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Welcome to this second lecture on concept
of image impedance Now I hope you agree with
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me that any 2 port network . can always be
represented by either a T section or a PIE
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section without losing a generality I take
that my 2 port network that means that ab
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issh transmission matrix characterization
is ABCD that is a T section the same analysis
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hold for PIE section so now I have a T section
this is Z1 this is Z2 this is Z3 all this
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are impedances complex impedances so this
is many port 2 this is my port 1 this is the
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internal description of the network now I
want when I will excide the port 1 with that
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voltage BS and I have some internal impedance
let us call that internal impedance Zi1last
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time I called it ZS this time I am calling
it ZI1 is simply change of numb…ain this
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is my port 1 this is my port 2 now you see
all of you are familiar with maximum power
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transfer theorem maximum power transfer theorem
says that if the load impedance complex is
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a complex conjugate of the source impedance
then maximum power gets transferred I think
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that you have noticed that in low frequency
particularly this VLSI people etc when they
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work upto gigahertz range they do not give
any consideration to these maximum power transfer
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theorem because in baseband unless you go
upto radio frequency and transmit it you have
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plenty of power so you are more concerned
with your voltage maximization thats why you
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design a good C amplifier with very high voltage
gain but voltage gain does not necessarily
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mean a maximum power gain but we when we go
to radio frequency we know power microwave
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power is very precious to produce microwave
power lots of complete complicated circuits
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are required So also when power is received
by a receiver in radio frequency it is very
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small amount so its life and death for RF
engineer or a RF circuits to maximize power
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So maximum power transfer theorem always is
the design of RF circuits always aa ba we
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try to pay regard to the maximum power transfer
theorem So can I have this whole thing suppose
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this I will terminate by some load impedance
let that load impedance is called ZI2 this
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one I called ZI1 source impedance is ZI2 now
the idea of image impedance is at this point
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obviously if I look at I will get some impedance
Now if this impedance is equal to ZI1 then
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I can from the source I can have maximum power
transfer ok now according to maximum power
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transfer I suppose these I am looking at some
Zin or something Zin now if ZI is complex
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conjugated ZI1 star then I know that maximum
power transfer will takes place But as I said
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that our consideration now is filter which
is lossless network so this Z1 Z2 Z3 there
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is no R involved ideally So they are complex
but they are generally they will be either
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real or not real either all of them will be
pure imaginary term there will be pure reactance
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So in that case I say that now I say that
I will look into here ZI1 then ZI1 is equal
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to ZI1 not comp not ness suppose any complex
conjugate for pure imaginary things it is
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equal as you know that suppose I have two
complex number A is equal to B star suppose
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A is complex number B is a complex number
if A is pure real and B is pure real then
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I can say A is equal to B Similarly if A is
pure imaginary and B is pure imaginary then
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only I can say A is equal to B Since we know
that this will be my all these are pure reactances
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this may be a pure resistance so that is why
I can call that my demand is ZI1 here should
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be equal to ZI1 here That means source impedance
and these input impedance looking at this
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port should be equal but you see this ZI1
is a function of this load resistance ZI2
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But now so why it is called this ZI1 is called
image impedance if I can find an impedance
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load impedance Zi2 so that if I terminate
this network this is already given network
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and I look at here and see that the input
impedance is ZI1 then I know that I can transfer
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maximum power from the source to this network
this to this port one of this network now
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why ZY ZI1 is called an image impedance because
if at this plane I look so I looking at this
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side I am getting to the right side and getting
and ZI1 looking at left side I am seeing the
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impedances ZI1 So the this side is image of
these that is why this is an image impedance
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it so ZI1 is an image impedance Same thing
here and here I want that if I look at here
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I should look at some output impedance that
should be equal to Zi2 So at port 2 if I look
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to this side I am getting ZI2 if I look this
side I should get ZI2 So if I can find PR
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to ZI1 ZI2 so that if I terminate by ZI2 then
I get here ZI1 impedance similarly here if
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I terminate by ZI1 and excite here I should
see here is ZI2 these two pairs are called
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image impedance Since this is an a symmetrical
network because ZI1 is not equal to ZI2 I
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will have two impedance ZI1 and ZI2 Let us
see that whether this image impedance can
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be represented in terms of this impedance
of this network . so the same diagram I can
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write that ZI1 or do like this so this ZI1
I can write as what is ZI1 obviously it is
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Z1 plus you see Z3 parallel to Z2 plus ZI2
Likewise what is ZI2 it is Z2 plus Z3 parallel
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to Z1 plus ZI1 Now these two equations you
see ZI1 here I have ZI2 I have ZI2 here I
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have ZI1 I have two equations so I can solve
for ZI1 and ZI2 in terms of Z1 Z2 Z3 If I
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do that upon solving this I get ZI1 is equal
to root over Z1 plus Z3 into Z1 Z2 plus Z2
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Z3 plus Z3 Z1by Z2 plus Z3 and ZI2 is Z2 plus
Z3 into Z1 Z2 plus Z2 Z3 plus Z3 Z1 So you
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see that this image impedance can be represented
in terms of the component impedances of the
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T section Now always we won’t be knowing
Z1 Z2 Z3 as I said that let us consider two
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port network as a black box But we can do
measurements and always find these image impedances
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How you know that any measurement requires
either an open circuit or short circuit of
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the one of the port So for any impedance measurement
you need to do this also you have seen that
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if you want to find any 2 port parameter you
need some port condition either short or open
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etc etc . So if we measure measurement of
image impedance
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let us say that one port one
we measure impedance
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when port 2 open We call that measurement
as Z1 since we are doing at port 1 open circuit
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Z1OC means I am measuring the impedance input
impedance at port 1 with port 2 open So if
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we look at the circuit if I open circuit this
what will be Z1OC it will be simply Z1 plus
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Z3 Similarly if we measure impedance at port
2 with port 2 with sorry ah imp the imp again
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port 1 measure impedance when port 2 is short
it now let me short this port So I call Z1SC
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second port is shortened you look at the circuit
If I short it it will be Z1 plus Z2 Z3 parallel
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So Z1 plus Z2 parallel Z3 Ok now what is this
this is Z1 Z2 plus Z2 Z3 plus Z3 Z1 by Z2
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plus Z3 Now you observe the image impedance
terms already I have solved ZI1 can I just
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compare can I say that Zi1 is equal to Z1OC
into Z1SC So by measurement I can always find
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Z1 OC I can find Z1 SC I know what image impedance
is immediately I can calculate from these
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Similarly instead of port 1 if i measure in
port 2 by once open circuiting port 1 find
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the input impedance at port 2 and then again
you short the port 1 and measure the input
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impedance and port 2 . So port 2 things if
we do you will see the same thing that ZI2
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can be expressed as Z2 OC into Z2 SC So this
shows that in image impedances can be always
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obtained from short or open circuit measurements
on any network So we can easily do this suppose
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I am given a network I can always find this
Z1 OC Z1 SC Z2 OC Z2 SC and find out tis Zi1
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Zi 2 and then I choose a source with that
internal impedance that I want and terminate
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or choose the load as Zi 2 I know I can achieve
maximum power transfer
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Already I said so that means I can have maximum
power transfer is guaranteed if I use image
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impedance as the terminating impedance at
both the ports Now
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now we know we have said that we will be using
the two port network as only lossless components
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that means you do not use any hard so their
own many internal loss there So by terminating
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with image impedances I assume maximum power
transfer no loss in the circuit lossless So
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image impedance is an important thing performance
measure of the power transport power transmission
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that is taking place to a network so you see
that we can specify something on it later
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when we will design a filter So instead of
ABCD we can specify image impedances and that
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will solve one many of our problems but think
one point that I have image impedance here
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I have image impedance here also you see with
this I require to know that ok by this Zi
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1 and Z Zi 2 terminations Zi 1 here and Zi
2 here I have to ensure that I am giving maximum
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power am delivering to this load But I am
assuming that here there is no loss but the
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power is flowing in this direction it may
so happen since I am using reactive elements
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power may be locally confined that is not
flowing there So I need to also see how propagation
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is taking place inside this 2 port network
So we need to have the transmission of power
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to the network also that we will next see
that this is called propagation of power . So
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what we define that again two port network
I have this V1 I have V2 I have I1 I have
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I 2 as before now let me define V1 I1 by V2
I2 VI is volt ampere the concept you have
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learnt in you electrical circuit class So
input volte ampere these are complex quantities
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input volte ampere and output volt ampere
What is the what is this ratio that will be
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something now I want to ensure that that is
fully making the power transmission possible
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So we call this I can name it any number this
will be some number you see input by output
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volt ampere but we have certain advantage
if we instead of defining any number here
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we write it as some exponential factor e to
the power two gamma why because you see when
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I will cascade many such networks this one
will have some this ratio e to the power 2
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gamma 1 this will have E to the power 2 gamma
2 another will have E to the power 2 gamma
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3 etc Now from this input to this input if
I want to find what is this transmission ratio
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of all the volte ampire if I express it exponential
factor the final thing will E to the power
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2 gamma 1 plus 2 gamma 2 plus 2 gamma 3 but
if I do not use this exponential factor If
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I just write it as gamma suppose then I will
have to work out and I will have to work out
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and I will have to find out what is the magnitude
and phase all these things here But exponential
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factor makes simply be an addition in if it
is an absolute value it would have been some
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multiplication We always prefer addition to
multiplication that is why it initially people
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did like that they put this as propagation
constant But now with after some learning
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people understood that if we represent this
ratio is it exponential factors and also you
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see I have taken a factor two here why because
many times will be interested to see what
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is the voltage ratio what is the current ratio
But this is actually a volt ampere ratio which
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is for that it is actually product of voltage
and current So I have taken two gamma this
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gamma is called propagation constant So what
is the definition of propagation constant
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you see gamma is equal to half LN V1 I1 by
V2I2 a very important definition propagation
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constant you see it shows that how input power
is propagating through the network inside
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the 2 port network . so my job is now to find
out what is this e to the power 2 gamma ratio
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is equal to V1 I1 by V2 I2 is equal to in
terms of ABCD parameters AV2 plus BI2 into
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CV2 plus DI2 by V2 I2 also I know V2 is equal
to ZI 2I image impedance it is terminated
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with image impedance so if you do that finally
you can solve that this ratio will turn out
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to be this simple manipulation put thus and
you know the value of Zi1ZI2 So you will get
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this will be simply this or e to the power
gamma is equal to root AD plus root BC Now
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here you see this propagation constant I have
expressed in terms of ABCD parameters One
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more thing is remaining I have already said
about characteristic impedances is characteristic
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impedances also expressible in terms of ABCD
parameters . let us see I have the same 2
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port network I have I1 here I have V1 here
and I want this should be ZI1 and here this
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should be terminated by ZI2 and this is V2
this is my I2 So I can write I know this is
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ABCD so V1I1 is equal to ABCD the definition
of transmission parameters also I have V2
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is equal to I2 ZI2 and V1 is equal to I1 ZI1
So put these equations and find out what the
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Zi1 you will see you will get AZI2 plus B
by CZi2 plus D let me call this for timing
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equation 1 . Now reverse the picture that
same transmission line this time I am putting
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the excitation here So I am looking at it
here I will get ZI2 and this is my V2 dashed
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as before this is my I2 dashed as before and
from here I am taking terminating you to Zi
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1 this is my V1dashed this is my I1 dashed
so here again I can write that V1 dashed I1
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dashed is equal to ABCD V2 dashed minus I2
dashed and what about the ports V1dashed is
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equal to ZI1 I1 dashed V2 dashed is equal
to ZI2 I2 dashed Then find out that what is
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your ZI2 So or you find what is your ZI1 which
is nothing but V1 dash by I1 dash that will
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turn out to be AZi2 minus B by minus C ZI2
plus D So this let me call equation 2 you
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have equation 1 you have equation 2 to solve
for Zi1.
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If you solve even so from 1 & 2 you can solve
for ZI1 and that will be equal to
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or ZI1 will be equal to AB by CD and ZI2 will
be equal to root over BD by AC Now I am happy
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because I know that ZI1 one of the image impedance
can be expressed in terms of four ABCD parameters
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ZI2 also I can express in terms of ABCD parameters
and I have already seen that propagation constant
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gamma you see this propagation constant that
also I can express in terms of ABCD parameters
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ABCD parameters completely characterization
2 port network I say equivalently I can say
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two image impedance ZI1 ZI2 and e to the and
gamma these three also characterizes a network
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But what is the beauty if I have ZI ZI 2 I
know what is i impedance level of the excitations
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of the network that means what is the source
impedance what is the load impedance they
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are according to the power matching So that
no maximum power sorry they are according
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to the maximum power will flow and by putting
conditions and gamma I will be able to say
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whether these frequency will pass or not So
instead of ABCD parameters this is a better
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description of a 2 port network if I want
to design a filter And already I have seen
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that I can that I can do the yes I can do
the measurement of image impedances that time
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I said in terms of the by opening and shorting
the port I will also have to prove that I
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can do this for propagation constant also
because this is a new thing that time I didn’t
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say these So that I will do now that measurement
of image impedance and propagation constant
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. So what we will do the same network this
is port 2 this is open circuit and I am looking
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here at let me call this Z01 So I know V1equal
to AV2 plus BI2 I1 is equal to CV2 plus Di2
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etc and open circuit means I2 is equal to
0 So if you enforce that is ZO1 that will
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be A by C then you short circuit so or open
circuit this port purpose you open circuit
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and measure here ZO2 so ZO2 that will be turn
out to be D by C then you do that short circuit
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port one and measure the from this port you
measure ZS2 you will see ZS2 will turn out
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to be B by A And
which one I missed ZS2 ZS1 this so you short
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circuit this port and measure here ZS1 ZS1
will be B by D . Once you have that you can
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immediately write because already we have
seen ZI1 is equal to the ZO1 into ZO2 etc
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So you will get that is equal to AB by CD
and that is nothing but ZO1 ZSL similary Zi2
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is equal to root over BD by AC and that is
D by C into B by A that is nothing but ZO2
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ZS2 And you see what is TAN gamma TAN hyperbolic
gamma all of you are familiar with hyperbolic
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functions So this is E to the power gamma
minus E to the power minus gamma by E to the
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power gamma plus and that is nothing but BC
by AD that is ZS1 by ZO1 and or ZS2 by ZO2
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So you see that gamma can be expressed completely
in terms of short circuit and open circuit
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measurement So I can measure image impedance
by open circuit short circuit measurements
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I can also measure gamma by open circuit short
circuit measurements Previously I showed that
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Zi1 Zi2 and gamma the completely characterizes
the network reciprocal 2 port network Now
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I have now I have shown that they also can
be measured so you do not have a difficulty
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any 2 port reciprocal network lossless network
you can represent like this So an alternate
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description for characterization of 2 port
network is in terms of ZI1 and ZI2 and gamma
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I think in the next class will introduce another
criteria all symmetrical network and we will
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simplify this procedure Thank you