Characterization Parameters of Microwave Circuit
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Transcript of Characterization Parameters of Microwave Circuit
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8/12/2019 Characterization Parameters of Microwave Circuit
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Characterization Parameters of Microwave CircuitImpedances and Admittances :Consider an arbitrary N-port microwave network
+ += nnn V V V
+ = nnn I I I n N
nV , n I are tota vo ta!e and current at port n "#he impedance matri$%z& of the microwave network re ates thesevo ta!es and currents is as fo ows ,
=
N NN N N
N
N
N I
I I
Z Z Z
Z Z Z Z Z Z
V
V V
'
(
'(
''''(
(('((
'
(
or%)&*%+&%I&
j fork I j
iij
k I V
Z =
=,
ij Z is obtained by drivin! port with the current j I openin! circuit
a other ports and measurin! the open . circuit vo ta!e at port i"/eciprocity princip e revea s that
jiij Z Z =
#he admittance matri$ %0& is
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=
N NN N N
N
N
N V
V V
Y Y Y
Y Y Y Y Y Y
I
I I
'
(
'(
''''(
(('((
'
(
or
%I&*%0&%)& where %y&* (&% z
Property of matri$ e ements for oss ess network
&12%&%&/e3%'
(&12%&4&%/e35%
'
(&12%&/e3%
'
( I Z I I Z I I V P t t t t ave ===
"""41115'( ('(''('((((( +++= I Z I I Z I I Z I
= =
= N
n
N
mnmnm I Z I
( (
1'(
6ince the net power de ivered to the network is zero , we have,23 =avee P R#his !ives us
,2/e32/e3 '1 ==
nmnnnmn Z I I Z I ,23 =nme Z R 5(4 indicates that :
2435 11 =+ mnnmmne Z I I I I R nmmn Z Z =
45 11 nmmn I I I I + is a rea 7uantity8e therefore have
,23 =mne Z R 5'45(4 and 5'4 show that the e ements of impedance matri$ are pure y
ima!inary "
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6catterin! Matri$#he scatterin! matri$ re ates the vo ta!e waves incident on
the ports to those ref ected from the ports" 8e assume that+
nV and
nV are the amp itudes of incident and ref ected waves at port n"
=
+
+
+
N NN N N
N
N
N V
V
V
S S S
S S S
S S S
V
V
V
'
(
'(
''''(
(('((
'
(
or% V &*%6&% +V &A specific e ement of the %6& matri$ can be determined as
j fork V j
iij
k V
V S
=
+
+
=
ijS is found by drivin! port with an incident wave of
vo ta!e+
jV and measurin! the ref ected wave amp itude
iV
comin! out of port i , whi e terminatin! a other ports inmatched oads"
#ransformation amon! %6& , %+& , and %0& matrices8e assume that the characteristic impedances on Z of athe ports are identica " 9or convenience , we set (=on Z
+ += nnn V V V
+ = nnn
I I I *
+ nn
V V
%+&%I&* %+& % +V &-%+& % V &*%)&* % +V & % V &
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5%+& %;&4 % V &*5%+&-%;&4 % +V &%;& is the unit matri$
%s&* &%&%&%&%
&%&%
U Z U Z
V V
+=+
&%&%
&%&%&%
S U
S U Z
+=
&%&%&%&%
&%&% (S U S U
Z Y +==
6ome properties of %6& matri$5(4 9or a reciproca network , %s&* t s &%
45'(
nnn I V V +=+
% +V &*'(
5%+& %;&4%I&
45'(
nnn I V V =
% V &*'(
5%+&-%;&4%I&
% V &* 5%+&-%;&4 -(%;&45%+&+ % +V &
%6&* &%&%
+
V
V *5%+&-%;&4 -(%;&45%+&+
t%6&* t t %;&45%+&2%;&435%+& -( +
If the network is reciproca , %+& is symmetrica and t%+& *%+&t%6&* -(%;&45%+&+ 5%+&-%;&4
6ince both -(%;&45%+&+ and 5%+&-%;&4 are symmetrica , we then !et%6&* t%6&
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5'49or a oss ess network , %6& is a unitary matri$#he power de ivered to the network can be e$pressed as
42&%&45%&%&/e35%'(
&12%&/e3%'( 11 ++
+== V V V V I V P t t t ave
42&%&%&%&%&%&%&%&/e35%'( 1111 ++++ += V V V V V V V V t t t t
11 &%&%'(&%&%
'( ++ = V V V V t t
If the network is oss ess , no rea power can be de ivered to thenetwork " 8e then !et
11 &%&%&%&% ++ = V V V V t t
Note that we have &&%%&% + = V S V #his !ives us
111 &%&%&%&%&%&% ++++ = V S S V V V t t t
8e therefore !et&%&%&% 1 U S S t =1&%S *3 (2&3% t S 2
#herefore %6& is a unitary matri$ and
ijkj
N
k ki S S =
=1
( for a i*
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6ince %6& is symmetrica , the network is reciproca "(?@",>",(", ''
'
'(
'
(( =+=+ S S
#he network is not oss ess "+++ =+= '('((('('(((( V S V S V S V S V +++ =+= '''('('''('(' V S V S V S V S V
+
+=
(''
'(' (
V S
S V
''
'(('((
(
'('((
(
(
( S S S
S V V
S S V V
+=== +
+
* ?AA",'",(4>",45>",5
(", =+ j j
#he return oss ess is
/"B"* dB
=C"Ao!' =
5>?"(E(4@,@?">5>?"(E(
@?">(,in Z
((S * * ''S
@?">@
@
4&@@?">FF5>?"(E(%@?">
4@@?">FF5>?"(E(('' +++
+== V V V
* += (( C C"C C" V V
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('
(
''( C C"
'
S V
V S
V
====
+
+
8e may use the fo owin! circuit to obtain the transmission coefficient'(
S
#ransmission ADC< matri$#his representation is especia y appro$imate for the confi!uration of
the Cascade connection of two or more two . port networks"#he ADCG matri$ is defined as''( BI AV V +=
''( DI CV I +=
Note that the direction of ' I is out of the network"
='
'
((
((
(
(
I V
DC B A
I V
Consider a cascade connection shown be ow
#he ADCG representation becomes==
A
A
''
''
((
((
'
'
((
((
(
(
I
V
DC
B A
DC
B A
I
V
DC
B A
I
V
Port (
DC
B A
I(
(V 'V
' I
- -
Port '
I(
DC
B A(
V - DC
B A'
V
A I
-AV
-
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=''
''
((
((
DC
B A
DC
B A
DC
B A
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Network Ana yzer Ca ibration : usin! thru . ref ect . ine6 parameters are ratios of comp e$ vo ta!e amp itudes" #he primaryreference p ane is !enera y at some point within the ana yzer itse f"8e consider the fo owin! schematic"
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A ca ibration procedure is used to characterize the error bo$es before the measurement of G;#"Consider the fo owin! error bo$
,
,
Z Z Z Z
L
L L +
=
LbS aS aS aS b +=+= '('((('('((((
LbS aS aS aS b +=+= '''('('''('('('(''' 4(5 aS S b L =
4(5 ''('(
' LS
aS b
=
4(5 ''('(
('(((( L
L
S aS
S aS b
+=
L
L
in S
S S S
a
b
+==
''
'(('
(((
(
(
Assume that the error bo$ is reciproca ,i"e", '((' Z Z =
L
Lin S
S S
+= ''
'('
(( (
#o ca ibrate the error bo$ means to find scatterin! parameters ijS ofthe error bo$" 8e consider the fo owin! three situations5(4 the oad is short . circuited , (= L
''
'
('((, ( S S
S sin += H 5a4
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8e want to find the fo owin! three power !ains
Power ain* *in
l
P P
the ratio of power dissipated in the oad to
the power de ivered to the input of two . port network"5independent of s
Z 4
Avai ab e ainavs
avn A P
P = the ratio of ma$imum power avai ab e to the
two - portnetwork to the ma$imum power avai ab e from the source "5dependenton s Z , but not l Z 4
#ransducer power ain*avs
l ! P
P = the ratio of power de ivered to the
oad to the power avai ab e from the source"5dependent on both s Z andl Z 4
#he ref ection coefficient ookin! from the network toward the oad is
+=
+=
'
'
,
,
V V
Z Z Z Z
l
l l
#he ref ection coefficient ookin! from the network toward the source
is +
=+=
(
(
,
,
V V
Z Z Z Z
s
s s
8e ne$t want to find in and o"t
+
= (
(
V V in , +
= '
'
V V o"t
+++ +=+= '('((('('(((( V S V S V S V S V l +++ +=+=
'''('('''('(' V S V S V S V S V l + =
('(''' 4(5 V S S V l
++
+= (
''
'(('(((( (
V S S S
V S V l
l
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,
,
''
'(('((
(
(
( Z Z Z Z
S S S
S V V
in
in
l
l in +
= +== +
Dy the same token , we obtain
s
so"t
S
S S S
+= ((
'(('''
(#he vo ta!e (V is
4(5(((( inin s
in s V V V Z Z
Z V V +=+=+
= ++
in
inin Z Z
+=((
4(5
(
((
(
( in
in
in s
in
in
s V Z Z
Z V +=
++
+
+
4(54(5( in sin s Z Z
Z V V
+ +=+
8e assume that Z *(
4(54(5(
((
( += + +=+
sin s s
in s sin s Z Z V Z Z V V
in s s s
s
sin
s s
V
Z Z Z
V =
+ +
=(
('
((
(
((
(
where ((
(((
'(
4((
(5'(
4(5'(
+=
+++=
+=
s s
s s
s
s s Z Z
Z Z Z Z
#he avera!e power de ivered to the network is
4(5(
(>
4(5'( '
'
''
'
'
(in
in s
s s
ininV
Z V P
= =
+
#he power de ivered to the oad is
4(5'
'
'
'l l
Z
V P =
4(5 ''('(
'l S
V S V
=+
''
''
'''
'(
'
'
''
''
'(
'
(
((
4(45(5
>(
4(5
'in sl
sl s
l
l
l S
S V
S
S
Z
V P
=
=
+
#he power !ain is
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*4(5(
4(5''
''
'''(
inl
l
in
l
S
S P
P
=
#he ma$imum power avai ab e to the network occurs when in * 1 s
8e then !et 4(5
(> '
'
,
'
1 s
sS
inavs Z
V P P sin
== =
6imi ar y , the ma$imum power avai ab e to the oad occurs when l *1
o"t " #herefore , we have
''''
''''(
,
'
1 (1(
4(45(5> in so"t
so"t S
inavsS
S
Z
V P P
o"t l
==
=
'''
''((
1
'
1(
4(5((
o"t
o"t s
in s S o"t l
=
8e !et
4(5(
(
> ''((
'''(
,
'
o"t s
sS avn
S
S
Z
V P
=
#he avai ab e power !ain A is
4(5(
4(5''
((
'''(
o"t s
s
avs
avn A
S
S
P
P
==
#ransducer power ain ! is
''''
''''(
((
4(45(5
in sl
l s
avs
l !
S
S P
P
==
#he matched 5or ma$imum4 transducer power !ain occurs when,= = l s
'
'(ma$, S
! =
#he uni ater power !ain is
'
''
'
((
'''
'(
,,((
4(45(5('((
l S
l s
S S ! "! S S
S
in
==
==
Chain 6catterin! Matri$
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='
'
'''(
('((
(
(
ab
! ! ! !
ba
chain scatterin! parametersscatterin! transfer parameters# parameters
1/e ationship between the 6 and # parameters
=
'(
''((('
'(
((
'(
''
'(
'''(
('((
(
S S S
S S S
S S
S ! ! ! !
and
=
((
('
((
((
(''(''
((
'(
'''(
('((
(
! !
!
! ! !
! ! !
S S
S S