<s>
In	O
electrical	O
engineering	O
and	O
control	O
theory	O
,	O
a	O
Bode	B-Algorithm
plot	I-Algorithm
is	O
a	O
graph	B-Application
of	O
the	O
frequency	O
response	O
of	O
a	O
system	O
.	O
</s>
<s>
He	O
developed	O
the	O
graphical	O
design	O
technique	O
of	O
the	O
Bode	B-Algorithm
plots	I-Algorithm
to	O
show	O
the	O
gain	O
margin	O
and	O
phase	O
margin	O
required	O
to	O
maintain	O
stability	O
under	O
variations	O
in	O
circuit	O
characteristics	O
caused	O
during	O
manufacture	O
or	O
during	O
operation	O
.	O
</s>
<s>
The	O
Bode	B-Algorithm
plot	I-Algorithm
is	O
an	O
example	O
of	O
analysis	O
in	O
the	O
frequency	O
domain	O
.	O
</s>
<s>
The	O
Bode	B-Algorithm
plot	I-Algorithm
for	O
a	O
linear	O
,	O
time-invariant	O
system	O
with	O
transfer	O
function	O
(	O
being	O
the	O
complex	O
frequency	O
in	O
the	O
Laplace	O
domain	O
)	O
consists	O
of	O
a	O
magnitude	O
plot	O
and	O
a	O
phase	O
plot	O
.	O
</s>
<s>
The	O
Bode	O
magnitude	O
plot	O
is	O
the	O
graph	B-Application
of	O
the	O
function	O
of	O
frequency	O
(	O
with	O
being	O
the	O
imaginary	O
unit	O
)	O
.	O
</s>
<s>
The	O
Bode	O
phase	O
plot	O
is	O
the	O
graph	B-Application
of	O
the	O
phase	O
,	O
commonly	O
expressed	O
in	O
degrees	O
,	O
of	O
the	O
transfer	O
function	O
as	O
a	O
function	O
of	O
.	O
</s>
<s>
This	O
section	O
illustrates	O
that	O
a	O
Bode	B-Algorithm
Plot	I-Algorithm
is	O
a	O
visualization	O
of	O
the	O
frequency	O
response	O
of	O
a	O
system	O
.	O
</s>
<s>
These	O
quantities	O
,	O
thus	O
,	O
characterize	O
the	O
frequency	O
response	O
and	O
are	O
shown	O
in	O
the	O
Bode	B-Algorithm
plot	I-Algorithm
.	O
</s>
<s>
For	O
many	O
practical	O
problems	O
,	O
the	O
detailed	O
Bode	B-Algorithm
plots	I-Algorithm
can	O
be	O
approximated	O
with	O
straight-line	O
segments	O
that	O
are	O
asymptotes	B-Language
of	O
the	O
precise	O
response	O
.	O
</s>
<s>
The	O
effect	O
of	O
each	O
of	O
the	O
terms	O
of	O
a	O
multiple	O
element	O
transfer	O
function	O
can	O
be	O
approximated	O
by	O
a	O
set	O
of	O
straight	O
lines	O
on	O
a	O
Bode	B-Algorithm
plot	I-Algorithm
.	O
</s>
<s>
The	O
premise	O
of	O
a	O
Bode	B-Algorithm
plot	I-Algorithm
is	O
that	O
one	O
can	O
consider	O
the	O
log	O
of	O
a	O
function	O
in	O
the	O
form	O
:	O
</s>
<s>
The	O
method	O
for	O
drawing	O
amplitude	O
plots	O
implicitly	O
uses	O
this	O
idea	O
,	O
but	O
since	O
the	O
log	O
of	O
the	O
amplitude	O
of	O
each	O
pole	O
or	O
zero	O
always	O
starts	O
at	O
zero	O
and	O
only	O
has	O
one	O
asymptote	B-Language
change	O
(	O
the	O
straight	O
lines	O
)	O
,	O
the	O
method	O
can	O
be	O
simplified	O
.	O
</s>
<s>
At	O
every	O
value	O
of	O
s	O
where	O
(	O
a	O
zero	O
)	O
,	O
increase	O
the	O
slope	O
of	O
the	O
line	O
by	O
per	O
decade	B-Application
.	O
</s>
<s>
At	O
every	O
value	O
of	O
s	O
where	O
(	O
a	O
pole	O
)	O
,	O
decrease	O
the	O
slope	O
of	O
the	O
line	O
by	O
per	O
decade	B-Application
.	O
</s>
<s>
The	O
initial	O
value	O
of	O
the	O
graph	B-Application
depends	O
on	O
the	O
boundaries	O
.	O
</s>
<s>
Draw	O
a	O
smooth	O
curve	O
through	O
those	O
points	O
using	O
the	O
straight	O
lines	O
as	O
asymptotes	B-Language
(	O
lines	O
which	O
the	O
curve	O
approaches	O
)	O
.	O
</s>
<s>
At	O
every	O
(	O
for	O
stable	O
zeros	O
)	O
,	O
increase	O
the	O
slope	O
by	O
degrees	O
per	O
decade	B-Application
,	O
beginning	O
one	O
decade	B-Application
before	O
(	O
E.g.	O
</s>
<s>
At	O
every	O
(	O
for	O
stable	O
poles	O
)	O
,	O
decrease	O
the	O
slope	O
by	O
degrees	O
per	O
decade	B-Application
,	O
beginning	O
one	O
decade	B-Application
before	O
(	O
E.g.	O
</s>
<s>
To	O
create	O
a	O
straight-line	O
plot	O
for	O
a	O
first-order	O
(	O
one-pole	O
)	O
lowpass	B-Algorithm
filter	I-Algorithm
,	O
one	O
considers	O
the	O
transfer	O
function	O
in	O
terms	O
of	O
the	O
angular	O
frequency	O
:	O
</s>
<s>
The	O
Bode	B-Algorithm
plot	I-Algorithm
is	O
shown	O
in	O
Figure	O
1(b )	O
above	O
,	O
and	O
construction	O
of	O
the	O
straight-line	O
approximation	O
is	O
discussed	O
next	O
.	O
</s>
<s>
Then	O
plotted	O
versus	O
input	O
frequency	O
on	O
a	O
logarithmic	O
scale	O
,	O
can	O
be	O
approximated	O
by	O
two	O
lines	O
and	O
it	O
forms	O
the	O
asymptotic	O
(	O
approximate	O
)	O
magnitude	O
Bode	B-Algorithm
plot	I-Algorithm
of	O
the	O
transfer	O
function	O
:	O
</s>
<s>
The	O
second	O
line	O
for	O
angular	O
frequencies	O
above	O
is	O
a	O
line	O
with	O
a	O
slope	O
of	O
−20dB	O
per	O
decade	B-Application
since	O
at	O
high	O
frequencies	O
the	O
term	O
dominates	O
and	O
the	O
decibel	O
gain	O
expression	O
above	O
simplifies	O
to	O
which	O
is	O
a	O
straight	O
line	O
with	O
a	O
slope	O
of	O
per	O
decade	B-Application
.	O
</s>
<s>
Figures	O
2-5	O
further	O
illustrate	O
construction	O
of	O
Bode	B-Algorithm
plots	I-Algorithm
.	O
</s>
<s>
Figure	O
2	O
shows	O
the	O
Bode	O
magnitude	O
plot	O
for	O
a	O
zero	O
and	O
a	O
low-pass	B-Algorithm
pole	O
,	O
and	O
compares	O
the	O
two	O
with	O
the	O
Bode	O
straight	O
line	O
plots	O
.	O
</s>
<s>
The	O
straight-line	O
plots	O
are	O
horizontal	O
up	O
to	O
the	O
pole	O
(	O
zero	O
)	O
location	O
and	O
then	O
drop	O
(	O
rise	O
)	O
at	O
20dB/decade	O
.	O
</s>
<s>
The	O
phase	O
plots	O
are	O
horizontal	O
up	O
to	O
a	O
frequency	O
factor	B-Application
of	I-Application
ten	I-Application
below	O
the	O
pole	O
(	O
zero	O
)	O
location	O
and	O
then	O
drop	O
(	O
rise	O
)	O
at	O
45°	O
/decade	O
until	O
the	O
frequency	O
is	O
ten	O
times	O
higher	O
than	O
the	O
pole	O
(	O
zero	O
)	O
location	O
.	O
</s>
<s>
Notice	O
in	O
Figure	O
4	O
that	O
the	O
20dB/decade	O
drop	O
of	O
the	O
pole	O
is	O
arrested	O
by	O
the	O
20dB/decade	O
rise	O
of	O
the	O
zero	O
resulting	O
in	O
a	O
horizontal	O
magnitude	O
plot	O
for	O
frequencies	O
above	O
the	O
zero	O
location	O
.	O
</s>
<s>
Notice	O
also	O
in	O
Figure	O
5	O
that	O
the	O
range	O
of	O
frequencies	O
where	O
the	O
phase	O
changes	O
in	O
the	O
straight	O
line	O
plot	O
is	O
limited	O
to	O
frequencies	O
a	O
factor	B-Application
of	I-Application
ten	I-Application
above	O
and	O
below	O
the	O
pole	O
(	O
zero	O
)	O
location	O
.	O
</s>
<s>
Where	O
the	O
phase	O
of	O
the	O
pole	O
and	O
the	O
zero	O
both	O
are	O
present	O
,	O
the	O
straight-line	O
phase	O
plot	O
is	O
horizontal	O
because	O
the	O
45°	O
/decade	O
drop	O
of	O
the	O
pole	O
is	O
arrested	O
by	O
the	O
overlapping	O
45°	O
/decade	O
rise	O
of	O
the	O
zero	O
in	O
the	O
limited	O
range	O
of	O
frequencies	O
where	O
both	O
are	O
active	O
contributors	O
to	O
the	O
phase	O
.	O
</s>
<s>
Bode	B-Algorithm
plots	I-Algorithm
are	O
used	O
to	O
assess	O
the	O
stability	O
of	O
negative	O
feedback	O
amplifiers	O
by	O
finding	O
the	O
gain	O
and	O
phase	O
margins	O
of	O
an	O
amplifier	O
.	O
</s>
<s>
Bode	B-Algorithm
plots	I-Algorithm
are	O
used	O
to	O
determine	O
just	O
how	O
close	O
an	O
amplifier	O
comes	O
to	O
satisfying	O
this	O
condition	O
.	O
</s>
<s>
Although	O
these	O
restrictions	O
usually	O
are	O
met	O
,	O
if	O
they	O
are	O
not	O
another	O
method	O
must	O
be	O
used	O
,	O
such	O
as	O
the	O
Nyquist	B-Application
plot	I-Application
.	O
</s>
<s>
For	O
a	O
three-pole	O
amplifier	O
,	O
Figure	O
6	O
compares	O
the	O
Bode	B-Algorithm
plot	I-Algorithm
for	O
the	O
gain	O
without	O
feedback	O
(	O
the	O
open-loop	O
gain	O
)	O
AOL	O
with	O
the	O
gain	O
with	O
feedback	O
AFB	O
(	O
the	O
closed-loop	O
gain	O
)	O
.	O
</s>
<s>
The	O
Bode	B-Algorithm
plotter	I-Algorithm
is	O
an	O
electronic	O
instrument	O
resembling	O
an	O
oscilloscope	O
,	O
which	O
produces	O
a	O
Bode	B-Algorithm
diagram	I-Algorithm
,	O
or	O
a	O
graph	B-Application
,	O
of	O
a	O
circuit	O
's	O
voltage	O
gain	O
or	O
phase	O
shift	O
plotted	O
against	O
frequency	O
in	O
a	O
feedback	O
control	O
system	O
or	O
a	O
filter	O
.	O
</s>
<s>
For	O
education/research	O
purposes	O
,	O
plotting	O
Bode	B-Algorithm
diagrams	I-Algorithm
for	O
given	O
transfer	O
functions	O
facilitates	O
better	O
understanding	O
and	O
getting	O
faster	O
results	O
(	O
see	O
external	O
links	O
)	O
.	O
</s>
<s>
Two	O
related	O
plots	O
that	O
display	O
the	O
same	O
data	O
in	O
different	O
coordinate	O
systems	O
are	O
the	O
Nyquist	B-Application
plot	I-Application
and	O
the	O
Nichols	B-Application
plot	I-Application
.	O
</s>
<s>
The	O
Nyquist	B-Application
plot	I-Application
displays	O
these	O
in	O
polar	O
coordinates	O
,	O
with	O
magnitude	O
mapping	O
to	O
radius	O
and	O
phase	O
to	O
argument	O
(	O
angle	O
)	O
.	O
</s>
<s>
The	O
Nichols	B-Application
plot	I-Application
displays	O
these	O
in	O
rectangular	O
coordinates	O
,	O
on	O
the	O
log	O
scale	O
.	O
</s>
