<s>
In	O
electronics	O
,	O
a	O
two-port	B-Algorithm
network	I-Algorithm
(	O
a	O
kind	O
of	O
four-terminal	O
network	O
or	O
quadripole	B-Algorithm
)	O
is	O
an	O
electrical	O
network	O
(	O
i.e.	O
</s>
<s>
In	O
a	O
two-port	B-Algorithm
network	I-Algorithm
,	O
often	O
port	O
1	O
is	O
considered	O
the	O
input	O
port	O
and	O
port	O
2	O
is	O
considered	O
the	O
output	O
port	O
.	O
</s>
<s>
The	O
two-port	B-Algorithm
network	I-Algorithm
model	O
is	O
used	O
in	O
mathematical	O
circuit	O
analysis	O
techniques	O
to	O
isolate	O
portions	O
of	O
larger	O
circuits	O
.	O
</s>
<s>
A	O
two-port	B-Algorithm
network	I-Algorithm
is	O
regarded	O
as	O
a	O
"	O
black	B-Device
box	I-Device
"	O
with	O
its	O
properties	O
specified	O
by	O
a	O
matrix	B-Architecture
of	O
numbers	O
.	O
</s>
<s>
For	O
example	O
,	O
transistors	O
are	O
often	O
regarded	O
as	O
two-ports	B-Algorithm
,	O
characterized	O
by	O
their	O
-parameters	O
(	O
see	O
below	O
)	O
which	O
are	O
listed	O
by	O
the	O
manufacturer	O
.	O
</s>
<s>
Any	O
linear	O
circuit	O
with	O
four	O
terminals	O
can	O
be	O
regarded	O
as	O
a	O
two-port	B-Algorithm
network	I-Algorithm
provided	O
that	O
it	O
does	O
not	O
contain	O
an	O
independent	O
source	O
and	O
satisfies	O
the	O
port	O
conditions	O
.	O
</s>
<s>
Examples	O
of	O
circuits	O
analyzed	O
as	O
two-ports	B-Algorithm
are	O
filters	O
,	O
matching	O
networks	O
,	O
transmission	O
lines	O
,	O
transformers	B-Algorithm
,	O
and	O
small-signal	O
models	O
for	O
transistors	O
(	O
such	O
as	O
the	O
hybrid-pi	O
model	O
)	O
.	O
</s>
<s>
The	O
analysis	O
of	O
passive	O
two-port	B-Algorithm
networks	I-Algorithm
is	O
an	O
outgrowth	O
of	O
reciprocity	O
theorems	O
first	O
derived	O
by	O
Lorentz	O
.	O
</s>
<s>
In	O
two-port	B-Algorithm
mathematical	O
models	O
,	O
the	O
network	O
is	O
described	O
by	O
a	O
2	O
by	O
2	O
square	O
matrix	B-Architecture
of	O
complex	O
numbers	O
.	O
</s>
<s>
At	O
high	O
frequencies	O
(	O
e.g.	O
,	O
microwave	O
frequencies	O
)	O
,	O
the	O
use	O
of	O
power	O
and	O
energy	O
variables	O
is	O
more	O
appropriate	O
,	O
and	O
the	O
two-port	B-Algorithm
currentvoltage	O
approach	O
is	O
replaced	O
by	O
an	O
approach	O
based	O
upon	O
scattering	O
parameters	O
.	O
</s>
<s>
There	O
are	O
certain	O
properties	O
of	O
two-ports	B-Algorithm
that	O
frequently	O
occur	O
in	O
practical	O
networks	O
and	O
can	O
be	O
used	O
to	O
greatly	O
simplify	O
the	O
analysis	O
.	O
</s>
<s>
A	O
network	O
that	O
consists	O
entirely	O
of	O
linear	O
passive	O
components	O
(	O
that	O
is	O
,	O
resistors	O
,	O
capacitors	O
and	O
inductors	O
)	O
is	O
usually	O
reciprocal	O
,	O
a	O
notable	O
exception	O
being	O
passive	O
circulators	O
and	O
isolators	B-Algorithm
that	O
contain	O
magnetized	O
materials	O
.	O
</s>
<s>
All	O
the	O
Y-parameters	B-Algorithm
have	O
dimensions	O
of	O
siemens	O
.	O
</s>
<s>
The	O
negative	O
sign	O
for	O
reflects	O
the	O
convention	O
that	O
are	O
positive	O
when	O
directed	O
into	O
the	O
two-port	B-Algorithm
.	O
</s>
<s>
The	O
negative	O
sign	O
for	O
reflects	O
the	O
convention	O
that	O
are	O
positive	O
when	O
directed	O
into	O
the	O
two-port	B-Algorithm
.	O
</s>
<s>
This	O
representation	O
is	O
preferred	O
because	O
when	O
the	O
parameters	O
are	O
used	O
to	O
represent	O
a	O
cascade	O
of	O
two-ports	B-Algorithm
,	O
the	O
matrices	O
are	O
written	O
in	O
the	O
same	O
order	O
that	O
a	O
network	O
diagram	O
would	O
be	O
drawn	O
,	O
that	O
is	O
,	O
left	O
to	O
right	O
.	O
</s>
<s>
The	O
negative	O
sign	O
of	O
arises	O
to	O
make	O
the	O
output	O
current	O
of	O
one	O
cascaded	O
stage	O
(	O
as	O
it	O
appears	O
in	O
the	O
matrix	B-Architecture
)	O
equal	O
to	O
the	O
input	O
current	O
of	O
the	O
next	O
.	O
</s>
<s>
Consequently	O
,	O
the	O
input	O
voltage/current	O
matrix	B-Architecture
vector	O
can	O
be	O
directly	O
replaced	O
with	O
the	O
matrix	B-Architecture
equation	I-Architecture
of	O
the	O
preceding	O
cascaded	O
stage	O
to	O
form	O
a	O
combined	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
terminology	O
of	O
representing	O
the	O
parameters	O
as	O
a	O
matrix	B-Architecture
of	O
elements	O
designated	O
etc	O
.	O
</s>
<s>
as	O
adopted	O
by	O
some	O
authors	O
and	O
the	O
inverse	O
parameters	O
as	O
a	O
matrix	B-Architecture
of	O
elements	O
designated	O
etc	O
.	O
</s>
<s>
In	O
this	O
respect	O
-parameters	O
fill	O
the	O
same	O
role	O
as	O
parameters	O
and	O
allow	O
the	O
-parameters	O
of	O
cascaded	O
networks	O
to	O
be	O
calculated	O
by	O
matrix	B-Architecture
multiplication	O
of	O
the	O
component	O
networks	O
.	O
</s>
<s>
When	O
two	O
or	O
more	O
two-port	B-Algorithm
networks	I-Algorithm
are	O
connected	O
,	O
the	O
two-port	B-Algorithm
parameters	I-Algorithm
of	O
the	O
combined	O
network	O
can	O
be	O
found	O
by	O
performing	O
matrix	B-Architecture
algebra	O
on	O
the	O
matrices	O
of	O
parameters	O
for	O
the	O
component	O
two-ports	B-Algorithm
.	O
</s>
<s>
The	O
matrix	B-Architecture
operation	O
can	O
be	O
made	O
particularly	O
simple	O
with	O
an	O
appropriate	O
choice	O
of	O
two-port	B-Algorithm
parameters	I-Algorithm
to	O
match	O
the	O
form	O
of	O
connection	O
of	O
the	O
two-ports	B-Algorithm
.	O
</s>
<s>
This	O
difficulty	O
can	O
be	O
overcome	O
by	O
placing	O
1:1	O
ideal	O
transformers	B-Algorithm
on	O
the	O
outputs	O
of	O
the	O
problem	O
two-ports	B-Algorithm
.	O
</s>
<s>
This	O
does	O
not	O
change	O
the	O
parameters	O
of	O
the	O
two-ports	B-Algorithm
,	O
but	O
does	O
ensure	O
that	O
they	O
will	O
continue	O
to	O
meet	O
the	O
port	O
condition	O
when	O
interconnected	O
.	O
</s>
<s>
When	O
two-ports	B-Algorithm
are	O
connected	O
in	O
a	O
series-series	O
configuration	O
as	O
shown	O
in	O
figure	O
10	O
,	O
the	O
best	O
choice	O
of	O
two-port	B-Algorithm
parameter	O
is	O
the	O
-parameters	O
.	O
</s>
<s>
The	O
-parameters	O
of	O
the	O
combined	O
network	O
are	O
found	O
by	O
matrix	B-Architecture
addition	O
of	O
the	O
two	O
individual	O
-parameter	O
matrices	O
.	O
</s>
<s>
A	O
simple	O
example	O
is	O
a	O
two-port	B-Algorithm
consisting	O
of	O
a	O
-network	O
of	O
resistors	O
and	O
.	O
</s>
<s>
The	O
total	O
-parameters	O
predicted	O
by	O
matrix	B-Architecture
addition	O
are	O
;	O
</s>
<s>
The	O
discrepancy	O
is	O
explained	O
by	O
observing	O
that	O
of	O
the	O
lower	O
two-port	B-Algorithm
has	O
been	O
by-passed	O
by	O
the	O
short-circuit	O
between	O
two	O
terminals	O
of	O
the	O
output	O
ports	O
.	O
</s>
<s>
This	O
problem	O
can	O
be	O
resolved	O
by	O
inserting	O
an	O
ideal	O
transformer	B-Algorithm
in	O
the	O
output	O
port	O
of	O
at	O
least	O
one	O
of	O
the	O
two-port	B-Algorithm
networks	I-Algorithm
.	O
</s>
<s>
While	O
this	O
is	O
a	O
common	O
text-book	O
approach	O
to	O
presenting	O
the	O
theory	O
of	O
two-ports	B-Algorithm
,	O
the	O
practicality	O
of	O
using	O
transformers	B-Algorithm
is	O
a	O
matter	O
to	O
be	O
decided	O
for	O
each	O
individual	O
design	O
.	O
</s>
<s>
When	O
two-ports	B-Algorithm
are	O
connected	O
in	O
a	O
parallel-parallel	O
configuration	O
as	O
shown	O
in	O
figure	O
13	O
,	O
the	O
best	O
choice	O
of	O
two-port	B-Algorithm
parameter	O
is	O
the	O
-parameters	O
.	O
</s>
<s>
The	O
-parameters	O
of	O
the	O
combined	O
network	O
are	O
found	O
by	O
matrix	B-Architecture
addition	O
of	O
the	O
two	O
individual	O
-parameter	O
matrices	O
.	O
</s>
<s>
When	O
two-ports	B-Algorithm
are	O
connected	O
in	O
a	O
series-parallel	O
configuration	O
as	O
shown	O
in	O
figure	O
14	O
,	O
the	O
best	O
choice	O
of	O
two-port	B-Algorithm
parameter	O
is	O
the	O
-parameters	O
.	O
</s>
<s>
The	O
-parameters	O
of	O
the	O
combined	O
network	O
are	O
found	O
by	O
matrix	B-Architecture
addition	O
of	O
the	O
two	O
individual	O
-parameter	O
matrices	O
.	O
</s>
<s>
When	O
two-ports	B-Algorithm
are	O
connected	O
in	O
a	O
parallel-series	O
configuration	O
as	O
shown	O
in	O
figure	O
15	O
,	O
the	O
best	O
choice	O
of	O
two-port	B-Algorithm
parameter	O
is	O
the	O
-parameters	O
.	O
</s>
<s>
The	O
-parameters	O
of	O
the	O
combined	O
network	O
are	O
found	O
by	O
matrix	B-Architecture
addition	O
of	O
the	O
two	O
individual	O
-parameter	O
matrices	O
.	O
</s>
<s>
When	O
two-ports	B-Algorithm
are	O
connected	O
with	O
the	O
output	O
port	O
of	O
the	O
first	O
connected	O
to	O
the	O
input	O
port	O
of	O
the	O
second	O
(	O
a	O
cascade	O
connection	O
)	O
as	O
shown	O
in	O
figure	O
16	O
,	O
the	O
best	O
choice	O
of	O
two-port	B-Algorithm
parameter	O
is	O
the	O
-parameters	O
.	O
</s>
<s>
The	O
-parameters	O
of	O
the	O
combined	O
network	O
are	O
found	O
by	O
matrix	B-Architecture
multiplication	O
of	O
the	O
two	O
individual	O
-parameter	O
matrices	O
.	O
</s>
<s>
A	O
chain	O
of	O
two-ports	B-Algorithm
may	O
be	O
combined	O
by	O
matrix	B-Architecture
multiplication	O
of	O
the	O
matrices	O
.	O
</s>
<s>
Suppose	O
we	O
have	O
a	O
two-port	B-Algorithm
network	I-Algorithm
consisting	O
of	O
a	O
series	O
resistor	O
followed	O
by	O
a	O
shunt	O
capacitor	O
.	O
</s>
<s>
The	O
transmission	O
matrix	B-Architecture
for	O
the	O
entire	O
network	O
is	O
simply	O
the	O
matrix	B-Architecture
multiplication	O
of	O
the	O
transmission	O
matrices	O
for	O
the	O
two	O
network	O
elements	O
:	O
</s>
<s>
The	O
admittance	B-Algorithm
parameters	I-Algorithm
are	O
the	O
matrix	B-Architecture
inverse	O
of	O
the	O
impedance	O
parameters	O
,	O
the	O
inverse	O
hybrid	O
parameters	O
are	O
the	O
matrix	B-Architecture
inverse	O
of	O
the	O
hybrid	O
parameters	O
,	O
and	O
the	O
form	O
of	O
the	O
-parameters	O
is	O
the	O
matrix	B-Architecture
inverse	O
of	O
the	O
form	O
.	O
</s>
<s>
While	O
two	B-Algorithm
port	I-Algorithm
networks	I-Algorithm
are	O
very	O
common	O
(	O
e.g.	O
,	O
amplifiers	O
and	O
filters	O
)	O
,	O
other	O
electrical	O
networks	O
such	O
as	O
directional	O
couplers	O
and	O
circulators	O
have	O
more	O
than	O
2	O
ports	O
.	O
</s>
<s>
However	O
the	O
following	O
representations	O
are	O
necessarily	O
limited	O
to	O
two-port	B-Algorithm
devices	O
:	O
</s>
<s>
A	O
two-port	B-Algorithm
network	I-Algorithm
has	O
four	O
variables	O
with	O
two	O
of	O
them	O
being	O
independent	O
.	O
</s>
<s>
The	O
two-port	B-Algorithm
becomes	O
a	O
one-port	O
impedance	O
to	O
the	O
remaining	O
independent	O
variable	O
.	O
</s>
<s>
The	O
negative	O
sign	O
is	O
because	O
the	O
positive	O
direction	O
for	O
is	O
directed	O
into	O
the	O
two-port	B-Algorithm
instead	O
of	O
into	O
the	O
load	O
.	O
</s>
<s>
So	O
,	O
in	O
effect	O
,	O
sees	O
an	O
input	O
impedance	O
and	O
the	O
two-port	B-Algorithm
'	O
s	O
effect	O
on	O
the	O
input	O
circuit	O
has	O
been	O
effectively	O
collapsed	O
down	O
to	O
a	O
one-port	O
;	O
i.e.	O
,	O
a	O
simple	O
two	O
terminal	O
impedance	O
.	O
</s>
