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Fundamentals of Alternating Current - Engineering

1 12 Fundamentals of Alternating Current In this chapter , we lead you through a study of the mathematics and physics of Alternating Current (AC) circuits. After completing this chapter you should be able to: Develop a familiarity with sinusoidal functions. Write the general equation for a sinusoidal signal based on its amplitude, frequency, and phase shift. Define angles in degrees and radians. Manipulate the general equation of a sinusoidal signal to determine its amplitude, frequency, phase shift at any time. Compute peak, RMS, and average values of voltage and Current . Define root-mean-squared amplitude, angular velocity, and phase angle.

12 Fundamentals of Alternating Current In this chapter, we lead you through a study of the mathematics and physics of alternating current (AC) circuits. After completing this chapter you should be able to: Develop a familiarity with sinusoidal functions. Write the general equation for a sinusoidal signal based on its amplitude,

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Transcription of Fundamentals of Alternating Current - Engineering

1 1 12 Fundamentals of Alternating Current In this chapter , we lead you through a study of the mathematics and physics of Alternating Current (AC) circuits. After completing this chapter you should be able to: Develop a familiarity with sinusoidal functions. Write the general equation for a sinusoidal signal based on its amplitude, frequency, and phase shift. Define angles in degrees and radians. Manipulate the general equation of a sinusoidal signal to determine its amplitude, frequency, phase shift at any time. Compute peak, RMS, and average values of voltage and Current . Define root-mean-squared amplitude, angular velocity, and phase angle.

2 Convert between time domain and phasor notation. Convert between polar and rectangular form. Add, subtract, multiply, and divide phasors. Discuss the phase relationship of voltage and Current in resistive, inductive, and capacitive loads. Apply circuit analysis using phasors. Define components of power and realize power factor in AC circuits. Understand types of connection in three-phase circuits. FOCUS ON MATHEMATICS This chapter relates the application of mathematics to AC circuits, covering complex numbers, vectors, and phasors. All these three concepts follow the same rules. REFERENCES Stephan J.

3 Chapman, Electric Machinery Fundamentals , Third Edition, McGraw-Hill, 1999. Stephan J. Chapman, Electric Machinery and Power System Fundamentals , McGraw-Hill, 2002. Bosels, Electrical Systems Design, Prentice Hall. James H. Harter and Wallace D. Beitzel, Mathematics Applied to Electronics, Prentice Hall. 2 chapter 12 INTRODUCTION The majority of electrical power in the world is generated, distributed, and consumed in the form of 50- or 60-Hz sinusoidal Alternating Current (AC) and voltage. It is used for household and industrial applications such as television sets, computers, microwave ovens, electric stoves, to the large motors used in the industry.

4 AC has several advantages over DC. The major advantage of AC is the fact that it can be transformed, however, direct Current (DC) cannot. A transformer permits voltage to be stepped up or down for the purpose of transmission. Transmission of high voltage (in terms of kV) is that less Current is required to produce the same amount of power. Less Current permits smaller wires to be used for transmission. In this chapter , we will introduce a sinusoidal signal and its basic mathematical equation. We will discuss and analyze circuits where currents i(t) and voltages v(t) vary with time.

5 The phasor analysis techniques will be used to analyze electric circuits under sinusoidal steady-state operating conditions. Single-phase power will conclude the chapter . SINUSOIDAL WAVEFORMS AC unlike DC flows first in one direction then in the opposite direction. The most common AC waveform is a sine (or sinusoidal) waveform. Sine waves are the signal whose shape neither is nor altered by a linear circuit , therefore, it is ideal as a test signal. In discussing AC signal, it is necessary to express the Current and voltage in terms of maximum or peak values, peak-to-peak values, effective values, average values, or instantaneous values.

6 Each of these values has a different meaning and is used to describe a different amount of Current or voltage. Figure 12-1 is a plot of a sinusoidal wave. The correspondence mathematical form is ()() +=wtVtvpcos ( ) Where Vp is the peak voltage, = 2 f is the angular speed expressed in radians per second (rad/s), f is the frequency expressed in Hertz (Hz), t is the time expressed in second (s), and is phase of the sinusoid expressed in degrees. The function (Figure 12-1) starts at a value of 0 at 0o, and rise smoothly to a maximum of 1 at 90o. They then fall, just as they rose, back to 0o at 180o.

7 The negative peak is reached three quarters of the way at 270o. The function then returns symmetrically to 0o at 360o. Fundamentals of Alternating Current 3 Figure 12-1 Sinusoidal wave values. Radian and Degree A degree is a unit of measurement in degree (its designation is or deg), a turn of a ray by the 1/360 part of the one complete revolution. So, the complete revolution of a ray is equal to 360 deg.

8 A radian is defined as the central angle, for which lengths of its arc and radius are equal (AB = A0). An arc length is the distance along the arc of a circle from the origin to the end of the angle. These terms are shown in Figure 13-8. Following Equation ( ), a length of a circumference C and its radius r can be expressed as: RC 2= ( ) So, a round angle, equal to 360 in a degree measure, is simultaneously 2 in a radian measure. Hence, we receive a value of one radian: 2360 rad 1 = ( ) and, rad 3602 deg 1 = ( ) Peak-to-peak Peak value RMS value 1 cycle 4 chapter 12 The following comparative table of degree and radian provides measure for some angles we often deal with: Figure 12-2 Radian and arc length.

9 Table 12-1 Angles in Degree and Radian Angle (deg) 0 45 90 180 270 360 Angle (rad) 0 /4 /2 3 /2 2 Peak and Peak-to-Peak Values During each complete cycle of AC signal there are always two maximum or peak values, one for the positive half-cycle and the other for the negative half-cycle. The peak value is measured from zero to the maximum value obtained in either the positive or negative direction. The difference between the peak positive value and the peak negative value is called the peak-to-peak value of the sine wave.

10 This value is twice the maximum or peak value of the sine wave and is sometimes used for measurement of ac voltages. The peak value is one-half of the peak-to-peak value. Instantaneous Value The instantaneous value of an AC signal is the value of voltage or Current at one particular instant. The value may be zero if the particular instant is the time in the cycle at which the polarity of the voltage is changing. It may also be the same as the peak value, if the selected instant is the time in the cycle at which the voltage or Current stops increasing and starts decreasing.


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