Download - RF Circuit Design - [Ch3-2] Power Waves and Power-Gain Expressions

Chapter 3-2

Power Waves and

Power-Gain Expressions

Chien-Jung Li

Department of Electronics Engineering

National Taipei University of Technology

Department of Electronic Engineering, NTUT

Maximum Power Transfer

LZ

sE

sZV

I

source

impedance

load

impedance

Phasor

s

s L

EI

Z Z

• The average power dissipated in the load

2 2

2 2

2 2

1 1 1

2 2 2

s s L

L rms L L L

s L s L s L

E E RP I R I R R

Z Z R R X X

• The maximum power dissipated in the load when s LX X s LR R

s LZ Z

• Maximum power transfer theorem

and

that is (conjugate matched)

• Can we link up the “conjugate matched impedances” and “reflection coefficients” ?

2/31


Power Waves

• In this section we discuss the analysis of lumped circuits in

terms of a new set of waves, called power waves.

LZ

sE

sZV

I

source

impedance

load

impedance

Since there is no transmission line, and therefore the characteristic

impedances is not defined.

oZ

d l

LZ

0

0d

IN d

0

L o

L o

Z Z

Z Zhas no meaning.

No transmission line in between

Can we define the reflection coefficient

w/o transmission lines?

s sV E Z I

3/31


Normalized Impedances (I)

Reference:

[1] K. Kurokawa, “Power waves and the scattering matrix.” IEEE Trans. Microwave Theory and techniques, vol. 13, pp.194-202,

Mar. 1965.

LZ

sE

sZV

I

s s sZ R jX

L L LZ R jX

• Normalize the impedances with respect to Rs

1 ss s s

s

Xz r jx j

R

L LL L L

s s

R Xz r jx j

R R

4/31


Normalized Impedances (II)

1

1s

s

z rz

z z r

U jV Γ-plane

U

V 1

1

z

z

0

1 1

• Recall the Smith Chart (Γ-plane)

1 ss s s

s

Xz r jx j

R

L LL L L

s s

R Xz r jx j

R R

L s L s L L s ss L s L s

s L s L s L L s s L s L s

r j x x r r jx r jxz r z z Z Z

z r r j x x r r jx r jx z z Z Z

z should contains the resistance and reactance of the load

(rL and xL), and the reactance of the source (xs)

• When , the reflection coefficient (maximum power delivering to

the load) L sZ Z 0

5/31


Power-waves Representation of One-port Network (I)

1

2p s

s

a V Z IR

1

2p s

s

b V Z IR

Res sR Z

• Reflected power wave is equal to zero when the load impedances is

conjugately matched to the source impedance, i.e., . pb

L sZ Z

where

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 ZZ

I

p pa b

• Normalized power waves

pL s

L L p

bZ Z

Z Z a

and

6/31


Available Power From Source

1

2 2

sp s s s

s s

Ea E Z I Z I

R R

2

2

4s

p

s

Ea

R

1

2p s

s

a V Z IR

s sV E Z I• For and

22 2

,

1

2 8

s

AVS p p rms

s

EP a a

Ris the power available from the source.

• Maximum power is delivered to the load when L sZ Z

2

21 1Re Re

2 2s

L L L

s L

EP I Z Z

Z Z

PL attains its maximum value when , and is given by L sZ Z ,maxL AVSP P

2

,max

1

8

s

L AVS

s

EP P

R

7/31


Impedance Mismatch

2 2

*1 1 1 1 1Re

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

s 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

2p AVS Lb P P

Power dissipated in the load = Available power from source – Reflected power

• When the impedances are mismatched, the power delivering to the

load is

Reflected power = Available power from source – Power dissipated in the load

8/31


Generalized Scattering Parameter (I)

1 11 1 12 2p p p p pb S a S a

2 21 1 22 2p p p p pb S a S a

1 1 1 1

1

1

2pa V Z I

R

2 2 2 2

2

1

2pa V Z I

R

1 1 1 1

1

1

2pb V Z I

R

Two-port

Network

[Sp]

2pa

2pb

1pa

1pb

Port 1 Port 2

1E

1Z

2I1I

1V

2V

2E

2Z

1 2 2 2

2

1

2pb V Z I

R

• Considering a two-port network, the generalized scattering matrix [Sp]

is found with respect to a reference impedance Re{Z1} at port 1 and

to Re{Z2} at port 2. If Z1 = Z2 = Zo, [Sp] = [S].

9/31


Generalized Scattering Parameter (II)

2

1

11

1 0p

p

p

p a

bS

a

Two-port

Network

[Sp]

2 0pa

2pb

1pa

1pb

Port 1 Port 2

1E

1Z

2I1I

1V

2V

2Z

1 11 1 12 2p p p p pb S a S a

2 21 1 22 2p p p p pb S a S a

1 111

1 1

Tp

T

Z ZS

Z Z

2 2 2

1 1 11

1 11

2 2IN p p AVS pP a b P S

1TZ

• Can we find the power by using [S] but not [Sp] ? Sure! We will talk

about this later.

10/31


Example

• Calculate the power waves and the power delivered to the load in the

circuit.

100 50 LZ j

10 0sE

100 50 sZ j V

I

100 50

10 5.59 26.57100 50 100 50

Ls

L s

Z jV E

Z Z j j

10

0.05 A100 50 100 50

s

L s

EI

Z Z j j

1 1 10

0.5 2 2 2 100

p s s s s

s s

a V Z I E Z I Z IR R

1

1 1 110 0.05 100 50 0.05 100 50 0

2 2 2 100p s s s s

s

b V Z I E Z I Z I j jR R

2 21 10.125 W

2 2L p pP a b (Try )

1Re

2LP VI

11/31


Example (I)

• Calculate the generalized parameter Sp11 and Sp21 at 1 GHz in the

lossless, reciprocal, two-port network. Then calculate Sp22 and Sp12.

2 10 Z

1.59 nHL

1E

1 50 50 Z j

1V

2V

10 LZ j

1TZ1I 2I

11 1 1

1 1

0.167 0T

T

ZV E E

Z Z

11 1

1 1

0.0118 45T

EI E

Z Z

2 1 0.118 45V E

2 1 0.0118 45I E

1 1 1 1

1

1

2p

a V Z IR

2 2 2 2

2

1

2p

a V Z IR

1 1 1 1

1

1

2p

b V Z IR

2 2 2 2

2

1

2pb V Z I

R

1 0.071 0pa

1 0.061 78.69pb

2 0pa

2 0.037 45pb

For Sp11 and Sp21

12/31


Example (II)

2

1 1 111

1 1 10

10 10 50 500.85 78.69

10 10 50 50p

p Tp

p Ta

b j jZ ZS

a Z Z j j

2

2

21

1 0

0.037 450.525 45

0.071 0p

p

p

p a

bS

a

2 10 Z

1.59 nHL

2E

1 50 50 Z j

1V

2V

10 LZ j

2TZ1I 2I

For Sp22 and Sp12

1 2 0.833 0V E 1 2 0.0118 45I E 2 2 0.92 5.19V E 2 2 0.0118 45I E

1 0pa 1 0.083 45pb 2 0.158 0pa 2 0.134 11.32pb

1

2 2 222

2 2 20

0.85 11.3

p

p Tp

p Ta

b Z ZS

a Z Z

1

1

12

2 0

0.083 450.525 45

0.158 0p

p

p

p a

bS

a

13/31


Power-Gain Expressions (I)

Transistor

[S]

2a

2b

1a

1b

Port 1 Port 2

sE

sZ

out

LZ

in

s L

s os

s o

Z Z

Z Z

L oL

L o

Z Z

Z Z

1 11 1 12 2b S a S a

2 21 1 22 2b S a S a

• Consider a microwave amplifier with the source and load reflection

coefficients and measured in a Zo system: s L

• For the transistor, the input and output traveling waves measured in a

Zo system (this is very practical) :

14/31


Power-Gain Expressions (II)

sE

sZ

s

LZ

L

Transistor

[S]

The reflection coefficients and S-parameters are separately measured

in a Zo (usually 50 Ω) system

Transistor

[S]

2a

2b

1a

1b

sE

sZ

out

LZ

in

s L

After connecting them all together The goal is to find the input and output

power relations.

1b

1a 2a

2b

15/31


Input Reflection Coefficient

1

1

in

b

a

2 2La b

2 21 1 22 2Lb S a S b 21 12

221 L

S ab

S

Transistor

[S]

2a

2b

1a

1b

sE

sZ

out

LZ

in

s L

• After connecting the circuits together, the first step is to find the new

input coefficient , which is the result coming from and . in S L

1 11 1 12 2b S a S a

2 21 1 22 2b S a S a

1 12 2111

1 221L

in

L

b S SS

a S

12 21

1 11 1 12 2 11 1 1

221L

L

L

S Sb S a S b S a a

S

a1 is your input, so the goal here is to find the reflected wave b1

1 11 1 12 2b S a S a a1 is your input, to find b1 = you need to find a2

to find a2 = you need to find b2

the relationship between b2 and a1

16/31


Output Reflection Coefficient

2

2 0s

out

E

b

a

1 1sa b

1 11 1 12 2sb S b S a 12 21

111 s

S ab

S

12 212 21 1 22 2 2 22 2

111s

s

s

S Sb S b S a a S a

S

12 21222

2 1101

s

sout

sE

S SbS

a S

Transistor

[S]

2a

2b

1a

1b

sE

sZ

out

LZ

in

s L

• After connecting the circuits together, the second step is to find the

new output coefficient , which is the result coming from and . out S s

1 11 1 12 2b S a S a

2 21 1 22 2b S a S a

The same procedure as finding is applied. in

17/31


The Available Power and Input Power (I)

sE

sZ

s

1a

1b

• After finding out the input/output refection coefficients, let’s now deal

with the power.

in

Since we have got , we can discard the circuits

connected after the source right here.

in

1 1s sV E I Z

1V

1I

1 11 1 1 1 1s s s s

o

V VV V V E I I Z E Z

Z

1 1 1 11 1 1s s s s s

o o o

V V V VV E Z V E Z Z V

Z Z Z

1 1o s o

s

o s s o

Z Z ZV E V

Z Z Z Z

• Use the normalized power waves

1 11 1

s o s os s

o s s oo o o

E Z Z ZV Va a b

Z Z Z ZZ Z Z

where , , and s o

s

o s

E Za

Z Z

11

o

Vb

Z

s os

s o

Z Z

Z Z

18/31


The Available Power and Input Power (II)

1 1inb a

1 1 1s s s s ina a b a a 11

s

s in

aa

2

2 2 2 2 2

1 1 1 2

11 1 1 11

2 2 2 2 1

in

in in s

s in

P a b a a

• The available power from source

2 2

2 2 2

2 2 22 2

1 11 1 1 1

2 2 2 11 1in s

s s

AVS in s s s

ss s

P P a a a

2 22

2

2 2

1 111

2 1 1

s inin

in s AVS AVS s

s in s in

P a P P M

• Ms is known as the source mismatch factor (or mismatch loss).

sE

sZ

s

1a

1b

in

1V

1I

Pin

19/31


The Available Power and Output Power (II)

LZ

L

out Since we have got , the circuits looking into the output

port (with source) can be simplified as a Thevenin’s

equivalent circuit.

out

thE

outZ 2a

2b

LV

LI

LZ

L

out

2 2 2 2

2 2 2

1 1 11

2 2 2L LP b a b

• The power delivered to the load ZL

2

2

2

11

2 1

L

L th

out L

P b

• The available power from the network

2

2

1 1

2 1L outAVN L th

out

P P b

2 2

2

1 1

1

L out

L AVN AVN L

out L

P P P M

• ML is known as the load mismatch factor (or mismatch loss).

20/31


Definition of the Power Gains

Transistor

[S]

sE

sZ

LZ

PAVN PAVS PL Pin

Ms

interface interface

ML

• The power gain Lp

in

PG

P

• The transducer power gain LT p s

AVS

PG G M

P

• The available power gain AVN TA

AVS L

P GG

P M

p TG G

A TG G

• When the Input and output are matched: p T AG G G

From the amplifier input to load

From the source to load

21/31


Power Gain

2 2

2

2 2

1

11

21

12

LL

p

inin

bP

GP

a

21 12

221 L

S ab

S

2

2

212 2

22

11

1 1

L

p

in L

G SS

• The Power Gain Gp

where

Transistor

[S]

sE

sZ

LZ

PAVN PAVS PL Pin

Ms

interface interface

ML

22/31


Transducer Power Gain

• The Transducer Power Gain GT

L L in inT p p s

AVS in AVS AVS

P P P PG G G M

P P P P

2 2 2 2

2 2

21 212 2 2 2

22 11

1 1 1 1

1 1 1 1

s L s L

T

s in L s out L

G S SS S

2 2

2

1 1

1

s in

s

s in

M

where

Transistor

[S]

sE

sZ

LZ

PAVN PAVS PL Pin

Ms

interface interface

ML

23/31


Available Power Gain

• The Available Power Gain GA

AVN L AVN AVN TA T

AVS AVS L L L

P P P P GG G

P P P P M

2

2

212 2

11

1 1

1 1

s

A

s out

G SS

Transistor

[S]

sE

sZ

LZ

PAVN PAVS PL Pin

Ms

interface interface

ML

2 2

2

1 1

1

L out

L

out L

M

where

24/31


Two-port Network Matrices

• Several ways that are commonly used to represent the

two-port network:

Impedance matrix : z-parameter

Admittance matrix : y-parameter

Hybrid matrix : h-parameter

ABCD matrix : ABCD parameters

Scattering matrix : S-parameter

• These matrices describe the relationship between the

input/output voltages and currents except the scattering

matrix which describes the relationship between the

input/output traveling waves (or power waves).

25/31


Two-port Network Representation

z-parameter

y-parameter

h-parameter

ABCD parameters

1 11 12 1

2 21 22 2

v z z i

v z z i

1 11 1 12 2v z i z i

2 21 1 22 2v z i z i

1 11 12 1

2 21 22 2

i y y v

i y y v

1 11 12 1

2 21 22 2

v h h i

i h h v

1 2

1 2

v vA B

i iC D

Two-port

network

1v

1i 2i

2v

Port 1 Port 2

26/31


Conversion Between the Network Parameter

• This table is provided at page 62 in the textbook.

27/31


Series Connection

• Series Connection: use z-parameter

1 11 1 11 11 12 12

2 22 2 21 21 22 22

a b a b a b

a b a b a b

v iv v z z z z

v iv v z z z z

28/31


Shunt Connection

• Shunt Connection: use y-parameter

1 11 1 11 11 12 12

2 22 2 21 21 22 22

a b a b a b

a b a b a b

i vi i y y y y

i vi i y y y y

29/31


Cascade Circuits

• Cascade Circuits : use ABCD parameters (chain)

1 1 2 2

1 1 2 2

a a ba a a a b b

a a ba a a a b b

v v v vA B A B A B

i i i iC D C D C D

30/31


Summary

• The power delivered to the load can be calculated by using three

methods:

(1) Real power dissipated at load ( )

(2) Power waves (generalized [Sp], linked with reflections)

(3) Traveling waves ([S], it’s practical and useful in amplifier design)

Re 2L L LP V I

• Available power from source (maximum average power the source can

provide when matched) :

22 2

,

1

2 8

s

AVS p p rms

s

EP a a

R

2 2 21 1 1

2 2 2L p p AVS pP a b P b

• When mismatch occurs:

Power wave

Power wave

L p inP G PL T AVSP G P

• Power gains (defined with traveling waves, circuitries are separately

measured in a Zo system) :

31/31

Download - RF Circuit Design - [Ch3-2] Power Waves and Power-Gain Expressions

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