Biomedical Engineering Reference
In-Depth Information
An important metric of a sinusoid is its
rms value
(square
r
oot of the
m
ean value of the
quared function), given by
s
t
1
T
Z
T
0
V
V
rms
¼
2
m
cos
2
o
ð
t
þ
f
Þ
dt
ð
9
:
32
Þ
V
rms
¼
V
m
which reduces to
p :
To appreciate the response to a time-varying input,
v
s
¼
V
m
cos
ð
o
t
þ
f
Þ
, consider the
circuit shown in Figure 9.33, in which the switch is closed at
t
¼
0 and there is no initial
energy stored in the inductor. Applying KVL to the circuit gives
L
di
dt
þ
iR
¼
V
m
cos
ð
o
t
þ
f
Þ
and after some work, the solution is
i
¼
i
n
þ
i
f
0
@
1
A
e
0
@
1
A
R
L
t
V
m
R
o
L
R
V
m
R
o
L
R
¼
p
þ
p
cos f
cos o
t
þ
f
2
þ
o
2
L
2
2
þ
o
2
L
2
The first term is the natural response that goes to zero as
goes to infinity. The second term
is the forced response that has the same form as the input (i.e., a sinusoid with the same
frequency o, but a different phase angle and maximum amplitude). If all you are interested
in is the steady-state response, as in most bioinstrumentation applications, then the only
unknowns are the response amplitude and phase angle. The remainder of this section deals
with techniques involving the
t
phasor
to efficiently find these unknowns.
9.12.1 Phasors
The phasor is a complex number that contains amplitude and phase angle information of
a sinusoid and for the signal in Eq. (9.30) is expressed as
ð
9
:
33
Þ
t = 0
i
R
+
−
v
s
L
FIGURE 9.33
An RL circuit with sinusoidal input.
