Lecture 7
Lecture 7
RF Amplifier Design
• 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
Electrical and Information Technology
Johan Wernehag, EIT
Johan Wernehag, EIT
Stability Analysis by S-Parameters
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
•
2-port
A unilateral two-port (S12 = 0) is
unconditionally stable if
and
A bilateral two-port is
1) unconditionally stable if
where
2) conditionally stable if
some S gives out| < 1
and some
L gives
in| < 1
to decide how “unilateral” a specified two-port is.
S
and load
L
where
•
IF the input and the output are supposed to be conjugate matched, i.e.
*
*
S = S11 and L = S22
where
NOTE – this discussion is not necessary if you have access to computer analysis
when you easily may calculate bilateral!
Johan Wernehag, EIT
Johan Wernehag, EIT
1
Power Gain Definitions
Amplifier Design
General design case
ZS
ZS’
Input
matching
network
ES
Output
matching
network
Two-port
network
ES
ZL
PAVS
PIN
G
PAVN
PL
ZL
• Given this:
– two-port (S-parameters) and
– source S’ and load L’
available gain
•
The stability analysis gives allowed values of
•
After a proper choice of S and L
the matching networks may be designed
S
and
L
operating gain
transducer gain
Johan Wernehag, EIT
Johan Wernehag, EIT
Case 1
Design Cases - Gain and Noise Figure
Amplifier Design
Approximation
suitable for
hand calculation
Maximum
Gain
Case 2
Specific
Gain
There is hardly no reason
to use these methods for
computer analysis.
Johan Wernehag, EIT
• Choose
and
i.e. apply conjugate match to both input and output
Unilateral Two-Port
Case 1
Unilateral Two-Port, Maximum Gain
• The gain is then
Bilateral Two-Port
Case 3
Maximum
Gain
Case 4
Specific
Gain
Minimum
Noise Figure
Compromise
Gain - Noise Figure
Maximum Unilateral Transducer Gain:
Assumption:
and
Johan Wernehag, EIT
2
Case 2
Case 2
Unilateral Two-Port, Specific Gain
Unilateral Two-Port, Specific Gain
For gS < 1 and gL < 1 there are lot of solutions. Described by
circles in the S-plane and the L-plane
• The gain is expressed by
Input:
(Unilateral Transducer Gain)
• Split up
Gain circle
input,
S -plane
note that
Gain circle
output,
L -plane
L
S
Output:
• Result:
L
where
S
• When
only one solution exists:
• If
and
circles in the
and
there are lot of solutions. Described by
S-plane and the L-plane
Johan Wernehag, EIT
How do you select
“smart” S and L?
Constant
conductance
circle
Johan Wernehag, EIT
Case 3
Bilateral Two-Port, Maximum Gain
• Maximum gain is achieved when
and
Bilateral Two-Port
• At “simultaneous conjugate match” the
maximum transducer gain is:
i.e. apply conjugate match to both input and output
and
• These equations need to be solved simultaneously,
that’s why it’s called “simultaneous conjugate match”
•
With K set to 1 the quantity Maximum Stable Gain is derived:
•
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
• Explicit solution:
and
where
• NOTE! The solution only exists when
the two-port is unconditionally stable, i.e. | | < 1 and K > 1!
Johan Wernehag, EIT
–
–
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.
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3
Case 4
Bilateral Two-Port, Specific Gain
Bilateral Two-Port
• The procedure will be like this:
1.
Calculate the “maximum stable gain”:
• At the unilateral case it was possible to handle
the input and output port separately.
2.
Back off a few dB or so to set a safety margin.
• BUT at the bilateral case the conditions at the input port
depends on the load and vice versa.
3.
Use the reduced gain as specific gain and design
according to case 4.
Is it possible to
disengage the ports
from each other?
• 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
• Then
Yes it is:
Assume conjugate match at one
of the ports and the other is
mismatched to obtain the
specified gain (GT)!
Johan Wernehag, EIT
Case 4
Case 4
Bilateral Two-Port, Specific Gain
Bilateral Two-Port, Specific Gain
Assume conjugate match at one port and mismatch is applied to the other!
Design by “operating gain”
If a mismatch is wanted at the output:
may be written as
1. use “operating gain”
Can we affect S21?
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
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
If a mismatch is wanted at the input:
S
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
Stable area
=
*
in
Stable
area
L
in
S
L
3. select a “smart”
L!
4. calculate in and conjugate
match the input
gP
Constant
conductance
circle
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4
Case 4
Case 4
Bilateral Two-Port, Specific Gain
Bilateral Two-Port, Specific Gain
summary
Design by “available gain”
• If a two-port is conditionally stable:
may be written as
1. Calculate stability circles
2. Calculate a gain circle to obtain the wanted GP, (GA)
3. Select
L,
(
S)
at the gain circle in the stable area
1. S21 is known, determine gA to obtain the wanted gain
2. what
•
S complies
4. Calculate
with the selected gA?
there are a number of solutions at a circle in the
S-plane
IN,
(
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
Johan Wernehag, EIT
Noise in a Two-Port
ZS
GA
PNo
ES
•
The signal-to-noise ratio
Noise in Cascaded Two-Ports
• The total noise figure is
ZL
is deteriorated due to noise
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
but
• Friis’ formula:
No decibels
here!
leads to
• NOTE: all variables must be denoted in linear quantities!
Johan Wernehag, EIT
Johan Wernehag, EIT
5
Design of Low Noise Amplifiers (LNA)
Design of Low Noise Amplifiers (LNA)
• The noise figure:
• The noise power from a transistor depends on
– the source impedance
– the quiescent point (IC , VCE)
Fmin - the minimum noise figure
RN - determines how much F increases
when YS deviates from Yopt
Yopt - the source admittance providing Fmin
• 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
where
The noise figure denoted by normalised parameters:
where
•
The noise figure denoted by reflection coefficients:
power mismatch at the input
Johan Wernehag, EIT
Johan Wernehag, EIT
Design of Low Noise Amplifiers (LNA)
•
There are a number of
•
These are found at circles in the
S
Summary of Amplifier Design
that provides a specified noise figure
1. Decide if the transistor is unconditionally stable
S-plane
2. Calculate stability circles if necessary
GT
GT , F
3. Choose a method for specific or
3.
Choose the method for specific gain
using available gain
maximum gain
4. Assume conjugate match at the
4.
Assume conjugate match at the
output
input (or the output)
opt
5. Calculate a gain circle to obtain the
SO
F
wanted GP (or GA)
6. Select
L
(or
S)
5.
Draw noise and gain circles
6.
Select
Constant
noise figure
circle
in the stable area that
noise and gain
stable area
7. Calculate
S
provides a suitable compromise of
at the gain circle in the
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
Johan Wernehag, EIT
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