Alternating Current Analysis 227
where j = √
___
−1 (Euler’s identity) and
e
n
n
(^) 1
23
23
!!
,,
!
(Taylor s series)’
cos( )
!!!
1
246
246
−
sin( )
!!!
357
357
Recall that cos(θ) = real[ejθ] (from Chapter 5 of the book titled Practical MATLAB®
Basics for Engineers), sin(θ) = imaginary [e−jθ], and ejθ = √
_____________
co s^2 θ + si n^2 θ = 1.
R.3.6 The average value of a given signal x(t) is given by
X
T
AVG xt dt
T
1
0
∫ ()
assuming that x(t) is a periodic wave with period T.
R.3.7 A DC voltmeter connected across an AC drop will indicate its average value.
R.3.8 The average value of a sinusoidal wave with period T is given by
X
T
AVG Xtdtm
T
1
0
∫ sin()
which is also equal to
X
X
AVGm td t
2
0
0
2
sin() ( )
π
∫
R.3.9 The average value of a sinusoidal wave over 1/2 period (T) is given by
X
T
AVG Xtdtm
T
1
(^20)
2
∫ sin()
or by
X
X
AVG m td t
sin() ( )
0
π
∫
which is equal to
(^) XXAVG^2064 m
Xm.