Characterization Parameters of Microwave Circuit

8/12/2019 Characterization Parameters of Microwave Circuit

1/16

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


2/16

=

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 "


3/16

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 &


4/16

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&


5/16

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*


6/16

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


7/16

('

(

''( 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

-


8/16

=''

''

((

((

DC

B A

DC

B A

DC

B A


9/16


10/16

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"


11/16

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


12/16


13/16

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


14/16

,

,

''

'(('((

(

(

( 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


15/16

*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$


16/16

='

'

'''(

('((

(

(

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

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Characterization Parameters of Microwave Circuit