EE194RF_L1 2 EE 194 RF: 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!
Cours Hyperfrequences
Oct 28, 2014
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
EE194RF_L1 2
EE 194 RF: 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!
EE194RF_L1 3
What do we mean by going from lumped to distributed theory?
• Example: INDUCTOR
Low-frequency
(lumped)
LjRZ ω+=
High-frequency
Z = ?
EE194RF_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
y
0
2
4
6
z
-1-0.5
00.5
1x
E (or V) and H (or I) fields
EE194RF_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
EE194RF_L1 6
Design FocusCell phone transceiver circuit
Typical frequencyrange:
• 950 MHz
• 1.9 GHz
EE194RF_L1 7
Implementation
• matching networks
• BJT/FET active devices
• biasing circuits
• printed circuit board
• mircostripline realization
• surface mount technology
EE194RF_L2 2
RF Behavior of Passive Components
• Conventional circuit analysis– R is frequency independent
– Ideal inductor:
– Ideal capacitor:
• Evaluation– Impedance chart
LjX L ω=
CXC ω1=
EE194RF_L2 3
Impedance Chart(impedance of C & L vs frequency)
ZC=1/(2πfC)
ZL=2πfL
EE194RF_L2 4
How does a wire behave at high frequency?
• Example: Resistorσπ 2a
lRDC =
δ2/
aRR DC =
δω
2/
aRL DC =
µσπδ
f
1=
High frequency results in skin-effect whereby current flow ispushed to the outside
EE194RF_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
EE194RF_L2 6
Impedance Measurement ExampleCapacitor going through resonance
CapacitorCharacteristics
EE194RF_L2 7
Equivalent Circuit Analysis
EEE194RF_L3 194 2
Transmission Line Analysis
• Propagating electric field
• Phase velocity
• Traveling voltage wave
)cos(0 kztEE XX −= ω
Time factor
Space factor
r
p
cfv
εεµλ ===
1
k
kztEtzV X
)sin(),( 0
−=
ω
EEE194RF_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
EEE194RF_L3 4
Generic way to measure spatial voltage variations
• For low frequency (1MHz)Kirchhoff’s laws apply
• For high frequency (1GHz)Kirchhoff’s laws do notapply anymore
EEE194RF_L3 5
Kirchhoff’s 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
EEE194RF_L3 6
Example of transmission line: Two-wire line
• Alternating electric fieldbetween conductors
• alternating magnetic fieldsurrounding conductors
• dielectric medium tendsto confine field insidematerial
EEE194RF_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
EEE194RF_L3 8
Example of transmission line: Microstip line
Cross-sectional view
Low dielectric medium High dielectric medium
EEE194RF_L3 9
Triple-layer transmission line
Conductor is completely shielded between twoground planes
Cross-sectional view
EEE194RF_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.
EEE194RF_L4 3
Kirchhoff’s 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
EEE194RF_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
EEE194RF_L4 5
)()()(
))()(
( zILjRdz
zdV
z
zVzzVLim ω+=−=
∆−∆+
−
Derivation of differential transmission line form
)()()()( zzVzzILjRzV ∆++∆+= ωKVL:
KCL:)()()()( zzIzzzVCjGzI ∆++∆+∆+= ω
)()()(
zVCjGdz
zdIω+=−
CoupledDE
EEE194RF_L4 6
Traveling Voltage and Current Waves
0)()( 2
2
2
=− zVkdz
zVd
where
))(( CjGLjRjkkk ir ωω ++=+=
kzkz eVeVzV +−−+ +=)( kzkz eIeIzI +−−+ +=)(
0)()( 2
2
2
=− zIkdz
zId
Left traveling wave
Right traveling wavePhasor expressions
EEE194RF_L4 7
General line impedance definition
)()(
)( kzkz eVeVLjR
kzI +−−+ −
+=
ω
−
−
+
+
−==++
=I
V
I
V
CjG
LjRZ
)()(
0 ωω
?
Characteristic line impedance
EEE194RF_L5 2
Lossless Transmission Line Model
• Line representation
)()(
0 CjG
LjRZ
ωω
++
=Characteristic impedance:
Note: R, L, G, C are given per unit length and depend on geometry
Lossless implies:R = G = 0!
EEE194RF_L5 3
Transmission Line Parameters for different line types
2-wire coax
σδπa
1
)2
(1
a
Dch−
πµ
R
L
G
C
)11
(2
1ba
+πσδ
parallel-plate
))2/((1 aDch−
πσ
))2/((1 aDch−
πε
σδw
2
w
dµ
d
wσ
d
wε
)ln(2 a
b
πµ
)/ln(
2
ab
πσ
)/ln(
2
ab
πε
EEE194RF_L5 4
Microstrip line
1/),4
8ln(2
/ 000 <+= hW
h
W
W
hZ
effεπεµ
])/1(04.0)/121[(2
1
2
1 22/1 hWWhrreff −++
−+
+= −εε
ε
EEE194RF_L5 5
What is a voltage reflection coefficient?
0
00 ZZ
ZZ
L
L
+−
=ΓReflection coefficientat the load location
)(10 ∞→=Γ LZ
)0(10 →−=Γ LZ
EEE194RF_L5 6
Standing Waves
)()( djdj eeVdV ββ −++ −=
)2/cos()sin(2),( πωβ += + tdVtdv
EEE194RF_L5 7
Standing wave ratio
||1
||1
||
||
||
||
0
0
min
max
min
max
Γ−Γ+
===I
I
V
VSWR
SWR is a measure of mismatch of theload to the line
SWR=1 (matched) or SWR ∞→ (total mismatch)
match
EEE194RF_L6 2
Special Termination Conditions
• Lossless transmission line
C
LZ =0
)tan(
)tan()(
0
00 djZZ
djZZZdZ
L
Lin β
β++
=
Characteristic impedance
EEE194RF_L6 3
Input impedance of short circuit transmission line
)tan()( 0 djZdZin β=Impedance
Voltage:
)sin(2)( djVdV β+=
Current:
)cos(2
)(0
dZ
VdI β
+
=
EEE194RF_L6 4
Input impedance of open circuit transmission line
Voltage:
Current:
Impedance
)cos(2)( dVdV β+=
)sin(2
)(0
dZ
jVdI β
+
=
)cot()( 0 djZdZin β−=
EEE194RF_L6 5
Quarter-wave transmission line
LL
Lin Z
Z
jZZ
jZZZZ
20
0
00 )4/tan(
)4/tan()4/( =
++
=βλβλ
λ
Quarter-wave transformer model:
given input and output impedances
Predict lineimpedance
inLZZZ =0
EEE194RF_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 87Matching works only forparticular frequencies
500 MHz 1.5 GHz
EEE194RF_L7 2
Sourced and Loaded Transmission Lines• Lossless transmission line with source
)()1(Gin
inGininin ZZ
ZVVV
+=Γ+= +
Voltage at the beginning of the transmission line iscomposed of an incident and reflected component!
0
0
ZZ
ZZ
G
G
+−
=0
0
ZZ
ZZ
L
L
+−
=
EEE194RF_L7 3
Power considerations
Re2
1 *ininin IVP =
)1( ininin VV Γ+= + )1(0
inin
in Z
VI Γ−=
+
)||1(||
2
1 2
0
2
inin
in Z
VP Γ−=
+
)||1(|1|
|1|||
8
1 22
2
0
2
ininS
SGin Z
VP Γ−
ΓΓ−Γ−
=
EEE194RF_L7 4
Two special cases:
Load and sourcematched line 00 =Γ=Γ S
0
2||
8
1
Z
VP G
in =
Mismatch at source,but match at load 00 =Γ 2
0
2
|1|||
8
1S
Gin Z
VP Γ−=
How to measure power?mW
WPdBmP
1
][log10][ =
EEE194RF_L7 5
Return and insertion losses
Return loss: ||log20||log10)log(10 2inin
i
r
P
PRL Γ−=Γ−=−= [dB]
Insertion loss: )||1log(10)log(10)log(10 2in
i
ri
i
t
P
PP
P
PIL Γ−−=
−−=−= [dB]
No reflection Full reflection
0
∞ dB
∞1
0 dB
1ÃRL
SWR
EEE194RF_L8 2
From Reflection Coefficient to LoadImpedance (Smith Chart)
• Reflection coefficient in phasor form
Ljir
L
L ejZZ
ZZ θ|| 0000
00 Γ=Γ+Γ=
+−
=Γ
The load reflectioncoefficient is identified inthe complex domain
0Γ
EEE194RF_L8 3
Normalized impedance
ir
irinin j
j
d
djxrzZdZ
Γ−Γ−Γ+Γ+
=Γ−Γ+
=+==1
1
)(1
)(1/)( 0
irdjj jeed L Γ+Γ=Γ=Γ − βθ 2
0 ||)(
22
22
)1(
1
ir
irrΓ+Γ−
Γ−Γ−=
22)1(
2
ir
ixΓ+Γ−
Γ=
Real part of normalizedimpedance
Imaginary part ofnormalized impedance
EEE194RF_L8 4
Inversion of complex reflection coefficient(constant normalized resistance)
222 )1
1()
1(
+=Γ+
+−Γ
rr
rir
EEE194RF_L8 5
Inversion of complex reflection coefficient(constant normalized reactance)
222 )1
()1
()1(xxir =−Γ+−Γ
EEE194RF_L8 6
Combined display: Smith Chart
EEE194RF_L9 2
Impedance Transformation(Smith Chart)
• Reflection coefficient in phasor form
Ljir
L
L ejZZ
ZZ θ|| 0000
00 Γ=Γ+Γ=
+−
=Γ
0Γ
ir
irinin j
j
d
djxrzZdZ
Γ−Γ−Γ+Γ+
=Γ−Γ+
=+==1
1
)(1
)(1/)( 0
EEE194RF_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 →
0Γ→Lz)(0 dΓ→Γ
)(dzin
)()( dZdz inin →
EEE194RF_L9 4
Graphical display
EEE194RF_L9 5
How to create ideal capacitors and inductors with atransmission line?
Start oftransformation
Capacitivedomain
Inductivedomain
EEE194RF_L9 6
Start oftransformation
EEE194RF_L10 2
Admittance Transformation(Smith Chart)
• impedance representation in Smith Chart
0Γ
)(1)(1
d
djxrzin Γ−
Γ+=+=
• admittance representation in Smith Chart
)(1
)(1
)(1
)(11
0 de
de
d
d
zY
Yy
j
j
in
inin Γ−
Γ+≡
Γ+Γ−
=== −
−
π
π
180 degreephase shift
EEE194RF_L10 3
Transformation21
21
11 jyjz inin −=→+=
EEE194RF_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.
EEE194RF_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.
EEE194RF_L11 2
Parallel Connection of R and L Elements(Smith Chart)
• parallel connection of R and L elements
0
1)(
LYjgy
LLin ω
ω −=
EEE194RF_L11 3
• Parallel connection of R and C elements
CjZgy LLin ωω 0)( +=
EEE194RF_L11 4
• Series connection of R and L elements
0
)(Z
Ljrz L
Lin
ωω +=
EEE194RF_L11 5
• Series connection of R and C elements
0
1)(
CZjrz
LLin ω
ω −=
EEE194RF_L11 6
Practical case: BJT connected viaa T-network
EEE194RF_L12 2
Single and Multi-Port Networks
• basic current and voltage definitions definitions
EEE194RF_L12 3
• Impedance and admittance networks
][ IZV = ][ VYI =
]][[ IYZV =
][][ 1 ZY =−
EEE194RF_L12 4
• Example Z-representation of Pi-network
+
+++
=)(
)(1][
PBPAPCPCPA
PCPAPCPBPA
PCPBPA ZZZZZ
ZZZZZ
ZZZZ
)(0| mkim
nnm ki
vz ≠==
EEE194RF_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)
EEE194RF_L13 2
Interconnecting Networks
• Certain networks are more advantageous tointerconnect.
Example: series connection
]"[]'[][ ZZZ +=
EEE194RF_L13 3
•Hybrid representation
]"[]'[][ hhh +=
Typical example
EEE194RF_L13 4
ABCD parameter representation
• Very useful when cascading networks
−
=
2
2
1
1
"
"
""
""
''
''
i
v
DC
BA
DC
BA
i
v
EEE194RF_L13 5
ABCD network is very useful for transmission linerepresentations
=
)cos(
)sin()sin()cos(
0
0
lZ
lj
ljZl
DC
BAβ
βββ
Example:
EEE194RF_L14 2
Scattering parameters
• There is a need to establish well-definedtermination 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
EEE194RF_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)
EEE194RF_L14 4
Basic configuration
1
1| 0
1
111 2 portatwavepowerincident
portatwavepowerreflected
a
bS a == =
1
2| 0
1
221 2 portatwavepowerincident
portatwavepowerdtransmitte
a
bS a == =
2
2| 0
2
222 1 portatwavepowerincident
portatwavepowerreflected
a
bS a == =
2
1| 0
2
11 portatwavepowerincident
portatwavepowerdtransmitte
a
bS a == =
EEE194RF_L14 5
Set-up for measuring S-parameters
• Properly terminated output
• Properly terminated input side
Load impedance =line impedance
input impedance =line impedance
EEE194RF_L15 2
Scattering parameters
• There is a need to establish well-definedtermination 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
EEE194RF_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)
EEE194RF_L15 4
Measurements of ScatteringParameters
01
111 2
| == aa
bS
01
221 2
| == aa
bS
02
222 1
| == aa
bS
02
112 1
| == aa
bS
Require proper terminationon port 2
Require proper terminationon port 1
EEE194RF_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
EEE194RF_L15 6
Example: S-parameters of T-network
Port 1 measurements Port 2 measurements
EEE194RF_L16 2
Working with S-parameters
• For network computations it is easier toconvert 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 ==
EEE194RF_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
EEE194RF_L16 4
Signal flow chart computations
Complicated networks can be efficiently analyzed in amanner identical to signals and systems and control.
in general
EEE194RF_L16 5
Arrangement for flow-chart analysis
GG
S VZZ
Zb
0
0
+=
EEE194RF_L16 6
Analysis of most common circuit
Sba1
Determination ofthe ratio
EEE194RF_L16 7
Important issue: what happens to the S11 parameter ifport 2 is not properly terminated?
LL
in S
SSS
a
bΓ
Γ−+==Γ
22
211211
1
1
1
Note: Only ΓL = 0 ensures that the S11 can be measured!
EEE 194RF_ L17 1
RF Filter Design – Basic Filter Types
EEE 194RF_ L17 2
Filter Attenuation Profiles
EEE 194RF_ L17 3
RF Filter Parameters• Insertion Loss:
• Ripple
• Bandwidth: BW 3dB = fu3dB – fL3dB
• Shape Factor:
• Rejection
( )210 10 1inin
L
PIL log log
P= = − − Γ
min
max
A
A
BWSF
BW=
EEE 194RF_ L17 4
Low-Pass Filter
Cascading four ABCD-networks.
( )
1 01 1 1 01 10 1 0 1 1
11
11
G
L
G G LL
L
A B R RC D j C R
R R j C R RR
j CR
ω
ω
ω
=
+ + + + =
+
EEE 194RF_ L17 5
RF Filter Parameters
( )
1 01 1 1 01 10 1 0 1 1
11
11
G
L
G G LL
L
A B R RC D j C R
R R j C R RR
j CR
ω
ω
ω
=
+ + + + =
+
Cascading four ABCD-networks.
EEE 194RF_ L17 6
Low-Pass Filter Frequency Response
• Frequency Response from the ABCD Definitions:
• So the Transfer Function is Simply:2
1
2 0i
vA
v =
=
( ) ( )1 1
1 G
HA j R R C
ωω
= =+ +
EEE 194RF_ L17 7
Low-Pass Filter Frequency Response
• Corresponding Phase is:
• Group Delay:
( )g
dt
d
φ ωω
=
( ) ( ) ( )
1 Im Htan
Re H
ωφ ω
ω−
=
EEE 194RF_ L17 8
High-Pass Filter
( )
1 01 01 1
11 110 1 0 1
1 11
1 11
G
L
G G LL
L
A B R RC D
Rj L
R R R Rj L R
j L R
ω
ω
ω
=
+ + + +
= +
EEE 194RF_ L17 9
High-Pass Filter Frequency Response• Frequency Response from the ABCD
Definitions:
• So the Transfer Function is Simply:2
1
2 0i
vA
v =
=
( )( )
1 11 1
1 GL
HA
R Rj L R
ω
ω
= =
+ + +
EEE 194RF_ L17 10
High-Pass Filter Frequency Response
• For ω → ∞:
• Inductive Influence Can Be Neglected
( )2 1
1
L
GG L G
L
V RR RV R R R
R
= =+ + +
+
EEE 194RF_ L17 11
Low-Pass Filter Realizations
EEE 194RF_ L17 12
Low-Pass Butterworth Filter Coefficients
EEE 194RF_ L17 13
Low-Pass Butterworth Filter Attenuation
EEE 194RF_ L17 14
Low-Pass Linear-Phase Filter Coefficients
EEE 194RF_ L17 15
Chebyshev-Type Filters
EEE 194RF_ L17 16
Chebyshev-Type Filters
EEE 194RF_ L17 17
Chebyshev-Type Filter Response
Response for 3 dB ripple Chebyshev LPF
EEE 194RF_ L17 18
Chebyshev-Type Filter Response
Response for 0.5 dB ripple Chebyshev LPF
EEE 194RF_ L17 19
Low-Pass Chebysev Filter Coefficients –3 dB Ripple
EEE 194RF_ L17 20
Low-Pass Chebysev Filter Coefficients –0.5 dB Ripple
EEE 194RF_ L17 21
Standard Low-Pass Filter Design
• The normalized inductors and capacitors (g1, g2 , ... , gN ) are denormalized using:
and
where Cn , Ln , are the gn normalized values from the tables
2n
C
CC
f R=
π 2n
C
L RL
f=
π
EEE 194RF_ L18 1
Low-Pass Filter Design Example
• Design a Low-Pass Filter with cut-off frequency of 900 MHz and a stop band attenuation of 18 dB @1.8 GHz.
• From the Butterworth Nomograph, Amax = 1 and Amin = 18. Amax = 1 since unity gain. And the order of the filter is N = 3.
• From Butterworth Tables, g1 = g3=1.0 and g2 = 2.
EEE 194RF_ L18 2
Low-Pass Filter Design Example
• De-Normalized Values For the Tee-Configuration Low-Pass Filter Are:
( )1
1 2 68 8
2 900 10Lg R
L L . nHπ
= = =×
( )2
1 67
2 900 10 L
gC pF
Rπ= =
×
EEE 194RF_ L18 3
Low-Pass Filter Design Example
EEE 194RF_ L18 4
Low- To High-Pass Transformation• Transform the Low-Pass Filter Normalized
Component Values to the Normalized High-Pass Values
• Inductors in Low-Pass Configuration Become Capacitors in High-Pass.
• Capacitors in Low-Pass Configuration Become Inductors in High-Pass
• 1HP _ norm
c LP _ norm
C ;Lω
= 1HP _ norm
c LP _ norm
LCω
=
EEE 194RF_ L18 5
RF Filter Parameters• Insertion Loss:
• Ripple
• Bandwidth: BW 3dB = fu3dB – fL3dB
• Shape Factor:
• Rejection
( )210 10 1inin
L
PIL log log
P= = − − Γ
min
max
A
A
BWSF
BW=
EEE 194RF_ L18 6
De-Normalizing Filter Component Values
• All Normalized Component Values Are De-Normalized Using the Following:
and
normalizedactual
g
CC
R=
actual normalized gL L R=
EEE 194RF_ L18 7
Transformation From Low-Pass Filter
EEE 194RF_ L18 8
Normalized Low- to Band-Pass Filter Transformation
• Normalized Band-Pass Shunt Elements from Shunt Low-Pass Capacitor:
2upper lower
BP _ norm_shunto LP _ norm
LC
ω ω
ω
−=
LP _ normBP _ norm_shunt
upper lower
CC
ω ω=
−
EEE 194RF_ L18 9
Normalized Low- to Band-Pass Filter Transformation
• Normalized Band-Pass Series Elements from Series Low-Pass Inductor:
LP _ normBP _ norm _ series
upper lower
LL
ω ω=
−
2upper lower
BP _ norm_serieso LP _ norm
CL
ω ω
ω
−=
EEE 194RF_ L18 10
Normalized Low- to Band-Stop Filter Transformation
• Normalized Band-Stop Shunt Component Values from Low-Pass Shunt Capacitor:
( )1
Stop _ norm_shuntupper lower LP_norm
LCω ω
=−
( )2
upper lower LP _ normStop_norm_shunt
o
CC
ω ω
ω
−=
EEE 194RF_ L18 11
Normalized Low- to Band-Stop Filter Transformation
• Normalized Band-Stop Series Component Values from Low-Pass Series Inductor:
( )2
upper lower LP_normStop _ norm_series
o
LL
ω ω
ω
−=
( )1
Stop_norm_seriesupper lower LP_norm
CLω ω
=−
EEE 194 RF 1
Stepped Impedance Low-Pass Filter• Relatively easy (believe that?) low-pass
implementation• Uses alternating very high and very low
characteristic impedance lines• Commonly called Hi-Z, Low-Z Filters• Electrical performance inferior to other
implementations so often used for filtering unwanted out-of-band signals
EEE 194 RF 2
Approximate Equivalent Circuits for Short Transmission line Sections
• Using Table 4-1, approximate equivalent circuits for a short length of transmission line with Hi-Z or Low-Z are found
EEE 194 RF 3
Approximate Equivalent Circuits for Short Transmission line Sections
• The equivalent circuits are:jX / 2 jX / 2
jB
XL=Zo βl
BC=Yo βl
T-Equialent circuit for transmission line sectionβ l << π / 2
Equialent circuit for small β l and large Zo
Equialent circuit for small β l and small Zo
EEE 194 RF 4
Approximate Equivalent Circuits for Short Transmission line Sections
• Series inductors of a low-pass prototype replaced with Hi-Z line sections (Zo= Zh)
• Shunt capacitors replaced with Low-Z line sections (Zo= Zl)
• Ratio Zh/Zl should be as high as possible
( )
( )
inductor
capacitor
g
h
l
g
LRl
Z
CZl
R
β
β
=
=
EEE 194 RF 5
Stepped Impedance Low-Pass Filter• Select the highest and lowest practical line
impedance; e.g. the highest and lowest line impedances could be 150 and 10 Ω, respectively
• For example, given the low-pass filter prototype, solve for the lengths of the microstriplines:
glowLn n Cn n
g high
RZl g ; l g
R Zβ β= =
EEE 194 RF 6
6th Order Low-Pass Filter Prototype
Stepped Impedance Implementation
Microstripline Layout of Filter
L1 L2
C2C1 C3
L3
Zo Zlow Zhigh ZoZlow ZlowZhigh Zhigh
l1 l2 l3 l4 l5 l6
Stepped Impedance Low-Pass Filter -Implementation
EEE 194 RF 7
Bandstop Filter• Require either maximum or minimal
impedance at center frequency fo
• Let line lengths l = λo /4• Let Ω = 1 cut-off frequency of the low-
pass prototype transformed into upper and lower cut-off frequencies of bandstopfilter via bandwidth factor :
( )1
2 2 2U LL
o o
sbwbf cot cot ; sbw
ω ωπ ω πω ω
− = = − =
EEE 194 RF 8
Bandstop Filter: Implementation1. Find the low-pass filter prototype2. The L’s and C’s replaced by open and short
circuit stubs, respectively as in Low-Pass filter design with
ZLn = (bf ) gn and YCn = (bf ) gn
3. Unit lengths of λo /4 are inserted and Kuroda’s Identities are used to convert all series stubs into shunt stubs
4. Denormalize the unit elements
EEE 194 RF 9
Coupled Filters: Bandpass• Even and Odd mode excitations resulting in
1 1Oe Oo
pe e po od
Z ; Zv C v C
= =
EEE 194 RF 10
Coupled Filters: Even & Odd Impedances
EEE 194 RF 11
Bandpass Filter Section
( ) ( ) ( ) ( )2 2 212in Oe Oo Oe OoZ Z Z Z Z cos l
sin lβ
β= − − +
EEE 194 RF 12
Bandpass Filter Section• According to Figure 5-47, the characteristic
bandpass filter performance attained when l = λ /4 or β l = π /2 .
EEE 194 RF 13
Bandpass Filter Section• The upper and lower frequencies are
( ) 11 21 2
Oe Oo,,
Oe Oo
Z Zl cos
Z Zβ θ − −
= = ± +
5th Order coupled line Bandpass Filter
EEE 194 RF 14
Bandpass Filter: Implementation1. Find the low-pass filter prototype2. Identify normalized bandwidth, uper, and lower
frequencies
• Allowing:
U L
O
BWω ω
ω−
=
0 1 1 11 11
1 1 12 22, i,i N ,N
O O O O N Ni i
BW BW BWJ ; J ; J
Z g g Z Z g gg gπ π π
+ +++
= = =
EEE 194 RF 15
Bandpass Filter: Implementation• This allows determination of the odd and
even characteristic line impedances:
• Indices i, i+1 refer to the overlapping elements and ZO is impedance at ends of the filter structure
( )
( )
21 11
21 11
1
and
1
Oo O O i,i O i,ii,i
Oe O O i,i O i,ii,i
Z Z Z J Z J
Z Z Z J Z J
+ ++
+ ++
= − +
= + +
EEE 194 RF 16
Bandpass Filter: Implementation• Determine line dimensions and S and W of
the coupled lines from the graph on Figure 5-45 p256.
• Length of each coupled line segment at the center frequency is λ /4.
• Normalized frequency is
c c
U L c
ω ωωω ω ω ω
Ω = − −
EEE 194RF_L19 1
Band-Pass Filter Design Example
Attenuation response of a third-order 3-dB ripple bandpass Chebyshev filter centered at 2.4 GHz. The lower cut-off frequency is f L = 2.16 GHz and the upper cut-off frequency is f U = 2.64 GHz.
EEE 194RF_L19 2
RF/µW Stripline Filters
• Filter components become impractical at frequencies higher than 500 MHz
• Can apply the normalized low pass filter tables for lumped parameter filters tostripline filter design
• Richards Transformation and Kuroda’s Identities are used to convert lumped parameter filter designs to distributed filters
EEE 194RF_L19 3
Richards Transformation: Lumped to Distributed Circuit Design• Open- and short-circuit transmission line
segments emulate inductive and capacitive behavior of discrete components
• Based on: • Set Electrical Length l = λ/8 so
( ) ( )in o oZ jZ tan l jZ tanβ θ= =
4 4o
fl
fπ π
θ β= = = Ω
EEE 194RF_L19 4
Richards Transformation: Lumped to Distributed Circuit Design• Richards Transform is:
and
• For l = λ/8, S = j1 for f = fo = fc
4L o ojX j L jZ tan SZπ
ω = = Ω =
4C o ojB j C jY tan SYπ
ω = = Ω =
EEE 194RF_L19 5
Richards Transformation: Lumped to Distributed Circuit Design
jXL
jBC
L
C
λ/ 8 at ωc
λ/ 8 at ωc
Zo = 1/(jω C)
Zo = jω L
EEE 194RF_L19 6
Unit Elements : UE
• Separation of transmission line elements achieved by using Unit Elements (UEs)
• UE electrical length: θ = πΩ /4• UE Characteristic Impedance ZUE
2
11
11
UE UE
UEUE UE
cos jZ sin j ZA B
j jsin cosC D
Z Z
θ θ
θ θ
Ω = = Ω + Ω
EEE 194RF_L19 7
The Four Kuroda’s Identities
EEE 194RF_L19 8
Kuroda’s Equivalent Circuit
=
l
ll
l
Z2
Z1
Z1 /N
Z2 /N
Short CircuitSeries Stub
Open CircuitShunt Stub
Unit Element Unit Element
EEE 194RF_L19 9
Realizations of Distributed Filters
• Kuroda’s Identities use redundant transmission line sections to achieve practical microwave filter implementations
• Physically separates line stubs • Transforms series stubs to shunt stubs or
vice versa• Change practical characteristic impedances
into realizable ones
EEE 194RF_L19 10
Filter Realization Procedure
• Select normalized filter parameters to meet specifications
• Replace L’s and C’s by λo /8 transmission lines
• Convert series stubs to shunt stubs using Kuroda’s Identities
• Denormalize and select equivalent microstriplines
EEE 194RF_L19 11
Filter Realization Example
• 5th order 0.5 dB ripple Chebyshev LPF• g1 = g5 = 1.7058, g2 = g4 = 1.2296, g3 =
2.5408, g6 =1.0
EEE 194RF_L19 12
Filter Realization Example
• Y1 = Y5 = 1.7058, Z2 = Z4 = 1.2296, Y3 = 2.5408; and Z1 = Z5 = 1/1.7058, Z3 = 1/2.5408
EEE 194RF_L19 13
Filter Realization Example
• Utilizing Unit Elements to convert series stubs to shunt stubs
EEE 194RF_L19 14
Filter Realization Example
• Apply Kuroda’s Identities to eliminate first shunt stub to series stub
EEE 194RF_L19 15
Filter Realization Example
• Deploy second set of UE’s in preparation for converting all series stubs to shunt stubs
EEE 194RF_L19 16
Filter Realization Example
• Apply Kuroda’s Identities to eliminate all series stubs to shunt stubs
• Z1 = 1/Y1 =NZ2 = (1+Z2/Z1)Z2=1+(1/0.6304); Z2 = 1 and Z1 = 0.6304
EEE 194RF_L19 17
Filter Realization Example
• Final Implementation
EEE 194RF_L19 18
Filter Realization Example
• Frequency Response of the Low Pass Filter
EEE 194RF_L20 1
Matching Networks
• MNs are critical for at least two critical reasons– maximize power transfer: – minimize
• Primary goal of a MN is to achieve
0=Γin
)||1( 2inirit PPPP Γ−=−=
||1||1
in
inSWRΓ−Γ+
=
EEE 194RF_L20 2
Matching 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
EEE 194RF_L20 3
Region of matching for shunt L, series C matching network
EEE 194RF_L20 4
Region of matching for series C shunt L matching network
EEE 194RF_L20 5
Region of matching for series L shunt C matching network
EEE 194RF_L20 6
Region of matching for shunt C and series L matching network
EEE 194RF_L20 7
There are two strategies
A) Source impedance -> conjugate complex load impedance
B) Load impedance -> conjugate complex source impedance
EEE 194RF_L20 8
General 2 Element Approach
EEE 194RF_L20 9
Load Impedance To Complex Conjugate Source Zs = Zs* = 50 Ω
EEE 194RF_L20 10
Art of Designing Matching Networks
EEE 194RF_L20 11
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)
EEE 194RF_L20 12
Quality Factor• Resonance effect has implications on design of
matching network.• Loaded Quality Factor: QL = fO/BW• If we know the Quality Factor Q, then we can find
BW• Estimate Q of matching network using Nodal
Quality Factor Qn
• At each circuit node can find Qn = |Xs|/Rs or Qn = |BP|/GP and
• QL = Qn/2 true for any L-type Matching Network
EEE 194RF_L20 13
Nodal Quality FactorsQn = |x|/r =2|Γi| / [(1- Γr)2 + Γi
2
EEE 194RF_L20 14
Matching Network Design Using Quality Factor
EEE 194RF_L20 15
T-Type Matching Networks
EEE 194RF_L20 16
Pi-Type Matching Network
EEE 194RF_L20 17
Microstripline Matching Network• Distributed microstip lines and lumped
capacitors• less susceptible to parasitics• easy to tune• efficient PCB implementation• small size for high frequency
EEE 194RF_L20 18
Microstripline Matching Design
EEE 194RF_L20 19
Two Topologies for Single-Stub Tuners
EEE 194RF_L20 20
Balanced Stubs
• Unbalanced stubs often replaced by balanced stubs
1 22
2S
SBl
l tan tanπλ
π λ− =
1 21
2 2S
SBl
l tan tanπλ
π λ− =
Open-Circuit Stub Short-Circuit Stub
lS is the unbalance stub length and lSB is the balanced stub length.
Balanced lengths can also be found graphically using the Smith Chart
EEE 194RF_L20 21
Balanced Stub Example
Single Stub Smith Chart
Balanced Stub Circuit
EEE 194RF_L20 22
Double Stub Tuners
• Forbidden region where yD is inside g = 2 circle
EEE 194RF_L21 1
General 2 Element Approach
EEE 194RF_L21 2
Load Impedance To Complex Conjugate Source Zs = Zs* = 50 Ω
EEE 194RF_L21 3
Art of Designing Matching Networks
EEE 194RF_L21 4
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)
EEE 194RF_L21 5
Quality Factor• Resonance effect has implications on design of
matching network.• Loaded Quality Factor: QL = fO/BW• If we know the Quality Factor Q, then we can find
BW• Estimate Q of matching network using Nodal
Quality Factor Qn
• At each circuit node can find Qn = |Xs|/Rs or Qn = |BP|/GP and
• QL = Qn/2 true for any L-type Matching Network
EEE 194RF_L21 6
Nodal Quality FactorsQn = |x|/r =2|Γi| / [(1- Γr)2 + Γi
2
EEE 194RF_L21 7
Matching Network Design Using Quality Factor
EEE 194RF_L21 8
T-Type Matching Networks
EEE 194RF_L21 9
Pi-Type Matching Network
EEE 194RF_L21 10
Microstripline Matching Network• Distributed microstip lines and lumped
capacitors• less susceptible to parasitics• easy to tune• efficient PCB implementation• small size for high frequency
EEE 194RF_L21 11
Microstripline Matching Design
EEE 194RF_L21 12
Two Topologies for Single-Stub Tuners
EEE 194RF_L21 13
Balanced Stubs
• Unbalanced stubs often replaced by balanced stubs
1 22
2S
SBl
l tan tanπλ
π λ− =
1 21
2 2S
SBl
l tan tanπλ
π λ− =
Open-Circuit Stub Short-Circuit Stub
lS is the unbalance stub length and lSB is the balanced stub length.
Balanced lengths can also be found graphically using the Smith Chart
EEE 194RF_L21 14
Balanced Stub Example
Single Stub Smith Chart
Balanced Stub Circuit
EEE 194RF_L21 15
Double Stub Tuners
• Forbidden region where yD is inside g = 2 circle
EEE 194RF 1
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
EEE 194RF 2
Passive biasingVCC
R1
RFCR2
IB
I1
RFOUT
RFIN
IC
RFC
CB
CB
• Simple two element biasing
• blocking capacitors CBand RFCs to isolate RF path
• Very sensitive to collector current variations
EEE 194RF 3
Passive biasingVCC
R1
RFCR2
IB
RFOUT
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
EEE 194RF 4
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
EEE 194RF 5
Active biasing in common base
VCC
RFC
RC1
RFOUT
RFIN
RFC
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
EEE 194RF 6
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
VG<0 and VD>0
EEE 194RF_L22 7
Matching to Self-Biased BJT Amp
• Design self-bias circuit as usual
• Design input and output matches to S11 and S22 respectively
RC
RE
RB1
RB2
RS
RL
Cin_match
0.1 uF
0.1 uF
Cout_match
CE0.1uF
VS
+VCC
Lout_match
Lin_match
EEE 194RF_L22 8
Equivalent RF Model of BJT Amp
• The equivalent RF model of the self-biased BJT amp is shown. Note that bias resistors do not affect RF performance
RS
RL
Cin_match
Cout_match
VSLout_match
Lin_match
EEE 194RF_L22 9
Matching to Self-Biased JFET Amp
• Design self-bias circuit as usual
• Design input and output matches to S11 and S22 respectively
RD
RS
RG1 M?
RS
RL
Cin_match
0.1 uF
0.1 uF
Cout_match
CS0.1uF
VS
+VCC
Lout_matchLin_match
EEE 194RF_L22 10
Equivalent RF Model of JFET Amp
• The equivalent RF model of the self-biased JFET amp is shown. Note that bias resistors do not affect RF performance
RS
RL
Cin_match
Cout_match
VSLout_match
Lin_match
EEE 194RF_L22 11
Matching Networks for Amplifiers
• Conjugate matching must be used for maximum power transfer
• Standard impedance matching using either two element L-C, Pi- or Tee-type network, or microstripline matching.
• Use Smith Charts with associated Node Quality Factor Qn to determine network
EEE 194RF_L22 12
Stub Tuner Matching for RF BJT Amp• Can implement impedance matching
network with microstriplines• Shown is single stub tuner with shorted stub
RC
RE
RB1
RB2
RSRL
CS0.1uF
0.1 uF
CE0.1uF
VS
+VCC
Cstub10.1uF
Cstub20.1uF
RFC
RFC
Shorted Stub
Shorted Stub
Xmission Line
Xmission Line
Stub Tuner Matched RF Amplifiers• Stub tuners can be used to match sources and load
to S11* and S22* of the RF BJT or FET• Either open or short circuit stubs may be used• When using short circuit stubs, place a capacitor
between the stub and ground to produce RF path to ground – Do not short directly to ground as this will affect transistor DC biasing
• High resistance λ/4 transformers or RFC’s may be used to provide DC path to transistor for biasing without affecting the RF signal path
Stub Tuner Matched RF Amplifier
01
resonant resonantL Cω =
Series Resonant Ckt at Operating Frequency:Short Ckt at Resonance, Open Circuit at DC
λ/4 Transformer: Transforms Short Circuit at Resonance to Open circuit at BJT Collector Thus Isolating RC from RF Signal Path
Stub tuners of two types:Base-Side: Open Circuit Stub w/ Isolation from DC Bias Circuit Using RFC.Collector-Side: RF Short Circuit Stub via By-Pass Capacitor
The BJT “Self-Bias” Configuration Is Shown Which Produces Excellent Quiescent Point Stability
Power Supplies Are Cap By-Passed and RF Input and Output are Cap Coupled
Stub Tuner Matched RF AmplifierSimpler method of bias isolation at BJT collector: CBP is RF short-circuit which when transformed by the Quarter-Wave Transformer is open circuit at the Single Stub Tuner and provides DC path for the Bias Network
Design Strategy: RF Amplifiers• Objective: Design a complete class A, single-stage
RF amplifier operated at 1 GHz which includes biasing, matching networks, and RF/DC isolation.
Design Strategy: RF Amplifier
• Design DC biasing conditions• Select S-parameters for operating frequency• Build input and output matching networks
for desired frequency response• Include RF/DC isolation• simulate amplifier performance on the
computer
Design Strategy: Approach
For power considerations, matching networks are assumed lossless
Power RelationshipsTransducer Power Gain
Stability of Active Device
Stability of Amplifiers
• In a two-port network, oscillations are possible if the magnitude of either the input or output reflection coefficient is greater than unity, which is equivalent to presenting a negative resistance at the port. This instability is characterized by
|Γin| > 1 or |Γout| > 1 which for a unilateral device implies |S11| > 1 or |S22| > 1.
Stability Requirements
• Thus the requirements for stability are
and
• These are defined by circles, called stability circles, that delimit |Γin | = 1 and | ΓL | = 1 on the Smith chart.
12 2111
22
S +in 1= < 1L
L
S SS
Γ− Γ
Γ
out 22| | = S +l < 1 Γ
Stability Regions: Stability Circles• Regions of amplifier stability can be
depicted using stability circles using the following:Output stability circle:
( )**22 1112 21
2 2222222
,out out
S SS Sr C
SS
− ∆= =
− ∆− ∆
Input stability circle:( )**
11 2212 212 222
1111
,in in
S SS Sr C
SS
− ∆= =
− ∆− ∆
Stability Regions: Stability Circles
Where:11 22 12 21S S S S∆ = −
Stability Regions: Output Stability Circles
Stability Regions: Input Stability Circles
Different Input Stability Regions
Dependent on ratio between rs and |Cin|
Unconditional Stability
Stability circles reside completely outside |ΓS| =1 and |ΓL| =1. Rollet Factor: 2 2 2
11 22
12 21
11
2S S
kS S
− − + ∆= >
Constant Gain Amplifier
Constant Gain Circles in the Smith ChartTo obtain desired amplifier gain performance
Circle Equation and Graphical Display
Gain Circles• Max gain Γimax =1/(1-|Sii|2) when Γi = Sii* ;
gain circle center is at dgi= Sii* and radius rgi =0
• Constant gain circles have centers on a line connecting origin to Sii*
• For special case Γi = 0, gi = 1-|Sii|2 and dgi = rgi = |Sii|/(1+|Sii|2) implying Γi = 1 (0 dB) circle always passes through origin of Γi plane
Trade-off Between Gain and Noise
What Does Stability Mean?• Stability circles determine what load or source
impedances should be avoided for stable or non-oscillatory amplifier behavior
• Because reactive loads are being added to amp the conditions for oscillation must be determined
• So the Output Stability Circle determine the ΓL or load impedance (looking into matching network from output of amp) that may cause oscillation
• Input Stability Circle determine the ΓS or impedance (looking into matching network from input of amp) that may cause oscillation
Criteria for Unconditional Stability• Unconditional Stability when amplifier
remains stable throughout the entire domain of the Smith Chart at the operating bias and frequency. Applies to input and output ports.
• For |S11| < 1 and |S22| < 1, the stability circles reside completely outside the |ΓS| = 1 and |ΓL| = 1 circles.
Unconditional Stability: Rollett Factor• |Cin| – rin | >1 and |Cout| – rout | >1 • Stability or Rollett factor k:
2 2 211 22
12 21
11
2S S
kS S
− − + ∆= >
with |S11| < 1 or |S22| < 1and
11 22 12 21 1S S S S∆ = − <
Stabilization Methods• Stabilization methods can be used to for
operation of BJT or FET found to be unstable at operating bias and frequency
• One method is to add series or shunt conductance to the input or output of the active device in the RF signal path to “move” the source or load impedances out of the unstable regions as defined by the Stability Circles
Stabilization Using Series Resistance or Shunt Conductance
Stabilization Method: Smith Chart
Constant Gain: Unilateral Design (S12= 0)• Need to obtain desired gain performance• Basically we can “detune” the amp
matching networks for desired gain• Unilateral power gain GTU implies S12 = 0
Unilateral Power Gain Equations• Unilateral Power gain
2 22
21 02 211 22
1 1
1 1S L
TU S LS L
G S G G GS S
− Γ − Γ= =
− Γ − Γ
• Individual blocks are: 2 2
20 212 2
11 22
1 1
1 1S L
S LS L
G ; G S ; GS S
− Γ − Γ= = =
− Γ − Γ
• GTU (dB) = GS(dB) + G0(dB) +GL(dB)
Unilateral Gain Circles
max max2 211 22
1 11 1
S LG ; GS S
= =− −
• If |S11| < 1 and |S22 |< 1 maximum unilateral power gain GTUmax when ΓS = S11* and ΓL = S22*
• Normalized GS w.r.t. maximum:
( )2
2112
max 11
11
1SS
SS S
Gg SG S
− Γ= = −
− Γ
Unilateral Gain Circles
• Results in circles with center and radii:
( )2
2222
max 22
11
1LL
LL L
Gg SG S
− Γ= = −
− Γ
• Normalized GL w.r.t. maximums:
( )( )( )
2
2 2
1 1
1 1 1 1i i
i iii iig g
ii i ii i
g Sg Sd ; rS g S g
− −= =
− − − −
ii = 11 or 22 depending on i = S or L
Gain Circle Observations• Gi max when Γi = Sii* and dgi = Sii* of radius
rgi = 0• Constant gain circles all have centers on
line connecting the origin to Sii* • For the special case Γi = 0 the normalized
gain is:gi = 1 - | Sii |2 and dgi = rgi = | Sii |/(1 + | Sii |2)
• This implies that Gi = 1 (0dB) circle always passes through origin of Γi - plane
Input Matching Network Gain Circles
ΓS is detuned implying the matching network is detuned
Bilateral Amplifier Design (S12 included)• Complete equations required taking into
account S12: Thus ΓS* ≠ S11 and ΓL* ≠ S22
12 21 1111
22 221 1* L LS
L L
S S SSS S
Γ − Γ ∆Γ = + =− Γ − Γ
12 21 2222
11 111 1* S SL
S S
S S SSS S
Γ − Γ ∆Γ = + =− Γ − Γ
Bilateral Conjugate Match• Matched source reflection coefficient
21 1 1
1 1 1
1 42 2
*
MSB B CC C C
Γ = − −
2 2 2
1 11 22 1 22 111*C S S ; B S S= − ∆ = − − ∆ +
• Matched load reflection coefficient2
2 2 2
2 2 2
1 42 2
*
MLB B CC C C
Γ = − −
2 2 22 22 11 2 11 221*C S S ; B S S= − ∆ = − − ∆ +
Optimum Bilateral Matching
12 2111
221MS
* ML
ML
S SSS
ΓΓ = +− Γ
12 2122
111ML
* MS
MS
S SSS
ΓΓ = +− Γ
Design Procedure for RF BJT Amps• Bias the circuit as specified by data sheet
with available S-Parameters• Determine S-Parameters at bias conditions
and operating frequency• Calculate stability |k| > 1 and |∆| < 1?• If unconditionally stable, design for gain• If |k| ≤ 1 and |∆| ≥1 then draw Stability
Circles on Smith Chart by finding rout, Cout, rin, and Cin radii and distances for the circles
Design Procedure for RF BJT Amps• Determine if ΓL ( S22* for conjugate match)
lies in unstable region – do same for ΓS• If stable, no worries. • If unstable, add small shunt or series
resistance to move effective S22* into stable region – use max outer edge real part of circle as resistance or conductance (do same for input side)
• Can adjust gain by detuning ΓL or ΓS
Design Procedure for RF BJT Amps• To design for specified gain, must be less than
GTU max (max unilateral gain small S12)• Recall that (know G0 = |S21|2)
GTU [dB] = GS [dB] + G0 [dB] + GL [dB]• Detune either ΓS or ΓL
• Draw gain circles for GS (or GL) for detuned ΓS (or ΓL) matching network
• Overall gain is reduced when designed for (a) Stability and (b) detuned matching netw0rk
Design Procedure for RF BJT Amps• Further circles on the Smith Chart include
noise circles and constant VSWR circles• Broadband amps often are feedback amps
RF Shunt-Shunt Feedback Amp Design
( )1 0 211R Z S= − 0
2
21
1
m
ZR
R g= −
Cm
T
IgV
= S21 calculated from desired gain G
Distortion: 1 dB Compression
Distortion: 3rd Order IntermodulationDistortion
Distortion: 3rd Order IMD[ ] ( )[ ] [ ]2 2 13 dB dBm (2 ) dBmout outIMD P f P f f= − −
[ ] [ ] [ ] [ ]( )0 ,2dB dBm dB dBm3f in mdsd IP G P= − −
Spurious Free Dynamic Range
Class C Amplifier
• Class C amplifier operates for less than half of the input cycle. It’s efficiency is about 75% because the active device is biased beyond cutoff.
• It is commonly used in RF circuits where a resonant circuit must be placed at the output in order to keep the sine wave going during the non-conducting portion of the input cycle.
Types of Signal Distortion
Types of distortion in communications:• harmonic distortion• intermodulation distortion• nonlinear frequency response• nonlinear phase response• noise• interference
Non-sinusoidal Waveform
• Any well-behaved periodic waveform can be represented as a series of sine and/or cosine waves plus (sometimes) a dc offset:
e(t)=Co+ΣAn cos nω t + ΣBn sin nω t (Fourier series)
External Noise
• Equipment / Man-made Noise is generated by any equipment that operates with electricity
• Atmospheric Noise is often caused by lightning
• Space Noise is strongest from the sun and, at a much lesser degree, from other stars
Internal Noise
• Thermal Noise is produced by the random motion of electrons in a conductor due to heat. Noise power, PN = kTB
where T = absolute temperature in oKk = Boltzmann’s constant, 1.38x10-23 J/KB = noise power bandwidth in Hz
Noise voltage, kTBR4VN =
Internal Noise (cont’d)
• Shot Noise is due to random variations in current flow in active devices.
• Partition Noise occurs only in devices where a single current separates into two or more paths, e.g. bipolar transistor.
• Excess Noise is believed to be caused by variations in carrier density in components.
• Transit-Time Noise occurs only at high f.
Noise Spectrum of Electronic DevicesDeviceNoise
Shot and Thermal Noises
Excess orFlicker Noise
Transit-Time orHigh-FrequencyEffect Noise
1 kHz fhcf
Noise Figure
• Noise Figure is a figure of merit that indicates how much a component, or a stage degrades the SNR of a system:
NF = (S/N)i / (S/N)o
where (S/N)i = input SNR (not in dB)and (S/N)o = output SNR (not in dB)
NF(dB)=10 log NF = (S/N)i (dB) - (S/N)o (dB)
Equivalent Noise Temperature and Cascaded Stages
• The equivalent noise temperature is very useful in microwave and satellite receivers.
Teq = (NF - 1)To
where To is a ref. temperature (often 290 oK)• When two or more stages are cascaded:
...AA
1NFA
1NFNFNF21
3
1
21T +−+−+=
Class C Amplifier
• Class C amplifier operates for less than half of the input cycle. It’s efficiency is about 75% because the active device is biased beyond cutoff.
• It is commonly used in RF circuits where a resonant circuit must be placed at the output in order to keep the sine wave going during the non-conducting portion of the input cycle.
Simple Oscillator Using Stability
L
EmitterBiasing,coupling,matching,
etc.
CollecterBiasing,coupling,matching,
etc.
LoadNetwork
TerminatingNetwork
Γ in ΓoutΓL ΓT
Choose transistor (BJT or FET) wisely so that common-base S11 > 1 and S22 >1 at oscillation frequency: This will cause instability.
NE021 npn High Frequency BJT
S22 >1: Potential Instability
Simple Oscillator Design: KISS!
• Select transistor that is potentially unstable at oscillation frequency
• Chose GT for terminating network that will make |GIN|>1
• Calculate GL for the load network that will resonate ZIN at oscillation frequency
• If ZIN = RIN + jXIN, then ZL = RL + jXL, where RL = |RIN| /3 and XL= –XIN
Hartley Oscillators
211
;2
1 LLLCL
f TT
o +==π1
21
LLLB +=
1
2
LLB =
Colpitts Oscillator
21
21
2
1
21
CCCCC;
LCf;
CCB T
To +
===π
Clapp Oscillator
The Clapp oscillator is a variation of the Colpitts circuit. C4 is added in series with L in the tank circuit. C2 and C3 are chosen
large enough to “swamp” out the transistor’s junction capacitances for greater stability. C4 is often chosen to be << either C2 or C3,
thus making C4 the frequency determining element, since CT = C4.
432
32
2
1111
21;
CCC
C
LCf
CCCB
T
To
++=
=+
=π
Mixers
• A mixer is a nonlinear circuit that combines two signals in such a way as to produce the sum and difference of the two input frequencies at the output.
• A square-law mixer is the simplest type of mixer and is easily approximated by using a diode, or a transistor (bipolar, JFET, or MOSFET).
Dual-Gate MOSFET Mixer
Good dynamic range and fewer unwanted o/p frequencies.
Balanced Mixers
• A balanced mixer is one in which the input frequencies do not appear at the output. Ideally, the only frequencies that are produced are the sum and difference of the input frequencies.
Circuit symbol:f1
f2
f1+ f2
Equations for Balanced Mixer
Let the inputs be v1 = sin ω1t and v2 = sin ω2t.A balanced mixer acts like a multiplier. Thusits output, vo = Av1v2 = A sin ω1t sin ω2t.Since sin X sin Y = 1/2[cos(X-Y) - cos(X+Y)]Therefore, vo = A/2[cos(ω1-ω2)t-cos(ω1+ω2)t].The last equation shows that the output of
the balanced mixer consists of the sum and difference of the input frequencies.
Balanced Ring Diode Mixer
Balanced mixers are also called balanced modulators.
Voltage-Controlled Oscillator
• VCOs are widely used in electronic circuits for AFC, PLL, frequency tuning, etc.
• The basic principle is to vary the capacitance of a varactor diode in a resonant circuit by applying a reverse-biased voltage across the diode whose capacitance is approximately:
b
oV V
CC21+
=
Basic Oscillator Model
• Oscillator has positive feedback loop at selected frequency
• Barkhausen criteria implies that the multiplication of the transfer functions of open loop amplifier and feedback stage is
HF (ω)HA (ω) = 1• Barkhausen criteria aka loop gain equation
LC Oscillators – Lower RF Frequencies
LC Oscillators – Lower RF Frequencies
LC Oscillators – Lower RF Frequencies
• Can also design with BJTs.
High RF & Microwave Oscillators
• Take advantage of our knowledge of stability
• Rollett Stability Factor k < 1
Microwave Oscillator Signal Flowb1/bs =Γin / (1- ΓsΓin )
Conditions of oscillation –
Unstable if:
ΓsΓin = 1 or ΓsΓL = 1
Creating Oscillator Condition
• Frequently begin with common-base or common-gate configuration
• Convert common-emitter s-parameters to common-base (similarly for FETs)
• Add inductor in series with base (or gate) as positive feedback loop network to attain unstable Rollett factor k <1
Unstable Condition – Oscillation
1. Convert transistor common-base [s] to [Z]tr
2. [Z]L =
3. [Z]Osc= [Z]L+[Z]tr
4. Convert [Z]Osc to [s]Osc
5. Plot stability circles
1 11 1
j Lω
Inductor Value for Oscillation• Must repeat
previous calculation ofRollet Factor for each value of L
• In this exampleL = 5 nH
s11 = -0.935613, s12 = -0.002108,
s21 = 1.678103 , s22 = 0.966101
Unstable Transistor Oscillator Design1. Select potentially unstable transistor at freq2. Select appropriate transistor configuration3. Draw output stability circle in ΓL plane4. Select appropriate value of ΓL to produce largest
possible negative resistance at input of transistor yielding |ΓL | >1 and Zin < 0
5. Select source tuning impedance Zs as if the circuit was a one-port oscillator by RS + RIN < 0 typically RS = |RIN|/3, RIN < 0 and XS = -XIN
6. Design source tuning and terminating networks with lumped or distributed elements
Dielectric Resonator Oscillator (DRO)
DRO Networks
DR-based input matching network of the FET oscillator.
Varactor Diodes (Voltage Variable Caps)
Gunn Elements For Oscillators
Gunn Oscillator with DRO
Mixer Basics
Heterodyne receiver system incorporating a mixer.
Basic mixer concept: two input frequencies are used to create new frequencies at the output of the system.
Mixing Process Spectrum
Simple Diode and FET Mixers
Compression Point and 3rd Order Intercept
Single-Ended BJT Mixer
Single-Ended BJT Mixer Design Biasing Network
Single-Ended BJT Mixer DesignLO and RF Connection
Single-Ended BJT Mixer DesignRF Input Matching Network
Single-Ended BJT Mixer DesignModified Input Matching for RF
Single-Ended BJT Mixer DesignCompleted Design
Related Documents
