Application of Differentiation and Integration: Creating RC circuits and using function
generator in MyDAQ to analyze the functions
Step-Up Lesson Plan 2015
Santhi Prabahar, Math Teacher
Johns Creek High School
Georgia
Title: Application of differentiation and Integration function in engineering field. Creating RC
Circuits to generate functions using function generator NI MyDAQ and then analyze the
functions using Calculus.
Problem:
Do we use calculus in everyday life? What is the relationship between electrical circuit and
calculus? What are differentiator and Integrator circuits?
Abstract:
High School calculus students often ask the question “When do we ever use this calculus in our
lives?” This lesson shows them how the electrical signal changing from differentiation to
integration and integration to differentiation. The concept of understanding integrating a
differential function gives the original function is very hard for a high school student. This lesson
basically explains when you send a function through the function generator it will show the
differentiation function and Integrated function.
State and National Standards:
According to College Board, students who are enrolled in AP Calculus AB are expected to
• Understand the relationship between the derivative and the definite integral as expressed
in both parts of the Fundamental Theorem of Calculus.
• Learn the concept of derivative as a function as in applications, and computations.
• Learn the concept of Integrals and its applications
TAG- Talented and Gifted Standards:
• The student uses a variety of multi-media and innovative technology to create
illustrations, models, charts, tables, and graphs as tools for communication.(ACS)
• The student develops and uses systematic procedures for recording and organizing
information(ARS)
• The student questions accepted practices, rules, and existing principles to discover new
knowledge.(CPCTS)
• The student designs, applies, evaluates, and adapts a variety of innovative strategies to
when problem solving (e.g., recognizes problems, defines problems, identifies possible
solutions, selects optimal solution, implements solution, and evaluates solution).(CPCTS)
• The student asks probing, insightful, and relevant questions.
• The student responds to questions with supporting information that reflects in-depth
knowledge of a topic
• The student identifies and illustrates basic principles and the foundational concepts that
are central to understanding the essence of a field of study.
Objectives:
• • • • • • • • • • Students will learn the applications of derivative and Integrals in engineering field.
Students will learn to graph both derivative and integral of a function on the same plane.
Students will recognize the given graph of f(x) draw graphs of f′(x) and f″(x)
Students will learn from given graph of f′(x) to find where the function is increasing,
decreasing, max, min, points of inflection, and concavity
Students will learn the connection to physics by learning Ohm’s law
Students will learn to create an RC circuit and use NI MyDAQ to analyze the graphs. Students should be able to communicate mathematics both orally and in well-written
sentences and should be able to explain solutions to problems.
Students should be able to model a written description of a physical situation with a
function, a differential equation, or an integral.
Students should be able to use technology to help solve problems, experiment, interpret
results, and verify conclusions.
Students should be able to determine the reasonableness of solutions, including sign,
size, relative accuracy, and units of measurement.
Anticipated Learner Outcomes:
• Students are able to understand the application of differentiation and integration.
• Students will be able to state Ohm’s law 𝑉 = 𝐼𝑅. Where V = voltage , I = Current ,and R
= resistance
• Students will be able to know that the Current is the derivative of Voltage and Voltage is
the integral of the current.𝐶𝑢𝑟𝑟𝑒𝑛𝑡, 𝐼 = 𝐶
• • • • • • ,,.
𝑎𝑛𝑑 𝑉𝑜𝑙𝑡𝑎𝑔𝑒 𝑉 =
5
6
𝐼 𝑑𝑡 where C is the
capacitor which is always a constant.
Students will be able to make simple RC circuits.
Students will able to graph both differentiation and integral of a function using
oscilloscope in MI myDAQ equipment
Students will be able to relate that calculus is used in electric signals.
Students will able to analyze the input and output of the graphs.
Students will be able to connect with antiderivative of the functions
Student will be able to connect with velocity, acceleration and position of the function
Assessment and Rubric:
This project is done by group of 2. The following documents need to be submitted for a test
grade. Each group submits one document with both names written on each document. (No name
no gradeJ)
1. Completed Applications of Integral worksheet.(10 points)
2. LAB 1 worksheet with calculations.(30 points)
3. LAB 2, Part A and Part B with pictures and screenshots copy pasted on a word document
and labeled correctly Make sure you write your names on the top with period and date.
(40 points)
4. Complete the Applications of integrals worksheet 2 problems.(20 points)
Background Knowledge:
Students should have learned the derivatives unit and antiderivatives. This lesson and labs
can be done right after the introduction of integrals.
If the students know about Ohm’s law and the simple circuit you can directly jump into the
LAB 1 and LAB 2
Materials Needed:
Day1 Application of Integrals WS1
Day2 and Day 3 LAB assignments packets
Day 4 Application of Integrals WS 2
Computer with NI myDAQ software installed. It is available on line. It is easy to use the CD that comes with the myDAQ, but you may also download it from the National
Instruments website at: http://joule.ni.com/nidu/cds/view/p/id/2157/lang/en
• NI myDAQ equipment with all the attachments.
• • • • Lesson Plan:
Day 1:
Warm-up: Introduce the Ohm’s law. Explain the relationship between Voltage, Current, and
resistance .https://www.youtube.com/watch?v=QwNSa_8ro_Y
Opening: Explain the Ohm’s law and the relationship between the differential and Integral
equations of Ohm’s law
Ohm’s law states that the electrical difference between the two points on a circuit Voltage
(V) is the product of the current between those two points (I) and the resistance (R) between
all electrical devices between those two points.
𝑉 = 𝐼 ∗ 𝑅, 𝑡ℎ𝑒 𝑢𝑛𝑖𝑡𝑠 𝑎𝑟𝑒 𝑣 = 𝑉 𝑣𝑜𝑙𝑡𝑠 , 𝑅 = 𝛀 𝑜ℎ𝑚𝑠 , 𝑎𝑛𝑑 𝐼 = 𝐴, 𝐴𝑚𝑝𝑒𝑟𝑒𝑠 The current 𝐼 =
,@
,.
=
ABCCDEDFGD BF 6HIEJD @
ABCCDEDFGD BF .BKD L
----------- (1)
The charge 𝑄 = 𝐶𝑉 𝑤ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑎𝑛𝑐𝑒 𝑎𝑛𝑑 𝑉 𝑖𝑠 𝑉𝑜𝑙𝑡𝑎𝑔𝑒 ------ (2)
Differentiating the equation (2) we get
Substituting (1) in (3), we get, 𝐼 = 𝐶
Integrating equation (4) we get
This implies 𝑣 =
5
6
,@
,.
,,.
𝐼𝑑𝑡 =
=𝐶
,,.
----- (3)
------- (4)
𝐶𝑑𝑣 𝐼𝑑𝑡 ----- (5)
From equation (4) and (5) we can see that Current is the derivative of the voltage and
Voltage is integral of the current.
Solve the following problems:
1. Find Voltage if the current I = 0.5m A and R = 5 k 𝛀.
Answer: V= 0.5 (5) = 2.5 volts
2. Find the Current in Amp if V= 12V and R = 2k 𝛀.
Answer: I= V/R = 12/2 = 6mA
3. Find the resistance if V= 20 volts and I = 4mA.
Answer: R = V/I = 20/4 = 5k 𝛀
4. Find the value of R in the given circuit.
Answer: R= 9 𝛀. Answer: I= 6 𝐀. Homework: Read the notes for circuit and pre lab assignments. Watch the videos
before come to lab tomorrow. Notes: Give the students LAB 1 materials and tell
them to do Pre-lab assignments
Day: 2
LAB 1:
Walk them through LAB1 in order for the students to understand NI myDAQ and introduction to
circuit.
Objective:
Learn about
• • • • NI myDAQ – How to use it?
Proto-boards or bread boards
Simple resistive circuits
Measure Volts across battery, resistor , and the circuits
Parts Needed
• 1kΩ resistor (5% tolerance)
• 2kΩ resistor (5% tolerance)
Pre-lab Assignments:
1) Every Student needs to install the NI ELVISmx software on their laptop ( if they are using
their laptops) and bring your laptop to class, whether you have a myDAQ or not. Failure to
do this may result in not being able to perform the laboratory exercise. It is easiest to use the
CD that comes with the myDAQ, but you may also download it from the National
Instruments website at: http://joule.ni.com/nidu/cds/view/p/id/2157/lang/en
2) View the National Instruments videos on MyDAQ.
a) Getting started myDAQ https://www.youtube.com/watch?v=oe8_GUMVFfU b) Lab View video https://www.youtube.com/watch?v=RsD2tHbuAF4
c) Building RC Circuit with myDAQ https://www.youtube.com/watch?v=z07vHmS8tkw
3) Read the Background section of this laboratory exercise.
Background
National Instruments MyDAQ
Digital Acquisition Boards (DAQs) are used to measure physical electrical signals and convert
them into a form that can be viewed and stored digitally. The National Instruments MyDAQ
also has analog and digital output signals. The MyDAQ comes with several software instrument
panels including a digital multimeter (DMM), an oscilloscope (Scope), and a function generator
(FGEN).
DMM: Digital multimeters measure voltage,
current, and resistance and can be purchased
as self-contained units from hardware stores.
The symbols typically used on DMMs to
represent different signal types are shown in
the table.
Symbol
Definition
V--
DC voltage
V~
AC voltage
A- -
DC current
A~
AC current
Ω
Resistance in ohms
Input/Output Interfaces:
VΩ A The DMM interface is on one end of the MyDAQ:
HI COM The left two connectors are for measuring voltage and resistance.
The right two are used for measuring the current.
The MyDAQ has an interface on one side with +15v, -15v, Analog Output (AO), Analog Input
(AI), and digital input/outputs. The part of the interface that will be used in this experiment is
shown below.
AO AI +15v -­‐15v AGND 0 1 AGND 0+ 0-­‐ 1+ 1-­‐ Resistors
The resistance of physical resistors is denoted by four color bands on the resistor. The color
code for bands 1-3 is
Color
Value 1st band and 2nd band give the first two significant numbers of the resistance
Black
0
3rd band gives the base 10 multiplier, x10n
Brown
1
4th band gives the tolerance (silver is ±10% and gold is ±5%)
Red
2
Orange 3
Yellow 4
Green
5
Blue
6
Purple
7
Grey
8
White
9
A resistor with bands (yellow, red, orange, silver) is a 42,000 Ω resistor with a
tolerance of ±10%.
Breadboard: Breadboards (also known as protoboards) make many connections under the board
in order to reduce the number of wires that you have to connect. Typically, groups of 5 holes are
connected. The side rails are lines of groups of 5 that are all connected. See Figure 1 below.
Be WARNED that many students report that some versions of the breadboards
have disconnect between the top and bottom portions of the Power and Ground bus (orientation
is as in Figure 1). If that’s your case, then be sure to jump wires to bridge the two portions
together; top portion of the power bus should be connected to the bottom portion of the power
bus and same with the ground bus portions.
Figure 1Breadboard Internal Connection. The breadboard on the right shows the internal
connections among the holes. Holes covered by the same yellow line are electrically connected
together.
Lab Procedures
The lab procedures familiarize the student with the MyDAQ system, the proto-boards, and
resistors.
Part A) MyDAQ
Learning how to use the DMM
1. To set up the DMM for measuring voltage:
• Plug the MyDAQ into a USB port on your computer
• Start the NI ELVISmx Instrument Launcher software.
• Click on the DMM icon to open the DMM screen shown below.
• Match the settings in the screen below (Run Continuously, and Auto mode).
• Click on the symbol V - - on the DMM screen (this is used to measure DC voltage).
• Plug the black DMM lead into the COM plug on side of the MyDAQ (this represents
the “common” reference, usually denoted as a negative sign when representing
voltage)
• Plug the red DMM lead into the red terminal to the left of the COM terminal, marked
with a VΩ label.
• Click on “Run”.
2. To use the DMM to measure the voltage difference between the +15 volt terminal and the
Analog Ground (AGND):
• Touch the red DMM lead to the red coated +15v wire and touch the black DMM lead
to the analog ground (AGND). Read the voltage.
Voltage measured by the DMM: _____________
• Reverse the leads so that the red DMM lead touches the black-coated analog ground
wire and the black lead touches the red-coated +15v wire.
Voltage measured by the DMM: _____________
Not working?
• Make sure you click on the
button to measure DC voltages.
• If the voltage measurement is slow to register, you may turn off the “Auto” mode and
select a range of 20V.
• Make sure the
button is pressed on the DMM.
Part B) Resistors
Measuring Resistance
1. Turn the DMM dial to Ohms (Ω) and touch the leads to both sides of a 1k Ω resistor.
Measure the resistance. It does not matter which color lead goes to which end of the
resistor. If the DMM is slow to register, switch the mode to “Specify Range” and select
20kΩ.
Measured Resistance = ______________
2. Measure the actual resistance of a 2k Ω resistor.
Measured Resistance = ______________
Not working?
Make sure you click on the
Make sure the
button to measure resistance.
button is pressed on the DMM.Determining Current by
Measuring Voltage across a Resistor
Current can be measured directly by placing an ammeter in series in the circuit. Current
can also be determined by using Ohm’s Law on the voltage measured across a known
resistor, as will be done in this part.
+
v
s -­‐
i
R
+
v
-­‐
R
1
2
2
3. Build the above circuit on the protoboard using the resistors measured in Parts 3 and 4,
where R1 = 2kΩ and R2 = 1 kΩ resistors.
4. Use the +15v and AGND wires from the MyDAQ for the voltage supply Vs
+15v
i
R
1
R
2
V
v
DMM Leads
2
AGN
D
COM
Measuring voltage, v2
5. Calculate the voltage across R2 using the nominal (ideal) resistance and source voltage.
v2= _______volts
6. Measure the voltage across R2 using the DMM, using the MyDAQ connections as shown
below.
v2= _______volts
7. Derive the current from Ohm’s Law: i=v2/R2 where R2 is the measured resistance and v2
is the voltage measured across R2.
i = _______ amps
Day 3:
For this LAB, Please follow all the directions carefully and provide all the documents needed.
LAB 2: In this LAB you will be using Function Generator to generate two sets of functions.
You will be using Differentiator Circuit and Integrator circuit to graph the functions on the same
screen using oscilloscope.
Materials Needed
• 1kΩ and 2 kΩ resistor
• 0.22 µf capacitor (224)
• 3.3mH inductor
Prelab
• Read the background section.
• Watch the video on the RC lab demo of oscilloscope
https://www.youtube.com/watch?v=E92mI4n0zvo
• Insert jumper wires into your myDAQ in the +15v, -15v, AGND, AO:A0, AI:0+, 0-, 1+,1-.
Use your longer jumper wires, and your screw driver to secure the wires tightly.
AO AI +15v -­‐15v AGND 0 1 AGND 0+ 0-­‐ 1+ 1-­‐ Background
Scope: An oscilloscope is a measurement instrument used to display how a measured voltage
varies with time. On the oscilloscope display, the y axis is voltage and the x axis is time. The
inputs to the oscilloscope are called channels. A two channel oscilloscope can simultaneously
display two measured voltages. The scales of the display axes may be varied to display the
voltage waveforms better by adjusting the volts per division (y axis) and time per division (x
axis) knobs. A trigger is an event that starts displaying the signal. In the picture below, the
trigger event is a rising edge with a trigger level of 0V. The signal is displayed from the moment
it increases to more than 0V.
A: Control the channel source. A B: Control the vertical scale and position. B C: Control the horizontal scale. C E D D: Control the trigger event and horizontal position. E:Enable/Select the cursors. FGEN: A function generator is a test instrument that generates a voltage that varies with time.
Typical generated functions are sine waves, square waves, and triangle waves.
Input/Output Interfaces:
The MyDAQ has an interface on one side with +15v, -15v, Analog Output (AO), Analog Input
(AI), and digital input/outputs. The part of the interface that will be used in this experiment is
shown below.
AO AI +15v Resistors
-­‐15v AGND 0 1 AGND 0+ 0-­‐ 1+ 1-­‐ The resistance of physical resistors is denoted by four color bands on the resistor. The color
code for bands 1-3 is 0-black, 1-brown, 2-red, 3-orange, 4-yellow, 5-green, 6-blue, 7-purple, 8grey, 9-white.
Capacitors
Like resistors, capacitor values are also encoded on capacitors. Use the table below to determine
Code
Value
(nF)
Code
Value
(nF)
Code
Value
(nF)
Code
Value
(nF)
100
0.01
331
0.33
472
4.7
154
150
150
0.015
471
0.47
502
5
254
200
220
0.022
561
0.56
562
5.6
224
220
330
0.033
681
0.68
682
6.8
334
330
470
0.047
751
0.75
103
10
474
470
101
0.1
821
0.82
153
15
684
680
121
0.12
102
1
223
22
105
1000
131
0.13
152
1.5
333
33
155
1500
151
0.15
202
2
473
47
205
2000
181
0.18
222
2.2
683
68
225
2200
221
0.22
332
3.3
104
100
335
3300
capacitor values from their codes or to determine the codes from their value. Nano is 10-9.
A capacitor labeled 154 has a value of 150 Nano Farads or 0.15µf.
LAB Procedure
In this LAB you will be doing following three circuits. Fill the blanks and make a screen shot
of all the three graphs and include in your LAB submission. Make sure you label the screen
shots with proper functions and values.
Part A: Differentiator Circuit: 1. Connect the myDAQ and then start the NI Instrument Launcher software. 2. On the protoboard, construct the RC circuit whose schematic is shown below. Scope Channel 0 FGEN Scope Channel 1
+
A1:1+
A1:0 A0:0
Vin
C(.22µ)
R 1K A1:1
-
Vout
A0: GND
A1:0 3. Take a picture of your RC circuit and label it Differentiator Circuit.
4. Set up the instruments: a. Function Generator • Match the settings in the screen shot to produce a 49.76Hz 1 V peak-­‐to-­‐peak square wave with a dc offset of 0.0V. b. Oscilloscope, enable Channel 1 then set: • the trigger type to edge • the trigger level to 0.0V • the horizontal position to 50% • the Volts/Div on channel 0 to 200 mV and Channel 1 to 20mv • the vertical position of both channels to 0 divisions • the Time/Div setting to 10ms. • the source for channel 1 to AI1 5. Click Run on both the function generator and the oscilloscope. 6. Verify that both the circuit input voltage and the voltage across the capacitor are
displayed properly:
• V in shown on scope channel 0 is a triangle wave
• V out displayed on channel 1 is a square wave which is a differentiated wave.
7. Zoom in on the measured voltages by changing the time base to 2ms per division. The
scope should now display the input voltage and the capacitor voltage for approximately
one period of the input. The voltages displayed should be similar to those shown in the
scope display figure above.
8. Make sure you Take the screen shot of the Functions and label it as “ Differentiated
circuit with Capacitor (c.22µ ) and resistor 1KΩ
Not Working:
• Make sure that the interface is pushed into to the myDAQ unit all the way. If it comes
loose, good electrical connections may not be present.
• Make sure that only the FGEN and SCOPE are connected to the circuit. The 15V
power supply should not be connected to this circuit. The circuit is being powered
by the function generator. (FGEN)
• Make sure that the rest of the circuit is connected correctly (according to the
schematic). It is generally good practice to use short wires and as few wires as
possible. Make sure the
the oscilloscope.
button is pressed on both the function generator and
Part B: Integrated Circuit
Scope Channel 0 FGEN Scope Channel 1
+
A1:1+
A1:0 A0:0
Vin
1KΩ C (.22µ) Vout A1:1 A0: GND A1:0 Note:
• Follow the above 8 steps just as PartA differentiated circuit. Remember since this
is an integrated circuit the settings are going to be different.
• Screen shot is not given in this case.
• Carefully analyze the adjustments in Part A. Change all the adjustments until you
receive the correct graph.
• HINT: Input a square wave this time to get a triangle wave as output.
• Take a picture of the circuit on the protoboard and take the screen shot of the
oscilloscope and function generator and label it as “Integrator circuit with
Resistor 1KΩ and Capacitor .22µF
Answer: Integrator circuit with capacitor and resistor
Picture of Integrator circuit
Oscilloscope Results
Extra credit: 5 points
• Try to include another resistor value in the circuit see the changes.
• Replace the capacitor by the inductor and see what happens.
• Include the screen shot and picture of the circuit and your analysis in complete sentence
to receive full credit.
Summary:
I believe this lesson plan is a great connection between calculus and engineering field. High
school students will learn the relationship between differentiation, antiderivative, and integrals.
Most of all they will use technology to create and simulate the circuits and functions. The
students will gain hands on experience and involve in experiments through LABs in traditional
math class. The students will realize that calculus plays an important role in everyday life.
AP Calculus AB
Application of Integrals
Name:____________ Date:______
Fill in the blanks and finish the problems at the end.
Ohm’s law states that the electrical difference between the two points on a circuit Voltage
(V) is the product of the current between those two points (I) and the resistance (R) between
all electrical devices between those two points.
𝑉 = 𝐼 ∗ 𝑅, 𝑡ℎ𝑒 𝑢𝑛𝑖𝑡𝑠 𝑎𝑟𝑒 𝑣 = 𝑉 𝑣𝑜𝑙𝑡𝑠 , 𝑅 = 𝛀 𝑜ℎ𝑚𝑠 , 𝑎𝑛𝑑 𝐼 = 𝐴, 𝐴𝑚𝑝𝑒𝑟𝑒𝑠 The current 𝐼 =
,@
,.
=
ABCCDEDFGD BF 6HIEJD @
ABCCDEDFGD BF .BKD L
----------- (1)
The charge 𝑄 = 𝐶𝑉 𝑤ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑎𝑛𝑐𝑒 𝑎𝑛𝑑 𝑉 𝑖𝑠 𝑉𝑜𝑙𝑡𝑎𝑔𝑒 ------ (2)
Differentiating the equation (2) we get
Substituting (1) in (3), we get, 𝐼 = 𝐶
Integrating equation (4) we get
This implies 𝑣 =
5
6
,@
,.
,,.
𝐼𝑑𝑡 =
=𝐶
,,.
----- (3)
------- (4)
𝐶𝑑𝑣 𝐼𝑑𝑡 ----- (5)
From equation (4) and (5) we can see that Current is the derivative of the voltage and
Voltage is integral of the current.
Solve the following problems:
5. Find Voltage if the current I = 0.5m A and R = 5 k 𝛀.
Answer: V= 0.5 (5) = 2.5 volts
6. Find the Current in Amp if V= 12V and R = 2k 𝛀.
Answer: I= V/R = 12/2 = 6mA
7. Find the resistance if V= 20 volts and I = 4mA.
Answer: R = V/I = 20/4 = 5k 𝛀
8. Find the value of R in the given circuit.
Answer: R= 9 𝛀. Homework: Read the notes for circuit and pre lab assignments.
AP Calculus AB
Application of Integrals WS 2 Name: _________________ Date:_______
Note: We learned from the LABs that the application of differentiation and integration occurs in
RC circuits. Graphing antiderivative functions, velocity, acceleration and position functions are
part of the AP Calculus requirements.
The problems here connects graphing the antiderivative functions and velocity, acceleration
Problem 1:
For each of the following voltage waveforms (Input), plot the corresponding capacitor current
waveform (output) as it would appear on the oscilloscope screen: remember the current 𝐼 = 𝐶
a) Input
b) Input
c) Input
Output
output
output
,,.
Problem 2: Based on what you learned in differentiator and Integrator in your LABS, what must
the signal look like in between the integrator and differentiator circuits to produce a final squarewave output? In other words, if we were to connect an oscilloscope in between these two
circuits, what sort of signal would it show us?
Problem 3: Given the graph of 𝑓 R 𝑥 draw the graph of 𝑓 𝑥 on the same graph.
Use the following to draw the graph.
Problem 4:
5
T
T
\
Let 𝑣 𝑡 = + sin (3𝑡)represent the velocity of an object moving on a line. At𝑡 = , the
position is 4.
a) Write the acceleration function.
b) Write the position function.
T
c) When𝑡 = , is the object is speeding up or slowing down? Explain your answer.
]
T
d) On the interval[ , 𝜋], what is the velocity when the acceleration is 3?
_
Answer:
Problem 1.
a) b) c) Problem:2 Problem 3: a) 𝑓(𝑥) is increasing at (1,6) and (8,9) and f(x) is
decreasing (0,1) and (6,8)
b) Local maxima or minima x=1,6,8
c) Concave up : (-2,2), (3,5), (7,9)
Concave down: (2,3), (5,7)
d) The points of inflection x= 2,3,5,7
Problem 4:
a) 𝑎 𝑡 = 𝑣 R 𝑡 = 3cos (3𝑡)
b) 𝑥 𝑡 = 𝑣 𝑡 𝑑𝑡
.
5
5e
T
\
\
So 𝑥 𝑡 = − cos 3𝑡 +
c) The object is slowing down because
𝑣
T
]
T
> 0 𝑎𝑛𝑑 𝑎( ) < 0.
]
d) When acceleration is 3 , 𝑡 =
𝑣
_T
\
_T
\
= 0.318
Extra Notes:
Notes on Basic Circuit system:
System of Units:
The SI system is used in Engineering field called – System International units, which is
composed of the basic units.
size
Units
10l5_
Pico(𝜌)
10lm
nano
(𝑛)
10ln
Micro(𝜇)
10l\
Milli
(m)
10\
1
Kilo
(k)
10n
Mega
(M)
10m Giga
(G)
105_
Tera
(T)
Electric Circuit:
It is an interconnection of electrical components, each of which will describe a mathematical
model.
An electric circuit is a pipe line that facilitates the transfer of charge from on point to another.
Electric current:
Electric current is the rate of change of a charge.
The mathematical relationship is
𝑑 𝑞 𝑡
𝑖 𝑡 =
𝑑𝑡
.
𝑜𝑟 𝑞 𝑡 = 𝑖(𝑥) 𝑑𝑥
lr
Where i and q represents current and charge. (lower case – dependent variable, capital letters –
constants). Unit of current Ampere (A) and 1 ampere = 1 coulomb per second.
Alternating Current: Used in common house hold items such as refrigerator, stove, washing
machine etc.
Direct current: It is the current in batteries which are found in automobiles and flash lights etc…
size
10l5_
10lm
10ln
10l\
10\
1
10n
10m Units
Pico(𝜌)
nano
(𝑛)
Micro(𝜇)
Milli
(m)
Kilo
(k)
Mega
(M)
Example
Current
magnitu
des in
amperes
(A)
Synaptic
current
in brain
cells
Speed
of the
digital
signal
Integrated
Circuit(IC)
memory
cell current
Human
Thresho
ld of
sensatio
n
Typical
Household
Appliance
current
Lightnin
g bolt
Antenna of
a radio
receiver
Voltage
between
two
points
of
human
scalp
(EEG)
Voltage
across
humon
chest
produce
d by the
heart
(EKG)
Car battery,
Voltage on
integrated
circuit
Flash light
batteries
Large
Lightni
industria ng bolt
l motors
AC
outlets
plug in
U.S
househol
ds
High
Voltage
transmis
sion
lines
Voltage
on TV
picture
tube
Example
of
Voltage
magnitu
des
In Volts
(V)
Giga
(G)
Resource: Basic Engineering Circuit Analysis by J. David Irwin (Auburn University) and R.
Mark Nelms (Auburn University)
Resistive Circuits:
Resistor: A circuit element whose electrical characteristic is primarily resistive is called resistor
Ohm’s Law:
105_
Ter
a
(T)
Ohm’s Law states that the voltage across a resistance is directly proportional to the current
flowing through it.
𝑣 𝑡 = 𝑅×𝑖 𝑡 , 𝑤ℎ𝑒𝑟𝑒 𝑅 ≥ 0
Since the resistor is constant the Voltage – current relationship is linear.
Capacitor: A capacitor is a circuit element that consists of two conducting surface separated by a
non-conducting (dielectric) material.
Suppose a source is connected to a capacitor then the positive charges are collected on one side
and negative charges to the other.
The charge (q) on the capacitor is proportional to the voltage (v).
𝑞=
𝐶𝑣, 𝑤ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑎𝑛𝑐𝑒 𝑖𝑛 𝐹𝑎𝑟𝑎𝑑𝑠 𝑜𝑟 𝑐𝑜𝑢𝑙𝑜𝑚𝑠 𝑝𝑒𝑟 𝑣𝑜𝑙𝑡
We know that the current is 𝑖 =
Since C is constant 𝑖 = 𝐶
,v
,.
then for a capacitor 𝑖 =
,
,.
(𝐶𝑣) ,,.
5
This can be written as 𝑑𝑣 = 𝑖𝑑𝑡
6
Integrating the above expression with respect to t from 𝑡 = −∞𝑡𝑜 𝑡 and assuming𝑣 −∞ = 0,
we get
1
𝑣 𝑡 =
𝑐
.
𝑖 𝑥 𝑑𝑥
lr