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ENEE 241 02*
READING ASSIGNMENT 2
Thu 02/05 Lecture
Topic:
n
th
root of a complex number; continuoustime sinusoids; phasors
Textbook References:
sections 1.3 and 1.4
Key Points:
•
The equation
z
n
=
e
jφ
, where
φ
is a given angle, has
n
roots of the form
z
=
e
jθ
; these are
obtained by setting
θ
= (
φ
+ 2
kπ
)
/n
, where
k
= 0
,...,n

1.
•
The functions cos
θ
and sin
θ
are both periodic with period 2
π
(radians), and are shifted
versions of each other.
•
The generic timedependent sinusoid
A
cos(Ω
t
+
φ
) has three parameters: amplitude
A
, an
gular frequency Ω (rad/sec) and initial phase
φ
. The cyclic frequency
f
(Hz) and period
T
(sec) are related to Ω by
f
= 1
/T
=
Ω
2
π
•
The stationary phasor of
A
cos(Ω
t
+
φ
) is the complex number
Ae
jφ
. The sum of two (or
more) sinusoids of arbitrary amplitudes and phases but of identical frequency Ω is a sinusoid
of frequency Ω. Sinusoids of identical frequency can be added together by taking the complex
sum of their stationary phasors.
Theory and Examples:
1
. The equation
z
n
=
v ,
where
z
is a complex variable and
v
is a complex constant, has
n
complex roots (this is true
for any polynomial of degree
n
). These roots can be found using the following observations:
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