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EE 40458Power Gain and Amplifier Design,
Stability Considerations
11/13/2018 – 11/15/2018
Motivation• Brief recap:– We’ve studied matching networks (several types, how
to design them, bandwidth, how they work, etc…)– Studied network analysis techniques (matrices with S,
ABCD, …; flow graphs, etc.)– But why? Ultimate goal: design a circuit that does
something useful• Amplifiers are a good starting point– Useful for many (all?) systems– Fundamental principles can be applied to other
circuits
Design/Analysis Approach
• Could approach design many ways– “Regular” circuit design: use equivalent circuit model
(hybrid-π, T model)– We’ll take an alternative: treat active device as
(known) 2-port S-matrix• This approach allows us to be more general
(results will apply more widely)– Example: we’ll be able to project what the highest
gain possible is, whether or not the circuit is stable (i.e., will it oscillate?); not always so obvious from equivalent circuits without a lot of work
– Side bonus: we’ll get to see what features of S-matrices are desirable for circuits; help us choose devices
Amplifiers – It’s (Mostly) About Gain
• From previous classes—of course gain is the key thing• But it isn’t quite so simple for high-frequency systems• Recall: from “regular” circuit design: gain was (mostly)
voltage gain Av; sometimes (more rarely) current gain– But you don’t really need an amplifier
for either of these!– A transformer can produce voltage
gain (and a lot of it…)– More generally, a matching network
trades voltage gain for current gain (or visa-versa)
• What do amplifiers really do, then? – Power gain is the key feature: not just
voltage or current in isolation
Power Gain
• Power gain: not quite so simple as it might look:
• OK—but what is Pout and Pin, exactly? There are several (useful) possibilities, we need to be clear which we mean–With voltage, current gain these ambiguities are
less troublesome
Gain = PoutPin
Power Gain – Definitions• Since input, output power can be defined different ways,
multiple definitions for power gain (not just one gain; there are several different alternatives!)
GP =PLPin
=Power delivered to load
Power delivered (absorbed by) network
GT =PLPavs
=Power delivered to load
Power available from source
GA =PavnPavs
=Power available from networkPower available from source
Power Gain - Comparisons• GP: “operating power gain,”
most obvious, but not all that useful
• GA: “available gain,” good baseline for comparisons, shows what performance is possible
• GT: “transducer gain,” most useful in system context, analysis• What’s the difference among these?
– Do you count what actually gets into the network (Pin) or what could have gotten in (Pavs)?
– Same idea at output (PL vs. Pavs)– Degree of mismatch is the
difference
GP =PLPin
=Power delivered to load
Power delivered (absorbed by) network
GT =PLPavs
=Power delivered to load
Power available from source
GA =PavnPavs
=Power available from networkPower available from source
Distinctions Among Power Gain Definitions
• How much power is transferred depends on the impedances/reflection coefficients:
• Pin: power carried by a1, minus that reflected off (b1)
• Pavs: what’s the most power you could get from the source (if it were conjugately matched)?
• Pavn: what’s the most power available from S?
• PL: power dissipated in ZL
Numerical Comparisons?• How do these three definitions compare in terms of
magnitude?
• GA > GT (because PL < Pavn from mismatch)• GP > GT (because Pin < Pavs…)• But not clear if GP, GA larger…depends on relative degree of
source, load mismatch• Can derive each gain from flow graph:
GT =PLPavs
; GP =PLPin; GA =
PavnPavs
Power Gain Expressions• Power gains:
• Where:
GT =1− ΓS
2
1−ΓinΓSS21
2 1− ΓL2
1− S22ΓL2 =
1− ΓS2
1− S11ΓSS21
2 1− ΓL2
1−ΓoutΓL2
GP =1
1− Γin2 S21
2 1− ΓL2
1− S22ΓL2
GA =1− ΓS
2
1− S11ΓS2 S21
2 11− Γout
2
Γin = S11 +S12S21ΓL1− S22ΓL
Γout = S22 +S12S21ΓS1− S11ΓS
Amplifier Design• With these definitions, we can start designing
for real• Several useful special cases for designs:–Maximum gain– Specified gain (i.e. trade lower gain for more
bandwidth)– Low noise
• Let’s do an example of each one to see how they work
Design for Maximum Gain
• For maximum gain: want to conjugately match both the input and output—simultaneously
• Mathematically:
– Note: can work with reflection coefficients since Z G as a 1-to-1 map
– Also, in this case, GT = GA; maximizing power in and out
Γin = ΓS*
Γout = ΓL*
Maximum Gain Design, Cont.
• So: how to design?– Basic idea:
– Our job:• Make input matching network present Gs*=Gin• Make output matching network present GL*=Gout• (So it is really just about making matching networks—
once the correct “targets” for GS, GL are found)
Maximum Gain Design, Cont.• Design approach: know input, output must be conjugately
matched. So:
• But we also know
• So: GS depends on GL (and visa-versa)– Must solve for both GS and GL simultaneously (a single solution that
satisfies all the constraints)– Book goes through the algebra in some detail…
Γin = ΓS*
Γout = ΓL*
Γin = S11 +S12S21ΓL1− S22ΓL
Γout = S22 +S12S21ΓS1− S11ΓS
Maximum Gain Design, Cont.• Bottom line: there is a unique solution for GS and GL
Where:
ΓS =B1 ± B1
2 − 4 C12
2C1
ΓL =B2 ± B2
2 − 4 C22
2C2
B1 =1+ S112− S22
2− Δ
2
B2 =1+ S222− S11
2− Δ
2
C1 = S11 −ΔS22*
C2 = S22 −ΔS11*
Δ = S11S22 − S12S21 (det(S))
Maximum Gain Design, Cont.• So: what to do with this?• Find [S] of
amplifying element(transistor, usuallyplus bias networks)
• Compute GS, GL from [S]• Design input matching
network to go from source to GS
• Design output matching network to go from load to GL
Input & Output Matching• So: maximum gain amplifier design is really designing two
matching networks:
Input & Output Matching• So: maximum gain amplifier design is really designing two
matching networks:
Specified Gain Design
• Key point: already know how to design an amplifier with maximum gain. Why would we ever go for less?– Need specific signal level– More likely: need more bandwidth than max. gain design
can deliver; to get bandwidth, need to give up something else
• Approach: look at GT expression:
• Note: 3 basic terms (source, S21, and load)
GT =1− ΓS
2
1−ΓinΓSS21
2 1− ΓL2
1− S22ΓL2 =
1− ΓS2
1− S11ΓSS21
2 1− ΓL2
1−ΓoutΓL2
GT =GS ⋅GO ⋅GL
Specified Gain Design, Cont.• But a little complicated: GS depends on load, GL
depends on source• Yuck. So let’s make a simplifying assumption—that
[S] is unilateral (e.g. S12 = 0). Then Gin=S11 and Gout=S22
• Net result:
• So how can we set/change the gain? Introduce (intentionally) mismatch at source and/or load– If the match is perfect, the gain will be maximized. We’re
intentionally mismatching to control the gain
GS =1− ΓS
2
1− S11ΓS2 ; GL =
1− ΓL2
1− S22ΓL2
Specified Gain Design, Cont.
• For maximum gain, conjugate match at both input and output: GS=S11*, GL=S22* (remember, we’re assuming unilateral)
• So the maximum GS, GL become:
• It turns out (read: after much algebra, you can show if you are lucky and factor things just exactly right) that:
constant |GS| and |GL| plot as circles in the GS and GL planes
GS,max =1
1− S112 ; GL,max =
11− S22
2
Specified Gain Design, Cont.• These are the “constant gain circles”• Mathematically: gs =
GSGS,max
; gL =GLGL,max
CS =gsS11
*
1− (1− gs ) S112
RS =1− gs (1− S11
2 )1− (1− gs ) S11
2
CL =gLS22
*
1− (1− gL ) S222
RL =1− gL (1− S22
2 )1− (1− gL ) S22
2
Source side: Load side:
C: coordinates (on imaginary plane) of center of circleR: radius
Specified Gain Design, Cont.• So: idea is to figure out how much mismatch to
introduce at input, output to get desired gain• Total gain:– Designer chooses GS, GL so that GT comes out at value
desired (but remember, GS≤ GS,max; GL≤GL,max)– In principle, any combination is ok (results in the correct
gain)– But usually want to get something in exchange for the
trade (e.g. bandwidth)– That additional trade is where design insight comes in
GT =GS ⋅GO ⋅GL
Example: Specified Gain
• Example 12.4 in Pozar is a good example• Summary: design an amplifier at 4 GHz with 11 dB
gain; try to maximize the bandwidth– Given: transistor S-parameters:
• Can compute GS,max; GL,max
Numerically: GS,max = 2.29 = 3.6 dB; GL,max = 1.56 = 1.9 dBGo = |S21|2 = 6.25 = 8 dB– Max gain: 13.5 dB. We can give up 2.5 dB in combination of GS, GL
S = 0.75∠−120 02.5∠80 0.6∠− 70
#
$%
&
'(
GS,max =1
1− S112 ; GL,max =
11− S22
2
Example: Specified Gain (cont.)• Pick a few choices for GS, GL• Plot the gain circles:
• OK…• Remember:
GS,max = 3.6 dBGL,max=1.9 dB
• Options:– GS=3 dB,
GL=0 dB – GS=2 dB
GL=1 dB – Both of these sets are 2.5 dB less than the maximum, give
us 11 dB total
Example: Specified Gain (cont.)
• How to choose?• Trade for something
else—like
bandwidth
• Recall: for largest
bandwidth,
matching
network should not “work hard”
• Pick GS, GL nearest center of Smith Chart—”laziest” matching network
• If pick roughly equal input, output matching networks, overall BW maximized (filters in cascade)
Example: Specified Gain (cont.)
Low Noise Amplifier Design• Last case: low noise amplifier• Noise limits many systems (radio, wireline, fiber
receivers, etc.) • Performance dominated (usually) by first amplifier in
cascade• Unfortunate fact: best noise performance not
obtained at maximum gain. Have to match for either noise or gain; a tradeoff is inevitable
Noise in Two-Port Networks• Noise in amplifiers (as represented by 2-port
networks) can be quantified by noise factor:
– Noise factor is a measure of how much worse the SNR (power) is at the output compared to the input, due to noise added by the network
– Noise factor depends on input termination (GS):
F = SNRinSNRout
F = Fmin +4RNZO
ΓS −Γopt2
(1− ΓS2 ) 1+Γopt
2
Zs
vsomgi.rsJIaina Igz'idtcannot tell SinfronNia
Signal amplifiesnoisetooSoma Also adds its ownnoise
Noise in Two-Port Networks• Noise factor depends on input termination (GS):
– Gopt, RN, Fmin are “noise parameters” of the device• Can show (if you are patient) that constant F
contours are (gasp) circlesin the GS plane:
• CF is center (complex plane), RF is radius
F = Fmin +4RNZO
ΓS −Γopt2
(1− ΓS2 ) 1+Γopt
2
!" =Γ%&'( + 1
+" =( ( + 1 − Γ%&'
-
( + 1
( = Γ. − Γ%&'-
1 − Γ. -= / − /0124+4/67
1 + Γ%&'-
Low Noise Amplifier Design
• Example 12.5 in Pozar:– Design an amplifier with 2 dB noise figure and maximum
gain (consistent with this noise level)– Plot noise figure circles– Note that noise figure is related to GS; but gain is too
(specified gain: GS term)– So we look at both constant gain and noise figure circles on
the same plot:
Low Noise Amplifier Design (cont.)• Noise figure: NF=2 dB circle• Several GS circles
1.0, 1.5, 1.7 dB• Need to satisfy
both: findintersectionbetweenNF, highestgain GScircle
• What about output? Conjugately match once input is finalized
Another Important Consideration: Stability
• In the design techniques so far, we’ve neglected an important issue, stability
• In microwave design, “stability” has a very specific meaning– Stability immunity to oscillation– A stable circuit won’t oscillate, and unstable one
might (or will, depending on the situation)• In “regular old” circuit design, stability usually
considered using feedback theory• Using s-parameters gives us an alternative
approach that can be helpful
S-parameter Based Stability Considerations
• If we treat the circuit (e.g. amplifier) as a black box with s-parameters, what would it mean if the circuit were oscillating?– An oscillator is a signal generator: produces signal power at
some frequency even when no input is present– Looking into the ports, “reflection coefficient” > 1: more power
“reflected” than put in– Not necessarily “reflected,” but “reflection coefficient” is just
the ratio of the outgoing wave (b) to incoming wave (a)– If a wave is zero, but b wave isn’t--circuit is oscillating
––
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S-parameter Based Stability Considerations
• What does this mean from a circuit perspective?– In terms of impedances, real part of input (Zin) and/or
output (Zout) impedances < 0– “negative resistances” can generate power—and
that’s what oscillators do
• S-parameter-based approach is sometimes easier– In terms of reflection coefficients (instead of
impedances): |Gin| and/or |Gout| > 1 (more reflected than incident) indicates instability
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Reflection Coefficient View• Let’s look at this with some analysis• Basic circuit:
• To check to see if the circuit can oscillate, choose GS and GL so that |GS|, |GL|
Reflection Coefficient View of Stability
• So for this circuit, what does this imply?• Mathematically:– Require: Γ" < 1, Γ& < 1– For stability (no oscillations):Γ'( = *++ + "-.".-/0+1"../0
Alternative Approach: Look for Edge of Stability
• Stability requires |Gin|, |Gout| < 1; conversely if > 1, then unstable
• “Edge of stability”: |Gin|, |Gout| = 1• Recall: Γ"# = %&& + ()*(*)+,&-(**+, (depends on GL), andΓ/01 = %22 + ()*(*)+3&-())+3 (depends on GS)
• Idea: look for set of GS, GL that make the corresponding |Gin|, |Gout| = 1– This will define the “border” between stability and
instability, and then we can just figure out which regions are which
Alternative Approach: Look for Edge of Stability
• Basic approach (math): solving
• Γ"# = %&& + ()*(*)+,&-(**+, = 1 à yields set of GL’s that satisfy the equation
• Math is messy, tedious. Bottom line: set of GL’s that solve the equation is a circle in the GL plane
• Can show that:
./ = (**-0())∗ ∗
(** *- 0 *and 2/ = ()*(*)(** *- 0 *
where CL is the center of the circle and rL is the radius
What does this mean?• GL’s that solve the equation is a circle in the GL plane• !" = $%%&'$((
∗ ∗
$%% %& ' %
• *" = $(%$%($%% %& ' %
• If GL is chosen on this circle, then |Gin|=1 – on edge of stability
• Circle divides GL plane into two regions; inside & outside– One region is stable, the other isn’t
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What does this mean?• Which region is stable, unstable?
– It could be either way– the circle is just the boundary• Quick check: check any point (choose some GL) not on the circle,
and see if Gin < or > 1– Easiest one? GL=0 (center of Smith chart)– For GL=0, Gin=S11 (Γ"# = %&& + ()*(*)+,&-(**+,)– If |S11|
How about source side?• So far, have focused entirely on effect of GL on Gin
– But Gout depends on GS (symmetry of the overall architecture)– Nothing special about the port numbers
• Can follow a very similar derivation for effect of GS• In the GS plane, GS’s that cause |Gout|=1 is a circle• !" = "$$%&"''
∗ ∗
"$$ '% & '
• )" = "$'"'$"$$ '% & '• To check region of stability or instability, look at |S22|
(just like |S11| check for the GL plane)• Bottom line: need to check both source and load sides;
circuit might be unstable at only one port, etc.
Careful: GL, GS are in Different Planes
• Warning: it is common to show both source & load circles on the same Smith chart
• Careful! These circles are in different planes; there isno significance to overlaps, etc.
• Remember: source stabilitycircle tells us what GS might cause problems (input); loadcircle tells us what GL might causeproblems (load)
• Treat them separately
Unconditional Stability• Special case: unconditional stability• In English:– A circuit is unconditionally stable if there is no combination
of passive source & load terminations that will cause oscillation
• In math:– |Gin|
Unconditional Stability• That’s a little tedious…there is an easier way.
Conditions on stability can be expressed mathematically
• Two main approaches• Classic approach: Rowlett stability test
! = 1 − %&&' − %'' ' + Δ '2 %&' %'&
> 1 ,-.Δ = %&&%'' − %'&%&' < 1
– Both must be satisfied simultaneously for unconditional stability
• Alternative: “µ test”. Circuit is unconditionally stable if
0 = 1 − %&&'
%'' − Δ%&&∗ + %&'%'&> 1
Stability Interpretation• A key point is that these expressions are frequency dependent
(since S is); circuits are often unconditionally stable at some frequencies and potentially unstable at others
• In general, stability must be checked/controlled for all frequencies
• Remember: the circuit doesn’t know (or care) what you want it to do; it does what it wants– The “frequency of interest” or the “design frequency” doesn’t mean
anything. A circuit that is “unconditionally stable” at a 1 GHz design frequency might well make an excellent oscillator at 10 MHz
– A circuit that is oscillating by accident will (usually) not work as intended; the large oscillation signals will corrupt the bias point, etc.
Stability Interpretation• The inequalities (K-D or µ) are the easiest way to quickly check
all frequencies for “yes/no” unconditional stability • Often the case that unconditional stability cannot be
achieved, have to “work around” unstable regions (i.e. operate within the stable region of a potentially unstable device)
• The stability circles give a lot more information about what regions are stable and unstable, and can help you avoid undesired areas (e.g. source, load impedances that would cause oscillations)
• Typically, check stability circles in regions that the µ or K-Dtests say are not unconditionally stable to see where your termination conditions are; often not at design frequency