Phasor

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  eBook - PDF

  The Analysis and Design of Linear Circuits

  • Roland E. Thomas, Albert J. Rosa, Gregory J. Toussaint(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  The Phasor V is written as V = V A e jϕ Polar = V A cos ϕ + jV A sin ϕ Rectangular (8 –5) Note that V is a complex number determined by the amplitude and phase angle of the sinusoid. Figure 8–1 shows a graphical representation commonly called a Phasor diagram. As shown in Eq. (8–5), a Phasor is a complex number that can be written in either polar or rectangular form. An alternative way to write the polar form is to replace the exponential e jϕ by the shorthand notation ∠ϕ. In subsequent discussions, we will often express Phasors as V = V A ∠ϕ, which is equivalent to the polar form in Eq. (8–5). When written this way, one states that the Phasor has an amplitude of V A at an angle of ϕ. Two features of the Phasor concept need emphasis: 1. Phasors are written in boldface type like V or I 1 to distinguish them from signal waveforms such as υ t and i 1 t . 2. A Phasor is determined by amplitude and phase angle and does not contain any information about the frequency of the sinusoid. The first feature points out that signals can be described in different ways. Although the Phasor V and waveform υ t are related concepts, they have different j Im Re V A cos ϕ V A V jV A sin ϕ ϕ FIGURE 8–1 Phasor diagram. 320 C H A P T E R 8 S INUSOIDAL S TEADY -S TATE R ESPONSE physical interpretations and our notation must clearly distinguish between them. The absence of frequency information in the Phasors results from the fact that in the sinus- oidal steady state, all currents and voltages are sinusoids with the same frequency. Carrying frequency information in the Phasor would be redundant, since it is the same for all Phasors in any given steady-state circuit problem. In summary, given a sinusoidal signal υ t = V A cos ωt + ϕ , the corresponding pha- sor representation is V = V A e jϕ = V A ∠ϕ. Conversely, given the Phasor V = V A e jϕ , the corresponding sinusoid is found by multiplying the Phasor by e jωt and reversing the steps in Eq.

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  Transformer Design Principles, Third Edition

  • Robert M. Del Vecchio, Bertrand Poulin, Pierre T. Feghali, Dilipkumar Shah, Rajendra Ahuja, Robert Del Vecchio, Pierre Feghali(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  5 Phasors, 3-Phase Connections, and Symmetrical Components 5.1 Phasors Phasors are essentially complex numbers with a built-in sinusoidal time dependence. They are usually written as a complex number in polar form. A complex number, C = C Re + jC Im, can be expressed in polar form as (5.1) C = C Re + jC Im = C e j θ = C cos θ + j sin θ where C = C Re 2 + C Im 2, θ = tan − 1 C Im C Re Here |C| is the magnitude of the complex number, with real and imaginary components, C Re and C Im. A sinusoidal voltage has the form V = V p cos(ωt + φ), where ω = 2πf is the angular frequency, with f being the frequency in Hz, and φ a-phase angle. V p is the peak value of the voltage. This can be written as (5.2) V = Re V p e j ω t + φ where Re[] means to take the real part of the expression following. In the complex plane, expression (5.2) is the projection on the real axis of the complex number given in the brackets, as illustrated in Figure 5.1. FIGURE 5.1 Voltage as the real projection of a complex number. As time increases, the complex voltage rotates counterclockwise around the origin, with the real projection varying as the cosine function. Currents and other voltages in a linear system, considered as complex numbers, will also rotate around the origin with time at the same rate since ω is the same for these. Since these numbers are all rotating together, at any snapshot in time, they will all have the same relative position with respect to each other. Therefore, it is convenient to factor out the common time dependence and consider only the phase relationships between the numbers. This is possible because the circuit equations are linear and (5.3) V = V p e j ω t + φ = V p e j ω t e j φ The complex number with time dependence is called a Phasor and written in boldfaced type. Usually, however, the time dependence is understood or factored out and the quantity without the time exponential factor is also called a Phasor

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  Protective Relaying

  Theory and Applications

  • Walter A. Elmore(Author)
  • 2003(Publication Date)
  • CRC Press

    (Publisher)

  2 Technical Tools of the Relay Engineer: Phasors, Polarity, and Symmetrical Components

  Revised by: W.A.ELMORE

  1 INTRODUCTION

  In addition to a general knowledge of electrical power systems, the relay engineer must have a good working understanding of Phasors, polarity, and symmetrical components, including voltage and current Phasors during fault conditions. These technical tools are used for application, analysis, checking, and testing of protective relays and relay systems.

  2 PhasorS

  A Phasor is a complex number used to represent electrical quantities. Originally called vectors, the quantities were renamed to avoid confusion with space vectors. A Phasor rotates with the passage of time and represents a sinusoidal quantity. A vector is stationary in space.

  In relaying, Phasors and Phasor diagrams are used both to aid in applying and connecting relays and for the analysis of relay operation after faults.

  Phasor diagrams must be accompanied by a circuit diagram. If not, then such a circuit diagram must be obvious or assumed in order to interpret the Phasor diagram. The Phasor diagram shows only the magnitude and relative phase angle of the currents and voltages, whereas the circuit diagram illustrates only the location, direction, and polarity of the currents and voltages. These distinctions are important. Confusion generally results when the circuit diagram is omitted or the two diagrams are combined.

  There are several systems and many variations of Phasor notation in use. The system outlined below is standard with most relay manufacturers.

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  Introduction to Energy, Renewable Energy and Electrical Engineering

  Essentials for Engineering Science (STEM) Professionals and Students

  • Ewald F. Fuchs, Heidi A. Fuchs(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  sinusoidal and cosinusoidal functions), the concept of a Phasor is based on complex number relations. Indeed, a Phasor represents a simplified complex number by ignoring the term e jωt. Phasor relations for resistor, inductor, and capacitor and the concept of impedance as well as admittance are specified. Also presented are Δ‐Y transformation, solutions based on Kirchhoff’s laws, nodal analysis, mesh/loop analysis, superposition, source transformation/exchange, Thévenin’s (principle of constant voltage source) and Norton’s (principle of constant current source) theorems, and steady‐state power analysis. Various power definitions are reviewed, such as instantaneous power p (t), average or real power P, imaginary power Q, apparent power S, and complex power. Amplitude excursions of voltages (V M) and current (I M) are defined, as well as root mean square (rms) error and effective (eff) values of signals. Lastly, nonsinusoidal steady‐state responses are treated with Fourier analysis, with its trigonometric and exponential forms. Problems 4.1 Complex numbers: conversion from rectangular to polar form and vice versa. Given three complex numbers:,, and. Note 1 radian ≡ 57.2958° or 1° ≡ π/180 radians = 0.017 45 rad. Find. Find in rectangular form, where * denotes the complex conjugate. Evaluate and place the results in polar form. Solve for if and express in polar form. Evaluate the following expressions and put your answer in polar form. What is the rms Phasor for v (t) = 5.2 V(100 t − 38°)? What time function is represented by the Phasor, if the frequency is f = 400 Hz? Determine the frequency of the following two currents and the phase angle between them: In the circuit of Figure P4.1.1, i s (t) =2 cos(377 t + 60°)A find, where “s” stands for source. Figure P4.1.1 Capacitor supplied by current i s (t). Find the equivalent impedance shown in Figure P4.1.2, where the angular frequency is ω = 377 rad/s. Figure P4.1.2 Impedance calculation. Find the

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  Engineering Circuit Analysis

  • J. David Irwin, R. Mark Nelms, Amalendu Patnaik(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  The Phasors are then simply transformed back to the time domain to yield the solution of the original set of differential equations. In addition, we note that the solution of sinusoidal steady-state circuits would be relatively simple if we could write the Phasor equation directly from the circuit description. In Section 8.4 we will lay the groundwork for doing just that. Note that in our discussions we have tacitly assumed that sinusoidal functions would be rep- resented as Phasors with a phase angle based on a cosine function. Therefore, if sine functions are used, we will simply employ the relationship in Eq. (8.7) to obtain the proper phase angle. In summary, while υ (t) represents a voltage in the time domain, the Phasor V represents the voltage in the frequency domain. The Phasor contains only magnitude and phase information, and the frequency is implicit in this representation. The transformation from the time domain to the frequency domain, as well as the reverse transformation, is shown in Table 8.1. Recall that the phase angle is based on a cosine function and, therefore, if a sine function is involved, a 90° shift factor must be employed, as shown in the table. However, if a network contains only sine sources, there is no need to perform the 90° shift. We simply perform the normal Phasor analysis, and then the imaginary part of the time-varying complex solution is the desired response. Simply put, cosine sources generate a cosine response, and sine sources generate a sine response. TABLE 8.1 Phasor representation TIME DOMAIN FREQUENCY DOMAIN A cos (ωt ± θ) A ±θ A sin (ωt ± θ) A ±θ − 90° PROBLEM-SOLVING STRATEGY STEP 1. Using Phasors, transform a set of differential equations in the time domain into a set of algebraic equations in the frequency domain. STEP 2. Solve the algebraic equations for the unknown Phasors. STEP 3. Transform the now-known Phasors back to the time domain.

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  Engineering Science

  • W. Bolton(Author)
  • 2015(Publication Date)
  • Routledge

    (Publisher)

  Currents and voltages in the same circuit will have the same frequency and thus the Phasors used to represent them will rotate with the same angular velocity and maintain the same phase angles between them at all times; they have zero motion relative to one another. For this reason, we do not need to bother about drawing the effects of their rotation but can draw Phasor diagrams giving the relative angular positions of the Phasors as though they were stationary.

  The following summarise the main points about Phasors:

  1. A Phasor has a length that is directly proportional to the maximum value of the sinusoidally alternating quantity or, because the maximum value is proportional to the root- mean- square value, a length proportional to the r.m.s. value.
  2. Phasors are taken to rotate anticlockwise and have an arrow- head at the end which rotates.
  3. The angle between two Phasors shows the phase angle between their waveforms. The Phasor which is at a larger anticlockwise angle is said to be leading, the one at the lesser anticlockwise angle lagging (Figure 16.4 ).
  4. The horizontal line is taken as the reference axis and one of the Phasors given that direction, the others have their phase angles given relative to this reference axis.

  Figure 16.4 Leading and lagging

  Note that, in textbooks, it is common practice where we are concerned with just the size of a Phasor to represent it using italic script, e.g. V , but where we are referring to a Phasor quantity with both its size and phase we use bold non- italic text, e.g. V. Thus we might say that Phasor V has size of V and a phase angle of φ .

  Example

  Draw the Phasor diagram to represent the voltage and current in a circuit where the current is described by i = 1.5 sin ωt A and the voltage by v = 20 sin (ωt + π /2) V.

  Figure 16.5 shows the Phasors with their lengths proportional to the maximum values of 1.5 A and 20 V.

  Figure 16.5 Example

  16.3 R, L, C in a.c. circuits

  In the following discussion the behaviour of resistors, inductors and capacitors are considered when each individually is in an a.c. circuit.

  16.3.1 Resistance in a.c. circuits

  Consider a sinusoidal current i = Im sin ωt passing through a pure resistance (Figure 16.6

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  eBook - PDF

  Basic Engineering Circuit Analysis

  • J. David Irwin, R. Mark Nelms(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  A review of the Appendix will indicate how these complex numbers or line segments can be added, subtracted, and so on. Since Phasors are complex numbers, it is convenient to represent the Phasor voltage and current graphically as line segments. A plot of the line segments representing the Phasors is called a Phasor diagram. This pictorial representation of Phasors provides immediate information on the relative mag- nitude of one Phasor with another, the angle between two Phasors, and the relative position of HINT 8.11 Current and voltage are in phase. FIGURE 8.6 Voltage–current relationships for a resistor. (a) (b) (c) (d) Im Re I V θ υ = θ i θ υ = θ i υ, i υ i ωt υ(t) = i(t) R i(t) R + − V = RI I R + − 278 CHAPTER 8 AC Steady-State Analysis one Phasor with respect to another (i.e., leading or lagging). A Phasor diagram and the sinusoidal waveforms for the resistor are shown in Figs. 8.6c and d, respectively. A Phasor diagram will be drawn for each of the other circuit elements in the remainder of this section. If the voltage υ ( t) = 24 cos (377t + 75°) V is applied to a 6-Ω resistor as shown in Fig. 8.6a, we wish to determine the resultant current. Solution Since the Phasor voltage is V = 24 /75° V the Phasor current from Eq. (8.22) is I = 24 /75° ______ 6 = 4 /75° A which in the time domain is i( t) = 4 cos (377t + 75°) A EXAMPLE 8.6 The voltage–current relationship for an inductor, as shown in Fig. 8.7a, is υ(t ) = L di(t ) ____ dt 8.23 Substituting the complex voltage and current into this equation yields V M e j(ωt + θ υ ) = L d __ dt I M e j(ωt + θ i ) which reduces to V M e jθ υ = jωLI M e jθ i 8.24 Equation (8.24) in Phasor notation is V = jωLI 8.25 Note that the differential equation in the time domain (8.23) has been converted to an algebraic equation with complex coefficients in the frequency domain [see HINT 8.12]. This relationship is shown in Fig. 8.7b. Since the imaginary operator j = 1e j 90° = 1 /90° = √ ___ −1 , Eq.

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  Electrical Trade Theory N3 Student's Book

  TVET FIRST

  • SA Chuturgoon(Author)
  • 2021(Publication Date)
  • Macmillan

    (Publisher)

  This means that it now has magnitude represented by the peak amplitude of the waveform, as well as direction given by the phase difference between the sinusoidal wave and the reference axis. It is important to note that we can draw a vector diagram to show the distribution of voltage and current vectors, provided that these quantities have the same frequency. In an AC circuit, the angle by which the current either leads or lags the voltage is called the phase angle (ϕ). vector: a quantity that has both direction and magnitude Phasor: rotating vector that represents a sinusoidal quantity such as alternating current and voltage angular velocity: rate of change of angular position of a rotating body vector diagram: a diagram that shows the distribution of voltage and current vectors of an AC circuit phase angle: the angle between voltage and current vectors 10 Module 1 TVET FIRST Also note that we draw vector diagrams using the peak or maximum values of the sinusoidal quantity, whereas Phasor diagrams are drawn using the rms values of the sinusoidal quantities. However, in both cases the phase angle and direction stay the same. 1.2.2 Drawing vector diagrams and adding vector quantities Figure 1.8 shows two athletes, A and B, running a race along a circular track. To establish which athlete is leading, we need to know two things: • The start and finish points. • The direction in which they are running. If a is the start and finish line and the athletes are running in an anticlockwise direction: • Athlete B leads athlete A. • Athlete A lags athlete B. a) A typical vector diagram representing AC quantities In a typical vector diagram (see Figure 1.9): • The reference vector is always drawn on the positive x-axis. • We always use an anticlockwise direction. • When commenting on lead or lag, always look at what the current vector is doing with respect to the voltage vector.

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  eBook - ePub

  Introduction to Electric Circuits

  • Ray Powell(Author)
  • 1995(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Chapter 6 .

  Example 4.9: In the circuit of Fig. 4.19 , R = 12 ω, L = 150 mH, C = 10 μF, ν = 100 sin 2πft and f = 100 Hz. Calculate (1) the impedance of the circuit, (2) the current drawn from the supply, (3) the phase angle of the circuit. Give an expression from which the current at any instant could be determined.

  Solution

  1. The impedance Z = √(R 2 + [X L – X C ]2 ). Now

      and     so

  2. The current drawn from the supply is given by I = V /Z where I and V are rms values. The peak value of the supply voltage is 100 V so that the rms value is 100/V2 = 70.7 V, soI = V /Z = 70.7/65.9 = 1.07 A

  3. The phase angle of the circuit is φ = cos−1 (R /Z ) = cos−1 (12/65.9) = 79.5°. Because the capacitive reactance is greater than the inductive reactance, the circuit is predominantly capacitive and so the phase angle is a leading one (i.e. the current leads the voltage).

      The expression for the current is i = I m sin (2πft – φ) with I m = √2I and φ = 79.5° = 79.5π/180 rad. Thus i = 1.5 sin (200πt − 0.44π) A.

  4.4 COMPLEX NOTATION

  We have seen that a Phasor quantity is one for which both magnitude and direction is important. These quantities may be represented by Phasor diagrams in the manner shown earlier in the chapter. In Fig. 4.21 (a), the Phasor V 1 is shown to be leading the reference Phasor by φ, degrees, whereas the Phasor V 2 is shown as lagging the reference by φ2 degrees. It is conventional to take the horizontal axis as the reference direction.

  Figure 4.21

  V 1 may be represented by

  In this notation |V 1 | indicates the magnitude of the quantity and is represented by the length of the Phasor, while ∠φ1 indicates that it is φ1

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