1190948120898
DESCRIPTION
1190948120898TRANSCRIPT
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EE3113_L1 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 1
Prof. R. LudwigDepartment of Electrical and Computer Engineering
Worcester Polytechnic InstituteWorcester, MA
Copyright, 1998 R. Ludwig
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EE3113_L1 2
EE 3113: Lecture 1
Importance of RF circuit design wireless communications (explosive growth of
cell phones) global positioning systems (GPS) computer engineering (bus systems, CPU,
peripherals exceeding 600 MHz)
Why this course??? lumped circuit representation no longer applies!
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EE3113_L1 3
What do we mean by going from lumped to distributed theory?
Example: INDUCTOR
Low-frequency
(lumped)
LjRZ w+=
High-frequency
Z = ?
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EE3113_L1 4
Current and voltage vary spatially over the component size
Upper MHz to GHz range
-1-0.5
00.5
1x-1
-0.5
0
0.5
1
y0
2
4
6
z
-1-0.5
00.5
1x
E (or V) and H (or I) fields
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EE3113_L1 5
Frequency spectrum RadioFrequency (RF)
TV, wireless phones, GPS 300 MHz 3 GHz operational frequency 1 m 10 cm wavelength in air
MicroWave (MW) RADAR, remote sensing 8 GHz 40 GHz operational frequency 3.75 cm 7.5 mm wavelength in air
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EE3113_L1 6
Design FocusCell phone transceiver circuit
Typical frequencyrange:
950 MHz
1.9 GHz
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EE3113_L1 7
Implementation
matching networks
BJT/FET active devices
biasing circuits
printed circuit board
mircostripline realization
surface mount technology
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EE3113_L2 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 2
Prof. R. LudwigDepartment of Electrical and Computer Engineering
Worcester Polytechnic InstituteWorcester, MA
Copyright, 1998 R. Ludwig
-
EE3113_L2 2
RF Behavior of Passive Components
Conventional circuit analysis R is frequency independent Ideal inductor: Ideal capacitor:
Evaluation Impedance chart
LjXL w=
CXC w1=
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EE3113_L2 3
Impedance Chart(impedance of C & L vs frequency)
ZC=1/(2pfC)
ZL=2pfL
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EE3113_L2 4
How does a wire behave at high frequency?
Example: Resistorsp 2a
lRDC =
d2/
aRR DC = d
w2
/a
RL DC =
mspd
f
1=
High frequency results in skin-effect whereby current flow ispushed to the outside
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EE3113_L2 5
How exactly is the current distribution as a function offrequency?
Low frequency showsuniform currentdistribution
medium to highfrequency pushescurrent to the outside
RF sees currentcompletely restrictedto surface
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EE3113_L2 6
Impedance Measurement ExampleCapacitor going through resonance
CapacitorCharacteristics
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EE3113_L2 7
Equivalent Circuit Analysis
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EE3113_L3 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGNLecture Notes for A-term 1999
LECTURE 3Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01619copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L3 2
Transmission Line Analysis
Propagating electric field
Phase velocity
Traveling voltage wave
)cos(0 kztEE XX -= w
Time factor
Space factor
rp
cfv
eeml ===
1
kkzt
EtzV X)sin(
),( 0-
=w
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EE3113_L3 3
High frequency implies spatial voltage distribution
Voltage has a time andspace behavior
Space is neglected for lowfrequency applications
For RF there can be a largespatial variation
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EE3113_L3 4
Generic way to measure spatial voltage variations
For low frequency (1MHz)Kirchhoffs laws apply
For high frequency (1GHz)Kirchhoffs laws do notapply anymore
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EE3113_L3 5
Kirchhoffs laws on a microscopic level
Over a differential sectionwe can again use basiccircuit theory
Model takes into accountline losses and dielectriclosses
Ideal line involves only Land C
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EE3113_L3 6
Example of transmission line: Two-wire line
Alternating electric fieldbetween conductors
alternating magnetic fieldsurrounding conductors
dielectric medium tendsto confine field insidematerial
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EE3113_L3 7
Example of transmission line: Coaxial cable
Electric field iscompletely containedwithin both conductors
Perfect shielding ofmagnetic field
TEM modes up to acertain cut-off frequency
E
H
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EE3113_L3 8
Example of transmission line: Microstip line
Cross-sectional view
Low dielectric medium High dielectric medium
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EE3113_L3 9
Triple-layer transmission line
Conductor is completely shielded between twoground planes
Cross-sectional view
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EE3113_L4 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 4
Prof. R. LudwigDepartment of Electrical and Computer Engineering
Worcester Polytechnic InstituteWorcester, MA 01609
copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L4 2
General Transmission Line Equations
Detailed analysis of a differential section
Note: Analysis applies to all types of transmission lines such ascoax cable, two-wire, microstrip, etc.
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EE3113_L4 3
Kirchhoffs laws on a microscopic level
Over a differentialsection we can againuse basic circuit theory
Model takes intoaccount line losses anddielectric losses
Ideal line involvesonly L and C
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EE3113_L4 4
Advantages versus disadvantages ofelectric circuit representation
Clear intuitivephysical picture
yields a standardizedtwo-port networkrepresentation
serves as buildingbocks to go frommicroscopic tomacroscopic forms
Basically a one-dimensional representation(cannot take into accountinterferences)
Material nonlinearities,hysteresis, and temperatureeffects are not accountedfor
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EE3113_L4 5
)()()(
))()(
( zILjRdz
zdVz
zVzzVLim w+=-=
D-D+
-
Derivation of differential transmission line form
)()()()( zzVzzILjRzV D++D+= wKVL :
KCL :)()()()( zzIzzzVCjGzI D++D+D+= w
)()()(
zVCjGdz
zdIw+=-
CoupledDE
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EE3113_L4 6
Traveling Voltage and Current Waves
0)()( 2
2
2
=- zVkdz
zVd
where
))(( CjGLjRjkkk ir ww ++=+=
kzkz eVeVzV +--+ +=)( kzkz eIeIzI +--+ +=)(
0)()( 2
2
2
=- zIkdz
zId
Left traveling wave
Right traveling wavePhasor expressions
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EE3113_L4 7
General line impedance definition
)()(
)( kzkz eVeVLjR
kzI +--+ -
+=
w
-
-
+
+
-==++
=IV
IV
CjGLjR
Z)()(
0 ww
?
Characteristic line impedance
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EE3113_L5 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 5
Prof. R. LudwigDepartment of Electrical and Computer Engineering
Worcester Polytechnic InstituteWorcester, MA
copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L5 2
Lossless Transmission Line Model
Line representation
)()(
0 CjGLjR
Zww
++
=Characteristic impedance:
Note: R, L, G, C are given per unit length and depend on geometry
Lossless implies:R = G = 0!
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EE3113_L5 3
Transmission Line Parameters for different line types
2-wire coax
sdpa1
)2
(1aD
ch-pm
R
L
G
C
)11
(2
1ba
+psd
parallel-plate
))2/((1 aDch-ps
))2/((1 aDch-pe
sdw2
wd
m
dw
s
dw
e
)ln(2 a
bpm
)/ln(2
abps
)/ln(2
abpe
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EE3113_L5 4
Microstrip line
1/),4
8ln(2
/ 000
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EE3113_L5 5
What is a voltage reflection coefficient?
0
00 ZZ
ZZ
L
L
+-
=GReflection coefficientat the load location
)(10 =G LZ
)0(10 -=G LZ
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EE3113_L5 6
Standing Waves
)()( djdj eeVdV bb -++ -=
)2/cos()sin(2),( pwb += + tdVtdv
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EE3113_L5 7
Standing wave ratio
||1||1
||||
||||
0
0
min
max
min
max
G-G+
===II
VV
SWR
SWR is a measure of mismatch of theload to the line
SWR=1 (matched) or SWR (total mismatch)
match
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EE3113_L6 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 6
Prof. R. LudwigDepartment of Electrical and Computer Engineering
Worcester Polytechnic InstituteWorcester, MA 01609
copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L6 2
Special Termination Conditions
Lossless transmission line
CL
Z =0
)tan()tan(
)(0
00 djZZ
djZZZdZ
L
Lin b
b++
=
Characteristic impedance
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EE3113_L6 3
Input impedance of short circuit transmission line
)tan()( 0 djZdZin b=Impedance
Voltage:
)sin(2)( djVdV b+=
Current:
)cos(2
)(0
dZV
dI b+
=
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EE3113_L6 4
Input impedance of open circuit transmission line
Voltage:
Current:
Impedance
)cos(2)( dVdV b+=
)sin(2
)(0
dZjV
dI b+
=
)cot()( 0 djZdZin b-=
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EE3113_L6 5
Quarter-wave transmission line
LL
Lin Z
ZjZZjZZ
ZZ2
0
0
00 )4/tan(
)4/tan()4/( =
++
=blbl
l
Quarter-wave transformer model:
given input and output impedances
Predict lineimpedance
inLZZZ =0
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EE3113_L6 6
What should you know? Input impedance: Page 80, equation (2.71) Example 2.6 on page 82 Example 2.7 on page 84 Example 2.8 on page 87
Matching works only forparticular frequencies
500 MHz 1.5 GHz
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EE3113_L7 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 7
Prof. R. LudwigDepartment of Electrical and Computer Engineering
Worcester Polytechnic InstituteWorcester, MA 01609
copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L7 2
Sourced and Loaded Transmission Lines Lossless transmission line with source
)()1(Gin
inGininin ZZ
ZVVV
+=G+= +
Voltage at the beginning of the transmission line iscomposed of an incident and reflected component!
0
0
ZZZZ
G
G
+-
=0
0
ZZZZ
L
L
+-
=
-
EE3113_L7 3
Power considerations
}Re{21 *
ininin IVP =
)1( ininin VV G+=+ )1(
0in
inin Z
VI G-=
+
)||1(||
21 2
0
2
inin
in ZV
P G-=+
)||1(|1|
|1|||81 2
2
2
0
2
ininS
SGin Z
VP G-
GG-G-
=
-
EE3113_L7 4
Two special cases:
Load and sourcematched line 00 =G=G S
0
2||81
ZV
P Gin =
Mismatch at source,but match at load 00 =G
2
0
2
|1|||
81
SG
in ZV
P G-=
How to measure power?mWWP
dBmP1
][log10][ =
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EE3113_L7 5
Return and insertion losses
Return loss: ||log20||log10)log(10 2 inini
r
PP
RL G-=G-=-= [dB]
Insertion loss: )||1log(10)log(10)log(10 2ini
ri
i
t
PPP
PP
IL G--=-
-=-= [dB]
No reflection Full reflection
0
dB10 dB
1RL
SWR
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EE3113_L8 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 8
Prof. R. LudwigDepartment of Electrical and Computer Engineering
Worcester Polytechnic InstituteWorcester, MA 01609
copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L8 2
From Reflection Coefficient to LoadImpedance (Smith Chart)
Reflection coefficient in phasor form
Ljir
L
L ejZZZZ q|| 000
0
00 G=G+G=+
-=G
The load reflectioncoefficient is identified inthe complex domain
0G
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EE3113_L8 3
Normalized impedance
ir
irinin j
jdd
jxrzZdZG-G-G+G+
=G-G+
=+==11
)(1)(1
/)( 0
irdjj jeed L G+G=G=G - bq 20 ||)(
22
22
)1(1
ir
irrG+G-
G-G-=
22)1(2
ir
ixG+G-
G=
Real part of normalizedimpedance
Imaginary part ofnormalized impedance
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EE3113_L8 4
Inversion of complex reflection coefficient(constant normalized resistance)
222 )1
1()
1(
+=G+
+-G
rrr
ir
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EE3113_L8 5
Inversion of complex reflection coefficient(constant normalized reactance)
222 )1
()1
()1(xxir
=-G+-G
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EE3113_L8 6
Combined display: Smith Chart
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EE3113_L9 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGNLecture Notes for A-term 1999
LECTURE 9Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L9 2
Impedance Transformation(Smith Chart)
Reflection coefficient in phasor form
Ljir
L
L ejZZZZ q|| 000
0
00 G=G+G=+
-=G
0G
ir
irinin j
jdd
jxrzZdZG-G-G+G+
=G-G+
=+==11
)(1)(1
/)( 0
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EE3113_L9 3
Generic Smith Chart computation
Normalize load impedance find reflection coefficient rotate reflection coefficient record normalized input impedance de-normalize input impedance
LL zZ
0GLz)(0 dGG
)(dzin)()( dZdz inin
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EE3113_L9 4
Graphical display
-
EE3113_L9 5
How to create ideal capacitors and inductors with atransmission line?
Start oftransformation
Capacitivedomain
Inductivedomain
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EE3113_L9 6
Start oftransformation
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EE3113_L10 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 10Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L10 2
Admittance Transformation(Smith Chart)
impedance representation in Smith Chart
0G
)(1)(1
dd
jxrzin G-G+
=+=
admittance representation in Smith Chart
)(1)(1
)(1)(11
0 dede
dd
zYY
y jj
in
inin G-
G+
G+G-
=== --
p
p
180 degreephase shift
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EE3113_L10 3
Transformation21
21
11 jyjz inin -=+=
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EE3113_L10 4
Alternative: re-interpretation
Instead of rotating the reflection coefficient about180 degree, we keep the location fixed and rotate theentire Smith Chart by 180 degree.
-
EE3113_L10 5
Re-interpretation leads to ZY-Smith Chart
The Smith Chart inits original form iskept for impedancedisplay,
but a second SmithChart is rotated by180 degree foradmittance display.
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EE3113_L11 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGNLecture Notes for A-term 1999
LECTURE 11Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L11 2
Parallel Connection of R and L Elements(Smith Chart)
parallel connection of R and L elements
0
1)(
LYjgy
LLin w
w -=
-
EE3113_L11 3
Parallel connection of R and C elements
CjZgy LLin ww 0)( +=
-
EE3113_L11 4
Series connection of R and L elements
0
)(Z
Ljrz LLinw
w +=
-
EE3113_L11 5
Series connection of R and C elements
0
1)(
CZjrz
LLin w
w -=
-
EE3113_L11 6
Practical case: BJT connected viaa T-network
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EE3113_L12 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 12Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L12 2
Single and Multi-Port Networks
basic current and voltage definitions definitions
-
EE3113_L12 3
Impedance and admittance networks
}]{[}{ IZV = }]{[}{ VYI =
}]{][[}{ IYZV =
][][ 1 ZY =-
-
EE3113_L12 4
Example Z-representation of Pi-network
+
+++
=)(
)(1][
PBPAPCPCPA
PCPAPCPBPA
PCPBPA ZZZZZ
ZZZZZ
ZZZZ
)(0| mkim
nnm ki
vz ==
-
EE3113_L12 5
Additional networks
-
=
2
2
1
1
i
v
DC
BA
i
v Chain or ABCD network
(often used for cascading)
=
2
1
2221
1211
2
1
v
i
hh
hh
i
v Hybrid or h-network
(often used for active devices)
Typical exampleof h-network(small signal, lowfrequency model)
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EE3113_L13 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 13Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L13 2
Interconnecting Networks
Certain networks are more advantageous tointerconnect.
Example: series connection
]"[]'[][ ZZZ +=
-
EE3113_L13 3
Hybrid representation
]"[]'[][ hhh +=
Typical example
-
EE3113_L13 4
ABCD parameter representation
Very useful when cascading networks
-
=
2
2
1
1
"
"
""
""
''
''
i
v
DC
BA
DC
BA
i
v
-
EE3113_L13 5
ABCD network is very useful for transmission linerepresentations
=
)cos(
)sin()sin()cos(
0
0
lZ
lj
ljZl
DC
BAb
bbb
Example:
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EE3113_L14 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 14Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L14 2
Scattering parameters There is a need to establish well-defined
termination conditions in order to find thenetwork descriptions for Z, Y , h, andABCD networks
Open and short voltage and currentconditions are difficult to enforce
RF implies forward and backward travelingwaves which can form standing wavesdestroying the elements
-
EE3113_L14 3
Solution: S-parameters
Input-output behavior of network is definedin terms of normalized power waves
Ratio of the power waves are recorded interms of so-called scattering parameters
S-parameters are measured based onproperly terminated transmission lines (andnot open/short circuit conditions)
-
EE3113_L14 4
Basic configuration
11
| 01
111 2 portatwavepowerincident
portatwavepowerreflectedab
S a == =
12
| 01
221 2 portatwavepowerincident
portatwavepowerdtransmitteab
S a == =
22
| 02
222 1 portatwavepowerincident
portatwavepowerreflectedab
S a == =
21
| 02
11 portatwavepowerincident
portatwavepowerdtransmitteab
S a == =
-
EE3113_L14 5
Set-up for measuring S-parameters
Properly terminated output
Properly terminated input side
Load impedance =line impedance
input impedance =line impedance
-
EE3113_L15 1
EE 3113INTRODUCTION TO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 15Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L15 2
Scattering parameters There is a need to establish well-defined
termination conditions in order to find thenetwork descriptions for Z, Y , h, andABCD networks
Open and short voltage and currentconditions are difficult to enforce
RF implies forward and backward travelingwaves which can form standing wavesdestroying the elements
-
EE3113_L15 3
Solution: S-parameters
Input-output behavior of network is definedin terms of normalized power waves
Ratio of the power waves are recorded interms of so-called scattering parameters
S-parameters are measured based onproperly terminated transmission lines (andnot open/short circuit conditions)
-
EE3113_L15 4
Measurements of ScatteringParameters
01
111 2
| == aab
S
01
221 2
| == aab
S
02
222 1
| == aab
S
02
112 1
| == aab
S
Require proper terminationon port 2
Require proper terminationon port 1
-
EE3113_L15 5
Arrangement for measuring S-parameters
Properly terminated port 2 in order to makeS11 and S21 measurements
Properly terminated port 1 in order to makeS22 and S12 measurements
Load impedance =line impedance
input impedance =line impedance
-
EE3113_L15 6
Example: S-parameters of T-network
Port 1 measurements Port 2 measurements
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EE3113_L16 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 16Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L16 2
Working with S-parameters For network computations it is easier to
convert from the S-matrix representation tothe chain scattering matrix notation
=
2
1
2221
1211
2
1
a
a
SS
SS
b
b
=
2
2
2221
1211
1
1
a
b
TT
TT
b
a
.,,1 2111212111 etcSSTST ==
-
EE3113_L16 3
Advantage: cascading just like in the ABCDform
=
B
B
BB
BB
AA
AA
A
A
a
b
TT
TT
TT
TT
b
a
2
2
2221
1211
2221
1211
1
1
-
EE3113_L16 4
Signal flow chart computations
Complicated networks can be efficiently analyzed in amanner identical to signals and systems and control.
in general
-
EE3113_L16 5
Arrangement for flow-chart analysis
GG
S VZZ
Zb
0
0
+=
-
EE3113_L16 6
Analysis of most common circuit
Sba1
Determination ofthe ratio
-
EE3113_L16 7
Important issue: what happens to the S11 parameter ifport 2 is not properly terminated?
LL
in SSS
Sab
GG-
+==G22
211211
1
1
1
Note: Only GL = 0 ensures that the S11 can be measured!
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EE3113_L17 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 17Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L17 2
Semiconductor fundamentals Basic semiconductor lattice structure
intrinsic, n and p-type semiconductors
-
EE3113_L17 3
Space charge formation across the pn junction
Behavior of junction due to an applied voltage
-
EE3113_L17 4
Voltage-Current response of conventional diode
V-I current behavior
II =
)1( /0 -= TAVVeII
-
EE3113_L17 5
Key terms
Concentrations (nn, np, pp, pn) Band model, Fermi energy Barrier voltage Space charge, junction capacitance reverse and forward biasing V/I characteristics
-
EE3113_L18 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 18Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA
Copyright, 1998 R. Ludwig
-
EE3113_L18 2
RF diodes Key equation: Schottky diode equation
diode analysis involves typically threemodels: DC model small signal ac model small signal RF model
)1(0 -= TA
nVV
eII
-
EE3113_L18 3
Small signal ac model
Dynamic resistance (junction resistance)
dQ
dQA
iII
vVV
+=
+=
0IInV
RQ
Tj +
=
Note: dynamic resistance depends on biasand is strongly temperature dependent
-
EE3113_L18 4
Small signal RF model
Schottky diode metal-n semiconductor majority carrier only
+
+
+
Applications: mixers, detectors, rectifiers
+ -
-
EE3113_L18 5
PIN diode
P+ N+I
Intrinsic (low doped) n, p, orpn layer
Major property: Variable resistance behavior
Forward bias:
Isolation state
Zero bias:
Insertion loss state
Applications: attenuating, switching, modulating, limiting,and phase shifting.
-
EE3113_L18 6
Electric behavior of PIN diode
Forward bias (low resistance)Reverse bias (capacitance)
Circuitapplication
-
EE3113_L19 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 19Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L19 2
RF Transistors BJT: low noise, linear power amplification,
power applications (bipolar operation)
GaAs FET: very low noise, low power (monopolar operation)
HEMT (High electron mobility transistor): very high frequency (f > 20 GHz) (electron gas)
-
EE3113_L19 3
Major issue when dealing with RF transistors: NOISE shot noise in emitter-base shot noise in collector system thermal noise in base resistance
How to reduce noise minimize current flow across pn-junction minimize resistance
Solution Finger structure of base, emitter configuration
-
EE3113_L19 4
Finger structure of BJT
Cross-sectional view
Top-down view
(see pp. 325 - 342)
-
EE3113_L19 5
Functionality of BJT
-
EE3113_L19 6
Structure of MESFET
(see pp. 344 - 355)
Functionality of MESFET
-
EE3113_L19 7
Modeling efforts for BJT Non-linear models
Ebers-Moll Gummel-Poon
Low frequency h-parameter model
High frequency modified hybrid model with additional Cbeand
Cbecapacitances Miller effect is used to convert Cbc into input
capacitance
-
EE3113_L19 8
Non-linear BJT model
Dynamic Ebers-Moll chip model
RBLB C
E
Cbe
Ebers-Moll Model
RCL
REL
Cce
Cbc
LBL LCL
LEL
RF model with parasitic effects
See notes, pp. 374 - 397
-
EE3113_L19 9
Small-signal BJT model
Hybrid-PI Ebers-Moll model (p. 384)
RF-model (p. 387)
Decoupled RF-model (Miller effect)
-
EE3113_L19 10
Performance analysis
Hybrid Pi-model (see p. 392)
-
EE3113_L19 11
S-parameter modeling of BJT
Hybrid model can be converted into S-parameters via input/output impedance and voltage relations
S-parameters can also be directly measured for certain biasing and operating frequency condition (values are provided by manufacturer)
-
EE3113_L19 12
How to measure S-parameters?
Vector voltmeter
Dual directional coupler
-
EE3113_L19 13
Alternative: Network analyzer with S-parameter test set
-
EE3113_L20 1
EE 3113INTRODUCTION TO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 20Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L20 2
Matching Networks
MNsare critical for at least two critical reasons maximize power transfer: minimize
Primary goal of a MN is to achieve
0=Gin
)||1( 2inirit PPPP G-=-=
||1||1
in
inSWRG-G+
=
-
EE3113_L20 3
MN strategy
Pick an appropriate two-element MN for which matching is possible (based on a given load impedance or S-parameter)
find the L, C values from the ZY Smith Chart
convert discrete values into equivalent microstriplines
-
EE3113_L20 4
Region of matching for shunt L, series C matching network
-
EE3113_L20 5
Region of matching for series C shunt L matching network
-
EE3113_L20 6
Region of matching for series L shunt C matching network
-
EE3113_L20 7
Region of matching for shunt C and series L matching network
-
EE3113_L21 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 21Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L21 2
There are two strategies
A) Source impedance -> conjugate complex load impedance
B) Load impedance -> conjugate complex source impedance
-
EE3113_L21 3
A) General two-element approach
0.2
0.2
0.2
0.50.5
0.5
1.01.0
1.0
2.02.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.0
2.00.2
0.2
0.2
0.5
0.5
0.5
1.0
1.0
1.0
2.0
2.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.0
2.0
2.0
2.0
0.2
0.2
0.2
0.50.5
0.5
1.01.0
1.0
2.02.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.0
2.0
0010
-10
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
0.
01
0.01
0.02
0.02
0.03
0.03
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.19
0.19
0.2
0.20.21
0.210.22
0.220.23
0.23
0.24
0.24
0.25
0.25
0.26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
0.0
0.0
zS
zL*
A D
B C
Source impedance transformation to conj. comp. load
-
EE3113_L21 4
B) Load impedance to conjugate complex source impedance ( )W== 50*SS ZZ
ZLZS
ZLZSZLZS
ZLZS
(c)
(b)
0.2
0.2
0.2
0.50.5
0.5
1.0
1.0
1.0
2.02.0
2.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.0
2.0
0.2
0.2
0.2
0.5
0.5
0.5
1.0
1.0
1.0
2.0
2.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.0
2.0
2.0
2.0
0.2
0.2
0.2
0.50.5
0.5
1.0
1.0
1.0
2.02.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.0
2.0
0.2
0.2
0.2
0.50.5
0.5
1.01.0
1.0
2.02.0
2.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.02.0
0.2
0.2
0.2
0.5
0.5
0.5
1.0
1.0
1.0
2.0
2.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.0
2.0
2.0
2.0
0.2
0.2
0.2
0.50.5
0.5
1.01.0
1.0
2.02.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.02.0
0.2
0.2
0.2
0.50.5
0.5
1.0
1.0
1.0
2.02.0
2.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.0
2.0
0.2
0.2
0.2
0.5
0.5
0.5
1.0
1.0
1.0
2.0
2.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.0
2.0
2.0
2.0
0.2
0.2
0.2
0.50.5
0.5
1.0
1.0
1.0
2.02.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.0
2.0
0.2
0.2
0.2
0.50.5
0.5
1.0
1.0
1.0
2.02.0
2.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.0
2.0
0.2
0.2
0.2
0.5
0.5
0.5
1.0
1.0
1.0
2.0
2.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.0
2.0
2.0
2.0
0.2
0.2
0.2
0.50.5
0.5
1.0
1.0
1.0
2.02.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.0
2.0
(a)
(d)
-
EE3113_L21 5
The art of designing MNs
0.2
0.2
0.2
0.50.5
0.5
1.01.0
1.0
2.02.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.02.0
0.2
0.2
0.2
0.5
0.5
0.5
1.0
1.0
1.0
2.0
2.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.0
2.0
2.0
2.0
0.2
0.2
0.2
0.50.5
0.5
1.01.0
1.0
2.02.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.02.0
zS
zL
A
B
VS
RS=50W
RL
CL
L nH=10
C pF=2.6Vout
VS
RS=50W
RL
CLL nH=9.75
C pF=0.6
Vout
Frequency , GHzf
Inpu
t ref
lect
ion
coef
fici
ent |
|G Circuit in
Figure 8-8(b)
Circuit inFigure 8-8(c)
(b)
(c)
Frequency , GHzf
Tran
sfer
func
tion
, dB
H
Circuit inFigure 8-8(b)
Circuit inFigure 8-8(c)
2.16.1 jzL +=
2.16.1 jzL +=
f = 1GHz
Z0= 50 Ohm
-
EE3113_L21 6
More complicated networks
Three-element Pi and T networks permit the matching of almost any load conditions
Added element has the advantage of more flexibility in the design process (fine tuning)
Provides quality factor design (see Ex. 8.4)
0.2
0.2
0.2
0.50.5
0.5
1.01.0
1.0
2.02.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.0
2.0
0.2
0.2
0.2
0.5
0.5
0.5
1.0
1.0
1.0
2.0
2.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.0
2.0
2.0
2.0
0.2
0.2
0.2
0.50.5
0.5
1.01.0
1.0
2.02.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.0
2.0
Qn=2
zL
B
A
zin Qn=2
-
EE3113_L21 7
MN realizations in microstripline
TL1TL2TL3
C1C2 ZL
Zin
Distributed microstip lines and lumped capacitors
less susceptible to parasitics
easy to tune
efficient PCB implementation
small size for high frequency
-
EE3113_L21 8
Microstip line procedure
0.2
0.2
0.2
0.50.5
0.5
1.01.0
1.0
2.02.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.0
2.0
0.2
0.2
0.2
0.5
0.5
0.5
1.0
1.0
1.0
2.0
2.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.0
2.0
2.0
2.0
0.2
0.2
0.2
0.50.5
0.5
1.01.0
1.0
2.02.0
2.0
5.0
5.0
5.0
0.5
0.5
0.5
0.5
2.02.0
2.0
2.0
0010
-10
20-20
30-30
40-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
180
0.
01
0.01
0.02
0.02
0.03
0.03
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.190.19
0.2
0.20.21
0.210.22
0.220.23
0.23
0.24
0.24
0.2
5
0.2
5
0. 26
0.26
0.27
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49
0.49
0.0
0.0
zin
zL
A
B
l 1=0.0
55l
l2=0.26l
l1=0.055l
C1=4.37pF ZL
Zin
l2=0.26l
W+= )1030( jZL W+= )8060( jZin
-
EE3113_L22 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 22Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L22 2
Microstripline Matching Networks
Most commonly used in RF circuits Can be used up to approximately 20 GHz (for
TEM modes) Microstrip lines require typically 6 parameters
dielectric constant eer PCB board height h, strip width w, thickness t resistivity rr and loss tangent d
-
EE3113_L22 3
Key parameter designations
er , d
wt
h
r
???0 =++
=CjGLjR
ZwwDont use:
-
EE3113_L22 4
Please keep in mind, there are two issues A) phase velocity and B) characteristic impedance:
effrp
ccv
ee=
1/)41
8ln(60
00 += hwhw
wh
ZZeffe
1/)444.1/ln(667.0393.1/
/120600 +++
= hwhwhw
Z eff
eff
ep
e
Numerical evaluation (Manuscript, page 70):
Z0(er)=F1(w/h) and eeff(er)=F2(w/h)
-
EE3113_L22 5
Microstriplines have two sources of losses
zcdeZ
VzP )(2
0
2||21
)( aa +-+
+ =
Dielectric losses ad (which are typically small)
and
Conduction losses ac (which can be significant)
Depending on frequency, one may have to deal with radiation losses as well!
-
EE3113_L22 6
Classes of amplifier operationIC
VBECut-off region
Quiescentpoint
V*
Linear regionIdeal transferfunction
Input waveform
Output waveform
IC
VBE
Quiescentpoint
Q0=180o
IC
VBE
Quiescentpoint
IC
VBE
Quiescentpoint
Class AClass B
Class AB Class C
-
EE3113_L22 7
Efficiency of an amplifier
%100S
RF
PP
PowerSourcePowerRF
==h
ILI0
0
Q0
Qp 2p 3p
Current through load
ISIQ+I0
0Q0
Qp 2p 3p
IQ
Q0/2
Current from the power supply
-
EE3113_L22 8
)]2/sin(2)2/cos([2sin
000
00
Q-QQQ-Q
-=h
Conduction angle, Qo
Eff
icie
ncy
, %h h=78.5%
Q=180o
Class B
Class A
Class AB
Class C
-
EE3113_L23 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 23Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L23 2
Biasing networks Biasing networks are needed to set appropriate
operating conditions for active devicesThere are two types: Passive biasing (or self-biasing)
resistive networks drawback: poor temperature stability
Active biasing additional active components (thermally coupled) drawback: complexity, added power consumption
-
EE3113_L23 3
Passive biasingVCC
R1
RFCR2
IBI1
RFOUT
RFIN
IC
RFC
CB
CB
Simple two element biasing
blocking capacitors CBand RFCs to isolate RF path
Very sensitive to collector current variations
-
EE3113_L23 4
Passive biasingVCC
R1
RFCR2
IBRFOUT
RFIN
IC
RFCR3
R4
IX
VX
CB
CB
Voltage divider to stabilize VBE
Freedom to choose suitable voltage and current settings (Vx, Ix)
Higher component count, more noise susceptibility
IB~10 IX
-
EE3113_L23 5
Active biasingVCC
RFCRC1
RFOUT
RFIN
RFC
VC1Q2
Q1
I1
IB1
IB1
IC2
RB1 RB2
RE1
RC2
IC1
CB
CB
Base current of RF BJT (Q2) is provided by low-frequency BJT Q1
Excellent temperature stability (shared heat sink)
high component count, more complex layout
-
EE3113_L23 6
Active biasing in common base
VCC
RFC
RC1
RFOUT
RFINRFC
VC1Q2
Q1
I1
IB1
IB1
IC2
RB1 RB2
RE1
RC2
IC1
CB
CB
RFC
VCC
RFC
RC1
RFCQ2
Q1
RB1 RB2
RE1
RC2
CB
CB
RFC
VCC
RFC
RC1
RFOUT
RFINRFCQ2
Q1
RB1 RB2
RE1
RC2
CB
CB
RFC
DC path
RF path
-
EE3113_L23 7
FET biasingVDVG
CB
RFC
CB
RFC
RFOUTRFIN
VD
VS
CB
CB
RFC
RFOUTRFIN
RFCRFC
VD
RSCB
CB
RFC
RFOUTRFIN
RFC
Bi-polar power supply
Uni-polar power supply
VG0
-
EE3113_L24 1
EE 3113INTRODUCTION TO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 24Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L24 2
RF Amplifier Objective:Design a complete class A,
single-stage RF amplifier operated at 1 GHz which includes biasing, matching networks, and RF/DC isolation.
MN1 MN2BJTRFin RFout
biasing
-
EE3113_L24 3
Strategy Design DC biasing conditions Select S-parameters for given bias and
operating frequency Build input and output matching networks
for desired frequency response include RF/DC isolation simulate amplifier performance on the
computer (OptoteksMMICAD or HPsLibra package)
-
EE3113_L24 4
Overall approach
PA RFsource
DC bias
InputMatching Network (IMN)
OutputMatching Network (IMN)
LoadPL
GS GL
Gin Gout
PA PL
GS GL
Gin Gout
[ ]SZS
VSZL
b1` a1 b2b1` a2`
b2 a2a1` b2`
ZS
VSZL
b1`b1`
a1`
GS
Gin
For power considerations, matching networks are assumed lossless
-
EE3113_L24 5
Power Relations
222
2
2221
2
|1||1|)||1(||)||1(
LinS
SLT S
SG
G-GG-G-G-
=
Transducer Power Gain
Available Power Gain ( )*outL G=G
211
2
221
2
|1||||1|||)||1(
Sout
SA S
SG
G-G-G-
=
Operating Power Gain ( )*inS G=G
222
2
221
2
|1||||1|||)||1(
Lin
L
SS
GG-G-
G-=
-
EE3113_L25 1
EE 3113INTRODUCTION TO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 25Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
-
EE3113_L25 2
Stability of active device
1||,1||
-
EE3113_L25 3
Constant Gain AmplifierG L` =0
Z0
GSVS G0( =0)S12
GL ZL
G S` =0 GS GL
Gin=S11 Gout=S22
222
22
21211
2
|1|||1
|||1|
||1
L
L
S
STU S
SS
GG-
G-
G-G-
=
)()()()( 0 dBGdBGdBGdBG LSTU ++=
-
EE3113_L25 4
Constant gain circles in the SC
211
max ||11S
GS -=
222
max ||11S
GL -=
)||1(|1|
||1 2112
11
2
max
SSG
Gg
S
S
S
SS -G-
G-==
)||1(|1|
||1 2222
22
2
max
SSG
Gg
L
L
L
LL -G-
G-==
normalize
)||1(|1|
||1 22
2
maxii
iii
i
i
ii SSG
Gg -
G-G-
== This can be written as a circle equation
(see page 511)
-
EE3113_L25 5
Circle equation and graphical display222 )()(iii g
Ig
Ii
Rg
Ri rdd =-G+-G
)1(||1 2*
iii
iiig gS
Sgd
i --=
)1(||1
)||1(12
2
iii
iiig gS
Sgr
i ----
=
-1dB0dB
1dB
2dB2.6dB
S11*
GL=0.49dB
GL
S22*
Constant source gain circlesConstant load gain circle
See Ex. 9.7 (p. 512)
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EE3113_L25 6
Trade-off between gain and noise
0.2
0.5
1.0
2.0
5.0
+0.2
-0.2
+0.5
-0.5
+1.0
-1.0
+2.0
-2.0
+5.0
-5.0
0.0
Fk=1.6dB
G=8dB
VS W Rin
=2
Maximum gain and minimum noise figure are mutually exclusive
0 50 100 150 200 250 300 3501.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
Input and output VSWR as a function of GS position
Angle a , deg.
Input
and o
utp
ut
VSW
Rs
VSWRout
VSWRin
Noise figure
Constant gain
-
EE3113_L26 1
EE 3113INTRODUCTION INTO RF
CIRCUIT DESIGN
Lecture Notes for A-term 1999LECTURE 26Prof. R. Ludwig
Department of Electrical and Computer EngineeringWorcester Polytechnic Institute
Worcester, MA 01609copyright 1999, R. Ludwig
Copyright, 1998 R. Ludwig
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EE3113_L26 2
RF Amplifier Design Project
Amplifier design passive bias network stability analysis of BJT (k-factor) no feed-back (S12=0) -> unilateral design class A (low-efficiency) BJT configuration S-parameter description (given bias, frequency) discrete, two-element matching networks
-
EE3113_L26 3
Results of constant gain analysis0.2
0.5
1.0
2.0
5.0
+0.2
-0.2
+0.5
-0.5
+1.0
-1.0
+2.0
-2.0
+5.0
-5.0
0.0
-1dB
0dB0.2dB
GGS opt
= S11*
0.2
0.5
1.0
2.0
5.0
+0.2
-0.2
+0.5
-0.5
+1.0
-1.0
+2.0
-2.0
+5.0
-5.0
0.0
-1dB
0dB
0.5dB
GGL opt
= S22*
Input gain circles Output gain circles
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EE3113_L26 4
Required Matching Networks0.2
0.5
1.0
2.0
5.0
+0.2
-0.2
+0.5
-0.5
+1.0
-1.0
+2.0
-2.0
+5.0
-5.0
0.0
0.2dB
0.2
0.5
1.0
2.0
5.0
+0.2
-0.2
+0.5
-0.5
+1.0
-1.0
+2.0
-2.0
+5.0
-5.0
0.0
0.5dB
Input matching network Output matching network
L, C elements
-
EE3113_L26 5
What to do next?
Additional design improvements multi-stage (dual) configuration microstriplinerealization triple and higher order matching network electric circuit transistor model with feed-back
(bilateral design approach)
-
EE3113_L26 6
Most recent MQP project
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EE3113_L26 7
Actual realization
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EE3113_L26 8
Circuit board cutter
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EE3113_L26 9
EE 3113 COURSE GOALS MATCHING NETWORK STRATEGY
input matching conjugate complex output matching
TRANSISTOR & DIODE CHARACTERIZATION S-parameter description (supplied by manufacturer) small signal, RF circuit model (BJT, GaAs FET, PIN,
Schottkydiode) DC biasing
NETWORK DESCRIPTION two-port model (S-parameter, Z, Y, hybrid, etc.)
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EE3113_L26 10
SIGNAL FLOW CHART MODELING S-PARAMETER THEORY
definitions (based on matched port terminations) measurements with network analyzer
SMITH CHART reflection coefficient/impedance representation impedance transformation as a function of either
length or frequency
TRANSMISSION LINE FUNDAMENTALS distributed circuit theory loaded and source transmission line
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