Page 101 - Numerical Analysis Using MATLAB and Excel

Chapter 3 Sinusoids and Phasors

–
e – jθ = cos θ jsin θ (3.74)

are known as the Euler’s identities.
Multiplying (3.73) by the real positive constant C we get:

Ce jθ = Ccos θ + jCsin θ (3.75)

This expression represents a phasor, say a + jb , and thus

Ce jθ = a + jb (3.76)
Equating real and imaginary parts in (3.75) and (3.76), we get

a = Ccos θ and b = Csin θ (3.77)

Squaring and adding the expressions in (3.77), we get

2
2
2
)
(
a + b = ( Ccos θ ) 2 + ( Csin θ ) 2 = C cos 2 θ + sin 2 θ = C 2
Then,
2
2
C = a + b 2
or
2 2
C = a + b (3.78)
Also, from (3.77)
b Csin θ
-
-- = --------------- = tan θ
a Ccos θ
or

b
--
-
θ = tan – 1 ⎛⎞ (3.79)
⎝⎠
a
Therefore, to convert a phasor from rectangular to exponential form, we use the expression
⎛
1 b ⎞
–
2
2
⎝
a + jb = a + b e jtan -- a ⎠ - (3.80)
To convert a phasor from exponential to rectangular form, we use the expressions
Ce jθ = Ccos θ + jCsin θ
Ce – jθ = Ccos θ jCsin θ (3.81)
–

3−14 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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