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

6𝐼 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 ,@,.= 𝐶 ,-

,. ----- (3)

Substituting (1) in (3), we get, 𝐼 = 𝐶 ,-,.

------- (4)

Integrating equation (4) we get 𝐼𝑑𝑡 = 𝐶𝑑𝑣

This implies 𝑣 = 56

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:

The DMM interface is on one end of the MyDAQ:

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.

+15v -­‐15v AGND 0 1 AGND 0+ 0-­‐ 1+ 1-­‐

AO AI

VΩ A

HI COM

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

3rd band gives the base 10 multiplier, x10n

4th band gives the tolerance (silver is ±10% and gold is ±5%)

A resistor with bands (yellow, red, orange, silver) is a 42,000 Ω resistor with a tolerance of ±10%.

Black 0

Brown 1

Red 2

Orange 3

Yellow 4

Green 5

Blue 6

Purple 7

Grey 8

White 9

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 button to measure resistance.

Make sure the 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.

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

Measuring voltage, v2

5. Calculate the voltage across R2 using the nominal (ideal) resistance and source voltage.

R1 R

2

+15v V

v2

i

AGND

COM

DMM Leads

vs

R1 R

2

+

-­‐

+

-­‐ v2

i

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.

+15v -­‐15v AGND 0 1 AGND 0+ 0-­‐ 1+ 1-­‐

AO AI

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.

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.

A: Control the channel source.

B: Control the vertical scale and position.

C: Control the horizontal scale.

D: Control the trigger event and horizontal position.

E:Enable/Select the cursors.

C

D

B

A

E

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.

Resistors

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, 8-grey, 9-white.

Capacitors

Like resistors, capacitor values are also encoded on capacitors. Use the table below to determine

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.

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

+15v -­‐15v AGND 0 1 AGND 0+ 0-­‐ 1+ 1-­‐

AO AI

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.

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.

Scope Channel 0 FGEN Scope Channel 1

A1:0+ A0:0 A1:1+

Vin C(.22µ)

R 1K A1:1-

Vout

A0: GND

A1:0

• 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 button is pressed on both the function generator and the oscilloscope.

Part B: Integrated Circuit

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.

Scope Channel 0 FGEN Scope Channel 1

A1:0+ A0:0 A1:1+

Vin

1KΩ

C (.22µ)

Vout A1:1-

A0: GND

A1:0

• 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 ,@,.= 𝐶 ,-

,. ----- (3)

Substituting (1) in (3), we get, 𝐼 = 𝐶 ,-,.

------- (4)

Integrating equation (4) we get 𝐼𝑑𝑡 = 𝐶𝑑𝑣

This implies 𝑣 = 56

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 Output

b) Input output

c) Input 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 square-wave 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 𝑓R 𝑥 draw the graph of 𝑓 𝑥 on the same graph.

Use the following to draw the graph.

Problem 4:

Let 𝑣 𝑡 = 5T+ sin (3𝑡)represent the velocity of an object moving on a line. At𝑡 = T

\, the

position is 4.

a) Write the acceleration function. b) Write the position function. c) When𝑡 = T

], is the object is speeding up or slowing down? Explain your answer.

d) On the interval[T_, 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 = 𝑣R 𝑡 = 3cos (3𝑡) b) 𝑥 𝑡 = 𝑣 𝑡 𝑑𝑡

So 𝑥 𝑡 = .T− 5

\cos 3𝑡 + 5e

\

c) The object is slowing down because 𝑣 T

]> 0 𝑎𝑛𝑑 𝑎(T

]) < 0.

d) When acceleration is 3 , 𝑡 = _T\

𝑣 _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 10l5_ 10lm 10ln 10l\ 1 10\ 10n 10m 105_ Units Pico(𝜌) nano

(𝑛) Micro(𝜇) Milli

(m) Kilo

(k) Mega (M)

Giga (G)

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

𝑖 𝑡 =𝑑 𝑞 𝑡𝑑𝑡 𝑜𝑟 𝑞 𝑡 = 𝑖(𝑥)

.

lr

𝑑𝑥

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…

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:

size 10l5_ 10lm 10ln 10l\ 1 10\ 10n 10m 105_

Units Pico(𝜌) nano (𝑛)

Micro(𝜇) Milli (m)

Kilo (k)

Mega (M)

Giga (G)

Tera (T)

Example Current magnitudes in amperes (A)

Synaptic current in brain cells

Speed of the digital signal

Integrated Circuit(IC) memory cell current

Human Threshold of sensation

Typical Household Appliance current

Lightning bolt

Example of Voltage magnitudes In Volts (V)

Antenna of a radio receiver

Voltage between two points of human scalp (EEG) Voltage across humon chest produced by the heart (EKG)

Car battery, Voltage on integrated circuit Flash light batteries

Large industrial motors AC outlets plug in U.S households High Voltage transmission lines Voltage on TV picture tube

Lightning bolt

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 = ,v,.

then for a capacitor 𝑖 = ,,.(𝐶𝑣)

Since C is constant 𝑖 = 𝐶 ,-,.

This can be written as 𝑑𝑣 = 56𝑖𝑑𝑡

Integrating the above expression with respect to t from 𝑡 = −∞𝑡𝑜 𝑡 and assuming𝑣 −∞ = 0, we get

𝑣 𝑡 =1𝑐 𝑖 𝑥 𝑑𝑥

.

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