Models of Two-port Networks Z, Y, H, parameters
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Transcript of Models of Two-port Networks Z, Y, H, parameters
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Next: Active Circuits Up: Chapter 2: Circuit Principles Previous: NetworkTheorems
Two-Port Networks
Models of two-port networks
Many complex passive and linear circuits can be modeled by a two-port networkmodel as shown below A two-port network is represented by !our external variables:
volta"e and current at the input port# and volta"e and current at theoutput 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!!erentways to describe the relationships between these variables# dependin" on which twoo! the !our variables are "iven# while the other two can always be derived
• Z or impedance model: $iven two currents and !ind volta"es
and by:
http://fourier.eng.hmc.edu/e84/lectures/ch2/node5.htmlhttp://fourier.eng.hmc.edu/e84/lectures/ch2/node1.htmlhttp://fourier.eng.hmc.edu/e84/lectures/ch2/node3.htmlhttp://fourier.eng.hmc.edu/e84/lectures/ch2/node3.htmlhttp://fourier.eng.hmc.edu/e84/lectures/ch2/node1.htmlhttp://fourier.eng.hmc.edu/e84/lectures/ch2/node3.htmlhttp://fourier.eng.hmc.edu/e84/lectures/ch2/node3.htmlhttp://fourier.eng.hmc.edu/e84/lectures/ch2/node5.html
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%ere all !our parameters # # # and represent impedance &n
particular# and are transfer impedances # de!ined as the ratio o! a
volta"e 'or ( in one part o! a network to a current 'or ( in another
part is a 2 by 2 matrix containin" all !our parameters
• Y or admittance model: $iven two volta"es and # !ind currents
and by:
%ere all !our parameters # # # and represent admittance &n
particular# and are transfer admittances is the correspondin" parameter matrix
• A or transmission model: $iven and # !ind and by:
%ere and are dimensionless coe!!icients# is impedance and
is admittance A ne"ative si"n is added to the output current in the model# sothat the direction o! the current is out-ward# !or easy analysis o! a cascade o!multiple network models
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• H or !"rid model: $iven and # !ind and by:
%ere and are dimensionless coe!!icients# is impedance andis admittance
#enerali$ation to nonlinear circuits
The two-port models can also be applied to a nonlinear circuit i! the variations o! thevariables are small and there!ore the nonlinear behavior o! the circuit can be piece-
wise lineari)ed Assume is a nonlinear !unction o! variables and &!
the variations and are small# the !unction can be approximated by a linearmodel
with the linear coe!!icients
%indin& t e model parameters
*or each o! the !our types o! models# the !our parameters can be !ound !rom
variables # # and o! a network by the !ollowin"
• *or +-model:
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• *or ,-model:
•
*or A-model:
• *or %-model:
&! we !urther de!ine
then the +-model and ,-model above can be written in matrix !orm:
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'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! :
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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:
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The input volta"e is with an internal impedance and the load
impedance is *ind the two volta"es # and two currents #
Met od ):
• *irst# accordin" the +-model# we have
• econd# two more e.uations can be obtained !rom the circuit:
• ubstitutin" the last two e.uations !or and into the !irst two# we "et
• olvin" these we "et
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• Then we can "et the volta"es
Met od *: /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 and
resistance # and the output volta"e and current can be !ound
• *ind with volta"e short-circuit:
o The +-model:
o Also due to the short-circuit o! volta"e source # we have
o e.uatin" the two expressions !or # we "et
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o ubstitutin" this into the e.uation !or above# we "et
o *ind :
• *ind open-circuit volta"e with :
o ince the load is an open-circuit# # we have
o *ind :
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olvin" this to "et
o *ind open-circuit volta"e :
• *ind load volta"e :
•
*ind load volta"e :
Principle of reciprocit! :
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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 is inthe branch on the ri"ht# which can be !ound to be 'current divider(:
/e next interchan"e the positions o! the volta"e source and the current# so that thevolta"e source is in the branch on the ri"ht and the current to be !ound is in the branchon 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 inbranch 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 is equal to the reciprocal
transfer impedance .
ased on this reciprocity principle# any complex passive linear network can bemodeled by either a T-network or a -network:
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• T-NetworkModel:
*rom this T-model# we "et
Comparin" this with the +-model# we "et
olvin" these e.uations !or # and # we "et
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• -NetworkModel:
*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
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+olution: $iven # # # we "et its +-model:
The +-model can be expressed in matrix !orm:
This +-model can be converted into a ,-model:
This ,-model can be converted to a network:
These admittances can be !urther converted into impedances:
The same results can be obtained by , to delta conversion
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'xample *: Consider the ideal trans!ormer shown in the !i"ure below
Assume # # and the turn ratio is 3escribe thiscircuit as a two-port network
• et up basic e.uations:
• 4earran"e the e.uations in the !orm o! a +-model The second e.uation is
ubstitutin" into the !irst e.uation# we "et
The +-model is:
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As # this is a reciprocal network
Alternatively# we can set up the e.uations in terms o! the currents:
•
• 4earran"e the e.uations in the !orm o! a ,-model The !irst e.uation is
• ubstitutin" into the second e.uation# we "et
The ,-model is:
*inally# we can veri!y that
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Next: Active Circuits Up: Chapter 2: Circuit Principles Previous: Network Theorems uye !ang 2"1#$"2$1%
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