Problems of dynamic logic circuits and how it is solved by Domino logic circuits, is explained over here. Why it is called domino and how domino logic works, that also explained here.
System on Chip is a an IC that integrates all the components of an electronic system. This presentation is based on the current trends and challenges in the IP based SOC design.
Problems of dynamic logic circuits and how it is solved by Domino logic circuits, is explained over here. Why it is called domino and how domino logic works, that also explained here.
System on Chip is a an IC that integrates all the components of an electronic system. This presentation is based on the current trends and challenges in the IP based SOC design.
solar power satellite & microwave power transmissionbhavisha patel
In this seminar topic,I included all the things related SPS system & how microwave power transmission can done through magetron,retro directive beam controlling scheme & all.I also mentioned the design of optical rectenna & economic evolution of the topic.
Series and parallel connection of mosfetMafaz Ahmed
The parallel connection of MOSFETs allows higher load currents to be handled by sharing the current between the individual switches. Because MOSFETs have a positive temperature coefficient they can be parallel without the need for source resistors.
Impact of high level penetration of photovoltaics on Power systemMuwaf_5
Discusses the impact of high penetration levels of PV system, and its effects on the grid. It has detrimental effects on system stability and cause power quality issues. The remedial measure is discussed
Oscillators introduction and its types, phase shift oscillators and wein bridge oscillators,difference between phase shift and wein bridge, frequency stability, oscillators principle and conditions, block diagram of oscillators, block diagram of phase shift oscillators
Quantum computers are designed to perform tasks much more accurately and efficiently than conventional computers, providing developers with a new tool for specific applications.
It is clear in the short-term that quantum computers will not replace their traditional counterparts; instead, they will require classical computers to support their specialized abilities, such as systems optimization.
This presentation reviews the following paper.
Giannini, Vincenzo, Antonio I. Fernández-Domínguez, Susannah C. Heck, and Stefan A. Maier. "Plasmonic nanoantennas: fundamentals and their use in controlling the radiative properties of nanoemitters." Chemical reviews 111, no. 6 (2011): 3888-3912.
Network Theory Part 1 questions and answers for interviews, campus placements...Prasant Kumar
Networks Analysis, Networks Synthesis, MCQ, Questions & Answers, ohm’s law ,Sine function ,Linear,parabola,hyperbola, VIVA Questions, The voltage across R and L in a series RL circuit are found to be 200 V and 150 V respectively the rms value of the voltage across the series combination is _ V, Kirchhoff s current law ,reciprocity , duality, on-linearity ,linearity ,resistive elements, passive elements ,non-linear elements, linear bilateral elements , interviews, campus placements, online tests, aptitude tests, quizzes and competitive exams, electrical engineering
1.The curve representing ohm's law is
a) Sine function
b) Linear
c) parabola
d) hyperbola
Ans- b
2. The voltage across R and L in a series RL circuit are found to be 200 V and 150 V respectively the rms value of the voltage across the series combination is _ V.
a) 360.
b) 250.
c) 200.
d) 450.
Ans: b
V = √( VR²+ VL²) = √( 200² + 150² ) = 250 V.
3. Kirchhoff s current law states that
(a) net current flow at the junction is positive
(b) Algebric sum of the currents meeting at the junction is zero
(c) no current can leave the junction without some current entering it.
(d) total sum of currents meeting at the junction is zero
Ans: b
According to Kirchhoffs voltage law, the algebraic sum of all IR drops and
e.m.fs. in any closed loop of a network is always
(a) negative
(b) positive
(c) determined by battery e.m.fs.
(d) zero
Ans: d
5. Kirchhoffs current law is applicable to only
(a) junction in a network
(b) closed loops in a network
(c) electric circuits
(d) electronic circuits
Ans: a
6. Superposition theorem can be applied only to circuits having
(a) resistive elements
(b) passive elements
(c) non-linear elements
(d) linear bilateral elements
Ans: d
7. The concept on which Superposition theorem is based is
(a) reciprocity
(b) duality
(c) non-linearity
(d) linearity
Ans: d
Network Theory Part 2 questions and answers for interviews, campus placements...Prasant Kumar
Networks Analysis, Networks Synthesis, MCQ, Questions & Answers, ohm’s law ,Sine function ,Linear,parabola,hyperbola, VIVA Questions, The voltage across R and L in a series RL circuit are found to be 200 V and 150 V respectively the rms value of the voltage across the series combination is _ V, Kirchhoff s current law ,reciprocity , duality, on-linearity ,linearity ,resistive elements, passive elements ,non-linear elements, linear bilateral elements , interviews, campus placements, online tests, aptitude tests, quizzes and competitive exams, electrical engineering,
Brief 5AC RL and RC CircuitsElectrical Circuits Lab VannaSchrader3
Brief 5
AC RL and RC Circuits
Electrical Circuits Lab I
(ENGR 2105)
Dr. Kory Goldammer
Review of Complex Numbers and Transforms
Transforms
The Polar Coordinates / Rectangular Coordinates Transform
The Complex Plane
We can use complex numbers to solve for the phase shift in AC Circuits
Instead of (x,y) coordinates, we define a point in the Complex plane by (real, imaginary) coordinates
Real numbers are on the horizontal axis
Imaginary numbers are on the vertical axis
The Complex Plane (cont.)
Imaginary numbers are multiplied by j
By definition,
(Mathematicians use i instead of j, but that would confuse us since i stands for current in this class)
Imaginary Plane: Rectangular Coordinates
We can identify any point in the 2D plane using (real, imaginary) coordinates
Complex Plane Using Rectangular Coordinates
Imaginary Plane: Transform to Polar Coordinates
We can identify any point in the complex plane using (r,) coordinates.
The arrow is called a Phasor.
r is the length of the Phasor, and is the angle between the Positive Real Axis and the Phasor
is the Phase Angle we want to calculate
Complex Plane Using Rectangular Coordinates
r
Imaginary Plane: Transform to Polar Coordinates
Complex Plane Using Rectangular Coordinates
(we will discuss the meaning of r later)
Use either the sin or cos term to find :
But we need in radians:
r
=7.07
Complex Math
For addition or subtraction, add or subtract the real and j terms separately.
(3 + j4) + (2 – j2) = 5 + j2
To multiply or divide a j term by a real number, multiply or divide the numbers. The answer is still a j term.
5 * j6 = j30
-2 * j3 = -j6
j10 / 2 = j5
Complex Math (cont. 1)
To divide a j term by a j term, divide the j coefficients to produce a real number; the j factors cancel.
j10 / j2 = 5
-j6 / j3 = -2
To multiply complex numbers, follow the rules of algebra, noting that j2 = -1
Complex Math (cont. 2)
To divide by a complex number: Can’t be done!
The denominator must first be converted to a Real number!
Complex Conjugation
Converting the denominator to a real number without any j term is called rationalization.
To rationalize the denominator, we need to multiply the numerator and denominator by the complex conjugate
Complex Number Complex Conjugate
5 + j3 5 – j3
–5 + j3 –5 – j3
5 – j3 5 + j3
–5 – j3 –5 + j3
Complex Math (cont. 2)
Multiply the original equation by the complex conjugate divided by itself (again, j2 = -1):
Phase Shift
Time Domain - ω Domain Transforms
Transforming from the Time (Real World) Domain to the (Problem Solving Domain
Note that in the ω Domain, Resistance, Inductance and Capacitance all of units of Ohms!ElementTime Domainω Domain TransformApplied Sinusoidal AC Voltage
(Volts) (ω=2πf)Vp
(Volts)Series Current(Amps) (ω=2πf)(Amps)ResistanceR
(Ohms)R
(Ohms)InductanceL
(Henry’s)(Ohms) (ω=2πf)CapacitanceC
(Farads)(Ohms) (ω=2πf)
Solving For Current
...
solar power satellite & microwave power transmissionbhavisha patel
In this seminar topic,I included all the things related SPS system & how microwave power transmission can done through magetron,retro directive beam controlling scheme & all.I also mentioned the design of optical rectenna & economic evolution of the topic.
Series and parallel connection of mosfetMafaz Ahmed
The parallel connection of MOSFETs allows higher load currents to be handled by sharing the current between the individual switches. Because MOSFETs have a positive temperature coefficient they can be parallel without the need for source resistors.
Impact of high level penetration of photovoltaics on Power systemMuwaf_5
Discusses the impact of high penetration levels of PV system, and its effects on the grid. It has detrimental effects on system stability and cause power quality issues. The remedial measure is discussed
Oscillators introduction and its types, phase shift oscillators and wein bridge oscillators,difference between phase shift and wein bridge, frequency stability, oscillators principle and conditions, block diagram of oscillators, block diagram of phase shift oscillators
Quantum computers are designed to perform tasks much more accurately and efficiently than conventional computers, providing developers with a new tool for specific applications.
It is clear in the short-term that quantum computers will not replace their traditional counterparts; instead, they will require classical computers to support their specialized abilities, such as systems optimization.
This presentation reviews the following paper.
Giannini, Vincenzo, Antonio I. Fernández-Domínguez, Susannah C. Heck, and Stefan A. Maier. "Plasmonic nanoantennas: fundamentals and their use in controlling the radiative properties of nanoemitters." Chemical reviews 111, no. 6 (2011): 3888-3912.
Network Theory Part 1 questions and answers for interviews, campus placements...Prasant Kumar
Networks Analysis, Networks Synthesis, MCQ, Questions & Answers, ohm’s law ,Sine function ,Linear,parabola,hyperbola, VIVA Questions, The voltage across R and L in a series RL circuit are found to be 200 V and 150 V respectively the rms value of the voltage across the series combination is _ V, Kirchhoff s current law ,reciprocity , duality, on-linearity ,linearity ,resistive elements, passive elements ,non-linear elements, linear bilateral elements , interviews, campus placements, online tests, aptitude tests, quizzes and competitive exams, electrical engineering
1.The curve representing ohm's law is
a) Sine function
b) Linear
c) parabola
d) hyperbola
Ans- b
2. The voltage across R and L in a series RL circuit are found to be 200 V and 150 V respectively the rms value of the voltage across the series combination is _ V.
a) 360.
b) 250.
c) 200.
d) 450.
Ans: b
V = √( VR²+ VL²) = √( 200² + 150² ) = 250 V.
3. Kirchhoff s current law states that
(a) net current flow at the junction is positive
(b) Algebric sum of the currents meeting at the junction is zero
(c) no current can leave the junction without some current entering it.
(d) total sum of currents meeting at the junction is zero
Ans: b
According to Kirchhoffs voltage law, the algebraic sum of all IR drops and
e.m.fs. in any closed loop of a network is always
(a) negative
(b) positive
(c) determined by battery e.m.fs.
(d) zero
Ans: d
5. Kirchhoffs current law is applicable to only
(a) junction in a network
(b) closed loops in a network
(c) electric circuits
(d) electronic circuits
Ans: a
6. Superposition theorem can be applied only to circuits having
(a) resistive elements
(b) passive elements
(c) non-linear elements
(d) linear bilateral elements
Ans: d
7. The concept on which Superposition theorem is based is
(a) reciprocity
(b) duality
(c) non-linearity
(d) linearity
Ans: d
Network Theory Part 2 questions and answers for interviews, campus placements...Prasant Kumar
Networks Analysis, Networks Synthesis, MCQ, Questions & Answers, ohm’s law ,Sine function ,Linear,parabola,hyperbola, VIVA Questions, The voltage across R and L in a series RL circuit are found to be 200 V and 150 V respectively the rms value of the voltage across the series combination is _ V, Kirchhoff s current law ,reciprocity , duality, on-linearity ,linearity ,resistive elements, passive elements ,non-linear elements, linear bilateral elements , interviews, campus placements, online tests, aptitude tests, quizzes and competitive exams, electrical engineering,
Brief 5AC RL and RC CircuitsElectrical Circuits Lab VannaSchrader3
Brief 5
AC RL and RC Circuits
Electrical Circuits Lab I
(ENGR 2105)
Dr. Kory Goldammer
Review of Complex Numbers and Transforms
Transforms
The Polar Coordinates / Rectangular Coordinates Transform
The Complex Plane
We can use complex numbers to solve for the phase shift in AC Circuits
Instead of (x,y) coordinates, we define a point in the Complex plane by (real, imaginary) coordinates
Real numbers are on the horizontal axis
Imaginary numbers are on the vertical axis
The Complex Plane (cont.)
Imaginary numbers are multiplied by j
By definition,
(Mathematicians use i instead of j, but that would confuse us since i stands for current in this class)
Imaginary Plane: Rectangular Coordinates
We can identify any point in the 2D plane using (real, imaginary) coordinates
Complex Plane Using Rectangular Coordinates
Imaginary Plane: Transform to Polar Coordinates
We can identify any point in the complex plane using (r,) coordinates.
The arrow is called a Phasor.
r is the length of the Phasor, and is the angle between the Positive Real Axis and the Phasor
is the Phase Angle we want to calculate
Complex Plane Using Rectangular Coordinates
r
Imaginary Plane: Transform to Polar Coordinates
Complex Plane Using Rectangular Coordinates
(we will discuss the meaning of r later)
Use either the sin or cos term to find :
But we need in radians:
r
=7.07
Complex Math
For addition or subtraction, add or subtract the real and j terms separately.
(3 + j4) + (2 – j2) = 5 + j2
To multiply or divide a j term by a real number, multiply or divide the numbers. The answer is still a j term.
5 * j6 = j30
-2 * j3 = -j6
j10 / 2 = j5
Complex Math (cont. 1)
To divide a j term by a j term, divide the j coefficients to produce a real number; the j factors cancel.
j10 / j2 = 5
-j6 / j3 = -2
To multiply complex numbers, follow the rules of algebra, noting that j2 = -1
Complex Math (cont. 2)
To divide by a complex number: Can’t be done!
The denominator must first be converted to a Real number!
Complex Conjugation
Converting the denominator to a real number without any j term is called rationalization.
To rationalize the denominator, we need to multiply the numerator and denominator by the complex conjugate
Complex Number Complex Conjugate
5 + j3 5 – j3
–5 + j3 –5 – j3
5 – j3 5 + j3
–5 – j3 –5 + j3
Complex Math (cont. 2)
Multiply the original equation by the complex conjugate divided by itself (again, j2 = -1):
Phase Shift
Time Domain - ω Domain Transforms
Transforming from the Time (Real World) Domain to the (Problem Solving Domain
Note that in the ω Domain, Resistance, Inductance and Capacitance all of units of Ohms!ElementTime Domainω Domain TransformApplied Sinusoidal AC Voltage
(Volts) (ω=2πf)Vp
(Volts)Series Current(Amps) (ω=2πf)(Amps)ResistanceR
(Ohms)R
(Ohms)InductanceL
(Henry’s)(Ohms) (ω=2πf)CapacitanceC
(Farads)(Ohms) (ω=2πf)
Solving For Current
...
This chapter provides complete solution of of first, Second order differential equations of series & parallel R-L, R-C, R-L-C circuits, bu using different methods.
Fleet management these days is next to impossible without connected vehicle solutions. Why? Well, fleet trackers and accompanying connected vehicle management solutions tend to offer quite a few hard-to-ignore benefits to fleet managers and businesses alike. Let’s check them out!
In this presentation, we have discussed a very important feature of BMW X5 cars… the Comfort Access. Things that can significantly limit its functionality. And things that you can try to restore the functionality of such a convenient feature of your vehicle.
"Trans Failsafe Prog" on your BMW X5 indicates potential transmission issues requiring immediate action. This safety feature activates in response to abnormalities like low fluid levels, leaks, faulty sensors, electrical or mechanical failures, and overheating.
5 Warning Signs Your BMW's Intelligent Battery Sensor Needs AttentionBertini's German Motors
IBS monitors and manages your BMW’s battery performance. If it malfunctions, you will have to deal with an array of electrical issues in your vehicle. Recognize warning signs like dimming headlights, frequent battery replacements, and electrical malfunctions to address potential IBS issues promptly.
Things to remember while upgrading the brakes of your carjennifermiller8137
Upgrading the brakes of your car? Keep these things in mind before doing so. Additionally, start using an OBD 2 GPS tracker so that you never miss a vehicle maintenance appointment. On top of this, a car GPS tracker will also let you master good driving habits that will let you increase the operational life of your car’s brakes.
Core technology of Hyundai Motor Group's EV platform 'E-GMP'Hyundai Motor Group
What’s the force behind Hyundai Motor Group's EV performance and quality?
Maximized driving performance and quick charging time through high-density battery pack and fast charging technology and applicable to various vehicle types!
Discover more about Hyundai Motor Group’s EV platform ‘E-GMP’!
Symptoms like intermittent starting and key recognition errors signal potential problems with your Mercedes’ EIS. Use diagnostic steps like error code checks and spare key tests. Professional diagnosis and solutions like EIS replacement ensure safe driving. Consult a qualified technician for accurate diagnosis and repair.
Ever been troubled by the blinking sign and didn’t know what to do?
Here’s a handy guide to dashboard symbols so that you’ll never be confused again!
Save them for later and save the trouble!
Why Is Your BMW X3 Hood Not Responding To Release CommandsDart Auto
Experiencing difficulty opening your BMW X3's hood? This guide explores potential issues like mechanical obstruction, hood release mechanism failure, electrical problems, and emergency release malfunctions. Troubleshooting tips include basic checks, clearing obstructions, applying pressure, and using the emergency release.
Comprehensive program for Agricultural Finance, the Automotive Sector, and Empowerment . We will define the full scope and provide a detailed two-week plan for identifying strategic partners in each area within Limpopo, including target areas.:
1. Agricultural : Supporting Primary and Secondary Agriculture
• Scope: Provide support solutions to enhance agricultural productivity and sustainability.
• Target Areas: Polokwane, Tzaneen, Thohoyandou, Makhado, and Giyani.
2. Automotive Sector: Partnerships with Mechanics and Panel Beater Shops
• Scope: Develop collaborations with automotive service providers to improve service quality and business operations.
• Target Areas: Polokwane, Lephalale, Mokopane, Phalaborwa, and Bela-Bela.
3. Empowerment : Focusing on Women Empowerment
• Scope: Provide business support support and training to women-owned businesses, promoting economic inclusion.
• Target Areas: Polokwane, Thohoyandou, Musina, Burgersfort, and Louis Trichardt.
We will also prioritize Industrial Economic Zone areas and their priorities.
Sign up on https://profilesmes.online/welcome/
To be eligible:
1. You must have a registered business and operate in Limpopo
2. Generate revenue
3. Sectors : Agriculture ( primary and secondary) and Automative
Women and Youth are encouraged to apply even if you don't fall in those sectors.
What Does the PARKTRONIC Inoperative, See Owner's Manual Message Mean for You...Autohaus Service and Sales
Learn what "PARKTRONIC Inoperative, See Owner's Manual" means for your Mercedes-Benz. This message indicates a malfunction in the parking assistance system, potentially due to sensor issues or electrical faults. Prompt attention is crucial to ensure safety and functionality. Follow steps outlined for diagnosis and repair in the owner's manual.
2. Lecture 03: Agenda
1. Complex Numbers
1. Types
2. Arithmetic Operations
2. The Complex Forcing Function
1. Real & Imaginary Sources
2. Real Sources Lead to… Real Responses
3. Imaginary Sources Lead to… Imaginary Responses
4. Applying a Complex Forcing Function
5. AnAlgebraicAlternative to Differential Equations
2
3. 1. Complex Numbers (1/7)
3
Rectangularform
Polar form
Exponential form
A a jb
A| A|
A| A| ej
Three forms
•Arithmetic operations of complex numbers
4. 1.1 Complex Numbers (2/7)
4
Rectangular form
A a jb
j 1
a Re[A]
b Im[ A]
a
b
Im
Re
0
|A|
A
Imaginary unit
Real part
Imaginary part
Magnitude
Angle
Conjugate
a2
b2
A
2
A a2
b2
a
arctan
b
A a jb A A AA A2
5. 1. 2 Complex Numbers (3/7)
5
a
Re
0
|A|
Im
b
A
A a2
b2
a
arctan
b
A| A|
a | A|cos
b | A|sin
Conversion between Rectangular and polar
Polar Form
Conversion
A| A|
A A cos j A sin
A cos jsin
A a jb
A a2
b2
arctan
b
a
6. 1. 3 Complex Numbers (4/7)
6
A Acos jsin
Aej
Euler’s Identity
ej
cos jsin a
b
Re
0
|A|
Im
A
Exponential form
7. 1. 4 Complex Numbers (5/7)
7
Arithmetic Operations
A a jb
A B
Given
Identity
B c jd
a c,b d
Adding and subtracting
C A B e1 je2
e1 a c e2 b d
8. 1. 5 Complex Numbers (6/7)
8
Arithmetic Operations
A a jb B c jd
Given
Product
C AB AejA
BejB
C ejC
C A B
ejC
ej(AB )
C A B
9. 1. 6 Complex Numbers (7/7)
9
B ejB
AejA
D A B
B
A
D A B
ejD
ej(AB ) D A B
D D ejD
B
A
ej(A B )
Arithmetic Operations
A a jb B c jd
Given
Dividing
10. 2. The Complex Forcing Function (1/2)
• Finding current in RL circuit with sinusoidal input involved
solution of non-homogenous differential equation
• It would be impractical to solve non-homogenous differential
equation for every circuit, especially a complex one
• Aim is to find algebraic relationship between inductor
voltage and current; and capacitor voltage and current
• Of course resistor voltage current relationship is algebraic
10
11. 2. The Complex Forcing Function (1/2)
• We introduce complex forcing function to emulate a
sinusoidal forcing function that will result into linear
algebraic equation instead of differential equation
• For linear circuits Steady State Response will follow the
shape of Forcing Function, therefore:
• DC source will produce constant response
• Sinusoidal source will result into Sinusoidal Response
• Exponential and Ramp input will produce exponential and
Ramp response respectively, in Steady State
11
12. 2. The Complex Forcing Function (2/2)
• The basic idea is that sinusoids and exponentials are related
through complex numbers
• Euler’s identity tells us that
12
13. 2.1.1 Real & Imaginary Sources (1/4)
• Whereas taking the derivative of a cosine function yields a
(negative) sine function, the derivative of an exponential is
simply a scaled version of the same exponential
• If at this point the You are thinking, “All this is great, but there
are no imaginary numbers in any circuit I ever plan to build!”
that may be true
• What we’re about to see, however, is that adding imaginary
sources to our circuits leads to complex sources which
(surprisingly) simplify the analysis process
13
14. 2.1.2 Real & Imaginary Sources (2/4)
• It might seem like a strange idea at first, but a moment’s
reflection should remind us that superposition requires any
imaginary sources we might add to cause only imaginary
responses, and real sources can only lead to real responses
• Thus, at any point, we should be able to separate the two by
simply taking the real part of any complex voltage or current
14
15. 2.1.3 Real & Imaginary Sources (3/4)
15
• Areal sinusoid will produce a real sinusoidal response
•An imaginary forcing function will produce imaginary
response
• Acomplex forcing function will produce complex response
• Real part corresponds to real response
• Imaginary part corresponds to imaginary response
16. 2.1.4 Real & Imaginary Sources (4/4)
16
Trick of the trade is to:
•Apply complex forcing function and find complex response
then take only the real part of the complex response
•We also note that for sinusoidal input the output (voltage or
current) at any point on the circuit will also be sinusoidal of the
same frequency but differ only in:
Magnitude and phase
Therefore response at any point on the circuit can be
characterized by two parameters:
Magnitude and phase
(Of course frequency is same as that of the source function)
17. 2.2 Real Sources Lead to… Real
Responses (1/2)
17
The sinusoidal forcing function Vm cos(ωt + θ) produces the
steady-state sinusoidal response Im cos(ωt + φ)
18. 2.2 Imaginary Sources Lead to . . .
Imaginary Responses (2/2)
18
The imaginary sinusoidal forcing function
jVm sin(ωt + θ)
produces the imaginary sinusoidal response
jIm sin(ωt + φ) in the network
19. 2.3Applying a Complex Forcing Function (1/3)
• We have applied a real source and obtained a real response; we
have also applied an imaginary source and obtained an
imaginary response
• Since we are dealing with a linear circuit, we may use the
superposition theorem to find the response to a complex forcing
function which is the sum of the real and imaginary forcing
functions
19
20. • The complex source and response may be represented more
simply by applying Euler’s identity, i.e.,
cos(ωt + θ) + jsin(ωt + θ) = e j(ωt + θ)
• Thus the source can be written as: Vme j(ωt + θ) and response as
Ime j(ωt + ϕ)
20
The complex forcing function Vme j (ωt + θ) produces
the complex response Ime j (ωt + ϕ) in the network
2.3Applying a Complex Forcing Function (2/3)
21. 2.3Applying a Complex Forcing Function (3/3)
• Again, linearity assures us that the real part of the complex
response is produced by the real part of the complex forcing
function, while the imaginary part of the response is caused by
the imaginary part of the complex forcing function
• Our plan is that instead of applying a real forcing function to
obtain the desired real response, we will substitute a complex
forcing function whose real part is the given real forcing
function; we expect to obtain a complex response whose real
part is the desired real response
• The advantage of this procedure is that the integrodifferential
equations describing the steady-state response of a circuit will
now become simple algebraic equations
21
23. 2.4An AlgebraicAlternative to Differential
Equations
23
Which agrees with the response obtained earlier for the same circuit
24. Example
24
• Find the Complex voltage across the series combination of a 500 Ω
resistor and a 95 mH inductor, if the complex current 𝟖𝒆𝒋𝟑𝟎𝟎𝟎𝒕 mA flows
thru the two elements.
Solution:
• We have R = 500 Ω, and L = 95 mH
• The complex source shall be of the form: 𝑉𝑚𝑒𝑗(𝜔𝑡+𝜃)
• From the current expression, we find that ω = 3000 rad/s
• Writing the KVL equation for the series combination, we have
𝑚 𝑑
𝑡
𝑉 𝑒𝑗(3000𝑡+𝜃) = 𝑅𝑖 + 𝐿 𝑑𝑖
= 0.008𝑒𝑗3000𝑡
𝑑
𝑡
500 + (0.095) 𝑑
(0.008𝑒𝑗3000𝑡)
𝑉𝑚𝑒𝑗(3000𝑡+𝜃) = 4𝑒𝑗3000𝑡 + (0.095 × 0.008 × 𝑗3000)𝑒𝑗3000𝑡
or
𝑉𝑚𝑒𝑗(3000𝑡+𝜃) = 4𝑒𝑗3000𝑡 + 𝑗2.28𝑒𝑗3000𝑡 𝑉𝑚𝒆𝒋𝟑𝟎𝟎𝟎𝒕𝑒𝑗𝜃 = (4 + 𝑗2.28)𝒆𝒋𝟑𝟎𝟎𝟎𝒕
• Suppressing the 𝒆𝒋𝟑𝟎𝟎𝟎𝒕 on both sides of the equation we are left with,
𝑚
𝑉 𝑒𝑗𝜃 = 4 + 𝑗2.28 = 4.60𝑒𝑗29.7𝑜
,
• The complex source can be expressed as 𝟒. 𝟔𝟎𝒆𝒋(𝟑𝟎𝟎𝟎𝒕+𝟐𝟗.𝟕𝒐) V
• The real part of the complex source is 𝒗𝒔 = 𝟒. 𝟔𝟎 𝐜𝐨𝐬 𝟑𝟎𝟎𝟎𝒕 + 𝟐𝟗. 𝟕𝒐 V