycles displayed on the oscilloscope screen by adjusting both the vertical and horizontal scale knobs in order to obtain an accurate reading.) What do you read for frequency? What do you read for amplitude voltage?

11. Adjust the ‘level’ knob on oscilloscope. What do you observe when the little arrow is within the magnitude range of the waveform? How about when it moves outside the magnitude range? Move it back in so that it is slightly below the peak and the waveform just becomes stable.

12. Keeping the level the same, slowly decrease the amplitude of the input signal on the function generator by adjusting the knob above the numeric keypad when ‘Amplitude’ is high-lighted on the screen (if not, click on the ‘Amplitude/High’ button). What do you see on the screen? Describe what you observed in step 11 and 12, and explain why.

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13. Adjust the level arrow back to the zero and the signal amplitude back to 5 V. 14. Adjust the ‘DC offset’ on the function generator by clicking the ‘Offset/Low’ button

and set the value to 1 V. What do you observe and read on the screen? 15. Set CH1 coupling to AC (refer to step 6) and repeat step 14. What do you observe

and read? Keep the measurement on the botton of the screen so that mean (offset) reading is available. Make comments on the difference of step 14 and 15.

16. Set CH1 coupling back to DC and set ‘DC offset’ back to zero.

B. Voltagemeasurement

1. With a sine wave of 5 V magnitude from function generator, set DMM to measure AC voltage and place it in parallel with the probe of the oscilloscope. Is the reading what you expect? Which measurement on the oscilloscope is this DMM reading equivalent to, Amplitude or RMS?

2. Switch the waveform to a square waveform first, and then to a triangle (ramp) waveform. In each case what is the AC voltage on DMM? Are all the measurements consistent with those calculated in PREPARATION?

C. Frequencymeasurement

1. Set the waveform back to sine wave on the function generator, record the frequency and period readings from the oscilloscope.

2. Change the frequency on the function generator to 100 Hz and 100 kHz. What are the frequency and period readings? Click ‘Autoset’ button or manually adjust the horizontal scale when needed. Are all the measurements consistent with those calculated in PREPARATION?

REPORT


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EXPERIMENT#3 NETWORKANALYSISMETHODS

OBJECTIVES

To analyze a resistive circuit using node or mesh analysis. To understand Thevenin’s and Norton’s theorems. To verify the superposition principle.

EQUIPMENT

Breadboard DC power supply Digital multimeter (DMM)

BACKGROUND

Electrical circuit analysis is the process of finding the voltages across and the currents through every component in the network. A number of techniques are frequently used for resistive circuits.

Nodal analysis is a method of determining the voltage at the nodes in an electrical circuit with respect to a reference node, using Kirchoff’s current law. Mesh analysis is a method that is used to solve for the current through any component in a planar circuit using Kirchoff’s voltage law. In some cases one method is clearly preferred over another. For example, when the circuit contains only voltage sources (or current sources), it is probably easier to use mesh analysis (or node analysis). It is often helpful to consider which method is more appropriate for the problem solution and make the selection.

Thevenin’s theorem, also called Thevenin equivalent, states that if we identify a pair of terminals in any circuit made up of both independent and dependent sources and resistors, the circuit can be replaced by an independent voltage source Voc in series with a resistor Rt. This series combination of Voc, the Thevenin voltage, and Rt, the Thevenin resistance, is equivalent to the original circuit in the sense that if we connect a same load across the terminals, we would get the same voltage and current at the terminals of the load as we would have with the original circuit. This equivalence holds for all possible values of load resistances. Voc is the open-circuit voltage of the original circuit across the terminals. Rt can be found by either of the two methods listed below. One is to find the short-circuit current isc, then find Rt using

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The other method usually used in less complicated circuits involves deactivating the sources in the circuit, i.e. replacing all independent voltage sources with short circuits and all independent current sources with open circuits, and finding the equivalent resistance, which is Rt. Dependent current and voltage sources are not replaced with open circuits or short circuits.

Norton’s theorem, also referred as Norton equivalent, is a dual of Thevenin’s theorem. If we identify a pair of terminals in any circuit made up of both independent and dependent sources and resistors, the circuit can be replaced by a parallel combination of an ideal current source isc and a conductance Gn, where isc is the short-circuit current at the terminals in the original circuit and Gn is the ratio of the short-circuit current to the open-circuit voltage at the terminals in the original circuit.

The four parameters Voc, Rt, isc and Gn are related by

1

For linear circuits containing two or more independent sources, the superposition principle can also be used for circuit analysis. The voltage across (or the current through) any element can be obtained by adding algebraically all the individual voltages (or currents) caused by each independent source acting alone, with all the other independent voltage sources replaced by short circuits and all the other independent current sources replaced by open circuits.

PREPARATION

For the circuit in Figure 3 - 1, pick any resistor values between 1 kΩ and 50 kΩ, such that not all resistors are the same. Refer to APPENDIX II for available resistors and capacitors.

- 30 -

Figure 3 - 1 Circuit

- 31 -

1. Short AB, as shown in Figure 3 - 2 (a). Use mesh analysis to calculate the voltage across each resistor and the current through AB.

2. Leave AB open, as shown in Figure 3 - 2 (b). Use nodal analysis to calculate the voltage across each resistor and that across AB.

3. Find Thevenin’s and Norton’s Equivalent (Voc, Isc and Rt) using the results from step 1 and 2.

4. If a resistor is connected between nodes A and B, use Thevenin’s theorem to calculate the current through this resistor using the following values: 1 kΩ, 2.2 kΩ and 4.7 kΩ.

5. Leave AB open. Find the voltage across AB caused by E1 only. In order to do so, E2 and E3 need to be set to zero by simply replacing each of them with a wire (short). Repeat this step and find the VAB caused by E2 only and E3 only. Verify that the sum of these three voltages found using each source individually equals to that found in step 2.

Figure 3 - 2 Mesh and nodal analysis. (a) mesh analysis; (b) nodal analysis (a) (b)

+_E =12V

R

R

R

R

RR

R

A B

+_

E =10V +_E =5V

V+ _

- 32 -

SIMULATION

Build and simulate the circuit in Figure 3 - 1 using Multisim. Verify all the results you calculated in PREPARATION section. Refer to the BACKGROUND section in EXPERIMENT #1 for instructions on how to use DMM and Multisim.

EXPERIMENT

Build the circuit in Figure 3 - 1 on the breadboard. Refer to Section III in Experiment #1 to set the voltages sources in the circuit.

A. Meshanalysisandnodalanalysis

1. Short AB by connecting a wire across nodes A and B. Measure the voltage across each resistor and the current through AB. Refer to the BACKGROUND section in Experiment #1 for how to use DMM to read the voltage and current values.

2. Leave AB open. Measure the voltage across each resistor and that between AB. 3. Compare the results from step 1 and 2 with those obtained from PREPARATION

and SIMULATION sections.

B. Thevenin’sandNorton’stheorems

1. Set all voltage sources to zero by simply replacing them with wires. With AB open, measure the resistance between AB using DMM. How does this compare to Rt values obtained earlier in the PREPARATION and SIMULATION sections?

2. Connect all voltages sources back into the circuit. With a resistor between A and B, measure the current through this resistor using DMM. Use the following resistance values:

a. 1 kΩ, b. 2.2 kΩ, c. 4.7 kΩ.

How does these current values compare with those obtained from PREPARATION and SIMULATION sections.

C. Superpositionprinciple

1. With AB open, measure the voltage across AB caused by each voltage source individually. E.g. Using source E1 only, set E2 and E3 to zero and measure the voltage VAB. Repeat for sources E2 and E3. Add these three voltages, what do you get? Compare the results with those obtained from PREPARATION and SIMULATION sections.

2. With AB open, measure the voltage across AB caused by: a. Using source E1 only,

- 33 -

b. Using sources E2 and E3 together with E1 set to zero.

Add these voltages, what do you get? Is it what you expect? Explain.

3. With AB open, measure the voltage across AB caused by: a. Using sources E1 and E2 together with E3 set to zero, b. Using sources E2 and E3 together with E1 set to zero, c. Using sources E3 and E1 together with E2 set to zero.

Add these voltages, what do you get? Is it what you expect? Explain.

REPORT


- 34 -

EXPERIMENT#4 FIRSTORDERCIRCUITS

OBJECTIVES

To study the step response of first order circuits. To understand the concept of the time constant.

EQUIPMENT

Breadboard Function generator Oscilloscope Digital multimeter (DMM)

BACKGROUND

First-order transient circuits are described by a first order differential equation. First-order circuits contain a resistor and only one type of storage element, either an inductor or a capacitor, i.e. RL or RC circuits.

For a step voltage/current source input, the output can be expressed as

∞ 0 ∞

Where, X(0) is the circuit response at t = 0, and X(∞) is the response at t = ∞. The parameter

is called time constant of the circuit and gives the time required for the response (i) to rise

from zero to 63% (or 1 ) of its final steady value as shown in Figure 4 - 1 (a), or (ii) to fall to

37% (or ) of its initial value as shown in Figure 4 - 1 (b). Therefore, the smaller the value of ,

the faster the circuit response is.

For a RC circuit

For a RL circuit

- 35 -

Applying the equations above, the voltage responses across the capacitor and the resistor in Figure 4-1 can be written as:

1 , for 0

, for 0

- 36 -

Figure 4 - 1 A first order circuit and its responses. (a) voltage over the capacitor; (b) voltage over the resistor.

(a) (b) τ

R

τ

C

C

R

in+_

- 37 -

PREPARATION

Figure 4 - 2 and Figure 4 - 3 show an RC and an RL circuit.

For all circuits, R = 1 kΩ, C = 0.1 uF, L = 100 mH.

A. Stepvoltageinput

1. For the circuits in Figure 4 - 2 using step voltage sources, derive the analytical expression for 0, when .

2. Sketch or plot for each circuit.

B. Squarewaveinput

1. For the circuits in Figure 4 - 3 use a square wave input. Assume that is a symmetric square wave with amplitude of and period of 10 .

2. Sketch or plot for each circuit using superposition. (Show at least five cycles.)

Hint: the square wave can be broken up into a series of step functions with displacement of /2 and alternate polarities. Each of these step function inputs generates an output. Thus the total output response is the summation of all the individual output.

- 38 -

Figure 4 - 2 Circuits with step voltage sources

(a) (b)

(c) (d)

- 39 -

Figure 4 - 3 Circuits with square wave input

(a) (b)

(c) (d)

in out

+

_

in out

+

_

in out

+

_

in out

+

_

- 40 -

SIMULATION

Build and simulate the circuits in Figure 4 - 3 using Multisim. Set the input voltage to 5 with a frequency of 1 kHz. Display on the oscilloscope. Compare this result with the plot from PREPARATION step B.

EXPERIMENT

Use the same component values as in the PREPARATION and the same input settings used in the SIMULATION, and build the circuits shown in Figure 4 - 3 (a) – (d). Complete the measurements described below. Refer to Experiment #2 for how to use function generator and oscilloscope.

A. Squarewaveoutput

On the oscilloscope, connect Ch1 to the input and Ch2 to the output so that both the input and the output are displayed on the screen. Save the screen image for both the input and the output, preferably to a USB drive. Use the ‘Menu’ button on the ‘Save/Recall’ section on the bottom of the oscilloscope screen, then use ‘File Utilities’ to select or create a folder to save the image. Press the ‘Save Screen Image’ button and use the associated buttons next to the screen to select the format or edit file name etc. Compare these waveforms with the results from PREPARATION and SIMULATION.

B. Timeconstantmeasurement

1. Turn off the input (Channel 1) by pressing the channel number button. Now only the output is displayed on the screen.

2. Zoom in on the output curve on the oscilloscope such that a large portion (at least half a cycle) of the rise/drop of a cycle is displayed on the screen. Consider the rise and drop over one half cycle only for each circuit. Use cursors to determine the maximum voltage difference E for the output. The time it takes for the output to rise from the minimum value to 63% of E or to drop from the maximum to 37% of E is the time constant of the circuit. Record this time constant for each circuit.

REPORT


- 41 -

EXPERIMENT#5 SECONDORDERCIRCUITS

OBJECTIVES

To study the step response of second order circuits. To understand the difference between overdamped, critically damped and

underdamped responses. To determine theoretically and experimentally the damped natural frequency in the

under-damped case.

EQUIPMENT


BACKGROUND

Second-order circuits are RLC circuits that contain two energy storage elements. They can be represented by a second-order differential equation. A characteristic equation, which is derived from the governing differential equation, is often used to determine the natural response of the circuit. The characteristic equation usually takes the form of a quadratic equation, and it has two roots s1 and s2.

0

42

42

When these roots are rewritten as,

- 42 -

then the natural response of the circuit is determined by:

(i) , there are two real and distinct roots Overdamped, as shown in Figure 5 - 1 (a) and (d).

(ii) , there are two real equal roots Critically damped, as shown in Figure 5 - 1 (b) and (e).

(iii) , there are two complex roots Underdamped, as shown in Figure 5 - 1 (c) and (f).

- 43 -

Figure 5 - 1 Second order circuits natural responses

(a) (d)

(b) (e)

(c) (f)

- 44 -

PREPARATION

For all circuits, C = 0.01 uF, L = 100 mH.

A. Stepvoltageinput

1. For both circuits in Figure 5 - 2, write the characteristic equation. 2. Calculate the resistance range for R for the following cases:

a. Over-damped response, b. Critically damped response, c. Under-damped response.

3. Plot or sketch the response due to a step voltage input, when:

For the circuit in Figure 5 - 2 (a),

a. R = 22 kΩ b. R = 6.3 kΩ c. R = 2.2 kΩ

For the circuit in Figure 5 - 2 (b),

a. R = 680 Ω b. R = 1.6 kΩ c. R = 4.7 kΩ

- 45 -

B. Squarewaveinput

1. Set R = 470 Ω (for the circuit in Figure 5 - 3 (a)) or R = 22 kΩ (for the circuit in Figure 5 - 3 (b)). Calculate α, ωo, and ωd.

2. Plot or sketch the output voltage due to a square wave input with a frequency of 400 Hz and amplitude of 4 V.


(a) (b)

- 46 -

SIMULATION

Build and simulate the circuits in Figure 5 - 3 using Multisim. Set the input voltage to 4 with a frequency of 400 Hz. Display on the oscilloscope. Compare this result with that from PREPARATION step B.

EXPERIMENT

On the function generator use the same square wave input settings as in SIMULATION. Build the circuits shown in Figure 5 - 3. Complete the measurements described below.

A. Naturalresponses

1. Use a resistor box, and set R at the values given below. Use the DMM to check the resistance values before connecting them into the circuit.

For Figure 5 - 3 (a)

a. R = 22 kΩ, b. R = 6.3 kΩ, c. R = 2.2 kΩ.

For Figure 5 - 3 (b)


(a) (b)

- 47 -

a. R = 680 Ω, b. R = 1.6 kΩ, c. R = 4.7 kΩ.

2. On the oscilloscope, connect Ch1 to the input and Ch2 to the output so that both the input and the output are displayed on the screen.

3. For each case, save the screen image with the associated measurements for both the input and the output on to a USB drive. (Follow the steps as explained in Experiment #4 to do this.)

4. For each case, indicate if the output response is overdamped, critically damped or underdamped.

B. Dampednaturalfrequencymeasurement

1. Set R = 470 Ω for the circuit in Figure 5 - 3 (a) or R = 22 kΩ for the circuit in Figure 5 - 3 (b), save the screen image for both the input and the output, and compare it with the results from PREPARATION and SIMULATION.

2. Zoom in on the output curve so that at least two whole oscillations (ripples) of the output from the beginning of an output cycle are displayed. Use the cursors to measure the time period Td between the first two peaks (or between two zero phases). ωd is calculated using:

2 ∙

REPORT


- 48 -

EXPERIMENT#6 SINUSOIDALSTEADYSTATE

OBJECTIVES

To understand and calculate the power factor of a passive circuit. To verify that resistive components dissipate power while reactive components do not.

EQUIPMENT


BACKGROUND

The steady-state response is the response that exists after the initial conditions and transient or natural response die out. AC steady state analysis determines the steady state response of a circuit when the inputs are sinusoidal functions. The steady state voltages and currents in the circuit will also be sinusoidal, with the same frequency as the input signal. The maximum amplitude and phase angle of the steady state response will, in general, differ from that of the source.

The angle referred to as the power factor angle, is involved in the calculation of the average and reactive power. The power factor is the cosine of this angle

cos cos

A lagging power factor implies that the currents lags the voltage, hence an inductive load. A leading power factor implies current leads the voltage, hence a capacitive load. The power factor (PF) is the ratio of the average power to the apparent power. The average power absorbed by the element is calculated by,

∗2

cos ∗ ∗ cos Eq 6-1

and the apparent power is given by,

| |∗2

∗ Eq 6-2

- 49 -

where Vm and Im are the magnitudes of the voltage and current respectively, VRMS and IRMS are the rms RMS values voltage and current respectively, θV and θI are the phase angles of the voltage and current respectively.

The average power (often simply called ‘power’) dissipated in a circuit with a periodic input signal of period T is defined as

1 Eq 6-4

The average power dissipated by a resistor is given by

∗ Eq 6-5

and the average power dissipated by reactive components, such as inductors and capacitors, is equal to zero. In other words, reactive components are storage elements and do not dissipate power.

PREPARATION

Consider the circuit in Figure 6 - 1 with Vin magnitude of 5 V and frequency of 3 kHz.

1. Calculate the total impedance of the circuit. 2. Calculate the power factor of the circuit. 3. Calculate the voltage and current (they are both complex numbers) in each element in

the circuit. 4. Calculate the average power dissipated in each element in the circuit. 5. Calculate the total average power provided by the source. 6. Verify that the power generated by the source equals the total power dissipated in all

the components in the circuit.

7. Assume that and ∅ ( and are the magnitudes of Vin and Vx respectively, ∅ is the phase angle between Vin and Vx, and they can be read on the oscilloscope). Prove that the following equation is valid,

tansin∅

cos ∅ Eq 6-6

where is the power factor angle of the source. Hence the power factor PF=cos(θ) can be found.

- 50 -

SIMULATION

1. In Multisim, build and simulate the circuits in Figure 6 - 1. Set the input voltage to 5 and frequency to 3 kHz.

2. Determine the voltage and current in each element using a DMM. Remember, the numbers you obtained are in RMS.

3. Determine the average power dissipated in each element using a power meter. 4. Determine the total power provided by the voltage source (or the total power consumed

by the circuit) using a power meter.

Figure 6 - 1 Circuit

- 51 -

5. Compare all the results from step 2 to 4 with those obtained in PREPARATION. 6. Verify the law of conservation of energy.

7. Determine , and ∅ (as in ∅) using oscilloscope and cursors. Calculate the power factor using the equation in PREPARATION step 7. Compare this result with that from PREPARATION step 2.

EXPERIMENT

1. Build the circuit on breadboard, and use the same input settings as in SIMULATION. Place the Ch1 probe at and the Ch2 probe at . Display both and on the screen. These two curves should have the same frequency but with a phase shift between each other.

2. Measure and using the oscilloscope. Note that ∅ is measured indirectly. First measure the time difference ∆ between the two peaks of and . Expand the time scale in order to get a better reading with the cursors. Then use the equation ∅ 2∆ (rad) to calculate ∅. Pay attention to the unit. Convert ∅ to degrees if needed.

3. Calculate the power factor using the equation in PREPARATION step 7. 4. Measure the voltage and current in each element using a DMM. How does this result

compare with that from PREPARATION and SIMULATION? 5. Calculate the average power dissipated in each element using the results (voltage

and/or current) from the last step. Compare the result with that from PREPARATION and SIMULATION.

6. Measure the total voltage and current provided by the voltage source. 7. Calculate the total power provided by the voltage source. Compare this result with that

from PREPARATION and SIMULATION. 8. Compare the total power delivered by the voltage source with the total power dissipated

by all the elements in the circuit. Explain your result (e.g. what element contributes to the power dissipation and what element doesn’t).

REPORT


- 52 -

EXPERIMENT#7 SERIESANDPARALLELRESONANCE

OBJECTIVES

To study the behavior of series and parallel LC circuits at resonance. To understand the resonance frequency, cut-off frequency, bandwidth and quality factor

of a resonance circuit. To determine if a circuit is inductive or capacitive. To understand the circuit behavior at resonance.

EQUIPMENT


BACKGROUND

Resonant circuits form the basis for filters that have better performance than first order (RL, RC) filters in passing desired signal or rejecting undesired signals that are relatively close in frequency. The resonance frequency is defined as the frequency at which the impedance of the circuit is purely real, that is, with zero reactance. For the reactance to be zero, impedance of the inductor must equal that of the capacitor. At resonance, the impedance of a branch with LC in series is equal to zero, which is equivalent to a short, and the admittance of a branch with LC in parallel is equal to zero, which is equivalent to an open. As the frequency increases, the magnitude of an inductive reactance increases, while the magnitude of a capacitive reactance decreases. A circuit is said to be inductive if the total reactance is positive, and a circuit is said to be capacitive if the total reactance is negative.

A bandpass, RLC circuit, will have two cut-off frequencies and where the amplitude

is √

of the maximum value. The cut-off frequency is also called the half-power frequency or 3-

dB frequency in some cases.

The bandwidth BW (or passband bandwidth) is defined as the difference between the upper and lower cutoff frequencies. In case of a low-pass filter or baseband signal, the bandwidth is equal to its upper cutoff frequency.

- 53 -

Examples of the resonance frequency, cutoff frequencies and the bandwidth are shown in Figure 7 - 1 for a bandpass and a bandreject filter

The quality factor or Q factor of the frequency response is described quantitatively in terms of the ratio of the resonance frequency to the bandwidth,

with both and are in radians. This definition lends itself to laboratory measurement because it is possible to measure both the resonance frequency ( ) and the bandwidth ( ). It is also defined as an energy ratio,

2

Figure 7 - 1 Resonance frequency, cutoff frequency and bandwidth

- 54 -

The steady response of a circuit will in general have a maximum amplitude and phase angle that is different from that of the source. In some cases, the magnitude of the voltage response may exceed that of the voltage source.

PREPARATION

A. RLCcircuitbasicmeasurement

For circuits (a) through (d) in Figure 7 - 2, use C = 0.1 uF, L = 100 mH, R = 1 kΩ. 1. Derive the transfer function. 2. Find the resonance frequency, cutoff frequencies, bandwidth and Q factor for each

circuit. 3. What is the phase relation between the total voltage and current, is it leading or

lagging when the frequency is (i) below resonance, and (ii) above resonance? What is the nature of the circuit in those two regions, ie, is it capacitive or inductive?

B. RLCcircuitatresonance

For circuit (e) in Figure 7 - 2, use C = 0.1 uF, L = 100 mH. 1. At resonance frequency, calculate VC if:

a. R = 3 kΩ b. R = 300 Ω

2. Find out whether the magnitude of VC is larger or smaller than Vin. Explain your result.

SIMULATION

1. Build and simulate the circuits (a) through (d) in Figure 7 - 2 using Multisim. Find the resonance frequency, half-power frequencies bandwidth and Q factor for each circuit. Compare the result with that from PREPARATION.

2. Build and simulate the circuit (e) in Figure 7 - 2. Find the magnitude for VC for the following cases.

a. R = 3 kΩ b. R = 300 Ω

Compare the result with that from PREPARATION.

EXPERIMENT

1. On the breadboard, build circuit (a) in Figure 7 - 2. Connect Ch1 to input and Ch2 to output so that both the input and the output are displayed on the oscilloscope.

- 55 -

2. Set the input voltage to 5 and frequency to the value calculated in PREPARATION. For circuits (a) and (c), vary the frequency from the theoretical value to get a maximum output voltage. The frequency at this maximum voltage Vmax is the resonance frequency. For circuits (b) and (d), vary the frequency from the theoretical value to get a minimum output voltage. The frequency at this minimum voltage Vmin is the resonance frequency.

3. If /√2, vary the frequency again until the output voltage equals VCutoff.

This frequency is the cut-off frequency. There should be 2 cut-off frequencies for each case.

4. Calculate the bandwidth by subtracting the 2 cut-off frequencies. 5. Calculate the Q factor using the equation from PREPARATION. 6. Repeat step 1 through 5 for circuits (b) to (d) in Figure 7 - 2. 7. Compare your result with that from PREPARATION and SIMULATION. 8. Build circuit (e) in Figure 7 - 2. Connect Ch1 to input and Ch2 to output so that both the

input and the output are displayed on the oscilloscope. Set the input voltage to 5 and frequency to the resonance frequency found in step 1 for circuit (b) in Figure 7 - 2.

9. Find the magnitude for VC for the following cases. a. R = 3 kΩ, b. R = 300 Ω.

Compare the result with that from PREPARATION and SIMULATION. Explain what you observe.

- 56 -


(a) (b)

(c) (d)

(e)

- 57 -

REPORT


- 58 -

EXPERIMENT#8 TRANSFERFUNCTIONS

OBJECTIVES

To study the transfer function of a circuit. To use the transfer function to find the specified frequency specified in the different

cases.

EQUIPMENT


BACKGROUND

The transfer function of a circuit is defined in the s domain as the ratio of the output (response) Y(s) of the circuit to an excitation X(s). All the initial conditions in the circuit are set to zero while computing the transfer function. The transfer function, denoted by H(s), is then expressed as

The circuit output (response) can be written as

which says that the Laplace transform of the output (response) is equal to the product of the transfer function and Laplace transform of the input function.

The transfer function is a complex quantity with a magnitude and phase that are functions of frequency. A plot of the magnitude and the phase of a transfer function is shown in Figure 8 - 1.

Keep in mind that the transfer function applies to a single source. If more than one source exists in the circuit, a transfer function for each source needs to be determined defined first and the total response can be found using superposition. Also, it is important to note that a

- 59 -

single circuit can have many transfer functions depending upon the output (response) of interest.

PREPARATION

For all circuits, R = 1kΩ, C = 0.1 μF.

Figure 8 - 1 Frequency response plot

- 60 -

A. Single‐stageRCcircuit

For circuits (a) and (b) in Figure 8 - 2, 1. Derive the transfer function. 2. Find the frequency where the output voltage is 45o out of phase with the input

voltage. Find the amplitude of the output voltage at this frequency.

3. Find the frequency where the amplitude of the output voltage is √

that of the input

voltage. Find the phase difference between the input and output voltages. 4. Are the frequencies found in step 2 and 3 the same? Please explain.

B. Two‐stageRCladdernetwork

For circuits (c) and (d) in Figure 8 - 2, 1. Derive the transfer function. 2. Find the frequency where the output voltage is 90o out of phase with the input

voltage. Find the amplitude of the output voltage at this frequency.

3. Find the frequency where the amplitude of the output voltage is that of the input

voltage. Find the phase difference between the input and output voltages. 4. Are the frequencies found in step 2 and 3 the same? Please explain.

- 61 -


(a) (b)

(c)

(d)

V C

R

V V V

C

R

V V

C

R

C

R

V VC

R

C

R

- 62 -

SIMULATION

In Multisim, build and simulate the circuits (a) through (d) in Figure 8 - 2.

1. For circuits (a) and (b), find the frequency where the output voltage is 45o out of phase with the input voltage and find the amplitude of the output voltage at this frequency.

2. Find the frequency where the amplitude of the output voltage is √

that of the input

voltage and find the phase difference between the input and output voltages. 3. For circuits (c) and (d), find the frequency where the output voltage is 90o out of phase

with the input voltage. Find the amplitude of the output voltage at this frequency.

4. Find the frequency where the amplitude of the output voltage is that of the input

voltage and find the phase difference between the input and output voltages.

EXPERIMENT

1. On breadboard, build circuits (a) through (d) in Figure 8 - 2. Connect Ch1 to input and Ch2 to output so that both the input and the output are displayed on the oscilloscope.

2. Set the input voltage to 5 and frequency to that calculated in the corresponding step in PREPARATION. Measure the phase difference between the input and output. Adjust the frequency of the input signal appropriately so that the phase difference equals the values given below and measure the output voltage at that frequency for each case.

a. 45o for circuit (a), b. 45o for circuit (b), c. 90o for circuit (c), d. 90o for circuit (d).

The way to measure the phase difference is as follows. First measure the time difference ∆ between the two peaks of and . Then use the equation ∅ 2∆ (rad) to calculate ∅. Pay attention to the unit. Convert ∅ to degrees if needed.

3. Set the input voltage frequency to the one calculated in the corresponding step in PREPARATION. Measure the output voltage. Adjust the frequency of the input signal appropriately so that the output voltage equals the values given below and measure the phase different between the input and output at this frequency for each case.

a. √

that of the input voltage for circuit (a),

b. √

that of the input voltage for circuit (b),

c. that of the input voltage for circuit (c),

d. that of the input voltage for circuit (d).

- 63 -

4. Compare the results in steps 2 through Error! Reference source not found. with those from PREPARATION and SIMULATION.

REPORT


- 64 -

EXPERIMENT#9 FREQUENCYRESPONSE

OBJECTIVES

To study the frequency response of RC circuits. To determine the frequency response using time domain time domain measurements. To study different types of filters (low pass, high pass, band pass and band reject).

EQUIPMENT


BACKGROUND

The frequency response of a circuit is a measure of the output in comparison to the input, as a function of frequency. The function used to characterize this is the transfer function, with its It magnitude or gain, typically expressed in dB, and the phase shift, expressed in radians or degrees. The frequency response is important in the analysis and design of filters, tuners, amplifiers, etc.

A filter is a network designed to pass signals with a specific frequency range (passband) and reject or attenuate signals whose frequencies lie outside of this passband. The most common filters are low pass filters, Figure 9 - 1 (a), which pass low frequencies and reject high frequencies, high pass filters, Figure 9 - 1 (b), which pass low frequencies and reject high frequencies, bandpass filters Figure 9 - 1 (c), which pass a select band of frequencies and reject those outside this range and band reject filters, Figure 9 - 1 (d), which reject a specific band of frequencies and pass all other frequencies.

- 65 -

Figure 9 - 1 Basic types of filters. (a) low pass filter, (b) high pass filter, (c) band pass filter, and (d) band reject filter.

(a) (b)

(c) (d)

- 66 -

PREPARATION

For circuits (a) and (b) in Figure 9 - 2, R = 1 kΩ, C = 0.1 μF. For circuit (c) in Figure 9 - 2, R1 = 1 kΩ, R2 = 10 kΩ, C1 = 0.01 μF, C2 = 0.0056 μF.

1. Derive the transfer function. 2. Plot or sketch the magnitude vs. frequency and the phase vs. frequency curves in a

linear or log scale. 3. Indicate if the circuit is a low pass, high pass, band pass or band reject filter.

SIMULATION

1. In Multisim, build and simulate the circuits (a) through (c) in Figure 9 - 2. Find the frequency response for each circuit. This can be accomplished indirectly. Set the input voltage magnitude to 5 V (pk-to-pk of 10 V). Vary the frequency of the input voltage source from 100 Hz to 100 kHz, record the output voltage magnitude and the phase difference between the input and the output voltages at each frequency. At least 10 or more points are needed in order to completely describe the frequency response. Plot the voltage vs. frequency curve and the phase vs. frequency curve in either linear or log scale. Compare the results of circuit (a) and (b) with those from PREPARATION.

2. Specify whether it’s a low pass, high pass or band pass filter for the frequency response curves obtained above.

- 67 -

(a) (b)

(c)


- 68 -

EXPERIMENT

1. On breadboard, build circuit (a) in Figure 9 - 2. Connect Ch1 to input and Ch2 to output so that both the input and the output are displayed on the oscilloscope.

2. Set the input voltage to 5 . Vary the frequency from 100 Hz to 100 kHz. Refer to SIMULATION for frequency response measurement. Plot the voltage vs. frequency curve and the phase vs. frequency curve in either linear or log scale by selecting at least ten frequency values and determining the amplitude and phase from the oscilloscope.

3. Compare the result with that from PREPARATION and/or SIMULATION. What kind of filter is it?

4. Repeat step 1 to 3 for circuit (b) and (c).

REPORT


- 69 -

APPENDIXI. STANDARDRESISTORCOLORCODE

The color code for the resistor value utilizes two digits and a multiplier digit in that order, as shown in Figure A - 1. A fourth band designates the tolerance.

The resistance of a resistor with the four bands of color may be written as

10

Where A and B are the values of the first and second bands, respectively, C is a multiplier and D is the tolerance. The color code together with multipliers and tolerances are listed in Table A - 1.

Table A - 1 Electronic Color Code

Color Significant

figures Multiplier Tolerance

Black 0 ×100 -- Brown 1 ×101 -- Red 2 ×102 ±2%

Orange 3 ×103 --

Figure A - 1 Resistor with four color bands.

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Yellow 4 ×104 -- Green 5 ×105 -- Blue 6 ×106 -- Violet 7 ×107 -- Gray 8 ×108 -- White 9 -- -- Gold -- ×10-1 ±5% Silver -- ×10-2 ±10% None -- -- ±20%

For example, if a resistor has four color bands, red-violet-green-gold, as shown in Figure A - 1, then

2 10 7 10 5% 2.7 10 5%

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APPENDIXII. LISTOFAVAILABLERESISTORSANDCAPACITORS

Availableresistors:

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Availablecapacitors:

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APPENDIXIII. GUIDELINESTOWRITEAFINALLABORATORYREPORT

Lab reports are an essential part of all laboratory courses. The goal of lab reports is to document your findings and communicate their significance. The ability to report technical information in a clear and concise manner is one of the most important practical skills that a technically trained person can develop.

A good lab report does more than present data. It demonstrates the writer’s comprehension of the concepts behind the data. Simply recording the expected and observed results is not sufficient. You should also identify how and why differences occurred, explain how they affect your experiment and show your understanding of the principles the experiment was designed to examine. Even though following a format is helpful, it cannot replace clear thinking, organized writing and proper usage of engineering language.

The way an electrical networks lab report is written may be different from other subjects, such as chemistry and biology. However, it should include at least the following sections: objectives, equipment, simulations, experiments and conclusions.

Here are a few tips that could be used when writing up a lab report.

The title of the lab should be straightforward and informative. For example, instead of using ‘Lab #3’, write ‘Lab #3 Network Analysis Methods’.

Since each circuit can be considered as one independent experiment, the procedure, data, analysis and conclusion for that particular circuit should be placed in the same area. Once everything about this circuit is finished, move on to the next one.

In most cases, your experimental data need to be compared with either calculated data or simulated data, or both. In the case when the two don’t match up, an explanation is needed.

Numerical data should always be presented in a table with proper label and unit. Graphics need to be clear and well labeled as well.

The conclusion section for the entire lab is general conclusion of what you learned, explanation of the results obtained, your understanding of the whole lab, etc.

A printed version is always preferred.

Attached is a format example for a lab report. It may be used as a reference, and doesn’t need to be followed strictly.

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Lab # 1 DC MEASUREMENT [Title] OBJECTIVE

[Objective list]

To understand the basics of DC (direct current) circuits. To use a digital multimeter (DMM) to measure DC voltage, current and resistance. To verify the valid measurement condition for a digital multimeter.

EQUIPMENT

[Equipment list]

Breadboard DC Power supply Digital multimeter

SIMULATION

1. Circuit 1 is simulated in Multisim. [Describe what is beingsimulated.]

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Table 1

[Numerical data]

I (mA) V1 (V) V2 (V)

2. …… [Second circuit (if any), brief description]

[Screen shot of the circuit simulated in Multisim]

Figure 1

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Table 2

[Numerical data]

EXPERIMENT

1. Measure the actual values of all resistors, and compare them with nominal values. [A brief description for this step.]

Table 3

[Numerical data] Nominal value Measured value

R1 (Ω) R2 (Ω) R3 (Ω)

[Screen shot of the circuit simulated in Multisim]

Figure 2

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2. Connect the circuit in Figure 3 and measure V1, V2 and I. Compare measured results with calculated and simulated ones. [A brief description for this step.]

[Circuit is displayed in Figure 3, data is recorded in Table 4, and data comparison is done in Table 5. Verify rules and laws if any.]

Table 4

[Numerical data] Measured result

I (mA) V1 (V) V2 (V)

Table 5

[ Compare Data ] Calculated result Simulated result Measured result

I (mA) V1 (V) V2 (V)

[Circuit for this step]

Figure 3

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Verify Ohm’s Law:

Verify Voltage Divider Rule:

Verify KVL:

…… [Conclude, comment on what you have observed.]

3. …… [Same as step 1 or 2]

CONCLUSIONS

[Provide all the conclusions you have arrived at from the calculated, simulated and measured results.]

eel 3123 networks and systems

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