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RF Circuit Design - [Ch3-2] Power Waves and Power-Gain Expressions
1. Chapter 3-2
Power Waves and
Power-Gain Expressions
Chien-Jung Li
Department of Electronics Engineering
National Taipei University of Technology
2. Department of Electronic Engineering, NTUT
Maximum Power Transfer
LZ
sE
sZ
V
I
source
impedance
load
impedance
Phasor
s
s L
E
I
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 R
P 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
3. Department of Electronic Engineering, NTUT
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
sZ
V
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 Z
has no meaning.
No transmission line in between
Can we define the reflection coefficient
w/o transmission lines?
s sV E Z I
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4. Department of Electronic Engineering, NTUT
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
sZ
V
I
s s sZ R jX
L L LZ R jX
• Normalize the impedances with respect to Rs
1 s
s s s
s
X
z r jx j
R
L L
L L L
s s
R X
z r jx j
R R
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5. Department of Electronic Engineering, NTUT
Normalized Impedances (II)
1
1
s
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 s
s s s
s
X
z r jx j
R
L L
L L L
s s
R X
z 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
6. Department of Electronic Engineering, NTUT
Power-waves Representation of One-port Network (I)
1
2
p s
s
a V Z I
R
1
2
p s
s
b V Z I
R
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
s
p s L s
p s L s
s
V
Zb 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
7. Department of Electronic Engineering, NTUT
Available Power From Source
1
2 2
s
p s s s
s s
E
a E Z I Z I
R R
2
2
4
s
p
s
E
a
R
1
2
p s
s
a V Z I
R
s sV E Z I• For and
2
2 2
,
1
2 8
s
AVS p p rms
s
E
P a a
R
is the power available from the source.
• Maximum power is delivered to the load when
L sZ Z
2
21 1
Re Re
2 2
s
L L L
s L
E
P 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
E
P P
R
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8. Department of Electronic Engineering, NTUT
Impedance Mismatch
2 2 *1 1 1 1 1
Re
2 2 8 8 2
L p p s s s s
s s
P a b V Z I V Z I V Z I V Z I V I
R R
2 2 21 1 1
2 2 2
L p p AVS pP a b P b
21
2
p 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
9. Department of Electronic Engineering, NTUT
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
2
pa V Z I
R
2 2 2 2
2
1
2
pa V Z I
R
1 1 1 1
1
1
2
pb 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
2
pb 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
10. Department of Electronic Engineering, NTUT
Generalized Scattering Parameter (II)
2
1
11
1 0p
p
p
p a
b
S
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 1
11
1 1
T
p
T
Z Z
S
Z Z
2 2 2
1 1 11
1 1
1
2 2
IN 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
11. Department of Electronic Engineering, NTUT
Example
• Calculate the power waves and the power delivered to the load in the
circuit.
100 50LZ j
10 0sE
100 50sZ j
V
I
100 50
10 5.59 26.57
100 50 100 50
L
s
L s
Z j
V E
Z Z j j
10
0.05 A
100 50 100 50
s
L s
E
I
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 I
R R
1
1 1 1
10 0.05 100 50 0.05 100 50 0
2 2 2 100
p s s s s
s
b V Z I E Z I Z I j j
R R
2 21 1
0.125 W
2 2
L p pP a b (Try ) 1
Re
2
LP VI
11/31
12. Department of Electronic Engineering, NTUT
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 10Z
1.59 nHL
1E
1 50 50Z j
1V
2V
10LZ j
1TZ
1I 2I
1
1 1 1
1 1
0.167 0T
T
Z
V E E
Z Z
1
1 1
1 1
0.0118 45
T
E
I E
Z Z
2 1 0.118 45V E
2 1 0.0118 45I E
1 1 1 1
1
1
2
pa V Z I
R
2 2 2 2
2
1
2
pa V Z I
R
1 1 1 1
1
1
2
pb V Z I
R
2 2 2 2
2
1
2
pb 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
13. Department of Electronic Engineering, NTUT
Example (II)
2
1 1 1
11
1 1 10
10 10 50 50
0.85 78.69
10 10 50 50
p
p T
p
p Ta
b j jZ Z
S
a Z Z j j
2
2
21
1 0
0.037 45
0.525 45
0.071 0
p
p
p
p a
b
S
a
2 10Z
1.59 nHL
2E
1 50 50Z j
1V
2V
10LZ j
2TZ
1I 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 2
22
2 2 20
0.85 11.3
p
p T
p
p Ta
b Z Z
S
a Z Z
1
1
12
2 0
0.083 45
0.525 45
0.158 0
p
p
p
p a
b
S
a
13/31
14. Department of Electronic Engineering, NTUT
Power-Gain Expressions (I)
Transistor
[S]
2a
2b
1a
1b
Port 1 Port 2
sE
sZ
out
LZ
in
s L
s o
s
s o
Z Z
Z Z
L o
L
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
15. Department of Electronic Engineering, NTUT
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
16. Department of Electronic Engineering, NTUT
Input Reflection Coefficient
1
1
in
b
a
2 2La b
2 21 1 22 2Lb S a S b 21 1
2
221 L
S a
b
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 21
11
1 221
L
in
L
b S S
S
a S
12 21
1 11 1 12 2 11 1 1
221
L
L
L
S S
b 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
17. Department of Electronic Engineering, NTUT
Output Reflection Coefficient
2
2 0s
out
E
b
a
1 1sa b
1 11 1 12 2sb S b S a 12 2
1
111 s
S a
b
S
12 21
2 21 1 22 2 2 22 2
111
s
s
s
S S
b S b S a a S a
S
12 212
22
2 110
1
s
s
out
sE
S Sb
S
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
18. Department of Electronic Engineering, NTUT
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 1
1 1 1 1 1s s s s
o
V V
V V V E I I Z E Z
Z
1 1 1 1
1 1 1s s s s s
o o o
V V V V
V E Z V E Z Z V
Z Z Z
1 1
o s o
s
o s s o
Z Z Z
V E V
Z Z Z Z
• Use the normalized power waves
1 1
1 1
s o s o
s s
o s s oo o o
E Z Z ZV V
a a b
Z Z Z ZZ Z Z
where , , ands o
s
o s
E Z
a
Z Z
1
1
o
V
b
Z
s o
s
s o
Z Z
Z Z
18/31
19. Department of Electronic Engineering, NTUT
The Available Power and Input Power (II)
1 1inb a
1 1 1s s s s ina a b a a 1
1
s
s in
a
a
2
2 2 2 2 2
1 1 1 2
11 1 1 1
1
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 1
in s
s s
AVS in s s s
ss s
P P a a a
2 2
2
2
2 2
1 111
2 1 1
s in
in
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
20. Department of Electronic Engineering, NTUT
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 1
1
2 2 2
L 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 out
AVN 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
21. Department of Electronic Engineering, NTUT
Definition of the Power Gains
Transistor
[S]
sE
sZ
LZ
PAVNPAVS PLPin
Ms
interface interface
ML
• The power gain L
p
in
P
G
P
• The transducer power gain L
T p s
AVS
P
G G M
P
• The available power gain AVN T
A
AVS L
P G
G
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
22. Department of Electronic Engineering, NTUT
Power Gain
2 2
2
2 2
1
1
1
2
1
1
2
L
L
p
in
in
bP
G
P a
21 1
2
221 L
S a
b
S
2
2
212 2
22
11
1 1
L
p
in L
G S
S
• The Power Gain Gp
where
Transistor
[S]
sE
sZ
LZ
PAVNPAVS PLPin
Ms
interface interface
ML
22/31
23. Department of Electronic Engineering, NTUT
Transducer Power Gain
• The Transducer Power Gain GT
L L in in
T p p s
AVS in AVS AVS
P P P P
G 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 S
S S
2 2
2
1 1
1
s in
s
s in
M
where
Transistor
[S]
sE
sZ
LZ
PAVNPAVS PLPin
Ms
interface interface
ML
23/31
24. Department of Electronic Engineering, NTUT
Available Power Gain
• The Available Power Gain GA
AVN L AVN AVN T
A T
AVS AVS L L L
P P P P G
G G
P P P P M
2
2
212 2
11
1 1
1 1
s
A
s out
G S
S
Transistor
[S]
sE
sZ
LZ
PAVNPAVS PLPin
Ms
interface interface
ML
2 2
2
1 1
1
L out
L
out L
M
where
24/31
25. Department of Electronic Engineering, NTUT
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
26. Department of Electronic Engineering, NTUT
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
27. Department of Electronic Engineering, NTUT
Conversion Between the Network Parameter
• This table is provided at page 62 in the textbook.
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28. Department of Electronic Engineering, NTUT
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
29. Department of Electronic Engineering, NTUT
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
30. Department of Electronic Engineering, NTUT
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
31. Department of Electronic Engineering, NTUT
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) :
2
2 2
,
1
2 8
s
AVS p p rms
s
E
P a a
R
2 2 21 1 1
2 2 2
L p p AVS pP a b P b
• When mismatch occurs:
Power wave
Power wave
L p inP G P L T AVSP G P
• Power gains (defined with traveling waves, circuitries are separately
measured in a Zo system) :
31/31