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AC Electrical Circuit AnalysisAC Electrical Circuit Analysis
A Practical ApproachA Practical Approach
by
James M. Fiore
Version 1.1.10, 30 March 2023
3
A Practical ApproachA Practical Approach
by
James M. Fiore
Version 1.1.10, 30 March 2023
3
This AC Electrical Circuit Analysis, by James M. Fiore is copyrighted under the terms of
a Creative Commons license:
This work is freely redistributable for non-commercial use, share-alike with attribution
Device data sheets and other product information are copyright by their respective owners
and have been obtained through publicly accessible manufacturer's web sites.
Published by James M. Fiore via dissidents
ISBN13: 979-8605022282
For more information or feedback, contact:
James Fiore, Professor
Electrical Engineering Technology
Mohawk Valley Community College
1101 Sherman Drive
Utica, NY 13501
jfiore@mvcc.edu
oer@jimfiore.org
For the latest revisions, related titles, and links to low cost print versions, go to:
www.mvcc.edu/jfiore or my mirror sites www.dissidents.com and www.jimfiore.org
YouTube Channel: Electronics with Professor Fiore
Cover art, Chapman's Contribution Redux, by the author
4
a Creative Commons license:
This work is freely redistributable for non-commercial use, share-alike with attribution
Device data sheets and other product information are copyright by their respective owners
and have been obtained through publicly accessible manufacturer's web sites.
Published by James M. Fiore via dissidents
ISBN13: 979-8605022282
For more information or feedback, contact:
James Fiore, Professor
Electrical Engineering Technology
Mohawk Valley Community College
1101 Sherman Drive
Utica, NY 13501
jfiore@mvcc.edu
oer@jimfiore.org
For the latest revisions, related titles, and links to low cost print versions, go to:
www.mvcc.edu/jfiore or my mirror sites www.dissidents.com and www.jimfiore.org
YouTube Channel: Electronics with Professor Fiore
Cover art, Chapman's Contribution Redux, by the author
4
IntroductionIntroduction
Welcome to the AC Electrical Circuit Analysis, an open educational resource (OER). The goal of this text is to
introduce the theory and practical application of analysis of AC electrical circuits. It assumes familiarity with DC
circuit analysis. If you have not studied DC circuit analysis, it is strongly recommended that you read the
companion OER text, DC Electrical Circuit Analysis before continuing. Both texts are offered free of charge
under a Creative Commons non-commercial, share-alike with attribution license. For your convenience, along
with the free pdf and odt files, print copies are available at a very modest charge. Check my web sites for links.
This text is based on the earlier Workbook for AC Electrical Circuits, which it replaces. The original expository
text has been greatly expanded and includes many examples along with computer simulations. For the
convenience of those who used the Workbook, many of the problem sets are the same, with some re-ordering
depending on the chapter.
If you are already familiar with DC Electrical Circuit Analysis, the format of this title is similar. This text picks
up where the DC text leaves off; beginning with AC concepts such as sinusoidal waveforms, basic Fourier
decomposition of complex waveforms, complex numbers and the like. Also, reactance and impedance are
introduced along with phasor diagrams. Chapters on series, parallel and series-parallel RLC circuits commence.
Following these, network theorems along with nodal and mesh analysis are discussed for the AC case. The text
completes with chapters on AC power, resonance, and introductions to polyphase systems and magnetic circuits.
Each chapter begins with a set of learning objectives and concludes with practice exercises that are generally
divided into four major types: analysis, design, challenge and simulation. Many SPICE-based circuit simulators
are available, both free and commercial, that can be used with this text. The answers to most odd-numbered
exercises can be found in the Appendix. A table of standard resistor sizes is also in the Appendix, which is useful
for real-world design problems. If you have any questions regarding this workbook, or are interested in
contributing to the project, do not hesitate to contact me.
This text is part of a series of OER titles in the areas of electricity, electronics, audio and computer
programming. It includes three other textbooks covering semiconductor devices, operational amplifiers, and
embedded programming using the C language with the Arduino platform. There is a text covering DC electrical
circuits similar to this one, and also seven laboratory manuals; one for each of the five texts plus individual titles
covering computer programming using the Python language, and the science of sound. The most recent versions
of all of my OER texts and manuals may be found at my MVCC web site as well as my mirror site:
www.dissidents.com
This text was created using several free and open software applications including Open Office, Dia, SciDAVis,
and XnView.
Special thanks to the following individuals for their efforts in reviewing and proofreading the DC and AC
Electrical Circuit Analysis texts: Glenn Ballard, John Markham, João Nuno Carvalho, Mark Steffka and Jim
Noon.
5
Welcome to the AC Electrical Circuit Analysis, an open educational resource (OER). The goal of this text is to
introduce the theory and practical application of analysis of AC electrical circuits. It assumes familiarity with DC
circuit analysis. If you have not studied DC circuit analysis, it is strongly recommended that you read the
companion OER text, DC Electrical Circuit Analysis before continuing. Both texts are offered free of charge
under a Creative Commons non-commercial, share-alike with attribution license. For your convenience, along
with the free pdf and odt files, print copies are available at a very modest charge. Check my web sites for links.
This text is based on the earlier Workbook for AC Electrical Circuits, which it replaces. The original expository
text has been greatly expanded and includes many examples along with computer simulations. For the
convenience of those who used the Workbook, many of the problem sets are the same, with some re-ordering
depending on the chapter.
If you are already familiar with DC Electrical Circuit Analysis, the format of this title is similar. This text picks
up where the DC text leaves off; beginning with AC concepts such as sinusoidal waveforms, basic Fourier
decomposition of complex waveforms, complex numbers and the like. Also, reactance and impedance are
introduced along with phasor diagrams. Chapters on series, parallel and series-parallel RLC circuits commence.
Following these, network theorems along with nodal and mesh analysis are discussed for the AC case. The text
completes with chapters on AC power, resonance, and introductions to polyphase systems and magnetic circuits.
Each chapter begins with a set of learning objectives and concludes with practice exercises that are generally
divided into four major types: analysis, design, challenge and simulation. Many SPICE-based circuit simulators
are available, both free and commercial, that can be used with this text. The answers to most odd-numbered
exercises can be found in the Appendix. A table of standard resistor sizes is also in the Appendix, which is useful
for real-world design problems. If you have any questions regarding this workbook, or are interested in
contributing to the project, do not hesitate to contact me.
This text is part of a series of OER titles in the areas of electricity, electronics, audio and computer
programming. It includes three other textbooks covering semiconductor devices, operational amplifiers, and
embedded programming using the C language with the Arduino platform. There is a text covering DC electrical
circuits similar to this one, and also seven laboratory manuals; one for each of the five texts plus individual titles
covering computer programming using the Python language, and the science of sound. The most recent versions
of all of my OER texts and manuals may be found at my MVCC web site as well as my mirror site:
www.dissidents.com
This text was created using several free and open software applications including Open Office, Dia, SciDAVis,
and XnView.
Special thanks to the following individuals for their efforts in reviewing and proofreading the DC and AC
Electrical Circuit Analysis texts: Glenn Ballard, John Markham, João Nuno Carvalho, Mark Steffka and Jim
Noon.
5
For the Have-nots and the Has-beens
“Well, the people, I would say. There is no patent. Could you patent the sun?”
— Jonas Salk, inventor of the polio vaccine,
when asked who owns the patent to it.
6
“Well, the people, I would say. There is no patent. Could you patent the sun?”
— Jonas Salk, inventor of the polio vaccine,
when asked who owns the patent to it.
6
Table of ContentsTable of Contents
Chapter 1: Fundamentals . . . . . . . 10
1.0 Chapter Objectives . . . . . . . . 10
1.1 Introduction . . . . . . . . . 10
1.2 Sinusoidal Waveforms . . . . . . . 11
1.3 Basic Fourier Analysis . . . . . . . . 25
1.4 Complex Numbers . . . . . . . . 29
1.5 Reactance and Impedance . . . . . . . 31
1.6 Phasor Diagrams . . . . . . . . 37
Summary . . . . . . . . . 38
Exercises . . . . . . . . . 39
Chapter 2: Series RLC Circuits . . . . . . 44
2.0 Chapter Objectives . . . . . . . . 44
2.1 Introduction . . . . . . . . . 44
2.2 The Series Connection . . . . . . . 44
2.3 Series Impedance . . . . . . . . 45
2.4 Series Circuit Analysis. . . . . . . . 48
2.5 Concerning Practical Inductors . . . . . . 61
Summary . . . . . . . . . 63
Exercises . . . . . . . . . 64
Chapter 3: Parallel RLC Circuits . . . . . 80
3.0 Chapter Objectives . . . . . . . . 80
3.1 Introduction . . . . . . . . . 80
3.2 The Parallel Connection . . . . . . 81
3.3 Parallel Impedance . . . . . . . . 81
3.4 Parallel Circuit Analysis . . . . . . . 85
Summary . . . . . . . . . 100
Exercises . . . . . . . . . 101
Chapter 4: Series-Parallel RLC Circuits . . . . 110
4.0 Chapter Objectives . . . . . . . . 110
4.1 Introduction . . . . . . . . . 110
4.3 The Series-Parallel Connection . . . . . . 110
4.4 Series-Parallel Impedance . . . . . . . 111
4.5 Series-Parallel Circuit Analysis . . . . . . . 114
Summary . . . . . . . . . 132
Exercises . . . . . . . . . 133
7
Chapter 1: Fundamentals . . . . . . . 10
1.0 Chapter Objectives . . . . . . . . 10
1.1 Introduction . . . . . . . . . 10
1.2 Sinusoidal Waveforms . . . . . . . 11
1.3 Basic Fourier Analysis . . . . . . . . 25
1.4 Complex Numbers . . . . . . . . 29
1.5 Reactance and Impedance . . . . . . . 31
1.6 Phasor Diagrams . . . . . . . . 37
Summary . . . . . . . . . 38
Exercises . . . . . . . . . 39
Chapter 2: Series RLC Circuits . . . . . . 44
2.0 Chapter Objectives . . . . . . . . 44
2.1 Introduction . . . . . . . . . 44
2.2 The Series Connection . . . . . . . 44
2.3 Series Impedance . . . . . . . . 45
2.4 Series Circuit Analysis. . . . . . . . 48
2.5 Concerning Practical Inductors . . . . . . 61
Summary . . . . . . . . . 63
Exercises . . . . . . . . . 64
Chapter 3: Parallel RLC Circuits . . . . . 80
3.0 Chapter Objectives . . . . . . . . 80
3.1 Introduction . . . . . . . . . 80
3.2 The Parallel Connection . . . . . . 81
3.3 Parallel Impedance . . . . . . . . 81
3.4 Parallel Circuit Analysis . . . . . . . 85
Summary . . . . . . . . . 100
Exercises . . . . . . . . . 101
Chapter 4: Series-Parallel RLC Circuits . . . . 110
4.0 Chapter Objectives . . . . . . . . 110
4.1 Introduction . . . . . . . . . 110
4.3 The Series-Parallel Connection . . . . . . 110
4.4 Series-Parallel Impedance . . . . . . . 111
4.5 Series-Parallel Circuit Analysis . . . . . . . 114
Summary . . . . . . . . . 132
Exercises . . . . . . . . . 133
7
Chapter 5: Analysis Theorems and Techniques . . . . 144
5.0 Chapter Objectives . . . . . . . . 144
5.1 Introduction . . . . . . . . . 144
5.2 Source Conversions . . . . . . . . 145
5.3 Superposition Theorem . . . . . . . 153
5.4 Thévenin's and Norton's Theorems . . . . . . 160
5.5 Maximum Power Transfer Theorem . . . . . . 170
5.6 Delta-Y (Pi-T) Conversions . . . . . . . 179
Summary . . . . . . . . . 185
Exercises . . . . . . . . . 187
Chapter 6: Nodal and Mesh Analysis . . . . . 204
6.0 Chapter Objectives . . . . . . . . 204
6.1 Introduction . . . . . . . . . 204
6.2 Nodal Analysis . . . . . . . . 205
6.3 Mesh Analysis . . . . . . . . 222
6.4 Dependent Sources . . . . . . . . 234
Summary . . . . . . . . . 240
Exercises . . . . . . . . . 242
Chapter 7: AC Power . . . . . . . 260
7.0 Chapter Objectives . . . . . . . . 260
7.1 Introduction . . . . . . . . . 260
7.2 Power Waveforms . . . . . . . . 261
7.3 Power Triangle . . . . . . . . 273
7.4 Power Systems . . . . . . . . 280
Summary . . . . . . . . . 286
Exercises . . . . . . . . . 287
Chapter 8: Resonance . . . . . . . 294
8.0 Chapter Objectives . . . . . . . . 294
8.1 Introduction . . . . . . . . . 294
8.2 Series Resonance . . . . . . . . 295
8.3 Parallel Resonance . . . . . . . . 310
Summary . . . . . . . . . 329
Exercises . . . . . . . . . 331
8
5.0 Chapter Objectives . . . . . . . . 144
5.1 Introduction . . . . . . . . . 144
5.2 Source Conversions . . . . . . . . 145
5.3 Superposition Theorem . . . . . . . 153
5.4 Thévenin's and Norton's Theorems . . . . . . 160
5.5 Maximum Power Transfer Theorem . . . . . . 170
5.6 Delta-Y (Pi-T) Conversions . . . . . . . 179
Summary . . . . . . . . . 185
Exercises . . . . . . . . . 187
Chapter 6: Nodal and Mesh Analysis . . . . . 204
6.0 Chapter Objectives . . . . . . . . 204
6.1 Introduction . . . . . . . . . 204
6.2 Nodal Analysis . . . . . . . . 205
6.3 Mesh Analysis . . . . . . . . 222
6.4 Dependent Sources . . . . . . . . 234
Summary . . . . . . . . . 240
Exercises . . . . . . . . . 242
Chapter 7: AC Power . . . . . . . 260
7.0 Chapter Objectives . . . . . . . . 260
7.1 Introduction . . . . . . . . . 260
7.2 Power Waveforms . . . . . . . . 261
7.3 Power Triangle . . . . . . . . 273
7.4 Power Systems . . . . . . . . 280
Summary . . . . . . . . . 286
Exercises . . . . . . . . . 287
Chapter 8: Resonance . . . . . . . 294
8.0 Chapter Objectives . . . . . . . . 294
8.1 Introduction . . . . . . . . . 294
8.2 Series Resonance . . . . . . . . 295
8.3 Parallel Resonance . . . . . . . . 310
Summary . . . . . . . . . 329
Exercises . . . . . . . . . 331
8
Chapter 9: Polyphase Power . . . . . . 338
9.0 Chapter Objectives . . . . . . . . 338
9.1 Introduction . . . . . . . . . 338
9.2 Polyphase Definition . . . . . . . . 339
9.3 Three-Phase Connections . . . . . . . 341
9.4 Power Factor Correction . . . . . . . 354
Summary . . . . . . . . . 364
Exercises . . . . . . . . . 365
Chapter 10: Decibels and Bode Plots . . . . . 372
10.0 Chapter Objectives . . . . . . . . 372
10.1 Introduction . . . . . . . . 372
10.2 The Decibel . . . . . . . . . 372
10.3 Bode Plots . . . . . . . . . 383
10.4 Combining the Elements - Multi-Stage Effects . . . . . 393
Summary . . . . . . . . . 395
Exercises . . . . . . . . . 396
Appendices
A: Standard Component Sizes . . . . . . . 399
B: Methods of Solution of Linear Simultaneous Equations . . . 400
C: Equation Proofs . . . . . . . . 405
D: Answers to Selected Odd-Numbered Problems . . . . 408
E: Questions for Selected Odd Answers . . . . . 422
9
9.0 Chapter Objectives . . . . . . . . 338
9.1 Introduction . . . . . . . . . 338
9.2 Polyphase Definition . . . . . . . . 339
9.3 Three-Phase Connections . . . . . . . 341
9.4 Power Factor Correction . . . . . . . 354
Summary . . . . . . . . . 364
Exercises . . . . . . . . . 365
Chapter 10: Decibels and Bode Plots . . . . . 372
10.0 Chapter Objectives . . . . . . . . 372
10.1 Introduction . . . . . . . . 372
10.2 The Decibel . . . . . . . . . 372
10.3 Bode Plots . . . . . . . . . 383
10.4 Combining the Elements - Multi-Stage Effects . . . . . 393
Summary . . . . . . . . . 395
Exercises . . . . . . . . . 396
Appendices
A: Standard Component Sizes . . . . . . . 399
B: Methods of Solution of Linear Simultaneous Equations . . . 400
C: Equation Proofs . . . . . . . . 405
D: Answers to Selected Odd-Numbered Problems . . . . 408
E: Questions for Selected Odd Answers . . . . . 422
9
11 基础基础
1.0 章节 学习目标1.0 章节 学习目标
学完本章后，您应该能够：
• 用数学术语描述正弦波形。
• 使用基本的傅里叶分析来表达复杂的波形。
• 使用复数执行数学运算。
• 绘制复数的相量图。
• 计算并绘制电容和电感电抗与频率的函数关系。
• 计算并绘制复阻抗作为频率的函数。
• 描述电阻、电抗、阻抗、电导、电纳和
导纳
1.1 介绍1.1 介绍
在本章中，我们开始研究交流电，或 交流、电路分析。因此，本章
》中充满了将在本文其余部分使用的基础材料。这些概念至关重要
为了在后面的章节中取得成功，请充分理解这里介绍的内容。我们从数学开始
最简单的描述 交流 waveform，即正弦波。这包括振幅、
频率、周期、相位和 直流 抵消。从那里，我们发现如何根据
正弦波的组合，以及如何确定有效的 或 直流 equivalent， 正弦波的值。 交流
如果不使用复数，几乎不可能进行分析，接下来是讨论
他们的描述和适当的操作。最后，我们介绍了电抗和阻抗的概念，以及
它们与抵抗的关系。这包括使用频域图和相量进行检查
图。相量图是矢量图，可用于显示各种电压之间的关系
在电路中，以及电流或电阻/电抗值之间。
本文中的许多主题将与您的研究相呼应 直流 电路分析，例如欧姆定律、基尔霍夫定律
电压和电流定律、串并联分析、节点分析等。因此，许多概念将是
熟悉。主要的实际区别在于 直流 系统是 标量，也就是说，他们只有
大小。相比之下， 交流 数量为 向量，因此同时具有大小和方向（或者更准确地说，
phase） 的后果;y，则加法或乘法等数学运算必须遵循向量规则
而不是标量规则。忘记这样做对于刚接触该学科的学生来说是一个常见的错误，而且会
带来无尽的悲伤。这些规则将在介绍复数的部分中进行回顾。确保
在继续之前，您已经掌握了这些材料。
10
1.0 章节 学习目标1.0 章节 学习目标
学完本章后，您应该能够：
• 用数学术语描述正弦波形。
• 使用基本的傅里叶分析来表达复杂的波形。
• 使用复数执行数学运算。
• 绘制复数的相量图。
• 计算并绘制电容和电感电抗与频率的函数关系。
• 计算并绘制复阻抗作为频率的函数。
• 描述电阻、电抗、阻抗、电导、电纳和
导纳
1.1 介绍1.1 介绍
在本章中，我们开始研究交流电，或 交流、电路分析。因此，本章
》中充满了将在本文其余部分使用的基础材料。这些概念至关重要
为了在后面的章节中取得成功，请充分理解这里介绍的内容。我们从数学开始
最简单的描述 交流 waveform，即正弦波。这包括振幅、
频率、周期、相位和 直流 抵消。从那里，我们发现如何根据
正弦波的组合，以及如何确定有效的 或 直流 equivalent， 正弦波的值。 交流
如果不使用复数，几乎不可能进行分析，接下来是讨论
他们的描述和适当的操作。最后，我们介绍了电抗和阻抗的概念，以及
它们与抵抗的关系。这包括使用频域图和相量进行检查
图。相量图是矢量图，可用于显示各种电压之间的关系
在电路中，以及电流或电阻/电抗值之间。
本文中的许多主题将与您的研究相呼应 直流 电路分析，例如欧姆定律、基尔霍夫定律
电压和电流定律、串并联分析、节点分析等。因此，许多概念将是
熟悉。主要的实际区别在于 直流 系统是 标量，也就是说，他们只有
大小。相比之下， 交流 数量为 向量，因此同时具有大小和方向（或者更准确地说，
phase） 的后果;y，则加法或乘法等数学运算必须遵循向量规则
而不是标量规则。忘记这样做对于刚接触该学科的学生来说是一个常见的错误，而且会
带来无尽的悲伤。这些规则将在介绍复数的部分中进行回顾。确保
在继续之前，您已经掌握了这些材料。
10
1.2 正弦波形1.2 正弦波形
交流或交流电之所以如此命名，是因为电流交流或反转
在两个极性之间。换句话说，当前（以及因此的
voltage） 是时间的函数。这与直流电有根本的不同
的极性是固定的，并且通常随时间变化而保持不变。实验室 直流 电压源 /
例如，理想情况下，在其端子上保持设定电压并且不会变化
随着时间的推移。相比之下，作为 交流 波形随时间来回摆动，其
shape 可以表现出广泛的变化，从简单、规则的路径
实验室标准，如正弦波、三角波和方波，到 FAR
乐器产生的更复杂和起伏的波形，以及
人声。
正弦波是可能产生的最简单的波。它表示
以恒定速度旋转的简单矢量，例如
时钟秒针。图 1.1 显示了一个例子。水平轴图
时间。当我们从左向右移动时，它会增加（即，如果点 A 位于点的右侧
B，则 A 出现的时间晚于 B）。纵轴在此处表示为
general 作为 maximum 的百分比，但通常是
电压、电流、声压等。
Note the smooth variation that starts at zero, rises to a positive peak one quarter way
through, falls back to zero when halfway through, continues to a negative peak three
quarters through, and then rises again to where it started. This process then repeats.
Each repeat is referred to as a cycle. In Figure 1.1, one complete cycle is shown.
11
图 1.1
正弦波。
正弦波
峰值百分比
-100
-80
-60
-40
-20
0
20
40
60
80
100
Time (cycles)
0 0.25 0.5 0.75 1
交流或交流电之所以如此命名，是因为电流交流或反转
在两个极性之间。换句话说，当前（以及因此的
voltage） 是时间的函数。这与直流电有根本的不同
的极性是固定的，并且通常随时间变化而保持不变。实验室 直流 电压源 /
例如，理想情况下，在其端子上保持设定电压并且不会变化
随着时间的推移。相比之下，作为 交流 波形随时间来回摆动，其
shape 可以表现出广泛的变化，从简单、规则的路径
实验室标准，如正弦波、三角波和方波，到 FAR
乐器产生的更复杂和起伏的波形，以及
人声。
正弦波是可能产生的最简单的波。它表示
以恒定速度旋转的简单矢量，例如
时钟秒针。图 1.1 显示了一个例子。水平轴图
时间。当我们从左向右移动时，它会增加（即，如果点 A 位于点的右侧
B，则 A 出现的时间晚于 B）。纵轴在此处表示为
general 作为 maximum 的百分比，但通常是
电压、电流、声压等。
Note the smooth variation that starts at zero, rises to a positive peak one quarter way
through, falls back to zero when halfway through, continues to a negative peak three
quarters through, and then rises again to where it started. This process then repeats.
Each repeat is referred to as a cycle. In Figure 1.1, one complete cycle is shown.
11
图 1.1
正弦波。
正弦波
峰值百分比
-100
-80
-60
-40
-20
0
20
40
60
80
100
Time (cycles)
0 0.25 0.5 0.75 1
