Home > Articles > Hardware

Introduction to AC Circuits

This chapter's topics are of practical importance in ac systems. The fundamental terminologies used in ac circuits are introduced, and a number of custom-written VIs, considering both single-phase and three-phase ac circuits, are provided.
This chapter is from the book

Electric power is generated, distributed, and used as sinusoidal voltages and currents in a great variety of commercial and domestic applications. Furthermore, in the industrial world a vast number of small-, medium-, or large-size ac power generators and loads are interlinked. Therefore, the design, operation, maintenance, and management of such systems very much depend on a good understanding of ac circuit theory.

The analysis of ac circuits involves the study of the behavior of the system under both normal and abnormal operating conditions. However, this book is not intended to include abnormal conditions. Instead, it focuses on the foremost fundamental issues and normal conditions and presents visual aids and interactive tools. Additionally, it is assumed that a steady-state sinusoidal condition is reached, which means that all transient effects in ac circuits have disappeared.

This chapter's topics are of practical importance in ac systems. The fundamental terminologies used in ac circuits are introduced, and a number of custom-written VIs, considering both single-phase and three-phase ac circuits, are provided.

The chapter begins with essential definitions of such terms as power factor, phasor, impedance, and per-unit value.

Section 3.2 describes the topological analysis of five basic ac electric circuits containing impedances and ac supplies. Then a reverse study is presented where an unknown impedance is determined by user-defined ac voltage and currents.

Section 3.3 is concerned with a description and visual demonstration of the powers in ac circuits and is followed by a discussion of power factor correction in Section 3.4.

The remaining sections describe various other aspects of three-phase circuits, accompanied by interactive VI modules. The chapter ends with a comprehensive study of real-time three-phase data logging.

Educational Objectives

The chapter develops appropriate relationships and visual aids for describing ac systems using ac voltage, ac current, impedance, ac power, and phasors. After completing this chapter, students should be able to

  • plot and interpret the characteristics of ac voltage, current, and power waveforms.

  • understand the definitions of peak to peak; peak and rms values; and phase/line voltage and current, phase angle, power factor, complex impedance, phasor and base (per-unit) values in ac circuits.

  • state the equations for series, parallel, and combination ac circuits that contain impedances and ac supplies, understand the effects on current caused by changes in impedances, and state the meaning of the term equivalent impedance.

  • analyze the sinusoidal steady-state behavior of single- and three-phase ac circuits using phasors, and study the effect of resistive, inductive, and capacitive loads in single-phase ac circuits.

  • understand the concept of complex power, power measurement methods, and power factor correction.

  • recognize the data logging techniques associated with ac circuits.

3.1 Fundamental Definitions

Steady-state sinusoidal time-varying voltage and current waveforms can be given by

Equation 3.1

03equ01.gif


Equation 3.2

03equ02.gif


where v and i are the time-varying voltage and current, and Vm and Im are the peak values (magnitudes or amplitudes) of the voltage and current waveforms. In equation 3.2, θ is known as the phase angle, which is normally defined with reference to the voltage waveform.

The term cos θ is called a power factor. Remember that we assumed a voltage having a zero phase. In general, the phase of the voltage may have a value other than zero. Then θ should be taken as the phase of the voltage minus the phase of the current.

In a linear circuit excited by sinusoidal sources, in the steady-state, all voltages and currents are sinusoidal and have the same frequency. However, there may be a phase difference between the voltage and current depending on the type of load used.

The three basic passive circuit elements, the resistor (R), the inductor (L), and the capacitor (C), are considered in this chapter. An ac load may be a combination of these passive elements, such as R + L or R + C.

Note that the current and voltage waveforms in the resistor are in phase, while inductances and capacitors both have a 90° phase shift between voltage and current. The inductor current waveform lags the inductor voltage waveform by 90°, while in the capacitor, the current leads the voltage by 90°.

The peak-to-peak value is also used in the analysis of ac circuits; it is the difference between the highest and lowest values of the waveform over one cycle. This can easily be visualized in the ac waveforms generated.

It might seem difficult to describe an ac signal in terms of a specific value, since an ac signal is not constant. However, as shown in Chapter 2, these sinusoidal signals are periodic, repeating the same pattern of values in each period. Therefore, when a voltage or a current is described simply as ac, we will refer to its rms or effective value, not its maximum value, which simplifies the description of ac signal.

Power Factor

In general, for nonsinusoidal systems (distorted waveforms), the power factor PF is equal to

Equation 3.3

03equ03.gif


where Stotal is the total apparent (or complex) power in VA, which is equal to Vrms · Irms.

If the voltage and current waveforms of an ac system are measured in real time, it is much easier to calculate the power factor simply by using the definitions of average and rms values (given in equations 2.3 and 2.4).

However, in many utility applications, the distortion in ac voltage is usually small, hence the voltage waveform can be assumed as an ideal sine wave at fundamental frequency. This assumption simplifies the analysis, which results in an analytical solution of power factor for the nonsinusoidal systems as

Equation 3.4

03equ04.gif


where Is is the rms value of the current (as in equation 2.4 for sinusoidal quantities), Is1 is the rms Fourier fundamental component of the current, and cos θ is the power factor in linear circuits with sinusoidal voltages and currents.

From equation 3.4, we can conclude that a large distortion in the current waveform will result in a small value of the current ratio, Is1/Is, which means a small value of power factor, even if θ = 0 (a unity power factor, cos θ = 1).

As seen, since the Fourier analysis is involved for the estimation of Is1, a quantity called total harmonic distortion (THD) can be defined in the current, which simplifies the estimation of PF.

Equation 3.5

03equ05.gif


Equation 3.6

03equ06.gif


Note that the definitions given for the PF will be utilized in the transformer experiment later since the excitation current of transformers is highly distorted.

Phasors

In most ac circuit studies, the frequency is fixed, so this feature is used to simplify the analysis. Sinusoidal steady-state analysis is greatly facilitated if the currents and voltages are represented as vectors in the complex number plane known as phasors. The basic purpose of phasors is to show the rms value (or the magnitude in some cases) and phase angle between two or multiple quantities, such as voltage and current.

The phasors can be defined in many forms, such as rectangular, polar, exponential, or trigonometric:

Equation 3.7

03equ07.gif


where R is the real part and X is the imaginary part of the complex number, and |Z| is the absolute value of Z.

To understand the phasors theoretically, let us consider a sinusoidal voltage function. If the rotating term at angular frequency ω is ignored, the phasor function can be given by using the real part of a complex function in polar form.

Equation 3.8

03equ08.gif


and the voltage phasor function is

Equation 3.9

03equ09.gif


This phasor is visualized as a vector of length Vrms that rotates counterclockwise in the complex plane with an angular velocity of ω. As the vector rotates, its projection on the real axis traces out the voltage as a function of time. In fact, the “phasor” is simply a snapshot of this rotating vector at t = 0 and will be shown in the phasor graphs of the VIs implemented.

In addition, when the periodic voltage, current, and power waveforms are considered, each data point in these waveforms can be represented in the complex plane. This feature is also demonstrated in the chapter's phasor VIs.

Impedance

The impedance Z in ac circuits is defined as the ratio of voltage function to current function. Hence, the impedance is a complex number and can be expressed in the rectangular form as

Equation 3.10

03equ10.gif


The real component of the impedance is called the resistance R and the imaginary component is called the reactance X, both of which are in ohms. The reactance is a function of ω in L and C loads, and for an inductive load, X is positive, whereas for a capacitive load, X is negative (Fig. 3-1).

03fig01.gifFigure 3-1. Equivalent circuits of an impedance and impedance phasor in the complex plane.

The impedance can also be displayed in the complex plane as phasors, the voltage, and the current waveforms. However, since the resistance is never negative, only the first and fourth quadrants are required. This restriction is implied in the associated VIs by limiting the upper and lower values of the controls.

Per-Unit Values

The per-unit system of measurement and computation is used in electrical engineering for two principal reasons:

  • To display multiple quantities (such as voltages and currents) on the same scale for comparison purposes

  • To eliminate the need for conversion of the voltages, currents, and impedances in the circuits and to avoid using transformation from three-phase to single-phase, and vice versa

The quantity subject to conversion is normalized in terms of a particularly convenient unit, called the per-unit base of the system. Note that whenever per-unit values are given, they are always dimensionless. To calculate the actual values of the quantities, the magnitude of the base of the per-unit system must be known. In electrical circuits, voltage, current, impedance, and power can be selected as base quantities.

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

Overview


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information


To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

Newsletters

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information


Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.

Security


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

Children


This site is not directed to children under the age of 13.

Marketing


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information


If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

Choice/Opt-out


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information


Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents


California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure


Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

Links


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact


Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice


We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020