CHAPTER 1
Basic Concepts

1.1 Introduction

The twentieth century began with the electric power industry in its infancy; Thomas Edison and Nikola Tesla were locked in battle with Edison advocating direct current (dc) and Tesla alternating current (ac). The century ended with the electric power industry expanding rapidly from the traditional power generation, transmission, and utilization into propulsion of air, ground, and sea transportation. The advent of the computer and the silicon-controlled rectifier in the mid-1900s brought about an expansion of the power area to include the smart grid, microgrids, efficient and robust electric drives, more-electric aircraft, ships, and land vehicles. This growth is likely to continue into the foreseeable future.

Before the advent of the computer, engineers were essentially limited to steady-state analysis and therefore unable to conveniently deal with the analytical challenges of the expanding power industry. This chapter sets forth some of the basic concepts and analysis tools that are part of the present-day power and electric drives area. Although not inclusive, the material covered in this chapter is representative and common to all disciplines of the power area.

1.2 Phasor Analysis and Power Calculations

Since the early twentieth century, we have lived in an alternating current (ac) world. Thanks to George Westinghouse and Nikola Tesla, power systems are predominately ac; power is generated by large ac generators, transmitted by high voltage transmission lines, and transformed to a low voltage and distributed to homes and factories. The evolution of the ac power system brought about many engineering challenges and, as we look back, it is difficult to comprehend how these problems were solved without a computer. Even steady-state ac-circuit analysis posed a problem until the early 1900s when Charles Stienmetz, who was a less flamboyant colleague of Edison and Tesla, came up with the concept of what is now known as phasors. Some may consider the phasor a casualty of the computer age along with the slide rule. It is, however, still a very useful means for understanding and portraying the steady-state performance of electric machines, power systems, and electric drives. Moreover, the phasor concept provides a means of visualizing sinusoidal variations from different frames of reference and in Chapter 2 we will find that the voltage and current phasors combined with Tesla's rotating magnetic field provides a straightforward means of analyzing and portraying the steady-state operation of ac machines.

The phasor can be established by expressing a steady-state sinusoidal variable as

where the a subscript is used here to denote sinusoidal quantities. The sinusoidal variations are expressed as cosines, capital letters are used to denote steady-state quantities, and Fp is the peak value of the sinusoidal variation. Generally, F or f represents voltage (V or v) or current (I or i) in circuit analysis, but it could be any sinusoidal variable. For steady-state conditions, ?ef may be written as

where ?e is the electrical angular velocity in radians/second (2p times the frequency) and ?ef(0) is the time-zero position of the electrical variable. Substituting (1.2-2) into (1.2-1) yields

Now, Euler's Identity is

and since we are expressing the sinusoidal variation as a cosine, (1.2-3) may be written as

where Re is the shorthand for "real part of." Equations (1.2-3) and (1.2-5) are equivalent. Let us rewrite (1.2-5) as

We need to take a moment to define what is referred to as the root-mean-square (rms) of a sinusoidal variation. In particular, the rms value is defined as

where F is the rms value of Fa(t) and T is the period of the sinusoidal variation. It is left to the reader to show that the rms value of (1.2-3) is . Therefore, we can express (1.2-6) as

By definition, the phasor representing Fa(t), which is denoted with a raised tilde, is

which is a complex number. The reason for using the rms value as the magnitude of the phasor will be addressed later in this section. Equation (1.2-6) may now be written as

A shorthand notation for (1.2-9) is

Equation (1.2-11) is commonly referred to as the polar form of the phasor. The Cartesian form is

When using phasors to calculate steady-state voltages and currents, we think of the phasors as being stationary at t = 0; however, we know that a phasor is related to the instantaneous value of the sinusoidal quantity it represents. Let us take a moment to consider this aspect of the phasor and thereby, give some physical meaning to it. From (1.2-4), we realize that is a line of unity length rotating counterclockwise at an angular velocity of ?e. Therefore, backing up for a minute

is a line with a constant amplitude of rotating counterclockwise in the real-imaginary plane at an angular velocity of ?e with a time-zero displacement from the positive real axis of ?ef(0). Since is the peak value of the sinusoidal variation, the instantaneous value of Fa(t) expressed as a cosine is the real part of (1.2-13). In other words, the real projection of the phasor rotating counterclockwise at ?e is the instantaneous value of . Thus, with ?ef(0) = 0 in (1.2-3)

the phasor representing (1.2-14) is

For

the phasor is

We will use degrees and radians interchangeably when expressing phasors. Although there are several ways to arrive at (1.2-17) from (1.2-16), it is helpful to ask yourself where the rotating phasor must be positioned at time-zero so that, when it rotates counterclockwise at ?e, its real projection is . It follows that a phasor of amplitude F positioned at 90° represents .

In other words, we are viewing a sinusoidal variation as the real projection in the real-imaginary plane of a rotating line equal in magnitude to the positive peak value of the variation and rotating at the electrical angular velocity of the sinusoidal variation. Since we are dealing with a steady-state variation, we can stop the rotation at any time and view it as a fixed line, but knowing full well that it, in fact, represents a sinusoidal variation and to represent the sinusoidal variation we must rotate it counterclockwise at ?e and take the real projection. Please understand that if we ran at ?e in unison with the rotating line it would appear as a constant to us; therefore, in viewing a sinusoidal variation in this manner it would appear to us as a constant. This is no different than stopping the phasor at some arbitrary time-zero; but realizing that it actually represents a sinusoidal variation. We will talk more about this important aspect as we go along; in particular, see Example 1A.

In order to show the facility of the phasor in the analysis of steady-state performance of ac circuits and devices, it is useful to consider a series circuit consisting of a resistance, an inductance L, and a capacitance C. Thus, using uppercase letters to indicate steady-state variables

Throughout the text, we will use either R or r to represent resistance. For steady-state operation, let

where we have dropped the functional notation and the subscript a helps to distinguish the instantaneous value from the rms value of the steady-state variables. The steady-state voltage equation may be obtained by substituting (1.2-19) and (1.2-20) into (1.2-18), whereupon we can write

The second term on the right-hand side of (1.2-21), which is , can be written

Since , from (1.2-21), we can write

Since , (1.2-23) may be written

If we follow a similar procedure, we can show that

Differentiation of a steady-state sinusoidal variable rotates the phasor counterclockwise by or j; integration rotates the phasor clockwise by or - j.

The steady-state voltage equation given by (1.2-21) can now be written in phasor form as

We can express (1.2-26) compactly as

where the impedance, Z, is a complex number; it is not a phasor. It may be expressed...