c
, Contents
Electrical Circuit Analysis is one of the most fundamental subjects of Electrical
Engineering which is taught in two courses in successive semesters under the
names of “Electrical Circuit Analysis I” and “Electrical Circuit Analysis II” or under
the names of “DC Electrical Circuit Analysis” and “AC Electrical Circuit Analysis” in
universities and colleges all over the world. This textbook, like the previously
published DC Electrical Circuit Analysis, includes basic and advanced exercises of
AC Electrical Circuit Analysis with very detailed and multiple methods of solutions.
The textbook can be used as a practicing textbook by students and as a
supplementary teaching source by instructors.
To help students study the textbook in the most efficient way, the exercises
have been categorized in nine different levels. In this regard, for each problem of the
textbook, a difficulty level (easy, normal, or hard) and a calculation amount (small,
normal, or large) have been assigned. Moreover, in each chapter, problems have
been ordered from the easiest problem with the smallest calculations to the most
difficult problem with the largest calculations. Therefore, students are suggested to
start studying the textbook from the easiest problems and continue practicing until
they reach the normal and then the hardest ones. On the other hand, this
classification can help instructors choose their desired problems to conduct a quiz
or a test. Moreover, the classification of computation amount can help students
manage their time during future exams and instructors give the appropriate problems
based on the exam duration.
Since the problems have very detailed solutions and some of them include multiple
methods of solutions, the textbook can be useful for the underprepared students. In
addition, the textbook is beneficial for knowledgeable students because it includes
advanced exercises.
In the preparation of problem solutions, use of typical methods of Electrical Circuit Analysis
has been tried to present the textbook as an instructor-recommended one. In other
words, the heuristic methods have never been used as the first method of problem
solution. By considering this key point, the textbook is in the direction of instructors’
lectures, and the instructors will not see any untaught problem solutions in their
students’answer sheets.
The Iranian University Entrance Exams for the Master’s and PhD degrees of
Electrical Engineering major is the main reference of the textbook; however, all the
problem solutions have been provided by me. The Iranian University Entrance Exam
is one of the most competi- tive university entrance exams in the world that allows
only 10% of the applicants to get into prestigious and tuition-free Iranian universities.
Buffalo, NY, USA Mehdi Rahmani-Andebili
,Contents
1 Problems: Sinusoidal Steady-State Analysis.......................................................1
2 Solutions of Problems: Sinusoidal Steady-State Analysis.................................. 37
3 Problems: Sinusoidal Steady-State Analysis of Circuits Including
Transformers and Magnetically Coupled Inductors149
4 Solutions of Problems: Sinusoidal Steady-State Analysis of Circuits
Including Transformers and Magnetically Coupled Inductors ........................... 169
Index ................................................................................................................... 225
vii
, About the Author
Mehdi Rahmani-Andebili is an Assistant Professor in the Engineering Technology
Department at State University of New York, Buffalo State. He received his first
M.Sc. and Ph.D. degrees in Electrical Engineering (Power System) from Tarbiat
Modares University and Clemson University in 2011 and 2016, respectively, and his
second M.Sc. degree in Physics and Astronomy from the University of Alabama in
Huntsville in 2019. Moreover, he was a Postdoctoral Fellow at Sharif University of
Technology during 2016–2017. As a professor, he has taught many courses such as
Essentials of Electrical Engineering Technology, Electrical Circuits Analysis I,
Electrical Circuits Analysis II, Electrical Circuits and Devices, Industrial Electronics,
and Renewable Distributed Generation and Storage. Dr. Rahmani-Andebili has
more than 100 single-author publications including textbooks, books, book chapters,
journal papers, and conference papers. His research areas include Smart Grid, Power
System Operation and Planning, Integration of Renewables and Energy Storages
into Power System, Energy Scheduling and Demand-Side Management, Plug-in
Electric Vehicles, Distributed Generation, and Advanced Optimization Techniques in
Power System Studies.
ix
, Problems: Sinusoidal Steady-State Analysis
1
Abstract
This chapter helps both groups of underprepared and knowledgeable students taking courses in AC
electrical circuit analysis. In this chapter, the basic and advanced problems of important subjects of AC
circuit analysis, that is, sinusoids and phasors, sinusoidal steady-state analysis, and AC power analysis,
are presented. The problems of sinusoids and phasors include complex numbers; rectangular, polar, and
exponential forms of phasors; phasor relationships for circuit elements; impedance and admittance and their
combinations; resonance frequency; and bandwidth of frequency response of series and parallel RLC circuits.
The problems of sinusoidal steady-state analysis include Kirchhoff’s laws in frequency domain; nodal and
mesh analyses in frequency domain; sinusoidal steady-state response; superposition theorem; source
transformation theorem; Thevenin and Norton theorems; and maximum average power transfer theorem. The
problems of AC power analysis are concerned with root mean square (rms) and peak quantities; average
power, active and reactive powers, apparent power, complex power; and lagging, unity, and leading power
factors. In this chapter, the problems are categorized in different levels based on their difficulty levels (easy,
normal, and hard) and calculation amounts (small, normal, and large). Additionally, the problems are ordered
from the easiest problem with the smallest computations to the most difficult problems with the largest
calculations.
1.1. In the circuit of Figure 1.1, what must be the resistance of the purely resistive load to absorb the maximum average
power [1]? Difficulty level ● Easy ○Normal ○Hard
Calculation amount ●Small ○ Normal ○ Large
1) 5 Ω
2) 7 Ω
3) 7.5 Ω
4) 4 Ω
Figure 1.1 The circuit of problem 1.1
1.2. In the circuit of Figure 1.2, calculate the resistance of R so that it can absorb the maximum average power.
Difficulty level ●Easy ○ Normal ○ Hard
Calculation amount ● ○ Normal ○ Large
Small
# Springer Nature Switzerland AG 2021 1
M. Rahmani-Andebili, AC Electrical Circuit Analysis, https://doi.org/10.1007/978-3-030-60986-3_1
,2 1 Problems: Sinusoidal Steady-State
Analysis
1) 1 Ω
2) 3 Ω
3) 9 Ω
pffiffiffi
4) 1 þ 2 2
Ω
Figure 1.2 The circuit of problem 1.2
1.3. In the circuit of Figure 1.3, calculate the value of vi(t) if the sinusoidal steady-state response of vo(t) is
equal to 2 cos (0.5t 30 ) V.
Difficulty level ●Easy ○ Normal ○ Hard
Calculation amount ●Small ○ Normal ○
Largepffi ffiffi
1) 2 2 cos 0:5t 15 V
pffi ffiffi
2) 2 2 cos 0:5t þ 15 V
3) 2 cos (0.5t + 30 ) V
pffi ffiffi
4) 2 2 cos 0:5t 30 V
Figure 1.3 The circuit of problem 1.3
1.4. In the circuit of Figure 1.4, calculate the sinusoidal steady-state
response of v(t ). Difficulty level ●Easy ○Normal ○Hard
Calculation amount ●Small ○ Normal ○
Large
2
1) 5 cos 2t þ 45 V
2) 5 cos (2t 45 ) V
3) 5 cos (2t 45 ) V
pffiffi
4) 5 2 2 cos 2t 45
V
Figure 1.4 The circuit of problem 1.4
,1 Problems: Sinusoidal Steady-State Analysis 3
1.5. The circuit of Figure 1.5 is in its sinusoidal steady state. Calculate the phasor of the voltage of the
inductor. Difficulty level ● Easy ○Normal ○Hard
Calculation amount ●Small ○ Normal ○ Large
1) pffi2ffiffie j135 V
p
2) ffiffi j135 V
22
ffieffip
2 j45
3) e V
pffi2ffiffi j45
4) 2 e
V
Figure 1.5 The circuit of problem 1.5
1.6. In the circuit of Figure 1.6, what impedance must be connected to terminal a–b to absorb the maximum
average power in sinusoidal steady state?
Difficulty level ●Easy ○ Normal ○ Hard
Calculation amount ●Small ○ Normal ○
Large
1) 4
þj2 Ω
45
2) 52
j 25 Ω
3) þ j 54 Ω
4) 52 j54 Ω
5 5
Figure 1.6 The circuit of problem 1.6
1.7. In the circuit of Figure 1.7, calculate the sinusoidal steady-state voltage of the
inductor. Difficulty level ●Easy ○Normal ○Hard
Calculation amount ●Small ○ Normal ○ Large
1) 2 sin (2t) A
2) 4 sin (2t) A
3) 2 sin (2t) A
4) 4 sin (2t) A
,4 1 Problems: Sinusoidal Steady-State Analysis
Figure 1.7 The circuit of problem 1.7
1.8. In the circuit of Figure 1.8, calculate the internal impedance of the voltage source (ZS) that can absorb the
maximum average power in sinusoidal steady state.
Difficulty level ●Easy ○ Normal ○ Hard
Calculation amount ●Small ○ Normal ○
Large
1) 4 Ω
2) 5 Ω
3) (4 j10) Ω
4) (5 + j10) Ω
Figure 1.8 The circuit of problem 1.8
1.9. Calculate the average power that the current of 2 cos (10t) 3 cos (20t) A delivers to a 4 Ω
resistor. Difficulty level ● Easy ○Normal ○Hard
Calculation amount ●Small ○Normal ○
Large 1) 26 W
2) 13 W
3) 6.5 W
4) 18 W
1.10. Determine the resonance frequency of the circuit shown in
Figure 1.9. Difficulty level ●Easy ○Normal ○ Hard
Calculation amount ●Small ○ Normal ○ Large
1) 5000 rad/sec
2) 1000 rad/sec
3) 500 rad/sec
4) 100 rad/sec
,1 Problems: Sinusoidal Steady-State Analysis 5
Figure 1.9 The circuit of problem
1.10
1.11. In the circuit of Figure 1.10, calculate the voltage of the capacitor in sinusoidal
steady state. Difficulty level ●Easy ○Normal ○ Hard
Calculation amount ●Small ○ Normal ○Large
1) 45.8 cos (100t 24.3 ) V
2) 36.3 cos (1000t 18.4 ) V
3) 4.58 cos (100t 24.3 ) V
4) 3.63 cos (1000t 18.4 ) V
Figure 1.10 The circuit of problem 1.11
1.12. In the circuit of Figure 1.11, v(t) ¼ Acos(t) V. Calculate the sinusoidal steady-state
response of i(t). Difficulty level ● Easy ○Normal ○Hard
Calculation amount ●Small ○ Normal ○ Large
1) Acos(t) A
2) Acos(t) A
3) 21 Acosðt Þ A
4) 41 Acosðt Þ A
Figure 1.11 The circuit of problem 1.12
, 1 Problems: Sinusoidal Steady-State Analysis 5
THOSE WERE PREVIEW PAGES
TO DOWNLOAD THE FULL PDF
CLICK ON THE L.I.N.K
ON THE NEXT PAGE
, Contents
Electrical Circuit Analysis is one of the most fundamental subjects of Electrical
Engineering which is taught in two courses in successive semesters under the
names of “Electrical Circuit Analysis I” and “Electrical Circuit Analysis II” or under
the names of “DC Electrical Circuit Analysis” and “AC Electrical Circuit Analysis” in
universities and colleges all over the world. This textbook, like the previously
published DC Electrical Circuit Analysis, includes basic and advanced exercises of
AC Electrical Circuit Analysis with very detailed and multiple methods of solutions.
The textbook can be used as a practicing textbook by students and as a
supplementary teaching source by instructors.
To help students study the textbook in the most efficient way, the exercises
have been categorized in nine different levels. In this regard, for each problem of the
textbook, a difficulty level (easy, normal, or hard) and a calculation amount (small,
normal, or large) have been assigned. Moreover, in each chapter, problems have
been ordered from the easiest problem with the smallest calculations to the most
difficult problem with the largest calculations. Therefore, students are suggested to
start studying the textbook from the easiest problems and continue practicing until
they reach the normal and then the hardest ones. On the other hand, this
classification can help instructors choose their desired problems to conduct a quiz
or a test. Moreover, the classification of computation amount can help students
manage their time during future exams and instructors give the appropriate problems
based on the exam duration.
Since the problems have very detailed solutions and some of them include multiple
methods of solutions, the textbook can be useful for the underprepared students. In
addition, the textbook is beneficial for knowledgeable students because it includes
advanced exercises.
In the preparation of problem solutions, use of typical methods of Electrical Circuit Analysis
has been tried to present the textbook as an instructor-recommended one. In other
words, the heuristic methods have never been used as the first method of problem
solution. By considering this key point, the textbook is in the direction of instructors’
lectures, and the instructors will not see any untaught problem solutions in their
students’answer sheets.
The Iranian University Entrance Exams for the Master’s and PhD degrees of
Electrical Engineering major is the main reference of the textbook; however, all the
problem solutions have been provided by me. The Iranian University Entrance Exam
is one of the most competi- tive university entrance exams in the world that allows
only 10% of the applicants to get into prestigious and tuition-free Iranian universities.
Buffalo, NY, USA Mehdi Rahmani-Andebili
,Contents
1 Problems: Sinusoidal Steady-State Analysis.......................................................1
2 Solutions of Problems: Sinusoidal Steady-State Analysis.................................. 37
3 Problems: Sinusoidal Steady-State Analysis of Circuits Including
Transformers and Magnetically Coupled Inductors149
4 Solutions of Problems: Sinusoidal Steady-State Analysis of Circuits
Including Transformers and Magnetically Coupled Inductors ........................... 169
Index ................................................................................................................... 225
vii
, About the Author
Mehdi Rahmani-Andebili is an Assistant Professor in the Engineering Technology
Department at State University of New York, Buffalo State. He received his first
M.Sc. and Ph.D. degrees in Electrical Engineering (Power System) from Tarbiat
Modares University and Clemson University in 2011 and 2016, respectively, and his
second M.Sc. degree in Physics and Astronomy from the University of Alabama in
Huntsville in 2019. Moreover, he was a Postdoctoral Fellow at Sharif University of
Technology during 2016–2017. As a professor, he has taught many courses such as
Essentials of Electrical Engineering Technology, Electrical Circuits Analysis I,
Electrical Circuits Analysis II, Electrical Circuits and Devices, Industrial Electronics,
and Renewable Distributed Generation and Storage. Dr. Rahmani-Andebili has
more than 100 single-author publications including textbooks, books, book chapters,
journal papers, and conference papers. His research areas include Smart Grid, Power
System Operation and Planning, Integration of Renewables and Energy Storages
into Power System, Energy Scheduling and Demand-Side Management, Plug-in
Electric Vehicles, Distributed Generation, and Advanced Optimization Techniques in
Power System Studies.
ix
, Problems: Sinusoidal Steady-State Analysis
1
Abstract
This chapter helps both groups of underprepared and knowledgeable students taking courses in AC
electrical circuit analysis. In this chapter, the basic and advanced problems of important subjects of AC
circuit analysis, that is, sinusoids and phasors, sinusoidal steady-state analysis, and AC power analysis,
are presented. The problems of sinusoids and phasors include complex numbers; rectangular, polar, and
exponential forms of phasors; phasor relationships for circuit elements; impedance and admittance and their
combinations; resonance frequency; and bandwidth of frequency response of series and parallel RLC circuits.
The problems of sinusoidal steady-state analysis include Kirchhoff’s laws in frequency domain; nodal and
mesh analyses in frequency domain; sinusoidal steady-state response; superposition theorem; source
transformation theorem; Thevenin and Norton theorems; and maximum average power transfer theorem. The
problems of AC power analysis are concerned with root mean square (rms) and peak quantities; average
power, active and reactive powers, apparent power, complex power; and lagging, unity, and leading power
factors. In this chapter, the problems are categorized in different levels based on their difficulty levels (easy,
normal, and hard) and calculation amounts (small, normal, and large). Additionally, the problems are ordered
from the easiest problem with the smallest computations to the most difficult problems with the largest
calculations.
1.1. In the circuit of Figure 1.1, what must be the resistance of the purely resistive load to absorb the maximum average
power [1]? Difficulty level ● Easy ○Normal ○Hard
Calculation amount ●Small ○ Normal ○ Large
1) 5 Ω
2) 7 Ω
3) 7.5 Ω
4) 4 Ω
Figure 1.1 The circuit of problem 1.1
1.2. In the circuit of Figure 1.2, calculate the resistance of R so that it can absorb the maximum average power.
Difficulty level ●Easy ○ Normal ○ Hard
Calculation amount ● ○ Normal ○ Large
Small
# Springer Nature Switzerland AG 2021 1
M. Rahmani-Andebili, AC Electrical Circuit Analysis, https://doi.org/10.1007/978-3-030-60986-3_1
,2 1 Problems: Sinusoidal Steady-State
Analysis
1) 1 Ω
2) 3 Ω
3) 9 Ω
pffiffiffi
4) 1 þ 2 2
Ω
Figure 1.2 The circuit of problem 1.2
1.3. In the circuit of Figure 1.3, calculate the value of vi(t) if the sinusoidal steady-state response of vo(t) is
equal to 2 cos (0.5t 30 ) V.
Difficulty level ●Easy ○ Normal ○ Hard
Calculation amount ●Small ○ Normal ○
Largepffi ffiffi
1) 2 2 cos 0:5t 15 V
pffi ffiffi
2) 2 2 cos 0:5t þ 15 V
3) 2 cos (0.5t + 30 ) V
pffi ffiffi
4) 2 2 cos 0:5t 30 V
Figure 1.3 The circuit of problem 1.3
1.4. In the circuit of Figure 1.4, calculate the sinusoidal steady-state
response of v(t ). Difficulty level ●Easy ○Normal ○Hard
Calculation amount ●Small ○ Normal ○
Large
2
1) 5 cos 2t þ 45 V
2) 5 cos (2t 45 ) V
3) 5 cos (2t 45 ) V
pffiffi
4) 5 2 2 cos 2t 45
V
Figure 1.4 The circuit of problem 1.4
,1 Problems: Sinusoidal Steady-State Analysis 3
1.5. The circuit of Figure 1.5 is in its sinusoidal steady state. Calculate the phasor of the voltage of the
inductor. Difficulty level ● Easy ○Normal ○Hard
Calculation amount ●Small ○ Normal ○ Large
1) pffi2ffiffie j135 V
p
2) ffiffi j135 V
22
ffieffip
2 j45
3) e V
pffi2ffiffi j45
4) 2 e
V
Figure 1.5 The circuit of problem 1.5
1.6. In the circuit of Figure 1.6, what impedance must be connected to terminal a–b to absorb the maximum
average power in sinusoidal steady state?
Difficulty level ●Easy ○ Normal ○ Hard
Calculation amount ●Small ○ Normal ○
Large
1) 4
þj2 Ω
45
2) 52
j 25 Ω
3) þ j 54 Ω
4) 52 j54 Ω
5 5
Figure 1.6 The circuit of problem 1.6
1.7. In the circuit of Figure 1.7, calculate the sinusoidal steady-state voltage of the
inductor. Difficulty level ●Easy ○Normal ○Hard
Calculation amount ●Small ○ Normal ○ Large
1) 2 sin (2t) A
2) 4 sin (2t) A
3) 2 sin (2t) A
4) 4 sin (2t) A
,4 1 Problems: Sinusoidal Steady-State Analysis
Figure 1.7 The circuit of problem 1.7
1.8. In the circuit of Figure 1.8, calculate the internal impedance of the voltage source (ZS) that can absorb the
maximum average power in sinusoidal steady state.
Difficulty level ●Easy ○ Normal ○ Hard
Calculation amount ●Small ○ Normal ○
Large
1) 4 Ω
2) 5 Ω
3) (4 j10) Ω
4) (5 + j10) Ω
Figure 1.8 The circuit of problem 1.8
1.9. Calculate the average power that the current of 2 cos (10t) 3 cos (20t) A delivers to a 4 Ω
resistor. Difficulty level ● Easy ○Normal ○Hard
Calculation amount ●Small ○Normal ○
Large 1) 26 W
2) 13 W
3) 6.5 W
4) 18 W
1.10. Determine the resonance frequency of the circuit shown in
Figure 1.9. Difficulty level ●Easy ○Normal ○ Hard
Calculation amount ●Small ○ Normal ○ Large
1) 5000 rad/sec
2) 1000 rad/sec
3) 500 rad/sec
4) 100 rad/sec
,1 Problems: Sinusoidal Steady-State Analysis 5
Figure 1.9 The circuit of problem
1.10
1.11. In the circuit of Figure 1.10, calculate the voltage of the capacitor in sinusoidal
steady state. Difficulty level ●Easy ○Normal ○ Hard
Calculation amount ●Small ○ Normal ○Large
1) 45.8 cos (100t 24.3 ) V
2) 36.3 cos (1000t 18.4 ) V
3) 4.58 cos (100t 24.3 ) V
4) 3.63 cos (1000t 18.4 ) V
Figure 1.10 The circuit of problem 1.11
1.12. In the circuit of Figure 1.11, v(t) ¼ Acos(t) V. Calculate the sinusoidal steady-state
response of i(t). Difficulty level ● Easy ○Normal ○Hard
Calculation amount ●Small ○ Normal ○ Large
1) Acos(t) A
2) Acos(t) A
3) 21 Acosðt Þ A
4) 41 Acosðt Þ A
Figure 1.11 The circuit of problem 1.12
, 1 Problems: Sinusoidal Steady-State Analysis 5
THOSE WERE PREVIEW PAGES
TO DOWNLOAD THE FULL PDF
CLICK ON THE L.I.N.K
ON THE NEXT PAGE
