射頻電子 - [第四章] 散射參數網路

Department of Electronic EngineeringNational Taipei University of Technology

•••• (Power waves)

• (Traveling waves) −

Department of Electronic Engineering, NTUT2/25

(Traveling Waves)

( ) ( ) ( ) j x j xV x V x V x Ae Beβ β+ − −= + = +

( ) ( ) ( ) ( ) ( )0 0

V x V xI x I x I x

Z Z

+ −+ −= − = −

( ) ( )( )

V xx

V x

−

+Γ =

•

•

• (Normalized traveling waves)

( ) ( )0

V xa x

Z

+

=

( ) ( )0

V xb x

Z

−

=

( ) ( ) 2

2

0

V xa x

Z

+ =

( ) ( )( )

210log 10log

20log

aP a x

a x

=

=( ) ( ) ( ) ( ) ( )0

0

1 1

2 2b x v x i x V x Z I x

Z= − = −

( ) ( ) ( ) ( ) ( )0

0

1 1

2 2a x v x i x V x Z I x

Z= + = +

( ) ( ) ( ) ( )0

V xv x a x b x

Z= + =

( ) ( ) ( ) ( )0i x a x b x Z I x= − =

( ) ( ) ( )b x x a x= Γ

slide 16, 30

3/25 Department of Electronic Engineering, NTUT

Two-portNetwork

( )2 2a l

( )2 2b l

( )2 2a x

( )2 2b x

( )1 1a l

( )1 1b l

( )1 1a x

( )1 1b x1oZ 2oZ

Input port Output port

Port 1

1 1x = l

Port 22 2x = l

• Port 1 ( )

• Port 2 ( )

( )1 1a l ( )1 1b l1 1x = l

( )2 2a l ( )2 2b l2 2x = l

Incident wave Reflected wave

• (Scattering Matrix)

( ) ( ) ( )1 1 11 1 1 12 2 2b S a S a= +l l l

( ) ( ) ( )2 2 21 1 1 22 2 2b S a S a= +l l l

( )( )

( )( )

1 1 1 111 12

2 2 2 221 22

b aS S

b aS S

=

l l

l l

xx

4/25 Department of Electronic Engineering, NTUT

( )( ) ( )2 2

1 111

1 1 0a

bS

a=

=l

l

lInput reflection coefficient with output properly terminated

( )( ) ( )1 1

2 222

2 2 0a

bS

a=

=l

l

lOutput reflection coefficient with input properly terminated

Forward transmission coefficient with output properly terminated

( )( ) ( )1 1

1 112

2 2 0a

bS

a=

=l

l

lReverse transmission coefficient with output properly terminated

(measured with port 2 properly terminated)

(measured with port 2 properly terminated)

(measured with port 1 properly terminated)

(measured with port 1 properly terminated)

( )( ) ( )2 2

2 221

1 1 0a

bS

a=

=l

l

l

5/25 Department of Electronic Engineering, NTUT

( )( ) ( )2 2

1 111

1 1 0a

bS

a=

=l

l

l

• Return Loss (RL)

( )( ) ( )2 2

2 221

1 1 0a

bS

a=

=l

l

l

( )( )

21 12 1

11 21 1 1

b

a

b PS

a P= =

l

l( )21

11 111

10log 10 log 20 log (dB)b

a

PS S

P

= =

11Return Loss (RL) 10 log 20 log (dB)in

reft

PS

P

= = −

( , )

( )( )

22 22 2

21 21 1 1

b

a

b PS

a P= =

l

l( )22

21 211

10log 10 log 20 log (dB)b

a

PS S

P

= =

21Insertion Loss (IL) 10 log 20 log (dB)transmit

receive

PS

P

= = −

( , )• Insertion Loss (IL)

|S11| −12 dB12 dB

6/25 Department of Electronic Engineering, NTUT

S11

Two-portNetwork

( )2 2 0a =l

( )2 2b l

( )1 1a l

( )1 1b l

1oZ 2oZ

Port 1

1 1x = l

Port 2

2 2x = l

2 2oZ Z=+

−1E

1 1oZ Z=

( )( ) ( )2 2

1 111

1 1 0a

bS

a=

=l

l

lOUTZ

• Z2=Zo2 (i.e. Zo1=Zo2)50Ω

( )2 2 0a =l

( )( ) ( )2 2

1 111

1 1 0a

bS

a=

=l

l

l( ) ( ) ( )1 1 11 1 1 12 2 2b S a S a= +l l l

0

port 2 ( )2 2 0a =l

7/25 Department of Electronic Engineering, NTUT

•Zoi (i=1 ton) n port

[ ] [ ][ ]b S a=

n-portNetwork

1oZ

Port 1Port 1'

1TZ

( )1 1a l

( )1 1b l

2oZ

Port 2Port 2'

( )2 2a l

( )2 2b l

onZ

Port nPort n'

( )n na l

( )n nb l

[ ]

11 12 1

21 22 2

1 2

n

n

n n nn

S S S

S S S

S

S S S

⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

8/25 Department of Electronic Engineering, NTUT

•

•

(i=1 for port 1 and i=2 for port 2)

( )0iP+ ( )0iP−

+

1oZ 2oZ

Port 1

1 1x = l

Port 2

2 2x = l

Port 1'

1 0x =

( )1 1I x

Port 2'

2 0x =

2l1l

−

( )1 1V x

( )2 2I x

+

−

( )2 2V x( )0iP+

( )0iP−

( )0iP+

( )0iP−

11 12

21 22

S S

S S

( )( )

( )( )

1 1 1 111 12

2 2 2 221 22

b aS S

b aS S

=

l l

l l

( )0iP+ ( )0iP−

( )a x

( )b x

9/25 Department of Electronic Engineering, NTUT

( ) ( ) 210 0

2i iP a+ =

( ) ( ) ( ) ( ) ( ) ( )2

22

,

01 1 10 Re 0 0 0 0

2 2 2i

i i i i i rmsoi

VP V I b b

Z

−∗− − − = ⋅ = = =

• ith port (x1=0, x2=0)

•

( ) ( )0i i iP P+ += l

( ) ( )0i i iP P− −= l

( ) ( )2 21 10

2 2i i ia a x=

( ) ( )2 21 10

2 2i i ib b x=

( ) ( ) 210 0

2i iP b− =

• ith port (x1=0, x2=0)

( ) ( ) ( ) ( ) ( ) ( )2

22

,

01 1 10 Re 0 0 0 0

2 2 2i

i i i i i rmsoi

VP V I a a

Z

+∗+ + + = ⋅ = = =

( ) ( )0

V xa x

Z

+

=

( ) ( )0

V xb x

Z

−

=

10/25 Department of Electronic Engineering, NTUT

( ) ( )2 2 20 0oV Z I= −

( ) ( ) ( ) ( ) ( )2 2 2 2 2 2 2 2

2 2

1 10 0 0 0 0 0

2 2o o o

o o

a V Z I Z I Z IZ Z

= + = − + =

Two-portNetwork

+

1oZPort 1

1 1x = l

Port 2

2 2x = l

Port 1'

1 0x =

( )1 0I

Port 2'

2 0x =

−

( )1 0V

( )2 0I

+

−( )2 2V l

2oZ

( )1 1I l ( )2 2I l

( )1 1V l

+

−

( )2 2a x

( )2 2b x

( )1 1a x

( )1 1b x

+

−

( )2 0V

+

−1E

1 1oZ Z=

2 2oZ Z=

• x2=0

( ) ( )1 1 1 10 0oV E Z I= −

( ) ( ) ( ) 11 1 1 1

1 1

10 0 0

2 2o

o o

Ea V Z I

Z Z= + = ( )

22 1

11

04 o

Ea

Z=• x1=0

Vpp

matched

• E1 (available power) x1=0 (Z1=Zo1 )

( ) ( )2

2 11 1

1

10 0

2 8AVSo

EP P a

Z+= = = ( ) ( )2 2

1 1 1

1 10

2 2a a= l

matched

11/25 Department of Electronic Engineering, NTUT

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 21 1 1 1 1 1 21 1 1 1 1 1 1 1 1 1

1 1

0 0 0 01 10 0 0 0 0 0 0

2 8 8o o

o o oo o

V Z I V Z Ia V Z I V Z V I Z I

Z Z

∗

∗ ∗+ + = = + + +

( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 221 1 1 1 1 1 1 1 1 1

1

1 10 0 0 0 0 0 0

2 8 o o oo

b V Z I V Z I V Z IZ

∗ ∗ = − − +

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2

1 1 1 1 1 1 1 1 1 1 1

1 1 1 10 0 0 0 0 0 0 0 0 Re 0 0

2 2 4 2P P P a b I V I V I V+ − ∗ ∗ ∗ = − = − = + =

• Z1 Zo1

• port 1' ( port 1 )

( ) ( )2

1 1

10 0

2 AVSb P P= − ( ) ( )2

1 1 1 1

1

2 AVSb P P= −l l• port 1 ( port 1')

( ) ( ) ( ) 2

1 1 1 1

10 0

2AVSP P P b= = −l

• ZT1=Zo1 0 ZT1≠Zo1

port 1

( )( )

( )

( )

2 2

2

2 1 1 1 111 2

1 1 0

AVS

AVSa

b P PS

Pa=

−= =

l

l l

l

( ) ( ) ( )2

1 1 1 110 1AVSP P P S= = −lor

12/25 Department of Electronic Engineering, NTUT

(Power Waves) (I)

•

• (Z0)Rs

s sV E Z I= −

LZ

+

−sE

sZV

I

+

−

sourceimpedance

loadimpedance

Department of Electronic Engineering, NTUT

L s

L s

Z Z

Z Z

∗−Γ =+

• ( )*L sZ Z= Γ = 0

slide 46

13/25

(Power Waves) (II)

( )1

2p s

s

a V Z IR

= + ( )1

2p s

s

b V Z IR

∗= −

LZ

+

−sE

sZ pa

pb

V

I

+

−

sp s L s

p s L ss

VZb V Z I Z ZI

Va V Z I Z ZZI

∗∗ ∗−− −Γ = = = =

+ ++p pa bΓ =

• (Normalized power waves)

0pb =L sZ Z ∗=

• (Available power)

( )1

2p s

s

a V Z IR

= + s sV E Z I= −

( )1

2 2s

p s s s

s s

Ea E Z I Z I

R R= − + =

22

4s

ps

Ea

R=

22 2

,

1

2 8s

AVS p p rmss

EP a a

R= = =

• ( )

2

21 1Re Re

2 2s

L L Ls L

EP I Z Z

Z Z= =

+

2

,max

1

8s

L AVSs

EP P

R= =

14/25 Department of Electronic Engineering, NTUT

( )( ) ( )( ) 2 2 *1 1 1 1 1Re

2 2 8 8 2L p p s s s ss s

P a b V Z I V Z I V Z I V Z I V IR R

∗∗ ∗ ∗= − = + + − − − =

2 2 21 1 1

2 2 2L p p AVS pP a b P b= − = −

21

2 p AVS Lb P P= −

= ( – )

•

= ( – )

15/25 Department of Electronic Engineering, NTUT

− (Traveling Waves)

0

0

ss

s

Z Z

Z Z

−Γ =+

0

0

LL

L

Z Z

Z Z

−Γ =+

1 11 1 12 2b S a S a= +

2 21 1 22 2b S a S a= +

Transistor[S]

2a

2b

1a

1b

Port 1 Port 2

+

−sE

sZ

outΓ

LZ

inΓ

sΓ LΓ

•[S] Z0

sΓ LΓ

?

+

−sE

sZ

sΓ

LZ

LΓ

Transistor[S]

1b

1a 2a

2b

16/25 Department of Electronic Engineering, NTUT

1

1in

b

aΓ =

2 2La b= Γ

2 21 1 22 2Lb S a S b= + Γ 21 12

221 L

S ab

S=

− Γ

• inΓ[ ]SLΓ

1 12 2111

1 221L

inL

b S SS

a S

ΓΓ = = +− Γ

12 211 11 1 12 2 11 1 1

221L

LL

S Sb S a S b S a a

S

Γ= + Γ = +− Γ

a1 b1

1 11 1 12 2b S a S a= + a1 b1 = a2

a2 = b2

Transistor[S]

2a

2b

1a

1b+

−sE

sZ

outΓ

LZ

inΓ

sΓ LΓ

1 11 1 12 2b S a S a= +2 21 1 22 2b S a S a= +

inΓ

17/25 Department of Electronic Engineering, NTUT

2

2 0s

out

E

b

a =

Γ =

1 1sa b= Γ

1 11 1 12 2sb S b S a= Γ + 12 21

111 s

S ab

S=

− Γ12 21

2 21 1 22 2 2 22 2111

ss

s

S Sb S b S a a S a

S

Γ= Γ + = +− Γ

12 21222

2 1101

s

sout

sE

S SbS

a S=

ΓΓ = = +− Γ

• outΓ[ ]S

sΓ

Transistor[S]

2a

2b

1a

1b+

−sE

sZ

outΓ

LZ

inΓ

sΓ LΓ

1 11 1 12 2b S a S a= +2 21 1 22 2b S a S a= +

outΓ

outΓ inΓ2 21 1 22 2b S a S a= +and

18/25 Department of Electronic Engineering, NTUT

+

−sE

sZ

sΓ

1a

1b

•

inΓ

+

−1V

1I

• 1 1s sa a b= + Γ

inΓ outΓ

Pin

1 1inb a= Γ

1 1 1s s s s ina a b a a= + Γ = + Γ Γ 1 1s

s in

aa =

− Γ Γ

( )2

2 2 2 2 2

1 1 1 2

11 1 1 11

2 2 2 2 1in

in in s

s in

P a b a a− Γ

= − = − Γ =− Γ Γ

•2 2

2 2 2

2 2 22 2

1 11 1 1 1

2 2 2 11 1in s

s sAVS in s s s

ss s

P P a a a∗

∗

Γ =Γ

− Γ − Γ= = = =

− Γ− Γ − Γ

( )( )2 222

2 2

1 111

2 1 1

s ininin s AVS AVS s

s in s in

P a P P M− Γ − Γ− Γ

= = =− Γ Γ − Γ Γ

• Ms (source mismatch factor) (mismatch loss)

inΓ s os

o so

E Za

Z ZZ=

+

19/25 Department of Electronic Engineering, NTUT

LZ

LΓ

outΓ

+

−thE

outZ 2a

2b

+

−LV

LI

LZ

LΓ

outΓ

( )2 2 2 2

2 2 2

1 1 11

2 2 2L LP b a b= − = − Γ

• ZL

22

2

11

2 1L

L th

out L

P b− Γ

=− Γ Γ

• ( source)

2

2

1 1

2 1L outAVN L th

out

P P b∗Γ =Γ= =

− Γ

( )( )2 2

2

1 1

1

L out

L AVN AVN L

out L

P P P M− Γ − Γ

= =− Γ Γ

• ML (load mismatch factor) (mismatch loss)

( source)outΓ

2 2La b= Γ

20/25 Department of Electronic Engineering, NTUT

Transistor[S]+

−sE

sZ

LZ

PAVNPAVS PLPin

Ms

interface interfaceML

• (power gain) Lp

in

PG

P=

• (transducer power gain) LT p s

AVS

PG G M

P= =

• (available power gain) AVN TA

AVS L

P GG

P M= =

p TG G>

A TG G>

• p T AG G G= =

21/25 Department of Electronic Engineering, NTUT

Gp (Operating Power Gain)

( )( )

2 2

2

2 2

1

11

21

12

LL

pin

in

bPG

P a

− Γ= =

− Γ

21 12

221 L

S ab

S=

− Γ

22

212 2

22

11

1 1L

p

in L

G SS

− Γ=

− Γ − Γ

• The Operating Power GainGp

where

Transistor[S]+

−sE

sZ

LZ

PAVNPAVS PLPin

Ms

interface interfaceML

slide 17

22/25 Department of Electronic Engineering, NTUT

GT (Transducer Power Gain)

• The Transducer Power GainGT

in inL LT p p s

AVS in AVS AVS

P PP PG G G M

P P P P= = = =

2 2 2 22 2

21 212 2 2 2

22 11

1 1 1 1

1 1 1 1s L s L

T

s in L s out L

G S SS S

− Γ − Γ − Γ − Γ= =

− Γ Γ − Γ − Γ − Γ Γ

( )( )2 2

2

1 1

1

s in

s

s in

M− Γ − Γ

=− Γ Γ

where

Transistor[S]+

−sE

sZ

LZ

PAVNPAVS PLPin

Ms

interface interfaceML

slide 19

23/25 Department of Electronic Engineering, NTUT

GA (Available Power Gain)

• The Available Power GainGA

AVN AVN AVNL TA T

AVS AVS L L L

P P PP GG G

P P P P M= = = =

22

212 2

11

1 1

1 1s

A

s out

G SS

− Γ=

− Γ − Γ

Transistor[S]+

−sE

sZ

LZ

PAVNPAVS PLPin

Ms

interface interfaceML

( )( )2 2

2

1 1

1

L out

L

out L

M− Γ − Γ

=− Γ Γ

where

slide 20

24/25 Department of Electronic Engineering, NTUT

•

(1) ( )

(2) (power waves, [Sp])

(3) (traveling waves, [S])

Re 2L L LP V I ∗=

• ( )

22 2

,

1

2 8s

AVS p p rmss

EP a a

R= = =

2 2 21 1 1

2 2 2L p p AVS pP a b P b= − = −

•

L p inP G P= L T AVSP G P=

• (defined with traveling waves, circuitries areseparately measured in aZo system):

25/25 Department of Electronic Engineering, NTUT