Frequency
Response
Plots
The
frequency response
of a fixed
linear
system
is
typically
represented graphically, using
one of
three
types
of
frequency response
plots.
A
polar plot
is
simply
a
plot
of the
vector
H(jcS)
in the
complex
plane,
where
Re(o>)
is the
abscissa
and
Im(cu)
is the
ordinate.
A
logarithmic plot
or
Bode
diagram
consists
of two
displays:
(1) the
magnitude
ratio
in
decibels
Mdb(o>)
[where
Mdb(w)
= 20 log
M(o))]
versus
log
w,
and (2) the
phase angle
in
degrees
<£(a/)
versus
log
a).
Bode
diagrams
for
normalized
first and
secondorder systems
are
given
in
Fig.
27.23.
Bode
diagrams
for
higherorder
systems
are
obtained
by
adding these
first
and
secondorder terms, appropriately scaled.
A
Nichols
diagram
can be
obtained
by
cross
plotting
the
Bode
magnitude
and
phase diagrams, eliminating
log
a).
Polar
plots
and
Bode
and
Nichols diagrams
for
common
transfer
functions
are
given
in
Table
27.8.
Frequency
Response Performance Measures
Frequency response
plots
show
that
dynamic
systems tend
to
behave
like
filters,
"passing"
or
even
amplifying
certain
ranges
of
input frequencies, while blocking
or
attenuating
other frequency ranges.
The
range
of
frequencies
for
which
the
amplitude
ratio
is no
less
than
3 db of
its
maximum
value
is
called
the
bandwidth
of the
system.
The
bandwidth
is
defined
by
upper
and
lower
cutoff
frequencies
o)c,
or by
o>
= 0 and an
upper cutoff frequency
if
M(0)
is the
maximum
amplitude
ratio.
Although
the
choice
of
"down
3 db"
used
to
define
the
cutoff frequencies
is
somewhat
arbitrary,
the
bandwidth
is
usually taken
to be a
measure
of the
range
of
frequencies
for
which
a
significant
portion
of the
input
is
felt
in the
system output.
The
bandwidth
is
also
taken
to be a
measure
of the
system speed
of
response, since attenuation
of
inputs
in the
higherfrequency ranges generally
results
from
the
inability
of the
system
to
"follow"
rapid changes
in
amplitude.
Thus,
a
narrow bandwidth generally
indicates
a
sluggish system response.
Response
to
General
Periodic Inputs
The
Fourier series provides
a
means
for
representing
a
general periodic
input
as the sum of a
constant
and
terms containing
sine
and
cosine.
For
this
reason
the
Fourier
series,
together with
the
super
position
principle
for
linear
systems, extends
the
results
of
frequency response analysis
to the
general
case
of
arbitrary
periodic inputs.
The
Fourier
series
representation
of a
periodic function
f(t)
with
period
2T on the
interval
t* + 2T
>
t
>
t*
is
jv
N
a°
^
i
n/Trt
i
•
n7rt\
/(O
=
T
+
Zr
I
an
cos
— +
bn
sin
— I
2,
n=l
\
i
i I
where
1
r+2^
nirt
j
an
=
~
J^
/(O
cos
—
dt
bn
=
J'L
f(f}
sin
T^dt
If
f(t)
is
defined outside
the
specified
interval
by a
periodic extension
of
period
27,
and if
f(t)
and
its
first
derivative
are
piecewise continuous, then
the
series
converges
to
/(O
if
f
is a
point
of
con
tinuity,
or to
l/2
[f(t+)
+
/(*)]
if t is a
point
of
discontinuity.
Note
that
while
the
Fourier
series
in
general
is
infinite,
the
notion
of
bandwidth
can be
used
to
reduce
the
number
of
terms required
for
a
reasonable approximation.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 Any
 Mechanical Engineering, Numerical Analysis, The Land, Runge–Kutta methods, Numerical ordinary differential equations

Click to edit the document details