Lecture 7
RF Amplifier Design
Johan Wernehag
Electrical and Information Technology
Johan Wernehag, EIT
Lecture 7
• Amplifier Design
– Stability Analysis by S-parameters
– Design Cases
• unilateral two-port, maximum gain
• unilateral two-port, specific gain
• bilateral two-port, maximum gain
– “simultaneous conjugate match”
• bilateral two-port, specific gain
– conjugate match at one port, mismatch the other port
– design method using “operating gain”
– design method using “available gain”
– Noise in a Two-Port
– Design of Low Noise Amplifiers (LNA)
Johan Wernehag, EIT
Stability Analysis by S-Parameters
Re zin
0 for any value of zL Re zL
in
<1 for all
L
1
S
zin ,
0
Re zout
zout ,
in
0 for any value of zS Re zS
out
out
L
<1 for all
0
1
S
S21
S112-port S22
S12
A unilateral two-port (S12 = 0) is
unconditionally stable if
A bilateral two-port is
1) unconditionally stable if
2) conditionally stable if
some S gives out| < 1
in
= S11 <1
1 and K
S
out
plane
1
1 where
and
S11S22
out
S12 S21 and K
and some
L gives
in| < 1
S22 <1
1
S11
2
S22
2 S12 S21
L
2
2
plane
in
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1
Unilateral Figure of Merit
•
S12 is hardly never zero, but can at some times be close enough to justify a
unilateral approximation
A tool to decide this is “Unilateral Figure of Merit”
•
Consider the ratio
•
For an arbitrary source
•
1
1
•
X
2
GT
GTU
GT
to decide how “unilateral” a specified two-port is.
GTU
S
and load
1
1
X
2
L
where
X
S12 S21 S L
1 S11 S 1 S22
L
IF the input and the output are supposed to be conjugate matched, i.e.
*
*
S = S11 and L = S22
1
1 U
2
GT
GTU
1
1 U
2
where
S12 S21 S11 S22
U
1
S11
2
1
S22
2
NOTE – this discussion is not necessary if you have access to computer analysis
when you easily may calculate bilateral!
Johan Wernehag, EIT
Amplifier Design
General design case
S
S
in
out
L
L
ZS’
ES
Input
matching
network
Two-port
network
Output
matching
network
ZL
• Given this:
– two-port (S-parameters) and
– source S’ and load L’
•
The stability analysis gives allowed values of
•
After a proper choice of S and L
the matching networks may be designed
Johan Wernehag, EIT
S
and
L
Power Gain Definitions
ZS
ES
PAVS
G
PIN
available gain
GA
PAVN
PAVN
PAVS
operating gain
transducer gain
Johan Wernehag, EIT
PL
GT
GP
PL
PAVS
PL
PIN
ZL
Design Cases - Gain and Noise Figure
Amplifier Design
Approximation
suitable for
hand calculation
Unilateral Two-Port
Case 1
Maximum
Gain
Case 2
Specific
Gain
There is hardly no reason
to use these methods for
computer analysis.
Johan Wernehag, EIT
Bilateral Two-Port
Case 3
Maximum
Gain
Case 4
Specific
Gain
Minimum
Noise Figure
Compromise
Gain - Noise Figure
Case 1
Unilateral Two-Port, Maximum Gain
• Choose
S11 and
S
S22
L
i.e. apply conjugate match to both input and output
• The gain is then
GT
PL
PAVS
2
1
1
S
2
in
S21
S
2
1
2
1
L
1 S22
2
1 S11
L
2
S
2
S21
S
Maximum Unilateral Transducer Gain:
GTUM
PL
PAVS
Assumption:
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1
S11
1
in
2
S21
1
2
1
= S11 <1 and
S22
2
out
S22 <1
2
1
2
1
L
2
out
L
Case 2
Unilateral Two-Port, Specific Gain
1
PL
PAVS
• The gain is expressed by GTU
2
S
1 S11
2
S21
GTU
S21
S
2
max
• Result:
GTU
GTU
• When gS
gL
S
max
max
max
S
S
S
GTUM
S
S
L
2
S21 max
L
max
1
S11
1
1
S22
L
max
2
2
L
0
where 0
L
S
S11 and
• If gS 1 and gL 1 there are lot of solutions. Described by
circles in the S-plane and the L-plane
Johan Wernehag, EIT
L
L
gS GTUM gL
1 only one solution exists:
2
1
note that
L
L
1 S22
(Unilateral Transducer Gain)
• Split up
1
2
S
max
2
L
gS 1
gL 1
S22
Case 2
Unilateral Two-Port, Specific Gain
GTU
For gS < 1 and gL < 1 there are lot of solutions. Described by
gL circles in the -plane and the -plane
S
L
gS GTUM
Input:
radius: rS
centre:
1 gS 1
1
S11
2
S11
2
1 gS
gS S11
SO
1
S11
2
Gain circle
output,
L -plane
Gain circle
input,
S -plane
1 gS
L
S
j arg S11
e
Output:
1 gL 1
radius: rL
centre:
1
S22
2
S22
2
gL S22
LO
1
S22
2
1 gL
How do you select
“smart” S and L?
Johan Wernehag, EIT
L
1 gL
j arg S22
e
S
Constant
conductance
circle
Case 3
Bilateral Two-Port, Maximum Gain
• Maximum gain is achieved when
S
IN
and
L
OUT
i.e. apply conjugate match to both input and output
S
S11 S12 S21
L
1 S22
and
S22 S12 S21
L
L
S
1 S11
S
• These equations need to be solved simultaneously,
that’s why it’s called “simultaneous conjugate match”
• Explicit solution:
B1
SM
B12
2C1
4 C1
B1 1 S11
2
and
B2
LM
B22
2C2
4 C2
2
where
B2
1 S22
C1 S11
C2 S22
• NOTE! The solution only exists when
the two-port is unconditionally stable, i.e. | | < 1 and K > 1!
Johan Wernehag, EIT
2
2
S22
S11
S22
2
2
S11
2
2
Bilateral Two-Port
• At “simultaneous conjugate match” the
maximum transducer gain is:
GTM
•
S21
S12
K
K
2
1
S11S22
S12 S21 and K
1 S11
2
S22
2 S12 S21
2
2
With K set to 1 the quantity Maximum Stable Gain is derived:
GMSG
S21
S12
•
At a conditionally stable two-port K may be altered by resistive loading of the
input or output without changing the ratio |S21|/|S12|.
•
But for a conditionally stable two-port
–
–
doesn’t give any sense to the quantity “maximum transducer gain” and
the simultaneous conjugate match doesn’t have any solution.
• Therefore, the method changes to case 4
when there is conditional stability.
Johan Wernehag, EIT
Bilateral Two-Port
• The procedure will be like this:
GMSG
S21
S12
1.
Calculate the “maximum stable gain”:
2.
Back off a few dB or so to set a safety margin.
3.
Use the reduced gain as specific gain and design
according to case 4.
• At a conditionally stable two-port
– may strictly any arbitrary gain be selected
– but as the gain increases the risk for self-oscillation
escalates
– GMSG is in this sense the absolute maximum level.
Johan Wernehag, EIT
Case 4
Bilateral Two-Port, Specific Gain
• At the unilateral case it was possible to handle
the input and output port separately.
• BUT at the bilateral case the conditions at the input port
depends on the load and vice versa.
Yes it is:
Assume conjugate match at one
of the ports and the other is
mismatched to obtain the
specified gain (GT)!
Is it possible to
disengage the ports
from each other?
• Then
GT
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1
1 S11
2
S
2
S
S21
2
1
2
1
1
L
2
out
L
L
*
OUT
1 S11
2
S
2
S
S21
1
2
1
2
out
GA
Case 4
Bilateral Two-Port, Specific Gain
Assume conjugate match at one port and mismatch is applied to the other!
If a mismatch is wanted at the output:
1. use “operating gain”
2. apply mismatch at the output
so that GP = GT and solve L
3. conjugate match the input ( in known)
then GP = PL /PIN = PL /PAVS = GT
If a mismatch is wanted at the input:
1. use “available gain”
2. apply mismatch at the input
so that GA = GT and solve S
3. conjugate match the output ( out known)
then GA = PAVN /PAVS = PL /PAVS = GT
Johan Wernehag, EIT
Case 4
Bilateral Two-Port, Specific Gain
Design by “operating gain”
may be written as
Can we affect S21?
1. S21 is known, determine gP to obtain the wanted gain
2. what
•
L complies
with the selected gP?
there are a number of solutions at a circle
in the L-plane
radius: rL
centre:
2
S12 S21 gP2
1 2K S12 S21 gP
1 gP S22
2
2
gPCL
LO
where CL
1 gP S22
S22
2
2
Stable area
4. calculate in and conjugate
match the input
*
in
Stable
area L
in
L
L!
=
S
S11
3. select a “smart”
Johan Wernehag, EIT
S
Constant
conductance
circle
gP
Case 4
Bilateral Two-Port, Specific Gain
Design by “available gain”
may be written as
1. S21 is known, determine gA to obtain the wanted gain
2. what
•
S complies
with the selected gA?
there are a number of solutions at a circle in the
radius: rS
centre:
2
S12 S21 gA2
1 2K S12 S21 gA
1 gA S11
2
2
gACS
SO
Johan Wernehag, EIT
1 gA S11
2
2
where CS
S11
S22
S-plane
Case 4
Bilateral Two-Port, Specific Gain
summary
• If a two-port is conditionally stable:
1. Calculate stability circles
2. Calculate a gain circle to obtain the wanted GP, (GA)
3. Select
L,
4. Calculate
(
S)
IN,
at the gain circle in the stable area
(
OUT )
5. Check if conjugate match is possible
• i.e. if S = *IN ( L = *OUT ) is located in the stable area
• if not, return to step 3 and make a new choice
• alternatively lower the demand for gain
Johan Wernehag, EIT
Noise in a Two-Port
ZS
ES
•
Psi
PNi
GA
PNo
The signal-to-noise ratio
SNR
G Psi
PNi
PNo
ZL
Ps
is deteriorated due to noise
PN added by the two-port
• The noise figure (F) denotes the increase of noise by the twoport, assumed a source noise temperature of T0 = 300 K
F
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PNo PNi GA
PNi GA
but
GA
PSo
leads to
PSi
F
PSi
PNi
PSo
PNo PNi GA
SNRi
SNRo
Noise in Cascaded Two-Ports
• The total noise figure is
FTOT
total available noise power at the output
available noise at the output originating from the source
FTOT
PNo1
1
PNi GA1
PNo2
PNi GA1GA2
• Friis’ formula:
FTOT
F1
F2 1 F3 1
GA1
GA1GA2
No decibels
here!
• NOTE: all variables must be denoted in linear quantities!
Johan Wernehag, EIT
Design of Low Noise Amplifiers (LNA)
• The noise power from a transistor depends on
– the source impedance
– the quiescent point (IC , VCE)
• There is an optimum source impedance
that gives the minimum noise figure for a
specified quiescent point
• The source impedance for minimum noise figure
does unfortunately NOT coincide with the source
impedance for maximum gain
noise match
Johan Wernehag, EIT
power mismatch at the input
Design of Low Noise Amplifiers (LNA)
• The noise figure:
F
Fmin
2
RN
YS Yopt
GS
where
GS Re YS
Fmin - the minimum noise figure
RN - determines how much F increases
when YS deviates from Yopt
Yopt - the source admittance providing Fmin
•
The noise figure denoted by normalised parameters:
F
•
Fmin
rN
yS yopt
gS
2
where
The noise figure denoted by reflection coefficients:
F
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Fmin
2
4rN
1
S
2
S
opt
1
2
opt
gS Re yS
Design of Low Noise Amplifiers (LNA)
•
There are a number of
•
These are found at circles in the
centre:
that provides a specified noise figure
S-plane
2
Ni2 Ni 1
radius: rS
S
opt
1 Ni
opt
SO
where Ni
1 Ni
F Fmin
opt
1
SO
2
F
opt
4rN
Constant
noise figure
circle
Johan Wernehag, EIT
Summary of Amplifier Design
1. Decide if the transistor is unconditionally stable
2. Calculate stability circles if necessary
GT
GT , F
3. Choose a method for specific or
3.
using available gain
maximum gain
4. Assume conjugate match at the
4.
5. Calculate a gain circle to obtain the
wanted GP (or GA)
L
(or
S)
5.
Draw noise and gain circles
6.
Select
S
in the stable area that
provides a suitable compromise of
at the gain circle in the
noise and gain
stable area
7. Calculate
Assume conjugate match at the
output
input (or the output)
6. Select
Choose the method for specific gain
IN
(or
OUT )
7.
Calculate
OUT
8. Check if conjugate match is possible
– i.e. if S = *IN ( L = *OUT ) is located in the stable area
– if not, return to step 6 and make a new choice
– alternatively lower the demand for gain and return to step 5
9. Design the matching networks and verify stability at all frequencies of interest
Johan Wernehag, EIT