Circuit Characterization

8/3/2019 Circuit Characterization

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Microwave circuit

characterization

1LEMA-EPFL

Microwave component

1

2

4

5

Portesports

2LEMA-EPFL

3 6

Elments de circuit

Circuit elements


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Reference planes

On each access line i of a

z

n

znz1

1

component, a coordinate axis zi

is defined. The origin of this axis

lies in the reference plane of the

port i.

Assumptions :

3LEMA-EPFL

i

They support only the dominant mode

The reference plane is distant enough

From discontinuities to ensure that the

Higher order modes are attenuated

Available models

Currents and voltages ill defined

Well known from low frequency circuit analysis

Not adapted to system approach

Scattering matrix

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Well suited to microwaves

Not known from low frequency circuit analysis

Adapted to system analysis


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Kirchhoffs model

i1 i4

1

2

3

4

5

6

i2

i3

i5

i6

u1

u2

u3

u4

u5

u6

5LEMA-EPFL

[Z]

Current, voltages and impedances

Current voltages and impedances

Current and voltage difficult to measure

Current and voltage not always uniquelydefined

6LEMA-EPFL


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Example : TEM line

V= E

+EH

V= E dl+

I= H dlC+

7LEMA-EPFL

-

Zc =V

I=

L

C

Example : TEM line

Fields identical to static fields

Zero cutoff frequency

Needs at least two separate conductors

8LEMA-EPFL


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Non-TEM line : the rectangular

waveguide

V j= E A

Ey x,y,z( )= E0ja

sin

x

ae

jz = E0ey x,y( )ejz

=ja x jz = jz

Example : mode TE10 of a rectangular

waveguide

9LEMA-EPFL

x , ,

ay ,

V=E0j a

sin

x

ae

jzdy

y

Thusa

b

Non TEM voltage and current definition

Onl defined for one mode

The voltage is proportional to the transverseelectric field

The current is proportional to the transversemagnetic field

The characteristic impedance is equal to U/I. Wechoose the characteristic im edance e ual to the

10LEMA-EPFL

wave impedance

The power flux is given by the product of thevoltage and the current


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Arbitrary line

Et x,y,z( ) = et x,y( ) Eo+e

jz + Eoe

jz( )=

et x,y( )C1

V+ejz + Vejz(

Ht x,y,z( ) = ht x,y( ) Eo+ejz Eo

ejz(

V z( ) = V+ejz + Vejz

I z( ) = I+ejz Iejz

Zc =V

+

I+=

V

I=

C1Eo+

C2Eo+ =

C1

C2

11LEMA-EPFL

= t x,yC2

I+ejz Iejz( )C1

C2= Zmod

Different impedances

Wave impedance :

Line characteristic impedance :

Impedance matrix of a circuit

o

Zmod =Et

HtZc = V

+

I+ =V

I =LC

12LEMA-EPFL


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Impedance matrix

13LEMA-EPFL

Single port element

V

Circuit une porteaccs

plan de rfrence

Zin

I

V

n

S

Ref. plane

Single port

Zin =I

Power delivered to the component

14LEMA-EPFL

port

P =1

2E H* ds= Pr+ 2j Wm We( )

s

(Poynting)


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Single port element

Et x,y,z( ) = V z( )

et x,y( )C1

ejz

Ht x,y,z( ) = I z( )ht x,y( )

C2e

jz

with1

et ht ds=1 thus P = 1 VI*et ht ds =1VI

*

15LEMA-EPFL

1 2 s 1 2 s

and

Zin = R + jX=V

I=

VI*

I2=

2P

I2

=2 Pr+ 2j Wm We( )( )

I2

Single port element

dissipated in the system (Losses)

X is proportional to the mean reactivestored energy in the system

16LEMA-EPFL


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Impedance and admittance matrix

v1+,i1+

v1-,-i1-

v2+,i2+

v2-,-i2-

v3+,i3+

v4+,i4+

v4-,-i4-

v5+,i5+

v5-,-i5-

v6+,i6+

t2

t3

et

17LEMA-EPFL

[Z]

v3-,-i3- v6-,-i6-

Vn = Vn+ + Vn

In = In+ In

At each port ti

Impedance and admittance matrix

obtained in open

circuit conditions

Admittance matrix is

obtained in short

circuit conditions

=

I[ ] = Y[ ] V[ ]

Zij =ViI

18LEMA-EPFL

k

Yij =Ii

Vj Vk=0 pour kj


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Properties of the impedance and

admittance matrix

Yi = Yi

Lossless circuit :

Zij = Zji

Re Zmn{ }= 0

19LEMA-EPFL

Example : transmission line

= =dI(0) I(d)

U(0)U(d

( ) ( )2

, ,

, 2U U d I I d = =

( )

( )

e e

e e

z z

z z

U z U U

I z I I

++

++

= +

=

20LEMA-EPFL

1 2

U1 U2( ) ( )

( ) ( )

0 e e

0 e e

d d

d d

U U U U d U U

I I I I d I I

++ +

++ +

= + = +

= =


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Example : transmission line

U Z Z I dI(0) I(d)

U(0) U(d ( ) ( )

( )( )

2 21 22 2

1

2

1coth

sinh

1coth

sinh

c

U Z Z I

dd I

ZI

dd

=

=

21LEMA-EPFL

1 2

U1 U2( )

( )

( )( )

2 21 22 2

1

2

1coth

sinh

1coth

sinh

c

I Y Y U

dd U

YU

dd

=

=

Equivalent T circuit of a reciprocal

two-port

11 12Z12 Z22

I1 I2

U1 U2

Za Zb

Zc

Za = Z11 Z12Zb = Z22 Z12

22LEMA-EPFL

c 12


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Equivalent T circuit of a transmisson

line

I1 I2

U1 U2

Za Za

Zc

Za = Zcaract coth d( )1

sinh d( )

= Zcaractch d( ) 1sinh d( )

= Zcaracttanh

d

2

23LEMA-EPFL

Zc = caractsinh d( )

Equivalent circuit of a reciprocaltwo-port

11 12Y12 Y22

I1 I2Yc Ya = Y11 + Y12

=

24LEMA-EPFL

1 2a b

Yc = Y12


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Equivalent circuit of atransmisson line

I1 I2

U1 U2Ya Ya

Yc

Ya = Ycaract coth d( )1

sinh d( )

= Ycaractch d( )1sinh d( )

= Ycaracttanh

d

2

25LEMA-EPFL

Yc = Ycaractsinh z( )

Scattering parameters

26LEMA-EPFL


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Normalized wave amplitudes :

definition

ai =vi + Zciii2 Zci

, bi =vi Zciii2 Zci

[W1/2]

27LEMA-EPFL

vi = Zci ai + bi( ) , ii =ai bi( )

Zci

Normalized wave amplitudes :

properties

Transmission lines

vi = vi+

e

jz

+ vi

e

+ jz

ii = ii+ejz + ii

e+ jz

ai

=vi

+

Zcie

jz

vi

+ z

28LEMA-EPFL

i =Zci

ai : progressive wave

bi : retrograde wave


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Power at one component port

Pi = Re viii*[ ]= Re ai + bi( ) ai* bi*( )[ ]= ai 2 bi 2

/a /2 : ower enterin ort i

29LEMA-EPFL

/bi/2 : power leaving port i

Signal model

Plans de rfrenceReference planes

a1

b1a2

b2

a4

b4a5

b

t1

t2

t4

t5

b[ ]= S[ ] a[ ]

=bi

30LEMA-EPFL

[S]

a3

b3

a6

b6

t3 t6 aj ak=0 , kj


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Scattering matrix : definition

ai =vi + Zciii2 Zci

, bi =vi Zciii2 Zci

orma ze wave amp u e

vi = Zci ai + bi( ) , ii =ai bi( )

Zci

31LEMA-EPFL

b[ ]= S[ ] a[ ] sij =bi

aj ak=0 , kj

Signal model

Signals are easy to measure

Well suited to a system approach (transferfunctions)

Drawbacks :

32LEMA-EPFL

Not known at low frequencies


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Scattering matrix : properties

Depends on the component and its

properties

To change the connecting lines implies to

chna e the scatterin matrix

33LEMA-EPFL

The scattering matrix is obtained termiating

the ports by matched loads

Reciprocity

ass ve, near, so rop c c rcu

The circuit is reciprocal

34LEMA-EPFL

zij = zji

sij = sji


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Lossless circuit

* tai = bi a[ ] a[ ] b b[ ] = 0 a[ ] = a

Moreoverb[ ] = S[ ] a[ ]

b[ ]= a[ ] S[ ]a[ ] a[ ] a[ ] S[ ]S[ ] a[ ] = 0a[ ] 1[ ] S[ ]S[ ]{ }a[ ] = 0S S[ ] = 1[ ]

35LEMA-EPFL

sij*sik = jk

i=1

N

jk =1 si j = k

0 si j k

Matched circuit

Sii = 0

36LEMA-EPFL


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Change of reference planeti

aib

[S].

.

.

tn

37LEMA-EPFL

[S'].

.

.

t i

t'n

z

b'i

a'i

Change of reference plane

a' i = aie i

b' i = bieji

i = izi

thus s' ii = siiej2i

And in general

a[ ] = diag ej[ ]a'[ ] j

38LEMA-EPFL

a'[ ] = diag e j[ ]a[ ]b[ ] = diag ej[ ]b'[ ]b'[ ] = diag ej[ ]b[ ]

diag ej[ ]=

...

0 ej2 :

:

0 ... ejn


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Change of reference plane

[ ] [ ] [ ]

[ ] [ ]

' e e '

' e e

j j

j j

b diag S diag a

S diag S diag

=

=

39LEMA-EPFL

' e i jjij ijs s +=

Passive circuits : questions

.

2. Reciprocity

3. Losslessness

4. Matched

40LEMA-EPFL

. ymme ry


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[S]->[Z][b] = [S] [a] ai= (Ui+ZciIi)/2Zci Ui= Zci(ai + bi)

[U] = [Z] [I] bi= (Ui ZciIi)/2Zci Ii= (ai bi)/Zci

We define two auxiliary matrices

[G] = [diag(Zci)] [F] = [diag(1/2Zci)]

And we can write

41LEMA-EPFL

[a] = [F] {[U] + [G] [I] } ([a] + [b]) = 2 [F] [U]

[b] = [F] {[U] [G] [I] } ([a] [b]) = 2 [F] [G] [I]

[S]->[Z]

[a] = [F] {[U] + [G] [I] } -> [a] = [F] {[Z] + [G] } [I]

-> = a

[b] = [F] {[U] [G] [I] } -> [b] = [F] {[Z] [G] } [I]

-> [b] = [F] {[Z] [G] } {[Z] + [G] }1 [F]1 [a] = [S] [a]

= 1 1

42LEMA-EPFL

In one dimension, we have = S11 = (Zt Zc)/(Zt + Zc)


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[Z]->[S]([a] + [b]) = 2 [F] [U] -> [U] = (2 [F])1([a] + [b]) = (2 [F])1([1] + [S]) [a]

a = = a-> a =

regrouping, we obtain:

[U] = (2 [F])1([1] + [S]) ([1] [S])1 2 [F] [G] [I] = [Z] [I]

1 1

43LEMA-EPFL

For a single acess component, we have

Zt = Zc(1 + r)/(1 r)

Flow charts

are represented as arrows

a sij b

44LEMA-EPFL


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Flow charts

Two port

a1

b1

b2

a2s11

s21

s22s12

45LEMA-EPFL

Flow charts

a1 b2s21

b

1

a

2

s11 s22

s

12

s13s32

s23s31

Three port

46LEMA-EPFL

a3b3

s33


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Flow chartsa1 b2s21

b1 a2

11 22

a3 b4

s13 s42

4-port

47LEMA-EPFL

b3 a4

s33

s34

s44

Flow chart reduction rules

1 multi lication

ab

s1 s2

a s1s2 b

48LEMA-EPFL


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2 addition

a b

s1

49LEMA-EPFL

s2

a b

s1+s2


3 retroaction

ab

s1

s2

50LEMA-EPFL

a b

1-s1s2

s1


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Example

a1 b'2s21 s'21b2=a'1

b1 a'2

s11 s22s12

s'11 s'22s'12

a2=b'1

51LEMA-EPFL

Example

a1 s21 b2='a1

b1

s11 s22

s12

s'11

52LEMA-EPFL

a2=b'1


27/44

Examplea1 s21

b1

s11

s12

a1

s'11

1-s'11s22

53LEMA-EPFL

s'11

1-s'11s22b1

s11+s21s12

Example

a1 b'2s21 s'21b2=a'1

s22 s'11

a2=b'1

54LEMA-EPFL


28/44

Example1

a1 b'2s21 s'21

1-s'11s22

a

s21

1-s'11s22

s'21

b'

55LEMA-EPFL

Example

b2=a'1

b1 a'2

s22

s12

s'11

s'12

a2=b'1

56LEMA-EPFL


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Example

''

1

a'2

b12s12 s 121-s22s'11

s12s'12

1-s s'

57LEMA-EPFL

Example

b'2s'21b2=a'1

a'2

s22 s'11 s'22s'12

58LEMA-EPFL

a2=b'1


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Exampleb'2s'21

a'2

s'22s'12

22

1-s'11

s22

59LEMA-EPFL

b'2

a'2

s'22+s'12s'21

s22

1-s'11

s22

Single port components

60LEMA-EPFL


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single portcircuitacces

Zin

I

V Single portcircuitacces

a

s11b

61LEMA-EPFL

[ ] [ ] [ ], , have only one term. They are scalars Z Y S


s11 =

Zin ZcZin + Zc

, Zin =1+ s111 s11

Equivalences

a1

62LEMA-EPFL

Flow chart b1s11, ,


32/44

Examples

short circuit : s11 = -1

open circuit : s11 = 1

total reflection : s11 = ej

matched sin le ort com onents :

63LEMA-EPFL

matched load

Others : antennas, loads, etc.

Short circuits

spring contact

64LEMA-EPFL

g/4


33/44

Matched loads

65LEMA-EPFL

Matched load

1

b1a

2

b2a3

4

b4

a5b5

a6

=0

=0

The measurement ofs21 has to be done

by terminating all non

concerned orts b

66LEMA-EPFL

[S]

3

s21 =b2

a1 ai =0, i1

matched loads


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Two port circuits

67LEMA-EPFL

Two port circuits

[S]

a1

b1

b2

a2

a1

b1

b2

a2s11

s21

s22s12

68LEMA-EPFL

[ ] 11 1221 22

s sS

s s

=


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Examples

, ,

Lossy : attenuators, isolators

69LEMA-EPFL

Phase shifters

matched

lossless

=0 e

j

70LEMA-EPFL

ej

o


36/44

Phase shifters

Matched transmission line

L

71LEMA-EPFL

S[ ] = 0 e jL

e jL 0

Phase shifters

Fox' phase shifter

Non reciprocal phase shifters

S[ ] =0 e

j1

ej2 0

1 2

72LEMA-EPFL

Gyrators

S[ ] =0 1

1 0

1 2 =


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Fox' phase shifters

/2

/4

73LEMA-EPFL

/4

Fox' phase shifters

mode 1mode 2pol. lin.pol. cir.

(1- 2)l= /2

74LEMA-EPFL

quarter wave line


38/44

Fox' phase shifters

mode 1mode 2

pol. cir. (1- 2)l=

pol. cir inverse

75LEMA-EPFL

rotation of + = phase shift of + for both components

Fox' phase shifters

mode 1mode 2

(1- 2)l= /2

pol. lin.pol. cir.

76LEMA-EPFL

rotation of - = phase shift of + for both components


39/44

Attenuators

Reciporcal

Lossy

Attenuation level given by:

S[ ] =0 s12

s12 0

s12


40/44

Attenuators using absorbers

79LEMA-EPFL

Rotative attenuator

lame mobile

lame fixefixed slab

rotatin slab

80LEMA-EPFL

lame fixefixed slab


41/44


42/44

Resistive attenuators (T

attenuators)

and R2 in order to be

matched

Then we can obtain

the attenuation

R2

83LEMA-EPFL

Resistive attenuators

R1 R1I

I2

R2

port 1 port 2

ZoZin

u1 u2

84LEMA-EPFL

1

2 1

1 1in o

o

Z R Z

R R Z

= + =+

+

12

12

oZ RRR

=


43/44

Resistive attenuatorsR1 R1I

II2a1 b2

R2

port 1 port 2

ZoZin

u1 u2

b1 a2

85LEMA-EPFL

12 221

1 1 1

o

o

Z Rb us

a u Z R

= = =

+

Other solution

Impedance matrix of the 2 port

[ ] 1 2 2

2 1 2

R R RZ

R R R

+ =

+

from which we obtain the scatterin arameter matrix

86LEMA-EPFL

[ ]( )

( )

( )

2 2 21 2 2 2

2 22 2 21 2 2 2 1 2 2

21

2

o o

o o o

R R Z R R Z S

R R Z R R Z R R Z R

+ = + + +


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Other solution

= =

2 21

212

oZ RRR

=

,

and1

21oZ Rs

Z R

=

+

87LEMA-EPFL

so finally, the attenuation level is given by

( ) 1 1211 1

[ ] 20log 20log 20logo o

o o

Z R Z R LA dB s

Z R Z R

+= = =

+

Isolators

non reciprocal

lossy

0 0

88LEMA-EPFL

=1 0

Circuit Characterization

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