Page 111 - Numerical Analysis Using MATLAB and Excel

Chapter 3 Sinusoids and Phasors

3.9 Summary

• An alternating current (or voltage) alternates between positive and negative values at regularly
recurring intervals of time.
• The period T of an alternating current or voltage is the smallest value of time which separates
recurring values of the alternating waveform.

• Sine and cosine waveforms and these are referred to as sinusoids.
ω
ω
• The angular velocity is commonly known as angular or radian frequency and T = 2π
• The term frequency in Hertz, denoted as Hz , is used to express the number of cycles per sec-
⁄
ond. The frequency is denoted by the letter and in terms of the period , f = 1T . The fre-
f
T
quency is often referred to as the cyclic frequency to distinguish it from the radian frequency
f
ω .
• The cosine function leads (is ahead of) the sine function by π 2⁄ radians or 90° , and the sine
function lags (is behind) the cosine function by π 2⁄ radians or 90° . Alternately, we say that
the cosine and sine functions are out-of-phase by 90° , or there is a phase angle of 90° between
the cosine and sine functions.

• Two (or more) sinusoids can be out-of-phase by a phase angle other than 90° .

• It is important to remember that when we say that one sinusoid leads or lags another sinusoid,
these are of the same frequency since two sinusoids of different frequencies can never be in
phase.

• It is customary to express the phase angle in degrees rather than in radians in a sinusoidal func-
⁄
tion. For example, we write vt() = 100sin ( 2000πt – π 6 ) as vt() = 100sin ( 2000πt – 30° )
• When two sinusoids are to be compared in terms of their phase difference, these must first be
written either both as cosine functions, or both as sine functions, and should also be written
with positive amplitudes.

•A negative amplitude implies 180° phase shift.
• The radian is a circular angle subtended by an arc equal in length to the radius of the circle,

whose radius is units in length. The circumference of a circle is 2πr .
r
y
• The notation cos – 1 y or arccos y is used to denote an angle whose cosine is . Thus, if
y = cos x , thenx = cos – 1 y . These are called Inverse Trigonometric Functions.

• A phasor is a rotating vector expressed as a complex number where is an operator that
j
rotates a vector by 90° in a counterclockwise direction.

3−24 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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