models of two-port networks z, y, h, parameters

8/17/2019 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:

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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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models of two-port networks z, y, h, parameters

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