Many complex passive and linear circuits can be modeled by a two-port network model as shown below A two-port network is represented by b y !our external variables: volta"e and current at the input port# and volta"e and current at the output port# so that the two-port network can be treated as a black box modeled by the relationships between the !our variables # # and There exist six di!!erent ways to describe the relationships between these variables# dependin" on which two o! the !our variables are "iven# while the other two can always be derived
•
Z or impedance model: $iven two currents
and
by:
and
!ind volta"es
%ere all !our parameters particular#
and
volta"e
'or
part
•
#
# and
( in one part o! a network to a current
'or
( in another
is a 2 by 2 matrix containin" all !our parameters and
# !ind currents
by:
%ere all !our parameters particular# and parameter matrix
#
and
#
# and
represent admittance &n
are transfer admittances
A or transmission model: $iven
%ere
represent impedance &n
are transfer impedances# de!ined as the ratio o! a
Y or admittance model: $iven two volta"es
and
•
#
and
# !ind
are dimensionless coe!!icients#
is the correspondin"
and
by:
is impedance and
is admittance A ne"ative si"n is added to the output current in the model# so that the direction o! the current is out-ward# !or easy analysis o! a cascade o! multiple network models
•
H or !"rid model: $iven
%ere and is admittance
and
# !ind
and
are dimensionless coe!!icients#
by:
is impedance and
#enerali$ation to nonlinear circuits
The two-port models can also be applied to a nonlinear circuit i! the variations o! the variables are small and there!ore the nonlinear behavior o! the circuit can be piecewise lineari)ed Assume the variations model
and
is a nonlinear !unction o! variables
%indin& te model parameters
*or each o! the !our types o! models# the !our parameters can be !ound !rom
•
#
#
*or +-model:
and
&!
are small# the !unction can be approximated by a linear
with the linear coe!!icients
variables
and
o! a network by the !ollowin"
•
*or ,-model:
•
*or A-model:
•
*or %-model:
&! we !urther de!ine
then the +-model and ,-model above can be written in matrix !orm:
'xample:
*ind the +-model and ,-model o! the circuit shown
•
*irst assume
# we "et
•
Next assume
# we "et
The parameters o! the ,-model can be !ound as the inverse o!
:
Note:
(om"inations of two-port models
•
eries connection o! two 2-port networks:
•
Parallel connection o! two 2-port networks:
•
Cascade connection o! two 2-port networks:
'xample: A The circuit shown below contains a two-port network 'e"# a !ilter circuit# or an ampli!ication circuit( represented by a +-model:
The input volta"e is impedance is
with an internal impedance *ind the two volta"es
#
and the load
and two currents
#
Metod ): •
*irst# accordin" the +-model# we have
•
econd# two more e.uations can be obtained !rom the circuit:
•
ubstitutin" the last two e.uations !or
•
olvin" these we "et
and
into the !irst two# we "et
•
Then we can "et the volta"es
Metod *: /e can also use Thevenin0s theorem to treat everythin" be!ore the load
impedance as an e.uivalent volta"e source with Thevenin0s volta"e resistance
•
# and the output volta"e
*ind
with volta"e
and current
and
can be !ound
short-circuit:
o
The +-model:
o
Also due to the short-circuit o! volta"e source
o
e.uatin" the two expressions !or
# we "et
# we have
•
o
ubstitutin" this
o
*ind
into the e.uation !or
above# we "et
:
*ind open-circuit volta"e
with
o
ince the load is an open-circuit#
o
*ind
:
:
# we have
olvin" this to "et
o
*ind open-circuit volta"e
•
*ind load volta"e
:
•
*ind load volta"e
:
Principle of reciprocit! :
:
Consider the example circuit on the le!t above# which can be simpli!ied as the network in the middle The volta"e source is in the branch on the le!t# while the current the branch on the ri"ht# which can be !ound to be 'current divider(:
is in
/e next interchan"e the positions o! the volta"e source and the current# so that the volta"e source is in the branch on the ri"ht and the current to be !ound is in the branch on the le!t# as shown on the ri"ht o! the !i"ure above The current can be !ound to be
The two currents and are exactly the same1 This result illustrates the !ollowin" reciprocit! principle# which can be proven in "eneral: In any passive (without energy sources), linear network, if a voltage applied in branch 1 causes a current in branch 2, then this voltage applied in branch 2 will cause the same current in branch 1. This reciprocity principle can also be stated as: In any passive, linear network, the transfer impedance transfer impedance
is equal to the reciprocal
.
ased on this reciprocity principle# any complex passive linear network can be modeled by either a T-network or a -network:
•
T-Network Model:
*rom this T-model# we "et
Comparin" this with the +-model# we "et
olvin" these e.uations !or
#
and
# we "et
•
-Network Model:
*rom this
-model# we "et:
Comparin" this with the ,-model# we "et
olvin" these e.uations !or
#
and
# we "et
'xample ): Convert the "iven T-network to a
network
+olution: $iven
#
#
# we "et its +-model:
The +-model can be expressed in matrix !orm:
This +-model can be converted into a ,-model:
This ,-model can be converted to a
network:
These admittances can be !urther converted into impedances:
The same results can be obtained by , to delta conversion
'xample *: Consider the ideal trans!ormer shown in the !i"ure below
Assume # circuit as a two-port network
# and the turn ratio is
3escribe this
•
et up basic e.uations:
•
4earran"e the e.uations in the !orm o! a +-model The second e.uation is
ubstitutin" into the !irst e.uation# we "et
The +-model is:
As
# this is a reciprocal network
Alternatively# we can set up the e.uations in terms o! the currents: •
•
4earran"e the e.uations in the !orm o! a ,-model The !irst e.uation is
