53
4.4 Mathematical Representation of Signals
Consider the following waveform:
()
cos
(
)
et
E
t
ω
φ
=+
e(t)
Δ
t
t
∂
=
∂
.
t
=−Δ
Note the sign convention where a negative phase difference is a positive time difference. (Things that
come later have phase < 0)
Here's a different way of showing the same thing. At any fixed point in time:
φ
E
R
i
Phasor.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document54
Properties of phasors
•
The phase angle
φ
is the angle between
E
and the Real axis. The amplitude at time
t
is the projection
of
E
onto R.
•
Oscillatory motion appears as counterclockwise rotation at a constant angular frequency,
ω
.
•
The usual variation in amplitude is seen as the variation in E
r
, the projected length of the vector of
fixed length E onto R.
•
When
= 0, E
r
= E
•
When
= 90
o
or
π
/2, E
r
= 0
•
When
= 180
o
or
π
, E
r
= E
An alternative to:
( )
()
cos
et
A
t
=+
is:
A
t
=
∠+
If the convention is adopted to draw the diagram at
t
=0, then we can write,
eA
=
∠
.
must be remembered.
Sometimes you will find,
A
=
∠
E
,
but we will keep this bold notation for standard complex variables. Finally, you may well encounter
This is the end of the preview.
Sign up
to
access the rest of the document.
 '07
 Pottebaum
 Complex number, fixed length, Complex Number Notation, negative phase difference, complex exponential Euler

Click to edit the document details