Biomedical Engineering Reference
In-Depth Information
Thus, the angle between the two sinusoids is 120 degrees
[30 - (-90)]. The period is given by:
Care must be taken in evaluating Eq.
2.4.10
to ensure
that q is determined to be in the correct quadrant on the
basis of the signs of
a
and
b.
If both
a
and
b
are positive, q
must be between 0 and 90 degrees; if
b
is positive and
a
is
negative, q must be between 90 and 180 degrees (a cal-
culator or MATLAB will not know this and will put any
negative product in the fourth quadrant); if both
a
and
b
are negative, q must be between 180 and 270 degrees
(calculators and MATLAB put positive arguments in the
first quadrant even if they result from two negative
numbers); and finally, if
b
is negative and
a
is positive, q
must be between 270 and 360 degrees. Again,
T ¼
1
1
u
=
2p
¼
1
4
=
2p
¼
1
:
57 seconds
f
¼
and the time delay is:
q
360
T ¼
120
t
d
¼
360
1
:
57
¼
0
:
523 seconds
it is
common to use degrees for phase angle.
To add sine waves, simply add their amplitudes. The
same applies to cosine waves:
2.4.1.1 Sinusoidal arithmetic
Equation
2.4.5
describes an intuitive way of thinking
about a sinusoid, as a sine wave with a phase shift. Al-
ternatively, Eq.
2.4.5
shows that a cosine could just as
well be used instead of the sine to represent a general
sinusoid, and in this text, we use both. Sometimes it is
mathematically convenient to represent a sinusoid as
a combination of a pure sine and a pure cosine. This
representation can be achieved using the well-known
trigonometric identity for the sum of two arguments of
a cosine function:
a
1
cos
ð
u
tÞþa
2
cos
ð
u
tÞ¼ða
1
þ a
2
Þ
cos
ð
u
tÞ
a
1
sin
ð
u
tÞþa
2
sin
ð
u
tÞ¼ða
1
þ a
2
Þ
sin
ð
u
tÞ
[Eq. 2.4.11]
To add two sinusoids
½
i
:
e
:; C
sin
ð
u
t þ
q
Þ
or
C
cos
ð
u
t
q
Þ;
convert them to sines and cosines using
Eq.
2.4.8
, add sines to sines and cosines to cosines, and
convert back to a single sinusoid if desired.
Example 2.4.2: Convert the sum of a sine and cosine
wave,
xðtÞ¼
5 cos
ð
10
tÞ
3 sin
ð
10
tÞ
cos
ðx yÞ¼
cos
ðxÞ
cos
ðyÞþ
sin
ðxÞ
sin
ðyÞ
into a single
sinusoid.
Solution:
Apply Eq.
2.4.9
and Eq.
2.4.10
:
[Eq. 2.4.7]
Based on this identity, the equation for a sinusoid can
be written as:
a ¼
5 and
b ¼
3
C ¼
p
¼
p
a
2
þ b
2
5
2
þ
3
2
¼
5
:
83
C
cos
ð
2p
ft
q
Þ
¼ C
cos
ð
q
Þ
cos
ð
2p
ftÞþC
sin
ð
q
Þ
sin
ð
2p
ftÞ
¼ a
cos
ð
2p
ftÞþb
sin
ð
2p
ftÞ
where :
q
¼
tan
1
b
a
¼
tan
1
3
5
¼
31 degrees
;
but q must be in the third quadrant since both
a
and
b
are
negative:
a ¼ C
cos
ð
q
Þ
;
b ¼ C
sin
ð
q
Þ
[Eq. 2.4.8]
To convert from a sine and cosine to a single sinusoid with
angle q, start with Eq.
2.4.8
.
q
¼
31
þ
180
¼
211 degrees
Therefore, the single sinusoid representation would be as
follows:
If
a ¼ C
cos
ð
q
Þ
and
b ¼ C
sin
ð
q
Þ;
then to determine
C
:
a
2
þ b
2
¼ C
2
ð
cos
2
q
þ
sin
2
q
Þ¼C
2
C ¼
p
xðtÞ¼C
cos
ð
u
t
q
Þ¼
5
:
83 cos
ð
10
t
211 degrees
Þ
a
2
þ b
2
[Eq. 2.4.9]
Equation
2.4.10
shows the calculation for q given
a
and
b
:
Analysis:
Using Equations
2.4.8 through 2.4.11
, any
number of sines, cosines, or sinusoids can be combined
into a single sinusoid if they are all at the same frequency.
This is demonstrated in Example 2.4.3.
C
cos
ð
q
Þ
¼
tan
ð
q
Þ;
q
¼
tan
1
b
b
a
¼
C
sin
ð
q
Þ
a
Example 2.4.3: Combine
xðtÞ¼
4 cos
ð
2
t
30 degrees
Þþ
3 sin
ð
2
tþ
60 degrees
Þ
into a single sinusoid.
[Eq. 2.4.10]
