ECE 231 Elements of Electrical Engineering
Laboratory Manual
Prepared by R. Frank Smith
California State Polytechnic University, Pomona
Reference Text – Student Reference Manual for Instrumentation Laboratories, Wolf and Smith, Prentice Hall, 2004
Revised 09/01/12
Table of Contents
Exercise 1 - Ohm's Law
1
Exercise 2 - Kirchhoff's Laws
7
Exercise 3 - Oscilloscope/Function Generator Operation
11
Exercise 4 - Thévenin's and Norton Theorems
16
Exercise 5A - Diode Characteristics
22
Exercise 5B - Diode Characteristics
28
Exercise 5C - Diode Characteristics
32
Exercise 6 – Frequency/Time Response of RL and RC Circuits
35
Exercise 7 - Resonant Circuits
42
Exercise 8 – Time Domain Response of 2nd Order Circuits
47
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page i
ECE 231 Laboratory Exercise 1
Ohm’s Law
ECE 231 Laboratory Exercise 1 – Ohm’s Law
Laboratory Group (Names) _______________ ______________ _______________
OBJECTIVES

Verify Ohm’s Law

Learn to read resistor color codes

Learn to use ohmmeter, voltmeter, and ammeter

Learn to calculate power loss in resistors
EQUIPMENT REQUIRED





ECE 231 Circuit Board (In Stock room)
One banana cable
One lot of clip leads (students must supply their own clip leads)
DMM (digital multimeter)
DC power supply
BACKGROUND
Resistors are used for many purposes such as electric heaters, voltage, and current dividing elements, and
current-limiting devices. As such, their resistance values and tolerances vary widely. Resistance tolerances may
range from +0.001 to +20%. The most common types of resistors are carbon composition, wire wound, metal
film, carbon film, steel, and liquid. Their ratings can range from microwatts to megawatts. Variable resistors are
called either potentiometers or rheostats. When used as a potentiometer their output is a variable voltage.
When used as a rheostat they are used to control current.
A good reference source is
http://en.wikipedia.org/wiki/Electronic_color_code. Review this website before you come to the laboratory.
Many types of resistors do not have a color code such as resistors made to military specifications and surface
mount resistors. You might remember the following mnemonic to remember the color versus number code:
Bad (0) Boys (1) Race (2) Our (3) Young (4) Girls (5) But (6) Violet(7) Generally (8) Wins (9).
Black
Brown
Red
Orange
Yellow
Green
Blue
Violet
Grey
White
Most resistors use either 4 or 5 bands of colors. The 5 band color is usually used for 1% and 0.1 % resistors.
This band represents 5% if gold, 1% if brown, and fire resistant if yellow.
When you observe a resistor it is not always possible to predict its wattage by just observing its size. There are
many variables that affect a resistor’s wattage. Some such parameters are size, mounting, encapsulation, and
cooling. There are three ways you can calculate the power being dissipated in a resistor in this laboratory. See
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 1
ECE 231 Laboratory Exercise 1
Ohm’s Law
Eq.1. In a thermodynamics' laboratory you could measure the rise in temperature of water in a calorimeter to
determine the power being dissipated by a resistor. Consider the following design problem. What size (ohms
and wattage) resistor would you use for the heating element in a coffee maker or toaster? Assume 120 VAC and
300 watts.
𝑃=
𝑉2
𝑅
= 𝐼 2 𝑅 = 𝑉∆𝑅 𝐼𝑅
(1)
The resistance of a resistor can be approximated by equation (2):
Resistance (R)=
𝜌𝐿
𝐴
(2)
Where 𝜌 =resistivity of the material; L = length of material; and A is the area of the material. The material may
be solid, liquid, or gaseous. Each of these parameters is often functions of temperature and stress. Liquid is
often used for low resistances rated in the megawatts.
Part 1. There are 7 resistors and one potentiometer on the BOARD. Determine and record the values of the 7
resistors and the potentiometer and their associated color code if appropriate. See your text or the internet for
the color code. Measure each resistor with an ohmmeter then see how that relates to the color code. We will
assume the color code is the Theoretical Value. See Figure 1.
Figure 1. Experimental board for ECE 231 experiments.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 2
ECE 231 Laboratory Exercise 1
Ohm’s Law
Table 1. Resistor color codes
Measured
Value
Color Code
Theoretical
Value (Color Code)
% error
Experimental
Discrepancy
Part 2. Connect a variable voltage supply to three different resistors and vary the voltage from 0 to 10 volts. See
Figure 2. If the overload light is illuminated you may have tripped the overload protective device. Press the red
reset button to reset the overload device.
Figure 2. Variable voltage supply. Use cable with banana plug. Notch side goes to black.
Plot the current versus the voltage in Figure 4 for each resistor. Label each curve with its resistance value.
There is both a Fluke and Beckman multimeter that can be used to measure the current. See Figure 3. How does
the plot verify Ohm’s Law? What can you say about the slope of the plots? Calculate the slopes and show that
they are equal to 1/R.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 3
ECE 231 Laboratory Exercise 1
Ohm’s Law
Hint: All of the curves go through zero so only one additional point for each resistor is required to generate the
Ohm’s Law curve. Simply set the voltage supply at one voltage (for example 10 volts) for all the resistors and
then measure the current in each resistor. Verify the current using Ohm’s Law.
Figure 3. Laboratory bench equipment.
𝟏
∆𝑰
Figure 4. Plot for verifying Ohm's Law, (𝑹 = ∆𝑽)
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 4
ECE 231 Laboratory Exercise 1
Ohm’s Law
How does the plot verify Ohm’s Law? What can you say about the slope of the plots? Calculate the slopes and
show that they are equal to 1/R.
What are the possible ways to measure the current through a resistor? There are several ways. How do you
calculate the current through a resistor under test without using an ammeter? For a circuit board with surface
mounted resistors you would usually use the calculation method. Calculation of the measure of uncertainty for
each method is different. A good reference source for error analysis is the Reference text or
http://www.lhup.edu/~dsimanek/errors.htm.
Part 3. Connect a small resistor (less than 100 ohms) to the variable power supply. Gradually increase the
voltage and feel, using your finger, the increase in the temperature of the resistor. Only increase the voltage so
that the wattage lost in the resistor is less than 1/2 watt. What voltage created a ¼ watt loss? At what wattage
does the resistor get too hot to touch? Comment on how hot the resistor gets when it is dissipating 1/4, 1/3, and
1/2 watt. Hint: Power = V2/R. Resistors are available on the 5th floor in the student work area and stock room.
CAUTION
Going beyond ¼ watt can cause the resistor to explode or ignite. A 100 ohm resistor will dissipate ¼ watt at 5
volts. You will usually see smoke or fire at ½ watt. Do NOT exceed 7 volts for a 100 ohm resistor.
Table 2. Wattage versus resistor temperature
Measured
Resistance
Voltage
Test Value
Wattage
Temperature
Check appropriate box
Ambient Warm
Hot
Comments
1/4
1/3
1/2
Voltage
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 5
ECE 231 Laboratory Exercise 1
Ohm’s Law
Part 4. Write a professional comprehensive laboratory report using a word processor. Show your
results, calculations, error analysis, and include a comprehensive conclusion. There are lots of sample
lab reports on the internet. Every figure must be sequentially numbered and referenced in the
preceding text. Your calculations may be handwritten and attached to the report if properly
referenced in the text. Number all pages.
On the cover page of your laboratory report include the number and tile of the experiment, date
performed, and laboratory partners.
Conclusion or comments.
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
___________________________________________________________________________________________
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 6
ECE 231 Laboratory Exercise 2
Oscilloscope/Function Generator Operation
ECE 231 Laboratory Exercise 2 – Kirchhoff’s Laws
Laboratory Group (Names) _______________ ______________ _______________
OBJECTIVE




Verify Kirchhoff’s voltage law
Verify Kirchhoff’s current law
Gain experience in using both an ammeter and voltmeter
Construct two (2) circuits as shown in Figure 1 and Figure 2.
Ammeter
B
A
Ammeter
R1
C
10Vdc
V1
10Vdc
V1
R1
R2
I
1
R2
I
2
R3
I
3
D
R3
0
Parallel Circuit
E
Series Circuit
0
Figure 1. Schematics for verifying Kirchhoff's Laws
EQUIPMENT REQUIRED





ECE 231 Circuit Board (In Stock room)
Two banana cables (one for DC power supply and one for DMM)
One lot of clip leads (students must supply their own clip leads)
DMM (digital multimeter)
DC power supply
BACKGROUND
Gustav Kirchhoff first described his laws in 1845. His first law KCL simply stated is that current into a
node must equal the current leaving a node where a node is the point where two or more components
are connected together. In Figure 1 above, the three currents I1, I2, and I3 leave the top node and go
through the three resistors and then merge on the ground circuit. The voltage across any parallel
resistors is always the same. Current through any resistor can be determined by using Ohm's law.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 7
ECE 231 Laboratory Exercise 2
Oscilloscope/Function Generator Operation
𝐼=
𝑉 𝑉𝑜𝑙𝑡𝑎𝑔𝑒 𝑎𝑐𝑟𝑜𝑠𝑠 𝑎 𝑟𝑒𝑠𝑖𝑠𝑡𝑜𝑟
=
𝑅
𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒
or by measuring the current through each resistor using an ammeter.
Kirchhoff's Current Law (KCL) ∑𝒏𝒌=𝟏 𝑰𝒌 = 𝟎 where is the n is the number of branches at a node.
Kirchhoff's Voltage Law (KVL)
and voltage sources) in a loop.
∑𝒏𝒌=𝟏 𝑽𝒌 = 𝟎 where is the n is the number of components (resistors
Kirchhoff's second law is like going on a hike from your car around a mountain (independent of path).
When you get back to your car, your net change in potential energy is zero. No matter how you
measure voltages around a circuit, when you return to your starting point the change in voltage is zero.
Figure 2. Protoboard connection for the series circuit
PROCEDURE
Part 1
Select three adjacent resistors and connect them in SERIES with your power supply. Now measure the
voltage at each node (A thru E) in your circuit.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 8
ECE 231 Laboratory Exercise 2
Oscilloscope/Function Generator Operation
1. Measure the voltages at all of the nodes relative to the power supply ground. Show that the
sum of the voltages ACROSS all of the components in a loop complies with Kirchhoff’s voltage
law. Complete Table 1.
Table 1. Voltages across each component and current through each resistor
VEA
Power
VAB
Power
VBC
VCD
VDE
Calculated IR1=
Calculated IR2=
Calculated IR3=
Measured IR1=
Measured IR2=
Measured IR3=
Power
Power
Power
2. Now measure the node voltages relative to node C. For example, Vca = -Vac which says that
the voltage from c to a = minus the voltage from a to c. The voltages at the nodes relative to
ground will not add to zero to prove Kirchhoff’s voltage law. It is the sum of the voltages across
each component in a series that add to zero NOT the sum of the node voltages. Remember, the
reference node in a circuit can be anywhere you want in a real circuit.
Table 2. Node voltages relative to node C (i.e. C is connect to the black meter lead)
VCA
VCB
VCC
VCD
VCE
0
NOTE
Remember, the reference node in a circuit can be anywhere you want in a real
circuit; therefore the voltage at a node will most likely change depending upon your
reference.
3. Calculate the power delivered by the power supply. Show that it is equal to the power
consumed by the resistors. Enter the power into Table 1.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 9
ECE 231 Laboratory Exercise 2
Oscilloscope/Function Generator Operation
CAUTION
Never connect an ammeter in parallel with the component you are trying to measure the current
through. The ammeter is in essence a short circuit and must be in series with the components through
which current is being measured. An error in the connection could seriously damage the ammeter and
other circuit components.
Part 2.
Select three resistors and connect them in PARALLEL with your power supply. Now measure the
current from the power supply. This procedure is NOT shown in Figure 2. It is up to you to figure out
the connection scheme since only the power supply ammeter connection is shown in Figure 1. You
only have one ammeter. Therefore, rewire each branch circuit with the ammeter in SERIES with the
branch circuit resistor. Verify the ammeter reading using the calculation method and a voltmeter.
1. Measure the source current and the branch currents I1, I2, and I3. Show that the currents
comply with Kirchhoff’s current law. If you read any negative currents with your ammeter,
what did you do wrong?
2. Calculate the power delivered by the power supply. Show that it is equal to the power
consumed by the three resistors.
Table 3. Currents in the parallel circuit of Figure 1.
I Source= I1+I2+I3
I1
I2
I3
Measured =
Measured=
Measured=
Measured=
Calculated=
Calculated=
Calculated=
Power=
Power=
Power=
Power=
Write a professional comprehensive lab report using a word processor. Show your results and
include a comprehensive conclusion. There are lots of sample lab reports on the internet.
Conclusion
_______________________________________________________________________
________________________________________________________________________
_______________________________________________________________________
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 10
ECE 231 Laboratory Exercise 3
Oscilloscope/Function Generator Operation
ECE 231 Laboratory Exercise 3 – Oscilloscope/Function Generator Operation
Laboratory Group (Names) _______________ ______________ _______________
OBJECTIVES




Gain experience in using an oscilloscope to measure time varying signals.
Gain experience in using a signal generator to create time varying test signals.
Gain experience in properly using an oscilloscope’s controls and soft keys.
Learn the frequency limitations of instruments.
EQUIPMENT REQUIRED







One banana cable
Three BNC cables
One lot of clip leads and/or jumper wires
DMM (digital multimeter)
Use the DC offset in signal generator for the DC power supply
Signal generator
BACKGROUND
The oscilloscope is primarily a voltmeter for observing time varying signals. It has a fairly low input
impedance of one megohm (1M ) so it cannot be used when a load impedance of this size would
distort the signal being measured. It is an excellent tool for measuring transient phenomenon such as
impact forces on a load cell. Modern oscilloscopes can operate in both a digital mode and analog
mode. They also have built-in computers for doing signal analysis such as Fourier transforms on the
incoming signal. This type of measurement and analysis would be very useful in measuring impact
response of a suspension system.
It is important that you do not indiscriminately turn the controls especially if you have not been
instructed in their use and function. This can prevent the oscilloscope from being able to properly
display an incoming signal.
An ideal meter will not disturb the circuit when taking measurements. Multimeters and oscilloscopes
are not ideal instruments. You can determine the root-mean-square (rms) value of a sine wave
displayed on an oscilloscope by the following equation: 𝑉𝑟𝑚𝑠 =
R. Frank Smith, California State Polytechnic University, Pomona, 2012
𝑉𝑝−𝑝 √2
2
2
= 0.3535𝑣𝑝−𝑝 = 0.707𝑣𝑝
Page 11
ECE 231 Laboratory Exercise 3
Oscilloscope/Function Generator Operation
If you are using one of the new digital oscilloscopes, you can read waveform parameters on the lower
menu which displays Vrms, Vp-p, and frequency. and phase The voltage from a household outlet is 120
VAC. This is the rms value. The peak value is 1.414 *120= 169.7 voltages. The heating value of 120
VAC rms is exactly equal to a 120 VDC voltage source such as a photovoltaic panel.
PROCEDURE
Part 1
1. Connect channel 1 of the oscilloscope to the signal generator and to the digital multimeter (set
to voltage). See Figure 1. Make sure that the ground on the oscilloscope and signal generator
are connected together. Both are internally grounded to the building ground system.
2. Set the signal generator to 1 KHz, 5 V pk-to-pk for each of the following waveforms: sine wave,
triangle wave, and square wave. Increase the frequency to 10 kHz, and then 100 kHz. Connect
a BNC cable to both the signal generator and the oscilloscope channel 1 (two cables required).
Connect the red clip leads together. Plug your banana cable into the multimeter then connect
the red clip to the red clip leads going to channel 1 and the signal generator. See Figure 1. The
instruments are internally connected to the black lead so you shouldn’t have to do anything
with the black lead. You can connect them all together if you want. The black lead should be at
earth ground potential. Make sure the trigger is set to channel 1.
3. Plot what you see on the oscilloscope screen. in Figure 2. You can copy the signal seen on the
oscilloscope and paste it into your lab report so that you don’t have to draw it by hand.
4. Compare the readings on the multimeter with what you see on the oscilloscope. Place the
results in Table 1. Add dc offset to your input signal and describe what happens on the
oscilloscope. Try to read just the offset using the multimeters and the oscilloscope. Change the
oscilloscope Vertical Mode from GND, to AC, and then to DC. On the dc setting you see both the
dc and ac signal. In the ac setting you only see the ac waveform. Describe what happens to
the waveform displayed on the oscilloscope with and without DC offset. The multimeter should
not be able to read the voltage as accurately as the oscilloscope. Record your readings in Table
1. The oscilloscope will automatically display the signal’s voltage value and frequency
automatically. Use the soft keys to select voltage and time measurements.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 12
ECE 231 Laboratory Exercise 3
Oscilloscope/Function Generator Operation
Figure 1. Test Setup
Time (sec.,msec., sec.)
Figure 2. Oscilloscope Display
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 13
ECE 231 Laboratory Exercise 3
Oscilloscope/Function Generator Operation
Table 1. Measured and calculated results
Waveform
Oscilloscope
reading Vp-p
Multimeter
reading
Frequency
Sine wave
Sine wave +5dc
Triangular
Square
1 k Hz
1 k HZ
1 kHz
1 kHz
Sine wave
Sine wave +5dc
Triangular
Square
10 kHz
10 kHz
10 kHz
10 kHz
Sine wave
Sine wave +5dc
Triangular
Square
100 kHz
100 kHz
100 kHz
100 kHz
Calculated RMS
voltage
5.
Now slowly increase the frequency of the function generator until the multimeter has an error of at
least 10%. The voltmeter reading will be less than the oscilloscope reading.
6. What is the frequency limitation of the multimeter. ______________ Hertz. Sine wave
7. What is the frequency limitation of the multimeter. ______________ Hertz. Square wave
8. What is the frequency limitation of the multimeter. ______________ Hertz. Saw tooth wave
2
Notes: If the amount of heat (joules) generated by a DC source (𝑖𝑑𝑐
𝑅𝑇) is equal to the heat generated
𝑇
by an ac source over the same period T ,(∫0 𝑅 ∗ 𝑖 2 𝑑𝑡). Equating the energies and solving results in
1
𝑇
𝐼𝐷𝐶 = 𝑖𝑟𝑚𝑠 = √𝑇 ∫0 𝑖𝑡2 𝑡 𝑑𝑡 . The following are 𝑉𝑟𝑚𝑠 equations for common waveforms:
Sine wave 𝑉𝑟𝑚𝑠 =
𝑉𝑝−𝑝 √2
2
2
; square wave 𝑉 𝑟𝑚𝑠 = 𝑉𝑝 ; triangle wave 𝑉𝑟𝑚𝑠 =
R. Frank Smith, California State Polytechnic University, Pomona, 2012
𝑉𝑝−𝑝
2√3
Page 14
ECE 231 Laboratory Exercise 3
Oscilloscope/Function Generator Operation
1
𝑝𝑒𝑟𝑖𝑜𝑑 𝑇 𝑜𝑓 𝑜𝑛𝑒 𝑐𝑦𝑐𝑙𝑒
= 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑤𝑎𝑣𝑒𝑓𝑜𝑟𝑚 (Hz) =
𝜔 𝑟𝑎𝑑𝑖𝑎𝑛𝑠/𝑠𝑒𝑐𝑜𝑛𝑑
2𝜋
9. Describe how you measure the frequency of a waveform from the oscilloscope display if you
didn’t have soft keys to measure it automatically
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
10. Why does the multimeter reading decrease as the frequency increases?
_____________________________________________________________________________
_____________________________________________________________________________
_____________________________________________________________________________
Hint: See Exercise 6. The input circuit topology to many analog voltmeters is usually a low pass filter.
Write a professional comprehensive lab report using a word processor. Show your results and include
a comprehensive conclusion. There are lots of sample lab reports on the internet.
Conclusion
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 15
ECE 231 Laboratory Exercise 4
Thévenin and Norton Theorems
ECE 231 Laboratory Exercise 4 – Thévenin and Norton Theorems
Laboratory Group (Names) _______________ ______________ _______________
OBJECTIVES


Learn various ways to measure Thévenin's voltage and resistance.
Validate the maximum power theorem.
EQUIPMENT REQUIRED







ECE 231 Circuit Board (In Stock room)
Two banana cables (one for DC power supply and one for DMM)
One lot of clip leads and/or jumper wires
DMM (digital multimeter)
One DC power supply
Four or more resistors (available in bins adjacent to stock room, 5th floor). Select various sizes
Potentiometer for variable load – 10K or larger. It should be twice as big as your largest resistor
BACKGROUND
Thévenin's Theorem (1883) states that any linear circuit can be replaced by a single voltage source and a
single series resistance. In 1926 Norton’s Theorem was shown to be equal to Thévenin’s Theorem, see
Figure 1. You might wonder why the 43 year delay between the two theorems. Batteries were easy to
construct and incorporate into a circuit. No one knew how to make a good constant current source.
We do not have current sources available in the lab to verify Norton's theorem, but it can be calculated
using Ohm’s Law. Constructing constant current sources is beyond the scope of this course.
Thévenin's
Resistance
Vout
Vout
Thévenin's
Voltage
Source
=
Norton
Current
Source
Thévenin's
Resistance
Figure 1. Thévenin's and Norton’s equivalent circuits for a Linear Circuit
Procedure for Finding the Thèvenin Equivalent Circuit Mathematically
A. Circuits with independent sources only. No Dependent sources.
Step 1. Find R Thèvenin
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 16
ECE 231 Laboratory Exercise 4
Thévenin and Norton Theorems
1. Deactivate all of the independent sources by shorting all batteries or DC supplies and opening all current
sources.
2. The equivalent resistance between the terminals for which you would like to know the Thèvenin
resistance is found by combining all of the resisters into one equivalent resistance between the
appropriate terminals. These are usually designated “a” and “b.”
3. A load resistor which is equivalent to the Thèvenin resistance will result in maximum power being
dissipated in the load resistor and of ½ the input voltage will be across the load.
Power maximum 
2
V th
4 Rth
Step 2. Find V Thèvenin between terminals “a” and “b.”
1. Use the original circuit and nodal analysis to find the voltage between terminals "a" and "b."
B. Circuits with independent and dependent sources.
Step 1. Find R Thèvenin
1. Deactivate all of the independent sources by shorting all batteries or DC supplies and opening all current
sources.
2. Connect a 1 amp your current source between terminals "a" and "b."
3. Find the voltage between terminals "a" and "b” using nodal analysis. This voltage will be the Thèvenin
resistance by the use of Ohm's law.
Resistance 
Voltage Vab

 Rab  RThevenin
Current
1
Step 2. Find V Thèvenin between terminals “a” and “b.”
1. Use the original circuit and nodal analysis to find the voltage between terminals "a" and "b."
C. Circuits with dependent sources only. No independent sources.
These circuits cannot output any power as such they reduce to a Thèvenin resistance only.
Step 1. Find R Thèvenin
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 17
ECE 231 Laboratory Exercise 4
Thévenin and Norton Theorems
1. Connect a 1 amp your current source between terminals "a" and "b."
2. Find the voltage between terminals "a" and "b.” This voltage will be the Thèvenin resistance by the use
of Ohm's law.
Resistance 
Voltage V ab

 V ab
Current
1
3. Find this voltage using nodal analysis.
Thévenin's and Norton’s Theorems are expressed mathematically by equation 1.
𝑟𝑇ℎ𝑒𝑣𝑒𝑛𝑖𝑛 =
𝑣𝑡ℎ𝑒𝑣𝑒𝑛𝑖𝑛
𝑖𝑁𝑜𝑟𝑡𝑜𝑛
𝑣
= 𝑖 𝑜𝑝𝑒𝑛 𝑐𝑖𝑟𝑐𝑢𝑖𝑡 =
𝑠ℎ𝑜𝑟𝑡 𝑐𝑖𝑟𝑐𝑢𝑖𝑡
𝑣𝑜𝑝𝑒𝑛 𝑐𝑖𝑟𝑐𝑢𝑖𝑡 − 𝑣𝑙𝑜𝑎𝑑𝑒𝑑
𝑣𝑙𝑜𝑎𝑑𝑒𝑑
𝑅𝑙𝑜𝑎𝑑
= ℎ𝑎𝑙𝑓 𝑝𝑜𝑤𝑒𝑟 𝑙𝑜𝑎𝑑
(1)
Measuring Vopen circuit just requires a single voltmeter measurement by definition.
CAUTION
Do not attempt to measure I short circuit by shorting your circuit under test. This can be hazardous to
both you and the circuit, especially when testing industrial power circuits.
Determining the short circuit current is extremely important in the design of power distribution
systems. When you examine the circuit breakers on your home power panel you will notice that the
manufacturer has the Short Circuit capacity prominently displayed on the circuit breaker. It will be
either 5000 A or 10,000 A. For industrial plants it can go as high as 200,000 A. Installing a circuit
breaker with a smaller short circuit rating than that which can be supplied by the utility company can
result in an explosion and fire. The short circuit capacity of a circuit determines the fuse size you use to
protect electronic circuits.
Small current sources are frequently used in many electronic circuits and integrated circuits; however,
they are rarely used in industrial power circuits. They are also commonly used to drive light emitting
diodes (LEDs).
The maximum power theorem states that the maximum power will be delivered to a load when the
load resistance is equal to the Thévenin's resistance. This is the basis for selecting the resistance of a
speaker system for a stereo. This assures that in the design stereo systems that maximize the power
will be delivered from the amplifier to the speakers.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 18
ECE 231 Laboratory Exercise 4
Thévenin and Norton Theorems
PROCEDURE
Part 1
1. Choose three resistors for R1, R2, and R3. Measure their resistance values using the
multimeter. Do not use the color code to determine the resistance value. Choose three
resistors that are reasonably close in value. Do not pick, for example, 10K, 300, and 100 ohms.
The 10 K resistor will make it difficult to get good experimental results. You should realize by
now that the resistor color codes are not an accurate way to determine resistor values.
2. Construct the circuit shown in Figure 1 on the protoboard. Use either a 5 V or 10 V source.
Using a multimeter, measure the voltage between points “a” and “b” with NO LOAD connected.
Record your measurement in column 6. This is Thévenin by definition.
3. Remove the Vdc power source and connect a jumper between “1” and “2.” This is the same as
shorting the supply voltage mathematically. Now measure the resistance between “a” and “b”
using your multimeter. By definition this is RThèvenin. Record this value in Table 1. Column 1.
4. Calculate RThèvenin by combining the series and parallel resistors with the source disabled
(shorted). Record this value in Table 1. Column 2. Now compare your measured value and
calculated values in order to perform an error analysis. Enter this value in column 3.
R1
"1"
R2
"a"
V1
Rload
10 Vdc
R3
"2"
"b"
0
Figure 1. Linear resistor circuit.
5. We are now going to determine RThevenin in another way. Connect a load to your circuit
constructed in step 2. For best results the load resistance should be in the same range as your
estimated RThevenin.
6. Now measure the output voltage between “a” and “b” in order to make the appropriate
calculation. Divide this voltage by Rload. This will be the current going through the Thévenin
equivalent circuit.
7. Simply apply Ohm’s Law to find rThevenin..
𝑟𝑇ℎ𝑒𝑣𝑒𝑛𝑖𝑛 =
𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑎𝑐𝑟𝑜𝑠𝑠 𝑟𝑇ℎ𝑒𝑣𝑒𝑛𝑖𝑛
𝑖𝑟𝑇ℎ𝑒𝑣𝑒𝑛𝑖𝑛
=
𝑣𝑜𝑝𝑒𝑛 𝑐𝑖𝑟𝑐𝑢𝑖𝑡 − 𝑣𝑙𝑜𝑎𝑑𝑒𝑑
𝑣𝑙𝑜𝑎𝑑𝑒𝑑
𝑅𝑙𝑜𝑎𝑑
(2)
8. How does this RThevenin compare to the value determined in column 2. Calculate % difference
between columns 2 and 4 then enter this value in Table 1, column 5.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 19
ECE 231 Laboratory Exercise 4
Thévenin and Norton Theorems
Table 1. Measured and calculated data.
1
2
3
4
5
6
7
8
RThevenin
RThevenin
% Error
RThevenin
% Difference
Vab
Vab
% Error
Measured
with sources
removed
(shorted)
Calculated 1
with sources
removed
between
measured
and
calculated
Calculated 2
using Ohm’s
Law and an
Rload
between
calculated 1
and
calculated 2
Measured
Calculated
Thévenin
voltage
measured
and
calculated
Part 2
1. Now construct the circuit shown in Figure 1, but replace Rload with a potentiometer
connected between “a” and “b.” The equivalent circuit is shown in Figure 2. We will now
determine rThèvenin using the potentiometer.
2. Measure the voltage between “a” and “b” as the potentiometer is adjusted.
3. Adjust the potentiometer wiper until the voltmeter reads VThèvenin/2 NOT Vsource/2. The
potentiometer is now set at the maximum power load which is equal to r Thèvenin
4. Calculate the maximum power delivered to the load using equation (3).
5. Measure the value of the potentiometer and determine how close it is to the value of
rThèvenin determined above.
𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑝𝑜𝑤𝑒𝑟 =
𝑉
( 𝑇ℎè𝑣𝑒𝑛𝑖𝑛 )
2
𝑅𝑙𝑜𝑎𝑑
2
=
2
𝑉𝑎𝑏
R Thevenin
V
R Thevenin
V1
V1
Rload
Thevenin
0
(3)
𝑅𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑜𝑚𝑒𝑡𝑒𝑟
V
Thevenin
Rload
Potentiometer
Wiper
0
Figure 2. Maximum power circuit.
6. Now prove that this is the load for maximum power. There are resistor values above and
below the maximum power point resistor that have equal powers since the solution to the
maximum power equation is a quadratic equation. Prove it by measuring the voltage Vab
across the potentiometer after the potentiometer is rotated 1 turn CW. Then measure the
potentiometer resistance at this position. Calculate the power delivered to the
potentiometer using equation (3).
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 20
ECE 231 Laboratory Exercise 4
Thévenin and Norton Theorems
𝑝𝑜𝑤𝑒𝑟 =
(𝑣𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑜𝑚𝑒𝑡𝑒𝑟 ) 2
𝑅𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑜𝑚𝑒𝑡𝑒𝑟
(3)
7. Repeat step 4, but this time rotate the potentiometer 2 turns CCW (1 turn to get back to the
maximum power resistance then one additional turn). Calculate the power delivered to the
potentiometer using equation (3). Compare results.
P1 turn CW= __________
Pmax= _________
This value must be less than Pmax
P2turns CCW= ___________
This value must be less than Pmax
Conclusion
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 21
ECE 231 Laboratory Exercise 5A
Diode Characteristics
ECE 231 Laboratory Exercise 5A – Diode Characteristics
The preferred Exercise is shown in Exercises 5B or 5C.
Laboratory Group (Names) ____________________ ___________________ __________________
OBJECTIVES



Validate the Schottky diode equation.
Calculate the dc and dynamic (ac) resistance of a diode.
Observe the rectifying characteristics of a diode.
EQUIPMENT REQUIRED






ECE 231 Circuit Board (In Stock room)
Two banana cables (one for DC power supply and one for DMM)
One lot of clip leads and/or jumper wires
DMM (digital multimeter)
One DC power supply
One diode

Four or more resistors (available in bins adjacent to the 5th floor stock
BACKGROUND
The Schottky diode equation (1) is a very good approximation of how an actual diode behaves in the
laboratory. The plot of this equation is shown in Figure 1. The experiment will be to investigate the
properties of a diode in quadrant I and III. Most diodes if operated in the breakdown region (far left)
will be destroyed. There are however diodes made to operate in this region, and they are called zener
diodes. The next region (center) is the reverse region. This is the normal region when a diode is
reverse biased. The next region, quadrant I, is the normal forward biased region.
𝐼𝑑𝑖𝑜𝑑𝑒 = 𝐼𝑠 (𝑒 𝑉𝐷/(𝑛𝑉𝑇 ) − 1)
Some definitions
Is is the reverse saturation current and is approximately equal to 10-12A. It is sometimes referred to as
Io or Ir. This current is proportional to the area of the diode. We will not measure this current in this
experiment.
VD = diode voltage, n is approximately 1 to 2. Use 1 for this lab. VT is 26 mV at 300oK.
VD should be in the range of 0.6 to 0.75 V for silicon diodes and 0.3 V for germanium diodes.is usually
taken as 1.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 22
ECE 231 Laboratory Exercise 5A
Diode Characteristics
Some industrial diodes can be several inches in diameter. Common ones used by students are shown
in Figure 2. The ones used on circuit boards with surface mounted components are only about 1 or 2
mm across. See Figure 3 for typical electronic schematic symbols for diodes. Photodiodes may receive
light or output light (Light Emitting Diode – LED). Zener diodes are designed to operate in the
breakdown region. There breakdown voltage can range from several volts to tens of volts.
Figure 1. Plot of diode characteristic equation. Source: Wikimedia Commons
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 23
ECE 231 Laboratory Exercise 5A
Diode Characteristics
Figure 2. An assortment of typical diodes.
+
-
Anode
+
Anode
-
Cathode
Zener diode
Cathode
General purpose diode
Photodiode or LED
Figure 3. Electronic symbols for diodes.
PROCEDURE
Part 1
1. Construct the circuit shown in Figure 4 on the protoboard. Identify three resistors on the
protoboards with values of 100, 1K, and 10K. Measure the resistor values using the
multimeter. Do not use the color code to determine the resistance value. You are going to
construct three circuits using these resistors.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 24
ECE 231 Laboratory Exercise 5A
Diode Characteristics
2. Measure and calculate the voltages, currents , and rac for Table 1.
𝑟𝑎𝑐 =
∆𝑉 𝑉𝑑𝑖𝑜𝑑𝑒 @12𝑉 − 𝑉𝑑𝑖𝑜𝑑𝑒 @ 8 𝑉
=
∆𝑖
𝑖𝑑𝑖𝑜𝑑𝑒 @ 12 𝑉 − 𝑖𝑑𝑖𝑜𝑑𝑒 @ 8 𝑉
D1
Vs
V1
V1
DIODE
R1
10Vdc
Multimeter
(volts)
R1 = 100
R1 = 1000
R1 = 10000
0
Figure 4. Linear resistor network.
Table 1. Diode Data
Vs = 10
Plot data in Figure 5
I = V1/R1
Vdiode =
Vs-V1
R1
100
I diode=
V1/R1
V diode=
Vs-V1
rac
Vs = 8
R1 = 1000
Vs = 12
R1 = 1000
1000
10000
r ac =
3. Plot the diode curve in Figure 5. Alternatively, you can copy the oscilloscope display and
paste it in your lab report when performing experiments 5B or 5C.
4. Connect the circuit shown in Figure 4, but replace the DC sources with a 1 KHz, 5 V sine
wave and replace the multimeter with the oscilloscope. Set R1 = 1000 ohms. Draw your
results in Figure 6.
10
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 25
ECE 231 Laboratory Exercise 5A
Diode Characteristics
9
Current -mA
8
7
6
5
4
3
2
1
0
0
1
2
4
3
5
6
7
Diode Voltage X 10
8
9
10
00
-1
Figure 5. Diode Characteristic Curve
5
4
3
2
Voltage
1
0
-1
-2
-3
-4
-5
0
1
2
3
4
5
6
7
8
9
Time
10
00
Figure 6. ½ Wave rectifier oscilloscope trace
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 26
ECE 231 Laboratory Exercise 5A
Diode Characteristics
1. Write a professional comprehensive lab report using a word processor. Show your results
and include a comprehensive conclusion. There are lots of sample lab reports on the
internet.
Conclusion
_____________________________________________________________________________
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 27
ECE 231 Laboratory Exercise 5B
Diode Characteristics
ECE 231 Laboratory Exercise 5B -Diode Characteristics
OBJECTIVES



Validate the Schottky diode equation.
Calculate the dc and dynamic (ac) resistance of a diode.
Observe the rectifying characteristics of a diode.
EQUIPMENT REQUIRED







ECE 231 Circuit Board (In Stock room)
Three BNC cables
One lot of clip leads and/or jumper wires
DMM (digital multimeter)
One AC power supply
Diode circuit (two diodes and two resistors) supplied by instructor or you can construct your own.
Oscilloscope with XY Mode capability
BACKGROUND
See experiment 5A.
This is an alternate method of measuring a diode characteristic using an oscilloscope. This is a much better
method and gives good results so long as the two diodes are reasonably matched. If you need better results
then use a commercial Curve Tracer. The method described in 5A is very labor intensive and is not
recommended. This method also allows the student to experiment with ½-wave rectification and ripple filtering.
A diode has two types of resistance, dc and ac.
𝑣
𝑅𝑑𝑐 = 𝑖 at a specific point on the curve. This value varies depending on the operating point that you select
but it should be in the neighborhood of 100 ohms. The diode curve may look like a straight line, but it is not. It
is an exponential curve.
𝑅𝑎𝑐 =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑎𝑏𝑜𝑢𝑡 𝑎 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑝𝑜𝑖𝑛𝑡
∆𝑣
=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑎𝑏𝑜𝑢𝑡 𝑎 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑝𝑜𝑖𝑛𝑡
∆𝑖
This is referred to as a diode’s dynamic resistance. It should be less than 10 ohms.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 28
ECE 231 Laboratory Exercise 5B
Diode Characteristics
Figure 1. Oscilloscope in X- Y Mode . Horizontal axis is diode voltage drop and the vertical axis is the diode
current (i=V/750). This analysis is done using National Instrument’s Multisim software.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 29
ECE 231 Laboratory Exercise 5B
Diode Characteristics
Figure 2. Diode used as a halfwave rectifier. Connect the largest capacitor across the resistor and see what
happens. This is called filtering. If you increase the frequency to 100 KHz and you will see that the capacitor
changes the pulses into a DC voltage with a small ripple voltage. Increasing either the frequency or the capacitor
size will reduce the ripple.
Use the original Exercise 5A for a theoretical background. Your lab report should discuss what you did and what
you observed. Diodes act like one way check valves in many electronic circuits. They allow current to go only
one way in a wire.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 30
ECE 231 Laboratory Exercise 5B
Diode Characteristics
Figure 3. Halfwave rectifier with filter capacitor. The input is a 10 sinewave (CH 2) and the output (CH 1) is a DC
value with about a 10% ripple.
As you increase the frequency, the ripple become less because the capacitor has to discharge for a shorter
period. You can also reduce the ripple by installing a larger capacitor. The minimum size capacitor is
𝐶𝑚𝑖𝑛𝑖𝑚𝑢𝑚 =
𝑖𝑙𝑜𝑎𝑑 ∗(𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒 𝑡𝑖𝑚𝑒 𝑜𝑓 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑜𝑟)
𝑅𝑖𝑝𝑝𝑙𝑒 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑝𝑒𝑎𝑘−𝑡𝑜−𝑝𝑒𝑎𝑘
(1)
This is a typical power supply design calculation.
2. Write a professional comprehensive lab report using a word processor. Show your results
and include a comprehensive conclusion. There are lots of sample lab reports on the
internet.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 31
ECE 231 Laboratory Exercise 5C
Diode Characteristics
ECE 231 Laboratory Exercise 5C -Diode Characteristics
OBJECTIVES



Validate the Schottky diode equation.
Calculate the dc and dynamic (ac) resistance of a diode.
Observe the rectifying characteristics of a diode.
EQUIPMENT REQUIRED







ECE 231 Circuit Board (In Stock room)
Three BNC cables
One lot of clip leads and/or jumper wires
DMM (digital multimeter)
One AC power supply
Diode and one resistor (10 ohms or less is best)
Oscilloscope with XY Mode capability
BACKGROUND
See experiment 5A.
This is an alternate method of measuring a diode characteristic using an oscilloscope and only one diode. Again,
if you need better results then use a commercial Curve Tracer. The method described in 5B ideally should use
two matched diodes. Even if the diodes aren’t perfectly matched, the results are pretty good.
This method requires an oscilloscope that has an XY mode. Most oscilloscopes have this option. Use the soft
keys to select the 1X and 2Y for channel 1 and 2. The x-axis is channel 1 and it will display the diode’s forward
voltage drop which should be between 0.6 and 0.75 volts. The y-axis is the current passing through the diode.
PROCEDURE
Part 1
Review experiments 5A for a discussion of diode theory and 5B for using diodes for generating a dc voltage from
an alternating current (ac) source by using a capacitor on the output of a rectifier circuit.
Construct the circuit shown in Figure 1 then observe the diode’s characteristic curve discussed in exercise 5A.
A diode has two types of resistance, dc and ac.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 32
ECE 231 Laboratory Exercise 5C
Diode Characteristics
𝑣
𝑅𝑑𝑐 = 𝑖 at a specific point on the curve. This value varies depending on the operating point that you select
but it should be in the neighborhood of 100 ohms. The diode curve may look like a straight line, but it is not. It
is an exponential curve.
𝑅𝑎𝑐 =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑎𝑏𝑜𝑢𝑡 𝑎 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑝𝑜𝑖𝑛𝑡
∆𝑣
=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑎𝑏𝑜𝑢𝑡 𝑎 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑝𝑜𝑖𝑛𝑡
∆𝑖
This is referred to as a diode’s dynamic resistance. It should be less than 10 ohms.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 33
ECE 231 Laboratory Exercise 5C
Diode Characteristics
Figure 1. Experimental determination of a diode’s operating characteristic with the oscilloscope in XY mode.
The 5 ohm resistor is used as a current sensor.
Determine both the dc and ac resistance of the diode using the oscilloscope cure.
You can download the diode curve into Microsoft word and paste it in your lab report.
A diode has two types of resistance, dc and ac.
𝑣
𝑅𝑑𝑐 = 𝑖 at a specific point on the curve. This value varies depending on the operating point that you select.
The diode curve may look like a straight line, but it is not. It is an exponential curve.
𝑅𝑎𝑐 =
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑎𝑏𝑜𝑢𝑡 𝑎 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑝𝑜𝑖𝑛𝑡
∆𝑣
=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑎𝑏𝑜𝑢𝑡 𝑎 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑝𝑜𝑖𝑛𝑡
∆𝑖
This is referred to as a diode’s dynamic resistance
Rdc = ________________
Rac= _______________
Perform the 1/2wave rectifier experiment described in Experiment 5B.
3. Write a professional comprehensive lab report using a word processor. Show your results
and include a comprehensive conclusion. There are lots of sample lab reports on the
internet.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 34
ECE 231 Laboratory Exercise 6
Frequency Response of RL and RC Circuits
ECE 231 Laboratory Exercise 6—Frequency / Time Response of RL and RC Circuits
Laboratory Group (Names)____________________ ___________________ __________________
OBJECTIVES



Observe and calculate the response of first-order low pass and high pass filters.
Gain experience in plotting Bode plots and calculating decibels.
Test your ability to design and properly test a circuit.
EQUIPMENT REQUIRED




ECE 231 Circuit Board (In Stock room)
Three BNC cables (one for input ac voltage and two for input/ output voltage to oscilloscope)
One lot of clip leads and/or jumper wires
Two-channel Oscilloscope
BACKGROUND
Both capacitors and inductors have reactances that are frequency dependent.
1
1
𝑋𝑐 = 2𝜋𝑓𝐶 = 𝜔𝐶
𝑋𝐿 = 2𝜋𝑓𝐿 = 𝜔𝐿 (1)
When measuring the capacitance and inductance of a component it is very important that you know
the frequency at which the measuring instrument is using. All components R, C, and L consist of all
three. The frequency at which they are operating is a predictor of which ones can be ignored in
calculations. This laboratory experiment will not examine these characteristics of R,C , and L.
Capacitors and inductors as received from manufacturers usually have high tolerances. For example it
is not uncommon for a capacitor to have a tolerance of + 20%.; therefore, measure the values of your
components on the protoboards using an RLC meter.
The voltage transfer function (voltage gain) of a filter is expressed as eq. (2)
𝐻𝑣(𝑗𝜔) =
𝑉𝑜𝑢𝑡(𝑗𝜔)
𝑉𝑖𝑛(𝑗𝜔)
(2)
The method used to calculate Vout is the voltage divider rule. The only difference is that resistances are
replaced by reactances which are complex vectors. Complex impedance is shown in equation (3).
𝑍 = √𝑅 2 + (𝑋𝐿 − 𝑋𝐶 )2
𝑋𝐿 −𝑋𝐶
𝑎𝑟𝑐𝑡𝑎𝑛 (
𝑅
) (3)
The transfer function will have to be plotted on semi log paper with the vertical axis in dB and the
horizontal axis in a logarithmic scale. The definition of dB is shown in eq. (4).
𝑑𝐵 = 20𝑙𝑜𝑔
𝑉𝑜𝑢𝑡
𝑉𝑖𝑛
(4)
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 35
ECE 231 Laboratory Exercise 6
Frequency Response of RL and RC Circuits
The corner frequency of the filters occurs when R=XL or XC. This is also called the -3dB corner
frequency, or ½ power frequency, eq. (5).
1
𝑅
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦𝑐𝑜𝑟𝑛𝑒𝑟 = 2𝜋𝑅𝐶 = 2𝜋𝐿
(5)
A negative slope on a Bode plot also functions as an integrator and a positive slope also
functions as a differentiator. This makes these circuits useful in signal conditioning as
well as filtering.
PROCEDURE
1. Construct the four circuits shown in Figure 1. Select components for a corner frequency
between 1 KHz and 5 KHz. Show all of your calculations. That is, design four circuits that
operate within the capabilities of the equipment in the laboratory.
L1
R1
1
2
V1
V1
C1
10 Vac
Low Pass Filters
0
R1
10 Vac
0
R1
C1
2
V1
10 Vac
V1
R1
L1
10 Vac
1
0
Hi Pass Filters
0
Figure 1. Low Pass and Hi Pass filter schematics. These same circuits can function as integrators or
differentiators.
The circuit simulations for low-pass and hi-pass circuits are shown in Figures 2 and 4. Notice that the
corner frequencies are approximately 1591 Hz.
Note
You cannot view the waveforms shown in Figures 2 and 4. You have to plot these by hand or with the
assistance of Excel. You can see the waveform shown in Figures 3 and 5 when the same circuits are
functioning as integrators and differentiators.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
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ECE 231 Laboratory Exercise 6
Frequency Response of RL and RC Circuits
Figure 2. Low Pass circuit simulation using National Instruments Multisim Software. At frequencies
above the corner frequency the circuit behaves as an integrator. Input a high frequency square wave
and you should see a triangular wave on the oscilloscope.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
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ECE 231 Laboratory Exercise 6
Frequency Response of RL and RC Circuits
Figure 3 is a plot of an integrator where the input square wave frequency is about 10 times the corner
frequency.
Figure 3. Low-Pass circuit functioning as an integrator.when the input frequency is above the corner
frequency. Input is 10 KHz square wave and output is a triangular wave.
2. Connect your signal generator to the input and channel 1 of the oscilloscope. Select a
reasonable input such as 5 volts peak. Sweep the frequency from about two decades below
the corner frequency to two decades above the corner frequency. You will know you are at
the corner frequency when the voltage output is 0.707 (-3 dB) lower than the input voltage.
Observe the output on channel 2 of the oscilloscope. Plot the output seen on channel 2 as
the input frequency is varied. We do not have Bode plotters as shown in Figure 2 in the lab
so the plots must be performed by hand. Record and tabulate all of your settings and
readings. The low-pass and hi-pass curves cannot be observed on the oscilloscope as shown
in Figures 2 and 4. The phase shift between the input and output can be read on the
oscilloscope using the soft keys.
3. Make plots of each low-pass filter shown in Figure 1. One plot with a capacitor/resistor and
one with an inductor/resistor. You can plot the phase shift on the same Bode plot by
adding a separate vertical scale for the phase.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
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ECE 231 Laboratory Exercise 6
Frequency Response of RL and RC Circuits
4. While one of the low-pass filters (you choose) is still connected change the input (channel 1)
to a high frequency square wave and observe that the output (channel 2) is a triangular
wave. The circuit is now functioning as an integrator. See Figure 3.
5. Make plots of each hi-pass filter shown in Figure 4.
6. While one of the hi-pass filters is still connected change the input (channel 1) to a low
frequency triangular wave and observe the output (channel 2) is a square wave. The circuit
is now functioning as a differentiator. The slope of the triangular wave is proportional to the
height of the square wave. See Figure 5. You can copy your oscilloscope trace and paste it
in your report.
Draw the schematic of the circuit used for your curves. Make sure the reader can tell which
schematic goes with which curve.
The Bode Plots of your data for the low-pass and hi-pass filters must be plotted on semi-log
graph paper. The vertical axis shall be in dBs and the horizontal axis shall be log frequency. You
can download semi log graph paper off the internet or buy it at the bookstore. Use two vertical
scales. One for dBs and one for phase.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 39
ECE 231 Laboratory Exercise 6
Frequency Response of RL and RC Circuits
Figure 4. Hi Pass circuit simulation using National Instruments Multisim Software. At frequencies
below the corner frequency the circuit behaves as a differentiator. Input a low frequency triangular
wave and you should see a square wave on the oscilloscope.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 40
ECE 231 Laboratory Exercise 6
Frequency Response of RL and RC Circuits
Figure 5. Hi-pass filter acting as an integrator. The input is a low frequency (100 Hz ) triangular wave
and the output is a square wave.
7. Write a professional comprehensive lab report using a word processor. Show your results
and include a comprehensive conclusion. There are lots of sample lab reports on the
internet.
Write a professional comprehensive lab report using a word processor. Show your results and
include a comprehensive conclusion. There are lots of sample lab reports on the internet.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 41
ECE 231 Laboratory Exercise 7
Resonant Circuits
ECE 231 Laboratory Exercise 7- Resonant Circuits
Laboratory Group (Names) ____________________ ___________________ __________________
OBJECTIVES



Observe and calculate the response of a resonant circuit.
Gain experience in plotting series and parallel resonant circuit response.
Gain experience in varying the bandwidth of a resonant circuit.
EQUIPMENT REQUIRED





ECE 231 Circuit Board (In Stock room) Use capacitors and resistors on board.
Three BNC cables (one for input ac voltage and two for input/ output voltage to oscilloscope)
One lot of clip leads and/or jumper wires
10 mH Inductor box from stockroom (provided by instructor)
Oscilloscope
BACKGROUND
Resonant circuits are used in many applications such as computer circuits, high voltage generators, and
communications devices such as radios. In this laboratory experiment you will construct and measure
the performance of both a series and parallel resonant circuits.
For a series resonant circuit the voltage of the voltage across the resistor will be the same as the
source voltage; however, the voltage across the inductors L and capacitor C will be considerably higher
depending upon the quality factor Q of the circuit. The series circuit is often called a bandpass circuit.
It usually provides voltage gain.
For a parallel circuit, just the opposite is true. The voltage across the inductor and capacitor will equal
the source voltage and the voltage across the resistor will approach zero. This type of circuit is often
called a notch filter. They are often used to drive induction heaters and welders. This circuit usually
provides current gain.
Use the inductor as the output load for the series circuit. If you use the capacitor and you are able to
build a circuit with a very high Q you could damage the capacitor. The series circuit you will construct
is shown in Figure 1. It has a resonant frequency of about 5 KHz.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 42
ECE 231 Laboratory Exercise 7
Resonant Circuits
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 43
ECE 231 Laboratory Exercise 7
Resonant Circuits
Figure 1. Series RLC Resonance circuit simulation using National Instruments Multisim software
Equations that predict the behavior of resonant circuits are as follows:
𝑉𝑜(𝑗𝜔)
=
𝑉𝑠(𝑗𝜔)
𝑅
𝜔𝐿
2
2
√( 1 − 𝜔 2 ) + (𝜔 𝑅 )
𝐿𝐶
𝐿
𝑜
𝜃(𝑗𝜔) = 90 − tan
−1
(
𝑅
𝜔𝐿
1
2
𝐿𝐶 − 𝜔
)
1
Vomax occurs at o (center frequency ) = √𝐿𝐶 where XL= Xc = 2 f
𝑅 2
𝑅
1
Cutoff frequency (1) = 𝜔𝑐1 = − 2𝐿 + √(2𝐿) + (𝐿𝐶)
𝑅 2
𝑅
1
Cutoff frequency (2) == 𝜔𝑐2 = 2𝐿 + √(2𝐿) + (𝐿𝐶)
Where 𝜔𝑜 = √𝜔𝑐1 ∗ 𝜔𝑐2 radian/second
The bandwidth is defined at the frequencies where Vout drops to 0.707 Vsource or–3dB.
That is bandwidth ccR/L (Series Ckt.) and = 1/RC (Parallel Ckt.)
Quality factor
𝜔𝑜
𝛽
=
𝑓𝑜
𝑓2 −𝑓1
𝐿
= √𝐶𝑅2 =
𝜔𝑜 𝐿
𝑅
(Series Ckt.) and  o RC (Parallel Ckt.)
PROCEDURE
1. Construct the circuit shown in Figure 1.
2. Connect your signal generator to the input. Select a reasonable input such as 5 volts peak.
Sweep the frequency from about two decades below the resonant frequency to two
decades above the resonant frequency. Observe the output on the oscilloscope. Record
and tabulate all of your settings and readings. You CANNOT see the curves shown in Figure
R. Frank Smith, California State Polytechnic University, Pomona, 2012
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ECE 231 Laboratory Exercise 7
Resonant Circuits
1 on the oscilloscope. These curves were generated using computer simulation and a Bode
Plotter. Therefore, you have to make the plots by hand.
3. Plot your results on semi log graph paper.
4. Construct the parallel resonant circuit shown in Figure 2.
Phase Angle
R. Frank Smith, California State Polytechnic University, Pomona, 2012
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ECE 231 Laboratory Exercise 7
Resonant Circuits
Magnitude
Figure 2. Parallel RLC Resonance circuit simulation using National Instruments Multisim software
5. Connect your signal generator to the input. Select a reasonable input such as 5 volts peak.
Sweep the frequency from about two decades below the resonant frequency to two
decades above the resonant frequency. Observe the output on the oscilloscope. Record
and tabulate all of your settings and readings.
6. Write a professional comprehensive lab report using a word processor. Show your results
and include a comprehensive conclusion. There are lots of sample lab reports on the
internet.
 Show your calculations and compare them to your measurements for f o,f1,f2, , and Q.
 What kind of errors did you get between what you calculated and what you measured?
 Draw the schematic of the circuit used for your curves. Make sure the reader can tell which
schematic goes with which curve.
 The plots of your data for resonant circuits must be plotted on semi-log graph paper. The
vertical axis shall be in dBs and the horizontal axis shall be log frequency. You can download
semi log graph paper off the internet or buy it at the bookstore. Use two vertical scales.
One for dBs and one for phase.
Write a professional comprehensive lab report using a word processor. Show your results and
include a comprehensive conclusion. There are lots of sample lab reports on the internet.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 46
ECE 231 Laboratory Exercise 8
Time Domain Response 2nd Order Circuits
ECE 231 Laboratory Exercise 8. Time Domain Response 2nd Order Circuits
Laboratory Group (Names) ____________________ ___________________ __________________
OBJECTIVES



Observe and calculate the time domain response of a 2nd order circuit.
Gain experience in plotting circuit response.
Gain experience in observing the time domain response of a 2nd order circuit on an
oscilloscope.
EQUIPMENT REQUIRED






ECE 231 Circuit Board (In Stock room) – use potentiometer and 0.1 f capacitor on board
Three BNC cables (one for input ac voltage and two for input/ output voltage to oscilloscope)
One lot of clip leads and/or jumper wires
400 mH inductor box (stockroom)
Signal generator Use square wave output for channel 1 input.
Two channel oscilloscope. Channel 1 input and channel 2 output (voltage across the capacitor).
BACKGROUND
The time domain response of a circuit is important to understanding the transient behavior of a circuit.
There are three cases that will be examined in this laboratory experiment. They are the under
damped, critically damped, and over damped cases. You will be using a series circuit similar to the one
used when determining the resonance behavior of a circuit (frequency response). Equations (6), (9),
and (12) show the basic form of the circuit behavior for the three cases. These equations are derived
beginning with a loop equation of the circuit shown in Figure 1. By knowing the boundary conditions
(initial and final values), the coefficients of the defining equations can be determined. The experiment
will determine the response of the circuit to a step input. This can be done by applying a square wave
to the circuit and observing the response on an analog oscilloscope or by apply in a step input voltage
to the circuit and observing the response on a digital storage oscilloscope. When applying a square
wave to the circuit the frequency of the square wave must be lower in frequency (longer in time) than
the time response you are trying to observe.
This experiment is similar to driving a car over a speed bump and observing the response of the shock
absorber system. An elevator control system is an example of a critical or over damped control
system. All structural systems are under damped and so are most mechanical systems. A mechanical
scale is designed to be critically damped. This is often accomplished by using an eddy current brake.
The pneumatic closure on a door is normally over damped. The damping ratio  is defined as the
cosine of the angle between the natural frequency vector,ωn, and the real axis in the complex plain
such that ωn=√𝛼 2 + 𝜔𝑑2 and where 𝛼 = 𝛿𝜔𝑛 and 𝜔𝑑 = 𝑗ωn√1 − 𝛿 2 . See Figure 1.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 47
ECE 231 Laboratory Exercise 8
Time Domain Response 2nd Order Circuits
(source: MIT Dept. of Mechanical Engineering)
Figure 1. Complex Plane where ζ is the dampening coefficient.
You will need to construct the circuit shown in Figure 2 for this exercise.
Figure 2. Circuit that you will construct to demonstrate 2nd order system operation.
The simulation results of an underdamped case is shown on the oscilloscope.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 48
ECE 231 Laboratory Exercise 8
Time Domain Response 2nd Order Circuits
The theoretical derivations of the equations that predict the behavior of a series RLC circuit are shown
below.
Write the general differential loop equation for the series (not a parallel) circuit is shown in Fig. 2. Start
by writing a loop (mesh) equation using Kirchhoff’s voltage law around the loop. The result is equation
(1). Voltage rises are negative and drops are positive.
𝑑𝑖
1
−𝑉𝑠 + 𝑅𝑖 + 𝐿 𝑑𝑡 + 𝐶 𝑖 = 0
(1)
Take the derivative of (1) and rearrange it.
𝑑𝑖 2
𝑑𝑡 2
𝑅 𝑑𝑖
1
+ 𝐿 𝑑𝑡 + 𝐿𝐶 𝑖 = 0
(2)
𝑑
Transfer to the frequency domain using the Laplace Transform where 𝑠 = 𝑑𝑡
R
1
s 2 + L s + LC = 0 = s2 + +2αs + ω2n = 0
(3)
Where (refer to Figure 1)
𝑅
𝛼 = 2𝐿 = 𝜁𝜔𝑛
𝑎𝑛𝑑 𝜔𝑛 =
1
√𝐿𝐶
(4)
Roots of the general solution are
𝑆1,2 = −𝛼 ± √𝛼 2 − 𝜔𝑛 2
(5)
The time domain solutions to the Laplacian Equation has three solutions we are interested in.
Over Damped solution 1 (two non-equal real roots)
𝑥(𝑡) = 𝑥𝑓 + 𝐴1′ 𝑒 𝑠1 𝑡 + 𝐴′2 𝑒 𝑠2 𝑡
(6)
𝑥(0) = 𝑥𝑓 + 𝐴1′ + 𝐴′2
(7)
𝑑𝑥
𝑑𝑡
(0) = 𝐴1′ 𝑠1 + 𝐴′2 𝑠2
(8)
The time domain simulation of the over damped solution is shown in Figure 3.
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 49
ECE 231 Laboratory Exercise 8
Time Domain Response 2nd Order Circuits
Figure 3. Circuit simulation of overdamped case using National Instruments Multisim software R = 10K
Critically Damped solution 2 (two equal real roots)
𝑥(𝑡) = 𝑥𝑓 + 𝐷1′ 𝑡𝑒 −𝛼𝑡 + 𝐷2′ 𝑒 −𝛼𝑡
(9)
𝑥(0) = 𝑥𝑓 + 𝛼𝐷2′
(10)
𝑑𝑥
𝑑𝑡
(0) = 𝐷1′ + 𝛼𝐷2′
(11)
The time domain simulation of the critically damped solution is shown in Figure 4.
Figure 4. Circuit simulation of critically damped case using National Instruments Multisim software R = 4K
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 50
ECE 231 Laboratory Exercise 8
Time Domain Response 2nd Order Circuits
Under Damped solution 3 ( two complex conjugate roots)
Type equation here.
𝑥(𝑡) = 𝑥𝑓 + (𝐵1′ 𝑐𝑜𝑠 𝜔𝑑 𝑡 + 𝐵2′ 𝑠𝑖𝑛 𝜔𝑑 𝑡)𝑒 −𝛼𝑡
(12)
𝑥(0) = 𝑥𝑓 + 𝐵1′
(13)
𝑑𝑥
𝑑𝑡
(0) = −𝛼𝐵1′ + 𝜔𝑑 𝐵2′
(14)
𝜔𝑑 = √𝜔𝑜2 + 𝛼 2
(15)
where
The time domain simulation of the underdamped solution is shown in Figure 5.
Figure 5. Circuit simulation of overdamped case using National Instruments Multisim software R =1K
Notice that the peak output voltage is 4 times the peak voltage of the input square wave.
PROCEDURE
1. Construct the circuit shown in Figure 1. You are going to vary the value of R 1 (the
potentiometer) from 10 K (overdamped case- two real roots) to 4 K (critically damped case- two
equal roots) to 1K (underdamped case- complex conjugate roots). Verify these numbers by
solving equation (5). The oscilloscope curve shown in Figure 3 is for R1 = 10 K.
2. Connect your signal generator (set to a square wave output) to the input of your circuit. Select
a reasonable input such as 5 volts. Adjust the frequency of the square wave until you can
observe the response of the circuit. Observe the output on the oscilloscope by connecting the
oscilloscope across the capacitor. Record and tabulate all of your settings and readings. For
R. Frank Smith, California State Polytechnic University, Pomona, 2012
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ECE 231 Laboratory Exercise 8
Time Domain Response 2nd Order Circuits
good characterization of the circuit behavior, the square wave pulse width must be sufficiently long in
order for the circuit to approximately reach a steady state condition. For this circuit a 200 Hz pulse
width square wave would be a good starting point. Set the oscilloscope vertical sensitivity to 5V per
division and the horizontal sensitivity to 1 ms per division.
3. Plot your results on linear graph paper. Estimate the damped frequency of oscillation. Compare
it to the value calculated using equation (15) or copy oscilloscope display using computer
software routine in Microsoft Word and paste into your lab report.
4. Write a professional comprehensive lab report using a word processor. Show your results and
include a comprehensive conclusion. There are lots of sample lab reports on the internet.
 What kind of errors did you get between what you calculated and what you measured?
 What is the comparison between the oscillations observed when R1 was set to 1k and n as
calculated using equation (4)?
 How does this exercise translate to the behavior of mechanical systems?
R. Frank Smith, California State Polytechnic University, Pomona, 2012
Page 52