<s>
Electronic	B-Algorithm
filter	I-Algorithm
topology	I-Algorithm
defines	O
electronic	B-Architecture
filter	I-Architecture
circuits	O
without	O
taking	O
note	O
of	O
the	O
values	O
of	O
the	O
components	O
used	O
but	O
only	O
the	O
manner	O
in	O
which	O
those	O
components	O
are	O
connected	O
.	O
</s>
<s>
Filter	O
design	O
characterises	O
filter	O
circuits	O
primarily	O
by	O
their	O
transfer	O
function	O
rather	O
than	O
their	O
topology	B-Architecture
.	O
</s>
<s>
Common	O
types	O
of	O
linear	O
filter	O
transfer	O
function	O
are	O
;	O
high-pass	O
,	O
low-pass	B-Algorithm
,	O
bandpass	O
,	O
band-reject	O
or	O
notch	O
and	O
all-pass	O
.	O
</s>
<s>
Once	O
the	O
transfer	O
function	O
for	O
a	O
filter	O
is	O
chosen	O
,	O
the	O
particular	O
topology	B-Architecture
to	O
implement	O
such	O
a	O
prototype	O
filter	O
can	O
be	O
selected	O
so	O
that	O
,	O
for	O
example	O
,	O
one	O
might	O
choose	O
to	O
design	O
a	O
Butterworth	O
filter	O
using	O
the	O
Sallen	O
–	O
Key	O
topology	B-Architecture
.	O
</s>
<s>
Filter	O
topologies	B-Architecture
may	O
be	O
divided	O
into	O
passive	O
and	O
active	O
types	O
.	O
</s>
<s>
Passive	O
topologies	B-Architecture
are	O
composed	O
exclusively	O
of	O
passive	O
components	O
:	O
resistors	O
,	O
capacitors	O
,	O
and	O
inductors	O
.	O
</s>
<s>
Active	O
topologies	B-Architecture
also	O
include	O
active	O
components	O
(	O
such	O
as	O
transistors	O
,	O
op	O
amps	O
,	O
and	O
other	O
integrated	O
circuits	O
)	O
that	O
require	O
power	O
.	O
</s>
<s>
Further	O
,	O
topologies	B-Architecture
may	O
be	O
implemented	O
either	O
in	O
unbalanced	O
form	O
or	O
else	O
in	O
balanced	O
form	O
when	O
employed	O
in	O
balanced	O
circuits	O
.	O
</s>
<s>
Most	O
are	O
built	O
from	O
simple	O
two-port	B-Algorithm
networks	I-Algorithm
called	O
"	O
sections	O
"	O
.	O
</s>
<s>
Sections	O
are	O
invariably	O
connected	O
in	O
a	O
"	O
cascade	O
"	O
or	O
"	B-Application
daisy-chain	I-Application
"	I-Application
topology	B-Architecture
,	O
consisting	O
of	O
additional	O
copies	O
of	O
the	O
same	O
section	O
or	O
of	O
completely	O
different	O
sections	O
.	O
</s>
<s>
All	O
are	O
built	O
upon	O
a	O
very	O
simple	O
"	O
ladder	O
"	O
topology	B-Architecture
(	O
see	O
below	O
)	O
.	O
</s>
<s>
The	O
chart	O
at	O
the	O
bottom	O
of	O
the	O
page	O
shows	O
these	O
various	O
topologies	B-Architecture
in	O
terms	O
of	O
general	O
constant	O
k	O
filters	O
.	O
</s>
<s>
Filters	O
designed	O
using	O
network	O
synthesis	O
usually	O
repeat	O
the	O
simplest	O
form	O
of	O
L-section	O
topology	B-Architecture
though	O
component	O
values	O
may	O
change	O
in	O
each	O
section	O
.	O
</s>
<s>
Image	O
designed	O
filters	O
,	O
on	O
the	O
other	O
hand	O
,	O
keep	O
the	O
same	O
basic	O
component	O
values	O
from	O
section	O
to	O
section	O
though	O
the	O
topology	B-Architecture
may	O
vary	O
and	O
tend	O
to	O
make	O
use	O
of	O
more	O
complex	O
sections	O
.	O
</s>
<s>
L-sections	O
are	O
never	O
symmetrical	O
but	O
two	O
L-sections	O
back-to-back	O
form	O
a	O
symmetrical	O
topology	B-Architecture
and	O
many	O
other	O
sections	O
are	O
symmetrical	O
in	O
form	O
.	O
</s>
<s>
Ladder	O
topology	B-Architecture
,	O
often	O
called	O
Cauer	O
topology	B-Architecture
after	O
Wilhelm	O
Cauer	O
(	O
inventor	O
of	O
the	O
elliptic	O
filter	O
)	O
,	O
was	O
in	O
fact	O
first	O
used	O
by	O
George	O
Campbell	O
(	O
inventor	O
of	O
the	O
constant	O
k	O
filter	O
)	O
.	O
</s>
<s>
Campbell	O
published	O
in	O
1922	O
but	O
had	O
clearly	O
been	O
using	O
the	O
topology	B-Architecture
for	O
some	O
time	O
before	O
this	O
.	O
</s>
<s>
There	O
are	O
two	O
forms	O
of	O
basic	O
ladder	O
topologies	B-Architecture
:	O
unbalanced	O
and	O
balanced	O
.	O
</s>
<s>
Cauer	O
topology	B-Architecture
is	O
usually	O
thought	O
of	O
as	O
an	O
unbalanced	O
ladder	O
topology	B-Architecture
.	O
</s>
<s>
In	O
low	B-Algorithm
pass	I-Algorithm
form	O
the	O
topology	B-Architecture
would	O
consist	O
of	O
series	O
inductors	O
and	O
shunt	O
capacitors	O
.	O
</s>
<s>
Other	O
bandforms	O
would	O
have	O
an	O
equally	O
simple	O
topology	B-Architecture
transformed	O
from	O
the	O
lowpass	B-Algorithm
topology	B-Architecture
.	O
</s>
<s>
Image	O
filter	O
design	O
commonly	O
uses	O
modifications	O
of	O
the	O
basic	O
ladder	O
topology	B-Architecture
.	O
</s>
<s>
These	O
topologies	B-Architecture
,	O
invented	O
by	O
Otto	O
Zobel	O
,	O
have	O
the	O
same	O
passbands	B-Algorithm
as	O
the	O
ladder	O
on	O
which	O
they	O
are	O
based	O
but	O
their	O
transfer	O
functions	O
are	O
modified	O
to	O
improve	O
some	O
parameter	O
such	O
as	O
impedance	O
matching	O
,	O
stopband	O
rejection	O
or	O
passband-to-stopband	O
transition	O
steepness	O
.	O
</s>
<s>
Usually	O
the	O
design	O
applies	O
some	O
transform	O
to	O
a	O
simple	O
ladder	O
topology	B-Architecture
:	O
the	O
resulting	O
topology	B-Architecture
is	O
ladder-like	O
but	O
no	O
longer	O
obeys	O
the	O
rule	O
that	O
shunt	O
admittances	O
are	O
the	O
dual	O
network	O
of	O
series	O
impedances	O
:	O
it	O
invariably	O
becomes	O
more	O
complex	O
with	O
higher	O
component	O
count	O
.	O
</s>
<s>
Such	O
topologies	B-Architecture
include	O
;	O
</s>
<s>
The	O
m-type	O
(	O
m-derived	O
)	O
filter	O
is	O
by	O
far	O
the	O
most	O
commonly	O
used	O
modified	O
image	O
ladder	O
topology	B-Architecture
.	O
</s>
<s>
There	O
are	O
two	O
m-type	O
topologies	B-Architecture
for	O
each	O
of	O
the	O
basic	O
ladder	O
topologies	B-Architecture
;	O
the	O
series-derived	O
and	O
shunt-derived	O
topologies	B-Architecture
.	O
</s>
<s>
Where	O
a	O
filter	O
is	O
being	O
designed	O
with	O
more	O
than	O
one	O
passband	B-Algorithm
,	O
the	O
m-type	O
topology	B-Architecture
will	O
result	O
in	O
a	O
filter	O
where	O
each	O
passband	B-Algorithm
has	O
an	O
analogous	O
frequency-domain	O
response	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
generalise	O
the	O
m-type	O
topology	B-Architecture
for	O
filters	O
with	O
more	O
than	O
one	O
passband	B-Algorithm
using	O
parameters	O
m1	O
,	O
m2	O
,	O
m3	O
etc.	O
,	O
which	O
are	O
not	O
equal	O
to	O
each	O
other	O
resulting	O
in	O
general	O
mn-type	O
filters	O
which	O
have	O
bandforms	O
that	O
can	O
differ	O
in	O
different	O
parts	O
of	O
the	O
frequency	O
spectrum	O
.	O
</s>
<s>
The	O
mm'-type	O
topology	B-Architecture
can	O
be	O
thought	O
of	O
as	O
a	O
double	O
m-type	O
design	O
.	O
</s>
<s>
Zobel	O
constant	O
resistance	O
filters	O
use	O
a	O
topology	B-Architecture
that	O
is	O
somewhat	O
different	O
from	O
other	O
filter	O
types	O
,	O
distinguished	O
by	O
having	O
a	O
constant	O
input	O
resistance	O
at	O
all	O
frequencies	O
and	O
in	O
that	O
they	O
use	O
resistive	O
components	O
in	O
the	O
design	O
of	O
their	O
sections	O
.	O
</s>
<s>
Topologies	B-Architecture
usually	O
associated	O
with	O
constant	O
resistance	O
filters	O
are	O
the	O
bridged-T	O
and	O
its	O
variants	O
,	O
all	O
described	O
in	O
the	O
Zobel	O
network	O
article	O
;	O
</s>
<s>
The	O
bridged-T	O
topology	B-Architecture
is	O
also	O
used	O
in	O
sections	O
intended	O
to	O
produce	O
a	O
signal	O
delay	O
but	O
in	O
this	O
case	O
no	O
resistive	O
components	O
are	O
used	O
in	O
the	O
design	O
.	O
</s>
<s>
Both	O
the	O
T-section	O
(	O
from	O
ladder	O
topology	B-Architecture
)	O
and	O
the	O
bridge-T	O
(	O
from	O
Zobel	O
topology	B-Architecture
)	O
can	O
be	O
transformed	O
into	O
a	O
lattice	O
topology	B-Architecture
filter	O
section	O
but	O
in	O
both	O
cases	O
this	O
results	O
in	O
high	O
component	O
count	O
and	O
complexity	O
.	O
</s>
<s>
Lattice	O
topology	B-Architecture
is	O
identical	O
to	O
the	O
more	O
familiar	O
bridge	O
topology	B-Architecture
,	O
the	O
difference	O
being	O
merely	O
the	O
drawn	O
representation	O
on	O
the	O
page	O
rather	O
than	O
any	O
real	O
difference	O
in	O
topology	B-Architecture
,	O
circuitry	O
or	O
function	O
.	O
</s>
<s>
Multiple	O
feedback	O
topology	B-Architecture
is	O
an	O
electronic	B-Algorithm
filter	I-Algorithm
topology	I-Algorithm
which	O
is	O
used	O
to	O
implement	O
an	O
electronic	B-Architecture
filter	I-Architecture
by	O
adding	O
two	O
poles	O
to	O
the	O
transfer	O
function	O
.	O
</s>
<s>
A	O
diagram	O
of	O
the	O
circuit	O
topology	B-Architecture
for	O
a	O
second	O
order	O
low	B-Algorithm
pass	I-Algorithm
filter	I-Algorithm
is	O
shown	O
in	O
the	O
figure	O
on	O
the	O
right	O
.	O
</s>
<s>
The	O
transfer	O
function	O
of	O
the	O
multiple	O
feedback	O
topology	B-Architecture
circuit	O
,	O
like	O
all	O
second-order	O
linear	O
filters	O
,	O
is	O
:	O
</s>
<s>
For	O
finding	O
suitable	O
component	O
values	O
to	O
achieve	O
the	O
desired	O
filter	O
properties	O
,	O
a	O
similar	O
approach	O
can	O
be	O
followed	O
as	O
in	O
the	O
Design	O
choices	O
section	O
of	O
the	O
alternative	O
Sallen	O
–	O
Key	O
topology	B-Architecture
.	O
</s>
<s>
Any	O
second-order	O
filter	O
topology	B-Architecture
can	O
be	O
referred	O
to	O
as	O
a	O
biquad	O
,	O
such	O
as	O
the	O
MFB	O
or	O
Sallen-Key	O
.	O
</s>
<s>
However	O
,	O
there	O
is	O
also	O
a	O
specific	O
"	O
biquad	O
"	O
topology	B-Architecture
.	O
</s>
<s>
Biquad	O
filters	O
are	O
typically	O
active	O
and	O
implemented	O
with	O
a	O
single-amplifier	O
biquad	O
(	O
SAB	O
)	O
or	O
two-integrator-loop	O
topology	B-Architecture
.	O
</s>
<s>
The	O
SAB	O
topology	B-Architecture
uses	O
feedback	O
to	O
generate	O
complex	O
poles	O
and	O
possibly	O
complex	O
zeros	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
feedback	O
moves	O
the	O
real	O
poles	O
of	O
an	O
RC	B-Algorithm
circuit	I-Algorithm
in	O
order	O
to	O
generate	O
the	O
proper	O
filter	O
characteristics	O
.	O
</s>
<s>
The	O
two-integrator-loop	O
topology	B-Architecture
is	O
derived	O
from	O
rearranging	O
a	O
biquadratic	O
transfer	O
function	O
.	O
</s>
<s>
The	O
SAB	O
topology	B-Architecture
is	O
sensitive	O
to	O
component	O
choice	O
and	O
can	O
be	O
more	O
difficult	O
to	O
adjust	O
.	O
</s>
<s>
Hence	O
,	O
usually	O
the	O
term	O
biquad	O
refers	O
to	O
the	O
two-integrator-loop	O
state	O
variable	O
filter	O
topology	B-Architecture
.	O
</s>
<s>
For	O
example	O
,	O
the	O
basic	O
configuration	O
in	O
Figure	O
1	O
can	O
be	O
used	O
as	O
either	O
a	O
low-pass	B-Algorithm
or	O
bandpass	O
filter	O
depending	O
on	O
where	O
the	O
output	O
signal	O
is	O
taken	O
from	O
.	O
</s>
<s>
where	O
low-pass	B-Algorithm
gain	O
.	O
</s>
<s>
If	O
a	O
noninverting	O
low-pass	B-Algorithm
filter	I-Algorithm
is	O
required	O
,	O
the	O
output	O
can	O
be	O
taken	O
at	O
the	O
output	O
of	O
the	O
second	O
operational	O
amplifier	O
,	O
after	O
the	O
order	O
of	O
the	O
second	O
integrator	O
and	O
the	O
inverter	O
has	O
been	O
switched	O
.	O
</s>
<s>
Figure	O
2	O
shows	O
a	O
variant	O
of	O
the	O
Tow-Thomas	O
topology	B-Architecture
,	O
known	O
as	O
Akerberg-Mossberg	O
topology	B-Architecture
,	O
that	O
uses	O
an	O
actively	O
compensated	O
Miller	O
integrator	O
,	O
which	O
improves	O
filter	O
performance	O
.	O
</s>
<s>
The	O
Sallen-Key	O
design	O
is	O
a	O
non-inverting	O
second-order	O
filter	O
with	O
the	O
option	O
of	O
high	O
Q	O
and	O
passband	B-Algorithm
gain	O
.	O
</s>
